A longitudinal crack in a prestressed thin elastic layer with free boundaries

The plane problem of a longitudinal crack loaded by a uniform pressure at its sides and symmetrically positioned in a prestressed thin layer with free boundaries is considered. The layer is prestressed in its plane by uniform forces applied at infinity. It is assumed that the material of a layer is described by an harmonic-type elastic potential. The additional stresses caused by the presence of the crack in the layer are considered to be small compared with the stresses of the main non-linear stress-strain state of the layer. This makes it possible to linearize the problem of determining the additional stresses on a background of the main stressed state. Such a linearized problem reduces to an integral equation of the first kind with a singular kernel with respect to a derivative of the function describing the crack opening. Asymptotic solutions of the integral equation for small values of the dimensionless parameter characterizing the layer thickness, are constructed for different values of the dimensionless parameter characterizing the prestressing of the layer. Examples are given. Similar problems concerning cracks in prestressed bodies were examined earlier (see, for example, [1, 2]). The problem is studied here for the first time.