A longitudinal crack in a prestressed physically non-linear elastic layer with free boundaries

The problem of a prestressed elastic layer with free boundaries, weakened by a longitudinal crack, situated symmetrically about its boundaries, is considered. In the initial state the layer is subject to a large deformation by uniform forces, applied at infinity. Two versions of the physical non-linearity of the material are investigated: a Mooney elastic potential and a harmonic-type potential. The perturbation of the initial stress-strain state is produced by a uniform pressure on the sides of the crack. It is assumed that the additional stresses and strains that arise are small on the background of the main stress state. This assumption enables the problem of determining the additional strains to be linearized. In both cases the problem is reduced to an integral equation of the first kind in the derivatives of the functions describing the opening of the crack. For different values of the parameters, characterising the initial stress state, approximate numerical and asymptotic solutions are constructed for a relatively thick layer.