Nonlinear Relations of Viscous Stress and Strain Rate in Nonlinear Viscoelasticity

Abstract We consider a Kelvin–Voigt model for viscoelastic second-grade materials, where the elastic and the viscous stress tensor both satisfy frame-indifference. Using a rigidity estimate by Ciarlet and Mardare (J. Math. Pures Appl. 104:1119–1134, 2015), existence of weak solutions is shown by means of a frame-indifferent time-discretization scheme. Further, the result includes viscous stress tensors which can be calculated by nonquadratic polynomial densities. Afterwards, we investigate the long-time behavior of solutions in the case of small external loading and initial data. Our main tool is the abstract theory of metric gradient flows (Ambrosio et al. in Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhäuser, Basel, 2005).