二阶对称张量谱表示及其导数的闭式精确公式
A closed form exact formulation of the spectral representation of a second-order symmetric tensor and of its derivatives
AI总结：
针对计算力学中对称二阶张量特征基及导数计算复杂、多特征值重合时求解困难的问题，提出一种简洁通用的闭式计算方法，可推导张量谱表示及其自身导数的闭式解，简化谱分解的工程实现。
摘要：
对称二阶张量的谱分解在计算力学的诸多领域中得到了广泛应用。例如，在大应变和大转动条件下的弹塑性力学中，给定柯西变形张量后，计算对数应变张量是一项核心步骤。\n 近年来，该方法也被应用于小应变各向同性塑性力学中，用于根据应力张量的特征值重构应力张量，从而能够在不变量空间中构建预测-校正返回算法。这类算法不仅减少了本构层面的未知量数量，还能正确处理塑性法向未定义的应力状态，因此相较于标准方法具备更好的收敛性。\n 虽然对称二阶张量的特征值可以通过张量不变量直接计算，但其特征基的计算难度更高，尤其是在两个或多个特征值相等的情况下。此外，当采用Newton-Rhapson（牛顿-拉夫逊）算法求解计算力学中的非线性问题时，还需要计算特征基的张量导数以组装切线矩阵，而该导数的计算复杂度更高。\n 本文提出了一种简洁且全面的方法，可用于推导二阶张量的闭式表示，以及该张量相对于自身的各阶导数的闭式表达，从而简化计算力学应用中张量谱分解的实现过程。
英文摘要：
The spectral decomposition of a symmetric, second-order tensor is widely adopted in many fields of Computational Mechanics. As an example, in elasto-plasticity under large strain and rotations, given the Cauchy deformation tensor, it is a fundamental step to compute the logarithmic strain tensor. Recently, this approach has been also adopted in small-strain isotropic plasticity to reconstruct the stress tensor as a function of its eigenvalues, allowing the formulation of predictor-corrector return algorithms in the invariants space. These algorithms not only reduce the number of unknowns at the constitutive level, but also allow the correct handling of stress states in which the plastic normals are undefined, thus ensuring a better convergence with respect to the standard approach. While the eigenvalues of a symmetric, second-order tensor can be simply computed as a function of the tensor invariants, the computation of its eigenbasis can be more difficult, especially when two or more eigenvalues are coincident. Moreover, when a Newton-Rhapson algorithm is adopted to solve nonlinear problems in Computational Mechanics, also the tensorial derivatives of the eigenbasis, whose computation is still more complicate, are required to assemble the tangent matrix. A simple and comprehensive method is presented, which can be adopted to compute a closed form representation of a second-order tensor, as well as their derivatives with respect to the tensor itself, allowing a simpler implementation of spectral decomposition of a tensor in Computational Mechanics applications.
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