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.
M.L. was partially supported by the NSF grants DMS-0707275 and DMS-0846996 and by the Center for Nonlinear Analysis (CNA) under
the NSF grants 0405343 and 0635983. L.M. was partially supported by the Harvard NSF-MRSEC and the MacArthur Foundation. R.P. was partially supported by the NSF grant DMS-0907844.
Lewicka, Marta; Mahadevan, L.; Pakzad, Reza.
The Foppl-von Karman equations for plates with incompatible strains.
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