Visco-hyperelastic law for finite deformations: a frequency analysis

Some biological tissues are repeatedly stimulated under cyclic loading, and this stimulation can be combined with large pressures, thus leading to large deformations. For such applications, visco-hyperelastic models have been proposed in the literature and used in finite-element studies. An extensively used quasi-linear model (QLVH), which assumes linear evolution equations, is compared with a nonlinear model (NLVH), which assumes a multiplicative split of the deformation gradient. The comparison is made here using sets of simulations covering a large frequency range. Lost and stored energies are computed, and the additional parameter of the NLVH model is set to two values found in the literature (NLVH-2 and NLVH-30 models). The predicted behaviour is very similar for all models at small strains, with each time constant (and corresponding viscous modulus) being associated with a damping peak and a stored-energy increase. When the strain amplitude is increased, the ratio of lost to stored energy increases for the QLVH model, but decreases for the NLVH models. The NLVH-30 model also displays a shift of the peak damping towards higher frequencies. Before reaching a steady state, all models display a decay of energy independent of the frequency, and the additional parameter of the NLVH model permits the modelling of complex types of evolution of the damping. In conclusion, this study compares the behaviour of two viscous hyper-elastic laws to allow an informed choice between them.