The phenomenological model of hyper viscoelasticity of elastomers with a constant orientation of the main stress axes in the accompanying frame of reference

The Bergström-Boyce hyper-viscoelasticity model, which is based on considerations of the microstructure of rubber-like materials and uses the multiplicative decomposition of the total strain gradient, is considered and analyzed. Special attention is paid to the choice of the single-valuedness condition, which determines the rotation of the intermediate (relaxed) configuration and ensures the uniqueness of the multiplicative decomposition of the gradient total strain to the gradient of elastic deformations and the gradient of viscous deformations. To verify the validity of the statement that this choice is not essential, we found a solution of the test problem of simple shear according to the Bergström-Boyce model is obtained for the three most frequently used single-valuedness conditions. The results of numerical calculations showed a significant discrepancy for dynamic stresses and a less significant discrepancy for total stresses. To separate the permissible single-valuedness conditions from the physically unacceptable single-valuedness conditions, it is proposed to use the principle of the material frame-indifference. For this purpose, of we studied the transformation of the elastic deformations and viscous deformations gradients when replacing the reference system, to which there is no single point of view in the scientific literature. For the most frequently proposed and used single-valuedness conditions, it is shown which of them do not depend on the reference system choice. Since the list of such kind of allowable ratios can be expanded many times, the choice a single-valuedness condition should be solved in the same way as it is done, e.g. for elastic potentials: a proper formulation and conducting experimental studies, or a theoretical study of the material microstructure. A phenomenological model of hyper-viscoelasticity is proposed, based on a one-dimensional Kelvin-Poynting model and limited to the case when the main axes of stresses and strains (full, elastic and viscous ones) coincide and do not change their orientation relative to the material lines (fibers). This ensures the uniqueness of the corresponding multiplicative decomposition. In order to expand the range of strain rates when describing experimental data, the dependence of the viscosity coefficient on the second invariant of the right Cauchy-Green viscous strain measure in the power law of the apparent viscosity of the Reiner-Rivlin model is taken into account, which generalizes the corresponding dependence of the Bergström-Boyce model. The developed mathematical model of hyper-viscoelasticity of rubber-like materials is intended for the stress-strain state calculation of highly elastic shells of rotation under symmetric loading.