The quasi-static and dynamic responses
of metallic sandwich structures
Luc St-Pierre
St. Catharine’s College
Submitted to the University of Cambridge
for the degree of Doctor of Philosophy
April 2012
Preface
This thesis is submitted for the degree of Doctor of Philosophy, at the University
of Cambridge. The work presented in this thesis was carried out at the Cambridge
University Engineering Department from October 2007 to October 2011, under the
supervision of Professor Norman A. Fleck. The research work was sponsored by the
Material Innovation Institute (M2i) under the project no. MC2.06261 and by the
Fonds Que´be´cois de la Recherche sur la Nature et les Technologies (FQRNT).
This dissertation is the result of my own work and includes nothing which is the
outcome of work done in collaboration except where specifically indicated in the
text. This document, in whole or in parts, has not been submitted for any other
degree, diploma or qualification.
This thesis contains approximately 44,000 words, 85 figures and 10 tables, which
does not exceed the requirements of the Degree Committee.
Luc St-Pierre
Cambridge, UK
April 2012
i
ii
Summary
Lattice materials are used as the core of sandwich panels to construct light and
strong structures. This thesis focuses on metallic sandwich structures and has two
main objectives: (i) explore how a surface treatment can improve the strength of a
lattice material and (ii) investigate the collapse response of two competing prismatic
sandwich cores employed in ship hulls.
First, the finite element method is used to examine the effect of carburisation and
strain hardening upon the compressive response of a pyramidal lattice made from
hollow tubes or solid struts. The carburisation surface treatment increases the yield
strength of the material, but its effects on pyramidal lattices are not known. Here,
it is demonstrated that carburisation increases the plastic buckling strength of the
lattice and reduces the slenderness ratio at which the transition from plastic to
elastic buckling occurs. The predictions also showed that strain hardening increases
the compressive strength of stocky lattices with a slenderness ratio inferior to ten,
but without affecting the collapse mode of the lattice.
Second, the quasi-static three-point bending responses of simply supported and
clamped sandwich beams with a corrugated core or a Y-frame core are compared
via experiments and finite element simulations. The role of the face-sheets is as-
sessed by considering beams with (i) front-and-back faces present and (ii) front face
present, but back face absent. These two beam designs are used to represent single
hull and double hull ship structures, and they are compared on an equal mass basis
by doubling the thickness of the front face when the back face is absent. Beams
with a corrugated core are found to be slightly stronger than those with a Y-frame
core, and two collapse mechanisms are identified depending upon beam span. Short
beams collapse by indentation and for this collapse mechanism, beams without a
back face outperform those with front-and back faces present. In contrast, long
beams fail by Brazier plastic buckling and for this collapse mechanism, the presence
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of a back face strengthens the beam.
Third, drop weight tests with an impact velocity of 5 m/s are performed on sim-
ply supported and clamped sandwich beams with a corrugated core or a Y-frame
core. These tests are conducted to mimic the response of a sandwich hull in a ship
collision. The responses measured at 5 m/s are found to be slightly stronger than
those measured quasi-statically. The measurements are in reasonable agreement
with finite element predictions. In addition, the finite element method is used to
investigate whether the collapse mechanism at 5 m/s is different from the one ob-
tained quasi-statically. The predictions indicate that sandwich beams that collapse
quasi-statically by indentation also fail by indentation at 5 m/s. In contrast, the
simulations for beams that fail quasi-statically by Brazier plastic buckling show that
they collapse by indentation at 5 m/s.
Finally, the dynamic indentation response of sandwich panels with a corrugated
core or a Y-frame core is simulated using the finite element method. The panels
are indented at a constant velocity ranging from quasi-static loading to 100 m/s,
and two indenters are considered: a flat-bottomed indenter and a cylindrical roller.
For indentation velocities representative of a ship collision, i.e. below 10 m/s, the
predictions indicate that the force applied to the front face of the panel is approxi-
mately equal to the force transmitted to the back face. Even at such low indentation
velocities, inertia stabilisation effects increase the dynamic initial peak load above
its quasi-static value. This strengthening effect is more important for the corrugated
core than for the Y-frame core. For velocities greater than 10 m/s, the force ap-
plied to the front face exceeds the force transmitted to the back face due to wave
propagation effects. The results are also found to be very sensitive to the size of the
flat-bottomed indenter; increasing its width enhances both inertia stabilisation and
wave propagation effects. In contrast, increasing the roller diameter has a smaller
effect on the dynamic indentation response. Lastly, it is demonstrated that material
strain-rate sensitivity has a small effect on the dynamic indentation response of both
corrugated and Y-frame sandwich panels.
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Acknowledgements
First, I want to thank my supervisor, Prof. Norman Fleck, and my advisor, Prof.
Vikram Deshpande. I am indebted for all their contributions of time, ideas and
funding that made my PhD possible. Their enthusiasm, dedication, intelligence and
the quality of their work will continue to inspire me during my future career.
I would like to thank Joep Broekhuijsen of Damen Schelde Naval Shipbuilding for
many insightful discussions on the Y-frame ship structure and for sharing his indus-
trial expertise. I am also grateful to other Dutch collaborators including Bert van
Haastrecht of M2i and Prof. Marc Geers and Dr. Ron Peerlings of TU Eindhoven
for their feedback on the project.
I am very grateful to all the technicians that made my experimental work possible,
especially Alan Heaver, Simon Marshall and Gareth Ryder. Thanks also to all
my colleagues at the Centre for Micromechanics for providing an enjoyable and
stimulating working environment, long-lasting friendship and numerous technical
and non-technical advices.
I would like to acknowledge many people outside the department, which have con-
tributed to balance, relaxation and fun during my studies. Thanks to my friends
from St. Catharine’s College — Amy, Catherine, Ivana, Julia, Nick and Rich — for
making me laugh and for making the college feels like home. I am also grateful to
my ice hockey teammates, especially Bill, Carl, Pete, Kevin and Owen. Playing
hockey with them has been a memorable experience to say the very least.
This thesis is dedicated to my parents, Aline and Claude, who have provided constant
support in all my pursuits. Thanks to Oli and all the guys from home as their
friendship has not been weakened by the distance. Lastly, I would like to thank
Rose for her understanding, encouragement and love that made this experience such
an incroyable journey.
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Table of Contents
Preface i
Summary iii
Acknowledgements v
Table of Contents vii
List of Figures xi
List of Tables xxi
1 Introduction 1
1.1 Lattice materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Sandwich structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Scope of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Literature review 7
2.1 Lattice materials used as core topologies . . . . . . . . . . . . . . . . 7
2.1.1 Metal foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.2 Truss cores . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.3 Prismatic cores . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.4 Hierarchical topologies . . . . . . . . . . . . . . . . . . . . . . 14
2.1.5 Multifunctionality . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Sandwich structures under quasi-static three-point bending . . . . . . 16
2.2.1 Collapse mechanisms . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Sandwich beams with a metal foam core . . . . . . . . . . . . 21
2.2.3 Sandwich beams with a truss core . . . . . . . . . . . . . . . . 21
2.2.4 Sandwich beams with a prismatic core . . . . . . . . . . . . . 22
2.2.5 Optimisation studies . . . . . . . . . . . . . . . . . . . . . . . 22
vii
2.3 Lattice materials and sandwich structures under dynamic loading . . 23
2.3.1 Dynamic compressive response of lattice materials . . . . . . . 24
2.3.2 Sandwich structures subjected to a low-velocity impact . . . . 26
2.3.3 Sandwich structures subjected to blast loading . . . . . . . . . 27
2.4 Ship hull design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.1 Regulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4.2 Evaluating the resistance of ship hulls . . . . . . . . . . . . . . 30
2.4.3 Design against collision and grounding . . . . . . . . . . . . . 31
2.4.4 Development of the Y-frame sandwich hull design . . . . . . . 32
2.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Compressive response of a carburised pyramidal lattice 37
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 Geometry and analytical collapse load of the pyramidal lattice . . . . 41
3.2.1 Relative density . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.2 Analytical collapse load . . . . . . . . . . . . . . . . . . . . . 42
3.3 Influence of strain hardening . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.1 Description of the finite element models . . . . . . . . . . . . 43
3.3.2 Results for an inclined tube . . . . . . . . . . . . . . . . . . . 45
3.3.3 Results for an inclined solid strut . . . . . . . . . . . . . . . . 48
3.3.4 Comparison between tube and solid strut . . . . . . . . . . . . 49
3.4 Influence of carburisation . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4.1 Description of the finite element models . . . . . . . . . . . . 54
3.4.2 Results for an inclined tube . . . . . . . . . . . . . . . . . . . 55
3.4.3 Results for an inclined solid strut . . . . . . . . . . . . . . . . 58
3.4.4 Comparison between tube and solid strut . . . . . . . . . . . . 61
3.4.5 Position of carburised lattices on the strength-density chart . . 62
3.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.A Influence of geometric imperfections . . . . . . . . . . . . . . . . . . . 64
3.A.1 Influence of the number of superimposed modes . . . . . . . . 65
3.A.2 Influence of amplitude . . . . . . . . . . . . . . . . . . . . . . 66
3.A.3 Influence of imperfection on the deformed meshes . . . . . . . 67
4 The influence of the back face on the bending response of prismatic
sandwich beams 69
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.1.1 Choice of test material . . . . . . . . . . . . . . . . . . . . . . 72
viii
4.1.2 Scope of study . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2.1 Specimen manufacture . . . . . . . . . . . . . . . . . . . . . . 74
4.2.2 Geometry of the three-point bending tests . . . . . . . . . . . 76
4.2.3 Finite element models . . . . . . . . . . . . . . . . . . . . . . 78
4.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3.1 Simply supported beams . . . . . . . . . . . . . . . . . . . . . 79
4.3.2 Clamped beams . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3.3 Collapse mechanisms . . . . . . . . . . . . . . . . . . . . . . . 82
4.4 Finite element predictions . . . . . . . . . . . . . . . . . . . . . . . . 88
4.4.1 Comparison between measurements and simulations . . . . . . 88
4.4.2 Sensitivity of the sandwich panel response to span and pro-
portion of mass in the core . . . . . . . . . . . . . . . . . . . 89
4.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.A Influence of parent material . . . . . . . . . . . . . . . . . . . . . . . 101
5 Drop weight tests on prismatic sandwich beams 103
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2.1 Geometry of the tested sandwich beams . . . . . . . . . . . . 105
5.2.2 Drop weight apparatus . . . . . . . . . . . . . . . . . . . . . . 106
5.2.3 Finite element models . . . . . . . . . . . . . . . . . . . . . . 106
5.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.3.1 Simply supported beams . . . . . . . . . . . . . . . . . . . . . 110
5.3.2 Clamped beams . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.3.3 Collapse mechanisms . . . . . . . . . . . . . . . . . . . . . . . 114
5.4 Finite element predictions . . . . . . . . . . . . . . . . . . . . . . . . 114
5.4.1 Comparison between simulations and measurements . . . . . . 116
5.4.2 Sensitivity of the peak load and collapse mechanism to the
loading velocity . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.A Finite element predictions with the projectile and roller modelled as
deformable parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.A.1 Simulations without the rubber pad . . . . . . . . . . . . . . . 121
5.A.2 Simulations with the rubber pad . . . . . . . . . . . . . . . . . 123
5.B Analytical prediction of the contact force . . . . . . . . . . . . . . . . 125
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6 Dynamic indentation of prismatic sandwich panels 127
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.1.1 Review of the dynamic uniform compressive response . . . . . 129
6.1.2 Scope of study . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.2 Finite element models . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.2.1 Geometric imperfections . . . . . . . . . . . . . . . . . . . . . 134
6.2.2 Material properties . . . . . . . . . . . . . . . . . . . . . . . . 135
6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.3.1 Indentation responses . . . . . . . . . . . . . . . . . . . . . . . 135
6.3.2 Deformed meshes . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.3.3 Influence of velocity . . . . . . . . . . . . . . . . . . . . . . . . 141
6.3.4 Influence of indenter size . . . . . . . . . . . . . . . . . . . . . 143
6.3.5 Force distribution on the back face . . . . . . . . . . . . . . . 146
6.3.6 Influence of material strain-rate sensitivity . . . . . . . . . . . 147
6.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.A Finite element model for uniform compression . . . . . . . . . . . . . 150
7 Conclusions and future work 151
7.1 Compressive response of a carburised pyramidal lattice . . . . . . . . 151
7.2 The influence of the back face on the bending response of prismatic
sandwich beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.3 Drop weight tests on prismatic sandwich beams . . . . . . . . . . . . 152
7.4 Dynamic indentation of prismatic sandwich panels . . . . . . . . . . . 153
7.5 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.5.1 Dynamic compressive response of a hollow pyramidal lattice . 154
7.5.2 Measured compressive response of a carburised pyramidal lattice154
7.5.3 Fracture of corrugated and Y-frame sandwich panels . . . . . 155
Bibliography 157
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List of Figures
1.1 Material property chart of strength versus density for lattice and fully-
dense engineering materials. Text boxes are used to distinguish lattice
materials from fully-dense materials. Adapted from Russell (2009). . 2
1.2 Examples of sandwich structures in nature: bird skulls of (a) a magpie
and (b) a long-eared owl. Adapted from Gibson et al. (2010). . . . . . 3
1.3 Examples of sandwich structures in industrial applications: (a) the
car body of a high speed train in Japan (Shinkansen 700 series) and
(b) Y-frame ship hull design developed by Damen Schelde Naval Ship-
building. Adapted from Matsumoto et al. (1999) and McShane (2007). 4
2.1 Classification of lattice materials proposed by Evans et al. (2001) and
Wadley (2002). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Aluminium metal foams with (a) closed-cell and (b) open-cell. Re-
produced from Paul and Ramamurty (2000) and Nieha et al. (2000). . 9
2.3 Relative strength as a function of relative density ρ¯ for different
core topologies. The relative strength is defined as the compressive
strength of the core σ¯pk divided by the yield strength of the material
σY . Examples of (b) bending- and (c) stretching-dominated lattices.
Adapted from Ashby (2005). . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Examples of truss cores: (a) tetrahedral, (b) pyramidal and (c) Kagome´
lattices. Reproduced from Kooistra and Wadley (2007). . . . . . . . . 10
2.5 Two manufacturing routes for truss cores: (a) perforating and folding
and (b) slitting, flattening and folding. Reproduced from Kooistra
and Wadley (2007). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.6 Lattices made from hollow tubes arranged in (a) square and (b) dia-
mond orientations. Reproduced from Queheillalt and Wadley (2005a). 12
xi
2.7 Measured normalised peak strength of stainless steel lattices loaded in
(a) compression and (b) shear. The lattice compressive strength σ¯pk
and shear strength τ¯pk are normalised by the relative density ρ¯ and
the yield strength of stainless steel σY . Data taken from Coˆte´ et al.
(2004); Coˆte´ et al. (2006); Queheillalt and Wadley (2011); Rubino
et al. (2008a); Zok et al. (2004). . . . . . . . . . . . . . . . . . . . . . 13
2.8 Examples of prismatic cores: (a) corrugated, (b) Y-frame, (c) square
honeycomb and (d) diamond cores. Reproduced from Zok et al. (2005)
and Coˆte´ et al. (2006). . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.9 Photographs showing the collapse mechanisms of (a) second order and
(b) first order corrugated cores, both with a relative density ρ¯ = 0.02.
Reproduced from Kooistra et al. (2007). . . . . . . . . . . . . . . . . 15
2.10 A sandwich beam with a metal foam core loaded in three-point bend-
ing that collapsed by (a) face yield, (b) core shear (mode B) and
(c) indentation. For each collapse mechanism, simply supported and
clamped sandwich beams are shown. Reproduced from Tagarielli and
Fleck (2005). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.11 (a) Simply supported and (b) clamped sandwich panels loaded in
three-point bending. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.12 Face yield collapse mechanism for (a) simply supported and (b) clamped
sandwich panels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.13 Core shear collapse mechanisms: (a) mode A and (b) mode B. . . . . 18
2.14 Indentation collapse mechanism. . . . . . . . . . . . . . . . . . . . . . 19
2.15 Collapse mechanism maps for a simply supported sandwich panel: (a)
contours of normalised collapse load F¯ = F/(2bLσfY ) for a/(2L) = 0.1
and (b) influence of the normalised indenter size a/(2L) upon the
operative collapse mechanism. In both cases, σcY /σfY = 0.005 and
τ cY /σfY = 0.005. Adapted from Ashby et al. (2000). . . . . . . . . . . . 21
2.16 A simply supported sandwich beam with a Y-frame core loaded in
three-point bending and collapsing by indentation: (a) experiment
and (b) finite element prediction. A side view showing half the beam
(left) and a view of the core deformation obtained by sectioning the
beam at mid-span (right) are included. Reproduced from Rubino
et al. (2010). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.17 Direct impact Kolsky bar setup in (a) back face and (b) front face
configurations. Adapted from Lee et al. (2006). . . . . . . . . . . . . 25
xii
2.18 Normalised maximum back face deflection w/L as a function of the
normalised blast impulse I/(M
√
σY /ρf ). Adapted from Rathbun
et al. (2006a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.19 Photographs and finite element simulations of (a) monolithic, (b) Y-
frame and (c) corrugated plates impacted by a metal foam projectile
with a momentum of 3 kNs/m2. All plates have the same mass.
Reproduced from Rubino et al. (2009). . . . . . . . . . . . . . . . . . 29
2.20 Four hull designs considered in full-scale collision tests: (a) a conven-
tional single hull, (b) a conventional double hull, (c) a double hull
with a Y-frame core and (d) a double hull made from two corrugated
panels. Adapted from ISSC (2006a). . . . . . . . . . . . . . . . . . . 33
2.21 Photographs of full-scale collision tests performed on the Y-frame
double hull structure. Adapted from Konter et al. (2004) and Wevers
and Vredeveldt (1999). . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.22 Construction of inland waterway tankers with a Y-frame single hull
structure. Adapted from Graaf et al. (2004) and Vredeveldt and
Roeters (2004). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1 (a) Unit cell of the hollow pyramidal lattice. (b) Top view of the lattice. 39
3.2 Collapse mechanism map for a hollow pyramidal lattice made from
AISI 304 stainless steel. Examples of the six collapse modes (A-F)
are also included. Representative geometries considered in this study
are indicated by filled black circles and contours of relative density ρ¯
are plotted as grey lines. Adapted from Pingle et al. (2011a). . . . . . 40
3.3 Finite element model used to simulate the compressive response of an
inclined tube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4 Uniaxial tensile responses of the two material models employed in the
finite element simulations analysing the influence of strain hardening. 45
3.5 Influence of the strain hardening modulus Et on the compressive re-
sponse of an inclined tube t/d = 0.1. Results are given for (a) l/d = 1,
(b) l/d = 3, (c) l/d = 20 (d) l/d = 100. . . . . . . . . . . . . . . . . . 46
3.6 Influence of the strain hardening modulus Et on the compressive re-
sponse of an inclined solid strut t/d = 0.5. Results are given for (a)
l/d = 3, (b) l/d = 20 and (c) l/d = 100. . . . . . . . . . . . . . . . . . 50
3.7 Influence of the strain hardening modulus Et on the compressive
strength of an inclined tube t/d = 0.1 and an inclined solid strut
t/d = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
xiii
3.8 Cross-sections of a carburised (a) tube and (b) solid strut. . . . . . . 53
3.9 Uniaxial tensile responses of annealed and carburised stainless steels
employed in the finite element simulations analysing the influence of
carburisation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.10 Influence of the carburisation depth h/d on the compressive response
of an inclined tube t/d = 0.1. Results are given for (a) l/d = 1, (b)
l/d = 3, (c) l/d = 20 (d) l/d = 100. . . . . . . . . . . . . . . . . . . . 56
3.11 Influence of the carburisation depth h/d on the compressive response
of an inclined solid strut t/d = 0.5. Results are given for (a) l/d = 3,
(b) l/d = 20 and (c) l/d = 100. . . . . . . . . . . . . . . . . . . . . . 59
3.12 Influence of the carburisation depth h/d on the compressive strength
of an inclined tube t/d = 0.1 and an inclined solid strut t/d = 0.5. . . 61
3.13 Strength versus density material chart. The simulated compressive
strength of a pyramidal lattice made from carburised tubes (t/d = 0.1,
h/d = 0.05) is also included. Al, aluminium; CRFP, carbon fibre
reinforced polymers; Ti, titanium; TMC, titanium matrix composites. 63
3.14 Influence of imperfection shape on the compressive response of an
inclined tube t/d = 0.1. In all cases, the imperfection amplitude is
ζ = 0.05t. Results are given for (a) l/d = 1, (b) l/d = 3, (c) l/d = 20
(d) l/d = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.15 Influence of imperfection amplitude on the compressive response of
an inclined tube t/d = 0.1. In all cases, the imperfection shape is in
the form of the 1st buckling mode. Results are given for (a) l/d = 1,
(b) l/d = 3, (c) l/d = 20 (d) l/d = 100. . . . . . . . . . . . . . . . . . 66
4.1 The Y-frame sandwich core in (a) double hull and (b) single hull designs. 71
4.2 The design space for mass distribution within a sandwich panel of
areal mass m. The proportion of mass in the core, in the front face
and in the back face are denoted by mc/m, mf/m and mb/m, respec-
tively. The mass distribution of the test geometries is indicated for
two choices of areal mass. . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3 Measured uniaxial tensile responses of as-brazed AISI 304 stainless
steel and Lloyd’s Grade A steel, at a strain rate of 10−3 s−1. . . . . . 73
4.4 Cross-sectional dimensions of the sandwich beams: (a) corrugated
core and (b) Y-frame core. (c) The chosen values of face-sheet thick-
ness used in the experimental study. All dimensions are in mm. . . . 75
xiv
4.5 The test fixtures used for (a) simply supported and (b) clamped
beams. A sandwich beam with a Y-frame core and without a back
face is shown. All dimensions are in mm. . . . . . . . . . . . . . . . . 77
4.6 Three-point bending responses of simply supported sandwich beams.
Sandwich beams with an areal mass m = 9.1 kg/m2 are shown with
(a) a corrugated core and (b) a Y-frame core. Likewise, sandwich
beams with an areal mass m = 13.8 kg/m2 are shown with (c) a
corrugated core and (d) a Y-frame core. . . . . . . . . . . . . . . . . 80
4.7 Three-point bending responses of clamped sandwich beams. Sand-
wich beams with an areal mass m = 9.1 kg/m2 are shown with (a)
a corrugated core and (b) a Y-frame core. Likewise, sandwich beams
with an areal mass m = 13.8 kg/m2 are shown with (c) a corrugated
core and (d) a Y-frame core. . . . . . . . . . . . . . . . . . . . . . . . 81
4.8 Photographs of the simply supported sandwich beams with a corru-
gated core (m = 13.8 kg/m2) (a) with front-and-back faces and (b)
without a back face. Deformed finite element meshes of the same
sandwich beam (c) with front-and-back faces and (d) without a back
face. A side view showing half of the beam and a view of the core
deformation at mid-span are given. To clarify the predicted deforma-
tion modes, the undeformed (dashed line) and deformed (solid line)
cross-sections at mid-span are included in (c) and (d). The images
are for beams loaded to δ = 0.2L and then unloaded. . . . . . . . . . 83
4.9 Photographs of the simply supported sandwich beams with a Y-frame
core (m = 13.8 kg/m2) (a) with front-and-back faces and (b) without
a back face. Deformed finite element meshes of the same sandwich
beam (c) with front-and-back faces and (d) without a back face. A
side view showing half of the beam and a view of the core deformation
at mid-span are given. To clarify the predicted deformation modes,
the undeformed (dashed line) and deformed (solid line) cross-sections
at mid-span are included in (c) and (d). The images are for beams
loaded to δ = 0.2L and then unloaded. . . . . . . . . . . . . . . . . . 84
xv
4.10 Photographs of the clamped sandwich beams with a corrugated core
(m = 13.8 kg/m2) (a) with front-and-back faces and (b) without a
back face. Deformed finite element meshes of the same sandwich
beam (c) with front-and-back faces and (d) without a back face. A
side view showing half of the beam and a view of the core deformation
at mid-span are given. To clarify the predicted deformation modes,
the undeformed (dashed line) and deformed (solid line) cross-sections
at mid-span are included in (c) and (d). The images are for beams
loaded to δ = 0.2L and then unloaded. . . . . . . . . . . . . . . . . . 85
4.11 Photographs of the clamped sandwich beams with a Y-frame core (m
= 13.8 kg/m2) (a) with front-and-back faces and (b) without a back
face. Deformed finite element meshes of the same sandwich beam
(c) with front-and-back faces and (d) without a back face. A side
view showing half of the beam and a view of the core deformation
at mid-span are given. To clarify the predicted deformation modes,
the undeformed (dashed line) and deformed (solid line) cross-sections
at mid-span are included in (c) and (d). The images are for beams
loaded to δ = 0.2L and then unloaded. . . . . . . . . . . . . . . . . . 86
4.12 Cross-sectional dimensions of the sandwich panels considered in the
numerical analysis: (a) corrugated core and (b) Y-frame core. (c)
The sandwich panels, shown here with a corrugated core, are simply
supported and loaded in three-point bending. . . . . . . . . . . . . . 91
4.13 Normalised peak load Fˆ = Fpk/(σY bc) as a function of the normalised
span 2L/c for simply supported sandwich panels and selected values
of mc/m (m/(ρc) = 0.052). Results are shown for sandwich panels
with (a) a corrugated core and (b) a Y-frame core. . . . . . . . . . . 94
4.14 The boundary conditions on finite element models to simulate (a)
indentation and (b) bending. A sandwich panel without a back face
is shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.15 (a) The predicted indentation response of sandwich panels with mc/m
= 0.5 resting on a rigid foundation. (b) Normalised indentation
strength FˆI = FI/(σY bc) as a function of mc/m (m/(ρc) = 0.052). . . 95
4.16 (a) The predicted bending response of sandwich panels with mc/m =
0.5. (b) Normalised Brazier buckling moment Mˆ = MB/(σY bc2) as a
function of mc/m (m/(ρc) = 0.052). . . . . . . . . . . . . . . . . . . . 97
xvi
4.17 Normalised peak load Fˆ = Fpk/(σY bc) as a function of the normalised
span 2L/c for simply supported sandwich panels and selected values
of mc/m (m/(ρc) = 0.052). The three-point bending results are repro-
duced from Fig. 4.13. The indentation and Brazier buckling strengths
are included as short and long dashed lines, respectively. Sandwich
panels with front-and-back faces are shown with (a) a corrugated core
and (b) a Y-frame core. Likewise, sandwich panels without a back
face are shown with (c) a corrugated core and (d) a Y-frame core. . . 98
4.18 Normalised indentation strength per unit mass ρcFˆI/m as a function
of mc/m for selected values of m/(ρc). Results are shown for sandwich
panels with (a) a corrugated core and (b) a Y-frame core. . . . . . . . 100
4.19 Normalised Brazier buckling moment per unit mass ρcMˆ/m as a func-
tion of mc/m for selected values of m/(ρc). Results are shown for
sandwich panels (a) with front-and-back faces present and (b) with-
out a back face. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.20 Sensitivity of the three-point bending response of a sandwich beam
with a Y-frame core to the choice of material. (a) Front-and-back
faces are present and (b) the back face is absent. . . . . . . . . . . . . 102
5.1 Cross-sectional dimensions of the tested sandwich beams with (a) a
corrugated core and (b) a Y-frame core. All dimensions are in mm. . 105
5.2 Experimental setup used to perform drop weight tests at 5 m/s on
(a) simply supported and (b) clamped sandwich beams. A sandwich
beam with a corrugated core is shown. All dimensions are in mm. . . 107
5.3 Finite element models used to simulate the drop weight tests on (a)
simply supported and (b) clamped sandwich beams. All dimensions
are in mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.4 (a) The quasi-static (ǫ˙p = 10−3 s−1) uniaxial tensile response of AISI
304 stainless steel and the estimated high strain-rate responses based
on the data of Stout and Follansbee (1986). (b) Dynamic strength-
ening ratio R as a function of plastic strain rate ǫ˙p. . . . . . . . . . . 109
5.5 Quasi-static and 5 m/s responses of simply supported sandwich beams
with (a) a corrugated core and (b) a Y-frame core. . . . . . . . . . . 111
5.6 Influence of the mid-span roller mass upon the measured 5 m/s re-
sponse of a simply supported sandwich beam with a Y-frame core.
The measured response with a steel mid-span roller is reproduced
from Fig. 5.5(b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
xvii
5.7 Quasi-static and 5 m/s responses of clamped sandwich beams with
(a) a corrugated core and (b) a Y-frame core. . . . . . . . . . . . . . 113
5.8 High-speed images captured during a drop weight test at 5 m/s on
(a) a simply supported and (b) a clamped sandwich beam with a Y-
frame core. The deformed beams are shown for two selected values
of mid-span roller displacement δ/L. All images are showing a side
view of the beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.9 FE predictions of the quasi-static and 5 m/s responses of simply sup-
ported and clamped sandwich beams. Beams with front-and-back
faces are shown with (a) a corrugated core and (b) a Y-frame core.
Likewise, beams with the back face absent are shown with (a) a corru-
gated core and (b) a Y-frame core. Beams with front-and-back faces
present collapse quasi-statically by indentation whereas those with
the back face absent fail quasi-statically by Brazier plastic buckling. . 118
5.10 Finite element model with the projectile and the assembly of the load
cell and mid-span roller modelled as two separate deformable parts.
The model is shown for a simply supported sandwich beam with a
corrugated core. All dimensions in mm. . . . . . . . . . . . . . . . . . 121
5.11 Simulated 5 m/s response of a simply supported beam with a corru-
gated core. The projectile and the mid-span roller are modelled as
separate deformable parts in the simulations. The contact forces be-
tween (a) the projectile and the mid-span roller and (b) the mid-span
roller and the beam are given. The simulations presented in Section
5.4.1 where the projectile and mid-span roller were modelled as one
rigid body are included in part (b) for comparison. . . . . . . . . . . 122
5.12 (a) Measured quasi-static (ǫ˙ = 10−3 s−1) compressive response of the
rubber pad used in the drop weight experiments. (b) Comparison
between the measured and simulated 5 m/s responses of a simply
supported beam with a corrugated core. The measured response is
reproduced from Fig. 5.5(a) whereas the simulations are using the
data shown in part (a) to model the contact between the projectile
and the mid-span roller. . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.13 Analytical model to predict the contact force between the projectile
and the assembly of the mid-span roller and load cell. . . . . . . . . . 125
xviii
6.1 Finite element models used to simulate (a) uniform compression, (b)
indentation by a flat-bottomed indenter and (c) indentation by a
cylindrical roller. The models are shown for a sandwich panel with a
Y-frame core. All dimensions are in mm. . . . . . . . . . . . . . . . . 130
6.2 Uniform compressive responses of sandwich panels with (a) a cor-
rugated core and (b) a Y-frame core. Results are given for quasi-
static loading and for crushing at 10 m/s. These simulations were
re-executed based on previous work done by Tilbrook et al. (2007). . 131
6.3 (a) The normalised peak stress and (b) the normalised average stress
up to δ/c = 0.2 for corrugated and Y-frame sandwich cores crushed
at a constant velocity V0. Those simulations were re-executed based
on previous work done by Tilbrook et al. (2007). . . . . . . . . . . . . 132
6.4 Cross-sectional dimensions of the sandwich panels: (a) corrugated
core and (b) Y-frame core. All dimensions in mm. . . . . . . . . . . . 133
6.5 Responses of sandwich panels indented by a flat-bottomed indenter
of normalised width a/L = 0.05. Results are shown at selected veloc-
ities: quasi-static and 1 m/s for (a) corrugated core and (b) Y-frame
core; 10 m/s for (c) corrugated core and (d) Y-frame core and 100
m/s for (e) corrugated core and (f) Y-frame core. . . . . . . . . . . . 137
6.6 Responses of sandwich panels indented by a cylindrical roller of nor-
malised diameter D/c = 0.41. Results are shown at selected veloci-
ties: quasi-static and 1 m/s for (a) corrugated core and (b) Y-frame
core; 10 m/s for (c) corrugated core and (d) Y-frame core and 100
m/s for (e) corrugated core and (f) Y-frame core. . . . . . . . . . . . 138
6.7 (a) The normalised initial peak load and (b) the normalised average
load up to δ/c = 0.2 for corrugated and Y-frame sandwich panels
indented at a constant velocity V0 by a flat-bottomed indenter of
normalised width a/L = 0.05. . . . . . . . . . . . . . . . . . . . . . . 142
6.8 (a) The normalised initial peak load and (b) the normalised average
load up to δ/c = 0.2 for corrugated and Y-frame sandwich panels
indented at a constant velocity V0 by a cylindrical roller of normalised
diameter D/c = 0.41. . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.9 Influence of the normalised width a/L of the flat-bottomed indenter
on the normalised initial peak load for (a) the corrugated core and (b)
the Y-frame core. Likewise, the influence of a/L on the normalised
average load up to δ/c = 0.2 is shown for (c) the corrugated core and
(d) the Y-frame core. . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
xix
6.10 Influence of normalised roller diameter D/c on the normalised ini-
tial peak load for (a) the corrugated core and (b) the Y-frame core.
