Incompatible Deformations of Elastic Plates

Abstract

This article considers the methods for mathematical modeling of incompatible finite deformations of elastic plates by using the principles of the differential geometry theory underlying continuously distributed defects. Equilibrium equations were derived by asymptotic expansions of the finite strain measures with respect to two small parameters. One parameter defines the order of smallness of displacements from the reference shape (self-stressed state), while the other specifies the thickness. Asymptotic orders were different for the deflections and displacements in the plate plane, as well as for their derivatives. They were selected in such a way that, with additional assumptions on the possibility of ignoring certain terms in the resulting expressions and the compatibility of deformations, the equations could be reduced to the system of F¨oppl–von Ka´rm´an equations.