A family of problems under consideration describes the contact of elastic plates situated at a given angle to each other and, in the natural condition, touching along a line. The plates are subjected only to bending. The limiting process from the elastic inclusion to the rigid one is studied. It is demonstrated that the limit problems precisely describe the contact of an elastic plate with a rigid beam and the problem of the equilibrium of an elastic plate with a rigid inclusion. The solvability of the problems is established; the boundary conditions holding on the possible contact set are found as well as their precise interpretation.

Under consideration is the equilibrium of a composite plate containing a through vertical crack of variable length at the interface between thematrix and the elastic inclusion. The deformation of the matrix is described by the Timoshenko model, and the deformation of the elastic inclusion, by the Kirchhoff–Love model. Some formula is obtained for the derivative of the energy functional with respect to the crack length.

Mathematical modeling and research of problems on the deformation of inhomogeneous bodies containing cracks along elastic inclusions involves setting the conjugation conditions at the interface between different materials. Difficulties are associated with the possibility of large stress values appearing near the inclusions. Determining the inhomogeneous bodies with the most optimal parameters is one of the most popular areas of theoretical and experimental research. In this paper, we study the problem of optimal control of the angle of the crack inclination to the median plane in the equilibrium problem for an elastic Timoshenko plate containing an oblique crack at the boundary of an elastic inclusion. The complexity of the problem is due to the fact that the non-penetration condition on the crack faces is given in the form of inequality. The quality functional characterizes the deviation from the specified displacements. We prove solvability of the optimal control problem and establish continuous dependence of the solutions on the value of the crack inclination angle.

We consider a nonlinear problem of equilibrium of an elastic body with periodically located cracks. On the edges of these cracks, non-penetration conditions are given. The nonlinear problem is formulated in the form of variational inequality. The period of distribution of the cracks, as well as their sizes, depends on a small parameter. The behavior of the solution to the problem with periodically located cracks is determined by the first two terms $u^0(x)$ and $u^1(x, y)$ of the asymptotic expansion. In this paper, we study the solution of the variational inequality on a periodicity cell (a local problem). For the first corrector $u^1(x, y)$, we construct a penalty equation and a linear iterative equation in integral form. We prove that the sequence of solutions of the problem with penalty converges to the solution of the problem on the cell when the small regularization parameter tends to zero. We show that the approximate solution of the iteration equation converges strongly to the solution of the penalty equation. Рассматривается нелинейная задача о равновесии упругого тела с периодически расположенными трещинами. При этом на краях трещин ставятся односторонние ограничения, что приводит к вариационному неравенству. Период распределения трещин, а также их размеры зависят от малого параметра. Поведение решения задачи с периодически расположенными трещинами определяется двумя первыми членами $u^0(x)$, $u^1(x, y)$ асимптотического разложения. В статье изучается решение вариационного неравенства на ячейке периодичности (локальная задача). Для первого корректора $u^1(x, y)$ строится уравнение со штрафом и линейное итерационное уравнение в интегральной форме. Доказано, что последовательность решений задачи со штрафом при стремлении малого параметра регуляризации к нулю сходится к решению задачи на ячейке. Показано, что приближенное решение итерационного уравнения сходится сильно к решению уравнения со штрафом.