# American Institute of Mathematical Sciences

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
##### 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. [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, (2012). [3] M. Bulíček, P. Gwiazda, J. Málek and A. Świerczewska-Gwiazda, On unsteady flows of implicitly constituted incompressible fluids,, SIAM J. Math. Anal., (2011). [4] E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,", McGraw-Hill Book Company, (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. [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. 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, (2004). [8] E. Feireisl and A. Novotný, "Singular Limits in Thermodynamics of Viscous Fluids,", Advances in Mathematical Fluid Mechanics, (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. [10] J. Frehse and M. Růžička, Non-homogeneous generalized Newtonian fluids,, Math. Z., 260 (2008), 355. 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. [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. 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, (1977). [15] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,", Dunod, (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. [17] J. Málek, J. Nečas, M. Rokyta and M. Růžička, "Weak and Measure-Valued Solutions to Evolutionary PDEs,", Applied Mathematics and Mathematical Computation, 13 (1996). [18] K. R. Rajagopal, A note on a reappraisal and generalization of the Kelvin-Voigt Model,, Mechanics Research Communications, 36 (2009), 232. 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). [20] W. Thompson, On the elasticity and viscosity of metals,, Proc. Roy. Soc. London A, 14 (1865), 289. doi: 10.1098/rspl.1865.0052. [21] B. Tvedt, Quasilinear equations for viscoelasticity of strain-rate type,, Arch. Ration. Mech. Anal., 189 (2008), 237. doi: 10.1007/s00205-007-0109-x. [22] W. Voigt, Ueber innere Reibung fester Körper, insbesondere der Metalle,, Annalen der Physik, 283 (1892), 671. 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. [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, (2012). [3] M. Bulíček, P. Gwiazda, J. Málek and A. Świerczewska-Gwiazda, On unsteady flows of implicitly constituted incompressible fluids,, SIAM J. Math. Anal., (2011). [4] E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,", McGraw-Hill Book Company, (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. [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. 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, (2004). [8] E. Feireisl and A. Novotný, "Singular Limits in Thermodynamics of Viscous Fluids,", Advances in Mathematical Fluid Mechanics, (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. [10] J. Frehse and M. Růžička, Non-homogeneous generalized Newtonian fluids,, Math. Z., 260 (2008), 355. 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. [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. 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, (1977). [15] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,", Dunod, (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. [17] J. Málek, J. Nečas, M. Rokyta and M. Růžička, "Weak and Measure-Valued Solutions to Evolutionary PDEs,", Applied Mathematics and Mathematical Computation, 13 (1996). [18] K. R. Rajagopal, A note on a reappraisal and generalization of the Kelvin-Voigt Model,, Mechanics Research Communications, 36 (2009), 232. 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). [20] W. Thompson, On the elasticity and viscosity of metals,, Proc. Roy. Soc. London A, 14 (1865), 289. doi: 10.1098/rspl.1865.0052. [21] B. Tvedt, Quasilinear equations for viscoelasticity of strain-rate type,, Arch. Ration. Mech. Anal., 189 (2008), 237. doi: 10.1007/s00205-007-0109-x. [22] W. Voigt, Ueber innere Reibung fester Körper, insbesondere der Metalle,, Annalen der Physik, 283 (1892), 671. doi: 10.1002/andp.18922831210.
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