Extraction of stress intensity factors of biharmonic equations with corner singularities corresponding to mixed boundary conditions of clamped, simply supported, and free (II)

Abstract

In [13], we derived stress intensity factors (SIF) extraction formulas of a biharmonic equation Δ2u=f in a cracked domain whose crack faces have clamped boundary conditions. In this paper, we extend our investigation to the derivation of the SIF extraction formulas of the biharmonic equation in non-convex polygonal domains containing cracks or reentrant corners when various BC such as clamped (CC), simply supported (SS), free (FF), or mixed conditions (CS, FC, SF) are imposed on the boundary. We prove that the SIF is expressed as the integral of fΨsk⁎−uΔ2Ψsk⁎ for a cut-off function Ψ and a dual singular function sk⁎ on a small neighborhood of the singularity. The dual singular function is determined in this paper. For a numerical approximation of u, we proposed an iteration method as well as a direct finite element method. For a finite element solution of Δ2u=f, we have to use C1-continuous basis functions. For continuous differentiable basis functions, we use either C1-continuous B-spline basis functions or the conventional Hermite basis function. Because of several advantages of B-spline functions in imposing complex boundary conditions, we choose B-spline basis functions for the finite element approximation of u. Moreover, in the direct numerical method, we propose to use implicitly enriched basis functions that resemble the singularities.