On delamination of a stipe along the boundary between two elastic layers. Part 1, problem formulation, the case of normal 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 with nonzero total force and moment is supposed to be applied at infinity. By means of Laplace transformation the problem is reduced to a homogeneous Riemann problem. Under the assumption of possibility to neglect the cross-terms related to the influence of the normal stresses to the shier displacements and the shier stresses to the normal displacements the problem is reduced to two scalar Riemann 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, when the two layers may slide but may not separate due to cohesion, e.g. by van-der-Waals forces. By means of factorization procedure the exact analytical solution has been obtained for one of the formulated scalar problems, namely, the problem of the normal separation. The asymptotical expression has been derived for the relative displacements of the crack faces far from its tip. It is shown that the leading asymptotic terms of these relative displacements correspond to a beam deflection under 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) was also derived.