On Modeling Thin Delaminated Inclusions with Local Damage in a Two-dimensional Elastic Body

The paper concerns the equilibrium problem for an elastic body containing a thin elastic inclusion with a local defect. The defect is modeled as a junction point between two separate inclusions, characterized by a positive damage parameter. The thin inclusion is described using the theory of thin elastic Timoshenko beam. The problem formulation involves the contact interaction of bodies of different dimensions (a 2D elastic matrix and 1D inclusions) and the junction of multiple inclusions. A variational statement of the problem and the corresponding differential formulation are presented. Delamination of the inclusion from the matrix is modeled as a crack with the inclusion bonded to one face. To prevent non-physical interpenetration of the crack faces, inequality-type boundary conditions (Signorini conditions) are imposed. The problem is formulated and analyzed using the variational inequality method, establishing the equivalence between the differential and variational statements. The main goal of the study is to develop an algorithm for the numerical solution of the problem. For this purpose, a domain decomposition method combined with the Lagrange multipliers approach is employed, reducing the problem to a saddle-point search. A Uzawa-type algorithm is constructed for this purpose. Numerical results for model problems, implemented using the FreeFem++ package, are presented.