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.

翻译：二阶对称张量的谱分解广泛应用于计算力学的多个领域。例如，在大应变和旋转条件下的弹塑性分析中，给定柯西变形张量，这是计算对数应变张量的基本步骤。近年来，该方法也被应用于小应变各向同性塑性理论中，通过特征值重构应力张量，从而允许在不变空间中制定预测-校正返回算法。这些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线未定义的应力状态，从而比标准方法具有更好的收敛性。虽然二阶对称张量的特征值可以简单地通过张量不变量计算，但特征基的计算可能更为困难，尤其是当两个或多个特征值相等时。此外，当采用牛顿-拉夫森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（其计算更为复杂），以组装切线矩阵。本文提出了一种简单且全面的方法，可用于计算二阶对称张量的封闭形式表示及其对张量本身的导数，从而在计算力学应用中更简洁地实现张量的谱分解。