On delamination of a strip along the boundary between two elastic layers. Part 2. Сase of shear crack

The problem of a strip, composed by two isotropic elastic layers of different elastic properties and thicknesses, separated by a semi-infinite crack located along the line between the layers is considered. The mechanical load is supposed to be applied at infinity. In the first part of the study [1] the mathematical formulation of the problem and its reduction to a homogeneous Riemann-Hilbert problem by application Laplace transform was presented. Under the assumption of possibility to neglect the cross-terms related to the influence of the normal stresses to the shear displacements and the shear stresses to the normal displacements, the problem is reduced to two scalar Riemann-Hilbert problems. Such a formulation may be considered as an approximation for the general case (which is not worse than the traditional beam or rode approximation) and as the exact one for the case, where the two layers may slide but may not separate due to cohesion. By means of factorization procedure the exact analytical solution has been obtained for one of the formulated in [1] scalar problems, namely, the problem of a shear crack. The asymptotical expression has been derived for the relative displacements of the crack faces far from the crack tip. It is shown, that the leading asymptotic terms of these relative displacements correspond to a rode under the boundary condition of the type of elastic clamping. i.e. the proportionality of the displacement of the clamping point to the applied force. The analytical expression for this coefficient has been obtained under the accepted assumptions. The asymptotical expression for the stress field near the crack tip (stress intensity factor and energy release rate) is also derived.