Likewise, the influence of D/c on the normalised average load up to
δ/c = 0.2 is shown for (c) the corrugated core and (d) the Y-frame
core. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.11 Distribution of the normal traction T on the back face at an inden-
tation depth δ/c = 0.2. Results are shown for a sandwich panel
indented by a cylindrical roller of normalised diameter D/c = 0.41:
(a) corrugated core and (b) Y-frame core. . . . . . . . . . . . . . . . 146
6.12 Influence of scale and material strain-rate sensitivity on the nor-
malised initial peak load for (a) the corrugated core and (b) the Y-
frame core. Likewise, the influence of scale and material strain-rate
sensitivity on the normalised average load up to δ/c = 0.2 is shown for
(c) the corrugated core and (d) the Y-frame core. Results are shown
for panels indented by a cylindrical roller of normalised diameter D/c
= 0.41. The force on the front and back faces are given by solid and
dashed lines, respectively. . . . . . . . . . . . . . . . . . . . . . . . . 148
xx
List of Tables
3.1 Influence of the strain hardening modulus Et on the deformed meshes
of an inclined tube t/d = 0.1. Results are given for selected values of
l/d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Influence of the strain hardening modulus Et on the deformed meshes
of an inclined solid strut t/d = 0.5. Results are given for selected
values of l/d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3 Influence of the carburisation depth h/d on the deformed meshes of
an inclined tube t/d = 0.1. Results are given for selected values of l/d. 57
3.4 Influence of the carburisation depth h/d on the deformed meshes of
an inclined solid strut t/d = 0.5. Results are given for selected values
of l/d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.5 Influence of a geometric imperfection on the deformed meshes of an
inclined tube t/d = 0.1. Results are given for selected values of l/d. . 67
4.1 The measured and predicted values of normalised peak load Fˆ =
Fpk/(σY bc). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.1 Equivalent plastic strain distribution for simply supported sandwich
beams with a Y-frame core. Beams with and without a back face
are shown. Results are given for quasi-static loading and 5 m/s. All
images are showing a side view of the beam focusing on a portion of
length 0.35L from the beam mid-span. . . . . . . . . . . . . . . . . . 119
xxi
6.1 Deformed meshes of sandwich panels with a corrugated core shown
at selected velocities. The results are given for uniform compres-
sion, indentation by a flat-bottomed indenter of normalised width
a/L = 0.05 and indentation by a cylindrical roller of normalised di-
ameter D/c = 0.41. For indentation, the cross-section underneath
the indenter is shown along with a side view of the panel. All images
are given for δ/c = 0.35. . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.2 Deformed meshes of sandwich panels with a Y-frame core shown
at selected velocities. The results are given for uniform compres-
sion, indentation by a flat-bottomed indenter of normalised width
a/L = 0.05 and indentation by a cylindrical roller of normalised di-
ameter D/c = 0.41. For indentation, the cross-section underneath
the indenter is shown along with a side view of the panel. All images
are given for δ/c = 0.35. . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.3 Positive and negative traction forces on the back face of a sandwich
panel indented by a cylindrical roller of normalised diameter D/c =
0.41. The forces are for an indentation depth δ/c = 0.2, the values
are integrated from the curves shown in Fig. 6.11. . . . . . . . . . . . 147
xxii
Chapter 1
Introduction
Minimising the mass without sacrificing the strength of a structure is an engineering
challenge that has led to the development of new materials, such as lattice materials,
and new structural designs, such as sandwich structures. These two examples con-
stitute fundamental components of this dissertation which aims to (i) improve the
strength of lattice materials and (ii) give a better understanding of the quasi-static
and dynamic behaviour of sandwich structures. The concepts of lattice materials
and sandwich structures are both introduced below, followed by the scope of this
thesis.
1.1 Lattice materials
A lattice is defined as “a connected array of struts or plates” (Ashby, 2005). Lat-
tices can be made from metals, ceramics or polymers. The term “material” here
emphasises that the lattice is considered at a macroscopic length scale, much larger
than the length scale of its constituent lattice elements. Hence, the macroscopic
properties of lattice materials, such as their density, stiffness and strength, can be
directly compared to those of fully-dense solid materials.
The strength of both lattice materials and fully-dense materials is plotted in Fig.
1.1 as a function of density. The figure indicate clearly that lattice materials (which
include polymer and metal foams, composite pyramidal and aluminium lattices)
occupy a region of the material property space that is left empty by fully-dense
materials; this region is the left hand side of the figure where the density is less than
1
Chapter 1. Introduction
Unattainable
Property Space
ecapS yt rep or P elb an iat tanU
0.01 0.1 1 10 100
Density (Mg/m )
gnireenignE
syollA
Engineering
Polymers
Natural
Materials
Metal
Foams
Polymer
Foams
Composite
Laminates
Engineering
Ceramics
Aluminium
Lattices
Composite
Pyramidal
0.01
10000
1000
100
10
1
0.1
3
Strength
(MPa)
Figure 1.1: Material property chart of strength versus density for lattice and fully-
dense engineering materials. Text boxes are used to distinguish lattice materials from
fully-dense materials. Adapted from Russell (2009).
0.1 Mg/m3. The low density of lattice materials makes them ideal candidates for
the core of sandwich structures, which are presented in the next section.
1.2 Sandwich structures
A sandwich structure usually consists of two thin but stiff face-sheets, made from a
fully-dense material, separated by a thick and light core. The result is a structure
with a high bending stiffness and strength with a low overall density. In fact, the
bending stiffness and strength of a sandwich structure is always superior to that of
a monolithic structure made from the same material and having the same mass.
Numerous examples of sandwich structures are found in nature. In this regard,
Galileo (1638) writes:
Art, and nature even more, makes use of these in thousands of operations
2
1.2 Sandwich structures
in which robustness is increased without adding weight, as is seen in the
bones of birds and in many stalks that are light and very resistant to
bending and breaking.
The skull of birds present a variety of sandwich topologies. For example, the skull
of a magpie is a double sandwich construction (see Fig. 1.2(a)) whereas the skull
of larger birds, such as owls, is a multiple sandwich construction (see Fig. 1.2(b)).
Another example is the stiff toucan beak, which constitutes one third of the bird’s
length, but only one twentieth of its weight (Seki et al., 2005).
(a) (b)
Figure 1.2: Examples of sandwich structures in nature: bird skulls of (a) a magpie
and (b) a long-eared owl. Adapted from Gibson et al. (2010).
The first industrial application of a sandwich construction is attributed to Fairbairn
(1849) who used iron face-sheets riveted to a wooden core in the construction of
bridges. However, the potential of sandwich structures was limited by the core ma-
terials available; restricted to Balsa and other types of wood. The popularity of
sandwich structures increased significantly when novel cores were introduced such
as the honeycomb core in the 1940s and polymer foams in the 1950s. The sandwich
construction was adopted rapidly by the aerospace and aeronautic industries (Her-
rmann et al., 2005) and more recently, it has been applied in high speed trains and
ship hulls (Vinson, 2005), as illustrated in Fig. 1.3.
A sandwich structure can be made from different materials; aluminium, steel or
carbon fibre reinforced polymers are some examples. This thesis focuses only on
metallic sandwich panels. These have two main advantages over their composite
counterparts: (i) the joining process for metals is well known, avoiding debonding
problems between the face-sheets and core and (ii) the ductility of metals enable the
sandwich structure to be used in energy absorption applications. This thesis will
focus on one particular industrial application where energy absorption is important,
ship hull design.
3
Chapter 1. Introduction
gas storage tank Y-frame core
(a)
(b)
Sandwich roof
and side panels
Figure 1.3: Examples of sandwich structures in industrial applications: (a) the car
body of a high speed train in Japan (Shinkansen 700 series) and (b) Y-frame ship hull
design developed by Damen Schelde Naval Shipbuilding. Adapted from Matsumoto
et al. (1999) and McShane (2007).
4
1.3 Scope of this thesis
1.3 Scope of this thesis
This thesis focuses on metallic sandwich structures and has two main objectives:
1. explore how a surface treatment can improve the strength of a lattice material
and
2. investigate the collapse response of two competing prismatic sandwich cores
employed in ship hulls.
A considerable amount of work has been published on metallic sandwich structures
and a review is presented in Chapter 2. First, lattice materials are introduced with
an emphasis on their manufacturing route and their compressive and shear strengths.
Second, the collapse mechanisms of a sandwich panel loaded quasi-statically under
three-point bending are presented. Third, the dynamic responses of sandwich struc-
tures subjected to (i) low-velocity impacts and (ii) blast loads are reviewed and
finally, the potential of using a sandwich construction to increase the resistance of
ship hulls against collision and grounding is also discussed.
As it will be shown in Chapter 2, the pyramidal lattice possesses a high compres-
sive strength. The potential of using a surface treatment, such as carburisation, to
increase its strength is investigated in Chapter 3. In this chapter, the finite ele-
ment method is used to predict the compressive response of a carburised pyramidal
lattice made from hollow tubes or solid struts. The effect of carburisation upon the
compressive strength and the collapse mode of the lattice is examined.
The pyramidal lattice is strong, but more difficult to manufacture than prismatic
lattices such as the corrugated core and Y-frame core. Consequently, prismatic cores
are more attractive for industrial applications like ship hull design. The quasi-static
three-point bending response of sandwich beams with a corrugated core or a Y-
frame core is investigated in Chapter 4. The role of the face-sheets is addressed by
considering beams with (i) front-and-back faces present and (ii) front face present,
but back face absent. Those two beam designs are used to represent double hull and
single hull ship structures, respectively. Experimental tests are complemented by
finite element simulations to gain additional insight into the collapse mechanisms.
Large vessels such as oil and chemical tankers are exposed to ship collisions that
occur at relatively low speeds, roughly 5 m/s. The response of a sandwich hull
construction to a ship collision is mimicked in Chapter 5 by performing drop weight
5
Chapter 1. Introduction
tests, with an impact velocity of 5 m/s, on sandwich beams with a corrugated core
or a Y-frame core. The beam response at 5 m/s is compared to its quasi-static
response to assessed the effect of velocity. This comparison is done via experiments
and finite element simulations.
Full-scale ship collision tests on a Y-frame sandwich hull have indicated that the
structure deforms by indentation (Wevers and Vredeveldt, 1999). For this reason,
the dynamic indentation response of sandwich panels with a corrugated core or a
Y-frame core is simulated in Chapter 6 using the finite element method. The
objective is to quantify the importance of (i) material strain-rate sensitivity, (ii)
inertia stabilisation effects and (iii) wave propagation effects upon the indentation
response.
Finally, Chapter 7 contains a summary of the conclusions reached in this thesis
and suggestions for future work.
6
Chapter 2
Literature review
Summary
An overview of the literature on metallic sandwich panels is presented in this chapter.
First, three families of lattice materials are introduced: metal foams, truss cores
and prismatic cores. The manufacturing route and the strength of each family of
lattices are reviewed. Second, the quasi-static three-point bending strength of a
sandwich panel is discussed with an emphasis on three collapse mechanisms: face
yield, core shear and indentation. Third, the dynamic behaviour of lattice materials
and sandwich structures is reviewed for both low-velocity impacts and blast loadings.
Finally, the potential of employing a sandwich construction to increase the structural
performances of ship hulls is addressed as it constitutes an important industrial
application of the work done in this thesis.
2.1 Lattice materials used as core topologies
The concept of lattice materials was introduced previously in Section 1.1. Lattice
materials possess a reasonably high strength at very low densities, which makes them
ideal candidates for the core of sandwich structures. The mechanical properties of
lattice materials are governed by three factors (Ashby, 2006):
1. the topology of the lattice,
2. the parent material and
7
Chapter 2. Literature review
3. the relative density ρ¯ defined as the volume fraction of solid material.
Several different topologies have been developed with the objective of maximising
the strength and minimising the density of the lattice. These topologies can be
classified in three families (Evans et al., 2001; Wadley, 2002): foams, truss cores and
prismatic cores, see Fig. 2.1. Foams have a random microstructure and accordingly
they fall in the stochastic category. In contrast, truss cores and prismatic cores are
constructed from a precise unit cell, which is repeated in an array. Consequently,
they form the category of periodic lattice materials. The research done for each of
these three families is presented below.
Lattice materials
Stochastic Periodic
Foams Truss cores Prismatic cores
- Open-cell
- Closed-cell
- Tetrahedral
- Pyramidal
- Kagomé
- Honeycomb
- Corrugated
- Y-frame
Figure 2.1: Classification of lattice materials proposed by Evans et al. (2001) and
Wadley (2002).
2.1.1 Metal foams
Metal foams are frequently made from aluminium filled with gas pores to form a
cellular structure. These pores can be either sealed (closed-cell, see Fig. 2.2(a)) or
interconnected (open-cell, see Fig. 2.2(b)). Numerous techniques have been devel-
oped to manufacture metal foams; the method employed influences the pore size and
their type (closed or open). For example, closed-cell foams can be manufactured by
bubbling gas through molten aluminium whereas open-cell foams can be fabricated
by casting using a polymer or a wax template. For more information about the
manufacturing process, the reader is referred to the textbooks of Gibson and Ashby
(1997) and Ashby et al. (2000).
Metal foams have poor structural performances. When a remote compressive stress
is applied, the ligaments forming the foam pores deform in bending. Consequently,
8
2.1 Lattice materials used as core topologies
(a) (b)
Figure 2.2: Aluminium metal foams with (a) closed-cell and (b) open-cell. Repro-
duced from Paul and Ramamurty (2000) and Nieha et al. (2000).
the stiffness and compressive strength of metal foams scale with ρ¯2 and ρ¯3/2, respec-
tively (Ashby, 2006). The yield behaviour of metal foams can be predicted using
the constitutive model of Deshpande and Fleck (2000a); this model has been imple-
mented in finite element simulations and the predictions were found to be in good
agreement with the measured deformation of metal foams (Bart-Smith et al., 2001;
Chen et al., 2001).
2.1.2 Truss cores
The bending-dominated behaviour of metal foams explains their poor compressive
strength. This led to the development of truss cores, which are designed to be
stretching-dominated rather than bending-dominated. Ashby (2006) demonstrated
that for stretching-dominated structures, the stiffness and strength of the lattice
both scale linearly with the relative density ρ¯. This implies that the strength of
a stretching-dominated structure is three times greater than that of a bending-
dominated structure when ρ¯ = 0.1. The behaviour of bending- and stretching-
dominated lattices is compared in Fig. 2.3.
Examples of truss cores include the tetrahedral, pyramidal and Kagome´ lattices, see
Fig. 2.4. The stiffness and strength of tetrahedral lattices was investigated ana-
lytically and experimentally by Wallach and Gibson (2001) and Deshpande et al.
(2001). In both studies, the authors note the superior performances of the tetrahe-
dral lattice compare to metal foams. Furthermore, experimental (Wang et al., 2003)
and numerical (Hyun et al., 2003) work on the Kagome´ lattice revealed that it is
slightly stronger than the tetrahedral lattice.
9
Chapter 2. Literature review
ρ
σpk
σY
Ideal
bending-dominated
behaviour
Ideal stretching-dominated behaviour
0.01 0.1 1
10
1
10-2
10-3
10-4
10-1
Foams
1.5
1Kagomé
HoneycombPyramidal
F
F
(a) (b)
(c) F
F
Figure 2.3: Relative strength as a function of relative density ρ¯ for different core
topologies. The relative strength is defined as the compressive strength of the core
σ¯pk divided by the yield strength of the material σY . Examples of (b) bending- and
(c) stretching-dominated lattices. Adapted from Ashby (2005).
(a) (b)
(c)
Figure 2.4: Examples of truss cores: (a) tetrahedral, (b) pyramidal and (c) Kagome´
lattices. Reproduced from Kooistra and Wadley (2007).
10
2.1 Lattice materials used as core topologies
In the experimental studies reported above, the tetrahedral and Kagome´ lattices
were fabricated by investment casting. This manufacturing route has important
limitations; low relative densities are difficult to manufacture because of their sus-
ceptibility to casting defects and the process requires an alloy with a high fluidity.
So far, aluminium-silicon and copper-beryllium alloys have been used, but their poor
ductility impairs the structural performances of the lattice. A novel manufacturing
route was proposed by Kooistra et al. (2004) to overcome those limitations. The
process involves perforating and folding a metallic sheet, as shown in Fig. 2.5(a).
Using this method, tetrahedral lattices with a relative density as low as ρ¯ = 0.02
were manufactured with ductile 6061 aluminium (Kooistra et al., 2004) and type
304 stainless steel (Sypeck and Wadley, 2002).
The fabrication method illustrated in Fig. 2.5(a) wastes a lot of material, especially
for lattices with a low relative density. This waste can be avoided by using the
Step 1: Slitting and expanding Step 2: Flattening Step 3: Folding
(a)
(b)
Figure 2.5: Two manufacturing routes for truss cores: (a) perforating and folding
and (b) slitting, flattening and folding. Reproduced from Kooistra and Wadley (2007).
11
Chapter 2. Literature review
manufacturing process proposed by Kooistra and Wadley (2007). The fabrication
method is illustrated in Fig. 2.5(b) and involves slotting and expanding a metal
sheet. Note that only the pyramidal core can be manufactured with this method.
The strength of truss cores can be significantly increased at low relative densities
if the solid struts are replaced by hollow tubes. Queheillalt and Wadley (2005a)
fabricated a lattice by brazing together an array of stainless steel tubes as shown
in Fig. 2.6. At low relative densities, the compressive strength of this lattice was
found to be higher than that of a pyramidal core.
Figure 2.6: Lattices made from hollow tubes arranged in (a) square and (b) diamond
orientations. Reproduced from Queheillalt and Wadley (2005a).
The potential of a lattice made from hollow tubes was extended further by Queheil-
lalt and Wadley (2005b, 2011) who fabricated a pyramidal core made from hollow
tubes. The measured compressive and shear strengths of this hollow pyramidal lat-
tice are compared to other core topologies in Fig. 2.7; the hollow pyramidal lattice
is significantly stronger than other lattices at low values of relative density. This
experimental work was extended by finite element simulations to develop collapse
mechanism maps for the hollow pyramidal lattice loaded in compression (Pingle
et al., 2011a) and in shear (Pingle et al., 2011b). In Chapter 3, the finite element
method will be used to examine the potential of using a surface treatment to increase
the compressive strength of the hollow pyramidal lattice.
2.1.3 Prismatic cores
Prismatic cores are composed of an assembly of plates; four examples are given in
Fig. 2.8. Prismatic cores are easier to manufacture than truss cores and consequently
12
2.1 Lattice materials used as core topologies
0.1
1
0.01 0.1 1
Hollow pyramidal
Pyramidal
(transverse)
Square
honeycombτ
σ
pk
Yρ
ρ
Shear
Y-frame
(longitudinal)
Corrugated
(longitudinal)
2
0.1
1
10
0.01 0.1 1
Square honeycomb
Hollow pyramidal
Compression
Corrugated
σ
σ
pk
Yρ
ρ
Y-frame
Pyramidal
(a) (b)
Figure 2.7: Measured normalised peak strength of stainless steel lattices loaded in (a)
compression and (b) shear. The lattice compressive strength σ¯pk and shear strength
τ¯pk are normalised by the relative density ρ¯ and the yield strength of stainless steel
σY . Data taken from Coˆte´ et al. (2004); Coˆte´ et al. (2006); Queheillalt and Wadley
(2011); Rubino et al. (2008a); Zok et al. (2004).
(a) (b)
(c) (d)
Figure 2.8: Examples of prismatic cores: (a) corrugated, (b) Y-frame, (c) square
honeycomb and (d) diamond cores. Reproduced from Zok et al. (2005) and Coˆte´ et al.
(2006).
13
Chapter 2. Literature review
they are more attractive for industrial applications. Square honeycomb and diamond
cores can be assembled by cutting and slotting metal sheets, whereas corrugated and
Y-frame cores require a folding operation. Finally, the sheets can be either brazed
(for lab-scale specimens) or welded together (for industrial applications) to make a
sandwich panel.
The compressive and shear strengths of the square honeycomb were measured by
Coˆte´ et al. (2004) and its performances were found to be similar to those of the
pyramidal core, see Fig. 2.7(a). On the other hand, the mechanical properties of
corrugated and diamond cores were measured by Coˆte´ et al. (2006). The compressive
and transverse shear strengths of the corrugated core are inferior to those of the
square honeycomb, see Fig. 2.7. In contrast, the longitudinal shear strength of the
corrugated core is similar to that of the square honeycomb (see Fig. 2.7(b)), making
the corrugated core a promising topology for sandwich panels.
The compressive response of the Y-frame core was studied numerically by Pedersen
et al. (2006). They found that the introduction of a horizontal flange between the
upper and lower sections of the Y-frame changed its behaviour from a stretching-
dominated to a bending-dominated structure. This reduces the compressive strength
of the Y-frame core, but increases its energy absorption capacities. The compressive
and shear strengths of the Y-frame core were measured by Rubino et al. (2008a) and
they were found to be similar to those of the corrugated core, as shown in Fig. 2.7.
2.1.4 Hierarchical topologies
At high relative densities, the compressive strength of prismatic and truss cores
is governed by yielding or plastic buckling of the core members. However, at low
relative densities, the core members are slender and collapse by elastic buckling.
This change in collapse mechanism reduces significantly the compressive strength of
the core. To increase the elastic buckling strength of the core members it is possible
to fabricate hierarchical cores, where the core members are made from a sandwich
construction of a smaller scale (Fleck et al., 2010).
This principle was investigated experimentally by Kooistra et al. (2007) who fabri-
cated a second order corrugated core with a relative density ρ¯ = 0.02, as shown in
Fig. 2.9(a). A comparison with a first order corrugated core of equal relative den-
sity (Fig. 2.9(b)) revealed that the compressive strength of the hierarchical core was
14
2.1 Lattice materials used as core topologies
about ten times stronger. However, the second order construction is only beneficial
for low values of relative density, ρ¯ < 0.05.
(a) (b)
Figure 2.9: Photographs showing the collapse mechanisms of (a) second order and
(b) first order corrugated cores, both with a relative density ρ¯ = 0.02. Reproduced
from Kooistra et al. (2007).
2.1.5 Multifunctionality
In previous sections, lattice materials were presented with a strong emphasis on their
compressive and shear strengths. However, the advantages of lattice materials are
not limited only to their structural performances. For example, closed-cell foams can
provide thermal and sound insulations (Gibson and Ashby, 1997; Ashby et al., 2000).
On the other hand, open-cell foams can be used as heat exchangers by pumping a
fluid through the foam pores (Lu et al., 1998; Evans et al., 2001). Prismatic and
truss cores can also be used as heat exchangers and, depending on the application,
they usually outperform metal foams (Lu et al., 2005; Valdevit et al., 2006a).
Truss cores, such as the Kagome´ lattice, have the potential to be actuated by elon-
gating or contracting the core members (Hutchinson et al., 2003; Symons et al.,
2005a,b). This work on the actuated Kagome´ lattice was extended to other core
topologies by Mai and Fleck (2009). Finally, another advantage of lattice materials
is their use as the core of sandwich structures, and this is covered in the next section.
15
Chapter 2. Literature review
2.2 Sandwich structures under quasi-static three-
point bending
Three families of lattice materials were introduced in the previous section: metal
foams, truss and prismatic cores. Their high compressive and shear strengths at low
densities make them ideal candidates for the core of sandwich structures. In this
section, the collapse mechanisms applicable to metallic sandwich structures loaded in
three-point bending are introduced along with simple analytical formulae to predict
their collapse strength. Subsequently, the research conducted on sandwich beams
with either a metal foam core, a truss core, or a prismatic core will be presented
in turn. For an extensive coverage of the design and the mechanics of sandwich
structures the reader is referred to the classic textbooks of Plantema (1966), Allen
(1969) and Zenkert (1995).
2.2.1 Collapse mechanisms
A sandwich panel loaded in three-point bending can collapse in different ways –
referred to as collapse mechanisms – depending on its geometry and material prop-
erties. The relevant collapse mechanisms for metallic sandwich panels are: face yield,
core shear and indentation (Ashby et al., 2000). These three collapse mechanisms
are illustrated in Fig. 2.10 for a sandwich beam with a metal foam core. For each
collapse mechanism, simply supported and clamped sandwich beams are compared
to show the effect of the boundary conditions. Analytical predictions for each of the
three collapse mechanisms are presented below.
(a) Face yield (b) Core shear (mode B) (c) Indentation
Figure 2.10: A sandwich beam with a metal foam core loaded in three-point bending
that collapsed by (a) face yield, (b) core shear (mode B) and (c) indentation. For
each collapse mechanism, simply supported and clamped sandwich beams are shown.
Reproduced from Tagarielli and Fleck (2005).
16
2.2 Sandwich structures under quasi-static three-point bending
Consider the simply supported and clamped sandwich panels loaded in three-point
bending and illustrated in Fig. 2.11(a) and (b), respectively. The geometry of the
panel is defined by: the span 2L, the overhang H , the face-sheet thickness t, the
core thickness c and the width b (normal to the plane). The core and face-sheets are
made from rigid perfectly plastic solids with a yield strength σcY and σfY , respectively.
In addition, the core is assumed to have a shear strength τ cY .
F, δ
2L
H H
F/2 F/2
t
σ , τcY
σY
f
c
Y
a
c
F/2 F/2
2L
F, δ
at
σ , τcY cY
σY
f
c
(a) (b)
Figure 2.11: (a) Simply supported and (b) clamped sandwich panels loaded in three-
point bending.
Face yield
Slender sandwich panels can collapse by face yield, see Fig. 2.10(a). Assuming the
formation of two global plastic hinges on each side of the indenter, see Fig. 2.12(a),
the collapse load for a simply supported panel is given by (Ashby et al., 2000):
Ffy =
2bt(c + t)
L σ
f
Y +
bc2
2Lσ
c
Y . (2.1)
For clamped beams, two additional plastic hinges are forming at the fixed ends, see
Fig. 2.12(b), which doubles the collapse load (Tagarielli and Fleck, 2005). Equation
(2.1) assumes that the face-sheets and core yields simultaneously; however, other
models proposed by Triantafillou and Gibson (1987a) and McCormack et al. (2001)
have neglected the contribution of the core in their estimation of the face yield
collapse load by setting σcY = 0 in Eq. (2.1).
17
Chapter 2. Literature review
(a) (b)
F, δ
a
Global plastic hinge
F, δ
a
Global plastic hinge
Figure 2.12: Face yield collapse mechanism for (a) simply supported and (b) clamped
sandwich panels.
Core shear
When a sandwich panel is loaded in three-point bending, the transverse shear force
is carried mostly by the core and consequently the panel can collapse by core shear.
Two modes of core shear are possible for simply supported panels and they are
illustrated in Fig. 2.13. Mode A assumes the formation of four plastic hinges in the
face-sheets and shearing of the core over a length 2(L + H). The collapse load for
mode A is given by (Ashby et al., 2000):
FA =
bt2
L σ
f
Y + 2bcτ cY
(
1 + HL
)
. (2.2)
On the other hand, mode B assumes four additional plastic hinges at the supported
ends and shearing of the core over a length 2L. The collapse load for mode B is
given by (Ashby et al., 2000):
FB =
2bt2
L σ
f
Y + 2bcτ cY . (2.3)
F, δ
2L
F/2 F/2
t Mp
a
H H
Core shear
(a) (b)
F, δ
2L
F/2 F/2
Mpa
H H
Core shear
Plastic hinge
Figure 2.13: Core shear collapse mechanisms: (a) mode A and (b) mode B.
18
2.2 Sandwich structures under quasi-static three-point bending
The operative mode is the one associated with the lowest collapse load; mode A is
operative for short overhangs H , whereas mode B is active for large values of H .
The transition between the two modes occurs at an overhang Ht given by:
Ht =
t2σfY
2cτ cY
. (2.4)
Tagarielli and Fleck (2005) noted that only mode B is applicable for clamped bound-
ary conditions, see Fig. 2.10(b). Core shear can also occur while the face-sheets are
loaded elastically (Chiras et al., 2002), in these cases, the contribution of the faces
can be neglected by setting σfY = 0 in Eq. (2.2) and (2.3).
Indentation
The indentation collapse mechanism consists of a localised failure of the core un-
derneath the mid-span indenter, see Fig. 2.10(c). Assuming the formation of four
plastic hinges in the front face and yielding of the core in compression, as shown
in Fig. 2.14, the collapse load can be obtained with an upper bound calculation
(Ashby et al., 2000):
F = 4Mpλ + (a+ λ)bσ
c
Y , (2.5)
where the full plastic moment of the face-sheet is Mp = σfY bt2/4. Minimising F with
respect to λ gives the collapse load:
FI = 2tb
√
σfY σcY + abσcY , (2.6)
F, δ
2L
F/2 F/2
t Mp
a
σ cY
θλ
Plastic hinge
Figure 2.14: Indentation collapse mechanism.
19
Chapter 2. Literature review
where λ = t
√
σfY /σcY . Note that a lower bound calculation gives the same collapse
load; therefore it can be concluded that Eq. (2.6) gives the exact solution for rigid
perfectly plastic materials. It is interesting to note that the indentation collapse
load is independent of the span 2L and of the boundary conditions.
The analysis above assumes that the front face and core both yield plastically. Ad-
ditional models have been developed in which the core yields plastically, but the
front face is loaded elastically, see for example Soden (1996), Shuaeib and Soden
(1997) and Steeves and Fleck (2004). In all studies cited above, the compressive
strength of the core is considered, but its longitudinal shear strength is neglected.
This assumption is not adequate for cores with a longitudinal shear strength com-
parable to their compressive strength, such as the corrugated and Y-frame cores.
To account for this, an analytical indentation model incorporating the longitudinal
shear strength of the core has been developed by Rubino et al. (2008a, 2010).
Collapse mechanism maps
The operative collapse mechanism is the one associated with the lowest collapse
load. Collapse mechanism maps provide a graphical representation of the opera-
tive collapse mechanism and associated collapse load as a function of two design
parameters. An example is given in Fig. 2.15(a) where the collapse mechanism and
contours of normalised collapse load F¯ = F/(2bLσfY ) are plotted as a function of t/c
and c/(2L). Those results were obtained for a normalised indenter size a/(2L) = 0.1
and the following material properties: σcY /σfY = 0.005 and τ cY /σfY = 0.005.
The normalised indenter size has a strong effect upon the collapse mechanism map;
reducing a/(2L) expands the indentation regime as shown in Fig. 2.15(b). The
map is also sensitive to the boundary conditions; the face yield domain decreases
when the boundary conditions are changed from simple support to fully-clamped
(Tagarielli and Fleck, 2005).
Collapse mechanism maps were developed for different core topologies including:
metal foams (McCormack et al., 2001; Bart-Smith et al., 2001), the tetrahedral core
(Deshpande and Fleck, 2001), the pyramidal core (Zok et al., 2004; Coˆte´ et al., 2007),
the square honeycomb core (Zok et al., 2005) and the corrugated core (Valdevit et al.,
2006a). The research done on each of these core topologies will be presented below
in more details.
20
2.2 Sandwich structures under quasi-static three-point bending
(a) (b)
0.01
0.1
0.01 0.1
Core shear
c
2L
t
c
0.5
0.5
Face
yield Indentation
a = 0.005
= 0.1
2L
a
2L
0.01
0.1
0.01 0.1
c
2L
t
c Core
shear
Face
yield
Indentation
0.5
0.5
1x10-2
3x10-3
1x10-3
3x10-4
1x10-4
3x10-5
F = 1x10-5
Figure 2.15: Collapse mechanism maps for a simply supported sandwich panel: (a)
contours of normalised collapse load F¯ = F/(2bLσfY ) for a/(2L) = 0.1 and (b) influ-
ence of the normalised indenter size a/(2L) upon the operative collapse mechanism.
In both cases, σcY /σ
f
Y = 0.005 and τ cY /σ
f
Y = 0.005. Adapted from Ashby et al. (2000).
2.2.2 Sandwich beams with a metal foam core
The three-point bending response of simply supported sandwich beams with a metal
foam core was investigated by Triantafillou and Gibson (1987b), McCormack et al.
(2001) and Bart-Smith et al. (2001). Chen et al. (2001) extended this work to simply
supported beams loaded in four-point bending whereas Tagarielli and Fleck (2005)
investigated the influence of the boundary conditions by comparing the responses
of simply supported and clamped beams. These studies revealed that the analytical
formulae presented in Section 2.2.1 gave an accurate prediction of the measured
collapse load. The measurements were also in good agreement with finite element
predictions, in which the metal foam core was treated as a homogeneous solid using
the constitutive model of Deshpande and Fleck (2000a).
2.2.3 Sandwich beams with a truss core
Sandwich beams with a tetrahedral core were first manufactured by investment
casting of an aluminium-silicon alloy (Deshpande and Fleck, 2001) or a beryllium-
copper alloy (Chiras et al., 2002). Subsequently, the technique illustrated in Fig.
2.5(a) was employed by Rathbun et al. (2004) to fabricate sandwich beams made
from stainless steel. In these three studies, the beams were tested under simply
supported boundary conditions and all geometries considered were found to collapse
21
Chapter 2. Literature review
by core shear. The measured collapse loads were also found to be in good agreement
with the analytical predictions of Wicks and Hutchinson (2001, 2004).
Stainless steel sandwich beams with a pyramidal core were tested by Zok et al. (2004)
and Coˆte´ et al. (2007). Zok et al. (2004) tested both simply supported and clamped
beams and proposed an orthotropic constitutive law to model the behaviour of the
pyramidal core. On the other hand, Coˆte´ et al. (2007) tested simply supported
beams only, but focused on their resistance to fatigue.
Sandwich beams with a truss core made from stainless steel tubes, as shown in Fig.
2.6, were tested by Rathbun et al. (2006b) under simply supported and clamped
boundary conditions. For both end conditions, the collapse load measured with
the diamond orientation (see Fig. 2.6) exceeded the one measured with the square
orientation. This is due to the fact that the longitudinal shear strength of the core
is greater in the diamond orientation than in the square orientation.
2.2.4 Sandwich beams with a prismatic core
Stainless steel sandwich beams with a square honeycomb core were tested by Zok
et al. (2005) under simply supported and clamped boundary conditions. The beams
collapsed by core shear or face yield depending upon their geometry, and the mea-
sured responses were used to calibrate an orthotropic constitutive law for the core.
Simply supported sandwich beams with a corrugated core were tested by Valdevit
et al. (2006a) and the measured collapse loads were found to be in good agreement
with analytical predictions. This work was extended by Rubino et al. (2010) who
compared the three-point bending strength of sandwich beams with a corrugated
core to those with a Y-frame core for both simply supported and clamped boundary
conditions. Both core topologies were found to have a similar three-point bending
strength. In all cases, the sandwich beams collapsed by indentation, and the mea-
surements were found to be in good agreement with finite element simulations, as
shown in Fig. 2.16.
