A new analytical procedure is developed for the deduction of the asymptotic series of the singular solutions in displacements and stresses near the vertex of the linear elastic isotropic corners with the Dirichlet–Robin (fixed-spring) and Neumann–Robin (free-spring) boundary conditions. Under the assumption of antiplane shear loading, the corresponding elastic problem reduces to the Laplace equation for the out-of-plane displacement. In the deduction of such singular solution, the complex variable is used to propose a harmonic function in the form of an asymptotic series including both power and logarithmic terms. This original procedure is suitable for its implementation in a computer algebra software which makes all the necessary symbolic computing, simplifications and rearrangements. This is a key issue due to the fact that the complexity of terms in these series may increase with increasing order of terms. These series are composed by the main terms (also called main singularities), solutions of the corresponding Dirichlet–Neumann or Neumann–Neumann problems, and the associated finite or infinite series of the so-called shadow terms (also called shadow singularities). These terms are determined by solving systems of recursive inhomogeneous Dirichlet–Neumann or Neumann–Neumann problems, respectively. A general classification of the behaviours of the asymptotic series covering all the considered corner problems is introduced. A few examples of the asymptotic series for corners with Dirichlet–Robin and Neumann–Robin boundary conditions are presented to illustrate the capabilities of this procedure.

中文翻译：

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反平面剪切下具有弹簧边界条件的拐角奇异弹性解

开发了一种新的分析程序，用于推导具有 Dirichlet-Robin（固定弹簧）和 Neumann-Robin（自由弹簧）边界的线弹性各向同性角的顶点附近的位移和应力的奇异解的渐近级数使适应。在反平面剪切载荷假设下，相应的弹性问题简化为平面外位移的拉普拉斯方程。在这种奇异解的推导中，复变量被用来提出一个包含幂和对数项的渐近级数形式的调和函数。这个原始程序适合在计算机代数软件中实现，该软件进行所有必要的符号计算、简化和重新排列。这是一个关键问题，因为这些系列中的术语的复杂性可能会随着术语顺序的增加而增加。这些级数由主项（也称为主奇点）、相应 Dirichlet-Neumann 或 Neumann-Neumann 问题的解以及所谓的影子项（也称为影子奇点）的相关有限或无限级数组成。这些项分别通过求解递归非齐次 Dirichlet-Neumann 或 Neumann-Neumann 问题的系统来确定。介绍了涵盖所有考虑的角问题的渐近级数行为的一般分类。给出了具有 Dirichlet-Robin 和 Neumann-Robin 边界条件的角的渐近级数的几个例子，以说明该过程的能力。