Boundary-Value Problems of Fracture Mechanics for a Nonlinear Anisotropic Body

The first fundamental problem for a nonlinear anisotropic body containing an arbitrary crack with a fracture-process zone near its front is stated in terms of the covariant components of the displacement vector. It is assumed that the body is described by a tensor linear constitutive equation. This equation is derived from the proposed relations between the covariant components of the strain tensor and the contravariant components of the stress tensor, which generalize Reiner’s relations. These relations are analyzed from the point of view of the first and second laws of thermodynamics. As a result, the algebraic invariants of the stress and tensors that appear in the constitutive equations are related. In the case of plane stress state, a system of equations for discretized variables is derived. A method for solving this system is proposed.