Singular asymptotic solution along an elliptical edge for the Laplace equation in 3-D

Abstract

a b s t r a c tExplicit asymptotic solutions are still unavailable for an elliptical crack or sharp V-notch in a three-dimensional elastic domain. Towards their derivation we first consider the Laplace equation. Both homogeneous Dirichlet and Neumann boundary conditions on the surfaces intersecting at the elliptical edge are considered. We derive these asymptotic solutions and demonstrate, just as for the circular edge case, that these are composed of three series with eigenfunctions and shadows depending on two coordinates.

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.…”