Circular edge singularities for the Laplace equation and the elasticity system in 3-D domains

Zohar Yosibash, Samuel Shannon, Monique Dauge, Martin Costabel

June 2010. Prépublication IRMAR, 10-38.

International Journal of Fracture (2011) 168:31-52 DOI: 10.1007/s10704-010-9553-y .

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Asymptotics of solutions to the Laplace equation with Neumann or Dirichlet conditions in the vicinity of a circular singular edge in a three-dimensional domain are derived and provided in an explicit form. These asymptotic solutions are represented by a family of eigen-functions with their shadows, and the associated edge flux intensity functions (EFIFs), which are functions along the circular edge. We provide explicit formulas for a penny-shaped crack for an axisymmetric case as well as a case in which the loading is non-axisymmetric.

For Neumann boundary conditions and axisymmetric loading, the solution $\tau$ has the following asymptotic expansion in local polar coordinates $(\rho,\varphi)$ in the meridian domain -- here the domain around the penny-shaped crack is determined by $-\pi<\varphi<\pi$ and $R$ is the radius of the crack $$ \eqalign{\tau \quad = \quad& A_0 \cr + & A_1\rho^{1/2} \left[\sin\frac{\varphi}{2} + \left(\frac{\rho}{R}\right) \frac{1}{4} \sin\frac{\varphi}{2} + \left(\frac{\rho}{R}\right)^2 \left( \frac{1}{12} \sin\frac{\varphi}{2} - \frac{3}{32} \sin\frac{3\varphi}{2} \right) + \cdots\right] \cr + & A_2\rho \left[\cos\varphi - \left(\frac{\rho}{R}\right) \frac{1}{4} + \left(\frac{\rho}{R}\right)^2 \frac{3}{16} \cos\varphi + \cdots\right] \cr + & A_3\rho^{3/2} \left[\sin\frac{3\varphi}{2} - \left(\frac{\rho}{R}\right) \frac{1}{4} \sin\frac{\varphi}{2} - \left(\frac{\rho}{R}\right)^2 \frac{1}{32} \left( 3 \sin\frac{\varphi}{2} - \frac{16}{5} \sin\frac{3\varphi}{2} \right) + \cdots\right] + \cdots } $$ Explicit formulas for other singular circular edges such as a circumferential crack and an external crack are also derived.

The mathematical machinery developed in the framework of the Laplace operator is extended to derive the asymptotic solution (three-component displacement vector) for the elasticity system in the vicinity of a circular edge in a three-dimensional domain. As a particular case we present explicitly the series expansion for a traction free or clamped penny-shaped crack in an axisymmetric or a non-axisymmetric situation.

The precise representation of the asymptotic series is required for constructing benchmark problems with analytical solutions against which numerical methods can be assessed, and to develop new extraction techniques for the edge flux/intensity functions which are of practical engineering importance in predicting crack propagation.

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