Stress Asymmetry in Hard Magnetic Soft Materials

TL;DR

This paper investigates the symmetry of Cauchy stress in hard magnetic soft materials, revealing that different modeling formulations can produce symmetric or asymmetric stresses, with implications for continuum magnetomechanical theories.

Contribution

It demonstrates that the choice of referential versus current magnetization formulations affects the symmetry of the resulting Cauchy stress tensor.

Findings

Referential magnetization formulation yields symmetric Cauchy stress.

Current magnetization formulation generally yields asymmetric Cauchy stress.

At energy-minimizing equilibrium, both formulations produce symmetric stresses with the same divergence.

Abstract

Hard magnetic soft materials -- soft polymers embedded with hard magnetic particles -- are modeled using continuum magnetomechanical formulations in which the deformation and the magnetization field are the primary kinematic variables. A recent question in such formulations is whether the Cauchy stress is symmetric, which is directly related to frame invariance and angular momentum balance. This note discusses energetically equivalent formulations, related by a change of variables between referential and current descriptions of the magnetization, and shows that they generally yield different Cauchy stresses, including a change in their symmetry. Specifically, the formulation based on a referential magnetization produces a symmetric Cauchy stress, while that based on a current magnetization generally yields an asymmetric Cauchy stress. We highlight that when the internal variable (magnetization field) is at the energy-minimizing equilibrium configuration, the divergences of these stresses are the same, and both stresses are symmetric.