R-函数理论在梯形截面柱弹性扭转问题中的应用

Application of R-function theory to the problem of elastic torsion with trapezium sections

将弹性扭转问题看成为泊松方程的边值问题,利用泊松方程的基本解构造了一个函数,推导出第二类Fredholm积分方程.应用R-函数理论,构建了一个规范化方程.通过寻找适当的规范化方程,来表示问题的边界,并证明积分方程核的奇异性被克服了.通过研究梯形截面弹性体的扭转问题,表明结果与有限元数值计算结果很接近,该方法具有较高的精度,为边值问题的研究和求解提供了一种新的数学方法.该方法同样可以解决位势问题,还可以用来讨论其他更为复杂的算子,并且适用于其他形状的情形.

Elastic torsion may be considered as boundary value problem of Poisson＇s equation. A function was set up to deduce a second kind of Fredholm integral equation by using the basic solution of Poisson＇s equation. According to R-function theory, a suitable form of the normalized function was build. It was proved that the irregularity of the kernel of integral equation could be overcome by choosing a suitable form of the normalized function. Via studying elastic torsion with trapezium sections, the results obtained by this method could approach the ones obtained by finite method. The numerical results showed that the method is of high accuracy and can be used to solve other boundary value problems in physics and mechanics. The method can be also used in the problems of stability in engineering since the problems were potential. Certainly this method may be also used in other complex operators and instances.