Design of Sandwich Structures Achilles Petras Robinson College Cambridge Supervisor Dr MPF Sutclie A dissertation submitted to Cambridge University Engineering Department for the degree of Doctor of Philosophy December  To my parents and bebita Declaration This dissertation presents the results of research carried out in the Engineering Department of the University of Cambridge between October  and October  Except where specic reference is made to the work of others this dissertation is the original result of my own work and includes nothing which is the outcome of work done in collaboration No part of this dissertation has been submitted for a degree at any other University This dissertation is approximately  words long and contains  gures Achilles Petras March   Robinson College Cambridge ii Acknowledgements I would like to express my gratitude to my supervisor Dr Michael Sutcli e for all his kind help encouragement and dedicated guidance without which this project would not have been possible He gave me exibility to follow my research interests but at the same time he kept me going in the right direction His patience with proof reading my thesis is highly appreciated I am also grateful to Professor Norman Fleck for his invaluable advice and fruitful discussions I would like to thank him for supporting nancially my project US O ce of Naval Research grant   J  and my attendance at the th International Conference on Sandwich Construction in Stockholm I acknowledge with gratitude the generous nancial support of the Greek State Scholarship Foundation IKY I would like to thank Professors Christos Panagopoulos and Constantinos Lascarides at the National Technical University of Athens and Dr Dimitris Niarchos at the National Centre for Scientic Research Demokritos in Greece for supporting and urging me to pursue my PhD studies at Cambridge I would like to thank Nigel Hookham and Peter Clayton from Hexcel Composites in Duxford for providing materials and valuable technical information I wish to ac knowledge the Cambridge University Engineering Department CUED and especially the Cambridge Centre for Micromechanics CCM for use of their facilities I am also grateful to Alan Heaver and Simon Marshall for building the test rigs and helping me during my experiments Special thanks to the CUEDs librarians for their kindest assistance I wish to thank my friend Antonios Zervos for helping me out on compu tational modelling topics Many thanks to Roxy Pia Ingo Sinisa Guy Dongquan Kieran Jide at the CCM for those stimulating discussions and entertaining moments when there was a chance to see them due to my night shifts Sta students and fellows at Robinson College have helped make my time at Cam bridge really enjoyable Thanks to all my friends here in Cambridge for the enjoyable moments we have shared during these three years Also to my friends in Athens Kostas and Stefanos for not forgetting me and for making my holidays in Greece a real fun I am extremely grateful to my parents Violetta and Alexandros for their constant understanding caring love and support     o o   Finally but most importantly I would like to thank Georgina for her love patience and the happy life we share together Te amo iii Abstract Failure modes for sandwich beams of GFRP laminate skins and Nomex honeycomb core are investigated Theoretical models using honeycomb mechanics and classical beam theory are described A failure mode map for loading under  point bending is constructed showing the dependence of failure mode and load on the ratio of skin thickness to span length and honeycomb relative density Beam specimens are tested in  point bending The e ect of honeycomb direction is also examined The experi mental data agree satisfactorily with the theoretical predictions The results reveal the important role of core shear in a sandwich beams bending behaviour and the need for a better understanding of indentation failure mechanism High order sandwich beam theory HOSBT is implemented to extract useful infor mation about the way that sandwich beams respond to localised loads under  point bending High order or localised e ects relate to the non linear patterns of the in plane and vertical displacements elds of the core through its height resulting from the unequal deformations in the loaded and unloaded skins The localised e ects are examined experimentally by Surface Displacement Analysis of video images recorded during  point bending tests A new parameter based on the intrinsic material and geometric properties of a sandwich beam is introduced to characterise its susceptibility to localised e ects Skin exural rigidity is shown to play a key role in determining the way that the top skin allows the external load to pass over the core Furthermore the contact stress distribution in the interface between the central roller and the top skin and its importance to an indentation stress analysis are investigated To better model the failure in the core under the vicinity of localised loads an Arcan type test rig is used to test honeycomb cores under simultaneous compression and shear loading The experimental measurements show a linear relationship between the out of plane compression and shear in honeycomb cores This is used to derive a failure criterion for applied shear and compression which is combined with the high order sandwich beam theory to predict failure caused by localised loads in sandwich beams made of GFRP laminate skins and Nomex honeycomb under  point bending loading Short beam tests with three di erent indenters size are performed on appropriately prepared specimens Experiments validate the theoretical approach and reveal the iv Abstract nature of pre and post failure behaviour of these sandwich beams HOSBT is used as a compact computational tool to reconstruct failure mode maps for sandwich panels Superposition of weight and sti ness contours on these failure maps provide carpet plots for design optimisation procedures Keywords composite structures sandwich structures Nomex honeycomb sandwich beams failure mode maps honeycomb anisotropy indentation localised e ects high order sandwich beam theory mixed failure criterion for honeycombs biaxial testing of honeycombs v Contents Preface and Declaration                              ii Acknowledgements                                 iii Abstract                                       iv Contents                                       vi List of Figures                                    viii List of Tables                                    xi Nomenclature                                    xii  Introduction and Overview   Introduction to Composite Structures                      Fibre Reinforced Polymer Composites                  Structural Optimisation                         Sandwich Structures                               Previous Work on Indentation Analyses                 Previous Work on Indentation Failure Prediction           Scope and Outline of the Thesis                         Failure Mode Maps for Honeycomb Sandwich Beams   Introduction                                    Beam Theory for Sandwich Panels                        Skin Failure                                Core Failure                                Honeycomb Mechanics                          Construction of a Failure Mode Map                      A Failure Mode Map for Beams with a GFRP Skin and Nomex Core                                    Experiments                                    Experimental Results                          vi Contents  Discussion                                     Skin Failure                                Core Failure                                E ect of Ribbon Direction                        Intra cell Buckling                            Conclusions                                    Indentation Resistance of Sandwich Beams   Introduction                                    High Order Sandwich Beam Theory                       Surface Displacement Analysis                          E ect of Spreading Stresses                           Contact Pressure Distribution                          Case Study                                Concluding Remarks                               Indentation Failure Analysis   Introduction                                    Failure Envelope for Nomex Honeycombs                    Failure Analysis with HOSBT                          Experimental Work                                Concluding Remarks                              Failure Maps using HOSBT   Introduction                                    Reconstruction of Failure Maps                         Optimisation Carpet Plots                            Concluding Remarks                               Conclusions and Future Work   Conclusions                                    Future Work                                   Appendix A Experimental Results Bibliography vii List of Figures  Sandwich construction with honeycomb core                  Sandwich panels with a corrugated b foam and c honeycomb core   Typical hexagonal honeycomb with a set of doubled walls t is the single wall thickness and is the honeycomb cell size                 Test rig for indentation fatigue tests on aircraft oor panels          a Simply supported beam b Cross section on A A             Failure modes in the skin                             Failure modes in the core                             Comparison of theoretical and experimentally measured out of plane a compressive modulus b compressive strength c shear modulus and d shear strength of Nomex honeycombs                    a Failure mode map Solid lines refer to beams in which the honey comb ribbon lies along the beam axis dashed lines for when the ribbon lies transverse to the beam axis The symbol identies experimen tal measurements described in section  b Failure load surface for ribbon lying along the beam axis                        Failure mode map for three typical values of tL and Rt          Layup details                                   Test setup                                     Photographs of the di erent failure modes for transverse ribbon direc tion and corresponding load de ection curves for both ribbon directions   Variation of failure load W o with skin thickness to span ratio tL      Photographs and results of specimens with  mm cell size          The behaviour of exible and rigid skins                    a Nonlinear displacement patterns b Beam geometry and stresses The origin of the z coordinate is always taken at the top of the beam element either skin or core which is being considered            viii List of Figures  Surface Displacement Analysis software window showing the reference image frame of a side cross section painted to have a random speckle pattern and the area of interest where the analysis takes place       Experimental setup                                Load de ection and core compression curves A and B correspond to the frames used for processing by SDA The dashed line indicates the total midspan core compression at a line load of  kNm           Vertical displacement eld in the core produced by a SDA and b HOSBT model All contour values are in mm and the scaling in both plots is the same                                 Parametric Study change of transmission coe cient C zz m with wave length Lm for variations in the parameters t E f c and  c  The boxed labels show each time the parameters and the arrows indicate the direc tion of each parameters increase The position of in ection is marked by a  symbol                                   Discrete contact pressure elements a uniform piecewise constant b overlapping triangles piecewise linear and c Geometric denitions    Distributions of contact stresses q t  cc and the corresponding normal stresses in the top skin core interface  zz  cc normalised by the cores out of plane compressive strength for beam A with   mm       Distributions of contact stresses q t  cc and the corresponding normal stresses in the top skin core interface  zz  cc normalised by the cores out of plane compressive strength for beam B with   mm       Dependence of spreading e ect on rollers radius R and determination on how exible or rigid are the skins with respect to the indentation resistance of the sandwich beam                         Combined indentation failure mechanism                    The Arcan type rig                                Load paths and determination of failure envelope               The three angle setups and the corresponding load de ection curves here for specimens with  kgm core density               Failure envelopes for Nomex honeycombs The dashed line corresponds to the linear failure criterion given by equation               Combined failure criterion                            Experimental setup                                Typical experimental results for sandwich beam with  kgm core den sity loaded by a roller with diameter of  mm Midspan bottom skin de ection midspan core compression and line load are plotted against midspan top skin de ection Lines legend  longitudinal and   transverse honeycomb ribbon direction                    ix List of Figures  The pre failure in uence of core density Video images captured just be fore failure during  point bending loading with a mm diameter central roller                                        The post failure in uence of core density Video images captured after failure  mm total de ection during  point bending loading with a mm diameter central roller                           The post failure in uence of core density on extent of damage Video im ages captured after failure  mm total de ection during  point bend ing loading with a mm diameter central roller                Theory lines vs experimental results symbols for failure line load W o   Improved failure map for Nomex honeycomb sandwich beams Each contour represents sandwich beams of equal strength in Nm         Comparison with the experimental results from Chapter  Lines show the predictions of HOSBT failure analysis and the symbols represent the experimental data cf Fig  Solid lines and symbols correspond to the longitudinal ribbon direction while dashed lines and hollow symbols correspond to the transverse ribbon direction                  Strength contour plot corresponding to Fig                 Sti ness contour plot                               Optimisation carpet plot for tL    with  c  s and cL as design parameters                                     Vertical displacements calculated by Abaqus nite element analysis    A For  mm diameter roller midspan bottom skin de ection midspan core compression and line load variation curves with respect to midspan top skin de ection Lines legend  longitudinal and   transverse honeycomb ribbon direction                          A For  mm diameter roller midspan bottom skin de ection midspan core compression and line load variation curves with respect to midspan top skin de ection Lines legend  longitudinal and   transverse honeycomb ribbon direction                          A For  mm diameter roller midspan bottom skin de ection midspan core compression and line load variation curves with respect to midspan top skin de ection Lines legend  longitudinal and   transverse honeycomb ribbon direction                          A Video snapshots during loading with a  mm diameter roller        A Video snapshots during loading with a  mm diameter roller        A Video snapshots during loading with a  mm diameter roller        x List of Tables  An example of structural e ciency of sandwich panels in terms of weight   Summary of failure criteria                            Material properties of Nomex and laminate skin                Experimental results Photographs of failure given in Fig  and  correspond to the entries in italics in the nal column of this table      The range of the geometric and material parameters used in the para metric study                                    Change of spreading length                          Expressions for peak failure loads                       xi Nomenclature Greek symbols Honeycomb cell size  Length of contact between the central roller and the top skin or length of top distributed load   Quantities dened in section  cxz Out of plane honeycomb Poissons ratio fxy Poissonss ratio of the skin material  c Honeycomb core density  s Density of honeycombs constituent material  cc Out of plane compressive strength of the honeycomb core  fi In plane compressive stress for intra cell buckling of the skin  fw In plane wrinkling strength of the skin  fx Maximum in plane stresses in the skins Chapter   fY In plane yield strength of the skins  txx   bxx In plane normal stresses in the skins  zz Out of plane normal stresses in the core      Out of plane shear strength of the honeycomb for transverse and longitudinal ribbon direction respectively  cs Out of plane shear strength of the honeycomb core  cxz Maximum shear stresses in the core Chapter   x Out of plane shear stresses in the core  Setup angle in Arcan test rig Latin symbols A t  A b In plane rigidity of top and bottom skin b Beam width C m Fourier coe cient c Core thickness D Flexural sti ness of the beam D t  D b Flexural rigidity of top and bottom skin Nomenclature d Distance between the midplanes of top and bottom skin E or E c Out of plane Youngs modulus of the honeycomb core E cx In plane Youngs modulus of the honeycomb core in the x direction E f E fx  Youngs modulus of the face material in the x direction E s Youngs modulus of the honeycombs solid material G   G  Out of plane shear moduli of the honeycomb for transverse and longitudinal ribbon direction respectively G c G cxz  Out of plane shear modulus of the core in the xz direction G s Shear modulus of the honeycombs solid material I Second moment of area of the sandwich beam I f Second moment of area of the skins with respect to their own cen troidal axes L Span of the sandwich beam long trans Longitudinal transverse honeycomb ribbon direction M Maximum beam bending moment P i Heights of triangular pressure elements Chapter  q i Discrete pressure elements Chapter  q t External distributed load applied on top skin or contact pressure distribution between the central indenter and top skin R Rollers radius t t t  t b  Skin thickness top bottom t Single wall thickness of honeycomb u t  u b In plane centroidal displacements of top or bottom skin v Base width of triangular pressure elements Chapter  W W o  Load per unit width at failure w t  w b Vertical displacements of top or bottom skin w c Vertical displacements in the core x y z Coordinates of sandwich beam Note Other symbols not mentioned are dened in the text or gure when appearing xiii Chapter Introduction and Overview One day towards the end of  a circus proprietor called George May called to see me at Farnborough After he had told me several Gerald Durrelltype stories about the diculties of keeping monkeys in travelling circuses he produced something which looked like a cross between a book and a concertina When he pulled on the ends of this invention the whole thing opened out like one of those colouredpaper festoons which people use for Christmas decorations It was in fact a sort of paper honeycomb of very light weight but of quite surprising strength and stiness Did I think that such a thing could be of any use in aircraft The snag as George May modestly admitted was that since it was only made from brown paper and ordinary gum it had no moisture resistance at all and would fall to bits if it got wet This must have been one of the relatively few occasions in history when a group of aircraft engineers have been seriously tempted to throw their collective arms around the neck of a circus proprietor and kiss him However we resisted the temptation and told May that there could be no serious diculty in waterproo ng the