Plane stress state of a uniformly piece-wise homogeneous plane with a periodic system of semi-infinite interphase cracks

Abstract

The plane stress state of a uniformly piecewise-homogeneous plane obtained by alternately joining two dissimilar strips is considered, which along the lines of joints of dissimilar strips is weakened by a periodic system of two semi-infinite interfacial cracks and is deformed using normal loads applied to the crack banks. The basic cell of the problem in the form of a two-component strip is considered and, using the generalized Fourier transform, a governing system of equations for the problem is obtained in the form of one singular integral equation of the second kind for a complex combination of contact stresses in the junction zone of the strips. As a special case, tending the height of the strips to infinity, the governing equation of the problem for a two-component plane of two dissimilar half-planes with two semi-infinite interfacial cracks is obtained and its exact solution is constructed. The governing equation for the stated problem is also obtained in the form of one singular integral equation of the first kind with respect to normal contact stresses in another particular case, when all strips are made of the same material, i.e. in the case of a homogeneous plane, a weakened periodic system of parallel, two semi-infinite cracks. In the general case, the behavior of the unknown function at the end points of the integration interval is determined and the solution of the problem by the numerical-analytical method of mechanical quadratures is reduced to solving a system of algebraic equations. Simple formulas are obtained to determine the intensity factors, the CherepanovRice integral and crack opening. A numerical calculation has been performed. Regularities of changes in contact stresses and the CherepanovRice integral at the endpoints of cracks are revealed, depending on the elastic characteristics of heterogeneous strips and the geometric parameters of the problem.

Keywords

• periodic problem
• mixed boundary value problem
• piece-wise homogeneous plane
• interface cracks