Asymptotics Without Logarithmic Terms for Crack Problems†

Abstract

We consider boundary value problems for elliptic systems in a domain complementary to a smooth surface M with boundary E . The same boundary conditions are prescribed on both sides of the surface M . The most important model behind this investigation is the crack problem in three-dimensional linear elasticity (either isotropic or anisotropic): there the boundary conditions are Neumann, i.e. tractions are prescribed on both faces of the crack surface M . We prove that the singular functions appearing in the expansion of the solution along the crack edge E all have the #(#) in local polar coordinates (r, #) : 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 with a smooth vector function on M . We present