The perturbation method in the problem on a nearly circular inclusion in an elastic body

The two-dimensional boundary value problem on a nearly circular inclusion in an infinity elastic solid is solved. It is supposed that the uniform stress state takes place at infinity. Contact of the inclusion with the matrix satisfies to the ideal conditions of cohesion. To solve this problem, Muskhelishvili’s method of complex potentials is used. Following the boundary perturbation method, this potentials are sought in terms of power series in a small parameter. In each-order approximation, the problem is reduced to the solving two independent Riemann – Hilbert’s boundary problems. It is constructed an algorithm for funding any-order approximation in terms of elementary functions. Based on the first-order approximation numerical results for hoop stresses at the interface are presented under uniaxial tension at infinity

Stress concentration in ultra-thin coating with undulated surface profile

The uniaxial loading of an isotropic film-substrate system with a sinusoidal surface profile and planar interface is considered under plain strain conditions. We for- mulate the corresponding boundary value problem involving two-dimensional constitutive equations for bulk materials and one-dimensional equations for membrane-type surface and interface with the extra elastic constants as well as the residual surface stresses. The mixed boundary conditions consist of the generalized Young–Laplace equations and rela- tions describing the continuous of displacements across the surface and interphase regions. Using the linear perturbation technique combined with the Goursat–Kolosov complex po- tentials and the superposition principle, the original boundary value problem is reduced to the analytical solution of the integral equations system

Effect of a Type of Loading on Stresses at a Planar Boundary of a Nanomaterial

Abstract A two-dimensional model of an elastic body at nanoscale is considered as a half-plane under the action of a periodic load at the boundary. An additional surface stress, and constitutive equations of the Gurtin-Murdoch surface linear elasticity are assumed. Using Goursat-Kolosov complex potentials and Muskhelisvili technique, the solution of the boundary value problem in the case of an arbitrary load is reduced to a hypersingular integral equation in an unknown surface stress. For the case of a periodic load, the solution of this equation is found in the form of Fourier series. The influence of the surface stress on the stresses at the boundary of the half-plane under the tangential and normal periodic loading is analyzed. In particular, it is found out the size effect which becomes apparent in the dependence of the stresses on a length of the load period of the order 10 nm. Moreover, the tangential stresses appear under the action of the normal loads

The perturbation method in the problem on a nearly circular inclusion in an elastic body

The two-dimensional boundary value problem on a nearly circular inclusion in an infinity elastic solid is solved. It is supposed that the uniform stress state takes place at infinity. Contact of the inclusion with the matrix satisfies to the ideal conditions of cohesion. To solve this problem, Muskhelishvili’s method of complex potentials is used. Following the boundary perturbation method, this potentials are sought in terms of power series in a small parameter. In each-order approximation, the problem is reduced to the solving two independent Riemann – Hilbert’s boundary problems. It is constructed an algorithm for funding any-order approximation in terms of elementary functions. Based on the first-order approximation numerical results for hoop stresses at the interface are presented under uniaxial tension at infinity