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Costabel, Martin; Dauge, Monique; Duduchava, Roland

Asymptotics without logarithmic terms for crack problems. (English) Zbl 1103.35321

Commun. Partial Differ. Equations 28, No. 5-6, 869-926 (2003).
Summary: 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 \(r^{1/2}\)with 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.

Cited in 36 Documents

MSC:

35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35Q72 Other PDE from mechanics (MSC2000)
47G30 Pseudodifferential operators
74G10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics
74R10 Brittle fracture
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