The Laplace equation in 3D domains with cracks: dual singularities with log terms and extraction of corresponding edge flux intensity functions

Abstract

The singular solution of the Laplace equation with a straight-crack is represented by a series of eigenpairs, shadows and their associated edge flux intensity functions (EFIFs). We address the computation of the EFIFs associated with the integer eigenvalues by the quasi-dual function method (QDFM).The QDFM is based on the dual eigenpairs and shadows, and we exhibit the presence of logarithmic terms in the dual singularities associated with the integer eigenvalues. These are then used with the QDFM to extract EFIFs from p-version finite element solutions. Numerical examples are provided.

Restrictions on the stress components in the edge points of the homogeneous elastic body

“…Publications reviews on the classical approach application are given in (Sinclair 2004, Paggi andCarpinteri 2008). The solution in the classical case is constructed by various methods: operational calculus (Williams, 1952, Cook & Erdogan, 1972, Sinclair, 2004, functions of a complex variable (Parton & Perlin, 1981), Erie functions and integral equations (Cook & Erdogan, 1972;Andreev, 2014), separation of variables and expansion in series into various functions (Shannon et al, 2014(Shannon et al, , 2015Galadzhiev et al, 2011;He & Kotousov 2016), etc. The authors who are using numerical methods: finite element method (Koguchi & Muramoto, 2000;Barut et al, 2001;Xu & Sengupta, 2004;Lee et al, 2006;Xu et al, 2016;Dimitrov et al, 2001), finite element method in combination with by searching for eigenvalues by the Arnold method (Apel et al, 2002), the method of boundary elements and the method of boundary states (Mittelstedt & Becker, 2006;Koguchi & Da Costa, 2010 ), implementing the asymptotic idea by unlimited refinement of the FE-grid at the region near the special points or by constructing special finite elements.…”