Bespoke two-dimensional elasticity and the nonlinear analogue of Cauchy’s relations

Is it possible to design an architectured material or structure whose elastic energy is arbitrarily close to a specified continuous function? This is known to be possible in one dimension, up to an additive constant (Dixon et al., Bespoke extensional elasticity through helical lattice systems, Proc. R. Soc. A. (2019)). Here, we explore the situation in two dimensions. Given (1) a continuous energy function E(C), defined for two-dimensional right Cauchy–Green deformation tensors C contained in some compact set and (2) a tolerance\epsilon > 0, can we construct a spring-node unit cell (of a lattice) whose energy is approximately E, up to an additive constant, with L∞-error no more than \epsilon? We show that the answer is yes for affine Es (i.e., for energies E that are quadratic in the deformation gradient), but that the general situation is more subtle and is related to the generalisation of Cauchy’s relations to nonlinear elasticity.