paper honeycomb by means of a synthetic resin JE Gordon Structures or why things dont fall down   Introduction to Composite Structures Innovative high performance design of load bearing components is always sought in high tech applications such as aircrafts spacecrafts satellites or F racing cars These structures should be as as light as possible while having high sti ness su cient strength and some damage tolerance This requires structurally ecient construc tion Structural e ciency can be maximised by using the most e cient materials and optimising the structures geometry To produce an optimum design both these factors need to be considered throughout the design process The catalogue of materials is one of forbidding length Thus a designer needs a systematic way to be guided through the maze of material classes so as to gradu ally narrow down the material choices and choose the optimum material Ashby  proposed a material selection procedure using material selection charts Birming ham et al  have introduced an integrated approach to the assessment of alternative  Chapter  Introduction and Overview  materials and structural forms at the concept stage of structural design based on the above methodology  Fibre Reinforced Polymer Composites Fibre Reinforced Polymer FRP composite materials are some of the most useful materials available to the designer of high performance structures in the aerospace or maritime industries because their high specic strength and sti ness can lead to signicant weight reductions The aerospace industry has proved the e ectiveness of composite materials in reducing component weight and increasing fuel economy In the marine eld the use of composites has been growing steadily since the early s particularly driven by the low construction costs of Glass FRPs Initial applications include small crafts lifeboats and pleasure crafts Potential applications are radar domes masts and piping ship hulls ship superstructures submersibles and o shore structure modules Yachts participating in the Americas Cup race have all composite hull keel and mast Racing car bodies are made of composites providing more safety per unit weight to drivers In the last decade composites have been used in applications outside the aerospace industry Sporting goods such as tennis and squash rackets golf shafts bicycles and oars are a major outlet for composites materials Future applica tions include high speed trains subway cars and surface ships  As well as weight reductions other benets of composites include their ability to cope with extreme en vironment reliability maintainability life cycle cost and service life extension Despite all their advantages composites are still regarded as expensive materials to purchase tool up for and work with When using composites a designer has the freedom to tailor the properties Al though this can improve the structural e ciency this also increases the complexity and number of design parameters Many design tools have been developed to predict their behaviour and so decrease the level of empiricism which has been used up to now in design procedures Finding an e cient composite structural design that meets the requirements of a given application can be achieved not only by sizing the cross sectional areas and members thicknesses but also by global or local tailoring of the material properties through selective use of orientation number and stacking sequence of laminae that make up the laminate composite  A comprehensive presentation of the equations which govern the mechanical behaviour of laminates can be found in Hulls book  Stress analysis for the design of composites laminates is commonly implemented by the use of computer programs ie Cambridge Composite Designer  based on laminated plate theory LPT Chapter  Introduction and Overview  This kind of software is convenient e cient and user friendly but does not help de termine the optimum laminate conguration or choose the best material system For this purpose there are procedures which are based on the use of carpet plots which are themselves generated using LPT principles but also provide a graphic illustration of the whole range of elastic properties available with chosen system  Miki  de scribes a highly practical tool for design optimisation of laminates based on a graphical procedure Tsai and Patterson  have introduced the laminate ranking method for selecting the optimum ply angles Recently Quinn  has published a design manual for composites which provides practical information for engineers to facilitate the de sign of GRP CFRP A ramideRP composites Quinn has also introduced a relevant nomogram   which allows the costs of the constituent materials bres matrix in a composite to be quickly assessed  Structural Optimisation The design of full scale composite materials structures usually requires a building block approach to testing and design  This approach involves increasing the testing com plexity and size from coupon tests to full structural tests The chemical and material tests in the rst stages are generally well dened tests As we move from the laminate level to sub element component and sub structure level the test become more appli cation dependent Neither standard test methods nor databases exist for these larger tests Thus there is a need to reduce cost and increase the e ciency of structural design and structural failure prediction by a globallocal testing and analysis approach The globallocal approach involves supporting the global tests and analyses used in tradi tional design approaches by critical local sub element tests These tests are intended to be in between a coupon and sub structure test They are cheaper to manufacture and test and most importantly they are fully representative of structural congurations undergoing the same manufacturing processes and may even be cut from an actual structure A major constraint to overcome in the design and construction of large bre rein forced plastic FRP structures eg in ship hulls or aircraft wings is the exibility of the panels Although this can be overcome by large increases in thickness this causes molding di culties and is also very uneconomical A more satisfactory alternative is the use of sti eners which are commonly in the form of top hats I beam tee joint or by use of sandwich constructions  The sti eners may be an integral part of the shell or they may be attached by adhesive bonding the latter is the usual case Bonded or laminated connections between two structural members represent a zone of Chapter  Introduction and Overview  potential weakness Recent studies have resulted in computer codes for design of specic sti ened panel congurations subject to simple loadings usually compression In the s a com puter code PASCO Panel Analysis and Sizing Code was developed by NASA  which has been widely used for composite optimization procedures  The code has been designed to have su cient generality in terms of panel conguration loading and practical constraints so that it can be used for nal sizing of panels in a realistic design situation The UWCODA University of Washington Composite Optimization and Design Algorithm design analysis optimization software tool was originally devel oped to optimise the lay up of at composite panels In the UWCODA analysis both ply orientation angles and sti ener geometries are treated as design variables Opti mum designs are sought  which minimise structural weight and satisfy mechanical performance requirements for example maximum strain and minimum strength Especially within the aircraft industry nite element based optimisation methods are used to size complex structures for minimum weight taking into account a va riety of constraints including strength sti ness and aeroelasticity However there is always a need for simple low cost direct methods which can assist in congurations and weight studies at the initial design stage Bartholomew    apply a simple optimization method known as geometric programming GP  In this method the minimisation of a function representing weight or cost subject to non linear constraints on the variables can sometimes be reduced to the solution of linear equations  Sandwich Structures Amongst all possible design concepts in composite structures the idea of sandwich construction has become increasingly popular because of the development of man made cellular materials as core materials Sandwich structures consist of  a pair of thin sti strong skins faces facings or covers  a thick lightweight core to separate the skins and carry loads from one skin to the other and  an adhesive attachment which is capable of transmitting shear and axial loads to and from the core Fig  The separation of the skins by the core increases the moment of inertia of the panel with little increase in weight producing an e cient structure for resisting bending and buckling loads Table  shows illustratively the exural sti ness and strength advantage of sandwich panels compared to solid panels using typical beam theory with typical values for skin and core density By splitting a solid laminate down the middle and separating the two halves with a core material the result is a sandwich panel The Chapter  Introduction and Overview Figure  Sandwich construction with honeycomb core new panel weighs little more than the laminate but its exural sti ness and strength is much greater by doubling the thickness of the core material the di erence is even more striking Thus sandwich panels are popular in high performance applications where weight must be kept to a minimum for example aeronautical structures high speed marine craft and racing cars In the most weight critical applications composite materials are used for the skins cheaper alternatives such as aluminium alloy steel or plywood are also commonly used Materials used for cores include polymers aluminium wood and composites To minimise weight these are used in the form of foams honeycombs or with a corrugated construction Fig  As well as mechanical requirements core t 2 t 4 t Relative Bending Stiffness Relative Bending Strength Relative Weight 1 1 1 7.0 3.5 1.03 37 9.2 1.06 Table  An example of structural eciency of sandwich panels in terms of weight Chapter  Introduction and Overview  materials may also be selected based on their re resistance or thermal properties (a) (b) (c) Figure  Sandwich panels with a corrugated b foam and c honeycomb core The most common and some unorthodox techniques employed to manufacture sand wich components for structural applications as well as the recent developments and future trends in terms of both materials and processing routes are comprehensively reviewed by Karlsson  Sandwich panels will have sti ness and strength criteria to meet The sti ness of honeycomb sandwich panels is straightforward to predict but it remains di cult to estimate the strength Typical modes of failure are face yielding face wrinkling intra cell dimpling core shear or local indentation where the load is applied to the panel These are described in detail in section  The critical failure mode and the corresponding failure load depend on the properties of the face and core materials on the geometry of the structure and on the loading arrangement Proper analysis of sandwich structures demands a thorough understanding of the mechanical behaviour of both the skins and the core The skins behave in a relatively simple manner and in case of composite laminates the aforementioned methods of Chapter  Introduction and Overview analysis ie laminated plate theory facilitate the modelling procedure However the mechanical modelling of the core material particularly for foams or honeycombs is less straightforward The response of the core to shear loading from the skins or loading normal to the plane of skins is required The behaviour depends both on the materials used in the core and on the core relative density which is the ratio of the core density to that of the solid material constituting the core Gibson and Ashby  give a thorough overview of the literature on cellular materials quoting many results for foam cores Figure  illustrates the honeycomb structure One of the most common core ma terials is Nomex honeycomb  because it possesses an extremely high strength to weight ratio It is also electrically and thermally insulating chemically stable self extinguishing and corrosion as well as shock and fatigue resistant As Nomex is used extensively in this thesis the make up of this core is described in detail here Nomex is constructed from ribbons of aramid paper running in the  direction the longitudinal ribbon direction These are glued together at intervals along the ribbon and the stack of ribbons is then expanded into a honeycomb by pulling in the  direction transverse The paper substrate is nally dipped into phenolic resin to build up the walls of the honeycomb Because of this construction method the honeycomb is anisotropic with respect to out of plane shear sti ness and strength Figure  Typical hexagonal honeycomb with a set of doubled walls t is the single wall thickness and is the honeycomb cell size Nomex is a Du Pont trademark Chapter  Introduction and Overview Zhang  and Ashby  model the elastic and collapse behaviour for Nomex hon eycomb materials under shear and out of plane compression Their models agree well with experiments that they made on a wide range of Nomex honeycombs Zhang and Ashby  have also investigated the in plane biaxial buckling behaviour of Nomex honeycombs Shi et al  and Grediac  model the transverse shear modulus of a honeycomb core Considerable e ort has been devoted to the analysis of beams panels and struts of sandwich construction and the results have been summarised in the books of Allen  and Plantema  Modelling a sandwich panel as a beam with the simplifying as sumptions that the skins are thin relative to the core and that the core material is ho mogeneous and much less sti than the skin material was presented by Allen  and developed by Gibson and Ashby  Triantallou and Gibson  have developed an optimisation procedure which can determine the optimum values of skins and cores thicknesses that satisfy the sti ness constraint at minimum weight Although most of research work in literature is concerned with bending loading of sandwich beams Kwon et al  or Pearce  have investigated the overall buckling and wrinkling of sandwich panels under in plane compression A recent comprehensive review to the subject of sandwich construction and the development of theoretical analyses up to now is given in the book of Zenkert  Holt and Webber  summarise recent developments and analyse the elastic behaviour of honeycomb sandwich beams assuming linear elastic behaviour for the skin and the core Mechanical thermal and hygrometric loading on a sandwich beam with a honeycomb core and laminated facings are included in reference  Failure mode maps have been derived by various authors for sandwich panels with exible cores    These authors have been particularly concerned with beams with ductile foam cores making appropriate assumptions about the elastic and plastic behaviour of the core and skin However there appears to be little work on failure of panels with honeycomb cores whose shear anisotropy can reveal the important role of core shear in the bending of sandwich beams Furthermore resistance to indentation failure is rarely considered as an important factor although it can be important in determining the durability and service life of the structure  Previous Work on Indentation Analyses The use of exible foams and non metallic honeycombs as core materials has intro duced a new design di culty they are transversely exible compared to the e ectively incompressible metallic honeycombs or rigid foam cores This leads to signicant lo Chapter  Introduction and Overview cal de ections of the loaded skin into the core material hence to indentation failure Indentation failure of sandwich structures has been primarily investigated for aircraft oor panels These panels need to be su ciently sti to avoid passengers perceiving that the oor is unsafe because of its de ection High heel shoes or dropped objects are typical causes for local indentations on the top skin of these panels in areas like the aisles thresholds toiletsgalley However a sandwich panel would continue to give good service until a number of such indentations had been made Therefore the time from incipient to nal failure should be su cient to allow maintenance operators to change a defective panel at the next convenient check A good source of information on indentation analyses of sandwich beams are the books by Allen  and Plantema  They cover the development of the theoretical analyses based on a so called splitted rigidity model This approach assumes that the beam consists of a component with only bending rigidity and a component with only shear rigidity They are connected through equilibrium assuming that the shear resultants in the two components are the same These models o ers an adequate level of accuracy for sandwich structures with incompressible cores but are inadequate for non metallic honeycomb sandwich panels since they neglect localised e ects which play an important role for these exible cores To model indentation failure of transversely exible cores in sandwich structures requires the use of models that take into account the compressibility of the core in the vicinity of the applied loads A common approach is the elastic foundation model Selvadurai  presents the principles of these models Their application in sandwich structures analysis is described in Zenkerts book  The simplest of these is the one parameter Winkler foundation model which treats the core material as a set of continuously distributed linear springs However its main drawback is that it neglects shear interactions between the loaded skin and the core The elastic foundation ap proach adopted by Thomsen   was based on the use of a two parameter elastic foundation model including the shear interaction between the skins and the core Typ ical two parameter models are those of Pasternak or Vlazov see Selvadurai  Nevertheless these elastic foundation models neglect the interaction between the top and bottom skin A di erent approach by Frostig and Baruch  consists of treating the beam as an ordinary incompressible sandwich substructure interconnected with a special elastic foundation substructure which provides the localised e ects due to the di erent displacements of the upper and lower skins The non planar deformed cross section of the sandwich beam which is observed by experiments suggested the need for a model which allows non linear variations of in plane and vertical displacement Chapter  Introduction and Overview  eld through the core Frostig et al   used variational principles to develop the high order sandwich panel theory which includes the transverse exibility of the core High order refers to the non linear way in which the in plane and vertical displace ments are allowed to vary through the height of the core in contrast to simple beam theory where the core in plane displacements are assumed to vary in a linear way through the depth and the out of plane displacements are assumed to be constant The high order theory enables the prediction of localised e ects under conditions such as concentrated loads delamination and diaphragms  curved sandwich panels  hygothermal environmental e ects and discontinuous skins  A comprehensive review and comparison between this method and conventional beam theories was pre sented recently by Frostig  In the literature the importance of the core behaviour in a ecting indentation failure has been considered however the in uence of the skins exural rigidity is generally overlooked This omission derives from the fact that in practical applications the skins are quite sti and the in uence of skin exural