Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A new representation formula for the logarithmic corotational derivative -- a case study in application of commutator based functional calculus

Michal Bathory, Miroslav Bulíček, Josef Málek, Vít Průša

TL;DR

The paper addresses obtaining a compact, non-spectral representation of the logarithmic spin underpinning the logarithmic corotational derivative, using a new commutator-based functional calculus. It derives the closed-form representation $\mathbb{\Omega}^{\mathrm{log}} = \mathbb{W} - \sigma(\mathrm{ad}_{\mathbb{H}})\mathbb{D}$ with $\sigma(x)=\coth x - \frac{1}{x}$ and $\mathrm{ad}$ the commutator with the Hencky strain $\mathbb{H}$, and extends the framework to matrix-logarithm identities and monotonicity of stress–strain relations. The contributions include the representation formula, derivative identities for the matrix logarithm, and a monotonicity analysis for isotropic tensor functions, illustrating the method's broader applicability to tensor/matrix analysis. The work suggests that commutator-based calculus can bypass spectral decompositions and remain meaningful beyond power-series convergence, with planned rigorous development in symmetric-tensor settings (Bathory-calculus).

Abstract

The logarithmic corotational derivative is a key concept in rate-type constitutive relations in continuum mechanics. The derivative is defined in terms of the logarithmic spin tensor, which is a skew-symmetric tensor/matrix given by a relatively complex formula. Using a newly developed commutator based functional calculus, we derive a new representation formula for the logarithmic spin tensor. In addition to the result on the logarithmic corotational derivative we also use the newly developed functional calculus to answer some problems regarding the matrix logarithm and the monotonicity of stress-strain relations. These results document that the commutator based functional calculus is of general use in tensor/matrix analysis, and that the calculus allows one to seamlessly work with tensor/matrix valued functions and their derivatives.

A new representation formula for the logarithmic corotational derivative -- a case study in application of commutator based functional calculus

TL;DR

The paper addresses obtaining a compact, non-spectral representation of the logarithmic spin underpinning the logarithmic corotational derivative, using a new commutator-based functional calculus. It derives the closed-form representation with and the commutator with the Hencky strain , and extends the framework to matrix-logarithm identities and monotonicity of stress–strain relations. The contributions include the representation formula, derivative identities for the matrix logarithm, and a monotonicity analysis for isotropic tensor functions, illustrating the method's broader applicability to tensor/matrix analysis. The work suggests that commutator-based calculus can bypass spectral decompositions and remain meaningful beyond power-series convergence, with planned rigorous development in symmetric-tensor settings (Bathory-calculus).

Abstract

The logarithmic corotational derivative is a key concept in rate-type constitutive relations in continuum mechanics. The derivative is defined in terms of the logarithmic spin tensor, which is a skew-symmetric tensor/matrix given by a relatively complex formula. Using a newly developed commutator based functional calculus, we derive a new representation formula for the logarithmic spin tensor. In addition to the result on the logarithmic corotational derivative we also use the newly developed functional calculus to answer some problems regarding the matrix logarithm and the monotonicity of stress-strain relations. These results document that the commutator based functional calculus is of general use in tensor/matrix analysis, and that the calculus allows one to seamlessly work with tensor/matrix valued functions and their derivatives.

Paper Structure

This paper contains 9 sections, 11 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Representation formula for the logarithmic spin
  3. Summary of known formulae for the commutator operator
  4. Proof of representation formula in Theorem \ref{['thr:1']}
  5. Applications
  6. Identities involving the derivative of matrix logarithm
  7. Monotonicity of stress--strain relations
  8. Conclusion
  9. Classical Lie algebra formulae from the perspective of power series based calculus

Key Result

Theorem 1

Let $\accentset{\medcircle_{\star}}{\overline{\left( \cdot \right)}}$ denote the corotational derivative eq:3 for which the derivative of the Hencky strain tensor $\mathbb{H}$, $\mathbb{H} =_{\mathrm{def}} \frac{1}{2} \ln \mathbb{B}$, yields the symmetric part of the velocity gradient $\mathbb{D}$, Then the spin $\mathbb{\Omega}^{\star}$ in eq:3 is the logarithmic spin $\mathbb{\Omega}^{\mathrm{l

Theorems & Definitions (22)

  • Theorem 1: Representation formula for the logarithmic spin
  • Lemma 1: Formulae for the commutator operator
  • Lemma 2: Identities for matrix logarithm
  • proof
  • proof : Proof of Theorem \ref{['thr:1']}
  • Lemma 3: Formulae for the derivative of matrix logarithm
  • proof
  • Lemma 4: Formula for the derivative of matrix logarithm
  • proof
  • Lemma 5: Commutativity of $\frac{\partial {{\mathfrak{f}}({\mathbb{A}})}}{\partial {{\mathbb{A}}}}$ and $\mathop{\mathrm{ad}}\nolimits_{{\mathbb{A}}}$ for isotropic tensorial functions
  • ...and 12 more