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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, Czech Republic, Czech Republic |
2. | Department of Mechanical Engineering, Texas A&M University, College Station, TX 77845, United States |
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). Google Scholar |
[3] |
M. Bulíček, P. Gwiazda, J. Málek and A. Świerczewska-Gwiazda, On unsteady flows of implicitly constituted incompressible fluids,, SIAM J. Math. Anal., (2011). Google Scholar |
[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). Google Scholar |
[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. |
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.
|
[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). Google Scholar |
[3] |
M. Bulíček, P. Gwiazda, J. Málek and A. Świerczewska-Gwiazda, On unsteady flows of implicitly constituted incompressible fluids,, SIAM J. Math. Anal., (2011). Google Scholar |
[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). Google Scholar |
[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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