2.2.5 Optimisation studies
The studies reported above have investigated the three-point bending response of
sandwich beams with different core topologies. However, one fundamental question
22
2.3 Lattice materials and sandwich structures under dynamic loading
(b) Finite Element Prediction
Experiment
F, δ
(a)
L
t
Figure 2.16: A simply supported sandwich beam with a Y-frame core loaded in three-
point bending and collapsing by indentation: (a) experiment and (b) finite element
prediction. A side view showing half the beam (left) and a view of the core deformation
obtained by sectioning the beam at mid-span (right) are included. Reproduced from
Rubino et al. (2010).
is left unanswered: for a given load, which core topology provides the lightest sand-
wich panel? This optimisation problem was investigated analytically by Valdevit
et al. (2004), Rathbun et al. (2005) and Wei et al. (2006). These studies revealed
that the differences between the mass of the optimised sandwich panels are very
small; about 15% variations between the different core topologies considered. Con-
sequently, the authors suggest that the choice of core topology should be based on
considerations other than strength, such as manufacturing costs and the potential
for multifunctionality (Rathbun et al., 2005).
2.3 Lattice materials and sandwich structures un-
der dynamic loading
So far, the behaviour of lattice materials and sandwich structures has been examined
for quasi-static loading only. In this section, their behaviour under dynamic loading
will be addressed. The dynamic collapse response of a structure can be significantly
different from its quasi-static response because of three effects described below:
23
Chapter 2. Literature review
Material strain-rate sensitivity. The yield strength of several alloys, such as
mild steel, increases with increasing strain-rate (Jones, 1989; Stout and Fol-
lansbee, 1986).
Inertia stabilisation. Take for example a column loaded dynamically in compres-
sion; its lateral inertia will stabilise it against buckling. This effect can be
important at low impact velocities for which wave propagation effects are neg-
ligible (Calladine and English, 1984; Karagiozova and Jones, 1996).
Wave propagation. As the impact velocity is increased, wave propagation effects
become important. Again, consider a column loaded dynamically in compres-
sion. An axial plastic wave propagates along the column and only the portion
of material engulfed by the plastic wave buckles (Vaughn et al., 2005; Vaughn
and Hutchinson, 2006). If the impact velocity is greater than the plastic wave
speed, the column does not buckle and material accumulates at the impacted
end (Taylor, 1948; McShane, 2007).
Three topics will be covered in this section. First, the literature on the dynamic
compressive response of lattice materials is reviewed. Second, the low-velocity im-
pact response of sandwich structures is presented, and finally, the use of sandwich
panels for blast protection is covered.
2.3.1 Dynamic compressive response of lattice materials
When a sandwich panel is loaded dynamically, some energy is absorbed by crushing
of the core. Consequently, the dynamic performances of sandwich panels are highly
dependent on the compressive response of the core (Liang et al., 2007; McShane
et al., 2007). The dynamic compressive response of a lattice material can be mea-
sured using a strain-gauged Kolsky bar, as shown in Fig. 2.17. The stresses on
front and back faces can be obtained from two independent tests. In the front face
configuration, the specimen is fixed to the striker and they are both fired on the
Kolsky bar. Alternatively, in the back face configuration, the specimen is fixed on
the stationary Kolsky bar and impacted by the striker.
This method was used by Deshpande and Fleck (2000b) to measure the dynamic
compressive response of closed-cell and open-cell aluminium foams, and their dy-
namic responses were found to be relatively similar to their quasi-static responses.
The same method was used to test truss and prismatic cores made from stainless
24
2.3 Lattice materials and sandwich structures under dynamic loading
Strain gauge
Kolsky bar
Kolsky bar
Specimen
Striker
Specimen
Striker
(a) Back face configuration
(b) Front face configuration
Figure 2.17: Direct impact Kolsky bar setup in (a) back face and (b) front face
configurations. Adapted from Lee et al. (2006).
steel, such as the square honeycomb core (Radford et al., 2007); the corrugated and
Y-frame cores (Tilbrook et al., 2007); the I-core (Ferri et al., 2006) and the pyra-
midal core (Lee et al., 2006). The measured dynamic compressive response of truss
and prismatic cores was stronger than its quasi-static response. This strengthening
is attributed to the three factors introduced above: (i) material strain-rate sensi-
tivity, (ii) inertia stabilisation of the core members against buckling and (iii) wave
propagation effects. In general, material strain-rate effects were found to have a
small influence on the results. On the other hand, inertia stabilisation effects were
predominant at low velocities (less than approximately 50 m/s, depending on the
lattice) whereas wave propagation effects appeared at high impact velocities.
In most practical engineering applications, a sandwich panel is not loaded in uni-
form compression. For example, during a ship collision, a sandwich hull structure
deforms by indentation. What will be the importance of the three dynamic strength-
ening effects mentioned above if the loading is changed from uniform compression
to localised indentation? This question will be addressed in Chapter 6.
25
Chapter 2. Literature review
2.3.2 Sandwich structures subjected to a low-velocity im-
pact
Sandwich structures used in aerospace, automotive and marine applications are ex-
posed to low-velocity impacts (below 10 m/s). These can be replicated in laboratory
using a drop weight apparatus. The literature on the low-velocity impact response
of sandwich structures is vast and can be divided in two categories aiming at:
1. quantifying the damage caused by a localised impact on a sandwich panel or
2. comparing the quasi-static and low-velocity responses of a sandwich structure.
The first category regroups most of the literature on low-velocity impacts and is
primarily interested in composite sandwich panels used in the aerospace industry.
These structures often have a Nomex or aluminium honeycomb core, which is glued
to the face-sheets that can be made from graphite, glass or carbon fibre reinforced
polymers. When these panels are subjected to a localised impact, delamination,
core crushing and debonding of the core and faces can occur. Several studies have
proposed methods to quantify and measure the damage inferred to the sandwich
panel (Hazizan and Cantwell, 2003; Meo et al., 2005; Castanie´ et al., 2008; Park
et al., 2008; Shin et al., 2008). These studies are not of primary interest in this
thesis; the second category is more relevant.
The second category includes a few papers only. The quasi-static and low-velocity
impact responses of simply supported sandwich beams with a metal foam core were
investigated by Yu et al. (2008). The authors showed that the collapse mechanism
obtained during drop weight tests at 5 m/s is the same as the one observed quasi-
statically. This finding was corroborated by other experimental studies on sandwich
beams with a metal foam core (Yu et al., 2003; Crupi and Montanini, 2007) or a
honeycomb core (Crupi et al., 2012).
The low-velocity impact response of sandwich beams with a corrugated core or a
Y-frame core will be investigated experimentally and numerically in Chapter 5. The
objective is to determine whether the collapse mechanism observed during a ship
collision at 5 m/s is the same as the one observed under quasi-static loading.
26
2.3 Lattice materials and sandwich structures under dynamic loading
2.3.3 Sandwich structures subjected to blast loading
Most of the literature on dynamic loading of metallic sandwich structures focuses
on their use for blast mitigation. Even though the emphasis of this thesis is on
the quasi-static and low-velocity impact response of sandwich structures used in
the construction of ship hulls, it is insightful to cover the blast response of sand-
wich structures to (i) determine the sensitivity of the dynamic response to the core
topology and (ii) evaluate the accuracy of the finite element method to simulate the
measured dynamic response.
Analytical and numerical studies
An analytical model was developed by Fleck and Deshpande (2004) to describe the
structural response of a clamped sandwich beam subjected to a blast. Their model
consists of three sequential stages:
Stage I is the one-dimensional fluid-structure interaction during which the front
face acquires an uniform velocity.
Stage II involves compression of the core until the velocities of the front and back
faces equalise.
Stage III is the dissipation of the beam’s kinetic energy through plastic bending
and stretching.
The analytical predictions of Fleck and Deshpande (2004) were found to be in good
agreement with the finite element simulations of Qiu et al. (2003). In the simulations
done by Qiu et al. (2003), the core was treated as a homogeneous solid based on the
metal foam constitutive model of Deshpande and Fleck (2000a). Xue and Hutchinson
(2004) performed finite element simulations on sandwich beams with three fully-
meshed core topologies: a square honeycomb core, a corrugated core and a pyramidal
core. The maximum back face deflection was used as a metric to compare the
performances of sandwich plates to those of a monolithic plate of the same mass.
The results are reproduced in Fig. 2.18 where the normalised maximum back face
deflection w/L is plotted as a function of the normalised blast impulse I/M
√
σY /ρf
for both air and water blasts. Note that the blast impulse is I, the back face
deflection is w, the beam has a span 2L, an areal mass M and is made from a
material with a yield strength σY and a density ρf . The main findings are:
27
Chapter 2. Literature review
1. Sandwich beams outperform monolithic beams on an equal mass basis.
2. The advantage of sandwich structures is sensitive to the core topology; pris-
matic cores, such as the corrugated and square honeycomb cores, are preferable
to truss cores because of their high longitudinal strength.
3. The benefit of using a sandwich construction is more important for a water
blast than for an air blast because the impulse transmitted to the front face of
the sandwich beam is lower than the one transmitted to a monolithic beam.
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.1 0.2 0.3 0.4
Air blast
Monolithic plate
Corrugated core
Pyramidal core
Square
honeycomb
core
0
0.1
0.2
0.3
0.4
0.5
0 0.2 0.4 0.6 0.8
Monolithic plate
Corrugated core
Pyramidal core
Square
honeycomb
core
Water blast
W
L
W
L
M√σ / ρY f
I
M√σ / ρY f
I
(a) (b)
Figure 2.18: Normalised maximum back face deflection w/L as a function of the
normalised blast impulse I/(M
√
σY /ρf ). Adapted from Rathbun et al. (2006a).
Experimental studies
Radford et al. (2005) proposed to use metal foam projectiles to produce dynamic
pressures representative of air and water blasts. This technique can generate peak
pressures around 100 MPa over a duration of about 100 µs. This approach was
used by several authors (Rathbun et al., 2006a; Radford et al., 2006a; Rubino et al.,
2008b) to compare, on an equal mass basis, the dynamic performances of clamped
sandwich beams with different core topologies to that of monolithic beams. Similar
work was done on circular plates (Radford et al., 2006b; McShane et al., 2006) and
on rectangular plates (Rubino et al., 2009). All studies have shown that sandwich
structures have a lower maximum back face deflection than their monolithic coun-
terparts. In addition, a good correlation was found between experiments and finite
element simulations; see for example Fig. 2.19 where the results of Rubino et al.
(2009) are reproduced.
28
2.4 Ship hull design
Experiments Simulations
(a)
(b)
(c)
Figure 2.19: Photographs and finite element simulations of (a) monolithic, (b) Y-
frame and (c) corrugated plates impacted by a metal foam projectile with a momentum
of 3 kNs/m2. All plates have the same mass. Reproduced from Rubino et al. (2009).
One disadvantage of using metal foam projectiles to simulate blast loading is that
the pressure pulse is localised over a central patch whereas a real shock wave would
spread over the entire front face. To address this issue, a few experiments have been
performed on rectangular sandwich plates loaded by an explosive charge detonated
in air (Dharmasena et al., 2008, 2011). Again, sandwich panels were found to have
lower back face deflections than monolithic plates.
2.4 Ship hull design
Commercial vessels need to resist accidental loads such as ship collisions and ground-
ings (when a ship hits the seabed). Recently, the growing threat of terrorist attacks
and the impressive growth of offshore oil and gas extraction systems have forced
commercial shipbuilders to also consider explosions as a potential accident scenario
(ISSC, 2006b). In this section, design strategies to minimise the consequences of
such accidents are presented with a focus on ship collisions and groundings. These
two accident scenarios are more likely to occur than explosions and are also more
relevant to this thesis.
29
Chapter 2. Literature review
2.4.1 Regulations
The design of ship hulls was influenced by major pollution catastrophes such as the
grounding of the Exxon Valdez of the coast of Alaska in 1989. When the single hull
construction of the tanker was breached, over ten million gallons of crude oil flooded
into a sensitive ecosystem within the first five hours only (Skinner and Reilly, 1989).
The environmental consequences of the oil spill attracted the public’s attention and
forced the political power to legislate on the maritime transport of oil and other
hazardous chemical products. The United States introduced the Oil Pollution Act
in 1990 (OPA 90) and the International Maritime Organization (IMO) followed with
a similar legislation in 1992. One important aspect of this legislation was that all
new tankers should be built with a double hull construction.
The OPA 90 and IMO regulations offer design guidelines for the construction of
double hull vessels; the plate thickness and the spacing between inner and outer
hulls are given as a function of the overall ship dimensions. From a structural point
of view, these design requirements are sub-optimal. However, the IMO regulations
allow shipbuilders to propose alternative designs as long as their crashworthiness is
proven to be equal or superior to that of a conventional double hull construction.
Only the ADNR regulations, regarding navigation on the Rhine River, quantify the
energy absorption capacity of a ship hull; the side structure must absorb 22 MJ if
the design differs from a conventional construction.
2.4.2 Evaluating the resistance of ship hulls
The issue of ship collisions and grounding events can be tackled from two different
angles. The first angle is to investigate how and why accidents occur. This leads to
the development of risk assessment methods to quantify the probability and severity
of different accident scenarios, see for example the work of Pedersen (2002) and
Friis-Hansen and Simonsen (2002).
The second perspective is more relevant to this thesis and examines the response
of the ship structure during an accident. Different methods have been developed
to evaluate the structural damage occurred during a ship collision or a grounding
event. Those methods can be divided in four categories: (i) empirical methods, (ii)
simplified analytical methods, (iii) finite element simulations and (iv) experimental
tests on full-scale and/or lab-scale structures.
30
2.4 Ship hull design
The most famous empirical method is certainly the one developed by Minorsky
(1959). By analysing several accidents, the author suggested that the energy ab-
sorbed by the ship structure E during an accident is given by:
E(MJ) = 47.2RT (m3) + 32.7 , (2.7)
where RT is the volume of deformed material. This empirical formula neglects several
parameters such the vessel speed, the structural arrangements and the material
properties. Several authors have proposed revised Minorsky formulae, for example
Pedersen and Zhang (2000).
Simplified analytical methods allow to study the motion of ships during a collision
(external dynamics) as well as the deformation of the structural components (internal
dynamics). The effect of the surrounding water, vessel speed and collision angle can
be used to estimate the contact forces (Simonsen, 1997; Pedersen and Zhang, 1998;
Zhang, 1999; Tabri et al., 2009).
Finite element simulations can capture accurately the deformation of a ship hull
under different loading scenarios. The method takes into account large deformations,
contact friction, non-linear material properties and even fracture. However, the
definition of an adequate fracture criterion is problematic due to the large mesh size
used to discretise the ship structure (Urban, 2002; To¨rnqvist, 2003; Simonsen and
To¨rnqvist, 2004).
Experimental tests are useful to validate the results of finite element simulations but
they are expensive when performed on full-scale structures. Tests on lab-scale struc-
tures are less expensive, but present scaling issues (Jones, 1979). See Pedersen et al.
(1993) and Wang et al. (2000) for a review of full-scale and lab-scale experiments,
respectively.
2.4.3 Design against collision and grounding
The regulations mentioned above have forced shipbuilders to migrate from a sin-
gle hull to a double hull construction. A double hull structure can absorb more
energy than a single hull construction of the same mass and overall dimensions
(Ozguc et al., 2005). The energy absorption capacity of a ship structure can also be
increased by using a high strength steel instead of traditional mild steel in its fabri-
31
Chapter 2. Literature review
cation (Lehmann and Peschmann, 2005). Recently, Paik (2003) proposed structural
design alternatives to increase the resistance of ship structures against collisions and
grounding events. A few examples are given here:
Soft bow. During a ship collision, the rigid bulbous bow of the striking ship can
penetrate the side of the stuck vessel. To limit the damage induced to the
struck ship, soft buffer bows can be designed to absorb a part of the collision
energy (Endo et al., 2004; Yamada and Endo, 2004).
Variable bottom height. About 65% of grounding accidents involve the front
part of the vessel. To account for this risk, the spacing between the inner
and outer hulls can be varied along the length of the ship to offer a better
protection of the inner hull (LR, 1991; Amdahl and Kavlie, 1995).
Mid-deck tanker. The mid-deck construction has double hull sides, but a single
hull bottom with a deck at mid-height. In the event of a side collision, the
mid-deck provides structural reinforcement to the side panels. On the other
hand, in the event of grounding, the design allows the oil to flow upward in
the mid-deck since the density of sea water is superior to the density of oil.
See Kawaichi et al. (1995) for more details.
Sandwich construction. The inner and outer hulls can be connected with a light
core to increase the structural performances of the ship hull. This concept was
proposed by Jones (1976), but it has been adopted by shipbuilders only re-
cently. Examples of core topologies considered for ship hulls include the X-core
(To¨rnqvist, 2003; To¨rnqvist and Simonsen, 2004) and the Y-core (Ludolphy,
2001; Konter et al., 2004). The Y-frame core is of particular interest in this
thesis and its development is reported in the next section.
2.4.4 Development of the Y-frame sandwich hull design
Full-scale collision tests were performed by Damen Schelde Naval Shipbuilding to
compare the resistance of different conventional and innovative hull designs. Four
structures were considered: (i) a conventional single hull, (ii) a conventional double
hull, (iii) a double hull with a Y-frame core and (iv) a double hull made from
two corrugated panels. These designs are illustrated in Fig. 2.20. The results
were striking; all structures were perforated except the Y-frame double hull, which
showed only a small dent after not only one, but two successive impacts, see Fig.
32
2.4 Ship hull design
2.21 (Wevers and Vredeveldt, 1999). Those experimental results were also supported
by finite element simulations, which revealed that the Y-frame double hull design
can absorb two times more energy than a conventional double hull before perforation
of the outer hull (Naar et al., 2002).
(a) (b) (c) (d)
Figure 2.20: Four hull designs considered in full-scale collision tests: (a) a conven-
tional single hull, (b) a conventional double hull, (c) a double hull with a Y-frame core
and (d) a double hull made from two corrugated panels. Adapted from ISSC (2006a).
Figure 2.21: Photographs of full-scale collision tests performed on the Y-frame dou-
ble hull structure. Adapted from Konter et al. (2004) and Wevers and Vredeveldt
(1999).
The examination of the Y-frame double hull structure after the full-scale collision
tests revealed that the inner hull played a minor role and underwent no visible plastic
deformation (Wevers and Vredeveldt, 1999). This motivated Damen Schelde Naval
Shipbuilding to develop a Y-frame single hull design where the Y-frame stiffeners
are fixed to the bulkhead with the back face absent. Full-scale collision tests were
also performed on this Y-frame single hull structure and it was found to be as
resistant as the Y-frame double hull design. However, the results did not allow a
direct comparison between the Y-frame single and double hull constructions; the
two designs had a slightly different geometry and a different mass. The strength of
33
Chapter 2. Literature review
Y-frame single and double hull designs will be compared on an equal mass basis in
Chapter 4.
The Y-frame single hull was approved by the classification society Germanischer
Lloyd and it is now a patented design1. So far, 24 inland waterway tankers have
been manufactured with the Y-frame single hull design. The Y-frame profile is
obtained by welding together folded mild steel plates, see Fig. 2.22 (Graaf et al.,
2004). The superior crashworthiness of the Y-frame single hull design (compared
to a conventional double hull design) allows Damen Schelde Naval Shipbuilding to
build the inline waterway tankers with four tanks of 550 m3 instead of six tanks
of 380 m3 as usually required by the ADNR regulations (Vredeveldt and Roeters,
2004). Using four tanks instead of six is a considerable competitive advantage for
Damen Schelde Naval Shipbuilding as it reduces the cost of piping equipment.
Figure 2.22: Construction of inland waterway tankers with a Y-frame single hull
structure. Adapted from Graaf et al. (2004) and Vredeveldt and Roeters (2004).
2.5 Concluding remarks
Metallic lattice materials have been introduced in this chapter and classified in three
families: foams, truss and prismatic lattices. They were shown to be ideal candidates
for sandwich cores on the account of their high strength and low density. Of all
stainless steel lattice materials tested, the pyramidal lattice made from hollow tubes
offered the highest compressive strength. The potential of using a surface treatment
to increase the strength of this lattice will be the subject of the next chapter.
Prismatic lattices, like the corrugated and Y-frame cores, are easier to manufacture
than truss lattices. Consequently they are more attractive for industrial applications
such as the construction of ship hulls. Extensive research has been conducted on the
1International Patent Application No.PCT/NL99/00757 with Publication No.WO 00/35746
34
2.5 Concluding remarks
quasi-static and the blast responses of both the corrugated core and the Y-frame
core. However, some aspects regarding their use in sandwich ship hulls remain
unclear and will be addressed in Chapters 4, 5 and 6. A comparison between the
strength of single and double hull designs will be presented in Chapter 4 for quasi-
static loading. Then, the response of a sandwich hull construction to a ship collision
at 5 m/s is compared to its quasi-static response in Chapter 5. Finally, the influence
of the loading velocity upon the indentation response of a sandwich ship hull is
addressed in Chapter 6.
35
Chapter 2. Literature review
36
Chapter 3
Compressive response of a
carburised pyramidal lattice
Summary
The finite element method was used to simulate the compressive response of a pyra-
midal lattice made from inclined tubes or solid struts. First, the response of both
lattices was compared for two levels of material strain hardening: (i) a perfectly plas-
tic solid and (ii) a strain hardening solid representative of stainless steel. The com-
pressive collapse mode of the lattice was relatively insensitive to the level of strain
hardening. In contrast, strain hardening increased the peak compressive strength of
both inclined tubes and struts with a slenderness ratio inferior to ten. Second, the
response of a carburised pyramidal lattice was simulated. Carburisation increases
the yield strength of the parent material and influences the collapse mode of the
lattice; the transition between plastic and elastic buckling occurred at a smaller
slenderness ratio when the lattice was carburised. Carburisation also increased the
peak compressive strength of the lattice, except for those collapsing by elastic buck-
ling. Finally, a comparison with other lattice materials revealed that the pyramidal
lattice made from carburised tubes is stronger than aluminium or titanium lattices
and as strong as those made from carbon fibre reinforced polymers.
37
Chapter 3. Compressive response of a carburised pyramidal lattice
3.1 Introduction
Lightweight metallic sandwich panels comprise of two face-sheets separated by a
low density core (Zenkert, 1995). The core has to be light but also strong as its
compressive and shear strengths have a significant influence on the overall bending
strength of the sandwich panel. The compressive and shear strengths of the core
depend on three parameters: (i) the topology, (ii) the relative density and (iii) the
mechanical properties of the material from which the core is made (Ashby, 2006).
Over the last decade, several different core topologies have been manufactured from
type 304 stainless steel, for example: the corrugated core (Coˆte´ et al., 2006), the
square honeycomb core (Coˆte´ et al., 2004) and the pyramidal core made from solid
struts (Zok et al., 2004) or hollow tubes (Queheillalt and Wadley, 2005b, 2011). The
measured compressive strength σ¯pk of each core topology is plotted in Fig. 2.7(a)
(on page 13) as a function of the relative density ρ¯. The results indicate clearly that
the hollow pyramidal core is stronger than other core topologies, especially for low
values of relative density.
The unit cell of a hollow pyramidal lattice is shown in Fig. 3.1(a); its geometry is
defined by the inclination angle ω, the tube length l, the tube outside diameter d and
the wall thickness t. Pingle et al. (2011a) used the finite element method to examine
the influence of the tube geometry upon the collapse mode of a hollow pyramidal
lattice with ω = 55◦. Their results are presented in the form of a collapse mechanism
map1 reproduced in Fig. 3.2. Six collapse modes are identified dependent upon the
tube slenderness ratio l/d and the normalised wall thickness t/d. This map was
developed for a hollow pyramidal lattice made from stainless steel, which possesses
an important strain hardening capacity.
In the first part of this study, the effect of strain hardening upon the collapse mode
and upon the compressive strength of a hollow pyramidal lattice will be evaluated.
In the second part, the effect of carburisation will be investigated. Carburisation is a
heat treatment process that hardens the surface of a metal. A low temperature car-
burisation treatment has been developed recently for stainless steel and, depending
on the duration of the treatment, carburisation depths of 25-70 µm can be achieved
(Cao et al., 2003; Michal et al., 2006). In this study, the effect of carburisation upon
the collapse mode and upon the compressive strength of a hollow pyramidal lattice
will be examined. The potential of carburisation to increase the strength of lattice
1Similar collapse mechanism maps were developed for vertical tubes made from aluminium by
Andrews et al. (1983) and Guillow et al. (2001).
38
3.1 Introduction
l t
d
ω = 55°
4 2 cos
2
k l
ω
+
4
2 co
s
2k
l
ω
+
,ε σ
1x
2x
3x
sin
d
ω
d
1x
2x
k
(a)
(b)
Figure 3.1: (a) Unit cell of the hollow pyramidal lattice. (b) Top view of the lattice.
39
Chapter 3. Compressive response of a carburised pyramidal lattice
dt /
d
l
Mode B
plastic
barrelling
Mode F elastic buckling
Mode D
two-lobe
diamond
Th
in
sh
el
l b
uc
kl
in
g
0.01 0.1 0.50.1
1
10
100
Mode A
axisymmetric
bulge
Mode C
ρ =
0.5
0.3
0.1
0.03
0.01
0.003
0.001
Mode E
global plastic
buckling
0.00030.0001
a
b
multi-lobe
diamond
c
e
f
d
a b c
d e f
Collapse mechanism map
Collapse modes
Figure 3.2: Collapse mechanism map for a hollow pyramidal lattice made from
AISI 304 stainless steel. Examples of the six collapse modes (A-F) are also included.
Representative geometries considered in this study are indicated by filled black circles
and contours of relative density ρ¯ are plotted as grey lines. Adapted from Pingle et al.
(2011a).
40
3.2 Geometry and analytical collapse load of the pyramidal lattice
materials made from stainless steel has not been investigated before; however, other
surface treatments, such as plasma electrolytic oxidation and electrochemincal an-
odizing, have been used recently to increase the compressive strength of aluminium
metal foams (Abdulla et al., 2011; Bele et al., 2011; Dunleavy et al., 2011).
The effect of strain hardening and carburisation will be studied using the finite
element method. The study will focus on two trajectories on the collapse mechanism
map shown in Fig. 3.2. The first trajectory is indicated by a dashed line and
represents tubes with a normalised wall thickness t/d = 0.1. The second trajectory
is the right hand side of the map and represents pyramidal lattices made from solid
struts, t/d = 0.5. For both trajectories, the slenderness ratio l/d will be varied from
1 to 100.
This chapter is organised as follows. First, the geometry and the analytical collapse
load of the hollow pyramidal lattice are presented in Section 3.2. Second, the effect
of strain hardening is addressed in Section 3.3 and then, the effect of carburisation
is analysed in Section 3.4. Both Sections 3.3 and 3.4 include a description of the
finite element models, an analysis of the compressive responses and a comparison
between the compressive strength of inclined tubes and solid struts.
3.2 Geometry and analytical collapse load of the
pyramidal lattice
3.2.1 Relative density
The unit cell of a hollow pyramidal lattice is shown in Fig. 3.1(a). Its geometry
is defined by the tube length l, the outside diameter d, the wall thickness t and
the inclination ω. A top view of the lattice, see Fig. 3.1(b), reveals that the tube
centres are offset by a distance k from the centre of the pyramid. The distance k is
constrained such that:
k ≥ kmin =
d
√
1 + sin2 ω
2 sinω . (3.1)
The tubes are touching each other at the face-sheets when k = kmin. The relative
density of the hollow pyramidal lattice is given by:
41
Chapter 3. Compressive response of a carburised pyramidal lattice
ρ¯ = 2π (d
2 − (d− 2t)2)
(4k + 2l cosω)2 sinω =
2π td
(
1− td
)
(
2γ + ld cosω
)2 sinω
, (3.2)
where γ is a function of the inclination ω and is given by:
γ =
√
1 + sin2 ω
2 sinω . (3.3)
In this study, the tube spacing k = kmin and the inclination angle ω = 55◦ in all
cases. With these two parameters fixed, Eq. (3.2) is used to plot contours of relative
density ρ¯ on the collapse mechanism map shown in Fig. 3.2.
3.2.2 Analytical collapse load
When the hollow pyramidal lattice is compressed by a downward displacement δ, a
vertical force P develops in each tube of the lattice. The nominal compressive stress
σ¯ on the front face-sheet can be expressed as:
σ¯ = 8P(4k + 2l cosω)2 , (3.4)
and the corresponding nominal compressive strain is:
ǫ¯ = δl sinω . (3.5)
Assuming that the lattice is made from an elastic perfectly plastic solid, two collapse
modes can be anticipated: (i) plastic collapse or (ii) elastic buckling. An analytical
expression for the plastic collapse load can be obtained by setting:
Ppl = σY
π
4
(
d2 − (d− 2t)2
)
sinω , (3.6)
where the yield strength of the material is σY . Substituting Eq. (3.6) in Eq. (3.4)
returns the plastic collapse stress of the lattice:
σ¯pl = ρ¯σY sin2 ω . (3.7)
42
3.3 Influence of strain hardening
Alternatively, an expression for the elastic buckling load can be obtained with:
Pel =
4π2EI
l2 sinω , (3.8)
where the Young’s modulus of the material is E and the second moment of area of
the tube is I = pi64(d4 − (d − 2t)4). Equation (3.8) assumes that the inclined tube
has both ends built in (Timoshenko and Gere, 1963). Substituting Eq. (3.8) in Eq.
(3.4) returns the elastic buckling stress of the lattice:
σ¯el =
π3E td
(
1− 3 td + 4
(
t
d
)2
− 2
(
t
d
)3
)
(
l
d
(
2γ + ld cosω
))2 sinω . (3.9)
Finally, the transition from plastic collapse to elastic buckling can be obtained by
setting Pel = Ppl. This transition occurs at a slenderness ratio:
l
d =
√
√
√
√
π2E
2σY
(
1− 2 td + 2
( t
d
)2)
. (3.10)
3.3 Influence of strain hardening
3.3.1 Description of the finite element models
All simulations were performed with the implicit solver of the commercially available
finite element software Abaqus (version 6.10). The boundary conditions, mesh,
geometric imperfections, material properties and dimensions employed are detailed
below.
Boundary conditions
It is sufficient to consider only one inclined tube to capture the compressive re-
sponse of the hollow pyramidal lattice. The boundary conditions employed in the
finite element simulations are illustrated in Fig. 3.3. Front and back face-sheets
were modelled as rigid surfaces and a perfect bonding was assumed between the
inclined tube and the face-sheets. The back face was clamped against translational
43
Chapter 3. Compressive response of a carburised pyramidal lattice
and rotational displacements whereas the front face had a prescribed downward dis-
placement δ, see Fig. Fig. 3.3. No lateral motion (in the x1 and x2 directions) and
no rotation were allowed for the front face. A hard frictionless contact was defined
between all surfaces of the model allowing the lattice to densify at large values of
nominal compressive strain ǫ¯.
ω = 55°
A
A
Section A-A
d
t
P,δ
L
x3
x2
x1
Figure 3.3: Finite element model used to simulate the compressive response of an
inclined tube.
Mesh and geometric imperfections
The inclined tubes were meshed using three-dimensional hexahedral elements (C3D8R
in Abaqus notation) with at least five elements through the wall thickness. A small
geometric imperfection was included in all simulations. The imperfection had the
shape of the first buckling mode with an amplitude ζ = 0.05t. The sensitivity of the
compressive response upon the choice of imperfection is discussed in Appendix 3.A.
Material properties
The material properties were chosen to be representative of AISI 304 stainless steel.
This material was used in previous experimental and numerical studies on the hollow
pyramidal lattice (Queheillalt and Wadley, 2005b, 2011; Pingle et al., 2011a,b). The
parent material of the lattice was modelled as a rate-independent elastic-plastic solid
in accordance with J2-flow theory. The elastic regime was linear and isotropic, as
characterised by a Young’s modulus E = 200 GPa and a Poisson’s ratio ν = 0.3.
The yield strength of the material was set to σAY = 200 MPa. In this section, two
levels of strain hardening are compared: (i) Et = 0, representing a perfectly plastic
44
3.3 Influence of strain hardening
solid and (ii) Et = 2 GPa, a realistic value for stainless steel. The uniaxial tensile
responses of these two material models are compared in Fig. 3.4.
0
100
200
300
400
500
600
0 0.05 0.1 0.15 0.2
Tr
ue
s
tre
ss
(M
Pa
)
Logarithmic strain
E = 0t
Perfectly plastic solid
E = 2 GPat
Stainless steel
Figure 3.4: Uniaxial tensile responses of the two material models employed in the
finite element simulations analysing the influence of strain hardening.
Dimensions analysed
In this study, the inclination angle was kept fixed at ω = 55◦ and two different cross-
sections were considered: (i) a tube with t/d = 0.1 and (ii) a solid strut corresponding
to t/d = 0.5. For these two values of t/d, the slenderness ratio l/d was varied from
1 to 100.
3.3.2 Results for an inclined tube
The collapse of an inclined stainless steel tube with t/d = 0.1 can be catalogued
into four distinct modes (A, D, E and F) depending on the slenderness ratio l/d,
see Fig. 3.2. Four selected geometries with l/d = 1, 3, 20 and 100 that collapse
in mode A, D, E and F, respectively, are marked on the map in Fig. 3.2 and their
compressive responses are shown in Fig. 3.5. The responses are plotted in terms
of the nominal compressive stress σ¯, normalised by the relative density ρ¯ and the
yield strength σAY , versus the nominal compressive strain ǫ¯. For each geometry, the
compressive response is given for two strain hardening moduli: (i) Et = 0 and (ii)
Et = 2 GPa. The deformed meshes corresponding to these responses are given in
Table 3.1 to exemplify the four collapse modes of the inclined tube.
45
Chapter 3. Compressive response of a carburised pyramidal lattice
ε
ρσYA
σ
ε
ρσYA
σ
ε
ρσYA
σ
ε
ρσYA
σ
(a) (b)
(c) (d)
0
0.5
1
1.5
2
2.5
0 0.1 0.2 0.3 0.4 0.5
E = 2 GPat
E = 0t
l/d = 1
ρ = 0.15
Mode A: axisymmetric bulge
0
0.5
1
1.5
2
0 0.1 0.2 0.3 0.4 0.5
E = 2 GPat
E = 0t
l/d = 3
Mode D: two-lobe diamond
ρ = 0.063
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.05 0.1 0.15 0.2
E = 2 GPat
E = 0t
l/d = 20
Mode E: global plastic buckling
ρ = 0.004
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.05 0.1 0.15 0.2
E = 2 GPa
E = 0
t
t
l/d = 100
Mode F: elastic buckling
ρ = 0.0002
Figure 3.5: Influence of the strain hardening modulus Et on the compressive response
of an inclined tube t/d = 0.1. Results are given for (a) l/d = 1, (b) l/d = 3, (c)
l/d = 20 (d) l/d = 100.
