构造极坐标中Airy应力函数的观察法

通过引入Airy应力函数，平面问题可以归结为在给定的边界条件下求解一个双调和方程．因此对双调和函数性质的研究将有利于平面问题的求解．首先给出一个有关双调和函数的引理，并分别从复变和微分两种角度提供该引理的证明．借助这个引理，提出了一种构造极坐标中Airy应力函数的观察法．最后，举例说明了该观察法在几个经典平面问题中的应用．这些例子说明，利用本的观察法可以将某些平面问题应力函数构造的过程简单化。

A DIRECT METHOD FOR CONSTRUCTING AIRY STRESS FUNCTION IN POLAR COORDINATES

With Airy stress function, a plane elasticity problem can be simplified to solve a biharmonic function equation. In this paper, we introduce a lemma concerning the biharmonic function. In order to prove the lemma, we present two different methods from two different angles in the first section. On the basis of the lemma, a direct method for constructing Airy stress functions in polar coordinates is derived. Finally, some examples are provided to illustrate how the direct method works in some classic plane cases and what we can benefit from this direct method for the stress function construction. These application examples indicate that the direct method derived in this paper simplifies the process of constructing the Airy function in specific plane problems.