On convergent series expansions of solutions of Navier’s equation near singular points

by · Sep 22, 2014 · 1 024 views ·

Hiromichi Itou, Tokyo University of Science, Tokyo, Japan Theory of partial differential equations has been thoroughly developed mainly in smooth domains. In non-smooth domains such as polyhedral or cracked domains, difficulties appear because domains have singular points. Then, it is important to analyze the precise behavior of the solution of the corresponding boundary value problems near singular points. And they have possibility of application in various fields of science and engineering such as Fracture problems, Inverse problems (nondestructive evaluation) and so on. The aim of this talk is to introduce some convergent expansion formulae of solutions of two dimensional linearized elasticity equation (called Navier’s equation) around singular points such as a crack tip and a tip of a rigid line inclusion, and furthermore, to clarify the relation between the order of singularities in series expansions and boundary conditions. Some formulae themselves may be known in engineering, however, the explicit formulae with rigorous convergent proof seem non-trivial. The derivation is based on complex analysis, especially, analytic continuation, Goursat-Kolosov-Muskhelishvili stress functions, Lekhnitskii formalism, Riemann-Hilbert problem, etc.

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