46
3.3 Influence of strain hardening
E = 0
ε = 0.2
l/d = 1
l/d = 3
l/d = 20
l/d = 100
ρ = 0.15
ρ = 0.064
ρ = 0.004
ρ = 0.0002
Geometry t E = 2 GPat
ε = 0.5ε = 0.5
ε = 0.5ε = 0.5
ε = 0.2
ε = 0.2ε = 0.2
Mode A
Mode D
Mode E
Mode F
Table 3.1: Influence of the strain hardening modulus Et on the deformed meshes of
an inclined tube t/d = 0.1. Results are given for selected values of l/d.
47
Chapter 3. Compressive response of a carburised pyramidal lattice
All compressive responses shown in Fig. 3.5 exhibit a peak stress σ¯pk. The axial
compressive stress in the tube reaches the yield strength when σ¯/(ρ¯σAY ) = sin2 ω ≈
0.67, see Eq. (3.7). Of the four collapse modes shown in Fig. 3.5, only mode F:
elastic buckling has a peak stress inferior to the yield limit. The influence of strain
hardening will be discussed below for each collapse mode.
Mode A: axisymmetric bulge is the collapse mode for l/d = 1, see Fig. 3.5(a).
Material strain hardening has a significant influence on the compressive response for
this collapse mode; the peak stress increases by a factor of three when Et is increased
from 0 to 2 GPa.
An inclined tube with l/d = 3 collapses by Mode D: two-lobe diamond, see Fig.
3.5(b). The two “bumps” in the compressive response represent the formation of
the two lobes. Again, material strain hardening increases the peak stress, but the
increase is slightly less than for l/d = 1.
Mode E: global plastic buckling is the operative collapse mode for an inclined tube
with l/d = 20 and its compressive response is shown in Fig. 3.5(c). Note that strain
hardening has no influence on the peak stress; in both cases the axial compressive
stress in the tube reaches the yield strength σ¯/(ρ¯σAY ) ≈ 0.67, then the tube buckles
and forms a plastic hinge at mid-length. However, the post-peak response is stronger
for Et = 2 GPa than for Et = 0.
Finally, an inclined tube with a slenderness ratio l/d = 100 collapses by Mode F:
elastic buckling, see Fig. 3.5(d). As expected, strain hardening as no influence on
the peak stress for this collapse mode. The stress drops sharply after the peak due
to the development of a plastic hinge at mid-length. Despite the formation of a
plastic hinge, strain hardening has a negligible effect on the post-peak response.
The deformed meshes, corresponding to the responses given in Fig. 3.5, are shown
in Table 3.1. The deformed meshes of simulations with a strain hardening solid
(Et = 2 GPa) have more diffuse plastic hinges than those obtained with a perfectly
plastic solid (Et = 0). Nevertheless, the collapse mode appears to be insensitive to
the strain hardening modulus.
3.3.3 Results for an inclined solid strut
The map in Fig. 3.2 indicates that the collapse of an inclined solid strut (t/d = 0.5)
made from stainless steel can be catalogued into three distinct modes (B, E and F)
48
3.3 Influence of strain hardening
depending on the slenderness ratio l/d. As marked in Fig. 3.2, one geometry was
selected in each of the three collapse modes: l/d = 3 for mode B, l/d = 20 for mode
E and l/d = 100 for mode F. The compressive responses of these three selected
geometries are shown in Fig. 3.6. In each plot, results are given for Et = 0 and
Et = 2 GPa. In addition, the deformed meshes corresponding to these compressive
responses are shown in Table 3.2.
An inclined solid strut with l/d = 3 collapses by mode B: plastic barrelling and its
compressive response is particularly sensitive to strain hardening, see Fig. 3.6(a).
Note that the response for this collapse mode does not exhibit a peak stress (below
ǫ¯ = 0.5). For this particular case, the peak stress σ¯pk will be defined as the stress at
a nominal compressive strain ǫ¯ = 0.5. The same definition was adopted by Pingle
et al. (2011a).
As the slenderness ratio of the inclined solid strut is increased, the collapse mode
changes to mode E: global plastic buckling, see Fig. 3.6(b), and subsequently to
mode F: elastic buckling, see Fig. 3.6(c). The influence of strain hardening on the
responses of these two collapse modes was discussed above for inclined tubes, and
the results for inclined struts are similar: (i) the peak stress for l/d = 20 and 100 is
insensitive to Et and (ii) strain hardening strengthens the post-peak response when
l/d = 20, but has minimal effect when l/d = 100.
The deformed meshes shown in Table 3.2 for inclined solid struts confirm the ob-
servations made above for inclined tubes: strain hardening results in more diffuse
plastic hinges, but the collapse mode is insensitive to Et. For the tube and the solid
strut, our simulations indicate that the collapse mechanism map shown in Fig. 3.2
is relatively insensitive to the strain hardening modulus, at least for the two values
of Et considered in this study.
3.3.4 Comparison between tube and solid strut
Above, the compressive responses of inclined tubes and solid struts were presented
separately for selected values of slenderness ratio l/d. To summarise these results,
the normalised peak stress σ¯pk/(ρ¯σAY ) is plotted in Fig. 3.7 as a function of the
relative density ρ¯ for both inclined tubes (t/d = 0.1) and solid struts (t/d = 0.5).
For both values of t/d, the collapse modes are identified and results are given for
Et = 0 and Et = 2 GPa. Three regimes can be identified in Fig. 3.7:
49
Chapter 3. Compressive response of a carburised pyramidal lattice
(a)
(b)
(c)
ρσYA
σ
ε
0
1
2
3
4
5
0 0.1 0.2 0.3 0.4 0.5
E = 2 GPat
E = 0t
l/d = 3
ρ = 0.18
Mode B: plastic barrelling
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.05 0.1 0.15 0.2
E = 2 GPat
E = 0t
l/d = 20
ρ = 0.011
Mode E: global plastic buckling
0
0.05
0.1
0.15
0.2
0 0.05 0.1 0.15 0.2
E = 2 GPa
E = 0
t
t
l/d = 100
ρ = 0.0006
Mode F: elastic buckling
ρσYA
σ
ρσYA
σ
ε
ε
Figure 3.6: Influence of the strain hardening modulus Et on the compressive response
of an inclined solid strut t/d = 0.5. Results are given for (a) l/d = 3, (b) l/d = 20
and (c) l/d = 100.
50
3.3 Influence of strain hardening
E = 0Geometry t E = 2 GPat
ε = 0.4ε = 0.4
ε = 0.2ε = 0.2
ε = 0.2ε = 0.2
l/d = 3
l/d = 20
l/d = 100
ρ = 0.18
ρ = 0.011
ρ = 0.0006
Mode B
Mode E
Mode F
Table 3.2: Influence of the strain hardening modulus Et on the deformed meshes of
an inclined solid strut t/d = 0.5. Results are given for selected values of l/d.
51
Chapter 3. Compressive response of a carburised pyramidal lattice
ρσYA
σ
ρ
pk
0.1
1
0.001 0.01 0.1
E = 2 GPa
E = 0
t/d = 0.5
t
tt/d = 0.1
EF B
EF
A
D
5
Collapse modes for t/d = 0.1
Collapse modes for t/d = 0.5
FE predictions
Eq. (3.9)
Eq. (3.7)
Analytical
predictions
Figure 3.7: Influence of the strain hardening modulus Et on the compressive strength
of an inclined tube t/d = 0.1 and an inclined solid strut t/d = 0.5.
1. The inclined tube with ρ¯ < 0.0005 and the inclined solid strut with ρ¯ <
0.002 both collapse by Mode F: elastic buckling. For this collapse mode, the
normalised peak stress increases with increasing relative density as suggested
by the analytical expression for elastic buckling, Eq. (3.9). This equation is
also plotted in Fig. 3.7 and is in excellent agreement with the simulations, for
both the tube and the solid strut. As mentioned above, strain hardening has
no influence on the peak stress for this collapse mode.
2. When the relative density of the inclined tube is in the range 0.0005 ≤ ρ¯ ≤
0.01, the collapse mode is E: global plastic buckling. The same collapse mode
is operative for inclined solid struts with relative densities 0.002 ≤ ρ¯ ≤ 0.04.
Equation (3.7) predicting the plastic collapse of the lattice is also included in
Fig. 3.7; there is a good agreement between the analytical formula and the
simulations. Again, for those intervals of relative density, the peak stress is
insensitive to the level of strain hardening.
3. Finally, when ρ¯ > 0.01 for the inclined tube and when ρ¯ > 0.04 for the inclined
solid strut, the peak stress becomes sensitive to strain hardening; values of
σ¯pk/(ρ¯σAY ) obtained with Et = 2 GPa exceed those obtained with Et = 0.
Note that an inclined tube with ρ¯ = 0.01 and a solid strut with ρ¯ = 0.04 both
52
3.4 Influence of carburisation
correspond to a slenderness ratio l/d = 10, see Fig. 3.2 or refer to Eq. (3.2).
It is clear from Fig. 3.7 that the tube outperforms the solid strut for low values of
relative density, ρ¯ < 0.002. This is because the transition from global plastic buckling
(Mode E) to elastic buckling (Mode F) occurs at a lower value of relative density
for the tube than for the solid strut. In contrast, at high values of relative density,
ρ¯ > 0.1, the solid strut collapses by plastic barrelling (Mode B) and outperforms
the tube, especially for Et = 2 GPa.
3.4 Influence of carburisation
Carburisation is a heat treatment process during which carbon is absorbed by a
metallic part making its surface harder. Different carburisation depths can be
achieved depending upon the duration and temperature of the heat treatment. The
cross-sections of a carburised tube and a carburised solid strut are illustrated in
Fig. 3.8. The carburisation depth h adds a third non-dimensional parameter to the
analysis; the compressive strength of the pyramidal lattice is now governed by l/d,
t/d and h/d (recall that the inclination angle is fixed at ω = 55◦).
h h
t
d
Carburised stainless steel
Annealed stainless steel
d
h
Annealed
stainless steel
Carburised
stainless steel
(a) (b)
Figure 3.8: Cross-sections of a carburised (a) tube and (b) solid strut.
53
Chapter 3. Compressive response of a carburised pyramidal lattice
3.4.1 Description of the finite element models
The boundary conditions, the mesh details and the geometric imperfections used in
this section were the same as those used previously, see Section 3.3.1.
Material properties
Both annealed and carburised stainless steels were modelled as rate-independent
elastic-plastic solids in accordance with J2-flow theory. The elastic regime of both
materials was linear and isotropic, as characterised by a Young’s modulus E = 200
GPa and a Poisson’s ratio ν = 0.3. Each material had a different plastic behaviour.
Annealed stainless steel had a yield strength σAY = 200 MPa and a linear strain
hardening response with a tangent modulus Et = 2 GPa. In contrast, carburised
stainless steel was considered to be perfectly plastic (Et = 0) with a yield strength
σCY = 2 GPa. The yield strength of carburised stainless steel is estimated from a
hardness test reported by Michal et al. (2006). The unixial tensile responses of
annealed and carburised stainless steels are compared in Fig. 3.9.
0
0.5
1
1.5
2
2.5
0 0.05 0.1 0.15 0.2
Tr
ue
s
tre
ss
(G
Pa
)
Logarithmic strain
Annealed stainless steel
Carburised stainless steel
Figure 3.9: Uniaxial tensile responses of annealed and carburised stainless steels
employed in the finite element simulations analysing the influence of carburisation.
Dimensions analysed
Again, the slenderness ratio l/d was varied from 1 to 100 for both the inclined tube
(t/d = 0.1) and the inclined solid strut (t/d = 0.5). Four values of normalised
54
3.4 Influence of carburisation
carburisation depth were considered h/d = 0, 0.02, 0.04 and 0.05. Note that for h/d
= 0.05, the entire cross-section of the tube (t/d = 0.1) is carburised.
3.4.2 Results for an inclined tube
The influence of carburisation upon the compressive response of an inclined tube
with t/d = 0.1 is shown in Fig. 3.10 for the four selected geometries identified in
Section 3.3.2. For each geometry, the results are given for a non-carburised tube,
h/d = 0, and for a tube with a normalised carburisation depth h/d = 0.05. The
deformed meshes corresponding to these responses are given in Table 3.3.
Carburisation significantly increases the peak compressive stress σ¯pk of short inclined
tubes with l/d = 1 and 3, see Fig. 3.10(a) and (b), respectively. Note that both
carburised tubes (h/d = 0.05) have the same peak stress σ¯pk = 6.7ρ¯σAY , which
corresponds to the plastic collapse stress evaluated by Eq. (3.7) by setting the
yield strength to σCY = 10σAY = 2 GPa (recall that for h/d = 0.05 the entire cross-
section of the tube is carburised). In addition, the peak stress occurs at a larger
value of nominal compressive strain for non-carburised tubes than for carburised
ones. Annealed stainless steel has a tangent modulus Et = 2 GPa and the response
of non-carburised tubes display a significant amount of plastic hardening before
reaching the peak stress. On the other hand, carburised stainless steel is modelled
as a perfectly plastic solid and consequently the response of carburised tubes do not
display any plastic hardening.
The non-carburised inclined tube with a slenderness ratio l/d = 20 collapses by
mode E: global plastic buckling and has a peak stress σ¯pk = 0.67ρ¯σAY , see Fig 3.10(c).
Carburising this tube increases the peak stress to σ¯pk = 5.7ρ¯σAY , which is inferior to
the plastic collapse load σ¯pl = 6.7ρ¯σAY prescribed by Eq. (3.7). Thus, carburisation
increases the peak stress, but also changes the collapse mode from mode E: plastic
buckling to mode F: elastic buckling.
The peak stress of long inclined tubes collapsing by mode F: elastic buckling is
insensitive to carburisation, see Fig. 3.10(d). This result was expected as both
annealed and carburised stainless steels have the same Young’s modulus E = 200
GPa. On the other hand, the post-peak response is sensitive to carburisation; the
carburised tube is significantly stronger than the non-carburised one.
Finally, the deformed meshes of non-carburised tubes (h/d = 0) are compared to
55
Chapter 3. Compressive response of a carburised pyramidal lattice
ε
ρσYA
σ
ε
ρσYA
σ
ε
ρσYA
σ
ε
ρσYA
σ
(a) (b)
(c) (d)
0
2
4
6
8
10
0 0.1 0.2 0.3 0.4 0.5
h/d = 0.05
l/d = 1
h/d = 0
ρ = 0.15
Mode A: axisymmetric bulge
0
1
2
3
4
5
6
7
0 0.1 0.2 0.3 0.4 0.5
l/d = 3
h/d = 0
h/d = 0.05
Mode D: two-lobe diamond
ρ = 0.063
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.05 0.1 0.15 0.2
l/d = 100
h/d = 0
h/d = 0.05
Mode F: elastic buckling
ρ = 0.0002
0
1
2
3
4
5
6
0 0.05 0.1 0.15 0.2
l/d = 20
h/d = 0
h/d = 0.05
Mode E: global
plastic buckling
ρ = 0.004
Mode F: elastic buckling
Figure 3.10: Influence of the carburisation depth h/d on the compressive response of
an inclined tube t/d = 0.1. Results are given for (a) l/d = 1, (b) l/d = 3, (c) l/d = 20
(d) l/d = 100.
56
3.4 Influence of carburisation
l/d = 1
l/d = 3
l/d = 20
l/d = 100
ρ = 0.15
ρ = 0.063
ρ = 0.004
ρ = 0.0002
Geometry h/d = 0
ε = 0.5
ε = 0.5
ε = 0.2 ε = 0.2
ε = 0.5
ε = 0.5
h/d = 0.05
Mode A
Mode D
ε = 0.2
Mode E
Mode F
ε = 0.2
Mode F
Table 3.3: Influence of the carburisation depth h/d on the deformed meshes of an
inclined tube t/d = 0.1. Results are given for selected values of l/d.
57
Chapter 3. Compressive response of a carburised pyramidal lattice
those of carburised tubes (h/d = 0.05) in Table 3.3. In general, carburisation has a
relatively small effect on the deformed meshes.
3.4.3 Results for an inclined solid strut
The compressive response of an inclined solid strut (t/d = 0.5) is shown in Fig. 3.11
for the three selected geometries identified in Section 3.3.3. For each geometry, the
response of a non-carburised strut, h/d = 0, is compared to that of a carburised strut
with h/d = 0.05. For completeness, the deformed meshes corresponding to these
responses are displayed in Table 3.4. In contrast with the carburised tube analysed
in the previous section, the cross-section of the carburised strut with h/d = 0.05 is
not entirely carburised; it has an outside layer of carburised stainless steel with an
inside core of annealed stainless steel.
The compressive response of a strut with l/d = 3, which collapses by mode B:
plastic barrelling, is shown in Fig. 3.11(a). Carburisation clearly increases the yield
stress of the lattice from 0.67ρ¯σAY to approximately 1.8ρ¯σAY . However, the slope of
the plastic hardening response, which is characteristic of plastic barrelling, is less
for carburised struts than for non-carburised ones. This can be explained by the
level of strain hardening of the two materials: carburised stainless steel is modelled
as a perfectly plastic solid (Et = 0) whereas annealed stainless steel has a strain
hardening modulus Et = 2 GPa.
Carburisation significantly increases the peak stress of an inclined strut with l/d =
20, see Fig. 3.11(b). The collapse mode of the carburised strut (h/d = 0.05) is
classified as mode F: elastic buckling because at σ¯pk, the axial compressive stress in
the strut is inferior to the yield strength of carburised stainless steel (but greater
than the yield strength of annealed stainless steel). A similar change in collapse
mechanism was observed in the previous section for an inclined tube with the same
slenderness ratio.
Similarly to the inclined tube analysed in the previous section, the peak stress of a
long inclined strut with l/d = 100, which collapses by mode F: elastic buckling, is
insensitive to carburisation, see Fig. 3.11(c). Nevertheless, carburisation strengthens
the post-peak response.
Finally, the deformed meshes of non-carburised struts (h/d = 0) are compared to
those of carburised struts (h/d = 0.05) in Table 3.4. It is clear from Table 3.4 that
58
3.4 Influence of carburisation
(a)
(b)
ε
ρσYA
σ
ε
ρσYA
σ
(c)
ε
ρσYA
σ
0
1
2
3
4
5
0 0.1 0.2 0.3 0.4 0.5
l/d = 3
h/d = 0
h/d = 0.05
ρ = 0.18
Mode B: plastic barrelling
0
0.05
0.1
0.15
0.2
0 0.05 0.1 0.15 0.2
l/d = 100
h/d = 0.05
h/d = 0
ρ = 0.0006
Mode F: elastic buckling
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.05 0.1 0.15 0.2
l/d = 20
h/d = 0.05
h/d = 0
ρ = 0.011
Mode E: global
plastic buckling
Mode F: elastic buckling
Figure 3.11: Influence of the carburisation depth h/d on the compressive response
of an inclined solid strut t/d = 0.5. Results are given for (a) l/d = 3, (b) l/d = 20
and (c) l/d = 100.
59
Chapter 3. Compressive response of a carburised pyramidal lattice
h/d = 0 h/d = 0.05
l/d = 3
l/d = 20
l/d = 100
ρ = 0.18
ρ = 0.011
ρ = 0.0006
Geometry
ε = 0.4
ε = 0.2
ε = 0.2
Mode B
Mode E
Mode F
ε = 0.4
ε = 0.2
ε = 0.2
Mode F
Table 3.4: Influence of the carburisation depth h/d on the deformed meshes of an
inclined solid strut t/d = 0.5. Results are given for selected values of l/d.
60
3.4 Influence of carburisation
carburisation has only a small effect on the deformed meshes.
3.4.4 Comparison between tube and solid strut
The normalised peak stress σ¯pk/(ρ¯σAY ), for both t/d = 0.1 and 0.5, is plotted as a
function of relative density in Fig. 3.12. The results are shown for four selected
values of normalised carburisation depth h/d ranging from 0 to 0.05. In addition,
the collapse modes are identified for both the tube and the solid strut. Note that the
collapse modes are different for non-carburised (h/d = 0) and carburised (h/d > 0)
lattices. Four regimes can be identified in Fig. 3.12:
1. Inclined tubes with ρ¯ < 0.0005 and inclined solid struts with ρ¯ < 0.002 both
collapse by mode F: elastic buckling. It is clear from Fig. 3.12 that carburisa-
tion has no effect on the peak stress for this particular collapse mode.
2. When the relative density of the inclined tube is in the range 0.0005 ≤ ρ¯ <
0.004 and that of the inclined strut is between 0.002 ≤ ρ¯ < 0.02, the non-
ρσYA
σpk Tube t/d = 0.1Strut t/d = 0.5
ρ
0.1
1
10
0.001 0.01 0.1
h/d = 0.05
0.04
0.02
0
0.05
0.04
0.02
0
EF B
EF AD
Collapse modes for t/d = 0.1
EF ADh/d > 0:
h/d = 0:
EF Bh/d > 0:
h/d = 0:
Collapse modes for t/d = 0.5
Figure 3.12: Influence of the carburisation depth h/d on the compressive strength
of an inclined tube t/d = 0.1 and an inclined solid strut t/d = 0.5.
61
Chapter 3. Compressive response of a carburised pyramidal lattice
carburised lattices collapse by mode E: global plastic buckling whereas the
carburised ones collapse by mode F: elastic buckling. In this regime, car-
burisation increases the peak stress of the lattice and this increase is more
important for the tube than for the strut.
3. The peak stress of inclined tubes with ρ¯ > 0.004 has reached the yield strength
of the material. Carburisation increases the peak stress of the lattice, but does
not change the transition between the plastic collapse modes A, D and E. There
is a similar regime for inclined struts with 0.02 ≤ ρ¯ < 0.1, but it covers only
mode E: global plastic buckling.
4. This regime is specific to inclined struts with ρ¯ ≥ 0.1, which collapse by
Mode B: plastic barrelling. Recall that the response for this collapse mode
does not exhibit a peak stress, see Fig. 3.11(a), and σ¯pk was defined as the
stress at a nominal compressive strain ǫ¯ = 0.5. Based on this definition, the
normalised peak stress appears to be insensitive to carburisation, but this
result is dependent upon the definition of σ¯pk.
3.4.5 Position of carburised lattices on the strength-density
chart
A material property chart allows us to position different materials on a figure where
each axis is a material property (Ashby, 2010). A chart of strength versus density is
presented in Fig. 3.13. The chart was generated using the software CES EduPack
20102. The strength of fully-dense materials such as metals, ceramics, composites
and polymers is compared to that of lattice materials such as metal and polymer
foams, tetrahedral lattices made from aluminium (Al) and pyramidal lattices made
from titanium (Ti) and carbon fibre reinforced polymer (CRFP). For comparison
purposes, the results of the finite element simulations for an inclined tube (t/d = 0.1)
with a normalised carburisation depth h/d = 0.05 are also plotted in Fig. 3.13.
Carburised stainless steel was assumed to have a density ρs = 8000 kg/m3, hence
the density of a carburised stainless steel lattice is given by ρl = ρ¯ρs.
The results indicate that carburised pyramidal lattices are stronger than their metal-
lic counterparts made from aluminium or titanium. For densities below 0.1 Mg/m3,
the carburised pyramidal lattices are positioned at the frontier of material space,
2Granta Design Limited, Rustat House, 62 Clifton Road, Cambridge, CB1 7EG, UK.
62
3.5 Concluding remarks
t/d = 0.1
h/d = 0.05
Natural
materials
Composites
Ceramics
PMMA
GFRP
Ti textile
TMC collinear
Al t
etra
hed
ral
CFR
P py
ram
idal
Ti pyramidal
TMC
Al2O3
C
SiC Ti alloys
Ni alloys
WC
Lead
alloys
Steels
Zinc
alloysAl
alloys
DiamondUnattainable material space
Un
at
ta
in
ab
le
m
at
er
ia
l s
pa
ce
Metals
Polymer
foams
Metal
foams
Polymers and
elastomers
Density (Mg / m3)
Co
m
pr
es
siv
e s
tre
ng
th
(M
Pa
)
10–2
10–1
100
101
102
103
104
105
101 10210–2 10–1 100
Figure 3.13: Strength versus density material chart. The simulated compressive
strength of a pyramidal lattice made from carburised tubes (t/d = 0.1, h/d = 0.05) is
also included. Al, aluminium; CRFP, carbon fibre reinforced polymers; Ti, titanium;
TMC, titanium matrix composites.
performing as well as the strongest pyramidal lattices made from carbon fibre rein-
forced polymer. Recall that carburised stainless steel was assumed to possess a yield
strength σCY = 2 GPa in this study. If carburisation (or another heat treatment)
is able to increase the yield strength above 2 GPa, this would expand the current
material space.
3.5 Concluding remarks
The finite element method was used to simulate the compressive response of a pyra-
midal lattice made from tubes (t/d = 0.1) or solid struts (t/d = 0.5), both with an
63
Chapter 3. Compressive response of a carburised pyramidal lattice
inclination angle ω = 55◦. First, the effect of material strain hardening was exam-
ined by comparing the compressive response of a lattice made from stainless steel
to that of a lattice made from a perfectly plastic solid. Strain hardening was found
to increase the compressive strength of lattices with a slenderness ratio l/d inferior
to ten, but had no influence on the compressive strength of lattices with l/d > 10.
Furthermore, strain hardening had a negligible effect on the collapse mode of the
pyramidal lattice. This holds true for both lattices made from tubes and those made
from solid struts.
Then, the influence of carburisation upon the compressive response of a pyrami-
dal lattice was analysed. The slenderness ratio l/d at which the collapse mode
changes from plastic to elastic buckling was less for carburised lattices than for their
non-carburised counterparts. Carburisation also increased the peak stress of the
lattice, except for geometries that collapse by elastic buckling. This increase of the
peak stress was more important for a lattice made from tubes than for one made
from solid struts. Finally, the performances of the pyramidal lattice made from car-
burised tubes were compared to other engineering materials and lattices on a chart of
strength versus density. The carburised lattice is stronger than other metallic lattices
made from aluminium or titanium and offers similar performances to pyramidal lat-
tices made from carbon fibre reinforced polymers. The simulations presented in this
chapter suggest that the carburisation surface treatment can significantly enhance
the strength of lattice materials, and this combination has the potential to expand
the current material space. However, the embrittlement that may be caused by the
carburisation surface treatment was neglected in the simulations presented above,
and experimental tests are necessary to validate this assumption and to evaluate the
accuracy of the finite element predictions.
3.A Influence of geometric imperfections
In this appendix, the sensitivity of the compressive response to the choice of geo-
metric imperfection is explored. The imperfection consists of one or multiple elastic
buckling modes, which can have different amplitudes. The effect of the number of
modes superimposed and the effect of amplitude will be addressed below. The sim-
ulations were done for an inclined tube t/d = 0.1 made from annealed stainless steel
(σAY = 200 MPa and Et = 2 GPa).
64
3.A Influence of geometric imperfections
3.A.1 Influence of the number of superimposed modes
The effect of the number of modes superimposed upon the compressive response
of an inclined tube with t/d = 0.1 is shown in Fig. 3.14 for selected values of
slenderness ratio l/d. In each plot, three cases are compared: (i) a perfect structure
(no imperfection), (ii) an imperfection of amplitude ζ = 0.05t in the form of the
first buckling mode and (iii) an imperfection of amplitude ζ = 0.05t in the form of
the first four buckling modes superimposed. Except for the case of l/d = 1, the
compressive response of an inclined tube is imperfection sensitive; case (i) differs
from cases (ii) and (iii). However, the results indicate that the compressive response
is relatively insensitive to the number of modes superimposed; cases (ii) and (iii)
are similar. For this reason, an imperfection in the shape of the first buckling mode
only was included in all simulations.
ε
ρσYA
σ
ε
ρσYA
σ
ε
ρσYA
σ
ε
ρσYA
σ
(a) (b)
(c) (d)
0
0.5
1
1.5
2
2.5
3
0 0.1 0.2 0.3 0.4 0.5
Perfect
1st Mode
4 Modes
l/d = 1
ρ = 0.15
Mode A: axisymmetric bulge
0
0.5
1
1.5
2
2.5
0 0.1 0.2 0.3 0.4 0.5
Perfect
1st Mode
4 Modes
l/d = 3
Mode D: two-lobe diamond
ρ = 0.063
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.05 0.1 0.15 0.2
Perfect
1st Mode
4 Modes
l/d = 20
Mode E: global plastic buckling
ρ = 0.004
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.01 0.02 0.03 0.04 0.05
Perfect
1st Mode
4 Modes
l/d = 100
Mode F: elastic buckling
ρ = 0.0002
Figure 3.14: Influence of imperfection shape on the compressive response of an
inclined tube t/d = 0.1. In all cases, the imperfection amplitude is ζ = 0.05t. Results
are given for (a) l/d = 1, (b) l/d = 3, (c) l/d = 20 (d) l/d = 100.
65
Chapter 3. Compressive response of a carburised pyramidal lattice
3.A.2 Influence of amplitude
The effect of imperfection amplitude upon the compressive response of an inclined
tube with t/d = 0.1 is shown in Fig. 3.15 for selected values of l/d. In each plot,
results are given for a perfect structure (no imperfection) and for an imperfection in
the form of the first buckling mode with three different amplitudes ζ = 0.01t, 0.05t
and 0.1t. Except for the case of l/d = 3, the compressive response of an inclined tube
is insensitive to the imperfection amplitude in the range ζ = 0.01t − 0.1t. Based
on the results of Fig. 3.15, an imperfection amplitude ζ = 0.05t was used in all
simulations.
ε
ρσYA
σ
ε
ρσYA
σ
ε
ρσYA
σ
ε
ρσYA
σ
(a) (b)
(c) (d)
0
0.5
1
1.5
2
2.5
3
0 0.1 0.2 0.3 0.4 0.5
Perfect
ζ = 0.01t
ζ = 0.05t
ζ = 0.10t
l/d = 1
ρ = 0.15
Mode A: axisymmetric bulge
0
0.5
1
1.5
2
2.5
0 0.1 0.2 0.3 0.4 0.5
Perfect
ζ = 0.01t
ζ = 0.05t
ζ = 0.10t
l/d = 3
Mode D: two-lobe diamond
ρ = 0.063
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.05 0.1 0.15 0.2
Perfect
ζ = 0.01t
ζ = 0.05t
ζ = 0.10t
l/d = 20
Mode E: global plastic buckling
ρ = 0.004
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.01 0.02 0.03 0.04 0.05
Perfect
ζ = 0.01t
ζ = 0.05t
ζ = 0.10t
l/d = 100
Mode F: elastic buckling
ρ = 0.0002
Figure 3.15: Influence of imperfection amplitude on the compressive response of an
inclined tube t/d = 0.1. In all cases, the imperfection shape is in the form of the
1st buckling mode. Results are given for (a) l/d = 1, (b) l/d = 3, (c) l/d = 20 (d)
l/d = 100.
66
3.A Influence of geometric imperfections
3.A.3 Influence of imperfection on the deformed meshes
The effect of including a geometric imperfection on the deformed meshes of an
inclined tube with t/d = 0.1 is shown in Table 3.5 for selected values of l/d. Results
are shown for a perfect structure (no imperfection) and for simulations with an
imperfection in the form of the first buckling mode with an amplitude ζ = 0.05t. For
l/d = 1, the deformed meshes are imperfection insensitive. In contrast, introducing
an imperfection has a strong influence on the deformed meshes of inclined tubes with
l/d ≥ 3. Note that for l/d = 20 and 100, a higher order buckling mode is obtained
when no imperfections are included.
Perfect
l/d = 1
l/d = 3
l/d = 20
l/d = 100
ρ = 0.15
ρ = 0.064
ρ = 0.004
ρ = 0.0002
Geometry First mode ζ = 0.05t
ε = 0.5
ε = 0.5
ε = 0.2
ε = 0.2
Mode A
Mode D
Mode E
Mode F
ε = 0.2
ε = 0.2
ε = 0.5
ε = 0.5
Table 3.5: Influence of a geometric imperfection on the deformed meshes of an
inclined tube t/d = 0.1. Results are given for selected values of l/d.
67
Chapter 3. Compressive response of a carburised pyramidal lattice
68
Chapter 4
The influence of the back face on
the bending response of prismatic
sandwich beams
Summary
Stainless steel sandwich beams with a corrugated core or a Y-frame core have been
tested in three-point bending and the role of the face-sheets has been assessed by
considering beams with (i) front-and-back faces present, and (ii) front face present
but back face absent. These competing beam designs were compared on an equal
mass basis by doubling the front face thickness when the back face is absent. The
quasi-static, three-point bending responses were measured under simply supported
and clamped boundary conditions. For both end conditions and for both core topolo-
gies, the beams containing front-and-back faces underwent indentation beneath the
mid-span roller whereas Brazier plastic buckling was responsible for the collapse of
beams without a back face. Three-dimensional finite element predictions were in
good agreement with the measurements and gave additional insight into the defor-
mation modes. The finite element method was also used to study the effect of (i)
mass distribution between core and face-sheets and (ii) beam span upon the collapse
response of a simply supported sandwich panel. Panels of short span are plastically
indented by the mid-span roller and the panels without a back face are stronger than
those with front-and-back faces present. In contrast, panels of long span undergo
Brazier plastic buckling, and the presence of a back face strengthens the panel.
69
Chapter 4. The influence of the back face on the bending response
4.1 Introduction
Oil tanker spills pose a significant environmental threat to the oceans and coast-
lines of the world: 60% of worldwide oil transportation is by tankers and many
heavily trafficked routes pass through regions of high marine biodiversity (Burgherr,
2007). The conventional double hull design, with minimal mechanical coupling be-
tween inner and outer hulls, is commonly used to safeguard oil tankers against spills.
