American Institute of Mathematical Sciences

Advanced
December  2012, 5(6): 1133-1145. doi: 10.3934/dcdss.2012.5.1133

The thermo-mechanics of rate-type fluids

K. R. Rajagopal 1,

1. 

Department of Mechanical Engineering, Texas A&M University, College Station, TX-77843, United States

Received  April 2012 Revised  June 2012 Published  August 2012

In this short paper a brief review is provided concerning the modeling of the thermo-mechanical response of rate type fluid models. Recently, two different approaches have been used to develop thermodynamically compatible rate type fluid models, one that assumes the Helmholtz potential and the other the Gibbs potential for the fluids. These two perspectives are complimentary, not all models that can be modeled within the first procedure can be obtained from the second one, and vice-versa. The two approaches greatly enlarge the arsenal of a modeler and most models that are used can be derived within the purview of these two approaches. More importantly, the two methodologies lead to interesting and useful new models which can be used to describe the behavior of materials that have hitherto defied proper description.
Keywords: Rate type fluid, incompressible fluid, thermo-mechanical response., Gibbs potential, Helmholtz potential.
Mathematics Subject Classification: Primary: 76A10, 76F05; Secondary: 81A1.
Citation: K. R. Rajagopal. The thermo-mechanics of rate-type fluids. Discrete and Continuous Dynamical Systems - S, 2012, 5 (6) : 1133-1145. doi: 10.3934/dcdss.2012.5.1133
References:
[1]

G. Barot, I. J. Rao and K. R. Rajagopal, A thermodynamic framework for the modeling of crystallizable shape memory polymers, International Journal of Engineering Science, 46 (2008), 325-351. doi: 10.1016/j.ijengsci.2007.11.008.

[2]

A. N. Beris and S. F. Edwards, "Thermodynamics of Flowing Systems with Internal Microstructure," Oxford Engineering Science Series 36, Oxford University Press, New York, 1994.

[3]

D. R. Bland, "The Linear Theory of Viscoelasticity," Pergamon Press, Oxford, 1960.

[4]

J. M. Burgers, "Mechanical Considerations-model Systems-Phenomenological Theories of Relaxation and Viscosity," In: First Report on Viscosity and Plasticity, Second ed. Nordemann Publishing Company, Inc., New York, Prepared by the committee of viscosity of the academy of sciences at Amsterdam, 1939.

[5]

A. L. Cauchy, Recherches sur lequilibre et le mouvement interieur des corps solides ou fluids, elastiques ou non elastiques, Bull. Soc. Philomath, (1823), 9-13.

[6]

J. D. Ferry, "Viscoelastic Properties of Polymers," Wiley, New York, 1980.

[7]

J. Finger, Über die allgemeisten Bezeihungen zwischen Deformationen und den zugehoringen Spannungenin aelotropen und isotropen substanzen, Akad. Wiss. Wien Sitzungsber, 103 (1894), 1073-1100.

[8]

G. Green, On the laws of reflexion and refraction of light at the common surface of two non-crystallized media, (1837), Trans. Cambr. Phil. Soc. 7 (1839-1842), 1-24. Papers, 245-269 (1839).

[9]

A. E. Green and P. M. Naghdi, On thermodynamics and nature of second law, Proc. Roy. Soc. Lond. A, 357 (1977), 253-270. doi: 10.1098/rspa.1977.0166.

[10]

M. Heida and J. Málek, On Korteweg-type compressible fluid-like materials, International Journal of Engineering Science, 48 (2010), 1313-1324. doi: 10.1016/j.ijengsci.2010.06.031.

[11]

M. Heida, J. Málek and K. R. Rajagopal, On the development and generalizations of Cahn-Hilliard equations within a thermodynamic framework, Zeitschrift für Angewandte Mathematik und Physik, 63 (2012), 145-169.

[12]

M. Heida, J. Málek and K. R. Rajagopal, On the development and generalizations of Allen-Cahn and Stefan equations within a thermodynamic framework, Zeitschrift für Angewandte Mathematik und Physik, (in print) (2012).

[13]

K. Kannan and K. R. Rajagopal, A thermodynamic framework for chemically reacting systems, Zeitschrift für Angewandte Mathematik und Physik, 62 (2011), 331-362.

[14]

J. M. Krishnan and K. R. Rajagopal, On the mechanical behavior of asphalt, Mech. of Materials, 37 (2005), 1085-1100. doi: 10.1016/j.mechmat.2004.09.005.

[15]

J. Málek and K. R. Rajagopal, Incompressible rate type fluids with pressure and shear-rate dependent material moduli, Nonlinear Anal. Real World Appl., 8 (2007), 156-164. doi: 10.1016/j.nonrwa.2005.06.006.

