Vol 212, No 1 (2016)

We establish and study sufficient conditions of solvability of the Neumann problem for fourth-order properly elliptic equations of the general form in a unit disk K in the space $C^4(K)\cap C^{3,\text{upalpha}}\left(\bar{K}\right)$.

We propose new procedures for the separation of singularities in the kernel and in the potential density for weakly singular Fredholm integral equations for the simple-layer potential in the case where the boundary surface has edges, ribs, and corner points. These procedures are based on the projection methods for the solution of these equations and finite-element approximations of the required potential density.

On the basis of the penalty method, we propose a number of continuous parallel domain decomposition schemes for the solution of the problems of perfect mechanical contact of elastic bodies. For some of these schemes, we prove the theorems on convergence. We study the problem of optimal choice of the iterative parameters. The relationship between the proposed schemes and the domain decomposition methods without penalty is established.

We study the problem of antiplane deformation of an isotropic medium containing a thin elastic curved inhomogeneity. The methods used for the solution of this problem are based on the application of the method of jump functions and the conditions of interaction of the matrix containing a thin curvilinear inclusion and the solution of the resultant system of singular integral equations with Cauchy-type kernels by the collocation method. Numerous examples are considered. The results of evaluation of the stress intensity factors for a crack and an absolutely rigid inclusion along a circular arc are compared with the corresponding analytic results. For a crack along a symmetric parabolic arc, the stressed state is thoroughly investigated. We also study the influence of the modulus of elasticity and the shape of the curvature of inhomogeneity (circular arc, parabola, or a half of a cosine curve) on the generalized stress intensity factors. It is shown that, for the stress intensity factors near the tip of the inhomogeneity, the inclination of the tangent at the tip to the plane of application of the shear forces is of determining importance.