Corner Singularities and Regularity of Weak Solutions for the Two-Dimensional Lamé Equations on Domains with Angular Corners

This paper is concerned with corner singularities of weak solutions of boundary value problems in the theory of plane linearized elasticity. The presence of angular corner points or points at which the type of boundary conditions changes yields generally local singularities in the solution. This singular behavior in the vicinity of such points can be described with the help of asymptotic singular representations for the solution, which essentially depend on the zeros of certain transcendental functions. These transcendental functions will be derived and analyzed for all ten possible combinations of boundary conditions, generated by the four basic ones, prescribing in the tangential and normal direction of the boundary, respectively, either the displacement or the tractions. The regularity of the corresponding weak solutions will be investigated.

[1]  Serge Nicaise,et al.  General Interface Problems-II , 1994 .

[2]  E. Sternberg,et al.  Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity , 1980 .

[3]  A. Kalandiia.  Remarks on the singularity of elastic solutions near corners , 1969 .

[4]  R. Rannacher,et al.  On the boundary value problem of the biharmonic operator on domains with angular corners , 1980 .

[5]  Serge Nicaise,et al.  About the Lamé system in a polygonal or a polyhedral domain and a coupled problem between the Lamé system and the plate equation. I : regularity of the solutions , 1992 .

[6]  V. A. Kondrat'ev,et al.  Boundary problems for elliptic equations in domains with conical or angular points , 1967 .

[7]  P. Grisvard.  Singularities in Boundary Value Problems , 1992 .

[8]  Monique Dauge,et al.  The Influence of Lateral Boundary Conditions on the Asymptotics in Thin Elastic Plates , 1999, SIAM J. Math. Anal..

[9]  F. Karal,et al.  The elastic-field behavior in the neighborhood of a crack of arbitrary angle , 1962 .

[10]  N. Muskhelishvili.  Some basic problems of the mathematical theory of elasticity , 1953 .

[11]  Andrzej Seweryn,et al.  Elastic stress singularities and corresponding generalized stress intensity factors for angular corners under various boundary conditions , 1996 .

[12]  Martin Costabel,et al.  Construction of Corner Singularities for Agmon‐Douglis‐Nirenberg Elliptic Systems , 1993 .

[13]  Monique Dauge,et al.  Stationary Stokes and Navier-Stokes systems on two-or three-dimensional domains with corners , 1989 .

[14]  M. Dauge.  Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions , 1988 .

[15]  V. Z. Parton,et al.  Integral equations in elasticity , 1982 .

[16]  I. Hlavácek,et al.  Mathematical Theory of Elastic and Elasto Plastic Bodies: An Introduction , 1981 .

[17]  Dominique Leguillon,et al.  Computation of singular solutions in elliptic problems and elasticity , 1987 .

[18]  M. Vishik,et al.  ELLIPTIC PROBLEMS WITH A PARAMETER AND PARABOLIC PROBLEMS OF GENERAL TYPE , 1964 .

[19]  P. Grisvard,et al.  Singularités en elasticité , 1989 .

[20]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .