Primal and Shadow functions, Dual and Dual-Shadow functions for a circular crack and a circular 90 degree V-notch with Neumann boundary conditions

TL;DR

This paper derives explicit analytical expressions for primal, shadow, dual, and dual-shadow functions related to the Laplace equation near circular singular edges with Neumann boundary conditions, focusing on penny-shaped cracks and V-notches.

Contribution

It provides novel explicit formulas for various functions associated with Laplace problems around circular singular edges with Neumann conditions.

Findings

Explicit formulas for primal and shadow functions

Formulas for dual and dual-shadow functions

Applicable to penny-shaped cracks and V-notches

Abstract

This report presents explicit analytical expressions for the primal, primal shadows, dual and dual shadows functions for the Laplace equation in the vicinity of a circular singular edge with Neumann boundary conditions on the faces that intersect at the singular edge. Two configurations are investigated: a penny-shaped crack and a 90^o V-notch.