rigidity can be neglected One of the main aims of this thesis is to focus on the role of skin rigidity in indentation failure and give a better insight in the mechanism of indentation  Previous Work on Indentation Failure Prediction The aircraft industry in collaboration with sandwich panel manufacturers has con ducted research to establish design codes for indentation resistant and cost e ective sandwich panels  However most of the design specications have been based on experimental results rather than accurate theoretical models In manufacturers data sheets  and handbooks  the prediction of indentation failure load is derived as a product of the out of plane compressive strength of the core and the area over which the load is applied However such an approach is at best approximate Special ex perimental setups have been introduced depending on the application static blunt indentation tests for freight oors or dynamic fatigue tests for cabin oors Bonded Structures Division CIBA now Hexcel Composites have devised the fatigue wheel machine Fig  to provide data on the relative performance of various sandwich panels Similar indentation damage accumulation can be observed in sandwich panels used as a part of control surfaces aps ailerons in this case because of hail or stone impacts A strength analysis of the damaged sandwich panels is essential to predict the in uence of the multi site damage Razi et al  have introduced an analytical method to determine the stress distribution in sandwich panels with damage that is Chapter  Introduction and Overview  Figure  Test rig for indentation fatigue tests on aircraft oor panels elliptical or circular in shape and at an arbitrary location The low velocity or quasi static impact behaviour of sandwich panels is also another aspect of the whole indentation problem that has been investigated by several authors The lack of accepted test methods for measuring impact damage resistance of com posite sandwich structures led Lagace et al  to propose a new methodology based on static indentation and impact tests Mines et al   have tested the dropped weight impact performance of sandwich beams with wovenchopped strand glass skins and polyester foam and aluminium honeycomb cores and have simulated the upper skin post failure energy absorption behaviour with an elastic plastic beam bending model Low velocity damage mechanisms and damage modes have been investigated in sand wich beams with Rohacell foam core by Wu and Sun  and with Nomex honeycomb core by Herup and Palazotto  The pure static indentation response of composite sandwich beams has been mod elled as linear elastic bending of the top skin on a rigid perfectly plastic foundation the core by Soden  and Shuaeib  Olsson and McManus  introduced a theory for contact indentation of sandwich panels the model is based on the assumption of axisymmetric indentation of an innite elastic face sheet bonded to an elastic ideally plastic core on a rigid foundation Despite the usefulness of such models for analysing sandwich beam indentation behaviour they are restricted only to the loading case where a sandwich beam is indented on a rigid foundation However this loading case is not normal in practice Most sandwich beams in service are simply supported or Chapter  Introduction and Overview  clamped and are indented whilst also su ering bending loads These loading conditions make the in uence of the bottom skin an important factor in the overall behaviour of a sandwich beam under localised loads  Scope and Outline of the Thesis The use of laminate composites as skins and low density cellular materials as cores in sandwich constructions has enabled a very good utilisation of the constituent materi als providing structural components with high sti ness and strength to weight ratios Non metallic honeycombs are popular in high performance applications because of their good mechanical properties excellent resistance to hostile environments and better fa tigue resistance than aluminium However because of the low sti ness of the core they are susceptible to indentation failure The need to take into consideration the indentation resistance of sandwich panels when optimising their design and the impor tance of having a robust computational tool to predict indentation failure in sandwich beams under bending have been the main motivations for this thesis To address design including indentation failure we rst need to investigate the other failure mechanisms described in Chapter  Failure maps for loading under  point bending are constructed and the region where indentation failure is important is identied This work uses sandwich panels made of GFRP laminates and Nomex honeycomb cores of various densities Experimental measurements are compared with simple theoretical predictions In Chapter  analytical modelling which can be used for indentation failure is addressed This uses an existing beam theory developed by Frostig et al   and applies it to the GFRP Nomex construction considered in the thesis The e ect of material properties and the beam and indenter geometry on the beam response are considered Chapter  presents the methodology to predict more accurately indentation failure loads This is achieved by introducing a new failure criterion for honeycomb core which takes into account both the out of plane normal compression stresses and the shear stresses in the core Theoretical predictions are compared with experimental data Rened failure maps including the new indentation analysis are described in Chapter  Carpet plots based on these failure mode maps provide a preliminary optimisation methodology for mimimum weight design of sand wich beams The thesis concludes in Chapter  by summarising the main results of this research together with a discussion of new research directions Chapter  Failure Mode Maps for Honeycomb Sandwich Beams  Introduction In this chapter we consider loading under three point bending of sandwich beams made with honeycomb and skins both of which fail in a brittle manner We apply the anal ysis framework and failure mode map technique used by Triantallou and Gibson  for ductile foam cores and ductile skin materials However here we investigate the failure modes of panels with honeycomb cores whose shear anisotropy can reveal the important role of core shear in the bending of sandwich beams Furthermore the con structed failure mode maps include indentation failure and identify for what beams this mechanism will be important In section  we review beam theory for sandwich panels Section  describes existing work on honeycomb mechanics By combining the analysis for sandwich beams with the honeycomb mechanics we derive in section  failure loads for the various failure mechanisms This information is used in section  to draw up failure mode maps in which the mechanisms of failure and the corresponding failure loads are plot ted as a function of the core relative density and skin thicknessspan ratio Theoretical results are illustrated using the commercially popular combination of laminated glass bre reinforced plastic GFRP skins and Nomex core Theory is compared in sec tion  with experiments on sandwich panels of various core relative densities and skin thicknessspan ratios  Chapter  Failure Mode Maps for Honeycomb Sandwich Beams   Beam Theory for Sandwich Panels In this section we outline the elastic analysis of sandwich beams in three point bending This will be used to evaluate the stresses in the core or skin and hence the failure loads due to the various mechanisms Consider a simply supported sandwich beam of span L and width b loaded in  point bending with a central load W per unit width as illustrated in Fig  The skins each have thickness t and are separated by a thick layer of honeycomb core of thickness c Figure  a Simply supported beam b Cross section on AA We assume that the skins remain rmly bonded to the core that the beam bends in a cylindrical manner with no curvature in the yz plane and that cross sections remains plane and perpendicular to the longitudinal axis of the beam The exural rigidity D of the sandwich beam is then given by D  E fx bt  E fx btd   E cx bc   where d is the distance between the midplanes of the upper and bottom skins E fx and E cx are the in plane Youngs moduli of the skin and core respectively for loading in the x direction along the axis of the beam Subscripts f  and c denote the face material and the honeycomb core respectively Subscript s is used in later expressions for the solid material from which the honeycomb is made The three terms on the right hand side of  correspond to bending of the skins about their centroidal axes bending of the skins about the centroid of the whole beam and bending of the core respectively We can simplify this equation by assuming that bending of the skins about the centroid of the beam is the dominant term The contributions of the rst Chapter  Failure Mode Maps for Honeycomb Sandwich Beams  and third terms amount to less than ! of this when d t   and E fx E cx t c d c      respectively so that  becomes D  E fx btd    E fx I  where I is second moment of area of the cross section of the sandwich beam With  point bending the maximum bending moment M is at the mid span and the corre sponding maximum stress  fx in the skins is given by  fx  ME fx D d   WL dt  However the above theoretical model neglects the e ect of shear de ection in the core which becomes signicant for low density cores Inclusion of this e ect also allows a prediction of observed di erences in beam strength for di erent orientations of the honeycomb ribbon see section  for further details For the above reasons we follow the suggestion of Allen  for the maximum axial stresses in the faces  fx  WbL  c t I WL  t I f     where   L c  G cxz E fx c t  d  t     I  bt  btd    I f  bt   G cxz is the out of plane shear modulus of the core I is the second moment of area of the sandwich with respect to its neutral axis and I f is the second moment of area of the faceplates with respect to their own centroidal axes   Equation  shows that  depends on the relative sti ness of the skin and the core Finally  gives W   fx  t L  where    t   t d   ht    t   t  d   As the span length L or the core shear sti ness G cxz approach innity  tends to the simple beam model  In the case study presented in section  maximum deviations from the simple beam model due to the nite thickness of the skins and the e ect of nite shear sti ness in the core amount to a maximum of ! Note that I f is not negligible in this approach Chapter  Failure Mode Maps for Honeycomb Sandwich Beams   Skin Failure Section  gives an expression for the maximum stress  fx in the skins This can be used to predict beam failure due to the skin failure modes of face yielding intra cell dimpling or face wrinkling as illustrated in Fig  a Face yielding b Intracell dimpling c Face wrinkling Figure  Failure modes in the skin Face Yielding Failure occurs in the top skin due to face yielding when the axial stress in either of the skins equation  reaches the in plane strength  fY of the face material for loading along the beam axis  fx   fY  It is assumed that the skin behaves in a brittle manner With a symmetrical beam the stress is the same in the tension and compression faces For composite face materials the compressive face is generally the critical one Intracell Dimpling A sandwich with a honeycomb core may fail by buckling of the face where it is unsup ported by the walls of the honeycomb Fig b Simple elastic plate buckling theory can be used to derive an expression for the in plane stress  fi in the skins at which intra cell buckling occurs as  fi  E fx   fxy t    where is the cell size ie the diameter of the inscribed circle of the honeycomb and E fx and fxy are the elastic modulus and Poissons ratio for the skin for loading in the axial direction A similar expression veried experimentally by Kuenzi  has been given by Norris  Equations  and  can be used to derive the value of cell Chapter  Failure Mode Maps for Honeycomb Sandwich Beams  size above which there is transition from face yielding to intra cell buckling as  t s    fxy E fx  fY  Face Wrinkling Face wrinkling is a buckling mode of the skin with a wavelength greater than the cell width of the honeycomb Fig c Buckling may occur either in towards the core or outwards depending on the sti ness of the core in compression and the adhesive strength In practice with  point bending inward wrinkling of the top skin occurs in the vicinity of the central load By modelling the skin as a plate on an elastic foundation Allen  gives the critical compressive stress  fw that results in wrinkling of the top skin as  fw     cxz    cxz     E  fx E   where cxz is the out of plane Poissons ratio and E the out of plane Youngs modulus of the honeycomb core see section   Core Failure Honeycomb sandwich structures loaded in bending can fail due to core failure Perti nent failure modes are shear failure or indentation by local crushing in the vicinity of the loads as illustrated in Fig  a Core shear b Local indentation Figure  Failure modes in the core Core Shear Assuming simple beam behaviour the shear stress varies through the face and core in a parabolic way under  point bending If the faces are much sti er and thinner than the core the shear stress can be taken as linear through the face and constant in the Chapter  Failure Mode Maps for Honeycomb Sandwich Beams  core Neglecting the contribution from the skins the mean shear stress in the core is given by  cxz  W d  Assuming brittle behaviour failure occurs when the applied shear stress  xz equals the shear strength  cs of the honeycomb core in this direction  cxz   cs  Low density Nomex cores are particular susceptible to this failure mode Due to the anisotropy of the honeycomb structure section  the shear strength of the core depends on the loading direction Local Indentation Failure of sandwich panels in  point bending can occur at the load point due to local indentation Failure is due to core crushing under the indenter The bending sti ness of the skin and the core sti ness determine the degree to which the load is spread out at the point of application It is important here to mention the main failure characteristic by which indentation di ers from skin wrinkling In indentation the top skin de ects after failure with a wavelength of the same scale as the indenter top skin contact length whereas in skin wrinkling the de ection of the top skin after failure exhibits wavelengths that are larger than the contact length between the indenter and the top skin Indentation failure has not been adequately modelled for honeycomb sandwich pan els To include this important failure mechanism we use a simple empirical approach used in handbooks on sandwich panel construction  In the next chapters a more accurate model will be presented for indentation failure prediction Here we assume that we know the length  of contact  between the central roller and the top skin It is further assumed that the load is transferred uniformly to the core over this contact length so that the out of plane compressive stress  zz in the core is given by  zz W  Failure is then predicted when this compressive stress equals the out of plane compres sive strength  cc of the honeycomb core  zz   cc   This length derives from the assumption that the skins are transparent enough to equate the contact length with the length of initial damage in the top skincore interface Chapter  Failure Mode Maps for Honeycomb Sandwich Beams  The above approach is decient in three respects i the contact area must be estimated in some way in the experiments described in section  this is measured ii load transfer from the roller to the core is over simplied this will depend on the relative skin and core sti nesses iii failure in the core will not be governed solely by the compressive stress in the core but will also be in uenced by the local shear stress A more rigorous stress analysis of the contact region can be found in Chapter  and its implementation to predict local failure in honeycomb panels is presented in Chapter   Honeycomb Mechanics To evaluate the failure mechanisms described in section  sti ness and strength prop erties for the honeycomb core are required In this section we use the results of refer ences   to express the properties of the honeycomb as a function of the properties of the solid material from which the honeycomb is made and the relative density  c  s of the honeycomb Although the theory is applicable to any honeycomb in practice we will focus on the Nomex honeycomb core used in the beam failure experiments of section  Expressions are compared with published experimental data to evaluate the applicability of the theoretical models to the Nomex honeycomb core and to nd the most suitable expressions for use in the beam calculations The following notations    for honeycombs main axes refer to those illustrated in Fig  The honeycomb Poissons ratio cxz required for the failure analysis section  is  or  for in plane Poisson strains due to out of plane loading in the  direction To a rst approximation its value can be taken as that of the solid material eq  in ref ie     s  The Youngs modulus of the honeycomb in the out of plane  direction is given by the rule of mixtures expression E E s   c  s  In honeycombs failure under out of plane compressive stresses occurs due to fracture of the cell walls or due to elastic or plastic buckling of the cell walls  For Nomex honeycombs failure is due to a crushing mechanism initiated by elastic buckling and developing as a plastic buckling process The relevant collapse strength  cc can be simply estimated using the rule of mixtures expression  cc  sc   c  s where  sc is the compressive strength of the solid from which the core is made Wierzbicki  gives an alternative expression for the failure stress based on a plastic collapse model For a Chapter  Failure Mode Maps for Honeycomb Sandwich Beams  honeycomb with regular hexagonal cells this approach predicts the collapse strength  cc   sc  c  s    Zhang and Ashby  show that the out of plane shear strength and sti ness of hon eycombs are independent of height and cell size Honeycomb cores exhibit slight anisotropy in their out of plane shear strength and sti ness due to the set of dou bled walls By using simple mechanics models based on an array of regular hexagons and considering the double wall e ect approximate expressions for the shear strengths   and   are derived as   E s    c  s  a   E s    c  s  b and for the shear moduli G  and G  as G  G s    c  s  a G  G s    c  s  b The core shear modulus G cxz used in equation  to calculate the skin stress should be taken as either G  or G  depending on the orientation of the ribbon direction in the honeycomb This anisotropy leads to a dependence of skin failure loads on the hon eycomb orientation Similarly the core shear strength  cs depends on the honeycomb orientation Experimental Evaluation of Honeycomb Mechanics The theoretical relations detailed in section  are those that we use for our cal culations in section  In this section we compare the theoretical expressions with experimental data from reference  and manufacturers data sheets  for hon eycombs made of Nomex aramid paper impregnated in phenolic resin Often it is observed that there is a wide variation amongst data from di erent sources re ecting the wide manufacturing tolerances in the constituent aramid sheet particularly  s  and the di culties in making accurate measurements  Hexcel Composites formerly Ciba Composites Chapter  Failure Mode Maps for Honeycomb Sandwich Beams  The measurements of out of plane compressive properties are made by testing hon eycombs under stabilised  compression As depicted in Fig  a the prediction of Equation  for Youngs modulus lie between the two sets of data and close to that