Recently, design alternatives have been proposed to improve the structural perfor-
mances of ship hulls over those normally achieved with a conventional double hull
construction, see for example the review by Paik (2003). One such alternative is
to employ a sandwich construction to increase the stiffness, strength and energy
absorption of the hull.
An example of sandwich construction is the Y-frame double hull design, as proposed
by Damen Schelde Naval Shipbuilding1 and as illustrated in Fig. 4.1(a). Full-scale
collision tests have been performed on this structure and its resistance to tearing was
found to exceed that of a conventional double hull design (Wevers and Vredeveldt,
1999). In these collision trials, the inner hull played a minor role and underwent
negligible plastic deformation. This motivated the development of a single hull
structure where the Y-frame stiffeners are welded directly to the bulkheads as shown
in Fig. 4.1(b). Full-scale collision tests have also been performed on this single hull
Y-frame structure. It has similar crashworthiness to the Y-frame double hull design,
but it is significantly simpler and cheaper to manufacture. Several inland waterway
tankers have been manufactured using the Y-frame single hull design (Graaf et al.,
2004). The corrugated core, under the trade-name Navtruss2, is a competing design
to the Y-frame. No large-scale collision tests on the Navtruss design have been
reported in the open literature, and little is known about its crashworthiness relative
to that of the Y-frame core.
The relative performance of corrugated and Y-frame cores has been explored re-
cently for a range of loadings in a laboratory setting. For example, the out-of-plane
compressive strength and longitudinal shear strength of the Y-frame core and cor-
rugated core have been investigated by Rubino et al. (2008a) and Coˆte´ et al. (2006),
respectively. The three-point bending response of sandwich beams with a corrugated
core was studied by Valdevit et al. (2006a); they proposed failure maps for simply
supported beams. This work was extended by Rubino et al. (2010) who compared
1Damen Schelde Naval Shipbuilding, Glacisstraat 165, 4381 SE Vlissingen, The Netherlands.
2Astech Engineering Products Inc., 3030 Red Hill Ave., Santa Ana, CA 92705, USA.
70
4.1 Introduction
(a) (b)
Figure 4.1: The Y-frame sandwich core in (a) double hull and (b) single hull designs.
the three-point bending responses of sandwich beams with a corrugated core and a
Y-frame core under both simply supported and clamped boundary conditions. It
was found that sandwich beams with a corrugated core or a Y-frame core have com-
parable responses on an equal mass basis. However, these studies have been limited
to sandwich beams with identical front-and-back faces.
The objective of this chapter is to explore the sensitivity of the three-point bending
response of a sandwich beam to the relative placement of material in the core, front
face and back face. The relative allocation of material can be represented in a
diagram resembling a triple phase diagram, as shown in Fig. 4.2. Any point on this
diagram corresponds to a sandwich structure of total areal mass m, with fraction
(mc/m) in the core, (mf/m) in the front face and (mb/m) = 1− (mf/m)− (mc/m)
in the back face.
This study focuses on two trajectories in the design space of Fig. 4.2. The first
one is indicated by the vertical dashed line and includes all sandwich beams with
identical front-and-back faces, mf = mb. The second trajectory is the left-hand edge
of the triangle and denotes all sandwich beams without a back face, mb = 0.
The three-point bending response of sandwich panels of geometry along these two
trajectories will be compared on an equal mass basis. Consider, as the reference de-
sign, a sandwich panel with identical front-and-back faces. If the back face material
is relocated to the front face or to the core, will the three-point bending strength
increase or decrease? This question will be addressed for a corrugated core and a
Y-frame core, and for both simply supported and clamped boundary conditions.
71
Chapter 4. The influence of the back face on the bending response
m / mc
m / mb
m / mf
1
1
1
Design space for all
sandwich structures
Tested sandwich
beams with:
m = 9.1 kg/m
m = 13.8 kg/m
2
2
0.8
0.6
0.4
0.2
0.20.40.60.8
0.0
0.01.0
0.8
0.6
0.4
0.2
0.0
1.0
1.0
m f
m
m c
m
m
m
b
m bm f =
Figure 4.2: The design space for mass distribution within a sandwich panel of areal
mass m. The proportion of mass in the core, in the front face and in the back face
are denoted by mc/m, mf/m and mb/m, respectively. The mass distribution of the
test geometries is indicated for two choices of areal mass.
4.1.1 Choice of test material
There is a need to select a pertinent test material, which in the as-manufactured
state has similar properties to that of commercial shipbuilding steel, such as Lloyd’s
Grade A steel. The uniaxial tensile response of Lloyd’s Grade A steel has been
measured by Broekhuijsen (2003) and is shown in Fig. 4.3. It is used by Damen
Schelde Naval Shipbuilding in the construction of tankers with a Y-frame sandwich
core.
In previous laboratory studies (Coˆte´ et al., 2006; Valdevit et al., 2006b; Rubino et al.,
2008a, 2010) corrugated cores and Y-frame cores have been manufactured by brazing
together AISI 304 stainless steel sheets. In order to compare the uniaxial properties
of this material with those of Lloyd’s Grade A steel, preliminary uniaxial tests have
been performed on dog-bone specimens cut from as-received AISI 304 stainless steel
sheets; these were subjected to the same braze cycle as that used in the manufacture
of sandwich beams (see Section 4.2.1). The uniaxial tensile response of the brazed
304 material, at an applied strain rate of 10−3 s−1, is included in Fig. 4.3. The
measured Young’s modulus E and 0.2% offset yield strength σY are 210 GPa and
72
4.1 Introduction
0
200
400
600
800
1000
0 10 20 30 40
Tr
ue
s
tre
ss
(M
Pa
)
Logarithmic strain (%)
As-brazed AISI 304
stainless steel
Lloyd's grade A steel
Figure 4.3: Measured uniaxial tensile responses of as-brazed AISI 304 stainless steel
and Lloyd’s Grade A steel, at a strain rate of 10−3 s−1.
210 MPa, respectively. The observed strain hardening response is close to linear,
with a tangent modulus of Et = 2.1 GPa. Lloyd’s Grade A steel has a slightly higher
yield strength of 280 MPa and a somewhat reduced ductility and strain hardening
capacity. In broad terms, however, the as-brazed stainless steel is representative of
Lloyd’s grade A steel at strain levels below 10%. To confirm this, a limited set of
finite element simulations have been performed on the three-point bending response
of sandwich beams made from as-brazed stainless steel and Lloyd’s grade A steel, as
summarised in Appendix 4.A. The simulations confirm that sandwich beams made
from as-brazed stainless steel or from Grade A steel have similar responses. Based
upon these exploratory findings, the sandwich beams of the present study were
manufactured by brazing together type 304 stainless steel sheets.
4.1.2 Scope of study
First, the methodology used to manufacture and test the sandwich beams is reported
along with a description of the finite element models. Second, the measured three-
point bending responses of sandwich beams, with and without a back face, are
compared for simply supported and clamped boundary conditions. Then, to gain
additional insight into the collapse mechanisms, the beam responses are simulated
73
Chapter 4. The influence of the back face on the bending response
by three-dimensional finite element simulations. Finally, the three-point bending
response of simply supported sandwich panels is explored numerically as a function
of span and of relative proportion of material in the core and face-sheets. The two
asymptotic responses of indentation at short span and a bending instability at long
span are analysed and used to determine the collapse load as a function of span.
4.2 Methodology
4.2.1 Specimen manufacture
Corrugated and Y-frame cores, of cross-section shown in Fig. 4.4, were used to con-
struct prismatic sandwich beams. These cores are approximately 1:20 scale models
of the cores used in a ship hull and had a relative density of 2.5%. Both cores were
made from AISI 304 stainless steel sheets of thickness 0.3 mm and density ρ = 7900
kg/m3.
The corrugated core was manufactured by alternately folding stainless steel sheets
at ±60◦ under computer-numerical-control (CNC). In contrast, the Y-frame core
was manufactured by CNC folding of stainless steel sheets and then assembling two
sections: the ±45◦ upper part of the Y-frame and the Y-frame leg. Slots were cut
periodically into the central flange of the upper part of the Y-frame and a matching
set of keys were cut into the top of the Y-frame leg to facilitate assembly, as described
by Rubino et al. (2008a).
Stainless steel face-sheets were brazed to the cores to produce two classes of sandwich
beam:
1. a beam, with front-and-back faces of thickness t and
2. a beam, with only a front face of thickness 2t.
Two different values of thickness t were considered, 0.3 mm and 0.6 mm, giving
sandwich beams of areal mass m = 9.1 and 13.8 kg/m2, respectively, as shown in
Fig. 4.4(c). These test geometries are also included in the design space of Fig. 4.2.
The proportion of mass in the core mc/m is 0.48 and 0.31 for sandwich beams with
an areal mass m of 9.1 and 13.8 kg/m2, respectively.
The sandwich beams were assembled as follows. First, the face-sheets were spot-
74
4.2 Methodology
x1
x2
= 0.3tc
b = 55
= 0.3 - 1.2tf
= 0tb= tf or tb
c = 22
26.5
b = 55
4
13
c = 22
45°
= 0.3
12.7
x1
x2 tc
= 0tb= tf or tb
= 0.3 - 1.2tf 26.5
Back face thickness t
0.3 0.6 0.9 1.2
0.3
0.6
0.9
1.2
Fr
on
t f
ac
e
th
ick
ne
ss
t
0
0
m = 13.8 kg/m2
9.1 kg/m2
b
f
(a)
(b)
(c)
Figure 4.4: Cross-sectional dimensions of the sandwich beams: (a) corrugated core
and (b) Y-frame core. (c) The chosen values of face-sheet thickness used in the exper-
imental study. All dimensions are in mm.
75
Chapter 4. The influence of the back face on the bending response
welded to the core, and second, a thin layer (of thickness 10 µm) of Ni-CR 25-P10
(wt.%) braze powder was applied over all sheets of the assembly. Third, brazing was
performed in a vacuum furnace (at 0.03-0.1 mbar) using a dry argon atmosphere at
1075◦C for one hour, followed by a slow furnace cool.
4.2.2 Geometry of the three-point bending tests
Simply supported and clamped sandwich beams were tested and their dimensions
are shown in Fig. 4.5. In all cases, the prismatic axis of the core was aligned with
the longitudinal direction of the beam (x3-axis). The span of the beams was held
fixed at 2L = 250 mm and load introduction at mid-span was via a steel roller of
diameter D = 9 mm.
Simply supported beams
Steel rollers of diameter D = 9 mm were also used to provide simple outer support
to the sandwich beams, see Fig. 4.5(a). For those specimens without a back face,
preliminary tests revealed that the core crushed and splayed out-of-plane (in the
x1-direction) at the outer supports. To prevent this mode of collapse, short sections
of back face were brazed to the core at both ends of the beam, see Fig. 4.5(a). These
additional face plates had the same thickness as that of the front face-sheet. No such
reinforcement was required for the sandwich beams with front-and-back faces.
Clamped beams
To achieve a fully-clamped boundary condition, the ends of the sandwich beams
were filled with an epoxy resin to make the core fully dense. Then, the end portions
of the sandwich beams were bolted to the test rig using steel clamping plates and
M6 bolts, as shown in Fig. 4.5(b). For those specimens without a back face, local
reinforcement was again achieved by brazing short sections of back face to the core
at each end of the beam.
76
4.2 Methodology
2L = 250
I-beam
D = 9
F , δ
Core filled with epoxy
c = 22
X3
X2
D = 9
D = 9
2L = 250
70
F , δ
c = 22
X3
X2
(a)
(b)
M6 bolts
View on A-A
A View on A-A
A
A
A
62.562.5
70
Figure 4.5: The test fixtures used for (a) simply supported and (b) clamped beams.
A sandwich beam with a Y-frame core and without a back face is shown. All dimen-
sions are in mm.
77
Chapter 4. The influence of the back face on the bending response
4.2.3 Finite element models
The commercial software Abaqus was used to develop three-dimensional finite el-
ement (FE) models for all sandwich beams tested. The geometries used in the
simulations were identical to those employed in the experimental investigation, re-
call Fig. 4.4 and 4.5. Perfect bonding between core and face-sheets was assumed
in all cases. Four noded, linear shell elements with reduced integration (S4R in
Abaqus notation) were used to discretise the sandwich beams using a mesh size of
0.5 mm. A convergence study showed that further mesh refinement did not improve
significantly the accuracy of the simulations.
Boundary conditions
Only one quarter of the sandwich beam was modelled in the simulations, with sym-
metric boundary conditions at mid-span (x3 = 0) and at mid-plane (x1 = 0). The
mid-span roller was modelled as a rigid body in the FE simulations and its displace-
ment was prescribed during the analysis. A frictionless hard contact condition was
used to model the interaction between the roller and front face. The same contact
properties were used between all potentially contacting surfaces of the sandwich
beam.
The overhang of the simply supported sandwich beams beyond the outer rollers
was included in the FE analysis. Alternatively, the clamped boundary condition
was enforced by imposing zero displacement on the nodes of the end face of the
sandwich beam (x3 = L).
Material properties
The as-brazed AISI 304 stainless steel was modelled as a rate-independent, elastic-
plastic solid in accordance with J2-flow theory. The elastic branch was linear and
isotropic, as characterised by a Young’s modulus E = 210 GPa and a Poisson’s ratio
ν = 0.3. The uniaxial yield strength was σY = 210 MPa, and the hardening response
was tabulated in Abaqus from the plot in Fig. 4.3.
78
4.3 Experimental results
4.3 Experimental results
The three-point bending tests were conducted using a 100 kN screw driven test
machine with a constant cross-head velocity of δ˙ = 5 × 10−3 mm/s. The load F
applied to the specimen was measured by the load cell of the test machine and the
mid-span roller displacement δ was measured via a laser extensometer.
The three-point bending responses of all sandwich beams tested are given in Fig.
4.6 and 4.7 for simply supported and clamped boundary conditions, respectively.
In each figure, results are shown for sandwich beams with a corrugated core and a
Y-frame core, and for an areal mass m = 9.1 and 13.8 kg/m2. The mid-span roller
displacement δ is normalised by the beam half-span L = 125 mm whereas the load
F is normalised by σY bc, where the yield strength is σY = 210 MPa, the width of
the sandwich beams is b = 55 mm and the core thickness is c = 22 mm.
4.3.1 Simply supported beams
The simply supported beam response is shown in Fig. 4.6(a) for the corrugated core
and in Fig. 4.6(b) for the Y-frame core, both at m = 9.1 kg/m2. Likewise, the
response is given in Fig. 4.6(c) and (d) for the corrugated core and Y-frame core,
respectively, at m = 13.8 kg/m2. In each plot, results are shown for sandwich beams
with both faces present and for sandwich beams with the back face absent.
All simply supported sandwich beams have an initial elastic regime. The elastic
stiffness is, however, sensitive to the distribution of face-sheet material: beams con-
taining both front-and-back faces are at least 40% stiffer than those with the back
face absent. In contrast, the peak load reduces by less than 20% when the back face
material is relocated onto the front face.
The peak load for sandwich beams with a corrugated core exceeds that of beams
with a Y-frame core by 15-25%. In all cases, the peak load is followed by a softening
response, with more pronounced softening for the corrugated core than for the Y-
frame core: the load drops to less than 35% of the peak load for sandwich beams
with a corrugated core when δ/L is increased to 0.15. In contrast, for the Y-frame
core the load at δ/L = 0.15 exceeds 55% of the peak load.
79
Chapter 4. The influence of the back face on the bending response
(a) (b)
(c) (d)
Corrugated core Y-frame core
0
1
2
3
4
5
6
7
0 0.05 0.1 0.15
Measured
FE
F
1 face
2 faces
δ
σ bc
L
Y
x10-3
m = 9.1 kg/m2 0
1
2
3
4
5
6
7
0 0.05 0.1 0.15
Measured
FE
1 face
2 faces
δ
L
F
σ bcY
x10-3
m = 9.1 kg/m2
0
2
4
6
8
10
0 0.05 0.1 0.15
Measured
FE
1 face
2 faces
δ
L
F
σ bcY
x10-3
m = 13.8 kg/m2 0
2
4
6
8
10
0 0.05 0.1 0.15
Measured
FE
1 face
2 faces
δ
L
F
σ bcY
x10-3
m = 13.8 kg/m2
Figure 4.6: Three-point bending responses of simply supported sandwich beams.
Sandwich beams with an areal mass m = 9.1 kg/m2 are shown with (a) a corrugated
core and (b) a Y-frame core. Likewise, sandwich beams with an areal mass m = 13.8
kg/m2 are shown with (c) a corrugated core and (d) a Y-frame core.
80
4.3 Experimental results
(a) (b)
(c) (d)
Corrugated core Y-frame core
0
5
10
15
20
0 0.05 0.1 0.15
Measured
FE
1 face
2 faces
δ
L
F
σ bcY
x10-3
m = 9.1 kg/m2
0
5
10
15
20
25
30
0 0.05 0.1 0.15
Measured
FE
1 face
2 faces
δ
L
F
σ bcY
x10-3
m = 13.8 kg/m2
0
5
10
15
20
0 0.05 0.1 0.15
Measured
FE
1 face
2 faces
δ
L
F
σ bcY
x10-3
m = 9.1 kg/m2
0
5
10
15
20
25
30
0 0.05 0.1 0.15
Measured
FE
2 faces
1 face
δ
L
F
σ bcY
x10-3
m = 13.8 kg/m2
Figure 4.7: Three-point bending responses of clamped sandwich beams. Sandwich
beams with an areal mass m = 9.1 kg/m2 are shown with (a) a corrugated core and
(b) a Y-frame core. Likewise, sandwich beams with an areal mass m = 13.8 kg/m2
are shown with (c) a corrugated core and (d) a Y-frame core.
81
Chapter 4. The influence of the back face on the bending response
4.3.2 Clamped beams
The measured three-point bending responses of clamped sandwich beams are shown
in Fig. 4.7. The layout of Fig. 4.7 is the same as that in Fig. 4.6: structures with a
corrugated core and a Y-frame core are shown in Fig. 4.7(a) and (b), respectively,
for an areal mass m = 9.1 kg/m2. Likewise, the responses for m = 13.8 kg/m2 are
given in Fig. 4.7(c) and (d) for the corrugated core and Y-frame core, respectively.
In each plot, the response of a sandwich beam with front-and-back faces present is
compared to that of a sandwich beam without a back face.
In all cases, an initial elastic regime is followed by a peak load Fpk. Subsequently,
the clamped beams soften and then re-harden due to longitudinal stretching of
the beam. The core topology has a similar influence upon the initial peak load
of clamped beams to that of the simply supported beams: sandwich structures
with a corrugated core are 10-25% stronger than their counterparts with a Y-frame
core. The initial peak load of sandwich beams with an areal mass m = 9.1 kg/m2 is
sensitive to the distribution of face-sheet material: beams without a back face are 25-
35% stronger than those with front-and-back faces. In contrast, for m = 13.8 kg/m2,
the sandwich beams with front-and-back faces present have comparable initial peak
strengths to those without a back face. For all clamped beams considered, the load
drop following the initial peak load Fpk is at most 20%. We note in passing that the
simply supported Y-frame core shows load drop of this order, whereas the corrugated
core exhibits much larger load drops, recall Fig. 4.6.
4.3.3 Collapse mechanisms
To gain additional insight into the collapse mechanisms, photographs of the deformed
sandwich beams with an areal mass m = 13.8 kg/m2 are shown in Fig. 4.8-4.11.
Simply supported sandwich beams with a corrugated core and with a Y-frame core
are given in Fig. 4.8 and 4.9, respectively. Likewise, photographs of clamped sand-
wich beams with a corrugated core and a Y-frame core are reported in Fig. 4.10 and
4.11, respectively. In part (a) of each figure, the deformed geometry is shown for
front-and-back faces present, whereas in part (b) the images are for the back face
absent. The photographs were taken after deforming the sandwich beam to δ = 0.2L
and then unloading. Two views are shown in the figures: on the left, a side view
along the x3-direction showing half of the sandwich beam and on the right, a view
of the core deformation after sectioning of the beam at mid-span.
82
4.3 Experimental results
(a)
(b)
(c)
(d)
CL
CL CL
CL
Figure 4.8: Photographs of the simply supported sandwich beams with a corrugated
core (m = 13.8 kg/m2) (a) with front-and-back faces and (b) without a back face.
Deformed finite element meshes of the same sandwich beam (c) with front-and-back
faces and (d) without a back face. A side view showing half of the beam and a view
of the core deformation at mid-span are given. To clarify the predicted deformation
modes, the undeformed (dashed line) and deformed (solid line) cross-sections at mid-
span are included in (c) and (d). The images are for beams loaded to δ = 0.2L and
then unloaded.
83
Chapter 4. The influence of the back face on the bending response
(a)
(c)
(d)
(b)
CL CL
CLCL
Figure 4.9: Photographs of the simply supported sandwich beams with a Y-frame
core (m = 13.8 kg/m2) (a) with front-and-back faces and (b) without a back face.
Deformed finite element meshes of the same sandwich beam (c) with front-and-back
faces and (d) without a back face. A side view showing half of the beam and a view
of the core deformation at mid-span are given. To clarify the predicted deformation
modes, the undeformed (dashed line) and deformed (solid line) cross-sections at mid-
span are included in (c) and (d). The images are for beams loaded to δ = 0.2L and
then unloaded.
84
4.3 Experimental results
(a)
(b)
(c)
(d)
CLCL
CLCL
Figure 4.10: Photographs of the clamped sandwich beams with a corrugated core (m
= 13.8 kg/m2) (a) with front-and-back faces and (b) without a back face. Deformed
finite element meshes of the same sandwich beam (c) with front-and-back faces and
(d) without a back face. A side view showing half of the beam and a view of the
core deformation at mid-span are given. To clarify the predicted deformation modes,
the undeformed (dashed line) and deformed (solid line) cross-sections at mid-span
are included in (c) and (d). The images are for beams loaded to δ = 0.2L and then
unloaded.
85
Chapter 4. The influence of the back face on the bending response
(a)
(b)
(c)
(d)
CLCL
CLCL
Figure 4.11: Photographs of the clamped sandwich beams with a Y-frame core (m
= 13.8 kg/m2) (a) with front-and-back faces and (b) without a back face. Deformed
finite element meshes of the same sandwich beam (c) with front-and-back faces and
(d) without a back face. A side view showing half of the beam and a view of the
core deformation at mid-span are given. To clarify the predicted deformation modes,
the undeformed (dashed line) and deformed (solid line) cross-sections at mid-span
are included in (c) and (d). The images are for beams loaded to δ = 0.2L and then
unloaded.
86
4.3 Experimental results
The photographs of sandwich beams with front-and-back faces present, as shown
in part (a) of Fig. 4.8-4.11, indicate that beam collapse is by indentation of the
core beneath the mid-span roller. This holds true for both corrugated and Y-frame
core topologies and for both simply supported and clamped beams. The normalised
initial peak loads, Fˆ = Fpk/(σY bc), for all sandwich beams tested are summarised
in Table 4.1. It is clear from the table that the initial peak load for indentation of
sandwich beams with both faces present has only minor sensitivity to the choice of
boundary conditions.
The images shown in part (b) of Fig. 4.8-4.11 reveal that the beams without a back
face collapse by plastic buckling at mid-span. This alternative mode is reminiscent of
the buckling of circular tubes by ovalisation of their cross-section, as first identified
by Brazier (1927). The progressive reduction of flexural plastic modulus of the
sandwich beams in bending induces a Brazier-type instability, and we shall refer to
this collapse mode by the generalised term Brazier plastic buckling. The mode of
Brazier plastic buckling is more diffuse than the highly localised indentation mode
beneath the central roller, compare the images as given in parts (a) and (b) of Fig.
4.8-4.11. For an introduction to Brazier buckling, see Calladine (1983).
Specimen Fˆ = Fpk/(σY bc) (10−3)
Boundary Areal mass Core Number Measured FEcondition (kg/m2) topology of faces
Simply
supported
9.1
Corrugated 1 6.5 5.82 6.5 4.9
Y-frame 1 4.8 4.92 4.9 4.6
13.8
Corrugated 1 7.0 6.62 8.3 7.8
Y-frame 1 5.7 5.42 7.1 7.4
Clamped
9.1
Corrugated 1 8.6 9.02 6.8 6.4
Y-frame 1 7.2 7.62 5.2 5.3
13.8
Corrugated 1 9.6 10.22 9.0 8.7
Y-frame 1 8.7 7.72 8.0 8.3
Table 4.1: The measured and predicted values of normalised peak load Fˆ =
Fpk/(σY bc).
87
Chapter 4. The influence of the back face on the bending response
The peak load Fpk associated with Brazier plastic buckling occurs at a significantly
larger value of δ/L than the indentation mode for simply supported beams, recall
Fig. 4.6. Also, the value of Fpk for Brazier plastic buckling is sensitive to the choice
of boundary condition: clamped beams are 30-50% stronger than simply supported
beams (see Table 4.1). This is consistent with the fact that for a given applied load
F , the bending moment at mid-span of clamped beams is less than that for simply
supported beams.
4.4 Finite element predictions
A Finite Element (FE) investigation of the three-point bending response of sandwich
beams with corrugated and Y-frame cores has been conducted with the following
objectives:
1. to obtain additional insight into the measured responses presented in Section
4.3,
2. to explore the influence of mass distribution between core and face-sheets upon
the three-point bending response of a sandwich panel and
3. to analyse the effect of beam span upon the collapse mechanism.
All computations were performed using the commercial software Abaqus (version
6.9). Most simulations were done with the implicit solver, but the explicit solver
was also used when convergence issues were encountered. The explicit solver can
handle more easily the complex contact conditions that arise within the sandwich
beam when the core is crushed beneath the mid-span roller. To ensure that a
quasi-static solution was obtained with the explicit solver, the kinetic energy of the
sandwich beam was monitored to ensure it never exceeds 10% of the strain energy,
as suggested within the Abaqus documentation3.
4.4.1 Comparison between measurements and simulations
The FE predictions for all sandwich beams tested are included in Fig. 4.6 and 4.7
for simply supported and clamped boundary conditions, respectively. In each figure,
3Dassault Syste`me Simulia Corp., Abaqus analysis users manual, version 6.9, Providence, RI,
USA.
88
4.4 Finite element predictions
the simulated response of sandwich beams with a corrugated core (part (a)) and
a Y-frame core (part (b)) is shown for m = 9.1 kg/m2. Likewise, the results for
sandwich beams with m = 13.8 kg/m2 are shown with a corrugated core (part (c))
and a Y-frame core (part (d)).
It is evident from Fig. 4.6 that the predicted peak loads of the simply supported
beams slightly underestimate the measured peak loads. This is attributed to the
fact that the FE analysis assumes frictionless contact between the sandwich beam
and rollers, and neglects the strengthening due to the presence of the braze alloy
over all surfaces of the sandwich beam. In contrast, the FE analysis somewhat
overpredicts the strength of the fully-clamped beams following the initial peak load.
This is traced to the fact that perfect clamping is assumed in the FE simulations
whereas the test rig was unable to achieve this. The finite additional compliance of
the test fixture is particularly significant for the sandwich beams of areal mass m
= 13.8 kg/m2 because the reaction force and moment at the supports is greater for
these specimens.
The predicted shapes of deformed sandwich beams of areal mass m = 13.8 kg/m2 are
compared with photographs of the as-tested specimens in Fig. 4.8-4.11. Recall that
simply supported beams with a corrugated core and a Y-frame core are shown in Fig.
4.8 and 4.9, respectively. Likewise, clamped beams with a corrugated core and a Y-
frame core are given in Fig. 4.10 and 4.11, respectively. In each figure, beams with
front-and-back faces present (part (c)) are compared with those without a back face
(part (d)). Additional views are included in parts (c) and (d) to show the predicted
cross-sections at mid-span.
The observed and predicted deformation of the sandwich beams with front-and-
back faces present is by indentation beneath the central roller. In contrast, for the
sandwich beams without a back face, the observed and predicted deformation mode
is by Brazier plastic buckling at mid-span.
4.4.2 Sensitivity of the sandwich panel response to span and
proportion of mass in the core
In the experimental investigation presented in Section 4.3, the proportion of mass
in the core and the span of the sandwich beams were held fixed. The sensitivity
of collapse strength to these geometric parameters is now explored using the FE
89
Chapter 4. The influence of the back face on the bending response
method. At this stage in the study, the perspective is changed from comparing
FE predictions with the measured responses of sandwich beams to predicting the
collapse response of sandwich panels in three-point bending. Sandwich panels are
more commonly used in engineering practice (such as ship hulls) than sandwich
beams, and it is of interest to evaluate the relative performance of corrugated cores
and Y-frame cores in the sandwich panel configuration. We shall limit our attention
to the simply supported case, and consider sandwich panels with identical front-
and-back faces and panels with the back face absent. Results are presented in
non-dimensional form so that they are applicable over a wide range of length scales;
from laboratory test to industrial application.
The cross-sections of the sandwich panels are given in Fig. 4.12 for the corrugated
core and Y-frame core. The panels are subjected to three-point bending, and are
idealised by unit cells in the width-direction, as defined in Fig. 4.12. Under three-
point bending, the panels will deform plastically over a limited portion along their
length, and display negligible straining in the width direction, x1. Consequently,
the behaviour of a panel of large width is adequately captured by considering the
response of a unit cell with symmetric boundary conditions imposed along the sides,
as shown in Fig. 4.12.
Dimensional analysis
In the simulations, the core shape is held fixed and parameterised in terms of the
core thickness c, as shown in Fig. 4.12. The relative mass distribution between
core and face-sheets is dictated by the thickness of the core members and of the
face-sheets according to the following prescription.
The areal mass of the core mc scales with the thickness of the core members tc
according to:
mc = Aρtc , (4.1)
where the constant of proportionality is A = 1.843 for both corrugated core and
Y-frame core. Likewise, the areal mass of the sandwich panel m scales with the
thickness of the face-sheets tf according to:
90
4.4 Finite element predictions
0.18c
0.59c
c
45°0.58c
x1
x2
tc
= 0tb= tf or tb
tf
b = 1.2c
x1
x2
tc
b = 1.2c
tf
= 0tb= tf or tb
c
(a) (b)
(c)
Figure 4.12: Cross-sectional dimensions of the sandwich panels considered in the
numerical analysis: (a) corrugated core and (b) Y-frame core. (c) The sandwich
panels, shown here with a corrugated core, are simply supported and loaded in three-
point bending.
91
Chapter 4. The influence of the back face on the bending response
m = ρtf + mc , (4.2)
when the back face is absent and as:
m = 2ρtf + mc , (4.3)
when both front-and-back faces are present. Now, Eq. (4.1) can be rewritten in
non-dimensional form as:
tc
c =
1
A
mc
m
m
ρc , (4.4)
and likewise Eq. (4.2) and (4.3) can be re-arranged to form:
tf
c =
(
1− mcm
) m
ρc , (4.5)
and
tf
c =
1
2
(
1− mcm
) m
ρc , (4.6)
respectively. Thus, the sheet thickness of the core and face-sheets can be expressed
directly in terms of the areal mass ratios mc/m and m/(ρc).
The three-point bending strength of a simply supported sandwich panel of width b,
core thickness c and span 2L scales as:
Fpk =
2Mp
L =
2σY bctf
L f1(tc, tf , c) , (4.7)
where Mp is the plastic moment of the cross-section and f1 is a function of the
cross-sectional geometry. Equation (4.7) can be rewritten in non-dimensional form
as:
Fˆ = FpkσY bc
= f2
(tc
c ,
tf
c ,
2L
c
)
, (4.8)
92
4.4 Finite element predictions
and using Eq. (4.4)-(4.6), the sheet thickness ratios can be expressed as areal mass
ratios giving:
Fˆ = FpkσY bc
= f3
(
m
ρc,
mc
m ,
2L
c
)
, (4.9)
Therefore, the non-dimensional collapse load Fˆ is a function of the normalised span
2L/c and of the areal mass ratios m/(ρc) and mc/m.
In the experimental study, and associated numerical simulations reported above, the
normalised span 2L/c was held fixed at 11.4. The mass ratios were m/(ρc) = 0.052
and mc/m = 0.48 for sandwich beams of areal mass m = 9.1 kg/m2, and were m/(ρc)
= 0.079 and mc/m = 0.31 for sandwich beams of areal mass m = 13.8 kg/m2.
We proceed by considering the sandwich panel response for corrugated cores and
Y-frame cores, first with m/(ρc) held fixed at 0.052 and second with varying mass
ratio m/(ρc). The simulations with m/(ρc) = 0.052 represent the case considered in
the above experimental study with m = 9.1 kg/m2 and c = 22 mm. Simulations were
performed for selected values of mc/m in the range 0.15 to 0.95 and of normalised
spans 2L/c in the range from 5 to 30. The overhang of the simply supported sand-
wich panels was 0.5L and the length of the face-plates added to the extremities of
the sandwich panels without a back face was 0.56L: again, these values were equal
to those used in the experimental investigation, recall Fig. 4.5(a). In all cases, the
central and support rollers had a diameter D = 9 mm, giving D/c = 0.41.
Peak loads
The normalised peak load Fˆ = Fpk/(σY bc) is plotted in Fig. 4.13 as a function of
normalised span 2L/c for four selected values of mc/m. The responses of sandwich
panels with a corrugated core and a Y-frame core are shown in Fig. 4.13(a) and
(b), respectively. In each plot, results are shown for sandwich panels with both faces
present and for sandwich panels with the back face absent.
The peak load of all sandwich panels increases with increasing proportion of mass
in the core, mc/m. This increase in strength is more significant for short panels
than for long panels. Also, with increasing mc/m, the peak strength becomes less
sensitive to whether the sandwich panel contains both face-sheets or only the front
face-sheet: this is consistent with the fact that the peak strength is dominated by
93
Chapter 4. The influence of the back face on the bending response
1
10
5 6 7 8 9 10 20 30
Corrugated core
F^
x10-3
2L
c
15
5
0.5
m / m = 0.15c
0.95
0.50
0.30
No back face
Front-and-back faces
1
10
5 6 7 8 9 10 20 30
x10-315
0.5
5
2L
c
Y-frame core
m / m = 0.15c
0.30
0.50
0.95
No back face
Front-and-back faces
F^
(a) (b)
Figure 4.13: Normalised peak load Fˆ = Fpk/(σY bc) as a function of the normalised
span 2L/c for simply supported sandwich panels and selected values of mc/m (m/(ρc)
= 0.052). Results are shown for sandwich panels with (a) a corrugated core and (b) a
Y-frame core.
the presence of the core rather than the relatively thin face-sheets at high mc/m.
