边界为三角形或四边形时反平面弹性或Laplace方程中的退化尺寸问题

Degenerate Scale Problem in Antiplane Elasticity or Laplace Equation for Contour Shapes of Triangles or Quadrilaterals

对于反平面弹性或Laplace方程的外部边值问题,给出了三角形或四边形周界时一系列退化尺寸问题的解,并利用了Schwarz-Christoffel保角映象.证实当某一尺寸"R"等于它的临界值或一个单位值时,一个形式上简明的复位函数满足单位圆上位移为0的条件,或w=0.这就意味着在实际平面上的退化尺寸已经得到.最后,退化尺寸可从某些特殊积分得出,这些积分依赖于保角映象中的诸参数.给出了三角形或四边形周界时一系列退化尺寸问题的数值结果.

Several solutions of the degenerate scale for shapes of triangles or quadrilaterals in an exterior boundary value problem of antiplane elasticity or Laplace equation were provided. The Schwarz-Christoffel mapping was used thoroughly. It is found that a complex potential with simple form in the mapping plane satisfies the vanishing displacement condition （ or w = 0） a- long the boundary of the unit circle when a dimension ＂R＂ reaches its critical value 1. This means the degenerate size in the physical plane is also achieved. Finally, the degenerate scales can be evaluated from some particular integrals that depend on some parameters in the mapping function. A lot of numerical results of degenerate sizes for shapes of triangles or quadrilaterals are provided.