Contour integrals in nonlinear fracture mechanics for mixed forms of deformation

Modern knowledge in the field of fracture mechanics is the first key knowledge in solving the problems of safety and strength of the objects with crack-like damages of the various origins. Nonlinear fracture mechanics in the analysis of the stress-strain state in the crack tip region is based on the one- and two-parameter approaches. The classical one-parameter studies involve the study of singular quantities, including a contour J -integral, independent of the path of integration, a stress intensity factor (SIF), etc. The values of the SIF and J -integral are interdependent. Combined methods are very popular, based on the union of numerical, experimental and analytical calculations, which make it possible to obtain the most clear description of the parameters of fracture mechanics. Calculation of the J -integral in finite element models, by the method of reactions or stresses, is very effective, but this requires sufficiently accurate analytical representations of the contour J -integral. There are certain limiting conditions when obtaining such formulas. In the numerous scientific works, it has been proved that J is an integral in the most cases does not depend on the path of integration, but is highly dependent on the methods of describing the parameters of the stress-strain state, as well as their derivatives, on the dimension of the problem and on the degree of distance of the contour of integration from the crack tip. In this paper, we review and present the author's conclusions of the contour integrals in nonlinear fracture mechanics for three cases: the classical Hutchinson - Rosengren - Rice solution (HRR), contour integrals in the gradient theory of plasticity, and the calculation of the J -integral for a general case when the components of stresses and displacements are the functions of three Cartesian coordinates. A generalized J- integral is derived and used to characterize a nonlinear amplitude fac.