On Kelvin-Voigt model and its generalizations

We consider a generalization of the Kelvin-Voigt model wherethe elastic part of the Cauchy stress depends non-linearly onthe linearized strain and the dissipative part of the Cauchystress is a nonlinear function of the symmetric part of thevelocity gradient. The assumption that the Cauchy stressdepends non-linearly on the linearized strain can be justifiedif one starts with the assumption that the kinematicalquantity, the left Cauchy-Green stretch tensor, is a nonlinearfunction of the Cauchy stress, and linearizes under theassumption that the displacement gradient is small. Long-timeand large data existence, uniqueness and regularity propertiesof weak solution to such a generalized Kelvin-Voigt model areestablished for the non-homogeneous mixed boundary valueproblem. The main novelty with regard to the mathematical analysis consistsin including nonlinear (non-quadratic) dissipation in the problem.