On delamination of a stripe along the boundary between two elastic layers part 3. Exact analytical solution for a particular case

The analytical solution has been obtained for a homogeneous problem of elasticity in plane strain setting of semi-infinite crack separating two isotropic layers of equal thicknesses and different but related with a particular restriction (corresponding to vanishing of the second Dundurs parameter) elastic constants. The problem under consideration is a particular case (allowing us to obtain the exact analytical solution) of the general problem on semi-infinite crack separating two isotropic layers of different thickness and arbitrary combination of elastic constants posed. The solutions for the cases of normal and shear crack were obtained under the assumption of the possibility to neglect cross-terms related to the influence of the normal stresses on the shear displacements and the shear stresses on the normal displacements. In this part of the work the solution for the case in question has been derived by means of Laplace transformation; and reducing it to a homogeneous Riemann-Hilbert problem with the matrix coefficient. The restriction imposed on elastic constants and the demand of equal thicknesses are determined by the used method allowing the factorization of the matrix coefficient. The asymptotical expression has been derived for the relative displacements of the crack faces far from its tip, corresponding to a beam deflection with the boundary condition of the type of generalized elastic clamping. i.e. the proportionality of the displacement and angle of rotation of the clamping point to the total vector and bending moment of the applied load by means of the matrix of coefficients of compliance. The analytical expressions for these coefficients have been obtained. The asymptotical expression for the stress field near the crack tip (stress intensity factor and energy release rate) has also been derived.