A continuum model for nonlinear lattices under large deformations

A continuum model is developed for hexagonal lattices, composed of a set of masses connected by linear axial and angular springs, with nonlinearity arising solely from geometric effects. For a set of lattice parameters, these lattices exhibit complex deformation patterns under uniform loading conditions due to instabilities. A continuum model accounting for these instabilities is developed from explicit expressions of the potential energy functional of a unit cell. This functional is non-convex, it captures the bistable nature of the lattice, and is used to derive its effective constitutive behavior. Finite element simulations of continuum medium illustrate the formation of microstructural patterns with discontinuous displacement gradients, similar to the features observed in nonlinear elasticity and finite deformation plasticity. A comparison of discrete lattice simulations and finite element analysis under general loading conditions illustrates that the continuum model captures the effective behavior due to instabilities within the lattice.