The explicit formulations for theL-integral

In this paper, a simple but inberent relation between theL-integral and the Bueckner work conjugate integral is proved for crack problems in isotropic, anisotropic, and dissimilar materials, respectively. It is found that, in the above-mentioned three cases, theL-integral, from the mathematical point of view as well as in principle, arises from Betti's reciprocal theorem. This means that the Bueckner work conjugate integral is a more general path-independent integral than the others since any other path-independent integrals could be derived by using the Bueckner integral while choosing a different subsidiary stress-displacement field.