Asymptotics Without Logarithmic Terms for Crack Problems†

Abstract

We consider boundary value problems for elliptic systems in a domain complementary to a smooth surface with boundary, which models a crack with its edge. The same boundary conditions are prescribed on both sides of the surface. We prove that the singular functions appearing in the expansion of the solution along the crack edge all have the form in local polar coordinates: The logarithmic shadow terms predicted by the general theory do not appear. Moreover, we obtain that, for a smooth right hand side, the jump of the displacement across the crack surface is the product of r 1/2with a smooth vector function. We present two different, but complementing, approaches leading to these results, and providing distinct generalizations. The first one is based on a Wiener–Hopf factorization of the pseudodifferential symbol on the surface obtained after reduction of the boundary value problem. The second approach concerns directly the boundary value problem and is based on a closer look at the Mellin symbol at each point of the crack edge. †Dedicated to Prof. W. L. Wendland for his 65th anniversary.