Next, consider the role of the back face upon the peak strength. For 2L/c less than
approximately 15, sandwich panels with a front face of double thickness but without
a back face are stronger than those with front-and-back faces present. This is due
to the fact that the thicker front face gives rise to a higher indentation strength.
In contrast, sandwich panels with front-and-back faces have higher peak loads than
panels without a back face for 2L/c > 15; this is consistent with the fact that the
Brazier buckling load is reduced when the back face is removed.
In order to determine the degree to which sandwich panel collapse is dictated by
core indentation or by Brazier buckling, a series of additional calculations have been
performed to obtain the collapse strength due to each of these mechanisms acting
in isolation. The details are as follows.
Collapse mechanisms
Indentation The FE method was also used to obtain the indentation strength of
the sandwich panels of geometry given in the previous section. To achieve this, the
boundary conditions were changed such that the panel was adhered to a rigid foun-
dation as shown in Fig. 4.14(a). This was achieved by constraining the translational
degrees-of-freedom to zero along the bottom face of the panel.
94
4.4 Finite element predictions
3/2 L
F, δ
(a)
X2
X3X1
L
Rigid surface
M, ω
(b)
f = 0t f = 0t
f = 0a
X2
X3X1
Figure 4.14: The boundary conditions on finite element models to simulate (a)
indentation and (b) bending. A sandwich panel without a back face is shown.
Representative collapse responses of sandwich panels resting upon a rigid foundation
are given in Fig. 4.15(a) for m/(ρc) = 0.052 and mc/m = 0.5. The predictions of
indentation strength are limited to 2L/c = 11.4, as used in the experimental study
on sandwich beams. Results are shown for corrugated and Y-frame cores, and for
sandwich panels with and without a back face. The responses exhibit a peak load FI
at a roller displacement δ of approximately 1% of the core thickness c. A small load
drop ensues and subsequent deformation occurs at almost constant load. These
simulations were repeated for other selected values of mc/m and the results are
summarised in Fig. 4.15(b): the normalised indentation strength FˆI is plotted as a
function of the proportion of mass in the core.
For all sandwich panels analysed, the indentation strength increases with increasing
0
2
4
6
8
10
12
0 0.01 0.02 0.03 0.04 0.05
Y-frame core
Corrugated core
F
σ bcY
δ
c
x10-3
No back face
Front-and-back faces
cm / m = 0.5
0
5
10
15
20
25
0 0.2 0.4 0.6 0.8 1
Front-and-back faces
No back face
F^
x10-3
m c
m
Corrugated core
Y-frame core
I
(a) (b)
Figure 4.15: (a) The predicted indentation response of sandwich panels with mc/m =
0.5 resting on a rigid foundation. (b) Normalised indentation strength FˆI = FI/(σY bc)
as a function of mc/m (m/(ρc) = 0.052).
95
Chapter 4. The influence of the back face on the bending response
mc/m. The indentation strength is also sensitive to topology:
1. sandwich panels with a corrugated core have higher indentation strengths than
their counterparts with a Y-frame core, and
2. the indentation strength of panels with a double thickness front face and with-
out a back face exceeds that of sandwich panels with front-and-back faces
present. These features have already been noted above in reference to Fig.
4.13.
Brazier plastic buckling The critical bending moment causing a sandwich panel
to collapse by Brazier plastic buckling was also obtained with the FE method. For
these simulations, all nodes (and corresponding degrees-of-freedom) at the right
end of the panel were tied to a rigid surface as illustrated in Fig. 4.14(b). The
rigid surface was rotated by an angle ω about the x1-axis, with the axis of rotation
positioned at mid-height of the panel. Otherwise, the rigid surface was free to
translate in the x2 and x3 directions to ensure that no axial or transverse forces
were applied to the panel. To prevent rigid body motion, the x2-component of
nodal displacement was constrained to equal zero for one node of the front face
(x1 = 0), at the left-hand end of the panel (x3 = 0).
The representative collapse response of sandwich panels with m/(ρc) = 0.052 and
mc/m = 0.5 is given in Fig. 4.16(a). Results are shown for both corrugated and
Y-frame core topologies and for sandwich panels with and without a back face. As
the angular displacement ω is increased, the reaction moment M increases up to a
peak value MB due to Brazier plastic buckling, and this is followed by a softening
response. These simulations have been repeated for selected values of mc/m and the
normalised Brazier buckling moment Mˆ = MB/(σY bc2) is plotted in Fig. 4.16(b)
as a function of the proportion of mc/m, with m/(ρc) = 0.052. The simulations
are done for sandwich panels with 2L/c = 11.4, but the peak moment is relatively
insensitive to this ratio.
It is clear from Fig. 4.16(b) that the Brazier buckling moment for a sandwich panel
without a back face increases with increasing mc/m. For these structures, the posi-
tion of the neutral axis is sensitive to the proportion of mass in the core; an increase
in mc/m moves the neutral axis closer to the centre of the core, increases the struc-
tural efficiency in plastic bending and leads to an increase in the Brazier buckling
strength. In contrast, for sandwich panels with front-and-back faces present, the
96
4.4 Finite element predictions
0
0.5
1
1.5
2
2.5
0 5 10 15 20 25
ω (degrees)
cm / m = 0.5
No back face
Front-and-back faces
M
σ bcY 2
Y-frame core
Corrugated core
x10-2
0
0.5
1
1.5
2
2.5
3
0 0.2 0.4 0.6 0.8 1
Front-and-back faces
No back face
m c
m
x10-2
M^
Corrugated core
Y-frame core
(a) (b)
Figure 4.16: (a) The predicted bending response of sandwich panels with mc/m
= 0.5. (b) Normalised Brazier buckling moment Mˆ = MB/(σY bc2) as a function of
mc/m (m/(ρc) = 0.052).
position of the neutral axis is independent of the proportion of mass in the core and
consequently Mˆ is relatively insensitive to the value of mc/m.
Interpretation of the three-point bending strength in terms of indentation
and Brazier buckling We anticipate that, at a sufficiently short span 2L, the
three-point bending strength Fpk is approximated by the indentation strength FI and
is independent of span. In contrast, the three-point bending strength of long panels
is dictated by Brazier plastic buckling; for a simply supported sandwich panel, the
collapse load associated with Brazier plastic buckling scales with the panel length
2L according to:
FB =
2MB
L . (4.10)
The indentation load and Brazier buckling moment, as given in Fig. 4.15(b) and
4.16(b), are now used to estimate the collapse load of a panel in three-point bending.
The lower value of FI and FB determines which collapse mechanism is active. These
asymptotic predictions of collapse loads are compared with the three-point bending
collapse loads in Fig. 4.17. Comparisons are made in Fig. 4.17(a) and (b) for
sandwich panels with front-and-back faces present, and in Fig. 4.17(c) and (d) for
sandwich panels with the back face absent.
In broad terms, there is excellent agreement between the predicted indentation load
97
Chapter 4. The influence of the back face on the bending response
and the three-point bending load at short spans, and between the predicted Brazier
buckling load and the three-point bending load at long spans. The deformation
mode of the panels in three-point bending confirms this (not shown). The switch
in response from indentation to Brazier buckling occurs at a transition value of
span 2Lt/c. For sandwich panels containing front-and-back faces, 2Lt/c decreases
with increasing mc/m. This is consistent with the feature that FI increases with
increasing mc/m whereas MB is relatively insensitive to mc/m for panels containing
front-and-back faces. In contrast, the transition span 2Lt/c for sandwich panels with
(a) (b)
(c) (d)
Corrugated core Y-frame core
1
10
5 6 7 8 9 10 20 30
No back face
2L
c
x10-315
0.5
5
m / m = 0.15c
0.95
0.50
0.30F^
1
10
5 6 7 8 9 10 20 30
x10-3
2L
c
Front-and-back faces15
5
0.95
m / m = 0.15c
0.50
0.30
Three-point bending
Indentation
Brazier buckling
F^
1
10
5 6 7 8 9 10 20 30
2L
c
x10-315
0.5
5
No back face
0.95
0.50
0.30m / m = 0.15c
F^
1
10
5 6 7 8 9 10 20 30
x10-3
2L
c
Front-and-back faces15
5
m / m = 0.15c
0.50
0.30
0.95
F^
Figure 4.17: Normalised peak load Fˆ = Fpk/(σY bc) as a function of the normalised
span 2L/c for simply supported sandwich panels and selected values of mc/m (m/(ρc)
= 0.052). The three-point bending results are reproduced from Fig. 4.13. The in-
dentation and Brazier buckling strengths are included as short and long dashed lines,
respectively. Sandwich panels with front-and-back faces are shown with (a) a corru-
gated core and (b) a Y-frame core. Likewise, sandwich panels without a back face are
shown with (c) a corrugated core and (d) a Y-frame core.
98
4.4 Finite element predictions
the back face absent is only mildly influenced by the value of mc/m. This arises
from the fact that FI and MB both increase with increasing mc/m for sandwich
panels without a back face.
Sensitivity of the three-point bending strength to the value of m/(ρc)
It has been demonstrated above that the three-point bending strength is adequately
represented by the two asymptotic behaviours of core indentation and Brazier buck-
ling, with the operative collapse mode dictated by the beam span. Here, the depen-
dence of the indentation strength and the Brazier buckling strength upon m/ρc is
explored.
The indentation strength and Brazier buckling strength are plotted as a function of
mc/m in Fig. 4.18 and 4.19, respectively, for selected values of m/(ρc) in the range
of 0.015 and 0.15. Indentation strengths are shown in Fig. 4.18(a) for panels with
a corrugated core and in Fig. 4.18(b) for panels with a Y-frame core; in each plot,
results are given for panels with both faces present, and for panels with the back
face absent. For all sandwich panels considered, the normalised indentation strength
per unit mass ρcFˆI/m increases with increasing value of m/(ρc). The observations
made previously for sandwich panels with m/(ρc) = 0.052 also hold true for other
values of m/(ρc):
1. sandwich panels with a corrugated core have higher indentation strengths than
those with a Y-frame core and
2. relocating the back face material onto the front face increases the indentation
strength of the sandwich panel.
The results for the Brazier buckling moment are given in Fig. 4.19(a) for panels
with front-and-back faces present and in Fig. 4.19(b) for panels with the back face
absent. In each plot, sandwich panels with a corrugated core are compared to those
with a Y-frame core. The limit of mc/m tending to zero is not included in Fig.
4.19(a) and (b) as this limit has no practical value and is not associated with a
peak moment. It is clear from Fig. 4.19(a) that the Brazier buckling moment is
relatively insensitive to mc/m when front-and-back faces are present. In contrast,
when the back face is absent, the Brazier buckling strength increases with increasing
mc/m. This was observed previously for sandwich panels with m/(ρc) = 0.052 (see
Fig. 4.16(b)) but the results of Fig. 4.19 demonstrate that it holds true for other
99
Chapter 4. The influence of the back face on the bending response
selected values of m/(ρc). Now consider the effect of m/(ρc) upon the normalised
Brazier buckling strength per unit mass ρcMˆ/m. Regardless of whether the back
face is present or absent (and regardless of the core topology), ρcMˆ/m increases by
a factor of about 3 when m/(ρc) is increased by a factor of 10 from 0.015 to 0.15.
(a) (b)
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
m
c = 0.15
0.052
0.015
F^I c
m
m c
m
Front-and-back faces
No back face
ρ
ρ
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
m
c = 0.15
0.052
0.015
m c
m
F^I
m
Front-and-back faces
No back face
ρ
cρ
Figure 4.18: Normalised indentation strength per unit mass ρcFˆI/m as a function
of mc/m for selected values of m/(ρc). Results are shown for sandwich panels with
(a) a corrugated core and (b) a Y-frame core.
(a) (b)
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
m
= 0.15
0.052
0.015
m c
m
Corrugated core
Y-frame core
M^
m
cρ
cρ
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.2 0.4 0.6 0.8 1
m c
m
M^
m
Corrugated core
Y-frame core m
= 0.15
0.052
0.015
cρ
cρ
Figure 4.19: Normalised Brazier buckling moment per unit mass ρcMˆ/m as a func-
tion of mc/m for selected values of m/(ρc). Results are shown for sandwich panels (a)
with front-and-back faces present and (b) without a back face.
4.5 Concluding remarks
Sandwich beams with corrugated and Y-frame cores have been manufactured by
brazing together AISI 304 stainless steel sheets. The dimensions of the cores were
100
4.A Influence of parent material
approximately 1:20 scale models of the cores used in a ship hull. In addition, the
uniaxial tensile response of as-brazed stainless steel was found to be representative
of shipbuilding steel up to strain levels of about 10%.
The three-point bending responses of sandwich beams with (i) front-and-back faces
present and (ii) front face present, but back face absent have been measured and
compared on an equal mass basis. The tests were done using simply supported and
clamped boundary conditions, with the prismatic axis of the core aligned with the
longitudinal axis of the beam. Sandwich beams with front-and-back faces present
collapsed by indentation whereas beams without a back face collapsed by Brazier
plastic buckling. Despite having different collapse mechanisms, sandwich beams
with front-and-back faces and those without a back face had comparable three-point
bending strengths for the choice of beam span employed.
Three-dimensional FE models were developed and the simulations were found to be
in good agreement with the measured responses. The FE method was also used
to study the influence of the mass distribution between the face-sheets and core.
Upon concentrating the mass of the sandwich panel within the core the three-point
bending strength of the structure increases. The analysis also showed the influence
of the span upon the collapse response of a sandwich panel; short panels failed
by indentation and long panels collapsed by Brazier plastic buckling. Sandwich
panels with a corrugated core, and without a back face have the highest indentation
strength and are thereby optimal for short spans, recall Fig. 4.18. In contrast, it
is clear from Fig. 4.19 that panels with front-and-back faces have greater Brazier
buckling strengths than their counterparts with the back face absent; consequently,
panels with front-and-back faces are optimal for long spans. However, the choice of
core topology plays only a minor role in the Brazier buckling regime: the corrugated
core is either stronger or weaker than the Y-frame core depending upon the precise
values of m/(ρc) and of mc/m and upon whether the sandwich panel has the back
face present or absent.
4.A Influence of parent material
The influence of the parent material on the three-point bending responses of simply
supported and clamped sandwich beams was investigated using the finite element
method. The models used to simulate the experiments (see Section 4.2.3) were used
101
Chapter 4. The influence of the back face on the bending response
for this analysis, except that the material properties of as-brazed stainless steel were
replaced by the ones of Lloyd’s grade A steel. The yield strength was taken to be
σY = 280 MPa and the hardening response of the material was tabulated in Abaqus
from the plot given in Fig. 4.3. Both grades of steel have a Young’s modulus of E
= 210 GPa and a Poisson’s ratio of ν = 0.3.
The three-point bending responses of beams with a Y-frame core are compared in
Fig. 4.20 for the two choices of material. The responses of sandwich beams with
front-and-back faces are shown in Fig. 4.20(a) whereas their counterparts without a
back face are shown in Fig. 4.20(b). In each figure, results are shown for both simply
supported and clamped boundary conditions, and the load F has been normalised
by the yield strength of the as-brazed stainless steel, σY = 210 MPa. The peak
load of sandwich beams made from as-brazed stainless steel are within 12% of those
made from grade A steel. In general, the results in Fig. 4.20 indicate that the three-
point bending response of a sandwich beam made from as-brazed stainless steel is
representative of one made from Lloyds grade A steel.
0
5
10
15
20
0 0.05 0.1 0.15
As-brazed stainless steel
Lloyds grade A steel
F
σ bcY
x10-3
δ
L
Simply supported beams
Clamped beams
2 faces 0.3 mm
0
5
10
15
20
0 0.05 0.1 0.15
As-brazed stainless steel
Lloyds grade A steel
δ
L
F
σ bcY
x10-3
Clamped beams
Simply supported beams
1 face 0.6 mm
(a) (b)
Figure 4.20: Sensitivity of the three-point bending response of a sandwich beam
with a Y-frame core to the choice of material. (a) Front-and-back faces are present
and (b) the back face is absent.
102
Chapter 5
Drop weight tests on prismatic
sandwich beams
Summary
To mimic the response of a sandwich hull design to a ship collision, drop weight
tests with an impact velocity of 5 m/s were performed on stainless steel sandwich
beams with a corrugated core or a Y-frame core. These tests were conducted on
both simply supported and clamped beams, and the responses measured dynamically
were compared to those measured quasi-statically. The dynamic peak load of the
beams could not be measured accurately due to an artifact of the experimental setup;
however, the instrumentation was able to capture precisely the post-peak response
of the beams. The post-peak response at 5 m/s was slightly stronger than the
one measured quasi-statically. Three-dimensional finite element simulations were
found to be in reasonable agreement with the measurements and gave additional
insight into the experiments. The finite element method was also used to investigate
whether the peak load and collapse mechanism at 5 m/s are different from those
obtained under quasi-static loading. The predictions indicated that sandwich beams
which collapse quasi-statically by indentation also fail by indentation at 5 m/s. In
contrast, the simulations for beams that fail by Brazier plastic buckling under quasi-
static loading indicated that they collapse by indentation at 5 m/s. Finally, for all
sandwich beams considered in the simulations, the dynamic peak load was found to
be higher than its quasi-static value, and the mass of the front face was found to be
an important factor contributing to this increase.
103
Chapter 5. Drop weight tests on prismatic sandwich beams
5.1 Introduction
More than 200 maritime accidents were recorded in the Gulf of Finland from 1997
to 2006 (Kujala et al., 2009). About 50% of those accidents were groundings and
another 20% were ship-ship collisions. For large vessels such as oil and chemical
tankers, ship collisions and groundings occur at low speeds, approximately 5 m/s1.
These accidents are in general considered as quasi-static loading scenarios (ISSC,
2006c). This assumption seems reasonable for vessels with a conventional hull de-
sign which can absorb energy only by bending and stretching of the outer hull.
The validity of this assumption is questionable for vessels with a sandwich hull con-
struction because additional energy is absorbed by crushing of the core, and the
crushing response of most core topologies is very sensitive to velocity. For exam-
ple, Tilbrook et al. (2007) have shown that when the corrugated core or the Y-frame
core are crushed dynamically, their dynamic strength is superior to their quasi-static
strength due to inertia stabilisation effects, even at low velocities between 1-10 m/s.
This strengthening effect was observed in compression by Tilbrook et al. (2007) and
here, its influence on the dynamic bending response will be investigated.
The objective of this chapter is to compare the dynamic response of sandwich beams
impacted at 5 m/s to their quasi-static response. Sandwich beams with a corrugated
core and a Y-frame core will be considered under simply supported and clamped
boundary conditions. Recall that their quasi-static responses have been measured
previously in Chapter 4, and two collapse mechanisms were identified: indentation
and Brazier plastic buckling. Will these collapse mechanisms change at 5 m/s? This
question will be addressed below via experiments and simulations.
A drop weight apparatus will be used to impact the sandwich beams at 5 m/s.
Similar tests were performed on aluminium sandwich beams with a metal foam core
(Crupi and Montanini, 2007; Yu et al., 2003, 2008) and more recently on aluminium
sandwich panels with a honeycomb core (Crupi et al., 2012). In both cases, the
responses measured dynamically were similar to those measured quasi-statically,
and no change in the collapse mechanism was observed when comparing the quasi-
static and dynamic deformation modes. No drop weight tests have been performed
on metallic sandwich beams with a corrugated core or a Y-frame core; hence, the
motivation for this chapter.
1This collision speed is based on discussions with Mr. Joep Broekhuijsen, research coordinator
at Damen Schelde Naval Shipbuilding.
104
5.2 Methodology
This chapter is organised as follows. First, the drop weight apparatus and the
dimensions of the tested sandwich beams are presented along with a description of
the finite element models. Second, the measured dynamic responses are compared to
the quasi-static responses for both simply supported and clamped sandwich beams.
Third, finite element predictions are compared to the experiments and finally, the
finite element method is used to explore if the quasi-static peak load and collapse
mechanism of a sandwich beam are different during an impact at 5 m/s.
5.2 Methodology
5.2.1 Geometry of the tested sandwich beams
Sandwich beams with a corrugated core and a Y-frame core were tested and their
cross-sectional dimensions are given in Fig. 5.1. The core and face-sheets were made
from AISI 304 stainless steel sheets of thickness t = 0.3 mm and density ρ = 7900
kg/m3. Both corrugated and Y-frame cores had a relative density ρ¯ = 0.025 and a
core thickness c = 22 mm. The face-sheets were brazed to the core to produce a
sandwich beam of areal mass m = ρ(2t + ρ¯c) = 9.1 kg/m2. The brazing cycle used
to manufacture the sandwich beams was detailed previously in Section 4.2.1.
b = 55
4
13
c = 22
45°
= 0.3
12.7
x1
x2 t
t = 0.3 26.5
t = 0.3
x1
x2
= 0.3t
b = 55
c = 22
26.5
t = 0.3
t = 0.3
(a) (b)
Figure 5.1: Cross-sectional dimensions of the tested sandwich beams with (a) a
corrugated core and (b) a Y-frame core. All dimensions are in mm.
Simply supported and clamped sandwich beams were tested, both with a span 2L =
250 mm. In all cases, the prismatic axis of the core was aligned with the longitudinal
direction of the beam (x3-axis). Steel rollers of diameter D = 9 mm were used to
provide simple support to the sandwich beams, see Fig. 5.2(a). On the other hand,
the fully-clamped boundary condition was achieved in two steps. First, the ends of
105
Chapter 5. Drop weight tests on prismatic sandwich beams
the sandwich beams were filled with an epoxy resin to make the core fully dense.
Second, the end portions of the sandwich beams were bolted to the testing rig using
steel clamping plates and M6 bolts, see Fig. 5.2(b).
5.2.2 Drop weight apparatus
The sandwich beams were impacted using a drop weight apparatus illustrated in
Fig. 5.2. A projectile of mass M was dropped from 1.3 m to achieve an impact
velocity of 5 m/s. The mass M was varied depending on the boundary conditions:
2 kg was used for simply supported beams whereas clamped beams were tested with
3 kg. Consequently, the kinetic energy of the projectile is 25 J for simply supported
beams and 37.5 J for clamped beams.
The contact force was measured by a piezoelectric load cell (PCB model 218C). The
load cell was mounted on a roller of diameter 2R = 20 mm. This assembly was
placed stationary at mid-span on the sandwich beam to be tested. During the test,
the projectile hits the load cell and the roller transfers the kinetic energy of the
projectile to the sandwich beam. To reduce ringing in the load cell, a rubber pad of
thickness tr = 1.5 mm, was added to the bottom surface of the projectile, see Fig.
5.2.
High-speed photography was used to capture the experiments with 10,000 frames per
second. The displacement of the mid-span roller as a function of time was inferred
from those images. For both simply supported and clamped beams, the duration
of the first impact was 8-10 ms. The first impact was followed by a succession of
rebounds and elastic impacts. The energy involved in these rebounds is negligible,
approximately 1 J or less, as estimated from the rebound height of the projectile.
5.2.3 Finite element models
Three-dimensional Finite Element (FE) models were developed to simulate the dy-
namic response of all sandwich beams tested. All simulations were done with the
explicit solver of the commercial software Abaqus (version 6.9).
106
5.2 Methodology
Transparent
guiding tube
Support roller
D = 9
2L = 250
Piezoelectric load cell
Projectile
Ø57
60
105
roller 2R = 20
M = 2 kg
Rubber pad
t = 1.5r
(a)
Steel (or aluminium)
Transparent
guiding tube
Steel clamping
plates and M6 bolts
2L = 250
Piezoelectric load cell
60
Projectile
Ø57
5
150
Aluminium roller
2R = 20
M = 3 kg
Rubber pad
t = 1.5r
(b)
5
x
x
x3 1
2
x
x
x3 1
2
Figure 5.2: Experimental setup used to perform drop weight tests at 5 m/s on
(a) simply supported and (b) clamped sandwich beams. A sandwich beam with a
corrugated core is shown. All dimensions are in mm.
107
Chapter 5. Drop weight tests on prismatic sandwich beams
Rigid mid-span roller
The FE simulations do not model the impact between the projectile and the assembly
of the mid-span roller and load cell. To simplify the analysis, only the contact
between the mid-span roller and the sandwich beam is simulated, see Fig. 5.3. The
mid-span roller is modelled as a rigid body in the simulations. It is given an initial
velocity vi = 5 m/s and then progressively slows down. The rigid mid-span roller is
also given a mass in the simulations corresponding to the mass of the projectile used
in the experiments. This ensures that the initial kinetic energy in the simulations
is the same as that in the experiments. The interaction between the rigid mid-span
roller and the front face of the sandwich beam was modelled as a hard frictionless
contact.
1/4 of the sandwich beam
Rigid mid-span roller
M / 4
v = 5 m/si
(b)
L = 125
R = 10
Rigid mid-span roller
M / 4
v = 5 m/si
L = 125
R = 10
(a)
D = 9
1/4 of the sandwich beam
62.5
x2
x3
x2
x3
Figure 5.3: Finite element models used to simulate the drop weight tests on (a)
simply supported and (b) clamped sandwich beams. All dimensions are in mm.
Geometry and mesh of the sandwich beams
The geometry of the sandwich beams used in the simulations was identical to those
employed in the experimental investigation, see Fig. 5.1. In all cases, the face-
sheets were assumed to be perfectly bonded to the core. The sandwich beams were
discretised using four noded linear shell elements (S4R in Abaqus notation) with
an average mesh size of 0.5 mm. Numerical experimentation revealed that further
mesh refinement did not improve significantly the accuracy of the calculations.
108
5.2 Methodology
Boundary conditions
It is sufficient to model only one quarter of the sandwich beam in the simulations,
applying symmetric boundary conditions at mid-span (x3 = 0) and at mid-plane
(x1 = 0). The overhang of simply supported beams beyond the outer rollers was
included in the simulations. Simply supported boundary conditions were modelled
by placing the sandwich beam on a rigid cylindrical roller with the same contact
properties as those mentioned above. On the other hand, clamped boundary con-
ditions were enforced by constraining to zero all degrees-of-freedom on the nodes of
the end face of the sandwich beam (x3 = L).
Material properties
The uniaxial tensile response of as-brazed AISI 304 stainless steel was measured in
Chapter 4 at a nominal strain-rate of 10−3 s−1 and is reproduced in Fig. 5.4(a). In
the simulations, the material was modelled as a rate-dependent, elastic-plastic solid
in accordance with J2-flow theory. The material has a density ρ = 7900 kg/m3, a
Young’s modulus E = 210 GPa, a Poisson’s ratio ν = 0.3 and a quasi-static (10−3
s−1) yield strength σY = 210 MPa. The hardening plastic behaviour was tabulated
in Abaqus from the data shown in Fig. 5.4(a).
The strain-rate sensitivity of stainless steel was investigated by Stout and Follansbee
0.8
1
1.2
1.4
1.6
1.8
R
Plastic strain rate ε (s-1)
10-4 104102110-2 .
p
pε = 0.1
(a) (b)
0
250
500
750
1000
1250
1500
0 0.1 0.2 0.3 0.4
Tr
ue
s
tre
ss
(M
Pa
)
Logarithmic strain
104
103
102
1
10-3
ε (s-1).p
Figure 5.4: (a) The quasi-static (ǫ˙p = 10−3 s−1) uniaxial tensile response of AISI 304
stainless steel and the estimated high strain-rate responses based on the data of Stout
and Follansbee (1986). (b) Dynamic strengthening ratio R as a function of plastic
strain rate ǫ˙p.
109
Chapter 5. Drop weight tests on prismatic sandwich beams
(1986) for strain-rates in the range 10−4 s−1 ≤ ǫ˙ ≤ 104 s−1. Their results are
reproduced in Fig. 5.4(b), where the dynamic strengthening ratio R is plotted as a
function of the plastic strain rate ǫ˙p. The ratio R is defined as the dynamic stress
σd(ǫp = 0.1) at an applied strain-rate ǫ˙p divided by the corresponding quasi-static
stress σqs(ǫp = 0.1) at ǫ˙p = 10−3 s−1. Stout and Follansbee (1986) also mentioned
that the ratio R is reasonably independent of ǫp. Consequently, the dynamic stress
σd versus plastic strain ǫp response can be expressed as:
σd(ǫp, ǫ˙p) = R(ǫ˙p)σqs(ǫp) , (5.1)
where R is given in Fig. 5.4(b). This prescription was employed in all simulations
with σqs(ǫp) given by the measured quasi-static response shown in Fig. 5.4(a).
To illustrate the influence of material strain-rate sensitivity, the estimated uniaxial
tensile responses of type 304 stainless steel at four selected additional values of
strain-rate are given in Fig. 5.4(a).
5.3 Experimental results
The dynamic responses at 5 m/s for all sandwich beams tested are given in Fig. 5.5
for simply supported beams and in Fig. 5.7 for clamped beams. In each figure, re-
sults are shown for sandwich beams with a corrugated core and a Y-frame core. For
each beam, the dynamic response is compared to the quasi-static response measured
previously in Chapter 4. In each plot, the mid-span roller displacement δ is nor-
malised by the beam half-span L = 125 mm whereas the load applied at mid-span
F is normalised by σY bc, where the quasi-static yield strength is σY = 210 MPa, the
width of the sandwich beams is b = 55 mm and the core thickness is c = 22 mm.
5.3.1 Simply supported beams
The quasi-static and 5 m/s responses of simply supported sandwich beams are given
in Fig. 5.5(a) for the corrugated core and in Fig. 5.5(b) for the Y-frame core. All
simply supported beams have an initial elastic regime up to a peak load, which is
followed by a softening response. For both core topologies, the dynamic peak load
is significantly higher than its quasi-static peak load. In contrast, the post-peak
110
5.3 Experimental results
response at 5 m/s is only slightly stronger than that under quasi-static loading. The
energy absorbed by the beams at 5 m/s is 24 and 25 J for the tests on the corrugated
core and on the Y-frame core, respectively. These values are in excellent agreement
with the kinetic energy of the projectile, which is 25 J.
0
5
10
15
20
25
0 0.05 0.1 0.15 0.2
δ
L
F
σ bcY
x10-3
Corrugated core
Measured FE
Quasi-static
5 m/s
0
5
10
15
20
25
0 0.05 0.1 0.15 0.2
δ
L
F
σ bcY
x10-3
Y-frame core
Measured FE
Quasi-static
5 m/s
(a)
(b)
Figure 5.5: Quasi-static and 5 m/s responses of simply supported sandwich beams
with (a) a corrugated core and (b) a Y-frame core.
One surprising result in Fig. 5.5 is that both core topologies have the same peak
loads at 5 m/s whereas their quasi-static peak loads are different. We anticipate
111
Chapter 5. Drop weight tests on prismatic sandwich beams
that the peak load measured during the drop weight tests is influenced by the mass
of the load cell and steel roller that are placed on the sandwich beam. To test this
hypothesis, an additional drop weight test was performed using aluminium roller
instead of a steel roller, and the influence on the measured response is shown in Fig.
5.6 for a simply supported sandwich beam with a Y-frame core. Note that a simply
supported sandwich beam has a total mass of approximately 200 g. In contrast,
the steel roller and load cell have a combined mass of 146 g, and replacing the steel
roller by an aluminium roller reduces the mass of this assembly to 64 g.
0
5
10
15
20
25
0 0.05 0.1 0.15
Steel mid-span roller
146g
Aluminium mid-span roller
64g
δ
L
F
σ bcY
x10-3
Y-frame core
Figure 5.6: Influence of the mid-span roller mass upon the measured 5 m/s response
of a simply supported sandwich beam with a Y-frame core. The measured response
with a steel mid-span roller is reproduced from Fig. 5.5(b).
It is clear from Fig. 5.6 that the mass of the mid-span roller has a strong influence
on the peak load measured at 5 m/s. Replacing the steel roller by an aluminium
roller reduces the mass of this component by about 50% and accordingly, the peak
load is also reduced by the same proportion. To minimise the effect of the mid-span
roller upon the measured dynamic peak load, the aluminium roller was used for the
tests on clamped beams, which are presented below.
5.3.2 Clamped beams
The responses of clamped sandwich beams are given in Fig. 5.7(a) and (b) for the
corrugated core and the Y-frame core, respectively. In each plot, the quasi-static
response is compared to the one measured at 5 m/s. All clamped sandwich beams
112
5.3 Experimental results
have an initial elastic regime followed by a peak load. This peak load is often followed
by a small load drop, and then the beam hardens due to longitudinal stretching. The
measured energy absorbed by clamped beams at 5 m/s is 34 J for the corrugated
core and 33 J for the Y-frame core. These values are slightly inferior to the kinetic
energy of the projectile, which is equal to 37.5 J.
0
5
10
15
20
0 0.05 0.1 0.15
δ
L
F
σ bcY
x10-3
Corrugated core
0
5
10
15
20
0 0.05 0.1 0.15
δ
L
F
σ bcY
x10-3
Y-frame core
Measured FE
Quasi-static
5 m/s(a)
(b)
Figure 5.7: Quasi-static and 5 m/s responses of clamped sandwich beams with (a)
a corrugated core and (b) a Y-frame core.
The initial peak load of clamped beams is significantly higher at 5 m/s than for quasi-
113
Chapter 5. Drop weight tests on prismatic sandwich beams
static loading; again, this is due to the inertia of the mid-span roller as demonstrated
in the previous section. For the corrugated core, the post-peak hardening response
measured at 5 m/s is stronger than the quasi-static response, see Fig. 5.7(a). In
contrast, the post-peak hardening response appears to be relatively insensitive to
the loading velocity for the Y-frame core, see Fig. 5.7(b). High-speed images taken
during the drop weight tests revealed that the test fixture was unable to ensure
perfectly clamped boundary conditions; a displacement of the order of 1-2 mm was
observed at the clamped ends. The compliance of the testing rig softens the post-
peak response of clamped beams and it is more important at 5 m/s than during
quasi-static tests.
