A delaminated inclusion in the case of adhesion and slippage
The author considers the plane problem of stress concentration near a thin absolutely rigid inclusion \((-a\leq x\leq a,\quad y =\pm 0)\) in a homogeneous elastic plane. A vertical force \(P\) is applied at the point \(x = 0\) to the upper edge of the inclusion which is bonded to the medium: \(u =0\), \(v =0\), \(- a\leq x\leq a\), \(y = +0\). The lower edge of the inclusion peels off. The median segment \((-b, b)\) of the crack \((-a< x <a,\quad y = - 0)\) is opened under the action of the force \(P\), \(\sigma_y=0\), \(\tau_{xy}=0\), \(-b<x<b\), \(y =-0\) and the conditions of frictionless slippage \(\tau_{xy}=0\), \(v =0\), \(b<| x|< a\), \(y =-0\) are satisfied in the end sections \(( - a, -b), \quad (b, a).\) The position of point \(b\) is determined from the condition of the smooth joining on the lower edge of the crack \(\partial v/\partial x(x,-0)\to 0\), \(x\to\pm b\mp 0\). It follows that one should accept as being physically correct the solution which eliminates tensile contact stresses in the slippage zone and the oscillations back and forth of crack edges in the separation zone \(\sigma_y(x,-0)\leq 0\), \(b<| x|<a\); \(v(x,-0)\leq 0\), \(-b<x<b\). The problem is equivalent to two Riemann vector boundary value problems for which analytic and asymptotic solutions are constructed. The zones of slippage and detachment, the angle of rotation of the inclusion, the normal displacements of the lower edge of the inclusion and the contact stresses in the slippage zone are found.
systems of singular integral equations with kernels of Cauchy type
scientific article; zbMATH DE number 1080325