Crack tip and associated domain integrals from momentum and energy balance

A unified derivation of crack tip flux integrals and their associated domain representations is laid out in this paper. Using a general balance statement as the starting point, crack tip integrals and complementary integrals which are valid for general material response and arbitrary crack tip motion are obtained. Our derivation emphasizes the viewpoint that crack tip integrals are direct consequences of momentum balance. Invoking appropriate restrictions on material response and crack tip motion leads directly to integrals which are in use in crack analysis. Additional crack tip integrals which are direct consequences of total energy and momentum balance are obtained in a similar manner. Some results on dual (or complementary) integrals are discussed. The study provides a framework for the derivation of crack tip integrals and allows them to be viewed from a common perspective. In fact, it will be easy to recognize that every crack tip integral under discussion can be obtained immediately from the general result by appropriately identifying the terms in the general flux tensor. The evaluation of crack tip contour integrals in numerical studies is a potential source of inaccuracy. With the help of weighting functions these integrals are recast into finite domain integrals. The latter integrals are naturally compatible with the finite element method and can be shown to be ideally suited for numerical studies of cracked bodies and the accurate calculation of pointwise energy release rates along a curvilinear three-dimensional crack front. The value of the domain integral does not depend on domain size and shape - this property provides an independent check on the consistency and quality of the numerical calculation. The success of the J-based fracture mechanics approach has led to much literature on pathindependent integrals. It will be shown that various so-called path-independent integrals (including path and area integrals) are but alternate forms of the general result referred to above and do not provide any additional information which is not already contained in the general result. Recent attempts to apply these 'newer' integrals to crack growth problems are discussed.