[16]

J. Málek and K. R. Rajagopal, A thermodynamic framework for a mixture of two liquids, 2008 Nonlinear Anal. Real World Appl., 9 (2008), 1649-1660. doi: 10.1016/j.nonrwa.2007.04.008.

[17]

J. C. Maxwell, On the dynamical theory of gases, Philosophical Transactions of the Royal Society, London A, 157 (1866), 26-78.

[18]

W. Noll, On the foundations of mechanics of continuous media, Carnegie Institute of Technology, Department of Mathematics Report 17 (1957).

[19]

J. G. Oldroyd, On the formulation of the rheological equations of state, Proc. Roy. Soc. Lond. A, 200 (1950), 523-541. doi: 10.1098/rspa.1950.0035.

[20]

S. C. Prasad, K. R. Rajagopal and I. J. Rao, A continuum model for the creep of single crystal nickel-base superalloys, Acta Mater., 53 (2005), 669-679. doi: 10.1016/j.actamat.2004.10.020.

[21]

S. C. Prasad, K. R. Rajagopal and I. J. Rao, A continuum model for the anisotropic creep of single crystal nickel-based superalloys, Acta Mater., 54 (2006), 1487-1500. doi: 10.1016/j.actamat.2005.11.016.

[22]

K. R. Rajagopal and A. R. Srinivasa, On the inelastic behavior of solids - Part 1: Twinning, Int. J. Plast., 11 (1995), 653-678. doi: 10.1016/S0749-6419(95)00027-5.

[23]

K. R. Rajagopal and A. R. Srinivasa, Inelastic behavior of materials - part II: Energetics associated with discontinuous twinning, Int. J. Plast., 13 (1997), 1-35. doi: 10.1016/S0749-6419(96)00049-6.

[24]

K. R. Rajagopal and A. R. Srinivasa, Mechanics of the inelastic behavior of materials - part I: Theoretical underpinnings, Int. J. Plast., 14) (1998), 945-967.

[25]

K. R. Rajagopal and A. R. Srinivasa, Mechanics of the inelastic behavior of materials - part II: Inelastic response, Int. J. Plast., 14) (1998), 969-995.

[26]

K. R. Rajagopal and A. R. Srinivasa, On the thermomechanics of shape memory wires, Zeitschrift für Angewandte Mathematik und Physik, 50 (1999), 459-496.

[27]

K. R. Rajagopal and A. R. Srinivasa, Thermodynamics of Rate type fluid model, Journal of Non-Newtonian Fluid Mechanics, 88 (2000), 207-227. doi: 10.1016/S0377-0257(99)00023-3.

[28]

K. R. Rajagopal and A. R. Srinivasa, Modeling anisotropic fluids within the framework of bodies with multiple natural configurations, Journal of Non-Newtonian Fluid Mechanics, 99 (2001), 109-124. doi: 10.1016/S0377-0257(01)00116-1.

[29]

K. R. Rajagopal and A. R. Srinivasa, On the thermomechanics of materials that have multiple natural configurations - part I: Viscoelasticity and classical plasticity, Zeitschrift für Angewandte Mathematik und Physik, 55 (2004), 861-893.

[30]

K. R. Rajagopal and A. R. Srinivasa, On the thermomechanics of materials that have multiple natural configurations - part II: Twinning and solid to solid phase transformation, Zeitschrift für Angewandte Mathematik und Physik, 55 (2004), 1074-1093.

[31]

K. R. Rajagopal and A. R. Srinivasa, On the response of non-dissipative solids, Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci., 463 (2007), 357-367.

[32]

K. R. Rajagopal and A. R. Srinivasa, On a class of non-dissipative materials that are not hyperelastic, Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci., 465 (2009), 495-500.

[33]

K. R. Rajagopal and A. R. Srinivasa, A Gibbs-potential-based formulation for obtaining the response functions for a class of viscoelastic materials, Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci., 467 (2011), 39-58.

[34]

K. R. Rajagopal and A. R. Srinivasa, "Restrictions Placed on Constitutive Relations by Angular Momentum Balance and Galilean Invariance," In Press, Zeitschrift für Angewandte Mathematik und Mechanik, 2012.

[35]

K. R. Rajagopal and A. S. Wineman, "Mechanical Response of Polymers," Cambridge University Press, Cambridge, 2001.

[36]

I. J. Rao and K. R. Rajagopal, A thermodynamic framework for the study of crystallization in polymers, Zeitschrift für Angewandte Mathematik und Physik, 53 (2002), 365-406.

[37]

C. Truesdell, Mechanical Foundations of Elasticity and Fluid Dynamics, Mechanics I, Gordon and Breach, New York (1966) (Reprinted from Journal of Rational Mechanics, 1, 125-300 (1952) as corrected in 2, 595-616 (1953), and 3, 801 (1954)).

