On the bifurcation of the equilibrium state of a three-dimensional isotropic elastic solid under large subcritical deformations: PMM vol. 34, n≗6, 1970, pp. 1113–1125

Abstract

General questions and the derivation of three-dimensional linearized equations for arbitrary subcritical deformations (linearized equilibrium equations in stresses) have been examined in a number of papers [1–11]. Particular forms of the elastic potetial [3,9, 12–16] have been applied in the derivation of the linearized equations in displacements for arbitrary subcritical deformations and in the solution of problems for 3 three-dimensional elastic isotropic solid in the majority of papers. For some of these problems, generala solutions have been constructed in the case of homogeneous subcritical states [3,14]. The linearized equations in displacements for an arbitrary form of the elastic potential have been obtained in [8], and a number of problems have been considered in such an approach [8,17,18]. Following [4], three-dimensional linearized equation in stresses are examined below, where components of the Green's finite strain tensor are chosen as strain characteristics. Linearized equations in displacements are obtained for an arbitrary form of the elastic potential. In the particular case of homogeneous subcritical states, general solutions are constructed which have been expressed in terms of the solution of second order equations for a solid with arbitrary cross-sectional outline. Proceeding from the general solutions, characteristics equations are obtained for a number of problems for an arbitrary form of the elastic potential. Certain forms of the elastic potential utilized in the literature are presented. Numerical examples are considered.