Path-independent integral

Rice J.R. (1968) A path-independent integral and the approximate analysis of strain concentration by notches and cracks. J. Appl. Mech. 35, 379-386. 

J. R. Rice, A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks, J. Appl. Mech., 1968, 35,379 386. 

T. Nishioka and S. N. Atluri, Path-Independent Integrals, Energy Release Rates, and General Solutions of Near-Tip Fields in Mixed-Mode Dynamic Fracture Mechanics, Engineering Fracture Mechanics, 18, 1-22 (1983). 

This is the case wherein the stress amplitude factor C in large-scale creep also governs the crack growth rate. This analogy occurs because C, as for the case of G or / at ambient temperature, is a path-independent integral. 

The path independent integral was presented for a plane problem as [11]... 

A second approach is that of Rice,(3) who defined a path-independent integral around a crack and denoted it J. It turns out that, for a nonlinear elastic solid, the integral is equivalent to the energy released from the stress field for unit propagation of the crack and is thus essentially the same as the more empirical parameter of Rivlin and Thomas.(2)... 

Here, is any scalar field quantity such as a Cartesian component of stress or strain, and d is the increment of crack advance in the xi direction. Finally, Hutchinson (1974) derived the following path-independent integral that can be use to determine the crack tip energy release rate Gtip during steady crack growth. 

Schapery R (1990) On some path independent integrals and their use in fracture of nonlinear viscoelastic media. Int J Fract 42 189-207... 

For the return path, the subscripts/and i in Eq. (1.10.16) must be interchanged and the subscript 0 must be replaced by 1. On adding this modified equation to Eq. (1.10.16) for the forward process, one obtains (in the approximation nb < V) the net contribution associated with (i) the path-independent integrals as... 

The connection with path-independent integrals is less evident. Consider the simplest J-integral. Let volume forces be absent, and the conditions of a plane stress-strain problem realized approximately in the vicinity of crack-tips. Equation (5) takes the form... 

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