Conservation laws from any conformal transformations and the parameters for a sharp V-notch in plane elasticity

Abstract

Based on the well known complex Kolosov–Muskhelishvili potentials, two new independent Lagrangian functions are presented and their variational problems lead to two independent harmonic equations, which are also the Navier’s displacement equations in plane elasticity. By applying Noether’s theorem to these Lagrangian functions, it is found that their symmetry-transformation in material space is a conformal transformation in planar Euclidean space. Since any analytic function is a conformal transformation in planar Euclidean space, the conservation law obtained from this kind of symmetry-transformation possesses universality and leads to a path-independent integral. By adjusting the conformal transformation or analytic function, a finite value can be obtained from calculating this kind of path-independent integral around a material point with any order singularity. By applying this path-independent integral to the tip of a sharp V-notch, unlike Rice’s J-integral, the parameters of Mode I and II problems are found, which remain invariant because of path independence for a fixed notch opening angle. That is, these two parameters are equivalent to the notch stress intensity factors (NSIFs), and two examples are presented to show the application.