Doubly Periodic Array of Thin Rigid Inclusions in An Elastic Solid

Abstract

An infinite elastic medium containing a doubly periodic array of strip-shaped thin rigid inclusions under the far-field uniform shear load in the plane perpendicular to the inclusions is considered. Employing oblique coordinates, this antiplane problem is formulated in the form of dual trigonometric series equations. Using a technique that is based on discontinuous trigonometric series involving Heun functions, the dual series equations are reduced to a single Fredholm integral equation of the second kind. It is shown that an approximate solution can be efficiently found with certain iterative, projective and asymptotic methods. Simple formulas for the stress fields at the inclusion tips and the stress intensity factor are derived. Exact solutions are established in a closed form for an array with a rectangular fundamental cell and its limiting cases. Solutions for periodic arrays of collinear or slant parallel inclusions and for a single inclusion are found in the limiting cases when the distance between stacks or rows of the array is large. The technique developed in the present paper provides us with ample opportunities for investigations of various mixed boundary-value problems which are analyzed with Fourier series.