In this paper we have derived constitutive equations for the stress tensor of a viscoelastic material with anisotropic rigid particles. We have assumed that the material has fading memory. The expressions are valid for slow and small deformations from equilibrium, and for systems that are nearly isotropic in the absence of shear. For viscoelastic materials with fading memory the free energy depends not only on the deformation and degree of orientation in the current configuration, but also on the deformation and orientation history of the system. We have incorporated this dependence by incorporating a dependence on the relative Finger tensor and the alignment difference history in the expression for the free energy. We have used the entropy balance and the expression for the free energy to arrive at a set of fluxes and corresponding driving forces for this type of material. We have used this set of fluxes and driving forces to derive expressions for the stress tensor. We have calculated the explicit form of these expressions for a simple shear deformation in the xy-plane with shear rate γ̇. For fully isotropic materials the expression for the xy-component of the stress reduces to an equation containing only odd powers of γ. The inclusion of a non-zero value for the alignment difference history leads to an additional set of terms in the equation, all proportional to Qdxy (the xy-component of the alignment history tensor), and all proportional to even powers of γ. In a Fourier transform rheometry experiment the even powers of γ introduce small even harmonics in the frequency spectrum of the stress response. The expression for the first normal stress contains only even powers of γ, and the inclusion of the alignment difference history does not lead to additional harmonics in the frequency spectrum. For the second normal stress, inclusion of the alignment history leads to additional terms in the expression proportional to Qdyy and Qdxy. For an isotropic material the spectrum of the second normal stress contains only even harmonics. The terms proportional to Qdxy are all proportional to odd powers of γ, and will generate additional odd harmonics in the frequency spectrum.
Sagis, L. M. C., & van der Linden, E. (2001). Viscoelastic materials with anisotropic rigid particles: stress-deformation behavior. Physica A, 297, 303-320. https://doi.org/10.1016/S0378-4371(01)00260-6