Residual stresses are self-equilibrated stresses on unloaded bodies. Owing to their complex origins, it is useful to develop functions that can be linearly combined to represent sufficiently regular residual stress field. In this work, we develop orthonormal sequences that span the set of all square-integrable residual stress fields on a given three-dimensional region. These sequences are obtained by extremizing the most general quadratic, positive-definite functional of the stress gradient on the set of all sufficiently regular residual stress fields subject to a prescribed normalization condition; each such functional yields a sequence. For the special case where the sixth-order coefficient tensor in the functional is homogeneous and isotropic and the fourth-order coefficient tensor in the normalization condition is proportional to the identity tensor, we obtain a three-parameter subfamily of sequences. Upon a suitable parameter normalization, we find that the viable parameter space corresponds to a semi-infinite strip. For a further specialized spherically symmetric case, we obtain analytical expressions for the sequences and the associated Lagrange multipliers. Remarkably, these sequences change little across the entire parameter strip. To illustrate the applicability of our theoretical findings, we employ three such spherically symmetric sequences to accurately approximate two standard residual stress fields. Our work opens avenues for future exploration into the implications of different sequences, achieved by altering both the spatial distribution and the material symmetry class of the coefficient tensors, toward specific objectives.

中文翻译：

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用于表示残余应力的一般三维状态的完整正交序列的推导、表征和应用


残余应力是空载物体上的自平衡应力。由于其复杂的起源，开发可以线性组合的函数来表示足够规则的残余应力场是有用的。在这项工作中，我们开发了跨越给定三维区域上所有平方可积残余应力场集合的正交序列。这些序列是通过在规定的归一化条件下对所有足够规则的残余应力场集上的应力梯度的最一般二次正定函数进行极值化而获得的；每个这样的函数都会产生一个序列。对于泛函中的六阶系数张量是齐次且各向同性且归一化条件下的四阶系数张量与恒等张量成正比的特殊情况，我们得到了序列的三参数子族。经过适当的参数归一化，我们发现可行的参数空间对应于半无限条带。对于进一步特殊的球对称情况，我们获得了序列和相关拉格朗日乘子的解析表达式。值得注意的是，这些序列在整个参数条上几乎没有变化。为了说明我们的理论发现的适用性，我们采用三个这样的球对称序列来精确近似两个标准残余应力场。我们的工作为未来探索不同序列的含义开辟了途径，通过改变系数张量的空间分布和材料对称性类别来实现特定目标。