Karel Tůma, Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, Poland Joint work with J. Hron, J. Málek, and K. R. Rajagopal For the description of complicated behavior and response of viscoelastic materials we derive a new thermodynamically compatible rate-type fluid model. We show that it is capable of capturing non-linear relaxation responses observed in the experimental data with asphalt binder while the standard linear models are not. Using this model we performed several computer simulations in time-varying domains. In particular, for a problem of pressing rectangular piece of viscoelastic material we show the influence of the material parameters on the behavior of viscoelastic material and we study the convergence with respect to the mesh size and time step. Further we compare the difference in the full simulaton of the linear and non-linear model for the same material parameters.