  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.
