MechE Colloquium: Bespoke Elasticity and the Nonlinear Analogue of Cauchy’s Relations

Abstract: 

Is it possible to design an architectured material or structure whose elastic energy is arbitrarily close to a specified continuous function? This is possible in one dimension, up to an additive constant [Dixon et al., Bespoke extensional elasticity through helical lattice systems, Proc. R. Soc. A. (2019) https://doi.org/10.1098/rspa.2019.0547]. After a review of that result, we explore the situation in two dimensions: Given (i) a continuous energy function E(C), defined for two-dimensional right Cauchy–Green deformation tensors C contained in some compact set, and (ii) a tolerance ϵ > 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 ϵ? 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. If time permits, we will also explore the three-dimensional situation.
[Reference]: Chenchiah IV. Bespoke two-dimensional elasticity and the nonlinear analogue of Cauchy’s relations. Mathematics and Mechanics of Solids. https://doi.org/10.1177/10812865231198204


Biography: 

After undergraduate education at IIT Madras (India), I received a PhD from Caltech (USA) and was a post-doctoral associate at the Max Planck Institute for Mathematics in the Sciences (Germany).
In addition to interests in mathematics, engineering and science, in my leisure time, I enjoy reading about ancient Greek Philosophy and Western medieval thought, especially Thomas Aquinas.