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

This note contains some observations on primary matrix functions and different notions of monotonicity with relevance toward constitutive relations in nonlinear elasticity. Focusing 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.