The particular solutions of a two-dimensional infinite wedge for various boundary conditions with $lnr$ weak singularity have been investigated in this paper. The relations of the weak singularities and the discontinuities of the first kind of the boundary variables at a corner of a two-dimensional elastic body have been established. By using the relations, the singular behaviors of the unknown boundary variables at a corner of an elastic body can be obtained before solving the boundary value problem by using the boundary element method (BEM). Especially, if the boundary conditions at a corner are displacements prescribed, the values of the unknown tractions at the corner can be determined in advance. Thus, the difficulty related to the multivalued tractions at a corner in BEM analysis for problems with boundary displacements prescribed has been overcome completely. In addition, more appropriate shape functions for the unknown boundary field variables of a corner element can be constructed, and the accuracy of the BEM may be greatly increased.

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