The Föppl-von Kármán equations for plates with incompatible strains

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 giv...

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Bibliographic Details
Published in:Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences Mathematical, physical, and engineering sciences, 2011-02, Vol.467 (2126), p.402-426
Main Authors: Lewicka, Marta, Mahadevan, L., Pakzad, Mohammad Reza
Format: Article
Language:English
Subjects:
Boundary conditions
Calculus Of Variations
Curvature
Elastic plates
Elasticity
Gamma Convergence
Matrices
Non-Euclidean Plates
Nonlinear Elasticity
Plant growth
Plasticity
Scalars
Swelling
Tensors
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Summary: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.
ISSN:1364-5021
1471-2946
DOI:10.1098/rspa.2010.0138