# American Institute of Mathematical Sciences

June  2012, 1(1): 17-42. doi: 10.3934/eect.2012.1.17

## On Kelvin-Voigt model and its generalizations

 1 Mathematical Institute of Charles University, Faculty of Mathematics and Physics, Sokolovská 83, 186 75 Prague, Czech Republic, Czech Republic 2 Department of Mechanical Engineering, Texas A&M University, College Station, TX 77845, United States

Received  October 2011 Revised  February 2012 Published  March 2012

We consider a generalization of the Kelvin-Voigt model where the elastic part of the Cauchy stress depends non-linearly on the linearized strain and the dissipative part of the Cauchy stress is a nonlinear function of the symmetric part of the velocity gradient. The assumption that the Cauchy stress depends non-linearly on the linearized strain can be justified if one starts with the assumption that the kinematical quantity, the left Cauchy-Green stretch tensor, is a nonlinear function of the Cauchy stress, and linearizes under the assumption that the displacement gradient is small. Long-time and large data existence, uniqueness and regularity properties of weak solution to such a generalized Kelvin-Voigt model are established for the non-homogeneous mixed boundary value problem. The main novelty with regard to the mathematical analysis consists in including nonlinear (non-quadratic) dissipation in the problem.
Citation: Miroslav Bulíček, Josef Málek, K. R. Rajagopal. On Kelvin-Voigt model and its generalizations. Evolution Equations & Control Theory, 2012, 1 (1) : 17-42. doi: 10.3934/eect.2012.1.17
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##### References:
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