Abstract We consider arbitrary preexisting residual stress states in arbitrarily shaped, unloaded bodies. These stresses must be self-equilibrating and traction free. Common treatments of the topic tend to focus on either the mechanical origins of the stress, or methods of stress measurement at certain locations. Here we take the stress field as given and consider the problem of approximating any such stress field, in a given body, as a linear combination of predetermined fields which can serve as a basis. We consider planar stress states in detail, and introduce an extremization problem that leads to a linear eigenvalue problem. Eigenfunctions of that problem form an orthonormal basis for all possible residual stress states of sufficient smoothness. In numerical examples, convergence of the approximating stress fields is demonstrated in the L2 norm for continuous stress fields as well as for a stress field with a simple discontinuity. Finally, we outline the extension of our theory to three dimensional bodies and states of stress. Our approach can be used to describe arbitrary preexisting residual stress states in arbitrarily shaped bodies using basis functions that are determined by the body geometry alone.

中文翻译：

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残余应力的基函数

摘要 我们考虑任意形状的无载荷物体中任意预先存在的残余应力状态。这些应力必须是自平衡和无牵引的。该主题的常见处理往往侧重于应力的机械起源或某些位置的应力测量方法。在这里，我们将应力场视为给定，并考虑将给定物体中的任何此类应力场近似为预定场的线性组合的问题，该组合可以作为基础。我们详细考虑了平面应力状态，并引入了导致线性特征值问题的极端化问题。该问题的特征函数形成了所有可能的足够平滑的残余应力状态的正交基础。在数值例子中，近似应力场的收敛在连续应力场以及具有简单不连续性的应力场的 L2 范数中得到证明。最后，我们概述了我们的理论对三维物体和应力状态的扩展。我们的方法可用于使用仅由主体几何形状确定的基函数来描述任意形状主体中任意预先存在的残余应力状态。