5.3.3 Collapse mechanisms
The simply supported and clamped sandwich beams tested both collapse by inden-
tation under quasi-static loading, recall Section 4.3.3 on page 82. To verify whether
the collapse mechanism is the same at 5 m/s, high-speed images of the dynamic
tests performed on a sandwich beam with a Y-frame core are shown in Fig. 5.8(a)
and (b) for simply supported and clamped boundary conditions, respectively. For
both end conditions, the images indicate clearly that the beam collapses by inden-
tation of the core underneath the mid-span roller. High-speed images of the drop
weight tests performed on the corrugated core (not shown here) also indicate that
the beam fails by indentation. Hence, for the limited number of experiments done,
it appears that sandwich beams which collapse quasi-statically by indentation also
fail by indentation at 5 m/s.
The number of experiments done was limited because the drop weight tests do not
allow us to measure accurately the peak load of the beam. No drop weight tests
have been performed on sandwich beams which collapse quasi-statically by Brazier
plastic buckling. Will these sandwich beams also fail by Brazier plastic buckling at
5 m/s? This question will be addressed below using the finite element method.
5.4 Finite element predictions
Finite element simulations were performed with the following objectives: (i) to gain
additional insight into the drop weight tests and (ii) to compare the peak load
114
5.4 Finite element predictions
δ/L = 0.05
δ/L = 0.10
(a)
δ/L = 0.05
δ/L = 0.09
(b)
Figure 5.8: High-speed images captured during a drop weight test at 5 m/s on (a)
a simply supported and (b) a clamped sandwich beam with a Y-frame core. The
deformed beams are shown for two selected values of mid-span roller displacement
δ/L. All images are showing a side view of the beam.
115
Chapter 5. Drop weight tests on prismatic sandwich beams
and collapse mechanism obtained at 5 m/s to the ones obtained under quasi-static
loading.
5.4.1 Comparison between simulations and measurements
The FE predictions for all sandwich beams tested are included in Fig. 5.5 and 5.7
for simply supported and clamped boundary conditions, respectively. In each figure,
results are shown for the corrugated core in part (a) and for the Y-frame core in
part (b). Simulations for the quasi-static beam responses are also included; they are
reproduced from Chapter 4.
The simulations underestimate the measured 5 m/s responses of simply supported
beams, see Fig. 5.5. The large discrepancy between the measured and predicted peak
load is attributed to the fact that impact between the projectile and the assembly of
the mid-span roller and load cell is not included in the simulations. Additional FE
simulations in which all parts used in the experiments are modelled as deformable
solids are presented in Appendix 5.A as an attempt to capture the measured peak
load more accurately. In addition, an analytical model given in Appendix 5.B gives
additional insight into the contact force generated by the impact of the projectile
on the assembly of the mid-span roller and load cell.
The post-peak response of simply supported beams measured at 5 m/s is also slightly
underestimated by the FE method. This discrepancy is traced to the fact that the
simulations assume a frictionless contact between the beam and the rollers. As
a result of underestimating the force, the simulations significantly over-predict the
maximum mid-span roller displacement of simply supported sandwich beams; for the
Y-frame core, the maximum mid-span roller displacement measured is δmax/L = 0.13
whereas the simulations predicts δmax/L = 0.18.
The initial peak load of clamped beams measured at 5 m/s is also underestimated by
the simulations for the same reason mentioned above for simply supported beams.
However, the post-peak hardening response of clamped beams at 5 m/s is over-
predicted by the FE calculations, see Fig. 5.7. This is attributed to the fact that
perfect clamping conditions were assumed in the simulations whereas the test fixture
was not able to achieve this. The maximum mid-span roller displacement predicted
by the FE analysis is in good agreement with the experiments, but this is the result
of first underestimating the initial peak load and then over-predicting the post-peak
measured response of clamped beams.
116
5.4 Finite element predictions
5.4.2 Sensitivity of the peak load and collapse mechanism
to the loading velocity
The experimental investigation reported in Section 5.3 has an important limitation;
the finite mass of the mid-span roller and load cell does not allow us to measure
accurately the peak load on the sandwich beam. To overcome this problem, the
finite element method is used in this section to compare the predicted peak loads
in the quasi-static and 5 m/s simulations of sandwich beams that collapse quasi-
statically by (i) indentation and (ii) Brazier plastic buckling. This comparison will
allow us to evaluate if one of the two collapse mechanisms is more sensitive to
velocity.
Simply supported and clamped sandwich beams with a corrugated core or a Y-frame
core are considered. Similarly to the work done in Chapter 4, two types of beams
are analysed:
(i) beams of dimensions shown in Fig. 5.1 with front-and-back faces present and
(ii) beams with cores of dimensions given in Fig. 5.1, but with a front face of
thickness 2t = 0.6 mm and no back face.
Both designs have an equal mass, but beams with front-and-back faces present
collapse quasi-statically by indentation whereas beams with the back face absent
collapse by Brazier plastic buckling, recall Chapter 4. In all cases, the beam span
was kept fixed at 2L = 250 mm as used in the experimental study. All other details
of the finite element models were the same as those prescribed in Section 5.2.3,
except that the mid-span roller was given a constant velocity of 5 m/s instead of an
initial velocity. Numerical experimentation revealed that this modification had only
a minor effect on the simulated response.
The simulated quasi-static and 5 m/s responses are compared in Fig. 5.9. Beams
with front-and-back faces present are shown in part (a) with a corrugated core and
in part (b) with a Y-frame core. Likewise, beams with the back face absent are given
in parts (c) and (d) for the corrugated and Y-frame cores, respectively. In each plot,
results are shown for simply supported and clamped boundary conditions.
The results in Fig. 5.9 indicate that the peak load predicted at 5 m/s is (i) insensitive
to the choice of boundary conditions and (ii) exceeds the quasi-static value in all
cases. This increase is more important for the corrugated core than for the Y-frame
117
Chapter 5. Drop weight tests on prismatic sandwich beams
0
5
10
15
20
0 0.02 0.04 0.06 0.08 0.1
Quasi-static
5 m/s
δ
L
F
σ bcY
Front-and-back faces
x10-3
Clamped beams
Simply supported beams
0
5
10
15
20
0 0.02 0.04 0.06 0.08 0.1
Quasi-static
5 m/s
No back face
δ
L
F
σ bcY
x10-3
Simply supported beams
Clamped beams
0
5
10
15
20
0 0.02 0.04 0.06 0.08 0.1
Quasi-static
5 m/s
δ
L
F
σ bcY
Front-and-back faces
Clamped beams
Simply supported beams
x10-3
0
5
10
15
20
0 0.02 0.04 0.06 0.08 0.1
Quasi-static
5 m/s
No back face
Clamped beams
Simply supported beams
δ
L
F
σ bcY
x10-3
(a) (b)
(c) (d)
Corrugated core Y-frame core
Figure 5.9: FE predictions of the quasi-static and 5 m/s responses of simply sup-
ported and clamped sandwich beams. Beams with front-and-back faces are shown
with (a) a corrugated core and (b) a Y-frame core. Likewise, beams with the back
face absent are shown with (a) a corrugated core and (b) a Y-frame core. Beams with
front-and-back faces present collapse quasi-statically by indentation whereas those
with the back face absent fail quasi-statically by Brazier plastic buckling.
core, compare Fig. 5.9(a) and (b). Moreover, the increase in peak load is sensitive
to the allocation of face-sheet material; the increase is greater for sandwich beams
with the back face absent than for those with front-and-back faces present. This can
be attributed to the finite mass of the front face; recall that beams without a back
face have a thicker front face than beams with front-and-back faces present.
The FE simulations revealed a change in collapse mechanism: beams that failed
quasi-statically by Brazier plastic buckling collapsed by indentation at 5 m/s. To
demonstrate this, contours of equivalent plastic strain ǫ¯pl are shown in Table 5.1
for simply supported sandwich beams with a Y-frame core. Predictions are shown
118
5.4 Finite element predictions
for sandwich beams with and without a back face. For each beam, results obtained
quasi-statically are compared to those at 5 m/s. In all cases, the distribution of
equivalent plastic strain is plotted on a side view of the beam focusing on a portion
of length 0.35L from the beam mid-span. The results are shown for a value of
mid-span roller displacement just after the peak load.
The results in Table 5.1 indicate clearly that sandwich beams with front-and-back
faces collapse by indentation for both quasi-static loading and at 5 m/s. In both
cases, the distribution of equivalent plastic strain is localised underneath the mid-
span roller. In addition, the maximum equivalent plastic strain obtained quasi-
statically is similar to that predicted at 5 m/s. In contrast, the equivalent plastic
strain distribution obtained quasi-statically for a beam with the back face absent is
significantly different from that obtained at 5 m/s; for quasi-static loading, a diffuse
plastic hinge is formed at mid-span whereas at 5 m/s, the plastic strain is localised
underneath the mid-span roller. This indicate a change in collapse mechanism:
beams that collapse quasi-statically by Brazier plastic buckling fail by indentation
at 5 m/s.
Quasi-
static
5 m/s
Front-and-back faces
δ/L = 0.02
No back face
δ/L = 0.05
F,δ F,δ
F,δF,δ
εpl
εplεpl
εpl
Table 5.1: Equivalent plastic strain distribution for simply supported sandwich
beams with a Y-frame core. Beams with and without a back face are shown. Re-
sults are given for quasi-static loading and 5 m/s. All images are showing a side view
of the beam focusing on a portion of length 0.35L from the beam mid-span.
119
Chapter 5. Drop weight tests on prismatic sandwich beams
5.5 Concluding remarks
The response of a sandwich hull to a ship collision was investigated in the laboratory
using a drop weight apparatus. Drop weight tests, with an impact velocity of 5 m/s,
were performed on stainless steel sandwich beams with a corrugated core and a Y-
frame core. The tests were conducted on simply supported and clamped sandwich
beams, with the prismatic axis of the core aligned with the longitudinal axis of the
beam. The experimental setup did not allow us to measure the dynamic peak load
accurately; the finite mass of the mid-span roller significantly increased the measured
peak load. This problem can be minimised by reducing the mass of the mid-span
roller and future work should consider using a polycarbonate roller instead of the
steel and aluminium rollers used in this study. Nevertheless, the measurements did
capture the post-peak force with adequate precision. For most beams tested, the
post-peak response measured at 5 m/s was slightly stronger than the one measured
quasi-statically.
Three-dimensional finite element models were developed to gain additional insight
into the experiments. The predicted post-peak forces were found to be in reasonable
agreement with the measurements for both simply supported and clamped beams.
The finite element method was also used to investigate whether the peak load and
collapse mechanism obtained quasi-statically are different at 5 m/s. A sandwich
beam, which collapses quasi-statically by indentation, was also found to fail by
indentation at 5 m/s. In contrast, a sandwich beam that fails quasi-statically by
Brazier plastic buckling was found to collapse by indentation at 5 m/s. Finally, the
peak loads predicted at 5 m/s were found to be (i) independent of the boundary
conditions and (ii) higher than those obtained for quasi-static loading. The finite
mass of the front face was identified as an important factor contributing to this
increase of the peak load.
5.A Finite element predictions with the projectile
and roller modelled as deformable parts
The impact between the projectile and the assembly of the load cell and mid-span
roller was not considered in the finite element simulations detailed above; only the
impact between the mid-span roller and the sandwich beam was modelled, see Sec-
120
5.A Finite element predictions with the projectile and roller
tion 5.2.3. The influence of this modelling assumption on the simulated responses
at 5 m/s is analysed in this appendix. This will be done by comparing the results
obtained previously to additional simulations in which the projectile and the assem-
bly of the mid-span roller and load cell are fully-meshed and modelled as separate
parts.
5.A.1 Simulations without the rubber pad
An additional finite element model was developed in which the projectile and the
assembly of the steel mid-span roller and load cell are modelled as two separate
parts as shown in Fig. 5.10. Note that the rubber pad, placed at the bottom of the
projectile in the experiments, was not modelled in the simulations.
Projectile with
Load cell and
mid-span roller
L = 125
x2
x1x3
vi = 5 m/s
Initial velocity
Figure 5.10: Finite element model with the projectile and the assembly of the load
cell and mid-span roller modelled as two separate deformable parts. The model is
shown for a simply supported sandwich beam with a corrugated core. All dimensions
in mm.
The projectile and the assembly of the mid-span roller and load cell had the same
dimensions as those used in the experiments, see Fig. 5.2(a). It is sufficient to
121
Chapter 5. Drop weight tests on prismatic sandwich beams
model only one quarter of these two parts, applying symmetric boundary conditions
on x1 = 0 and x3 = 0 planes, see Fig. 5.10. The projectile was meshed with
linear hexahedral elements (C3D8R in Abaqus notation) whereas the assembly of
the mid-span roller and load cell was discretised using tetrahedral elements (C3D4 in
Abaqus notation). Both parts were modelled as isotropic elastic solids with material
properties representative of steel: a density ρ = 7900 kg/m3, a Young’s modulus
E = 210 GPa and a Poisson’s ratio ν = 0.3. The projectile was given an initial
downward velocity vi = 5 m/s and the interaction between all parts was defined as
a hard frictionless contact. Finally, the sandwich beam and the support roller were
modelled according to the prescription detailed previously in Section 5.2.3.
The response predicted by this additional finite element model is shown in Fig. 5.11
for a simply supported sandwich beam with a corrugated core. Two contact forces
are plotted:
(i) the contact force between the projectile and the assembly of the load cell and
mid-span roller is given in part (a) and
(ii) the contact force between the mid-span roller and the sandwich beam is shown
in part (b).
The latter is compared to the results of simulations shown previously where the
0
50
100
150
200
0 0.05 0.1 0.15 0.2 0.25 0.3
Corrugated core
δ
L
F
σ bcY
x10-3
Contact force between the
projectile and mid-span roller
0
2
4
6
8
10
12
14
0 0.05 0.1 0.15 0.2 0.25 0.3
Corrugated core
δ
L
F
σ bcY
x10-3
Contact force between the
mid-span roller and beam
Simulation with the projectile and
roller modelled as one rigid body
(a) (b)
Figure 5.11: Simulated 5 m/s response of a simply supported beam with a corrugated
core. The projectile and the mid-span roller are modelled as separate deformable parts
in the simulations. The contact forces between (a) the projectile and the mid-span
roller and (b) the mid-span roller and the beam are given. The simulations presented
in Section 5.4.1 where the projectile and mid-span roller were modelled as one rigid
body are included in part (b) for comparison.
122
5.A Finite element predictions with the projectile and roller
projectile and mid-span roller were modelled as one rigid body. The results shown
in Fig. 5.11(b) indicate that the assumption of modelling the projectile and mid-
span roller as a single rigid body instead of two deformable entities has very little
influence on the simulated dynamic response.
The contact force between the projectile and the mid-span roller, see Fig. 5.11(a),
reveals that the two parts come in contact for a very short duration and then separate
in several occasions at the beginning of the simulation for δ/L < 0.1. This was not
observed in the experiments because of the additional damping introduced by the
rubber pad at the bottom of the projectile, see Fig. 5.5(a). The finite element model
was then modified to take into account the rubber pad in the simulations. This is
presented in the next section.
5.A.2 Simulations with the rubber pad
It was necessary to measure the compressive response of the rubber pad used in the
experiments to take it into account in the simulations. To do so, the projectile, with
the rubber pad, and the assembly of the mid-span roller and load cell were loaded
in compression using a screw-driven test machine. The quasi-static compressive
response measured at a strain-rate ǫ˙ = 10−3 s−1 is shown in Fig. 5.12(a), where
the contact pressure is plotted as a function of the compressive displacement. The
compressive displacement was measured using a laser extensometer whereas the
contact force was measured using the load cell of the testing machine. The contact
pressure was calculated by dividing the contact force by the contact area of the load
cell (127 mm2).
The loading and unloading compressive responses of the rubber pad are given in
Fig. 5.12(a); the rubber pad exhibit a pronounced hysteresis. For a compressive
displacement inferior to 1 mm, the response of the rubber pad is approximately
linear with a stiffness of 2.5 kN/mm. This stiffness is significantly softer than that
of the piezoelectric load cell which has a stiffness of 1050 kN/mm according to the
data sheet of the manufacturer.
The rubber pad used in the experiments was not modelled as a separate part in
the FE simulations. Instead, the “softened” contact option of Abaqus was used.
According to the documentation this contact option can be used to model a soft,
thin layer on one or both contact surfaces. The contact pressure versus penetra-
tion relationship was tabulated in Abaqus from the loading part of the measured
123
Chapter 5. Drop weight tests on prismatic sandwich beams
0
5
10
15
20
25
0 0.05 0.1 0.15 0.2
δ
L
F
σ bcY
x10-3
Corrugated core
Measured
5 m/s
FE: Contact projectile-roller
FE: Contact roller-beam
(a) (b)
0
5
10
15
20
25
30
35
40
0 0.5 1 1.5
Compressive displacement (mm)
Pr
es
su
re
(M
Pa
)
ε = 10 s. -1-3
Figure 5.12: (a) Measured quasi-static (ǫ˙ = 10−3 s−1) compressive response of
the rubber pad used in the drop weight experiments. (b) Comparison between the
measured and simulated 5 m/s responses of a simply supported beam with a corrugated
core. The measured response is reproduced from Fig. 5.5(a) whereas the simulations
are using the data shown in part (a) to model the contact between the projectile and
the mid-span roller.
compressive response of the rubber pad shown in Fig. 5.12(a). This change in the
contact option was the only modification done to the FE model detailed above in
Section 5.A.1.
The results predicted by the FE method using the softened contact option are pre-
sented in Fig. 5.12(b) for a simply supported beam with a corrugated core. Pre-
dictions of the contact force are given for (i) the contact between the projectile and
mid-span roller (where the rubber pad is) and (ii) the contact between the mid-span
roller and the beam. In addition, the measured response at 5 m/s, presented in
Fig. 5.5(a), is reproduced in Fig. 5.12(b) for comparison purposes. The peak load
of the simulated contact force, between the projectile and the mid-span roller, is in
satisfactory agreement with measured peak load. However, there is an important
discrepancy between the simulated and measured post-peak responses. This is due
to the fact that the simulations assume the contact response to be perfectly elas-
tic and does not take into account the pronounced hysteresis of the rubber pad as
observed in Fig. 5.12(a). Also, this analysis neglects the strain-rate sensitivity of
the rubber, which is likely to be stiffer at 5 m/s than under quasi-static loading.
Nevertheless, the simulations shown in Fig. 5.12(b) give a valuable insight into the
experiments; the results indicate that the peak load measured at 5 m/s is influenced
by (i) the mass of the mid-span roller and load cell and (ii) the damping of the
rubber pad.
124
5.B Analytical prediction of the contact force
5.B Analytical prediction of the contact force
An analytical model is presented here to estimate the contact force between the
projectile and the assembly of the mid-span roller and load cell. The model is
illustrated in Fig. 5.13 and has the following elements:
1. a rigid projectile of mass M ,
2. a spring of stiffness k representing the rubber pad used in the experiments and
3. a second rigid body of mass m representing the assembly of the mid-span roller
and load cell.
MProjectile
kRubber pad
m
vi
Mid-span roller
and load cell
M
k
m
vf
Before impact After impact
vf
Figure 5.13: Analytical model to predict the contact force between the projectile
and the assembly of the mid-span roller and load cell.
The projectile has a given initial velocity vi before hitting the assembly of the mid-
span roller and load cell, which is at rest before the collision. After the collision, the
two bodies are assumed to adhere to each other and travel together with a common
velocity vf . Conservation of momentum provides that:
vf =
Mvi
M + m . (5.2)
The change in kinetic energy is assumed to compress the spring by a displacement
u, which generates a force P = ku. Hence, conservation of energy dictates that:
Mv2i
2 =
P 2
2k +
(M + m)v2f
2 . (5.3)
Substituting Eq. (5.2) in Eq. (5.3) and rearranging gives:
P =
√
kMmv2i
M + m . (5.4)
125
Chapter 5. Drop weight tests on prismatic sandwich beams
This equation provides an estimation of the force P generated during the collision of
the projectile and the assembly of the mid-span roller and load cell. It is insightful
to estimate the value of P for parameters used in the experimental study. First, the
stiffness of the rubber pad can be estimated to k = 2.5 kN/mm from the measured
compressive response shown previously in Fig. 5.12(a). Tests on simply supported
beams were done with a projectile of mass M = 2 kg and an impact velocity vi = 5
m/s. Finally, the assembly of the steel mid-span roller and load cell has a combined
mass m = 0.146 kg. Using these values, the contact force P = 2.9 kN which corre-
sponds to P/(σY bc) = 0.0119. Replacing the steel mid-span roller by an aluminium
roller reduces m to 0.064 kg, and for this new value of m the force P = 2 kN, which
corresponds to P/(σY bc) = 0.008. These values of P are significantly lower that
the measured peak loads, see Fig. 5.6, because the finite mass and stiffness of the
sandwich beam tested is not included in this analytical model. However, this simple
model is able to capture the strong influence of the mass m upon the contact force
P generated during the collision of the projectile and the assembly of the mid-span
roller and load cell.
126
Chapter 6
Dynamic indentation of prismatic
sandwich panels
Summary
The dynamic indentation response of stainless steel sandwich panels with a corru-
gated core or a Y-frame core was simulated using the finite element method. The
effect of the loading velocity upon the indentation response is assessed by indenting
the panels with a constant velocity ranging from quasi-static loading to 100 m/s.
The influence of the indenter’s geometry is also addressed by considering two dif-
ferent indenters: a flat-bottomed indenter and a cylindrical roller. The predictions
indicated that the indentation load applied to the front face is equal to the load
transmitted to the back face for velocities below approximately 10 m/s. For such
low velocities, inertia stabilisation effects were found to increase the dynamic initial
peak load above its quasi-static value. This strengthening effect was more impor-
tant for the corrugated core than for the Y-frame core. For loading velocities greater
than 10 m/s, the indentation force applied to the front face exceeded the force trans-
mitted to the back face due to wave propagation effects. The dynamic indentation
response was found to be very sensitive to the size of the flat-bottomed indenter;
increasing its width increased the importance of both inertia stabilisation and wave
propagation effects. In contrast, increasing the roller diameter had a much smaller
effect of the dynamic indentation response. Finally, the simulations indicated that
material strain-rate sensitivity has only a minor effect on the dynamic indentation
response of both lab-scale and full-scale sandwich panels.
127
Chapter 6. Dynamic indentation of prismatic sandwich panels
6.1 Introduction
The quasi-static three-point bending response of sandwich beams with a corrugated
core or a Y-frame core was investigated in Chapter 4 of this thesis and two collapse
mechanisms were identified: short beams were found to collapse by indentation
whereas long beams failed by Brazier plastic buckling. Beams that failed quasi-
statically by indentation were also found to collapse by indentation when they were
subjected to an impact at 5 m/s, recall Chapter 5. In contrast, beams that collapsed
quasi-statically by Brazier plastic buckling were found to fail by indentation at 5 m/s.
These findings obtained in the laboratory are in line with the results of full-scale
collision tests performed on the Y-frame sandwich hull; these experiments revealed
that the structure deforms by indentation, with the inner hull undergoing negligible
plastic deformation (Wevers and Vredeveldt, 1999). Hence, these lab-scale and full-
scale results indicate that the deformation of a sandwich structure during a ship
collision at 5 m/s is adequately represented by its indentation response. However,
little is known about the effect of the loading velocity upon the dynamic indentation
response of the structure; is the response at 1 m/s different from the one at 10 m/s?
The work of Tilbrook et al. (2007) can help to answer this question. The authors
investigated the dynamic compressive response of corrugated and Y-frame sandwich
cores at velocities ranging from 1-100 m/s. Two dynamic strengthening mechanisms
were identified: (i) inertia stabilisation of the core members against buckling and
(ii) wave propagation effects. The first mechanism was predominant at low crush-
ing velocities and the second one was active for high velocities. What will be the
importance of those dynamic strengthening effects when the loading conditions are
changed from uniform compression to localised indentation? In this study, the finite
element method is used to address this question for both sandwich panels with a
corrugated core or a Y-frame core.
The objective of this chapter is to analyse the sensitivity of the indentation response
to (i) the loading velocity, (ii) the shape of the indenter and (iii) the size of the
indenter. Velocities varying from quasi-static loading to 100 m/s are considered.
Ship collisions are likely to occur below 10 m/s, but the range from 10-100 m/s is
also examined for two reasons: (i) it is of interest for other industrial applications
such as automotive or rail transport and (ii) to allow comparison with the results of
Tilbrook et al. (2007) which covered velocities ranging from 1-100 m/s. Two shapes
of indenters are used in this study: a flat-bottomed indenter and a cylindrical roller.
128
6.1 Introduction
For each indenter, two different sizes are considered. Note that if the size of the
indenter is infinitely large, the panel is loaded in uniform compression. As uniform
compression represents the limiting case in this study, the results of Tilbrook et al.
(2007) are reviewed below.
6.1.1 Review of the dynamic uniform compressive response
The dynamic compressive response of corrugated and Y-frame cores was studied
experimentally and numerically by Tilbrook et al. (2007). Their simulations were
repeated as part of this study and the key results are presented below. For a complete
discussion of the subject, the reader is referred to Tilbrook et al. (2007).
The boundary conditions used to simulate the dynamic uniform compressive re-
sponse are shown in Fig. 6.1(a). All degrees-of-freedom are constrained to zero on
the back face whereas the front face has a constant downward velocity V0 ranging
from 1 to 100 m/s. To simulate the quasi-static compressive response, the velocity
V0 was replaced by a prescribed downward displacement δ. A complete description
of the finite element model is given in Appendix 6.A.
The quasi-static and 10 m/s compressive responses are shown in Fig. 6.2(a) for the
corrugated core and in Fig. 6.2(b) for the Y-frame core. The nominal compressive
stress is plotted as a function of the nominal compressive strain δ/c, where the core
compression is δ and the core thickness is c = 22 mm. The nominal compressive
stress on the front face is defined as:
σf =
Ff
bL , (6.1)
and that on the back face is:
σb =
Fb
bL , (6.2)
where the width of the panel is b = 26.5 mm, the length of the panel (in the prismatic
direction x3) is L and the normal component of the front face and back face reaction
forces are Ff and Fb, respectively.
The quasi-static and dynamic compressive responses are both characterised by an
initial elastic regime up to a peak stress σpk, followed by a steeply softening response
129
Chapter 6. Dynamic indentation of prismatic sandwich panels
(b)
(c)
L = 125
F/2, V
RP
0
D
A
A
Z
View on A-A
(a)
Rigid front face with
constant velocity V0
1x
2x
3x
c = 22
3x
2x
1x
3x
2x
1x
c = 22
c = 22
L = 125
F/2, V
RP
0
a
A
A
View on A-A
b = 26.5
b = 26.5
b = 26.5
Figure 6.1: Finite element models used to simulate (a) uniform compression, (b)
indentation by a flat-bottomed indenter and (c) indentation by a cylindrical roller.
The models are shown for a sandwich panel with a Y-frame core. All dimensions are
in mm.
130
6.1 Introduction
(a) (b)
0
1
2
3
4
5
6
7
0 0.1 0.2 0.3 0.4 0.5
Front face
Back face
St
re
ss
(M
Pa
)
Corrugated core
δ
c
10 m/s
Quasi-static
(front and back faces)
0
1
2
3
4
5
0 0.1 0.2 0.3 0.4 0.5
Front face
Back face
St
re
ss
(M
Pa
)
Y-frame core
δ
c
10 m/s
Quasi-static
(front and back faces)
Figure 6.2: Uniform compressive responses of sandwich panels with (a) a corrugated
core and (b) a Y-frame core. Results are given for quasi-static loading and for crush-
ing at 10 m/s. These simulations were re-executed based on previous work done by
Tilbrook et al. (2007).
due to buckling of the core members. Both core topologies have a quasi-static peak
stress σqspk ≈ 1 MPa which corresponds to the elastic buckling strength of the core
members. Increasing the loading velocity to 10 m/s, increases significantly the peak
stress, and this increase is more important for the corrugated core than for the
Y-frame core.
The effect of velocity on the dynamic compressive peak stress σdpk is shown in Fig.
6.3(a), where the results are normalised by the quasi-static compressive peak stress
σqspk. Likewise, the dynamic average stress σdav, normalised by the quasi-static average
stress σqsav, is plotted in Fig. 6.3(b) as a function of velocity. The average stress, up
to a nominal core compression of 20%, is defined as:
σav =
∫ 0.2
0
σd(δ/c) , (6.3)
and gives a measure of the core crushing resistance after the peak stress. The results
in Fig. 6.3 are given for both corrugated and Y-frame cores and for both stresses on
the front and back faces.
Two regimes can be identified in Fig. 6.3:
1. At low crushing velocities, V0 ≤ 10 m/s, the front and back face stresses are ap-
proximately equal and increase with increasing velocity. The dynamic stresses
are higher than their corresponding quasi-static values and this strengthening
131
Chapter 6. Dynamic indentation of prismatic sandwich panels
1
10
1 10 100
σpk
pk
qs
d
σ
V0 (m/s)
30
Corrugated core
Y-frame core
Front face
Back face
1
10
1 10 100
30
av
av
qs
d
σ
V0 (m/s)
σ
Corrugated core
Y-frame core
Front face
Back face
(a) (b)
Figure 6.3: (a) The normalised peak stress and (b) the normalised average stress up
to δ/c = 0.2 for corrugated and Y-frame sandwich cores crushed at a constant velocity
V0. Those simulations were re-executed based on previous work done by Tilbrook et al.
(2007).
effect is mainly due to inertia stabilisation of the core members against elas-
tic buckling. In line with the results of Calladine and English (1984), inertia
effects are more important for the stretching-dominated corrugated core than
for the bending-dominated Y-frame core. In addition, inertia effects have a
greater influence on the peak stress (Fig. 6.3(a)) than on the average stress
(Fig. 6.3(b)).
2. For velocities greater than approximately 10 m/s, the peak stress on the back
face is roughly constant whereas the peak stress on the front face increases
with increasing velocity. This indicates that the peak stress is governed by
plastic wave propagation. Note that the normalised peak stress on the front
face is relatively insensitive to the core topology. In contrast, the normalised
peak stress transmitted to the back face is less for the Y-frame core than for
the corrugated core.
In this chapter, the dynamic indentation response will be simulated for a flat-
bottomed indenter of width 2a and a cylindrical roller of diameter D as shown
in Fig. 6.1(b) and (c), respectively. The results presented in this section for dy-
namic uniform compression represent the case of an infinitely large indenter with
a = L or D →∞. What will be the influence of decreasing a and D on the results
shown in Fig. 6.3? This question will be addressed for both corrugated and Y-frame
cores.
132
6.2 Finite element models
6.1.2 Scope of study
First, a description of the finite element models used to simulate the dynamic inden-
tation response is given in Section 6.2. Second, the dynamic indentation responses
and corresponding deformation modes are presented for selected loading velocities.
Finally, the effects of velocity, indenter size and material rate-sensitivity upon the
dynamic indentation response are considered in turn.
6.2 Finite element models
The commercial finite element code Abaqus (version 6.10) was used to simulate the
quasi-static and dynamic indentation responses of sandwich panels with a corrugated
core or a Y-frame core. The cross-sectional dimensions of each core are given in Fig.
6.4. Both cores have a relative density ρ¯ = 0.025 and are approximately 1:20 scale
models of the cores used in a ship hull. Both core topologies have a unit cell of
width b = 26.5 mm and a core thickness c = 22 mm. The core members and the
face-sheets have a thickness t = 0.3 mm. The core is assumed to be perfectly bonded
to the face-sheets to produce a sandwich panel of half-length L = 125 mm, see Fig.
6.1(b,c). Four noded linear shell elements with reduced integration (S4R in Abaqus
notation) were used to discretise the sandwich panels with an average mesh size of
0.5 mm. A convergence study indicated that further refinement of the mesh did not
improve significantly the results.
Z
(a) (b)
b = 26.5
c = 22
b = 26.5
c = 22t = 0.3
60°
13
45°
4
1.5
1x
2x
1x
2x
t = 0.3
Figure 6.4: Cross-sectional dimensions of the sandwich panels: (a) corrugated core
and (b) Y-frame core. All dimensions in mm.
The boundary conditions were applied as follows. All degrees-of-freedom were con-
133
Chapter 6. Dynamic indentation of prismatic sandwich panels
strained to zero on the back face of the panel, see Fig. 6.1(b,c). Symmetric boundary
conditions were applied underneath the indenter (x3 = 0) and at the right end of
the panel (x3 = L). Similarly, symmetric boundary conditions were also applied on
both sides of the panel unit cell, see Fig. 6.4.
The indentation response of the sandwich panels was simulated for two different
indenters:
1. A flat-bottomed indenter of width 2a, see Fig. 6.1(b). To simplify the analysis,
this loading condition was achieved by prescribing a constant velocity V0 over
a width a of the front face. Two values of width were considered, 2a = 12.5
and 50 mm corresponding to a/L ratios of 0.05 and 0.2, respectively.
2. A cylindrical roller of diameter D, as shown in Fig. 6.1(c). The roller was
modelled as a rigid body in the simulations and had a prescribed constant
velocity V0. Calculations were performed for two roller diameters, D = 9 and
66 mm corresponding to D/c ratios of 0.41 and 3, respectively.
The interaction between the roller and the front face, and between all potentially con-
tacting surfaces of the sandwich panel, was modelled as a hard frictionless contact.
Numerical experimentation revealed that the indentation response is insensitive to
the coefficient of friction used in the contact properties.
The dynamic indentation response was simulated for velocities V0 ranging from 1 to
100 m/s. Those simulations were performed using the explicit solver of Abaqus. On
the other hand, the implicit solver of Abaqus was used to predict the quasi-static
indentation response. For quasi-static simulations, the velocity V0 was replaced by
a prescribed displacement δ.
