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Evolution Equations and Control Theory (EECT)
 

On Kelvin-Voigt model and its generalizations

Pages: 17 - 42, Volume 1, Issue 1, June 2012      doi:10.3934/eect.2012.1.17

 
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Miroslav Bulíček - Mathematical Institute of Charles University, Faculty of Mathematics and Physics, Sokolovská 83, 186 75 Prague, Czech Republic (email)
Josef Málek - Mathematical Institute of Charles University, Faculty of Mathematics and Physics, Sokolovská 83, 186 75 Prague, Czech Republic (email)
K. R. Rajagopal - Department of Mechanical Engineering, Texas A&M University, College Station, TX 77845, United States (email)

Abstract: 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.

Keywords:  Kelvin-Voigt model, viscoelastic solid, non-linear wave equation, weak solution, large data existence, uniqueness, regularity.
Mathematics Subject Classification:  Primary: 74D10, 35Q74; Secondary: 35K20, 35B65.

Received: October 2011;      Revised: February 2012;      Available Online: March 2012.

 References