Geometric Charges and Nonlinear Elasticity of Two-Dimensional Elastic Metamaterials

Proc Natl Acad Sci U S A. 2020 May 12;117(19):10195-10202. doi: 10.1073/pnas.1920237117. Epub 2020 Apr 29.

Abstract

Problems of flexible mechanical metamaterials, and highly deformable porous solids in general, are rich and complex due to their nonlinear mechanics and the presence of nontrivial geometrical effects. While numeric approaches are successful, analytic tools and conceptual frameworks are largely lacking. Using an analogy with electrostatics, and building on recent developments in a nonlinear geometric formulation of elasticity, we develop a formalism that maps the two-dimensional (2D) elastic problem into that of nonlinear interaction of elastic charges. This approach offers an intuitive conceptual framework, qualitatively explaining the linear response, the onset of mechanical instability, and aspects of the postinstability state. Apart from intuition, the formalism also quantitatively reproduces full numeric simulations of several prototypical 2D structures. Possible applications of the tools developed in this work for the study of ordered and disordered 2D porous elastic metamaterials are discussed.

Keywords: elastic charges; mechanical metamaterials; nonlinear elasticity.