  The theory of quasi-linear viscoelasticity (QLV) is modified and developed
for transversely isotropic (TI) materials under finite deformation. For the
first time, distinct relaxation responses are incorporated into an integral
formulation of nonlinear viscoelasticity, according to the physical mode of
deformation. The theory is consistent with linear viscoelasticity in the small
strain limit and makes use of relaxation functions that can be determined from
small-strain experiments, given the time/deformation separability assumption.
After considering the general constitutive form applicable to compressible
materials, attention is restricted to incompressible media. This enables a
compact form for the constitutive relation to be derived, which is used to
illustrate the behaviour of the model under three key deformations: uniaxial
extension, transverse shear and longitudinal shear. Finally, it is demonstrated
that the Poynting effect is present in transversely isotropic, neo-Hookean,
modified QLV materials under transverse shear, in contrast to neo-Hookean
elastic materials subjected to the same deformation. Its presence is explained
by the anisotropic relaxation response of the medium.
