# 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 and Control Theory, 2012, 1 (1) : 17-42. doi: 10.3934/eect.2012.1.17
##### References:
 [1] M. Bulíček, F. Ettwein, P. Kaplický and D. Pražák, On uniqueness and time regularity of flows of power-law like non-Newtonian fluids, Math. Methods Appl. Sci., 33 (2010), 1995-2010. [2] M. Bulíček, P. Gwiazda, J. Málek, K. R. Rajagopal and A. Świerczewska-Gwiazda, On flows of fluids described by an implicit constitutive equation characterized by a maximal monotone graph, 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. [3] M. Bulíček, P. Gwiazda, J. Málek and A. Świerczewska-Gwiazda, On unsteady flows of implicitly constituted incompressible fluids, SIAM J. Math. Anal., revised version submitted, 2011. [4] E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. [5] L. Diening, M. Růžička and J. Wolf, Existence of weak solutions for unsteady motions of generalized Newtonian fluids, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 9 (2010), 1-46. [6] E. Emmrich and M. Thalhammer, Convergence of a time discretisation for doubly nonlinear evolution equations of second order, Found. Comput. Math., 10 (2010), 171-190. doi: 10.1007/s10208-010-9061-5. [7] E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford Lecture Series in Mathematics and its Applications, 26, Oxford University Press, Oxford, 2004. [8] E. Feireisl and A. Novotný, "Singular Limits in Thermodynamics of Viscous Fluids," Advances in Mathematical Fluid Mechanics, Birkhäuser Verlag, Basel, 2009. [9] J. Frehse, J. Málek and M. Růžička, Large data existence result for unsteady flows of inhomogeneous shear-thickening heat-conducting incompressible fluids, Comm. Partial Differential Equations, 35 (2010), 1891-1919. [10] J. Frehse and M. Růžička, Non-homogeneous generalized Newtonian fluids, Math. Z., 260 (2008), 355-375. doi: 10.1007/s00209-007-0278-1. [11] A. Friedman and J. Nečas, Systems of nonlinear wave equations with nonlinear viscosity, Pacific J. Math., 135 (1988), 29-55. [12] G. Friesecke and G. Dolzmann, Implicit time discretization and global existence for a quasi-linear evolution equation with nonconvex energy, SIAM J. Math. Anal., 28 (1997), 363-380. doi: 10.1137/S0036141095285958. [13] Y. Fung, "Biomechanics: Mechanical Properties of Living Tissues," Springer-Verlag, 1993. [14] A. Kufner, O. John and S. Fučík, "Function Spaces," Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis, Noordhoff International Publishing, Leyden, Academia, Prague, 1977. [15] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," Dunod, Gauthier-Villars, Paris, 1969. [16] J. Málek, J. Nečas and M. Růžička, On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: The case $p\geq2$, Adv. Differential Equations, 6 (2001), 257-302. [17] J. Málek, J. Nečas, M. Rokyta and M. Růžička, "Weak and Measure-Valued Solutions to Evolutionary PDEs," Applied Mathematics and Mathematical Computation, 13, Chapman & Hall, London, 1996. [18] K. R. Rajagopal, A note on a reappraisal and generalization of the Kelvin-Voigt Model, Mechanics Research Communications, 36 (2009), 232-235. doi: 10.1016/j.mechrescom.2008.09.005. [19] W. Ramberg and W. R. Osgood, Description of stress-strain curves by three parameters, Technical Notes Nat. Adv. Comm. Aeronaut., 1943 (1943), 13 pp. [20] W. Thompson, On the elasticity and viscosity of metals, Proc. Roy. Soc. London A, 14 (1865), 289-297. doi: 10.1098/rspl.1865.0052. [21] B. Tvedt, Quasilinear equations for viscoelasticity of strain-rate type, Arch. Ration. Mech. Anal., 189 (2008), 237-281. doi: 10.1007/s00205-007-0109-x. [22] W. Voigt, Ueber innere Reibung fester Körper, insbesondere der Metalle, Annalen der Physik, 283 (1892), 671-693. doi: 10.1002/andp.18922831210.

