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

On Kelvin-Voigt model and its generalizations

Miroslav Bulíček 1, , Josef Málek 1, and  K. R. Rajagopal 2,

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.
Keywords: viscoelastic solid, non-linear wave equation, large data existence, weak solution, uniqueness, Kelvin-Voigt model, regularity..
Mathematics Subject Classification: Primary: 74D10, 35Q74; Secondary: 35K20, 35B6.
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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show all references

References:
[1]

Math. Methods Appl. Sci., 33 (2010), 1995-2010.  Google Scholar

[2]

in "Mathematical Aspects of Fluid Mechanics" (eds. J. C. Robinson, J. L. Rodrigo and W. Sadowski), London Mathematical Society Lecture Note Series, Cambridge University Press, to appear, 2012. Google Scholar

[3]

SIAM J. Math. Anal., revised version submitted, 2011. Google Scholar

[4]

McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.  Google Scholar

[5]

Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 9 (2010), 1-46.  Google Scholar

[6]

Found. Comput. Math., 10 (2010), 171-190. doi: 10.1007/s10208-010-9061-5.  Google Scholar

[7]

Oxford University Press, Oxford, 2004.  Google Scholar

[8]

Advances in Mathematical Fluid Mechanics, Birkhäuser Verlag, Basel, 2009.  Google Scholar

[9]

Comm. Partial Differential Equations, 35 (2010), 1891-1919.  Google Scholar

[10]

Math. Z., 260 (2008), 355-375. doi: 10.1007/s00209-007-0278-1.  Google Scholar

[11]

Pacific J. Math., 135 (1988), 29-55.  Google Scholar

[12]

SIAM J. Math. Anal., 28 (1997), 363-380. doi: 10.1137/S0036141095285958.  Google Scholar

[13]

Springer-Verlag, 1993. Google Scholar

[14]

Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis, Noordhoff International Publishing, Leyden, Academia, Prague, 1977.  Google Scholar

[15]

Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[16]

Adv. Differential Equations, 6 (2001), 257-302.  Google Scholar

[17]

Applied Mathematics and Mathematical Computation, 13, Chapman & Hall, London, 1996.  Google Scholar

[18]

Mechanics Research Communications, 36 (2009), 232-235. doi: 10.1016/j.mechrescom.2008.09.005.  Google Scholar

[19]

Technical Notes Nat. Adv. Comm. Aeronaut., 1943 (1943), 13 pp.  Google Scholar

[20]

Proc. Roy. Soc. London A, 14 (1865), 289-297. doi: 10.1098/rspl.1865.0052.  Google Scholar

[21]

Arch. Ration. Mech. Anal., 189 (2008), 237-281. doi: 10.1007/s00205-007-0109-x.  Google Scholar

[22]

Annalen der Physik, 283 (1892), 671-693. doi: 10.1002/andp.18922831210.  Google Scholar

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