Some remarks on the monotonicity of primary matrix functions on the set of symmetric matrices

TL;DR

This paper examines various notions of monotonicity for primary matrix functions on symmetric matrices, with implications for nonlinear elasticity and constitutive inequalities, clarifying previous statements and comparing criteria.

Contribution

It provides a comparative analysis of monotonicity criteria for primary matrix functions and clarifies a potentially misinterpreted statement in the context of nonlinear elasticity.

Findings

Different criteria for monotonicity are compared and analyzed.

Results are applicable to the true-stress-true-strain monotonicity condition.

Clarification of a previous statement by Jog and Patil.

Abstract

This note contains some observations on primary matrix functions and different notions of monotonicity with relevance towards constitutive relations in nonlinear elasticity. Focussing on primary matrix functions on the set of symmetric matrices, we discuss and compare different criteria for monotonicity. The demonstrated results are particularly applicable to computations involving the true-stress-true-strain monotonicity condition, a constitutive inequality recently introduced in an Arch. Appl. Mech. article by C.S. Jog and K.D. Patil. We also clarify a statement by Jog and Patil from the same article which could be misinterpreted.