of reference  Fig  b shows that Wierzbickis  equation  for compres sive strength ts the experimental data better than the usual mixtures law re ecting the plastic collapse mechanism of Nomex under compression Figures  cd show plots of shear modulus and strength Manufacturers measure ments are signicantly higher than the measurements of Ashby and Zhang because of the di erent test setup Ciba or Hexcel use a short beam test where shear strength and sti ness are out of plane and measured indirectly see eq  in ref Zhang and Ashby has tested the honeycombs in in plane shear with an appropriate testing rig We believe that the latter source gives a more direct estimate of shear properties The major di erence between equations a b and the experimental data of  for  c  s   are due to debonding of honeycomb specimens from the rig We can use equations a b when  c  s   since in the material systems considered here no debonding occurs  Construction of a Failure Mode Map Sections  and  have described various mechanisms of failure which occur with honeycomb sandwich panels and the honeycomb mechanics needed to evaluate the failure loads for each of these mechanisms In this section we describe how a map can be constructed detailing which failure mechanism actually occurs for a given material combination and beam geometry This follows the work of Triantallou and Ashby  for foam core sandwich panels The failure mode map is illustrated by way of examples in section  The failure loads depend on the properties of the skin and honeycomb solid material the relative density  c  s of the core the thicknesses t and c of the skin and the core and the beam span L Because we include indentation failure failure also depends on the loading details In the experiments we use rollers to apply the load so that failure depends on the roller radius R The failure line load W o can then be expressed as a function of the material properties and the beam parameters  W o  f tL tR  c  s  To evaluate this function the expressions for skin and core stresses equations    are substituted into the various failure criteria equations    and   as described in section  to give the critical line loads as summarised in Table  The actual failure load and mode are  Stabilised compression means restriction of the cell walls from slipping between the specimen and the rig plates during the test Chapter  Failure Mode Maps for Honeycomb Sandwich Beams  0 .10 R e la tiv e D e n s ity , ρc /ρs 0 .10 0.03 1 .00 R el at iv e Y o u n g' s M o du lu s, E c /E s CIBA Ashby-Zhang eq  0 .10 R e la tiv e D en s ity , ρc /ρs 0.01 0.03 0.10 R el at iv e St ab ili se d Co m pr es siv e St re n gt h, σ cc /σ sc C IBA data H exce l data Ashby-Zhang mixture law eq  a b 0 .10 R e la tiv e D e n s ity , ρc /ρs 0 .01 0 .03 0 .10 R el at iv e Sh ea r M o du lu s, G c/ G s G 31 G 32 C IB A H E XC E L Ashby-Zhang eq a eq b 0.10 R e la tiv e D e n sity , ρc /ρs 0.03 0 .001 0 .010 R el at iv e Sh ea r St re n gt h, τ c /E s τ31 τ32 C IB A H E X C E L A sh by-Z hang eq a eq b c d Figure  Comparison of theoretical and experimentally measured outofplane a com pressive modulus b compressive strength c shear modulus and d shear strength of Nomex honeycombs Chapter  Failure Mode Maps for Honeycomb Sandwich Beams  given by the mode with the minimum failure load Maps of the failure mode and failure load can then be drawn as a function of the beam geometry for a given material system The Matlab  programming language is used to evaluate the equations Table  Summary of failure criteria Top skin yield W o   fY t L  Intracell buckling W o     f  t    E f t L  Face wrinkling W o  B  E  f E  s  t L    c  s    Core Shear W o  AE s d   c  s  Indentation W o   sc   c  s    Note The quantities and A change according to the honeycomb ribbon orientation  A Failure Mode Map for Beams with a GFRP Skin and Nomex Core The above section describes how a failure mode map can be constructed for a honey comb sandwich panel This is illustrated in this section using sandwich panels made of GFRP laminate skins and Nomex honeycomb cores of di erent densities The core and skin thicknesses c and t are  mm and  mm respectively and the nominal honey comb cell size is  mm Experimental results for this sandwich panel type are presented in section  Further details of the panel construction materials and material prop erties are given in that section Fig  b shows the dependence of line load W o at failure on core relative density  c  s and face thickness to span tL ratio for a value of the radius of the roller to the skin thickness Rt of  Each surface corresponds to a di erent failure mode The failure mode map Fig  a is given by the projection of the intersections between failure surfaces on the  c  s  tL plane Slightly di erent failure maps are calculated depending on whether the honeycomb ribbon lies in the longitudinal direction along the beam axis or transverse to the beam axis Di erences arise from the anisotropy in the strength and sti ness of honeycomb in shear cf equa tions  and ab As noted in section  failure by indentation is estimated using an empirical approach and relies on experimental measurements of the contact area as described in section  Indentation is the only mechanism which depends on Chapter  Failure Mode Maps for Honeycomb Sandwich Beams  the roller diameter This will be a more widespread mechanism of failure with smaller roller diameters For the honeycomb geometry chosen for these plots intra cell dim pling is not predicted At the map boundaries the failure loads for the mechanisms either side of the boundary are equal In practice failure near a boundary may be due to a combination of the two mechanisms and coupling between the two mechanisms may reduce the load below that predicted for each of the modes independently The failure mode map shown in Fig  a is useful where a designer has specied face and core thicknesses and wishes for example to select an appropriate span or relative density More commonly however the span is xed a standard skin construc tion and thickness is specied and only the core thickness or density can be relatively easily changed In this case it is more useful to plot a map of the failure modes and loads as a function of core relative density and core thickness to span ratio cL at a xed skin thickness to span ratio tL and roller radius to face thickness ratio Rt Figure  shows such maps for three typical values of tL with Rt   Note that for long spans Fig a the core crushing failure mode vanishes while for short spans face wrinkling is not predicted Fig c  Experiments Sandwich panels were made of Nomex honeycomb core and GFRP laminate skins and were supplied by Hexcel Composites All panels had the same skin cross ply laminate on either side of the core as depicted in Fig  Each laminate comprised two glass prepregs the outer with a resin content of ! and the inner with ! giving a skin thickness t of  mm The Nomex honeycomb cores used in the panels are designated by the manufacturer as Aeroweb r type A Panels with core densities of    and  kgm were used For most of the tests the honeycomb had a nominal cell size of  mm but for tests described in section  a cell size of  mm was used The core thickness c was  mm Mechanical properties of the skin and the honeycombs constituent solid material aramid paper resin are listed in Table  Since the compressive strength of the laminate has been inferred from bending tests of sandwich beams this is not an independent measurement Hexcel quote a value of  MPa based on simple beam theory equation  for long beams Using the beam model equation  with a correction for shear in the core a revised estimate for the compressive strength of the laminate of  MPa is inferred from the data at long spans Panels were cut into beams using a diamond wheel A width b of mm was chosen Chapter  Failure Mode Maps for Honeycomb Sandwich Beams  10−4 10−3 10−2 0.4 0.02 0.1 Face Yield Face Wrinkling Core Shear Core Crushing R e l a t i v e d e n s i t y  c   s Face thickness  span tL a 10−4 10−3 10−2 0.1 0.01 0.1 1 10 102 103 104 105 106 Relative density  c  s Face thickness  span tL F a i l u r e l i n e l o a d W o b Figure  a Failure mode map Solid lines refer to beams in which the honeycomb ribbon lies along the beam axis dashed lines for when the ribbon lies transverse to the beam axis The symbol identies experimental measurements described in section   b Failure load surface for ribbon lying along the beam axis Chapter  Failure Mode Maps for Honeycomb Sandwich Beams  10−3 10−2 10−1 0.01 0.1 1 Face Yield Face Wrinkling Core Shear R e l a t i v e d e n s i t y  c   s Core thickness c m a tL     10−3 10−2 10−1 0.01 0.1 1 Face Yield Face Wrinkling Core Shear Core Crushing R e l a t i v e d e n s i t y  c   s Core thickness c m b tL     10−3 10−2 10−1 0.01 0.1 1 Face Yield Core Shear Core Crushing R e l a t i v e d e n s i t y  c   s Core thickness c m c tL     Figure  Failure mode map for three typical values of tL and Rt Chapter  Failure Mode Maps for Honeycomb Sandwich Beams  Nomex  Laminate skin  Youngs modulus GNm   E s  E f  Shear modulus GNm   G s  G f  Compressive strength MNm    sc   fY  Poissons ratio  s   f  Density Mgm   s   Table  Material properties of Nomex and laminate skin so that it was greater than twice the sandwich height and three times the cell size as recommended in by ASTM standards  Beams were cut with the ribbon direction either in the longitudinal direction along the beam axis or in the transverse direction as depicted in Fig  Beams of varying spans and core relative densities were made to probe the various regions of the failure mode maps Fig  Specimens were tested in  point bending applying the load through rollers of diameter of  mm cf ASTM standard C   The crosshead speed was kept constant through the test and was chosen so that the maximum load occurred between  and  min after the start of the test Displacement of the central loading point relative to the end rollers was monitored using an LVDT and logged on a computer The central section of the beam where failure invariably occurred was also recorded on a video recorder Fig  as the test progressed To model failure due to local indentation section  it is necessary to dene the value the length of the contact area  between the central roller and the top skin This was estimated experimentally by putting carbon paper between the roller and the specimen For all the specimens tested  was between  and  mm Longitudinal Direction Transverse DirectionPrepreg Fibre Direction Honeycomb Ribbon Direction Figure  Layup details Chapter  Failure Mode Maps for Honeycomb Sandwich Beams  with a typical value of  mm This information was used to estimate the failure load due to indentation as described in section  Figure   Test setup  Experimental Results Figures  a d show photographs of the section under the central load just after fail ure illustrating four failure modes observed during the tests of those beams with  mm honeycomb cell size These were taken from videos recorded during the tests Beam de tails for each of these gures are given in Table  to scale the photographs note that the cell size is  mm for all the specimens of Fig  except Fig  ab which has the larger cell width of  mm For clarity the roller seen in the top half of the gures has been outlined in Fig  a Corresponding loadde ection curves are included in Fig  With all the failure mechanisms except intracell buckling Fig ab and core shear Fig d failure occurred in a brittle manner with little non linearity in the load displacement curves before failure and a sharp drop in load at failure For short span tests where indentation occurred Fig c the beam has a signicant post failure strength as the core progressively crushes Table  presents the average  line loads W o at failure and corresponding observed failure mechanisms The mode is described as Complex when failure appeared to occur simultaneously in the core and in the face Those mechanisms denoted as mode mode  ie Intracell dimpling core crushing indicates the appearance of mode  as elastic instability before the nal failure mode  The measured e ect of skin thickness to span ratio on the failure load is compared with theory in Figs  a d for the di erent core densities The theoretical graphs are sections through the load surface Fig  b at constant core relative density Experimental mechanisms of failure are indicated on these graphs to allow a comparison between the actual and predicted failure mechanisms  Most of tests were repeated and the dierence between failure loads was less than  Chapter  Failure Mode Maps for Honeycomb Sandwich Beams  Core Honeycomb Span Measured Observed Density ribbon length tL line failload failure modes kgm   direction mm kNm  E  Face wrinkling  E  Face wrinkling  E  Core shearface wrink L o n g   E  Core shearcrushing  E  Core shearcrushing  E  Face wrinkling  E  Face wrinkling    m m c e l l   E  Core shearface wrink T r a n s   E  Core shearcrushing  E  Core shearcrushing  E  Face wrinkling  E  Face wrinkling  E  Complex L o n g   E  Core crushing  E  Core crushing  E  Face wrinkling  E  Face wrinkling    m m c e l l   E  Complex T r a n s   E  Core crushing  E  Core crushing  E  Face yield  E  Face yieldwrinkling  E  Complex L o n g   E  Core crushing  E  Core crushing  E  Face yield  E  Face yieldwrinkling    m m c e l l   E  Complex T r a n s   E  Core crushing  E  Core crushing  E  Face yield  E  Face yield  E  Face yield  E  Face yield L o n g   E  Face yield  E  Complex  E  Core crushing  E  Face yield  E  Face yield     m m c e l l   E  Face yield  E  Face yield T r a n s   E  Face yield  E  Complex  E  Core crushing  E  Intracell dimplface peel  E  Intracell dimplface peel  E  Intracell dimplcore crush L o n g   E  Intracell dimplcore crush  E  Intracell dimplface peel  E  Intracell dimplface peel     m m c e l l   E  Intracell dimplcore crush T r a n s   E  Intracell dimplcore crush Table  Experimental results Photographs of failure given in Fig  and  corre spond to the entries in italics in the nal column of this table Chapter  Failure Mode Maps for Honeycomb Sandwich Beams  0 10 20 30 M idspan D eflection (m m ) 0 5 10 15 20 L in e L oa d, W ( kN /m ) Core Density = 128 kg/m3 Span Length = 345 mm Honeycomb Direction Long itud ina l T ransverse a Face yield  c   kgm  L   mm 0 5 10 15 20 M idspan D eflection (m m ) 0 2 4 6 8 L in e L oa d, W ( kN /m ) Core Density = 48 kg/m3 Span Length = 380 mm Honeycom b Direction Long itud ina l T ransverse b Face wrinkling  c  kgm  L  mm 0 2 4 6 M idspan D eflection (m m ) 0 20 40 60 L in e L oa d, W ( kN /m ) Core Density = 128 kg/m3 Span Length = 60 mm Honeycomb Direction Long itud ina l T ransve rse c Core crushing  c   kgm  L  mm 0 1 2 3 M idspan D eflection (m m ) 0 2 4 6 8 L in e L oa d, W ( kN /m ) Core Density = 29 kg/m3 Span Length = 50 mm H oneycom b D irection Longitud inal T ransverse d Core crushing  c   kgm  L   mm Figure   Photographs of the dierent failure modes for transverse ribbon direction and corresponding load deection curves for both ribbon directions Chapter  Failure Mode Maps for Honeycomb Sandwich Beams  0 .001 0 .010 F ace th ickness / span , t/ L 1 10 100 L in e L oa d, ( kN /m ) Core Density = 128 kg/m3 T he o re tic a l C urve s F ace Y ie ld C ore C rush ing Long T rans Long T rans E xperim enta l D a ta Face Y ie ld C ore C rush in g C om p lex W o 0 .001 0 .010 F ace th ickness / span , t/ L 10 L in e L oa d, ( kN /m ) Core Density = 64 kg/m3 Theore tica l C urves Face Y ie ld Face W rink ling C ore C rush ing Long T rans F ace Y ie ld C om plex F ace W rink ling C ore C rush ing Experim en ta l D a ta Long Trans W o a  c  s   b  c  s   0.001 0.010 F ace th ickness / span, t/ L Theoretica l C urves Face W rin k ling C ore C rush ing 10 L in e L oa d, ( kN /m ) Core Density = 48 kg/m3 Face W rinkling C om plex C ore C rushing Long T ran s E xperim enta l D ata Long T ran s W o 1 10 L in e L oa d, ( kN /m ) 0.001 0.010 F ace th ickness / span, t/ L Core Density = 29 kg/m3 Theore tica l C urves Face W rink ling C ore S hear C ore C rush ing Long T rans Face W rinkling C om plex C ore S hear + C rushing E xperim en ta l D a ta T rans Lo ng W o c  c  s   d  c  s   Figure   Variation of failure load W o with skin thickness to span ratio tL Chapter  Failure Mode Maps for Honeycomb Sandwich Beams   Discussion Figure  shows that in general the experimental failure loads agree satisfactorily with theoretical values and that the observed failure modes are generally the same as the predicted modes Fig  Where a transition from one failure mode to an other occurs the mechanism of failure is mixed but the failure load is still adequately predicted The main source of deviations arises from the errors in predicting the hon eycomb material properties section   Skin Failure The laminate face yielding strength has been chosen to t data at long spans as discussed in section  At the longest spans with the  kgm honeycomb face yield occurs although wrinkling is predicted but as shown in Fig  a these points are very close to the boundary between these two mechanisms  Core Failure Failure load predictions for core crushing are adequate for the higher density cores but predictions are poor for the lower density cores The photograph in Figure  d illus trates a beam with a low density core in transverse direction near the failure load where core shear failure is predicted Final failure is a mixture of core shear and crushing under the indenter The di erence in the initial slopes of the loadde ection curves  for the specimens with di erent ribbon directions re ects the anisotropy associated with the core shear sti ness Non linearity in the curve for the transverse direction indicates that there may be some ductile shear of the core prior to nal failure The signicant under prediction of the failure load Fig  d due to core shear is because nal failure does not occur when the core fails in shear Predictions of failure based on the core crushing strength would give a relatively good agreement with measurements in this case Again this highlights the need for more elaborate models of core failure near loading points  Eect of Ribbon Direction Specimens with ribbon running in the longitudinal direction fail at a signicantly higher load than those with ribbon in the transverse direction in all cases except where failure  The kinks in the load deection curves at a low load are an artefact of the measuring system Chapter  Failure Mode Maps for Honeycomb Sandwich Beams  is by intracell dimpling section  For face failure modes a slight di erence is predicted associated with the e ect of core shear sti ness on the stress in the skins although the measured di erences are somewhat greater than the prediction The the oretical model described for core crushing does not include any anisotropic material properties The substantial di erence between the strengths for the two ribbon ori entations suggest that an improved model is necessary looking in more detail at the stresses around an indenter and including the e ect of core shear  Intracell Buckling Section  denes a critical honeycomb cell size above which failure occurs by intra cell buckling instead of by top face yielding For the materials used here this transition is predicted at a cell size of approximately  mm This hypothesis was tested by using specimens with a cell size  mm a core density of  kgm and a range of skin thickness to span ratios All of the specimens with the larger cell size exhibited elastic intracell buckling Final failure did not occur immediately but at a higher load either by delamination of the top face or by core crushing for long or short spans respectively Fig  a shows the section under the central roller at the onset of intra cell buckling while Fig  b shows the section when nal failure occurs by delamination Figure  c compares the