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Active materials are media for which deformations can occur in absence ofloads, given an external stimulus. Two approaches to the modeling of suchmaterials are mainly used in literature, both based on the introduction of anew tensor: an additive stress $\mathsf{P}_\text{act}$ in the active stresscase and a multiplicative strain $\mathsf{F}_a$ in the active strain one. Aimof this paper is the comparison between the two approaches on simple shears.Considering an incompressible and transversely isotropic material, we designconstitutive relations for $\mathsf{P}_\text{act}$ and $\mathsf{F}_a$ so thatthey produce the same results for a uniaxial deformation along the symmetryaxis. We then study the two approaches in the case of a simple sheardeformation. In a hyperelastic setting, we show that the two approaches producedifferent stress components along a simple shear, unless some necessaryconditions on the strain energy density are fulfilled. However, such conditionsare 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 onthe stress-stretch response on uniaxial deformations are not enough toestablish which activation approach can capture better the mechanics of activematerials. We conclude that other types of deformations, beyond the uniaxialone, should be taken into consideration in the modeling of such materials.

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