Plane deformation of solid with periodic arrays of rigid elliptical inclusions (Q2759511)

An infinite solid containing an absolutely rigid inclusion or an array of them is considered. First, the problem for a single inclusion is solved. Stress concentration near inclusion is calculated through jump of stresses and displacements of the corresponding singular problem for a body with mathematical cut, on whose surfaces unknown stresses are applied. The singular integral equation for determination of the jump of normal stresses is derived and a formula for stress intensity factor is written. For infinite periodic array of coplanar inclusions, a singular integral equation in interval \((-a,a)\) with the Gilbert kernel \(\cot\pi(t-x)/d\) (\(2a\) is the crack length, \(d\) is the distance between cracks) is obtained. For the case \(2a/d<1\) the kernel is expanded into the Cauchy kernel plus the Taylor series and solution for the jump of tangential stresses is sought in the form \((A_0 +A_2x^2 +A_4x^4)x/\sqrt{a^2-x^2}\). Approximate analytic formulas for contact stresses and stress intensity factor are obtained. In a similar way the problem of periodic array of parallel cracks is approximately solved.