Title: Derivation, characterization, and application of complete orthonormal sequences for representing general three-dimensional states of residual stress Show Chinese title

Title: 残余应力一般三维状态的完整正交序列的推导、表征和应用

Abstract: 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 any 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. Show Chinese abstract

Abstract: 残余应力是未加载物体上的自平衡应力。 由于其复杂的起源，开发能够线性组合以表示任何足够规则的残余应力场的函数是有用的。 在本工作中，我们开发了正交序列，这些序列覆盖了给定三维区域上所有平方可积残余应力场的集合。 这些序列是通过在所有足够规则的残余应力场集合上极化应力梯度的最一般的二次正定泛函，并在规定的归一化条件下得到的；每个这样的泛函都会产生一个序列。 对于泛函中的六阶系数张量为均匀各向同性和归一化条件中的四阶系数张量与单位张量成比例的特殊情况，我们得到了一个三参数子序列族。 经过适当的参数归一化后，我们发现可行参数空间对应于一个半无限条带。 对于进一步简化的球对称情况，我们得到了序列和相关拉格朗日乘数的解析表达式。 值得注意的是，这些序列在整个参数条带上变化很小。 为了说明我们理论结果的适用性，我们采用三个这样的球对称序列来准确逼近两个标准残余应力场。 我们的工作为未来探索不同序列的影响开辟了道路，这些序列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。