An equilibrium penny-shaped crack in an inhomogeneous elastic medium

The problem of a penny-shaped tensile crack in a continuously-inhomogeneous space is considered. The problem reduces to a dual integral equation for which an approximate analytical solution is constructed. It is proved that the approximate solution of the integral equation is asymptotically exact for both small and large values of the dimensionless geometric parameter of the problem. The accuracy of the solution obtained is investigated. Expressions are presented for the stress intensity factor, the energy of the opening of the crack, the displacements of its sides and the normal components of the stress tensor in the neighbourhood of its contour. In the numerical analysis of the solution of the problem, special attention is paid to analysing of the problem when the first derivative of the change in the elastic properties of the material changes sign.     

The size-dependent elastohydrodynamic lubrication contact of a coated half-plane with non-Newtonian fluid

Based on the couple-stress theory, the elastohydrodynamic lubrication(EHL)contact is analyzed with a consideration of the size effect. The lubricant between the contact surface of a homogeneous coated half-plane and a rigid punch is supposed to be the non-Newtonian fluid. The density and viscosity of the lubricant are dependent on fluid pressure. Distributions of film thickness, in-plane stress, and fluid pressure are calculated by solving the nonlinear fluid-solid coupled equations with an iterative method. The effects of the punch radius, size parameter, coating thickness, slide/roll ratio, entraining velocity, resultant normal load, and stiffness ratio on lubricant film thickness, in-plane stress, and fluid pressure are investigated. The results demonstrate that fluid pressure and film thickness are obviously dependent on the size parameter, stiffness ratio, and coating thickness.     

Analytical solution of axisymmetric contact problem about indentation of a circular indenter into a soft functionally graded elastic layer

The paper addresses a contact problem of the theory of elasticity,i.e.,the penetration of a circular indenter with a flat base into a soft functionally graded elastic layer.The elastic properties of a functionally graded layer arbitrarily vary with depth,and the foundation is assumed to be elastic,yet much harder than a layer.Approximated analytical solution is constructed,and it is shown that the solutions are asymptotically exact both for large and small values of characteristic dimensionless geometrical parameter of the problem.Numerical examples are analyzed for the cases of monotonic and nonmonotonic variations of elastic properties.Numerical results for the case of homogeneous layer are compared with the results for nondeformable foundation.     

The deformation of a half-space with a gradient elastic coating under arbitrary axisymmetric loading

An analytical-numerical method of solving the Neumann boundary-value problem for an elastic half-space with a gradient elastic coating is proposed. The problem is formulated and the construction of the fundamental solution (Green’s function) is described. The method enables a solution of the problem to be obtained for a fairly wide class of types of non-uniformity of the medium, and effects related to the non-uniformity are investigated analytically. A procedure for calculating the displacement, stress and strain fields is described. Particular attention is devoted to analysing the mechanical characteristics in the transition region from the coating to the elastic substrate.     

A bilateral asymptotic method of solving the integral equation of the contact problem of the torsion of an elastic half-space inhomogeneous in depth

An approximate semi-analytical method for solving integral equations generated by mixed problems of the theory of elasticity for inhomogeneous media is developed. An effective algorithm for constructing approximations of transforms of the kernels of integral equations by analytical expressions of a special type is proposed, and closed analytical solutions are presented. A comparative analysis of the approximation algorithms is given. The accuracy of the method is analysed using the example of the contact problem of the torsion of a medium with a non-uniform coating by a stiff circular punch. The relation between the error of the approximation of the transform of a kernel by special analytical expressions, constructed using different algorithms and the error of approximate solutions of the corresponding contact problems is investigated using a numerical experiment.     

Approximated Analytical Solution of a Problem on Indentation of an Electro-Elastic Half-Space with Inhomogeneous Coating by a Conductive Punch

The paper addresses to the development of the bilateral asymptotic method of solution of contact problems taking into account electro-mechanical fields coupling. Approximated analytical solutions are provided for contact problems on indentation of an electroelastic piezoelectric half-space with inhomogeneous in depth piezoelectric coating by a rigid conductive circular flat, spherical or conical punch.