Точные решения задач теории многократного наложения больших деформаций для тел, образованных последовательным соединением деформированных частей

Abstract

Large strains of composite solids made of incompressible isotropic nonlinear-elastic materials are analyzed for the case in which the parts of these solids are preliminarily strained. The approaches to exact analytical solutions of these problems are given and developed in cooperation with V.An. Levin. He is a professor at the Lomonosov Moscow University. The solution of these problems is useful for stress analysis in members containing preliminarily stressed parts. The results can be used for the verification of industrial software for numerical modeling of additive technologies. The problems are formulated using the theory of repeated superposition of large strains. Within the framework of this theory these problems can be formulated as follows. Parts of a member, which are initially separated from one another, are subjected to initial strain and passes to the intermediate state. Then these parts are joined with one another. The joint is performed by some surfaces that are common for each pair of connected parts. Then the body, which is composed of some parts, is strained as a whole due to additional loading. The body passes to the final state. It is assumed that the ideal contact conditions are satisfied over the joint surfaces. In other words, the displacement vector in the joined parts is continuous over these surfaces. The exact solutions for isotropic incompressible materials are obtained using known universal solutions and can be considered as generalizations of these solutions for superimposed large strains. The following problems are considered in detail: — the problem of stress and strain state in two hollow circular elastic cylinders (tubes) one of which is preliminarily strained and inserted into another cylinder (the Lam´e-Gadolin problem); — the problem of torsion of a composite cylinder; — the problem of large bending strains of a composite beam consisting of some preliminarily strained parts (layers). The mathematical statements of these problems are given, the methods of solution are presented, and some results of solution are shown. The impact of preliminary strains on the state of stresses and strains is investigated, and nonlinear effects are analyzed.