6.2.1 Geometric imperfections
A geometric imperfection was introduced in both core topologies. The shape of the
imperfection had the form of the first mode of elastic buckling and the amplitude was
set equal to the sheet thickness t = 0.3 mm. The elastic buckling calculations were
performed under uniform compression and the face-sheets were considered rigid,
such that the imperfection affected the core only and not the face-sheets. The same
geometric imperfection was used by Tilbrook et al. (2007) to simulate the dynamic
compressive responses of corrugated and Y-frame cores, and their predictions were
found to be in good agreement with experiments. In addition, Tilbrook et al. (2007)
134
6.3 Results
mentioned that the dynamic stress versus strain response is relatively imperfection-
insensitive, but the deformed shape varies with the choice of imperfection.
6.2.2 Material properties
The material properties were chosen to be representative of AISI 304 stainless steel.
The material was modelled as a rate-dependent J2-flow theory solid with a density
ρ = 7900 kg/m3, a Young’s modulus E = 210 GPa, a Poisson’s ratio ν = 0.3
and a quasi-static (10−3 s−1) yield strength σY = 210 MPa. The hardening plastic
behaviour, at strain-rates in the range 10−3 s−1 ≤ ǫ˙ ≤ 104 s−1, was tabulated in
Abaqus using the prescription described previously in Section 5.2.3 and employing
the data shown in Fig. 5.4(a).
6.3 Results
The results of the finite element predictions are presented as follows. First, the
dynamic indentation responses and the deformed meshes of both corrugated and Y-
frame sandwich panels are presented for selected velocities. Second, the influence of
the loading velocity and of the indenter geometry upon the initial peak load and the
average indentation load is examined. Third, the load transmitted to the back face
of the panel is analysed in details and finally, the influence of material strain-rate
sensitivity upon the dynamic indentation response is assessed.
6.3.1 Indentation responses
The responses of sandwich panels indented by (i) a flat-bottomed indenter of nor-
malised width a/L = 0.05 and (ii) a roller of normalised diameter D/c = 0.41 are
shown in Fig. 6.5 and 6.6, respectively. Results are given for both corrugated and
Y-frame cores. The responses are shown for selected velocities V0: the quasi-static
and 1 m/s responses are both shown in parts (a,b); the responses at 10 m/s are
shown in parts (c,d) and the responses at 100 m/s are given in parts (e,f). In each
plot, the indentation depth δ is normalised by the core thickness c = 22 mm whereas
the load F is normalised by σY bc, where the quasi-static yield strength is σY = 210
MPa and the width of the panel is b = 26.5 mm. Both the load applied to the front
135
Chapter 6. Dynamic indentation of prismatic sandwich panels
face and the load transmitted to the back face of the sandwich panel are plotted in
Fig. 6.5 and 6.6. The total back face force is the summation of the normal reaction
force for all nodes on the back face.
At low velocities, V0 ≤ 10 m/s, the forces on the front and back faces are approx-
imately equal over the entire deformation history, see Fig. 6.5(a-d) and 6.6(a-d).
The indentation response is characterised by an elastic regime up to an initial peak
load Fpk. Subsequently, the panel softens and then re-hardens due to longitudinal
stretching of the front face. The initial peak load is sensitive to the core topology;
sandwich panels with a corrugated core are at least 12% stronger than those with
a Y-frame core. The initial peak load is also sensitive to the loading velocity; Fpk
increases with increasing V0.
When the velocity is increased to 100 m/s, the force on the front face largely exceeds
the force transmitted to the back face over the entire deformation history, see Fig.
6.5(e,f) and 6.6(e,f). At such a high velocity, the core topology has a minimal
influence on the force applied to the front face, but it has a strong effect on the load
transmitted to the back face; the force on the back face is significantly higher for
the corrugated core than for the Y-frame core. Note that for panels indented by a
cylindrical roller, the force on the front face is particularly noisy at the beginning of
the response (δ/c < 0.1), see Fig. 6.6(e,f). This is due to the fact that the roller and
front face come in contact and then separate on a few occasions before a permanent
contact is established. The contact noise is significantly less important at lower
velocities; see for example the responses at 10 m/s in Fig. 6.6(c,d).
6.3.2 Deformed meshes
The deformed meshes, associated with the responses shown in Fig. 6.5 and 6.6, are
given in Table 6.1 for panels with a corrugated core and in Table 6.2 for panels
with a Y-frame core. In each table, the deformed meshes of panels indented by
a flat-bottomed indenter (a/L = 0.05) and a cylindrical roller (D/c = 0.41) are
shown at selected velocities. The deformed cross-section underneath the indenter is
shown along with a side view of the sandwich panel. For comparison purposes, the
deformed meshes obtained under uniform compression are also included. All images
are shown for δ/c = 0.35.
136
6.3 Results
(a) (b)
(c) (d)
(e) (f)
Corrugated core Y-frame core
0
20
40
60
80
100
0 0.1 0.2 0.3 0.4 0.5
Front face
Back face
x10-3
δ
c
F
σ bcY
100 m/s
0
20
40
60
80
100
0 0.1 0.2 0.3 0.4 0.5
Front face
Back face
x10-3
δ
c
F
σ bcY
100 m/s
0
5
10
15
20
0 0.1 0.2 0.3 0.4 0.5
Quasi-static
(front and back faces)
1 m/s (front and back faces)
x10-3
δ
c
F
σ bcY
0
5
10
15
20
0 0.1 0.2 0.3 0.4 0.5
Quasi-static
(front and back faces)
1 m/s (front and back faces)
x10-3
δ
c
F
σ bcY
0
10
20
30
40
0 0.1 0.2 0.3 0.4 0.5
Front face
Back face
x10-3
δ
c
F
σ bcY
10 m/s
0
10
20
30
40
0 0.1 0.2 0.3 0.4 0.5
Front face
Back face
x10-3
δ
c
F
σ bcY
10 m/s
Figure 6.5: Responses of sandwich panels indented by a flat-bottomed indenter of
normalised width a/L = 0.05. Results are shown at selected velocities: quasi-static
and 1 m/s for (a) corrugated core and (b) Y-frame core; 10 m/s for (c) corrugated
core and (d) Y-frame core and 100 m/s for (e) corrugated core and (f) Y-frame core.
137
Chapter 6. Dynamic indentation of prismatic sandwich panels
0
5
10
15
20
0 0.1 0.2 0.3 0.4 0.5
Front face
Back face
x10-3
10 m/s
F
σ bcY
δ
c
Corrugated core Y-frame core
0
5
10
15
0 0.1 0.2 0.3 0.4 0.5
x10-3
δ
c
F
σ bcY
1 m/s (front and back faces)
Quasi-static
(front and back faces)
0
5
10
15
0 0.1 0.2 0.3 0.4 0.5
x10-3
δ
c
F
σ bcY Quasi-static(front and back faces)
1 m/s (front and back faces)
(a) (b)
(c) (d)
(e) (f)
0
20
40
60
80
100
0 0.1 0.2 0.3 0.4 0.5
Front face
Back face
x10-3
100 m/s
δ
c
F
σ bcY
0
20
40
60
80
100
0 0.1 0.2 0.3 0.4 0.5
Front face
Back face
x10-3
100 m/s
δ
c
F
σ bcY
0
5
10
15
20
0 0.1 0.2 0.3 0.4 0.5
Front face
Back face
δ
c
x10-3
F
σ bcY
10 m/s
Figure 6.6: Responses of sandwich panels indented by a cylindrical roller of nor-
malised diameter D/c = 0.41. Results are shown at selected velocities: quasi-static
and 1 m/s for (a) corrugated core and (b) Y-frame core; 10 m/s for (c) corrugated
core and (d) Y-frame core and 100 m/s for (e) corrugated core and (f) Y-frame core.
138
6.3
R
esu
lts
V0
Uniform
compression Indentation a/L = 0.05
Quasi-static
1 m/s
10 m/s
100 m/s
Indentation D/c = 0.41
Table 6.1: Deformed meshes of sandwich panels with a corrugated core shown at selected velocities. The results are given for
uniform compression, indentation by a flat-bottomed indenter of normalised width a/L = 0.05 and indentation by a cylindrical roller
of normalised diameter D/c = 0.41. For indentation, the cross-section underneath the indenter is shown along with a side view of the
panel. All images are given for δ/c = 0.35.
139
C
h
ap
ter
6.
D
y
n
am
ic
in
d
en
tation
of
p
rism
atic
san
d
w
ich
p
an
els
V0
Uniform
compression Indentation a/L = 0.05
Quasi-static
1 m/s
10 m/s
100 m/s
Indentation D/c = 0.41
Table 6.2: Deformed meshes of sandwich panels with a Y-frame core shown at selected velocities. The results are given for uniform
compression, indentation by a flat-bottomed indenter of normalised width a/L = 0.05 and indentation by a cylindrical roller of
normalised diameter D/c = 0.41. For indentation, the cross-section underneath the indenter is shown along with a side view of the
panel. All images are given for δ/c = 0.35.
140
6.3 Results
First, consider the influence of velocity on the deformation modes. The deformed
meshes at 1 and 10 m/s are very similar to those obtained quasi-statically. However,
the deformed meshes at 100 m/s are considerably different from the quasi-static
results; deformation is localised near the front face of the panel. This deformation
mode is indicative of plastic wave propagation effects and it is also consistent with
the indentation responses presented above; for V0 = 100 m/s, the force applied on
the front face exceeds the force transmitted to the back face, recall Fig. 6.5(e,f) and
6.6(e,f).
Second, consider the effect of the indenter geometry upon the deformation modes.
The deformed cross-sections of panels indented by a flat-bottomed indenter (a/L =
0.05) are very similar to those of panels indented by a cylindrical roller (D/c =
0.41). Furthermore, the deformed cross-sections obtained for localised indentation
(for both the flat-bottomed indenter and the cylindrical roller) are comparable to
those obtained for uniform compression. These observations hold true for both
corrugated and Y-frame core topologies.
6.3.3 Influence of velocity
The effect of velocity upon the initial peak load is shown in Fig. 6.7(a) for panels
indented by a flat-bottomed indenter with a/L = 0.05, and in Fig. 6.8(a) for panels
indented a cylindrical roller with D/c = 0.41. In both figures, the dynamic initial
peak load F dpk is normalised by the quasi-static initial peak load F qspk . The results are
plotted for velocities ranging from 1 to 30 m/s only because it is difficult to evaluate
accurately the initial peak load at higher velocities, see for example the indentation
responses at 100 m/s in Fig. 6.6(e,f).
The average load is also plotted as a function of velocity in part (b) of Fig. 6.7 and
6.8. Similarly to the average stress definied in Eq. (6.3), the average load up to
δ/c = 0.2 is defined as:
Fav =
∫ 0.2
0
Fd(δ/c) . (6.4)
The dynamic average load F dav is normalised by the quasi-static average load F qsav in
Fig. 6.7 and 6.8. In each plot, the results are given for both the corrugated core
and the Y-frame core and for both forces on the front face and on the back face.
141
Chapter 6. Dynamic indentation of prismatic sandwich panels
(a) (b)
1
10
1 10 100
Front face
Back face
0.3
av
av
qs
d
V0 (m/s)
F
F
Corrugated core
Y-frame core
1
2
3
4
5
1 10 100
Front face
Back face
F pk
pk
qs
d
F
V0 (m/s)
Corrugated core
Y-frame core
Figure 6.7: (a) The normalised initial peak load and (b) the normalised average load
up to δ/c = 0.2 for corrugated and Y-frame sandwich panels indented at a constant
velocity V0 by a flat-bottomed indenter of normalised width a/L = 0.05.
(a) (b)
0.1
1
10
1 10 100
Front face
Back face
V0 (m/s)
F av
av
qs
d
F
Corrugated core
Y-frame core
1
2
3
4
5
1 10 100
Front face
Back face
F pk
pk
qs
d
F
V0 (m/s)
Corrugated core
Y-frame core
Figure 6.8: (a) The normalised initial peak load and (b) the normalised average load
up to δ/c = 0.2 for corrugated and Y-frame sandwich panels indented at a constant
velocity V0 by a cylindrical roller of normalised diameter D/c = 0.41.
The normalised initial peak load and the normalised average load on the front and
back faces are approximately equal for velocities ranging from 1 to 10 m/s. Even
at such low velocities, the dynamic initial peak load is greater than its quasi-static
value, and increases slightly with increasing velocity due to inertia stabilisation
effects. Inertia effects are more important for the corrugated core than for the Y-
frame core, see Fig. 6.7(a). In addition, the normalised initial peak loads of panels
indented by a flat-bottomed indenter (Fig. 6.7(a)) display a greater sensitivity
to velocity than those indented by a cylindrical roller (Fig. 6.8(a)). In contrast,
inertia effects have no influence on the normalised average load; F dav/F qsav is relatively
insensitive to velocity and to the choice of core topology for indentation velocities
142
6.3 Results
between 1 and 10 m/s.
The average force applied on the front face exceeds the force transmitted to the back
face when the velocity is greater than roughly 10 m/s, see Fig. 6.7(b) and 6.8(b),
and this is due to wave propagation effects. For velocities between 10 and 100 m/s,
the normalised average load on the front face increases with increasing velocity and
display a mild sensitivity to the core topology; F dav/F qsav is slightly higher for the
corrugated core than for the Y-frame core. In contrast, the normalised average load
on the back face is highly sensitivity to the choice of core; the load transmitted to
the back face for the corrugated core significantly exceeds that for the Y-frame core.
In fact, the normalised average load transmitted to the back face for the Y-frame
core decreases with increasing velocity. To explain this result, the force distribution
on the back face will be analysed in Section 6.3.5.
6.3.4 Influence of indenter size
The effect of the indenter size on the normalised initial peak load and on the nor-
malised average load is shown in Fig. 6.9 for panels indented by a flat-bottomed
indenter and in Fig. 6.10 for panels indented by a cylindrical roller. In each figure,
the normalised initial peak loads are shown in parts (a) and (b) for the corrugated
core and the Y-frame core, respectively. Likewise, the normalised average loads
are given in part (c) for the corrugated core and in part (d) for the Y-frame core.
In each plot, the results are given for two values of indenter size and for uniform
compression.
It is clear from Fig. 6.9 and 6.10 that the dynamic uniform compression response is
more sensitive to velocity than the localised indentation response; values of F dpk/F qspk
and F dav/F qsav for uniform compression always exceed those obtained for indentation.
The width of the flat-bottomed indenter has a strong influence on the normalised
initial peak load; increasing a/L from 0.05 to 0.20 increases F dpk/F qspk by a factor
of approximately two for both the corrugated core and the Y-frame core. In con-
trast, the roller diameter has only a mild effect on the normalised initial peak loads;
increasing D/c from 0.41 to 3 increases F dpk/F qspk by 45% at the most.
The normalised average loads are less sensitive to the size of the indenter than the
normalised initial peak loads. Note that the normalised average load applied to the
front face starts to exceed the normalised average load transmitted to the back face
at approximately 10 m/s in all cases considered. Hence, the velocity at which the
143
Chapter 6. Dynamic indentation of prismatic sandwich panels
(a) (b)
(c) (d)
Corrugated core Y-frame core
1
10
1 10 100
Front face
Back face
30
F av
av
qs
d
F
V0 (m/s)
0.05
0.2
Uniform
compression
a/L = 1
1
10
1 10 100
Front face
Back face
F av
av
qs
d
F
20
V0 (m/s)
Uniform compression
a/L = 1
0.05
0.2
0.3
1
10
1 10 100
Front face
Back face
F pk
pk
qs
d
F
30
V0 (m/s)
Uniform
compression
a/L = 1
0.05
0.20
1
10
1 10 100
Front face
Back face
Uniform compression
a/L = 1 F pk
pk
qs
d
F
30
V0 (m/s)
0.2
0.05
Figure 6.9: Influence of the normalised width a/L of the flat-bottomed indenter on
the normalised initial peak load for (a) the corrugated core and (b) the Y-frame core.
Likewise, the influence of a/L on the normalised average load up to δ/c = 0.2 is shown
for (c) the corrugated core and (d) the Y-frame core.
144
6.3 Results
(a) (b)
(d)(c)
Corrugated core Y-frame core
0.1
1
10
1 10 100
Front face
Back face
F av
av
qs
d
F
20
V0 (m/s)
Uniform
compression
D/c = 0.41
D/c = 3
D/c → 8
1
10
1 10 100
Front face
Back face
F pk
pk
qs
d
F
30
V0 (m/s)
Uniform
compression
D/c = 3
D/c = 0.41
D/c → 8
1
10
1 10 100
Front face
Back face
D/c = 0.41
Uniform
compression
F pk
pk
qs
d
F
30
V0 (m/s)
0.5
D/c = 3
D/c → 8
1
10
1 10 100
Front face
Back face
30
F av
av
qs
d
F
V0 (m/s)
D/c = 0.41
D/c = 3
Uniform
compression
D/c → 8
Figure 6.10: Influence of normalised roller diameter D/c on the normalised initial
peak load for (a) the corrugated core and (b) the Y-frame core. Likewise, the influence
of D/c on the normalised average load up to δ/c = 0.2 is shown for (c) the corrugated
core and (d) the Y-frame core.
145
Chapter 6. Dynamic indentation of prismatic sandwich panels
force equilibrium between the front and back faces is lost appears to be relatively
insensitive to the geometry of the indenter.
6.3.5 Force distribution on the back face
In all simulations, the back face of the sandwich panel is fully-clamped against
translational and rotational displacements, see Fig. 6.1(b,c). It is anticipated that
the normal traction on the back face will be positive underneath the indenter, but
negative at the right end of the panel. Consequently, the total force transmitted to
the back face is the sum of positive and negative forces, and in this section their
relative proportions are analysed.
The distribution of the normal traction T on the back face, at an indentation depth
δ/c = 0.2, is plotted in Fig. 6.11 for sandwich panels indented by a cylindrical
roller with D/c = 0.41. Results are given for the corrugated core in part (a) and
for the Y-frame core in part (b). In each plot, the normal traction distributions are
shown for three selected values of velocity V0. As expected, the normal traction is in
general positive underneath the indenter (around x3 = 0), and negative at the right
end (around x3 = L). The transition between the positive and negative traction
occurs at a position xt3, which is sensitive to velocity; xt3 decreases with increasing
V0. The transition xt3 is also sensitive to the core topology; values of xt3 are lower
for the corrugated core than for the Y-frame core.
(a) (b)
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0 0.2 0.4 0.6 0.8 1
Corrugated core
x
L
3
T
σ Y
10 m/s50 m/s
100 m/s
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0 0.2 0.4 0.6 0.8 1
Y-frame core
10 m/s
50 m/s
100 m/s
T
σ Y
x
L
3
Figure 6.11: Distribution of the normal traction T on the back face at an indentation
depth δ/c = 0.2. Results are shown for a sandwich panel indented by a cylindrical
roller of normalised diameter D/c = 0.41: (a) corrugated core and (b) Y-frame core.
146
6.3 Results
The total force transmitted to the back face at δ/c = 0.2 is obtained by integrat-
ing the curves shown in Fig. 6.11. The total positive and negative traction forces
exerted on the back face are compared in Table 6.3. For the corrugated core, the
total positive and negative traction forces are both increasing with increasing ve-
locity. On the other hand, for the Y-frame core, the total positive traction force
is almost insensitive to velocity whereas the total negative traction force increases
with increasing velocity. Consequently, the total force on the back face decreases
with increasing velocity. This result explains why F dav/F qsav for the back face is going
below unity at high velocities for the Y-frame core in Fig. 6.9(d) and 6.10(d).
Back face traction force ( )YF bcσ (10-3)
V0 (m/s)
Corrugated core Y-frame core
Positive Negative Total Positive Negative Total
10 27.9 -17.2 10.7 17.8 -5.9 11.9
50 34.8 -20.3 14.5 22.2 -14.0 8.2
100 37.7 -24.7 13.0 19.3 -18.0 1.3
Table 6.3: Positive and negative traction forces on the back face of a sandwich panel
indented by a cylindrical roller of normalised diameter D/c = 0.41. The forces are for
an indentation depth δ/c = 0.2, the values are integrated from the curves shown in
Fig. 6.11.
6.3.6 Influence of material strain-rate sensitivity
The compressive strain-rate experienced by the core members underneath the inden-
ter scales with ǫ˙ ≈ V0/c. Therefore, increasing the size of the structure reduces the
strain-rate that it experiences. Tilbrook et al. (2007) demonstrated that material
strain-rate sensitivity has a negligible effect upon the dynamic compressive response
of corrugated and Y-frame cores. Note that the cores analysed by Tilbrook et al.
(2007) had the same dimensions as those considered in this study. In this section,
the influence of material strain-rate sensitivity on the dynamic indentation response
will be evaluated by comparing the results of three FE models:
(i) A lab-scale sandwich panel of dimensions presented in Section 6.2 and modelled
with a rate-dependent solid. This is the FE model used in all simulations
presented above.
(ii) The same sandwich panel as model (i), but modelled as a rate-independent
solid. The uniaxial tensile response of the material was tabulated from the
147
Chapter 6. Dynamic indentation of prismatic sandwich panels
quasi-static response (10−3 s−1) plotted in Fig. 5.4(a).
(iii) A full-scale sandwich panel with the dimensions presented in Section 6.2 in-
creased by a factor of 20. The material was modelled as a rate-dependent
solid.
These three sandwich panels were indented by a cylindrical roller of normalised
diameter D/c = 0.41 and the results are shown in Fig. 6.12. The normalised peak
loads for the three models are given in Fig. 6.12(a) and (b) for the corrugated
core and the Y-frame core, respectively. Likewise, the normalised average loads up
to δ/c = 0.2 are shown in Fig. 6.12(c) for the corrugated core and Fig. 6.12(d)
0.1
1
10
1 10 100
F av
av
qs
d
F
V0 (m/s)
Lab-scale
strain-rate
independent
Full-scale
strain-rate
dependent
Lab-scale
strain-rate
dependent
1
10
1 10 100
F av
av
qs
d
F
V0 (m/s)
Lab-scale
strain-rate
dependent
Lab-scale
strain-rate
independent
Full-scale
strain-rate
dependent
Corrugated core Y-frame core
(a) (b)
(c) (d)
1
2
3
4
5
1 10 100
F pk
pk
qs
d
F
V0 (m/s)
Lab-scale
strain-rate
dependent
Lab-scale
strain-rate
independent
Full-scale
strain-rate
dependent
1
2
3
4
5
1 10 100
F pk
pk
qs
d
F
V0 (m/s)
Lab-scale
strain-rate
independent
Full-scale
strain-rate
dependent
Lab-scale
strain-rate
dependent
Figure 6.12: Influence of scale and material strain-rate sensitivity on the normalised
initial peak load for (a) the corrugated core and (b) the Y-frame core. Likewise, the
influence of scale and material strain-rate sensitivity on the normalised average load up
to δ/c = 0.2 is shown for (c) the corrugated core and (d) the Y-frame core. Results are
shown for panels indented by a cylindrical roller of normalised diameter D/c = 0.41.
The force on the front and back faces are given by solid and dashed lines, respectively.
148
6.4 Concluding remarks
for the Y-frame core. Comparing the results of models (i) and (ii) indicate that
neglecting the material strain-rate sensitivity decreases the normalised average loads
by about 15%. As expected, the results for full-scale sandwich panels, model (iii),
show that the effect of material strain-rate sensitivity is less important when the
size of the panel is increased. Nevertheless, the normalised peak loads and the
normalised average loads for model (iii) are higher than those obtained for the strain-
rate independent simulations (model (ii)). The results presented in Fig. 6.12 confirm
that material strain-rate sensitivity has a small influence of the dynamic indentation
responses of corrugated and Y-frame sandwich panels. Consequently, the results
presented above for lab-scale specimens are adequate to represent the behaviour of
full-scale sandwich structures.
6.4 Concluding remarks
The finite element method was used to investigate the dynamic indentation response
of stainless steel sandwich panels with a corrugated core and a Y-frame core. The
panels were indented at a constant velocity ranging from quasi-static loading to 100
m/s, and two different indenters were considered; a flat-bottomed indenter and a
cylindrical roller.
The indentation force applied to the front face of the panel was equal to the force
transmitted to the back face for velocities below 10 m/s. Even for such low velocities,
inertia stabilisation effects were found to increase the dynamic initial peak load above
its quasi-static value, and this effect was more important for the corrugated core than
for the Y-frame core. At velocities greater than 10 m/s, the force applied to the
front face exceeded the force transmitted to the back face due to wave propagation
effects. The force applied to the front face was mildly sensitive to the core topology;
however, the force transmitted to the back face was significantly higher for panels
with a corrugated core than for those with a Y-frame core.
The ratio of the dynamic initial peak load divided by its quasi-static value was found
to be sensitive to the size of the indenter. Increasing the width of the flat-bottomed
indenter increased this ratio significantly whereas increasing the roller diameter had
a smaller effect on this quantity. Furthermore, a comparison of the deformed meshes
revealed that the deformation modes are sensitive to velocity, but relatively insen-
sitive to the indenter shape and size. Finally, the simulations demonstrated that
149
Chapter 6. Dynamic indentation of prismatic sandwich panels
material rate-sensitivity has only a mild effect on the dynamic indentation response
of lab-scale sandwich panels. Consequently, the results on lab-scale sandwich panels
are representative of full-scale panels.
6.A Finite element model for uniform compres-
sion
The dynamic uniform compressive responses of corrugated and Y-frame cores were
simulated using the commercial software Abaqus (version 6.10). The dimensions of
both cores were identical to those shown in Fig. 6.4, except that the Y-frame core
had a fillet, with a radius of 1.5 mm, between the Y-frame leg and the horizontal
flange, see Fig. 6.1(a). This local reinforcement was also present in the simulations
of Tilbrook et al. (2007); the authors found that it was necessary to obtain a good
agreement with their experimental results.
Both core topologies were meshed with four-noded plane strain quadrilateral ele-
ments (CPE4R in Abaqus notation). An average mesh size of t/8 = 0.0375 mm was
used in all calculations; additional mesh refinements did not improve significantly
the accuracy of the results. The geometric imperfection and the material proper-
ties were the same as those employed for the dynamic indentation simulations, see
Sections 6.2.1 and 6.2.2, respectively.
The front and back faces were modelled as rigid surfaces. All degrees-of-freedom
were constrained to zero on the back face whereas the front face had a constant
downward velocity V0, see Fig. 6.1(a). The crushing velocity V0 was varied from
1 to 100 m/s and the dynamic simulations were executed with the explicit solver
of Abaqus. For the quasi-static simulations, the constant velocity was replaced by
a prescribed displacement δ and the simulations were executed using the implicit
solver of Abaqus. A hard frictionless contact was defined between all surfaces of the
model.
150
Chapter 7
Conclusions and future work
As mentioned in Section 1.3, the objectives of this thesis were: (i) to explore how a
surface treatment can improve the strength of a lattice material and (ii) to investigate
the collapse response of two competing prismatic sandwich cores employed in ship
hulls. The first objective was treated in Chapter 3 whereas the second objective
was addressed in Chapter 4 for quasi-static loading and in Chapters 5 and 6 for
dynamic loading. The conclusions reached in Chapters 3 to 6 are summarised below
in relation to those two objectives. This chapter ends with recommendations for
future work.
7.1 Compressive response of a carburised pyrami-
dal lattice
• The finite element method was used to simulate the quasi-static compressive
response of a pyramidal lattice made from (i) tubes and (ii) solid struts.
• First, the influence of strain hardening was investigated by comparing the
response of a lattice made from a perfectly plastic solid (Et = 0) to one made
from stainless steel (Et = 2 GPa). Strain hardening was found to increase the
peak compressive stress of lattices with a slenderness ratio l/d < 10. However,
strain hardening had no influence on the collapse mode of the lattice.
• Second, the effect of carburisation was examined. Carburisation is a surface
treatment that increases the yield strength of the material. The collapse mode
151
Chapter 7. Conclusions and future work
of the pyramidal lattice was found to be sensitive to carburisation; the transi-
tion between plastic and elastic buckling occurred at a lower slenderness ratio
when the lattice was carburised. In addition, carburisation increased the peak
compressive stress of the lattice, except for those collapsing by elastic buckling.
• The pyramidal lattice made from carburised tubes was found to possess a
compressive strength superior to that of other metallic lattices made from
aluminium or titanium.
7.2 The influence of the back face on the bending
response of prismatic sandwich beams
• Sandwich beams with a corrugated core or a Y-frame core were manufactured
by brazing together stainless steel sheets. Their quasi-static three-point bend-
ing responses were measured under simply supported and clamped boundary
conditions. The role of the back face was assessed by comparing the response
of beams with (i) front-and-back faces present and (ii) front face present, but
the back face absent.
• The measured responses were in good agreement with finite element simula-
tions.
• Two collapse mechanisms were identified: short panels collapse by indentation
whereas long panels fail by Brazier plastic buckling. Panels without a back
face have a superior indentation strength than those with front-and-back faces
present. In contrast, the Brazier plastic buckling strength of panels with front-
and-back faces present exceeds that of panels without a back face.
• For both collapse mechanisms, concentrating the mass of the sandwich panel
in the core increased the three-point bending strength of the structure.
7.3 Drop weight tests on prismatic sandwich beams
• Simply supported and clamped sandwich beams with a corrugated core or
a Y-frame core were subjected to an impact at 5 m/s using a drop weight
152
7.4 Dynamic indentation of prismatic sandwich panels
apparatus. The responses measured at 5 m/s were compared to their quasi-
static responses to assess the influence of the loading velocity.
• The peak load at 5 m/s could not be measured accurately due to an artifact
of the experimental setup. However, the post-peak response was captured
precisely and it was found to be slightly stronger at 5 m/s than for quasi-
static loading.
• The measured post-peak response was in reasonable agreement with three-
dimensional finite element predictions.
• Experiments and simulations have shown that a sandwich beam which col-
lapses quasi-statically by indentation also fails by indentation at 5 m/s. In con-
trast, predictions have shown that a sandwich beam which fails quasi-statically
by Brazier plastic buckling collapses by indentation at 5 m/s.
• For all sandwich beams considered, simulations indicated that the peak load
at 5 m/s exceeds its quasi-static value. The mass of the front face was found
to be an important factor contributing to this increase.
7.4 Dynamic indentation of prismatic sandwich
panels
• The finite element method was used to simulate the dynamic indentation re-
sponse of stainless steel sandwich panels with a corrugated core or a Y-frame
core. The indentation response was simulated for velocities ranging from quasi-
static loading to 100 m/s, and two different indenters were considered: a flat-
bottomed indenter and a cylindrical roller.
• The indentation force applied to the front face of the panel was approximately
equal to the force transmitted to the back face for velocities below 10 m/s.
Even for such low indentation velocities, the dynamic initial peak load was
found to be higher than its quasi-static value due to inertia stabilisation effects.
This strengthening effect was more important for the corrugated core than for
the Y-frame core.
• For indentation velocities greater than 10 m/s, the force applied on the front
face exceeded the force transmitted to the back face due to wave propagation
153
Chapter 7. Conclusions and future work
effects. The force transmitted to the back face was higher for the corrugated
core than for the Y-frame core.
• Increasing the width of the flat-bottomed indenter was found to enhance both
inertia stabilisation and wave propagation effects. In contrast, increasing the
roller diameter had only a mild effect of the dynamic indentation response.
• Material strain-rate effects were found to have a small influence on the dynamic
indentation response of both lab-scale and full-scale sandwich panels.
7.5 Future work
7.5.1 Dynamic compressive response of a hollow pyramidal
lattice
The pyramidal lattice made from hollow tubes possesses a high quasi-static compres-
sive strength at low densities, recall Fig. 2.7(a). However, the dynamic compressive
response of the lattice has not been investigated yet, neither experimentally nor nu-
merically. The dynamic compressive response of a pyramidal lattice made from solid
struts has been measured by Lee et al. (2006) and similar tests should be repeated
on the hollow pyramidal lattice. In addition, finite element simulations should be
performed to capture the influence of geometry and crushing velocity upon the com-
pressive response of the hollow pyramidal lattice. A similar numerical investigation
was performed by McShane (2007) for the corrugated core.
7.5.2 Measured compressive response of a carburised pyra-
midal lattice
The finite element simulations shown in Chapter 3 demonstrated that carburisa-
tion can increase significantly the peak compressive strength of a pyramidal lattice.
Those simulations should be compared to experimental tests to evaluate their accu-
racy. In addition, the residual stress and the embrittlement caused by carburisation
were neglected in the simulations of Chapter 3. Experimental data is necessary to
determine if this hypothesis is adequate.
154
7.5 Future work
7.5.3 Fracture of corrugated and Y-frame sandwich panels
The experiments performed in Chapters 4 and 5 allowed us to investigate the struc-
tural collapse of corrugated and Y-frame sandwich panels. However, none of the
specimens were tested up to material failure, i.e. fracture. The accreditation of
a new ship hull design is often based on the energy that the structure can absorb
before perforation. Thus, it is important for shipbuilders to (i) understand the key
factors governing the onset of fracture and (ii) predict accurately the fracture pro-
cess in large scale structures. Fracture of monolithic plates has been investigated by
several authors, see for example Stoughton (2000), Wisselink (2000) and Balden and
Nurick (2005), but future studies should extend this work to sandwich structures.
First, quasi-static three-point bending tests should be performed on corrugated and
Y-frame sandwich panels up to the onset of fracture. These test specimens should
be made from shipbuilding steel and assembled by a conventional welding route.
It is known that the toughness of a metallic plate scales with its thickness; hence,
the dimensions of the test panels should be similar to those of a full-scale ship
structure. Finally, it is also important that the boundary conditions applied to the
test specimens are representative of a full-scale sandwich hull structure.
Second, these experimental tests should be compared to finite element simulations.
The simulations will require the calibration of a fracture criterion such as Johnson-
Cook (Johnson and Cook, 1985) or Cockcroft-Latham (Cockcroft and Latham,
1968). Finite element predictions of fracture are usually sensitive to the mesh size
and to the type elements. Consequently, an additional challenge is to determine if
accurate fracture predictions can be obtained for large scale ship structures discre-
tised with a coarse mesh of shell elements.
Published work
Chapter 4 has been published in an international scientific journal and the refer-
ence is given below. Additional publications on Chapters 3, 5 and 6 are also in
preparation.
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and Y-frame cores: does the back face contribute to the bending response? Journal
of Applied Mechanics, 79(1), 011002 (13 pages), 2012.
155
Chapter 7. Conclusions and future work
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