On the problem of a Gurtin–Murdoch cylindrical material surface embedded in an infinite matrix

Abstract

The problem of an infinite isotropic elastic matrix subjected to uniform far-field load and containing a Gurtin–Murdoch material surface of cylindrical shape is considered in plane strain setting. The governing equations and the boundary conditions for the problem, reduced to that of an infinite plane containing a material curve along a circular arc, are reviewed. The displacements inside the matrix are sought in the complex variables form of a single layer elastic potential whose density represents the jump in complex tractions across the curve. Exact complex integral representations for the elastic fields everywhere in the material are provided and the problem is further reduced to the system of real variables hypersingular boundary integral equations in terms of the strain and rotation components associated with the curve. The components are then approximated by the series of trigonometric functions that are multiplied by the square root weight functions to allow for automatic incorporation of the tip conditions. The unknown coefficients in the series are found from the system of linear algebraic equations that is solved using standard collocation method. The numerical examples are presented to illustrate the influence of dimensionless parameters with the main focus on the study of curvature-induced effects.