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Generalized complex variable boundary integral equation for stress fields and torsional rigidity in torsion problems

机译：扭转问题中应力场和扭转刚度的广义复变边界积分方程

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摘要

Theory of complex variables is a very powerful mathematical technique for solving two-dimensional problems satisfying the Laplace equation. On the basis of the conventional Cauchy integral formula, the conventional complex variable boundary integral equation (CVBIE) can be constructed. The limitation is that the conventional CVBIE is only suitable for holomorphic (analytic) functions, however. To solve for a complex-valued harmonic-function pair without satisfying the Cauchy-Riemann equations, we propose a new boundary element method (BEM) based on the general Cauchy integral formula. The general Cauchy integral formula is derived by using the Borel-Pompeiu formula. The difference between the present CVBIE and the conventional CVBIE is that the former one has two boundary integrals instead of only one boundary integral in the latter one. When the unknown field is a holomorphic function, the present CVBIE can be reduced to the conventional CVBIE. Therefore, the conventional Cauchy integral formula can be viewed as a special case applicable to a holomorphic function. To examine the present CVBIE, we consider several torsion problems in this paper since the two shear stress fields satisfy the Laplace equation but do not satisfy the Cauchy-Riemann equations. Using the present CVBIE, we can directly solve the stress fields and the torsional rigidity simultaneously. Finally, several examples, including a circular bar containing an eccentric inclusion (with dissimilar materials) or hole, a circular bar, elliptical bar, equilateral triangular bar, rectangular bar, asteroid bar and circular bar with keyway, were demonstrated to check the validity of the present method.
机译：复变量理论是一种非常有效的数学技术，用于解决满足Laplace方程的二维问题。在常规柯西积分公式的基础上，可以构造常规复变边界积分方程（CVBIE）。局限性在于，传统的CVBIE仅适用于全纯（解析）函数。为了解决不满足Cauchy-Riemann方程的复值谐波函数对，我们在一般Cauchy积分公式的基础上提出了一种新的边界元方法（BEM）。一般的柯西积分公式是使用Borel-Pompeiu公式导出的。当前CVBIE与常规CVBIE之间的区别在于，前者具有两个边界积分，而不是后者中的一个边界积分。当未知场是全纯函数时，当前的CVBIE可以简化为常规的CVBIE。因此，传统的柯西积分公式可以看作是适用于全纯函数的特殊情况。为了检验当前的CVBIE，我们在本文中考虑了几个扭转问题，因为两个切应力场满足Laplace方程，但不满足Cauchy-Riemann方程。使用当前的CVBIE，我们可以同时直接解决应力场和扭转刚度。最后，展示了几个示例，包括带有偏心夹杂物（具有不同材料）或孔的圆棒，圆棒，椭圆棒，等边三角形棒，矩形棒，小行星棒和带键槽的圆形棒，以检验其有效性。本方法。

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