<p>We study the incompressible axial extension of a cylindrical, fiber-reinforced non-linear elastic material, where fibers are aligned with the cylinder axis. We compare three transversely isotropic constitutive models that encode matrix-fiber coupling through different sets of invariants and a small number of material parameters. Under axial symmetry and the isochoric constraint, we derive closed-form relations between axial stretch and axial stress that can be used as a benchmark for model comparison and parameter sensitivity. The resulting expressions make transparent how distinct fiber terms affect the axial response and the calibration of material parameters from standard uniaxial tests. Finally, we report the corresponding linearized limits, connecting the finite-strain formulations to classical small-strain transversely isotropic elasticity and supporting parameter identification in the infinitesimal-strain regime.</p>