A closed form exact formulation of the spectral representation of a second-order symmetric tensor and of its derivatives

TL;DR

This paper presents a closed-form exact formulation for the spectral decomposition of symmetric second-order tensors and their derivatives, simplifying computations in computational mechanics applications such as elasto-plasticity and stress analysis.

Contribution

It introduces a comprehensive method to compute the spectral decomposition and derivatives of symmetric tensors in closed form, improving implementation and accuracy in computational mechanics.

Findings

Provides a closed-form solution for tensor spectral decomposition.

Simplifies computation of tensor derivatives for nonlinear algorithms.

Enhances convergence and robustness in stress and strain calculations.

Abstract

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.