failure load and failure mechanisms for this set of tests The theory equation  under predicts the nal failure load because it relates to the initial buckling instability and not the nal delamination or core failure Failure loads are independent of the honeycomb ribbon direction because the single and double walls paper walls make up a smaller proportion of the honeycomb for a given honeycomb density  Conclusions Previous research on honeycomb mechanics and the behaviour of sandwich beams in  point bending have been combined to model the behaviour of honeycomb sandwich panels It is assumed that the skin and core materials behave in a brittle manner The failure mechanisms considered were face yield face wrinkling intra cell buckling core shear and indentation at the load points leading to core crushing The latter is treated in an empirical way using measurements of the bearing area at the load points The failure loads for each region are estimated assuming that there is no coupling between failure mechanisms Following the work of Triantallou and Gibson  failure mode maps are derived with axes as the core relative density and the ratio of the face thickness Chapter  Failure Mode Maps for Honeycomb Sandwich Beams  to span length Each map is appropriate for a single value of the ratio of the radius of the roller to the skin thickness Regions in which the various failure mechanisms occur are identied on the maps Alternative axes for the map of core relative density and the core thickness are suggested as a more useful set of parameters for a beam designer or manufacturer Although the maps are generated for three point bending the method can straightforwardly be applied to other loading geometries for example four point bending The method is illustrated using the widely used combination of cross ply GFRP laminate skins with a Nomexphenolic resin honeycomb core Experimental tests showed that in general the maps predicted adequately failure modes and failure loads The transition from face yielding to intra cell buckling for long span beams was demonstrated by increasing the honeycomb cell size from  to  mm Failure near the load points due either to core shear or core indentation was not modelled well To model this behaviour more sophisticated models of the contact region and of the failure criteria in the core are needed Chapters  and  show how the implementation of the high order sandwich beam theory of Frostig  provides a better insight into the mechanism of localised e ects induced by concentrated loads and helps to produce an improved indentation failure analysis for honeycomb sandwich beams Chapter  Failure Mode Maps for Honeycomb Sandwich Beams  0 4 8 12 M idspan D eflection (m m ) 0 4 8 12 L in e L oa d, W ( kN /m ) Core Density = 64 kg/m3 Cell size = 13 mm Span Length = 230 mm Honeycom b Direction Long itud ina l T ransverse (a) (b) a Elastic intracell buckling b Final failure due to top skin delamination 0.001 0.010 F ace th ickness / span , t/ L 1 10 100 L in e L oa d, ( kN /m ) Core Density = 64 kg/m3 Cell Size = 13 mm Experim enta l D a ta Long Trans Intracell D im p l. + Face P ee ling Intracell D im p l. + C ore C rush ing Theore tica l C urves In tracell Buckling C ore C rushing W o c Experimental results compared with theory Figure  Photographs and results of specimens with  mm cell size Chapter  Indentation Resistance of Sandwich Beams  Introduction The experiments described in the previous chapter revealed the important role of the exural rigidity of the skins and the transverse shear sti ness of the core on the overall resistance of a sandwich beam to indentation In the literature the importance of the core behaviour in a ecting indentation failure has been considered however the in uence of the skins exural rigidity is generally overlooked This omission derives from the fact that in practical applications the skins are quite sti and the in uence of skin exural rigidity can be neglected One of the main aims of this chapter is to focus on the role of skin rigidity in indentation failure Before introducing any analytical methods it is instructive to consider two extreme cases as illustrated in Fig   For very exible skins Fig a there is large local deformation under the load which can easily lead to core failure  For very rigid skins Fig b indentation failure will be relatively hard as the skins spread the load Skin exibility is a prerequisite for indentation failure and the cores transverse exibil ity enhances this susceptibility In this chapter the high order sandwich beam theory HOSBT is used to analyse the bending behaviour of sandwich beams under localised loads and extract conclusions about their indentation resistance by examining the ex ibility of the top skin to the external applied loads The ability of HOSBT to cope with the localised e ects that appear in sandwich beams at the load points is experimen tally veried by Surface Displacement Analysis in section  Section  introduces a length scale that characterises the susceptibility of a sandwich beam to localised e ects  Chapter  Indentation Resistance of Sandwich Beams  In section  high order sandwich beam theory is used to predict the contact stress distribution between a cylindrical indenter and the top skin of a sandwich beam low flexural rigidity of skins high flexural rigidity of skins (a) (b) Figure  The behaviour of exible and rigid skins  HighOrder Sandwich Beam Theory The mathematical formulation of high order sandwich beam theory HOSBT is intro duced by Frostig and Baruch  High order refers to the non linear way in which the in plane and vertical displacements are allowed to vary through the height of the core The core vertical displacement is assumed to have a quadratic variation with height z Other core displacements also vary in a non linear way with the exact variation being expressed in terms of Fourier series This contrasts with simple beam theory where the core in plane displacements are assumed to vary in a linear way through the depth and the out of plane displacements are assumed to be constant These high order vari ations allows modelling of the more complicated changes in core geometry which occur at loading points Thus the basic assumptions of HOSBT are  The shear stresses in the core are uniform through the height of the core  The core vertical displacement variation is a quadratic polynomial in z see equa tion  allowing the core to distort and its height to change  The core is considered as a D elastic medium which has out of plane  compres sive and shear rigidity whereas its resistance to in plane  normal shear stresses is negligible Outofplane refers to  direction as illustrated in Fig  or the direction normal to the plane of the sandwich panel  Inplane refers to  plane as illustrated in Fig  or the plane of the sandwich panel Chapter  Indentation Resistance of Sandwich Beams  This high order analysis is based on variational principles the details of derivation of the governing equations and associated boundary conditions are presented in ref  In this study we examine the bending behaviour and the localised e ects of sandwich beams formulating a two dimensional model of a sandwich beam with unit width and span L which consists of a core with thickness c Youngs and shear modulus E c and G c respectively and two skins with the same thickness t  t t  t b Youngs modulus E f and Poissons ratio f as depicted in Fig  b An external distributed load q t is applied on the top skin The displacement and stress elds of the core are expressed in terms of the followings ve unknowns the in plane deformations u t and u b in the x direction of the centroid of the top and bottom skin respectively their corresponding vertical displacements w t and w b  and the shear stresses  x in the core The relevant geometric parameters and the notation of stresses and displacements appear in Fig  The governing equations see equations   and   in ref  are given below A t u tx  x   A b u bx   x   D t w tx  E c c w b  w t  ct   xx  q t D b w bx E c c w b  w t  ct   xx   u t  u b  ct  w tx  ct  w bx  c G c  x  c  E c  xx                where A t  A b  E f t   f  and D t  D b  E f t    f  are the in plane and exural rigidity of the skins The notation  x for example denotes the th partial derivative with respect to x For a simply supported sandwich beam the solution can be expressed as a Fourier series expressing the variation of the relevant variables in the x direction as u t x  M X m C ut m cos mx L  u b x  M X m C ub m cos mx L  w t x  M X m C wt m sin mx L  w b x  M X m C wb m sin mx L   x x  M X m C  x m cos mx L  where M  number of terms in the series and C ut m  C ub m  C wt m  C wb m and C  x m are constants to be determined We assume that the external loads are exerted only on the top skin Chapter  Indentation Resistance of Sandwich Beams  wt ut wb ub deformed top skin deformed bottom skin x z d deformed cross section wc (a) qt τx σzz(z=0) σbxx σtxx c tb tt core top skin bottom skin δ L x unit width W (b) Figure  a Nonlinear displacement patterns b Beam geometry and stresses The origin of the zcoordinate is always taken at the top of the beam element either skin or core which is being considered Chapter  Indentation Resistance of Sandwich Beams  So in terms of the Fourier series we set q t x  M X m C qt m sin mx L  where C qt m is a constant that depends on the distribution of the external load After substituting every term of the Fourier series   in the governing equations  the problem can be expressed in matrix form as D  C  Q  where D  A t m L        A b m L        D t m L   E c c  E c c ct  m L    E c c D b m L   E c c ct  m L    ct  m L  ct  m L c G c c  E c m L    C  C ut m C ub m C wt m C wb m C  x m  and Q    C qt m    By solving  with respect to matrix C for every m      M the Fourier coe cients in equations   can be determined These equations can then be used to calculate all the in plane and vertical displacements and core shear stresses The in plane normal stresses  txx in the top skin and the out of plane normal stresses  zz in the top skin core interface are calculated indirectly as  txx  E f u tx   zz  c   xx E c c w b  w t    Surface Displacement Analysis An experimental verication of high order sandwich beam theory HOSBT has been undertaken by Thomsen and Frostig  They conducted a series of photoelastic  Only top skin compressive failure is considered Chapter  Indentation Resistance of Sandwich Beams  experiments compared the results with analytical calculations and demonstrated a good agreement between the photoelastic measurements and the predictions based on the HOSBT In this section a similar comparative study is conducted with respect to the displacement eld in the core The objective is to compare the analytical and experimental displacement distributions in sandwich beams with a soft core as for example the Nomex cores used in Chapter  Special emphasis is given to the local out of plane compression of the core under the localised loads as this will be important in determining failure of the core For the analytical predictions the equations described in section  are implemented in a Matlab code to calculate the vertical displacement eld within the core by the following equation given in ref w c x z   xx z  cz E c w b  w t  z c w t  The experimental determination of vertical displacements in the core is achieved by Surface Displacement Analysis SDA is a Windows based software developed by Instron Corporation which detects and maps horizontal and vertical displacement of points on the surface of a specimen or structure There are two main parts to the procedure image capture and image analysis SDA captures images of the surface of a material or structure using a video camera and a frame grabber board The surface requires a random speckle pattern either natural or applied which has to be illuminated to ensure good contrast Image frames of the surface of the specimen are captured at intervals while it deforms under load Then SDA compares and analyses pairs of these image frames using a designated area of interest as depicted in Fig  Each area of interest contains one or more analysis cells During analysis the pattern in each cell of the current frame is compared with the pattern in the corresponding cell of the reference frame The software maps the deformation of the surface as a series of vectors displayed on the image one vector for each cell The software also calculates the strain for each cell SDA contains a graphics module that plots displacement and strain If the software is calibrated to the dimensions of the image absolute displacement values can be obtained In this study we tested sandwich beams cut out from the panels described in section  Testing was under  point bending at a constant displacement rate of  mmmin The beams width was mm the span  mm and the rollers diameter  mm The experimental setup is shown in Fig  The clip gauge measures the midspan core compression during loading while the video camera captures images of the other cross section of the beam To apply a random speckle pattern of that cross Chapter  Indentation Resistance of Sandwich Beams  Figure  Surface Displacement Analysis software window showing the reference image frame of a side cross section painted to have a random speckle pattern and the area of interest where the analysis takes place camera clip gauge beam measured core compression Figure  Experimental setup Chapter  Indentation Resistance of Sandwich Beams  section at rst it was painted black and then ne sprayed with white paint to produce a pattern of random white dots on a black background see the image in Fig  The three dimensional surface of the cross section arising from the honeycomb structure and the subsequent buckling of the cell walls causes errors in the image processing and mapping of displacements using SDA However in the initial stage of loading with little buckling the results are satisfactory and illustrative of the localised e ects that occur under the central load For a sandwich beam with Nomex honeycomb core with density of  kgm the load de ection curve is shown in Fig  The same gure also shows the clip gauge measurements of midspan core compression against top skin de ection For Surface Displacement Analysis two frames are chosen the rst reference frame just before loading point A in Fig  and the second when the applied line load reaches the arbitrary value of  kNm labelled B in Fig  The SDA graphics module after calibration plots contours of equal vertical dis placements within the core in mm as depicted in Fig  a It should be noted that the camera was xed relative to the central roller so that the specimen appears to move upwards with zero displacements at the roller The SDA measurements demonstrate the occurrence of localised de ections under the applied load due to the core local compression The contour of  mm close to the bottom skin indicates the total compression of the core in the midspan of the sandwich beam This value agrees well with that measured by the clip gauge see the dashed line in Fig  Fig  b shows a theoretical plot at the corresponding load of  kNm using equation  These calculations use the material properties of the skin and the honeycomb and the geometric attributes of the sandwich beam Table  The total applied load is taken as  kNm and applied uniformly over an area of  mm long   A comparison between the theory and experiments shown in Fig  show that HOSBT provides an analytical tool that is capable of predicting the localised e ects due to concentrated loads both qualitatively and quantitatively  Eect of Spreading Stresses After demonstrating the applicability of high order sandwich beam theory in this sec tion we use it to investigate the mechanism that controls the behaviour of a sandwich beam under localised loads One of the key factors that governs the behaviour of a sandwich beam under localised loads is the shape of the normal stress distribution  The reason for this is explained in section  Chapter  Indentation Resistance of Sandwich Beams  Speed = 0 .5 m m /m in C ore D en sity = 48 kg /m 3 Span L ength = 60 m m 0.0 0 .5 1 .0 1 .5 2 .0 T op sk in deflection (m m ) 0.0 0.1 0 .2 0 .3 M id -s pa n c or e co m pr es si on ( m m ) 0 4 8 12 L in e lo ad , W ( kN /m ) A B A B Figure  Loaddeection and core compression curves A and B correspond to the frames used for processing by SDA The dashed line indicates the total midspan core compression at a line load of kNm Chapter  Indentation Resistance of Sandwich Beams  a 24 26 28 30 32 34 36 0 1 2 3 4 5 6 7 8 9 0.035 0.04 0.045 0.06 0.1 0.1 0.15 0.15 V e r t i c a l c o o r d i n a t e z m m  Horizontal coordinate x mm b Figure  Vertical displacement eld in the core produced by a SDA and b HOSBT model All contour values are in mm and the scaling in both plots is the same Chapter  Indentation Resistance of Sandwich Beams  along the beam in the top skin core interface see Fig  Two extreme cases can be considered a very rigid top skin that spreads the external stress in the interface or a membrane skin allows the applied stresses to pass intact through the interface into the core In this section we extract a dimensionless quantity which characterises the behaviour of a sandwich beam under  point bending independently of the applied central load We do this by dening an index which measures the transparency of the top skin ie to what extent the top skin core interface feels the stresses exerted on the external surface of the top skin For this purpose we rewrite the matrices D and Q introduced in equation  as Q    C qt m    and D  A f      A f      D f E c C t   E c D f C t   C t C t G c  where                                      A f  A t m L   D f  D t m L   E c c E c   E c c C t  ct  m L G c  c G c c  E c m L   Using the Symbolic Toolbox of Matlab equation  is solved with respect to the matrix C C  QD   C ut m C ub m C wt m C wb m C  x m   C qt m  C t R R  C t R R  R D f E c R R   R  D f E c R R  A f C t R R    where R  D f A f G c   A f C t   and R   E c A f G c   A f C t   Substituting equations   and the above Fourier coe cients into  the normal stresses in the top skin core interface are given as  zz  M X m  c  m L C  x m E c c C wb m  C wt m   sin mx L  M X m  cm L A f C t R R  E c D f E c  C qt m sin mx L  Chapter  Indentation Resistance of Sandwich Beams  Equation  shows that the Fourier coe cients for the stress  zz can be separated into two parts one part containing the coe cient C qt m units of stress which depends on the distribution of the load and a second part represented by the dimensionless transmission coe cient C zz m which is entirely dependent on the geometric attributes and material properties of the sandwich beam where C zz m is given by C zz m   cm L A f C t R R  E c D f E c   A f C t   R R  E c D f E c if tn c  For any geometry and material combination this transmission coe cient is a discrete function of the parameter Lm which is the semi wavelength of every term in the Fourier series The value of C zz m varies from  small Lm to  large Lm  implies no transmission of this Fourier component of applied stress while  implies total transmission through the top skin Thus we conduct a parametric study in which we calculate the variation of C zz m with Lm for the range of material and geometric parameters given in Table  Fig  plots this variation Each sub plot shows curves for variations in the skin thickness t and core depth c Changing from one sub plot to the next represents a change in either skin sti ness E f or core density  c as indicated by the arrows to the left and top of the gure Parameter Low Value  High Value t mm    E f GPa     c kgm     c mm    Note The  c refers to the density of Nomex honey comb with  s   kgm  Table  The range of the geometric and material parameters used in the parametric study To characterise the transparency of the skin it