Задача нелинейной теории упругости о растяжении и кручении составного цилиндра с предварительно напряжённым включением, содержащим винтовую дислокацию

Abstract

The problem of large torsional and tension-compression deformations of a composite circular cylinder of incompressible Bartenev-Khazanovich material is considered. The cylinder contains a central circular cylindrical inclusion in which a concentrated screw dislocation is formed. It is pre-twisted and stretched (or compressed) along the axis and bonded to an unstressed external hollow cylinder. A single reference configuration is used when solving a problem for a composite body. This configuration is natural (unstressed) for the outer hollow cylinder and prestressed for the inner solid cylinder. The large deformation superposition theory is used to write the determining ratios of the material of the inner cylinder. The nonlinear theory of torsion of elastic cylinders containing a screw dislocation is used to solve the problem of the prestressed state of the internal inclusion. This theory is physically correct not for any models of isotropic elastic materials, but only for those for which the screw dislocation in the cylinder has a finite linear energy and creates a longitudinal force of finite value. The Bartenev-Khazanovich model belongs to this class. The problem for a composite cylinder is solved by a semi-inverse method. Using this method, it is reduced to nonlinear ordinary differential equations. The assumption of isotropy and incompressibility of the material makes it possible to find an exact solution to the problem.