Vol 220, No 2 (2017)

Under the conditions of plane deformation, by using the Wiener–Hopf method, we perform the numerical analysis of a narrow small-scale process zone in the elastic body subjected to shear near the tip of an interface crack whose faces are in contact with friction. The process zone originates at the crack tip and makes a certain angle with the interface of the media. It is modeled by the straight line of discontinuities of displacements and consists of two segments. In the segment adjacent to the crack tip, which models the region of destruction of the material, both normal and tangential displacements have discontinuities. At the same time, in the second segment, only the normal component of displacements suffers discontinuities. The angle between the process zone and the interface of the media is determined from the condition of maximum of circumferential tensile stresses. We determine the sizes of the entire process zone and the region of destruction, the crack-tip opening displacements, and maximum opening displacement of the initial process zone. We also investigate the influence of the friction coefficient on the parameters of the process zone. By using the deformation criterion of fracture, we analyze the role played by the zone of destruction in the onset of crack propagation. It is shown that the crack starts as a result of the relative shear of its faces near the tip. The comparison of the obtained results with the data of other researchers is presented.

We consider the problem of bending of a thin circular plate under the action of concentrated normal forces at its center in the case where the plate rests on two symmetrically located contour supports of finite length. The integral equation of a Prandtl-type problem is numerically solved by the method of mechanical quadratures and the method of orthogonal polynomials. Moreover, it is reduced to the Fredholm integral equation of the second kind. We compute the distribution of generalized shear forces in the support and the deflection of the plate.

We suggest a method for the determination of the thermoelastic state caused by plane axisymmetric temperature fields and surface loads in layered isotropic bodies with cylindrical interfaces. The temperature and coordinate dependences of the moduli of elasticity, coefficients of linear temperature expansion, and Poisson ratios are taken into account. The method is based on the solution of the systems of integral-algebraic equations for radial displacements. In the case of a cylinder, these systems are obtained from the integral representation of the solution of the problem for the ordinary differential equation with generalized derivatives. In this case, we use the Green function of the elasticity problem for a homogeneous cylinder. In the cases of a layered space with cylindrical cavity, a continuous cylinder, and the continuous space, the corresponding systems and the remaining relations required for the determination of the thermoelastic state are obtained as a result of the limit transitions. The relations for the determination of thermal stresses in the corresponding single-layer bodies are presented. The numerical investigations are performed for a three-layer cylinder with functionally gradient layer.

We consider contact problems of determination of the stress-strain state in the elastic half plane under the action of punches of different shapes (parabolic, cylindrical, elliptic, and hyperbolic). We study specific features of the distribution of contact pressure and stresses in the elastic half plane by using the developed software modules with the use of specially embedded libraries for the evaluation of elliptic integrals of the third kind and the construction of 3D -images and level lines.

To solve the nonlinear nonstationary problem of radiative interaction of a half space with an ambient medium, we use the methods of reduction to nonlinear integral equations of the Volterra type, simple iteration, successive approximations, and quasilinearization. We perform the comparative analysis of the efficiency of application of these approaches to the solution of the analyzed class of problems and show that the approach based on the method of quasilinearization guarantees the best possible convergence.