Using the method of boundary states with perturbations to solve physically nonlinear problems of the theory of elasticity

Abstract

Abstract The study looks upon the process of physically nonlinear deformation of isotropic and transversely isotropic homogeneous continuous solid bodies made from fibre composites where the reinforcing elements are far more rigid than the binder. The study offers an approach to writing out an explicit solution to a problem that effectively links a small parameter to the method of boundary states. Equations of the medium are presented as power series of small parameters. Each decomposition step calls for a solution of a linear elasticity problem, which is adequately addressed by the method of boundary states. Below are the results of solving test problems featuring an isotropic cube and a transversely isotropic cylinder with homogeneous boundary conditions. In both cases high accuracy is achieved as early on as the third iteration. Also presented is an axisymmetry problem for a transtropic cylinder with inhomogeneous boundary conditions. In this case accuracy depends on the values of small parameters. For all of the problems described we provide a detailed accuracy analysis and draw conclusions as to convergence.