The von Karman equations for plates with residual strain

TL;DR

This paper rigorously derives the Föppl-von Kármán equations for elastic plates with residual strains, providing bounds on the convergence from 3D elasticity and justifying models for growing leaves and active materials.

Contribution

It offers a rigorous derivation and convergence analysis of plate equations with residual strains, applicable to growth, swelling, and other active material behaviors.

Findings

Provides bounds on 3D to 2D elasticity convergence.

Justifies low-dimensional models for growing leaves.

Formalizes derivation procedures for active materials.

Abstract

We provide a derivation of the Foppl-von Karman equations for the shape of and stresses in an elastic plate with residual strains. These might arise from a range of causes: inhomogeneous growth, plastic deformation, swelling or shrinkage driven by solvent absorption. Our analysis gives rigorous bounds on the convergence of the three dimensional equations of elasticity to the low-dimensional description embodied in the plate-like description of laminae and thus justifies a recent formulation of the problem to the shape of growing leaves. It also formalizes a procedure that can be used to derive other low-dimensional descriptions of active materials.