Numerical Solution of the Equilibrium Problem for a Two-dimensional Elastic Body with a Delaminated Rigid Inclusion

In the paper, a multiphysics problem of equilibrium for a two-dimensional body containing a delaminated rigid inclusion is considered. The rectangular rigid inclusion delaminates from the elastic matrix, forming a crack; therefore, the problem is posed in a nonsmooth domain with a cut. The mathematical model of the rigid inclusion was developed on the assumption that the functions of its displacements has a prescribed structure. The problem is formulated both in a form of a minimization problem for the energy functional and in the form of variational inequality. The boundary condition on the crack faces has a form of inequality and as a result, the problem is non-linear. As a consequence, the construction of an algorithm for the numerical solution of the problem requires the use of additional analytical methods. The methods of domain decomposition, the method of Lagrange multipliers, and the finite element method are used. To calculate functions of rigid displacements, the method of decomposing space into a direct sum of orthogonal subspaces is used and the corresponding subalgorithm is implemented.