A comparison between active strain and active stress in transversely isotropic hyperelastic materials

TL;DR

This paper compares active strain and active stress modeling approaches in transversely isotropic hyperelastic materials, revealing they produce different stress responses in shear deformations, highlighting the need for broader experimental validation.

Contribution

It provides a detailed comparison of active strain and active stress methods, showing their differences in shear and emphasizing the importance of considering multiple deformation modes.

Findings

Active stress and active strain produce different shear stress components.

Conditions for equivalence are very restrictive and exclude common energy functions.

Uniaxial data alone cannot determine the better modeling approach.

Abstract

Active materials are media for which deformations can occur in absence of loads, given an external stimulus. Two approaches to the modeling of such materials are mainly used in literature, both based on the introduction of a new tensor: an additive stress $\mathsf{P}_\text{act}$ in the active stress case and a multiplicative strain $\mathsf{F}_a$ in the active strain one. Aim of this paper is the comparison between the two approaches on simple shears. Considering an incompressible and transversely isotropic material, we design constitutive relations for $\mathsf{P}_\text{act}$ and $\mathsf{F}_a$ so that they produce the same results for a uniaxial deformation along the symmetry axis. We then study the two approaches in the case of a simple shear deformation. In a hyperelastic setting, we show that the two approaches produce different stress components along a simple shear, unless some necessary conditions on the strain energy density are fulfilled. However, such conditions are very restrictive and rule out the usual elastic strain energy functionals. Active stress and active strain therefore produce different results in shear, even if they both fit uniaxial data. Our results show that experimental data on the stress-stretch response on uniaxial deformations are not enough to establish which activation approach can capture better the mechanics of active materials. We conclude that other types of deformations, beyond the uniaxial one, should be taken into consideration in the modeling of such materials.