is helpful to identify a semi wave length Lm below which the values of the coe cient C zz m are close to zero so that these wavelengths cannot be transmitted Examining the curves in Fig  it can be seen that the curves all have similar shapes and that the in ection point in each curve can be used to characterise where C zz m is small The semi wavelength at the in ection point characterises the susceptibility of sandwich beams to localised e ects and this semi wavelength is dened as the spreading length  In general the larger is the broader is the distribution of the normal out of plane stresses in the top skin core interface and vice versa Chapter  Indentation Resistance of Sandwich Beams  0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1  c    k g  m  c      k g  m E f   GPa E f    GPa t   mm t   mm t   mm t   mm t    mm t    mm t    mm t    mm c   mm c   mm c   mm c   mm c    mm c    mm c    mm c    mm Lm mmLm mm Lm mmLm mm C o e c i e n t C z z m C o e c i e n t C z z m C o e c i e n t C z z m C o e c i e n t C z z m Figure  Parametric Study change of transmission coecient C zz m with wavelength Lm for variations in the parameters t E f  c and  c  The boxed labels show each time the parameters and the arrows indicate the direction of each parameters increase The position of inection is marked by a  symbol Chapter  Indentation Resistance of Sandwich Beams  Spreading length is a function of the skin thickness t skin Youngs modulus E f core density  c and core thickness c but it is independent of span length L Table  shows the in uence of each parameter to spreading length  This table shows the in uence of changing the various parameters from the low values given in the left hand column of Table  Table  shows that the skin thickness t is the most signicant factor as expected the larger the skin thickness the stronger the e ect of spreading stresses A similar e ect is shown with increasing sti ness E f although given the big change in sti ness of  this e ect is less important Core density also a ects  the sti er is the core the less the external stresses spread Core thickness c seems to play a small role Table  Change of spreading length  Low values from Ta ble  for t c E f   c t t c c E f  E f  c   c      At this point it is important to note that is not directly related to the strength of the sandwich beam Instead it measures how the top skin is able to spread the external load over the core is property of the beam material and geometry In the following sections we show how it can be normalised and provide a dimensionless index to characterise the spreading e ect in sandwich beams under indentation loads  Contact Pressure Distribution Having used the high order sandwich beam theory to characterise the way in which contact loads are transmitted through the top skin in this section we investigate how this methodology can be applied to the contact between an indenter and a sandwich beam In the literature there are only a few references relating to the contact stresses exerted between the top skin of a simply supported sandwich beam and an indenter which applies  point bending on the beam Frostig " Baruch  examine analytically the e ect of four types of localised load distributions a point load uniform sinusoidal distributions or two concentrated loads These represent very exible intermediate and very sti indenters respectively Johnson  p investigates thin plates in contact with a rigid indenter and concludes that the contact stress distribution changes from having a maximum in the centre to one in which the pressure is concentrated at the edges when the thickness of the plate increases A di erent problem is examined by Keer and Ballarini  who investigate the problem of a rigid punch in smooth Chapter  Indentation Resistance of Sandwich Beams contact with an initially stressed transversely isotropic elastic beam In this study we investigate how sandwich beams behave under contact with a rigid central roller calculating the pressure distribution under the indenter and through the beam Fig  shows a rigid cylinder of radius R which is pressed into contact with a sand wich beam of unit width and length L such that the contact width is  Equation  can be used to calculate the top skin displacements for a given distribution of the ex ternal load Following standard contact methods  we can divide the contact into a number of discrete sections calculate the displacements resulting from a unit load on each of these sections and superimpose the results to nd de ections for any contact load In particular for a given pressure distribution q t x it is possible to nd the corresponding distribution of heights P i Fig ab along the top skin Conversely we shall show that for a given roller geometry we can evaluate the contact width and contact pressure by inverting this approach The shape of discrete pressure elements can be step functions Fig a or over lapping triangles Fig b The second choice is better because the total distributed load function is smoother along the x direction In addition the Fourier series converges more quickly for a triangle shape rather than for a rectangular one The pressure ele ments on the top skin can be described using the series q i  P i #q i  P i M X m C q i m sin mx L  where P i is the height of the triangular distribution and C q i m   L m    v sin m x i L    cos m v L   are the Fourier coe cients for an isosceles triangle of unit magnitude Each unit length has coordinate x i and base width v #q i is the distribution of pressure q i with unit amplitude For this and subsequent symbols the hat refers to variables relating to this unit pressure distribution Using the procedure described in section  the vertical displacements w i t x of the top skin associated with each of the triangular pressure elements are given by equation  w i t x  P i M X m C w t m  i sin mx L  P i #w i t  If N is the number of pairs of symmetrical triangular pressure elements covering a contact width  then the total overall top skin de ection is given by w t x  N X n w i t x  N X n P i #w i t  Chapter  Indentation Resistance of Sandwich Beams  v v/2 Pi Contact width δ top skin surface xi x=L/2 Pi (a) (b) End of beam R L δ W (c) Figure   Discrete contact pressure elements a uniform piecewise constant b over lapping triangles piecewise linear and c Geometric denitions Chapter  Indentation Resistance of Sandwich Beams  In the case where load is applied by a rigid indenter which is in perfect contact with the top skin the top skin de ection follows the shape of the indenter within the contact area In this section we examine the case of a cylindrical indenter for comparison with our experiments but the calculations can be applied to any indenter shape Having a cylinder of R radius as a central indenter and assuming perfect contact with the top skin within contact of width  then the top skin de ection for all x  L      L     is w R t x  s R   x L     R w t L  s R   x L     R N X n P i M X m C w t m  i sin m   s R   x L     R N X n P i #w i t L  where the superscript R denotes the de ection at the roller We can also write the equality only for x  L      L     as w t x  w R t x giving N X n P i #w i t x  #w i t L   s R   x L     R  If we write the above equality for N di erent values of x j  L      L   which represent di erent values of w t x j   w R t x j  along the left half  of the contact width  then we get a system of N equations with N unknowns which are the distributions of pressure P i  This system can be written in a matrix form P  W  R where  P  h P  P     P N i W  #w  t x  #w  t L #w  t x   #w  t L    #w  t x N  #w  t L #w  t x  #w  t L #w  t x   #w  t L    #w  t x N  #w  t L             #w N t x  #w N t L #w N t x   #w N t L    #w N t x N  #w N t L  R  h q R   x   L    R q R   x   L     R    q R   x N  L     R i  Or right half because of symmetry Chapter  Indentation Resistance of Sandwich Beams  Substitution of P i into equation  and summation for all N triangular pressure elements gives us the contact pressure distribution q t x  N X i q i  The same concept of superposition can be used to calculate the other stresses ie  zz  The total load is derived by integration along the x direction W  Z L q t x  N X i P i M X m C q i m Z L sin mx L  W  L  N X i P i M X m mC q i m  for m      M  In principle this approach can be used to determine the variation in load distribution and the contact arc  as the total load W increases In practice a problem arises at small  As it is necessary to span the whole beam length with elements the number of elements needed to deal with sharp variations in pressure becomes prohibitively large This method is used in the following section to answer the following questions  How accurate is it to assume that the load is distributed uniformly over the contact length  $  How transparent is the top skin to the external loads In other words how does the distribution of normal stress  zz x in the top skin core interface compare with the contact stress distribution q t x $  Case Study In this section the results from the aforementioned methodology for two extreme cases of skins exural rigidity are presented to illustrate the e ect of indenters size on the contact stress distribution and the top skins transparency to external loads We consider one sandwich beam denoted as beam A with very exible skins ie  ply  GFRP laminates with t   mm and E f   GPa which gives   mm and another beam B with very rigid skins ie  ply  laminates with t   mm and E f   GPa which corresponds to   mm Both beams have unit depth are  mm long have Nomex honeycomb with core density of  kgm are simply supported and loaded in  point bending by a central roller For each beam two radii for the central roller are considered R  mm and R   mm The methodology described in the previous section requires the contact width  as an input data To Chapter  Indentation Resistance of Sandwich Beams  compare results for each roller  is adjusted manually in a simple numerical algorithm so that for each beam the maximum normal stresses in the top skin core interface are approximately equal Fig  and Fig  show the distributions of contact stresses q t  cc and the cor responding normal stresses in the top skin core interface  zz  cc normalised by the cores out of plane compressive strength for beams A and B respectively For beam B the input value of  is chosen to give max zz  cc   However for beam A this is not feasible The much greater exibility of the beam means that it is not possible to generate stresses in the core greater than half the compressive strength of the core without the beam bending excessively Further loading by increasing the value of  induces large skin deformations which also make HOSBT incapable to model the be haviour of the sandwich beam Thus for beam A the input value of  is chosen to give max  zz  cc   However this di erence in peak stress between the two beams is immaterial as the calculations are elastic and the stress distributions are independent of the applied load A particular feature of the contact pressure distribution for the exible beam A with R   mm and for the sti er beam B with R   mm is the sharp peaks in pressure at the edges of the contact This feature is also noted by Johnson  for similar contacts These peaks however are not transmitted through the skins To characterise the degree to which the skins are able to spread the load we need to compare the width of the contact patch  and the width over which there are signicant pressures transmitted across the interface This latter width can be characterised by a length % given when the pressures have positive non zero values Results show that for the beam B with a rigid skin Fig  the pressure width % of the pressure distribution in the interface is considerably more than the contact width  and does not change signicantly with roller diameter or contact width The skin is su ciently rigid to spread the local pressure distribution out For beam A with a more exible skin Fig  the width of the pressure distribution in the interface corresponds to that of the contact The ratio % can be used to measure how well the contact load has been spread out For a very exible skin this will equal  while for a very rigid skin this will tend to a large value which will depend on the roller radius By considering a range of skin sti nesses changing the thickness t orand Youngs modulus E f this method can be used to nd the variation of % for a range of beam spreading lengths for the two roller diameters  and  mm This is illustrated in Fig  This gure shows that for very exible skins with small the width of the contact equals the width of the region in which there are signicant pressures across Chapter  Indentation Resistance of Sandwich Beams 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Load distribution −0.1 −0.05 0 0.05 0.1 0 0.2 0.4 0.6 0.8 1 Normal stresses in the interface δ ∆ N o r m a l i s e d q t   c c N o r m a l i s e d  z z   c c x LL R  mm R  mm Figure   Distributions of contact stresses q t  cc and the corresponding normal stresses in the top skincore interface  zz  cc normalised by the cores outofplane compressive strength for beam A with   mm Chapter  Indentation Resistance of Sandwich Beams  0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Load distribution −0.1 −0.05 0 0.05 0.1 0 0.2 0.4 0.6 0.8 1 Normal stresses in the interface ∆ δ N o r m a l i s e d q t   c c N o r m a l i s e d  z z   c c x LL R  mm R  mm Figure   Distributions of contact stresses q t  cc and the corresponding normal stresses in the top skincore interface  zz  cc normalised by the cores outofplane compressive strength for beam B with   mm Chapter  Indentation Resistance of Sandwich Beams the interface with %   as expected As the spreading length increases the distribution of pressure in the interface broadens with respect to the contact width with larger % This change depends on roller diameter If we take the transition from exible to rigid behaviour when % equals  then this occurs for R equal to  and  for the  and  mm rollers respectively This case study demonstrates the usefulness of the spreading length in character ising the beam exibility It also allows us to answer the questions posed in section  relating to the assumptions commonly used to model the contact Firstly for practical beams with large the distribution of pressure over the contact is not important and the assumption that the external load is applied uniformly across the beam will be reasonable However this contact distribution will not be transmitted to the in terface and the core unchanged By contrast Fig  shows that it is essential to model beams with sti skins accurately in order to get a reasonable estimate of the stress distribution in the core The commonly used assumption that the failure can be predicted using Failure line load  Cores out of plane compressive strength  cc   Contact width  will be poor unless a judicous choice of  is made based on ex perimental measurements In the nal section  we summarise the conclusions from this chapter including guidelines of a more accurate failure analysis  Concluding Remarks The benets of using the high order sandwich beam theory HOSBT to analyse the behaviour of sandwich beams under indentation are presented The capabilities of HOBST are veried experimentally by Surface Displacement Analysis The symbolic manipulation of the governing equations of HOSBT is used to extract a spreading length  This is a property of a sandwich beam depending mainly on the skins exural sti ness and characterising the susceptibility of the sandwich beam to indentation loads A further manipulation of HOSBT gives an insight into the contact mechanics for loading of a beam by a cylindrical indenter The way in which the contact pressure is transmitted through the core is examined A case study shows the fact that sandwich beams used as standard in industry have skins which are rigid enough to spread the external loads The most important conclusion from this section is that the maximum stresses  zz which are responsible for indentation failure cannot be predicted in a straightforward way for sandwich beams with rigid skins even if the contact width  is known For Chapter  Indentation Resistance of Sandwich Beams 0 1 2 3 4 0 5 10 15 20 ∆ / δ λ (mm) Flexible skins Rigid skins R = 5 mm R = 10 mm Figure  Dependence of spreading eect on rollers radius R and determination on how exible or rigid are the skins with respect to the indentation resistance of the sandwich beam Chapter  Indentation Resistance of Sandwich Beams sandwich beams with very exible skins the approximation of dividing the total line load W by  can give us reliable estimation of the failure stresses in the skin core interface However for commercially applied sandwich congurations the former case is the dominant Thus in practice one should follow the following steps for indentation failure analysis of sandwich beams  For a particular indenters size and shape do one test to measure the  just before failure since  develops with the increase ofW  This value for  can be assumed for other beam geometries  Use the approximation that the external stresses are uniform within this measured  length For sandwich beams with small this approximation is good while for sandwich beams with large the exact distribution of contact load does not a ect the core stresses signicantly  Use high order sandwich beam theory and apply the equations   and  to calculate the stress eld in the core Compare them with the maximum allowables to predict failure This approach is adopted in the next chapter to undertake an indentation failure anal ysis Chapter  Indentation Failure Analysis  Introduction If we assume that the behaviour of a sandwich beam is elastic up to failure the high order sandwich beam theory can be used to calculate the stresses in the core due to the indentation loading This can then be compared with the cores out of plane com pressive strength However a more accurate failure criterion for the core is determined from biaxial tests on Nomex honeycomb cores and used in this analysis Finally short beam  point bending tests with three di erent central roller diameters verify the the oretical predictions and show the importance of the skin exural rigidity on the beams overall strength  Failure Envelope for Nomex Honeycombs A straightforward way to calculate the failure load by indentation for honeycomb cores is by equating the maximum of the out of plane normal stresses  zz with the allowable stresses  cc  However the experiments in Chapter  showed that the interaction of core shear and out of plane stresses was important The concept of the indentation failure due to these combined stress components is depicted in Fig  Therefore before per forming an indentation analysis we investigate both theoretically and experimentally in this section the two dimensional failure envelope of Nomex honeycombs under shear and out of plane compression loading and we dene a new failure criterion References to biaxial loading tests for cellular materials are not common in the literature Gibson et al  have performed a series of tests on a variety of foams under biaxial axisymmetric and hydrostatic loading conditions Zhang and Ashby  have investigated the in plane biaxial buckling behaviour of Nomex honeycombs More  Chapter  Indentation Failure Analysis                            Figure  Combined indentation failure mechanism recently Stronge and Klintworth  have studied the yield surfaces for honeycombs under biaxial macroscopic stresses In this chapter Nomex honeycombs are tested under combined shear and compres sive out of plane loading For this purpose we used the Arcan test rig as illustrated in Fig  This geometry was adapted for shear testing of composite materials by Ar can et al  and extended to biaxial testing of butter y shaped specimens by Voloshin