Solution of the first basic physically nonlinear problem of elasticity theory for anisotropic bodies

Abstract

Abstract The study looks upon the process of the physically non-linear deformation of transversely isotropic homogeneous continuous solid bodies made from composites where the reinforcing elements are far more rigid than the binder. The solution of the physically non-linear problems employs the simplified theory of plasticity as proposed by B.E. Pobedra. The study proposes an approach to the writing out of an explicit solution that builds on the small parameter method. Ilyushin plasticity functions (that fall within the generalized Hooke’s law) are assigned discrete small values, and the resulting equations are then decomposed into power series in small parameters. The values of such small parameters are the measures of the deflection of the non-linear vs. the linear medium. Such decomposition produces an analytical coordinate & small parameters function enabling an immediate solution of the non-linear problem using just one linear elasticity field, which, in turn, is also created exclusively in reliance on the method of boundary states. Below are the results of solutions of test problems featuring a transversely isotropic cylinder with homogeneous boundary conditions. High precision in this case is achieved as early on as the third iteration. In problems with non-trivial boundary conditions, maximal precision is achieved at the first iteration and heavily depends on small parameter values. Each of the problems presented provides a detailed convergence analysis and graphic illustrations of the results.