Antiplane deformation of continuously-inhomogeneous half-plane with surface crack under the action of stamps (Q2753468)

In the elastic half-plane \(-\infty<x<\infty\), \(y>0\), which is inhomogeneous (the shear modulus is \(G=G(y)= G_\nu y^\nu\), \(0\leq \nu < 1\)) there is a crack \(x=0\), \(0<y<c\) generated by antiplane shear in opposite directions of two equal (\(-b\leq x\leq 0\), \(y=0\) and \(0\leq x\leq b\), \(y=0\)) stamps adhered to the half-plane. A load \(p_0\) is applied to each stamp. The mixed boundary value problem is solved by the Fourier transform in \(x\) coordinate and results in a Riemann boundary value problem. Using the Mellin transform, the problem is reduced to infinite systems of algebraic equations. Convergence of the approximate solution of the problem to the exact solution is examined. An expression for the stress intensity factor near the crack tip is derived.