Path-independent integral for the sharp V-notch in longitudinal shear problem

By applying Noether’s theorem to the elastic energy density in longitudinal shear problem, it is shown that its symmetry-transformations of material space can be expressed by the real and imaginary parts of an analytic function. This kind of the symmetry-transformations leads to the existence of a conservation law in material space, which does not belong to trivial conservation laws and whose divergence-free expression gives a path-independent integral. It is found that by adjusting the analytic function, a finite value can be obtained from this path-independent integral calculated around the material point with any order singularity. For a sharp V-notch placed on the edge of homogenous materials and/or the interface of bi-materials, application shows that the finite value obtained from this path-independent integral is directly related to the notch stress intensity factor (NSIF) and does not depend on the location of integral endpoints chosen respectively along two traction-free surfaces of which form a notch opening angle. Usability is presented in an example to estimate the NSIF of a bi-material plate.