期刊
PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
卷 467, 期 2126, 页码 402-426出版社
ROYAL SOC
DOI: 10.1098/rspa.2010.0138
关键词
non-Euclidean plates; nonlinear elasticity; gamma convergence; calculus of variations
资金
- NSF [DMS-0707275, DMS-0846996, DMS-0907844]
- Center for Nonlinear Analysis (CNA) through NSF [0405343, 0635983]
- Harvard NSF-MRSEC
- MacArthur Foundation
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1142369] Funding Source: National Science Foundation
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [0846996, 1338869] Funding Source: National Science Foundation
We provide a derivation of the Foppl-von Karman 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 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 with complex non-Euclidean geometries.
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