On numerical solving of junction problem for semirigid and Timoshenko inclusions in elastic body

Abstract

The paper considers an example of mathematical modeling of the process of deformation of a structure made of fiber-composite material. Timoshenko’s theory of thin beams is used to describe the reinforcing fiber model. The paper concerns a numerical solving of the equilibrium problem for a two-dimensional elastic body with a thin Timoshenko elastic inclusion and a thin semirigid inclusion. Both inclusions are of a rectilinear shape and delaminate from the elastic matrix. Therefore, the problem is posed in the domain with a cut and conditions of the form of inequalities are specified at the edges of the crack, as on a part of the boundary. These conditions exclude mutual penetration of the crack edges into each other. At the same time, such a formulation leads to the nonlinearity of the problem and the need to use additional mathematical methods to construct an algorithm for the numerical solution of the problem. The inclusions have a joint point at which junction conditions are written out. For the numerical solving of the problem in a domain with a cut, a variational formulation is used using the domain decomposition method and the Uzawa algorithm. To obtain an approximate solution that satisfies the conditions of semirigid inclusion, an additional algorithm is built using methods of mathematical analysis, an example with calculations is given.