The degree to which the toughness of a lattice material can be enhanced by the suitable placement of multiple phases is explored. To achieve this, the resistance to mode I and mode II crack growth in a two-dimensional (2D), elastoplastic, triangulated lattice is investigated using finite element (FE) simulations. The fully triangulated lattice is idealised as a pin-jointed truss, and each strut has an axial force versus elongation (or shortening) characteristic based on the uniaxial tensile response of an elastoplastic solid with power-law hardening. When the tensile force in the strut attains a critical value, a linear softening law is adopted for the force versus elongation response of the strut to simulate its failure. FE simulations of crack growth in the 2D lattice are performed under small-scale yielding conditions, and the sensitivity of the crack growth resistance curve (curve) to the cell wall strain hardening exponent and cell wall ductility is determined. Three concepts for enhancing the curve of a triangulated lattice are explored: (i) a brittle lattice reinforced by long ductile fibres transverse to the cracking plane, (ii) a bilattice such that a small scale brittle lattice is reinforced by a large scale ductile lattice, and (iii) a 2D version of an interpenetrating lattice wherein a large-scale ductile lattice is bonded at its joints to an underlying small-scale brittle lattice.