残余应力体平面应力状态的应力函数法
The plane stress state of residually stressed bodies: a stress function approach
摘要：
平面弹性薄板的应力状态，以及几何对称的三维固体的应力状态，实际上是二维问题。本文研究了度量不协调二维弹性材料的应力平衡态。该解决方案是由标量应力函数，推广艾里应力函数，这是由几何相容性条件。我们开发了一个微扰近似方法来解决这个应力函数，适用于任何本构关系。我们适用于一个胡克固体的情况下，解决典型的例子中，经典的艾里方法是不准确或不适用的方法。结果表明，在以前的作品中得到的数值结果吻合良好。
英文摘要：
The stressed state of flattened thin elastic sheet, as well as that of translationally symmetric 3D solids, are effectively 2D problems. This paper study equilibrium state-of-stress in metrically-incompatible 2D elastic materials. The solution is represented by a scalar stress function, generalizing the Airy stress function, which is determined by geometric compatibility conditions. We develop a perturbative approximation method for solving this stress function, valid for any constitutive relation. We apply the method for the case of a Hookean solid to solve prototypical examples in which the classical Airy approach is either inaccurate or inapplicable. Results are shown to agree well with numerical results obtained in previous works.
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