and Arcan  The rig consists of two pairs of plane circular S shaped parts with an tisymmetric cutouts To accommodate honeycomb specimens we added two loading platens The bottom pair of semicircular xtures are gripped to the frame of a servo hydraulic testing machine The upper pair are mounted to the load cell and actuator transferring the load to the specimen which is attached between the two rectangular plates To ensure a uniform stress through the specimen the grips are clamped rather than being pivoted since Marlo s nite element analysis  shows that stress uni formity under biaxial conditions is not good The perimetric holes in the semicircular plates allow a change of the angle  between the plane of the specimen and the loading direction and therefore the ratio tan of compression to shear applied to the speci men between the central plates The specimens are rectangular plates of no more than  mm in width or length The success of such tests depends on being able to attach the specimen to the loading platens e ectively In this case the at external surface of the skins can be easily attached with standard acrylic adhesive Specimens were prepared by cutting rectangular pieces from each of the available sandwich panels of  di erent densities as described in Chapter  Although the aim of this work is to examine the behaviour of the honeycomb core the presence of the skins does not pose any di culties on the contrary they restrict movement of the honeycomb walls relative to the grips and help attachment of the specimen to the rig For each core density tests were conducted for Chapter  Indentation Failure Analysis  P ϕ Fitted to crosshead Fitted to frame A A′ Section A-A′ Specimen Grip Pin∅200mm P 15mm Specimen Loading platens Loading platen 10mm 50mm Figure  The Arcantype rig Chapter  Indentation Failure Analysis  both orientations of the honeycomb ribbon longitudinal and transverse Loading was monotonic up to failure with a constant displacement rate of  mmmin In order to cover a wide range from nearly pure compression to nearly pure shear tests were made at three angles of    o   o and  o  Fig  shows the corresponding load paths Failure for each of the shear angles  denes one point on the failure surface with corresponding values of the core normal and shear stresses  zz and  x at failure In Fig  photos of the three angle setups and typical load de ection curves are presented The de ection is measured from the crosshead displacement of the machine ϕ = 81.5° C om pr es si ve s tr es se s, σ zz Shear stresses, τx Pure compressive strength, σcc Pure shear strength, τcs Failure surface ϕ = 51.5° ϕ = 21.5° σzz τx Load path Figure  Load paths and determination of failure envelope The failure peak loads give the combination of normal compressive load P sin and shear load P cos which leads to failure Assuming that these loads are distributed uniformly over the external surfaces of area bL of both skins the compressive and shear stresses in the core are given by P sin bL and P cos bL respectively The calculated failure stresses  zz and  x are normalised by the cores out of plane com pressive strength  cc and shear strength  cs respectively as given by equations  a and b Normalised stresses  zz  cc are plotted against  x  cs for every Chapter  Indentation Failure Analysis  0.0 0.5 1.0 D eflection (m m ) 0 4 8 12 L oa d (k N )    o 0.0 0 .5 1.0 D eflection (m m ) 0 2 4 6 8 L oa d (k N )    o 0.0 0 .5 1.0 D eflection (m m ) 0 2 4 6 L oa d (k N )    o Figure  The three angle setups and the corresponding loaddeection curves here for specimens with   kgm core density Chapter  Indentation Failure Analysis  angle core density and both honeycomb ribbon directions in Fig  Except for the honeycombs with density  kgm the failure envelopes are well approximated by a linear failure criterion given by  zz  cc  x  cs    as illustrated in Fig  by the dashed line The inconsistency observed in the  kgm honeycombs is due to an inconsistency in the measured shear stresses and does not a ect the accuracy of our calculations Thus equation  allows us to dene a failure criterion for combined loading cases as required for an indentation failure analysis  Failure Analysis with HOSBT In section  the basics of high order sandwich beam theorys implementation to sand wich beam bending behaviour analysis were presented Here the equations needed for a failure analysis for honeycomb sandwich beams loaded under  point bending are assembled To simplify our calculations we assume that the external loads applied by the central roller are uniform within a given width  in the midspan of the beam This assumption is discussed in section  So for a uniformly distributed load applied in the midspan of the beam the Fourier coe cient C qt m cf eqn  reads C qt m  W m sin m   sin m L  or C qt m W # C qt m  where W is the total line load For the failure analysis we need to know the stress eld in the core top skin interface ie the out of plane normal stresses  zz and  x see Fig b Substitution of the above coe cients C qt m into equation  gives the Fourier coe cients C wt m C wb m and C  x m for the top and bottom skin de ections and for core shear stresses C wt m  C qt m R D f  E c  R R   C wb m  C qt m R  D f  E c  R R   C  x m  C qt m A f C t R R  Chapter  Indentation Failure Analysis  0.0 0 .2 0.4 0.6 0.8 1 .0 R el. S hear S tren gth , τx/τcs 0.0 0.2 0.4 0.6 0.8 1.0 R el . C om pr es si ve S tr en gt h , σ zz / σ cc C ore density = 29 kg/m 3 Long Trans 0.0 0.2 0.4 0.6 0.8 1.0 1 .2 R el. S hear S tren gth , τx/τcs 0.0 0.2 0.4 0.6 0.8 1.0 R el . C om pr es si ve S tr en gt h , σ zz / σ cc Core dens ity = 48 kg/m 3 Long Trans 0.0 0 .2 0.4 0.6 0.8 1 .0 R el. S hear S tren gth , τx/τcs 0.0 0.2 0.4 0.6 0.8 1.0 R el . C om pr es si ve S tr en gt h , σ zz / σ cc C ore density = 64 kg/m 3 Long Trans 0.2 0.4 0 .6 0.8 1.0 1.2 1 .4 R el. S h ear S tren gth , τx/τcs 0.0 0.2 0.4 0.6 0.8 1.0 R el . C om pr es si ve S tr en gt h , σ zz / σ cc C ore dens ity = 128 kg/m 3 Long Trans Figure  Failure envelopes for Nomex honeycombs The dashed line corresponds to the linear failure criterion given by equation   Chapter  Indentation Failure Analysis  Then from equations   and  the required stresses are given as follows  zz x  W M X m # C qt m E c D f  E c  cm L A f C t R R   sin mx L W # zz x   x x  W M X m # C qt m A f C t R R  cos mx L W # x x  The variation of the stresses  zz x and  x x with position x along the beam and with load W can be represented in   space by lines such as that shown dashed in Fig  For every W the lowest part of the curve corresponds to the edge of the beam x   where shear stresses are dominant but still there are some out of plane normal stresses hence the curve does not touch the x axis Towards the midspan of the beam ie the upper section of the curve the out of plane normal stresses increase while the shear stresses decrease falling to zero at x  L As W increases the stresses increase until the failure envelope shown by a solid straight line is reached at a failure loadW o  This failure load will di er signicantly from that based on pure compressive loading N or m al s tr es se s, σ zz Shear stresses, τx Failure surface Location of failure, xf ⊆ [0, L/2] x = 0 (at end of beam) x = L/2 (at centre of beam) σcc τcs Increasing x Increasing W Locus of stresses in beam Figure  Combined failure criterion Chapter  Indentation Failure Analysis  W o can be calculated after substituting equations  and  into equation  and nding the minimum value of the following expression W   cc # zz x cc  cs # x x   Experimental Work To validate the theoretical predictions for the indentation failure loads  point bending tests were performed on short sandwich beams Beams  mm long and  mm wide were cut in both honeycomb ribbon directions from Nomex core sandwich panels Panels had nominal core densities    and  kgm with a  mm cell size and  kgm with a  mm cell size To observe in detail the vertical de ections during loading we painted one of the specimens long cross sections so that the surface of the honeycomb walls became black and the edges of both skins and of honeycomb walls became white The experimental setup is depicted in Fig  It consists of a video camera and recorder which keep track of the high contrast black and white cross sections of the specimens during loading The specimens are simply supported by two rollers of  mm diameter The distance between them is  mm  point bending tests were performed with  di erent diameters   and  mm of the central roller in a servohydraulic testing machine The loading rate was  mmmin As well as the midspan top skin de ection and the load measurements which are given by the machines transducers we attached in the midspan of every specimen a clip gauge to measure the relative displacement between the skins camera clip gauge beam measured core compression Figure  Experimental setup The experimental measurements of midspan core compression from the clip gauge of load from the load cell and of top skin de ection from the crosshead displacement Chapter  Indentation Failure Analysis  transducer are plotted for every central roller size and every core density in Figs A A and A in Appendix A A representative combined plot is shown here in Fig  The bottom skin de ection vs top skin de ection curves show clearly the di erent response of the two skins the top skins exhibit larger absolute de ections than the bottom skins This is due to the compression of the core as shown clearly by the core compression vs top skin de ection curves Core compression is almost linear up to failure For all core densities the core compression is more severe for the beams with longitudinal honeycomb ribbon direction due to the higher shear sti ness of the Nomex honeycomb in the longitudinal direction than in the transverse direction This di erence conrms the importance of the core shear sti ness in indentation failure The di erence between longitudinal and transverse ribbon directions is also re ected in the line load de ection curves in the former case the beam has a higher overall exural rigidity The letters A B C and D in the upper plot of Fig  indicate critical stages of the loading procedure ie before loading just before failure just after failure and well after failure At these points video snapshots were captured These are presented for every central roller size and core density in Figs A A and A in Appendix A Here we present some key photos for discussion Fig  shows two video images captured just before failure position B in Fig  during  point bending loading with a mm diameter central roller It is apparent how the low density core is deformed with signicant shear causing the honeycomb cell walls to buckle elastically Fig  shows the e ect of core density after failure when the total de ection is approximately  mm The high out of plane sti ness of the high density core does not allow the damage to propagate within the core and the skins fail eventually by compressive macrobuckling For the same reason the indented top skin follows tightly the curvature of the roller for high density cores while for low density cores the top skin is indented over a wider area without following the indenters curvature see Fig  After comparing the corresponding curves amongst Figs A A and A we realise that there are no signicant di erences due to the central rollers size This is in accor dance with the prediction made in Chapter  This insensitivity to the indenters size is conrmed by comparing the corresponding failure peak loads in Fig  The same gure also shows the satisfactory theoretical predictions using the high order sandwich beam theory in conjunction with the mixed failure criterion The values of out of plane core compressive strength and core shear strength calculated in section  for every core density and ribbon direction are substituted in equation  and the minimum value of this provides the theoretical predictions of failure loads which are represented by the lines in Fig  The advantage of using the mixed failure crite Chapter  Indentation Failure Analysis 0.0 0 .5 1 .0 1 .5 2 .0 T op skin deflection (m m ) 0.0 0.5 1.0 1.5 2.0 B ot to m s ki n d ef le ct io n ( m m ) 0.0 0.1 0.2 0.3 C or e co m pr es si on ( m m ) 0 2 4 6 8 L in e lo ad , W ( kN /m ) A B C D Figure   Typical experimental results for sandwich beam with  kgm core density loaded by a roller with diameter of mm Midspan bottom skin deection midspan core compression and line load are plotted against midspan top skin deection Lines legend  longitudinal and   transverse honeycomb ribbon direction Chapter  Indentation Failure Analysis  29 kg/m3 128 kg/m3 Figure   The prefailure inuence of core density Video images captured just before failure during point bending loading with a mm diameter central roller 29 kg/m3 128 kg/m3 Figure   The postfailure inuence of core density Video images captured after failure  mm total deection during point bending loading with a mm diameter central roller 29 kg/m3 64 kg/m3 Figure  The postfailure inuence of core density on extent of damage Video images captured after failure  mm total deection during point bending loading with a mm diameter central roller Chapter  Indentation Failure Analysis  rion instead of simply checking when the maximum out of plane stresses in the top skin core interface reach the out of plane compressive strength of the honeycomb core is illustrated in Fig  It is shown that without the correction of the mixed crite rion the theoretical predictions are larger than the experimental data measurements of the failure loads especially as the core density increases Also the use of the mixed failure criterion catches the di erent indentation resistance of sandwich beams due to the di erent honeycomb ribbon direction The sandwich beams with longitudinal honeycomb ribbon direction have slightly higher strengths than those with a transverse ribbon direction The di erence increases with the core density  Concluding Remarks A systematic approach has been developed to determine the failure load of sand wich honeycomb structures under indentation loading Firstly the failure envelope for Nomex honeycombs under simultaneous out of plane compression and shear has been determined by biaxial tests using an Arcan rig A linear dependence on pure compression and shear proves to be a good approximation for the failure envelope of Nomex honeycombs The implementation of high order sandwich beam theory allows a more accurate failure analysis for sandwich beams subjected to localised loads Using the failure envelope determined by the biaxial tests a mixed failure criterion has been introduced to predict the indentation failure caused by the simultaneous action of out of plane compressive and shear stresses exerted in the vicinity of a localised load This criterion can predict failure that ranges from pure core crushing to pure core shear The short beam bending tests validated the theoretical predictions of high order beam model Video captures of the deformed side cross section illustrate the involve ment of core shear in the elastic behaviour of sandwich beams with low density cores Also these tests showed the di erent post failure damage extent between sandwich beams with di erent core densities They showed that the mixed failure criterion of fers a signicant improvement in predicting indentation strength as compared with models which do not include combined loading This model is particularly needed in high core densities which induce more severe stress elds in the core Moreover this approach explains observed di erences in behaviour for longitudinal and transverse ribbon directions Chapter  Indentation Failure Analysis  0 40 80 120 C ore density , (kg/m 3) 0 20 40 60 L in e lo ad , W o (k N /m ) L ong Inden ter T rans d iam eter 20 m m 10 m m 6 m m Failure C riteria O nly com pression W ith m ixed shear an d com pression L ongitudinal Transverse Figure  Theory lines vs experimental results symbols for failure line load W o Chapter  Failure Maps using HOSBT  Introduction In Chapter  failure mode maps for Nomex honeycomb sandwich beams with GFRP laminate skins are produced using simplied beam models and honeycomb mechanics However the simplied model for indentation failure prediction used in chapter  can not be used with great condence especially where core crushing and shear interact In the previous chapter a theoretical model based on high order sandwich beam theory HOSBT and a mixed failure criterion for Nomex honeycombs has been shown to give accurate predictions of the peak loads when sandwich beams made of GFRP skins and Nomex honeycomb fail by indentation Here the use of high order beam theory is extended as a compact computational tool which determines the maximum stresses in the skins and the core Thus by comparing the calculated maximum stresses with the relevant allowable stresses we nd the range of design parameters where each failure mode is dominant Frostig and Shenhar  have conducted an analogous failure analysis for foam core sandwich panels by using the HOSBT but for failure patterns and criteria di erent than those we present here In particular the failure mode of skin wrinkling is not examined Furthermore specically for the failure mode of indentation the failure criterion for the core was based on the maximum principal stresses Although this approach is appropriate for foams that are isotropic in case of honeycombs which are orthotropic a more suitable failure criterion is that based on the normal and shear stresses exerted in the top skin core interface Thus in this chapter we make use of the mixed failure criterion introduced in the previous chapter Based on the above approach new failure mode maps for Nomex honeycomb sandwich beams are constructed and provide the basis for design optimisation carpet plots  Chapter  Failure Maps using HOSBT  Reconstruction of Failure Maps In chapter  we presented the basics of high order beam theorys implementation to sandwich beam bending behaviour analysis By assuming a linear elastic response for the sandwich beam up to failure it is feasible to use the appropriate equations derived by high order beam theory to calculate the maximum stresses in the skins and the core under  point bending The advantages of such approach are  use of a compact computational tool to calculate all the maximum stresses in the sandwich beam and  ability to use mixed failure criteria for the core to model more accurately the indentation failure as shown in the previous chapter The experimental results presented in Chapter  show that the short beams nally failed by core crushing under the central load but with an increased interaction of core shear as the core density gets smaller Thus in this chapter the core shear is not considered as a separate failure mode The failure in the honeycomb core is treated as one mixed failure mode and denoted as core indentation The theoretical determination of the peak load for this mode is presented thoroughly in the previous chapter and gives W o  min  cc # zz x cc  cs # x x  These calculations are based on the assumption that the external loads applied by the central roller are uniform within a measured width  in the midspan of the beam As explained in Chapter  the rigidity of the laminate skins used in the examined sandwich beams is high enough that the calculations are insensitive to the contact area between the indenter and the top skin It is assumed in this chapter that this remains true For the skin failure we need to know the maximum normal stresses  txx exerted on the cross section of the top skin Equation  after substitutions gives  txx x  W M X m # C qt m C t R R  m L sin mx L W # txx x  This is a function of the x coordinate and the maximum value is reached as expected in the midspan of the beam ie for x  L max txx W # txx xL  Now the failure loads for every failure mode are calculated by