The Foppl-von Karman equations for plates with incompatible strains
The Foppl-von Karman equations for plates with incompatible strains
M. Lewicka, L. Mahadevan, and M. Pakzad, Proceedings of the Royal Society, Lond. (A) , 467, 402-426, 2011.
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Abstract
We provide a derivation of the Föppl-von Kármán equations for the shape of and stresses
in an elastic plate with incompatible or 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 threedimensional 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 with complex non-Euclidean geometries