Tensor decomposition for modified quasi-linear viscoelastic models: Towards a fully non-linear theory

We discuss the decomposition of the tensorial relaxation function for isotropic and transversely isotropic (TI) modified quasi-linear viscoelastic (MQLV) models. We show how to formulate the constitutive equation using a convenient decomposition of the relaxation tensor into scalar components and tensorial bases. We show that the bases must be symmetrically additive, i.e., they must sum up to the symmetric fourth-order identity tensor. This is a fundamental property both for isotropic and anisotropic bases that ensures the constitutive equation is consistent with the elastic limit. We provide two robust methods to obtain such bases. Furthermore, we show that, in the TI case, the bases are naturally deformation-dependent for deformation modes that induce rotation or stretching of the fibres. Therefore, the MQLV framework allows to capture the non-linear phenomenon of strain-dependent relaxation, which has always been a criticised limitation of the original quasi-linear viscoelastic theory. We illustrate this intrinsic non-linear feature, unique to the MQLV model, with two examples (uni-axial extension and perpendicular shear).