equating these maxi mum stresses with the allowable stresses  fY  fw for skin compressive yield and skin wrinkling respectively Chapter  Failure Maps using HOSBT  As in Chapter  a failure map for this type of sandwich beam is constructed by nding the minimum failure loads for each of the failure modes in Table  for each value of the design parameters ie tL and  c  s  Then we plot the minimum load Table  Expressions for peak failure loads Top skin yield W o   fY # txx Face wrinkling W o   fw # txx Indentation W o  min cc zz x  cc  cs  x x W o against the skin thickness over span ratio tL and the core relative density  c  s  The projection of the lines where two modes intersect provides the boundaries for the corresponding failure mode map as depicted in Fig  By comparing this map with those given in Fig  one can see that the failure modes of pure core crushing or shear are now replaced by a core indentation region The upper part of the indentation area higher core densities corresponds to core crushing under the vicinity of the localised loads whereas the lower part lower densities is characterised mainly by core shear A comparison between the failure loads calculated with HOSBT and the exper imental results presented in Chapter  is depicted in Fig  for all four examined core densities The theoretical predictions for core crushing are more accurate for the new failure prediction especially for the lower core densities of  or  kgm cf Fig  Also this new approach predicts the observed di erences in behaviour for longitudinal and transverse honeycomb ribbon direction Although the high order sandwich beam theory provides a robust and compact computational tool for all the modes there are two drawbacks rstly when the skin fails the model calculates the same failure load for the longitudinal and transverse direction of the honeycomb core and secondly the predictions for the mode of skin wrinkling are overpredicted espe cially closer to the transition from skin wrinkling mode to indentation mode The latter is due to fact that HOSBT allows for local bending of the top skin before failure and therefore the calculated in plane normal stresses  txx in the midspan are lower than those calculated by simplied beam models Indeed the distribution of in plane normal stresses  txx along the beam is triangular when predicted by simple beam theory while HOSBT gives a curve tangential to this triangular distribution close to midspan This discrepancy causes the di erences in prediction Chapter  Failure Maps using HOSBT 10−4 10−3 10−2 10−2 10−1 100 102 103 104 105 106 Relative density  c  s Face thickness  span tL F a i l u r e l i n e l o a d W o N  m  10−4 10−3 10−2 10−2 10−1 100 336 695 1440 2980 6160 12700 26400 54600 skin thickness / span re la tiv e de ns ity newmap2.m Skin wrinkling Skin yielding Core indentation R e l a t i v e d e n s i t y  c   s Face thickness  span tL Figure  Improved failure map for Nomex honeycomb sandwich beams Each contour represents sandwich beams of equal strength in Nm Chapter  Failure Maps using HOSBT 10 100 P ea k L oa d, W o (k N /m ) 0.001 0.010 F ace th ickness / span, t/L Skin Y ie ld C ore Indenta tion Core D ensity = 128 kg/m 3 0.001 0.010 F ace th ickness / span, t/L 10 P ea k L oa d, W o (k N /m ) Skin Y ie ld or W rink ling C ore Indentation C ore D ensity = 64 kg /m 3 (a) ρc /ρs = 0.160 (b) ρc /ρs = 0.080 0 .001 0.010 F ace th ickn ess / span , t/ L 10 P ea k L oa d, W o (k N /m ) C ore Indentation Skin W rinkling C ore D ensity = 48 kg/m 3 1 10 P ea k L oa d, W o (k N /m ) 0.001 0 .010 F ace th ickness / span , t/L Skin W rinkling C ore Indentation C ore Density = 29 kg/m 3 (c) ρc /ρs = 0.066 (d) ρc /ρs = 0.040 Figure  Comparison with the experimental results from Chapter  Lines show the pre dictions of HOSBT failure analysis and the symbols represent the experimental data cf Fig  Solid lines and symbols correspond to the longitudinal rib bon direction while dashed lines and hollow symbols correspond to the trans verse ribbon direction Chapter  Failure Maps using HOSBT  Optimisation Carpet Plots When designing a load bearing structure the most important functional requirements are sti ness strength and weight In general these are attributes that we want to maximise or minimise Minimum weight design is considered by Triantallou and Gibson  for foam core sandwich beams and extended in Gibsons book  In both cases indentation is not examined as a possible core failure mode only core shear failure is considered They conclude that in minimum weight design of a foam core sandwich beam of a given strength the top skin and core must fail simultaneously In this work we use the failure maps to optimise the design of honeycomb sandwich beams with a graphical methodology Here it is assumed that the failure in the core is by indentation and is a result of the simultaneous e ect of core shear and out of plane compression Fig  Fig  shows a useful practical form of a failure map In this section we consider a typical case study seeking an optimum design for a sandwich beam which is loaded under  point bending and for which the material properties and skin thickness are given The ratio cL and the relative core density  c  s are variables to be optimised Hence it is most useful to present failure maps in terms of cL and  c  s  Fig  presents the data for Fig  in this form showing contours of equal failure line load By adding to this map contours of equal sti ness and mass carpet plots are produced which can be used to nd the optimum design conguration Such a carpet plot is shown in Fig  where for clarity the mass contours and only one strength and one sti ness contour are shown The full set of sti ness contours are plotted in Fig  Two optimisation cases are considered strength limited design or sti ness limited design These constraints are represented as hatched contours in Fig  In addition a geometric constraint is included In this case we suppose that the ratio of core thickness to span cannot exceed     Where a minimum strength constraint is applied the potential choices are restricted to the upper right area of the map By considering the mass contours it is clear that the design which minimises the mass lies at point A on the intersection of the skin wrinkling core indentation region on the strength contour However if other considerations impose a value of cL less than     the value of cL which corresponds to point A this conclusion would no longer hold It may also be desirable to choose a panel which fails by indentation because this is a safer failure mode than the alternatives For the material combination considered the optimum design point lies on the border between core indentation and skin failure for a range of strength constraints This accords with the relevant conclusion of ref  Chapter  Failure Maps using HOSBT 10-2 10-1 10 -2 10-1 10 0 Core thickness / span, c/L R el at iv e de ns it y, ρ c / ρ s skin yielding skin wrinkling increasing strength strength contours core indentation Figure  Strength contour plot corresponding to Fig  10 -2 10-1 10 0 Core thickness / span, c/L R el at iv e de ns it y, ρ c / ρ s skin yielding skin wrinkling stiffness contours increasing stiffness core indentation 10-2 10-1 Figure  Stiness contour plot Chapter  Failure Maps using HOSBT  and  that the minimum design is such that the skin and core fail at the same peak load The above illustration assumes that sti ness criteria are always satised Consider instead a sti ness limited design with a sti ness constraint shown by the hatched contour in Fig  Now the contours of mass suggest that the beam with minimun mass would be that with the minimum available relative density of the core In practice some other constraint for example a geometric constraint a strength constraint or a manufacturing constraint will limit the extent to which the core density can be reduced Indeed this sti ness limited case study illustrates the need to model strength of these panels as this will be necessary to determine the optimum design 10-2 10-1 100 Core thickness / span, c/L R el at iv e de n si ty , ρ c /ρ s decreasing mass st re n gt h co ns tra in t ge om et ric co ns tra in t A core indentation skin wrinkling mass contours skin yield st iff ne ss co n st ra in t 10-2 10-1 Figure  Optimisation carpet plot for tL     with  c  s and cL as design parameters Chapter  Failure Maps using HOSBT   Concluding Remarks The high order sandwich beam theory can be used not only to predict indentation failure loads but also the failure modes of skin failure enabling the construction of failure mode maps This approach o ers the advantage of using only one computa tional tool to calculate the failure loads for every failure mode The predictions follow satisfactorily the experimental data However the disadvantages of using the HOSBT for failure maps are overprediction of the failure loads for skin wrinkling and inability to predict the di erent behaviour between the two honeycomb ribbon directions when the top skin fails either by yield or wrinkling Finally it is shown that failure maps can help with the preliminary design of sandwich beams under bending by superimposing contours of mass sti ness and strength Chapter  Conclusions and Future Work This thesis addresses theoretical and experimental aspects of the design of honeycomb sandwich panels The research focuses on sandwich beams made of cross ply GFRP laminate skins with a Nomexphenolic resin honeycomb core a widely used combina tion for aircraft ooring panels and freight loading pallets Indentation resistance of such sandwich panels with transversely exible cores is an important factor in their design An accurate model of the elastic deformation of such sandwich beams under three point bending has been derived In particular this model includes the local behaviour under the central load allowing indentation failure to be examined  Conclusions Firstly the behaviour of sandwich beams of di ering lengths and core densities is con sidered to examine the importance of the possible failure modes including failure by skin yielding skin wrinkling intra cell buckling core shear and indentation At this initial treatment indentation is treated in an empirical way using measurements of the bearing area at the load points Previous research on honeycomb mechanics and simple beam models has been combined to derive failure mode maps for  point bending with axes as the core relative density and the ratio of the skin thickness to span length These maps are based on those of Triantallou and Gibson  who focused on sand wich beams with ductile aluminium skin and isotropic foam cores This thesis appears to be the rst attempt to construct maps for sandwich beams with laminate skins and honeycomb core Since commercial panels are generally provided with standard skin thicknesses but with di ering core thickness and density alternative maps with these two core variables are presented since these will be more useful for a beam designer or manufacturer Although the maps are generated for three point bending the method can straightforwardly be applied to other loading geometries for example four point  Chapter  Conclusions and Future Work  bending Experimental results for failure under three point bending are summarised as fol lows A transition from face yielding to intra cell buckling for long span beams was observed for a honeycomb cell size above a critical value The measured peak loads were dependent on the direction of the honeycomb ribbon This di erence is due to the shear anisotropy of the honeycomb illustrating the important role that core shear plays in the bending behaviour of these sandwich beams Experimental results veried satisfactorily the predicted failure loads The boundary between skin and core failure on the failure mode maps was also predicted with good accuracy However failure near the load points due either to core shear or core indentation was not modelled well A high order sandwich beam theory HOSBT was implemented to provide a better deformation model of localised e ects under concentrated loads and so to produce an improved indentation failure analysis for honeycomb sandwich beams The benets of using the high order sandwich beam theory to analyse the behaviour of sandwich beams under indentation are presented and veried experimentally by measurements of the core deformation close to an indenter HOSBT is used to extract a characteristic spreading length  This is a property of a sandwich beam depending mainly on the skins exural sti ness and characterising the susceptibility of the sandwich beam to indentation loads Small values of correspond to sandwich beam with very exible skins which are transparent to the external loads Large values of indicate rigid skins that restrict the transmission of localised loads to the core HOSBT is further used to give an insight into the contact mechanics for a beam loaded by a cylindrical indenter The way in which the contact pressure is transmitted through the core is examined A case study shows the fact that sandwich beams used as standard in industry have skins which are rigid enough to spread the external loads This spreading e ect allows to predict the indentation behaviour of a sandwich beam with rigid skins without having to model accurately the contact between the indenter and the top skin The assumption of a uniform distribution over a roughly estimated width which depends on the size of the indenter and the use of the high order sandwich beam theory can provide reliable predictions of the stress eld in the core The most important conclusion from this analysis is that the maximum normal stresses in the top skin core interface which are mainly responsible for indentation failure cannot be predicted in a straightforward way for sandwich beams with rigid skins even if the contact width is known For sandwich beams with very exible skins the approximation of dividing the total line loadW by  can give us reliable estimation of the failure stresses in the skin core interface However for commercially applied Chapter  Conclusions and Future Work sandwich congurations the skin is rather rigid and the exible skin approximation cannot be used A systematic approach has been developed to determine the failure load of sandwich honeycomb structures under indentation loading Biaxial tests of Nomex honeycombs using an Arcan rig show that a linear dependence on pure compression and shear proves to be a good approximation for the failure envelope of these honeycombs Using this failure envelope and the core stress eld determined by HOSBT a mixed failure crite rion has been introduced to predict the indentation failure caused by the simultaneous action of out of plane compressive and shear stresses exerted in the vicinity of a lo calised load This criterion can predict failure that ranges from pure core crushing to pure core shear Short beam bending tests validated the theoretical predictions of high order beam model and showed that the mixed failure criterion o ers a signicant improvement in predicting indentation strength as compared with models which do not include combined loading This new approach is particularly needed in high core densities which induce more severe stress elds in the core It also predicts observed di erences in behaviour for longitudinal and transverse ribbon directions Video captures of the deformed side cross section of the tested sandwich beams illustrate the involvement of core shear in the elastic behaviour of sandwich beams with low density cores Also they showed the di erent post failure damage extent between sandwich beams with di erent core densities the higher the core density the less damage propagates into the core and skins can su er eventually macrobuckling failure The high order sandwich beam theory can also be used to predict the failure modes of skin failure enabling the construction of failure mode maps This approach o ers the advantage of using only one computational tool to calculate the failure loads for every failure mode The predictions follow satisfactorily the experimental data However the disadvantages of using the HOSBT for failure maps are overprediction of the failure loads for skin wrinkling and inability to predict the di erent behaviour between the two honeycomb ribbon directions when the top skin fails either by yield or wrinkling Finally it is shown that failure maps can help with the preliminary design of sandwich beams under bending by superimposing contours of mass sti ness and strength  Future Work With this study a framework of failure analysis of Nomex honeycomb panels has been established and provides the basis for further investigation in the following areas Chapter  Conclusions and Future Work  Bending tests should be done with sandwich beams with di erent skin thicknesses and core densities than those tested in this research work in order to validate further the applicability of the proposed methodologies for failure analysis and for failure map construction The failure envelope for biaxial loading of Nomex honeycombs determined in section  should also be investigated for other core materials In section  a parametric study is conducted to dene the dependence of the spread ing length on the material and geometric properties of a sandwich beam The in u ence of each parameter has been determined qualitatively Further work on this area could attempt to derive a single normalised curve C zz m with respect to Lm instead of all the curves in Fig  that accurately captures the indentation resistance behaviour of a wide range of sandwich beam designs This approach would lead to simple formu lae relating to the material and geometric properties of a sandwich beam Miller  adopts such an approach to model the indentation behaviour of foamed metals A nite element analysis of a model see Fig  of a rigid indenter applying loads to a simply supported beam would be useful to determine the contact pressure between the indenter and the top skin and the corresponding normal stresses transmitted to the core These stresses can be compared with those calculated by the high order beam theory model in section  In Chapter  we showed the ability of the spreading length parameter to char acterise the static indentation resistance of sandwich beams It should be possible to compare values with experimental results on dynamic indentation resistance and investigate the applicability of as a quality factor in sandwich panel manufacturing For this purpose the software code used in section  to calculate could be provided to industry for assessment of its practical usefulness Finally the methodology for failure analysis proposed in this study should be used to explore the benets of unsymmetrical sandwich beams with bottom skins thinner than the top ones This option would exploit rstly the fact that composite laminates have approximately tensile strength twice as much their compressive and secondly the higher indentation resistance that a thicker top skin can provide Chapter  Conclusions and Future Work ABAQUS 1 2 3 DISPLACEMENT MAGNIFICATION FACTOR = 6.77 ORIGINAL MESH DISPLACED MESH RESTART FILE = indent1 STEP 1 INCREMENT 1 TIME COMPLETED IN THIS STEP 2.220E-16 TOTAL ACCUMULATED TIME .000E+00 ABAQUS VERSION: 5.6-1 DATE: 17-APR-98 TIME: 03:53:10 Figure  Vertical displacements calculated by Abaqus nite element analysis Appendix A Experimental Results  Appendix A Experimental Results Figure A For mm diameter roller midspan bottom skin deection midspan core com pression and line load variation curves with respect to midspan top skin deec tion Lines legend  longitudinal and   transverse honeycomb ribbon direction Appendix A Experimental Results Figure A For  mm diameter roller midspan bottom skin deection midspan core compression and line load variation curves with respect to 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