Explicit formulations of the J-integral considering higher order singular terms in eigenfunction expansion forms - Part I. Analytical treatments

A semi-infinite crack with a nonlinear zone around the crack tip is studied in detail in the following four cases: (i) antiplane shear deformation, (ii) plane deformation, (iii) plane anisotropic deformation with purely imaginary characteristic roots, and (iv) interface between dissimilar solids, respectively. The complete Williams eigenfunction expansion forms including both positive and negative powers of a distance from the crack tip in each of the four cases are considered, which could be used to describe the elastic state in an annulus around the nonlinear zone. Explicit formulations of the path independent J-integral are presented by utilizing the differential property and the so-called pseudo-orthogonal property of the complete Williams expansion forms in each of the four cases. It is shown that the complete Williams expansion forms in every case mentioned above have two kinds of contributions to the J-integral. The first one is similar to the traditional one arising from the well-known r(-1/2) singularity (or r(-1/2+i epsilon) singularity for the interface crack). The second one is a summation induced from the interaction of higher order singular terms and nonsingular terms of the expansion forms. Although the coefficients of the complete Williams expansion forms-in each of the four cases should be determined not only by the prescribed outer boundary conditions but by some specific material model for the nonlinear zone surrounding the crack tip, once they are determined by whatever method, the J-integral could be calculated by using the formulations derived in this paper without any difficulties. It is found also that the elastic T-term acting parallel to the crack plane has no direct interaction with the higher order singular terms such that has no direct effect to the J-integral, although the presence of the T-term will dramatically affect the size of the nonlinear zone and in this way affect the coefficients of the higher order singular terms and in turn the values of the J-integral. The present results in the four cases support the conclusion derived by Hui and Ruina (1995) that the nonsingular terms and the higher order singular terms in the complete Williams expansion forms are of equal importance. Thus, in order to improve and to confirm the small scale yielding description, not only the nonsingular terms, but also the higher order singular terms should be determined for a given crack configuration, body geometry, loading conditions and the prescribed material model in the nonlinear zone.