The von Karman equations for plates with residual strain

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.

26 pages

References in corpus (4)

• Elastic theory of unconstrained non-Euclidean plates
• Morphogenesis of growing soft tissues
• Scaling laws for non-Euclidean plates and the $W^{2,2}$ isometric immersions of Riemannian metrics
• Shell theories arising as low energy Γ-limit of 3d nonlinear elasticity

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