论文标题
残余应力的基础功能
Basis functions for residual stresses
论文作者
论文摘要
我们考虑在任意形状的卸载身体中任意存在的残留应力状态。这些应力必须是自我平衡的，没有牵引力。该主题的常见治疗方法倾向于关注应力的机械起源，或者在某些位置进行压力测量方法。在这里，我们将应力场作为给定的应力场进行，并考虑在给定的身体中近似任何此类应力场的问题，作为可以用作基础的预定场的线性组合。我们详细考虑平面应力状态，并引入一个极端的问题，导致线性特征值问题。该问题的本征函数构成了所有可能具有足够平稳性的残留应力状态的正顺序基础。在数值示例中，连续应力场的$ l^2 $ norm中以及对于具有简单不连续性的应力场中的近似应力场的收敛性。最后，我们概述了将理论扩展到三维身体和压力状态的。我们的方法可用于使用仅由身体几何形状确定的基础函数来描述任意形状的身体中的任意存在的残留应力状态。
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 $L^2$ 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.
