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