On Kelvin-Voigt model and its generalizations

Miroslav Bulíček; Josef Málek; K. R. Rajagopal

doi:10.3934/eect.2012.1.17

Article Contents

2012, Volume 1, Issue 1: 17-42. Doi: 10.3934/eect.2012.1.17

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On Kelvin-Voigt model and its generalizations

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

Received: September 30, 2011

Revised: January 31, 2012

Published: May 2012

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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.

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Mathematics Subject Classification: Primary: 74D10, 35Q74; Secondary: 35K20, 35B65.

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