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##### References:
 [1] M. Bulíček, F. Ettwein, P. Kaplický and D. Pražák, On uniqueness and time regularity of flows of power-law like non-Newtonian fluids, Math. Methods Appl. Sci., 33 (2010), 1995-2010. [2] M. Bulíček, P. Gwiazda, J. Málek, K. R. Rajagopal and A. Świerczewska-Gwiazda, On flows of fluids described by an implicit constitutive equation characterized by a maximal monotone graph, 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. [3] M. Bulíček, P. Gwiazda, J. Málek and A. Świerczewska-Gwiazda, On unsteady flows of implicitly constituted incompressible fluids, SIAM J. Math. Anal., revised version submitted, 2011. [4] E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. [5] L. Diening, M. Růžička and J. Wolf, Existence of weak solutions for unsteady motions of generalized Newtonian fluids, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 9 (2010), 1-46. [6] E. Emmrich and M. Thalhammer, Convergence of a time discretisation for doubly nonlinear evolution equations of second order, Found. Comput. Math., 10 (2010), 171-190. doi: 10.1007/s10208-010-9061-5. [7] E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford Lecture Series in Mathematics and its Applications, 26, Oxford University Press, Oxford, 2004. [8] E. Feireisl and A. Novotný, "Singular Limits in Thermodynamics of Viscous Fluids," Advances in Mathematical Fluid Mechanics, Birkhäuser Verlag, Basel, 2009. [9] J. Frehse, J. Málek and M. Růžička, Large data existence result for unsteady flows of inhomogeneous shear-thickening heat-conducting incompressible fluids, Comm. Partial Differential Equations, 35 (2010), 1891-1919. [10] J. Frehse and M. Růžička, Non-homogeneous generalized Newtonian fluids, Math. Z., 260 (2008), 355-375. doi: 10.1007/s00209-007-0278-1. [11] A. Friedman and J. Nečas, Systems of nonlinear wave equations with nonlinear viscosity, Pacific J. Math., 135 (1988), 29-55. [12] G. Friesecke and G. Dolzmann, Implicit time discretization and global existence for a quasi-linear evolution equation with nonconvex energy, SIAM J. Math. Anal., 28 (1997), 363-380. doi: 10.1137/S0036141095285958. [13] Y. Fung, "Biomechanics: Mechanical Properties of Living Tissues," Springer-Verlag, 1993. [14] A. Kufner, O. John and S. Fučík, "Function Spaces," Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis, Noordhoff International Publishing, Leyden, Academia, Prague, 1977. [15] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," Dunod, Gauthier-Villars, Paris, 1969. [16] J. Málek, J. Nečas and M. Růžička, On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: The case $p\geq2$, Adv. Differential Equations, 6 (2001), 257-302. [17] J. Málek, J. Nečas, M. Rokyta and M. Růžička, "Weak and Measure-Valued Solutions to Evolutionary PDEs," Applied Mathematics and Mathematical Computation, 13, Chapman & Hall, London, 1996. [18] K. R. Rajagopal, A note on a reappraisal and generalization of the Kelvin-Voigt Model, Mechanics Research Communications, 36 (2009), 232-235. doi: 10.1016/j.mechrescom.2008.09.005. [19] W. Ramberg and W. R. Osgood, Description of stress-strain curves by three parameters, Technical Notes Nat. Adv. Comm. Aeronaut., 1943 (1943), 13 pp. [20] W. Thompson, On the elasticity and viscosity of metals, Proc. Roy. Soc. London A, 14 (1865), 289-297. doi: 10.1098/rspl.1865.0052. [21] B. Tvedt, Quasilinear equations for viscoelasticity of strain-rate type, Arch. Ration. Mech. Anal., 189 (2008), 237-281. doi: 10.1007/s00205-007-0109-x. [22] W. Voigt, Ueber innere Reibung fester Körper, insbesondere der Metalle, Annalen der Physik, 283 (1892), 671-693. doi: 10.1002/andp.18922831210.
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