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.

show all references

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.

[1]

Mikhail Turbin, Anastasiia Ustiuzhaninova. Pullback attractors for weak solution to modified Kelvin-Voigt model. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022011

[2]

Fathi Hassine. Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1757-1774. doi: 10.3934/dcdsb.2016021

[3]

Serge Nicaise, Cristina Pignotti. Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 791-813. doi: 10.3934/dcdss.2016029

[4]

Ahmed Bchatnia, Nadia Souayeh. Eventual differentiability of coupled wave equations with local Kelvin-Voigt damping. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1317-1338. doi: 10.3934/dcdss.2022098

[5]

Miroslav Bulíček, Victoria Patel, Yasemin Şengül, Endre Süli. Existence of large-data global weak solutions to a model of a strain-limiting viscoelastic body. Communications on Pure and Applied Analysis, 2021, 20 (5) : 1931-1960. doi: 10.3934/cpaa.2021053

[6]

Kaïs Ammari, Thomas Duyckaerts, Armen Shirikyan. Local feedback stabilisation to a non-stationary solution for a damped non-linear wave equation. Mathematical Control and Related Fields, 2016, 6 (1) : 1-25. doi: 10.3934/mcrf.2016.6.1

[7]

Mohammad Akil, Ibtissam Issa, Ali Wehbe. Energy decay of some boundary coupled systems involving wave\ Euler-Bernoulli beam with one locally singular fractional Kelvin-Voigt damping. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021059

[8]

Faustino Sánchez-Garduño, Philip K. Maini, Judith Pérez-Velázquez. A non-linear degenerate equation for direct aggregation and traveling wave dynamics. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 455-487. doi: 10.3934/dcdsb.2010.13.455

[9]

Louis Tebou. Stabilization of some elastodynamic systems with localized Kelvin-Voigt damping. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7117-7136. doi: 10.3934/dcds.2016110

[10]

Ahmad El Hajj, Aya Oussaily. Continuous solution for a non-linear eikonal system. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3795-3823. doi: 10.3934/cpaa.2021131

[11]

Alain Hertzog, Antoine Mondoloni. Existence of a weak solution for a quasilinear wave equation with boundary condition. Communications on Pure and Applied Analysis, 2002, 1 (2) : 191-219. doi: 10.3934/cpaa.2002.1.191

[12]

Daniele Garrisi, Vladimir Georgiev. Orbital stability and uniqueness of the ground state for the non-linear Schrödinger equation in dimension one. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4309-4328. doi: 10.3934/dcds.2017184

[13]

Út V. Lê. Regularity of the solution of a nonlinear wave equation. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1099-1115. doi: 10.3934/cpaa.2010.9.1099

[14]

César E. Torres Ledesma. Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well. Communications on Pure and Applied Analysis, 2016, 15 (2) : 535-547. doi: 10.3934/cpaa.2016.15.535

[15]

Xiaobin Yao, Qiaozhen Ma, Tingting Liu. Asymptotic behavior for stochastic plate equations with rotational inertia and Kelvin-Voigt dissipative term on unbounded domains. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1889-1917. doi: 10.3934/dcdsb.2018247

[16]

Robert E. Miller. Homogenization of time-dependent systems with Kelvin-Voigt damping by two-scale convergence. Discrete and Continuous Dynamical Systems, 1995, 1 (4) : 485-502. doi: 10.3934/dcds.1995.1.485

[17]

Manil T. Mohan. On the three dimensional Kelvin-Voigt fluids: global solvability, exponential stability and exact controllability of Galerkin approximations. Evolution Equations and Control Theory, 2020, 9 (2) : 301-339. doi: 10.3934/eect.2020007

[18]

Manil T. Mohan. Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory". Evolution Equations and Control Theory, 2022, 11 (1) : 125-167. doi: 10.3934/eect.2020105

[19]

Zhong-Jie Han, Zhuangyi Liu, Jing Wang. Sharper and finer energy decay rate for an elastic string with localized Kelvin-Voigt damping. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1455-1467. doi: 10.3934/dcdss.2022031

[20]

Kim-Ngan Le, William McLean, Martin Stynes. Existence, uniqueness and regularity of the solution of the time-fractional Fokker–Planck equation with general forcing. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2765-2787. doi: 10.3934/cpaa.2019124

2020 Impact Factor: 1.081

Metrics

  • PDF downloads (707)
  • HTML views (0)
  • Cited by (22)

[Back to Top]