[38]

C. Truesdell and W. Noll, "Non-Linear Field Theories of Mechanics," 2nd edition, Springer, Berlin, 1992.

show all references

References:
[1]

G. Barot, I. J. Rao and K. R. Rajagopal, A thermodynamic framework for the modeling of crystallizable shape memory polymers, International Journal of Engineering Science, 46 (2008), 325-351. doi: 10.1016/j.ijengsci.2007.11.008.

[2]

A. N. Beris and S. F. Edwards, "Thermodynamics of Flowing Systems with Internal Microstructure," Oxford Engineering Science Series 36, Oxford University Press, New York, 1994.

[3]

D. R. Bland, "The Linear Theory of Viscoelasticity," Pergamon Press, Oxford, 1960.

[4]

J. M. Burgers, "Mechanical Considerations-model Systems-Phenomenological Theories of Relaxation and Viscosity," In: First Report on Viscosity and Plasticity, Second ed. Nordemann Publishing Company, Inc., New York, Prepared by the committee of viscosity of the academy of sciences at Amsterdam, 1939.

[5]

A. L. Cauchy, Recherches sur lequilibre et le mouvement interieur des corps solides ou fluids, elastiques ou non elastiques, Bull. Soc. Philomath, (1823), 9-13.

[6]

J. D. Ferry, "Viscoelastic Properties of Polymers," Wiley, New York, 1980.

[7]

J. Finger, Über die allgemeisten Bezeihungen zwischen Deformationen und den zugehoringen Spannungenin aelotropen und isotropen substanzen, Akad. Wiss. Wien Sitzungsber, 103 (1894), 1073-1100.

[8]

G. Green, On the laws of reflexion and refraction of light at the common surface of two non-crystallized media, (1837), Trans. Cambr. Phil. Soc. 7 (1839-1842), 1-24. Papers, 245-269 (1839).

[9]

A. E. Green and P. M. Naghdi, On thermodynamics and nature of second law, Proc. Roy. Soc. Lond. A, 357 (1977), 253-270. doi: 10.1098/rspa.1977.0166.

[10]

M. Heida and J. Málek, On Korteweg-type compressible fluid-like materials, International Journal of Engineering Science, 48 (2010), 1313-1324. doi: 10.1016/j.ijengsci.2010.06.031.

[11]

M. Heida, J. Málek and K. R. Rajagopal, On the development and generalizations of Cahn-Hilliard equations within a thermodynamic framework, Zeitschrift für Angewandte Mathematik und Physik, 63 (2012), 145-169.

[12]

M. Heida, J. Málek and K. R. Rajagopal, On the development and generalizations of Allen-Cahn and Stefan equations within a thermodynamic framework, Zeitschrift für Angewandte Mathematik und Physik, (in print) (2012).

[13]

K. Kannan and K. R. Rajagopal, A thermodynamic framework for chemically reacting systems, Zeitschrift für Angewandte Mathematik und Physik, 62 (2011), 331-362.

[14]

J. M. Krishnan and K. R. Rajagopal, On the mechanical behavior of asphalt, Mech. of Materials, 37 (2005), 1085-1100. doi: 10.1016/j.mechmat.2004.09.005.

[15]

J. Málek and K. R. Rajagopal, Incompressible rate type fluids with pressure and shear-rate dependent material moduli, Nonlinear Anal. Real World Appl., 8 (2007), 156-164. doi: 10.1016/j.nonrwa.2005.06.006.

[16]

J. Málek and K. R. Rajagopal, A thermodynamic framework for a mixture of two liquids, 2008 Nonlinear Anal. Real World Appl., 9 (2008), 1649-1660. doi: 10.1016/j.nonrwa.2007.04.008.

[17]

J. C. Maxwell, On the dynamical theory of gases, Philosophical Transactions of the Royal Society, London A, 157 (1866), 26-78.

[18]

W. Noll, On the foundations of mechanics of continuous media, Carnegie Institute of Technology, Department of Mathematics Report 17 (1957).

[19]

J. G. Oldroyd, On the formulation of the rheological equations of state, Proc. Roy. Soc. Lond. A, 200 (1950), 523-541. doi: 10.1098/rspa.1950.0035.

[20]

S. C. Prasad, K. R. Rajagopal and I. J. Rao, A continuum model for the creep of single crystal nickel-base superalloys, Acta Mater., 53 (2005), 669-679. doi: 10.1016/j.actamat.2004.10.020.

[21]

S. C. Prasad, K. R. Rajagopal and I. J. Rao, A continuum model for the anisotropic creep of single crystal nickel-based superalloys, Acta Mater., 54 (2006), 1487-1500. doi: 10.1016/j.actamat.2005.11.016.

[22]

K. R. Rajagopal and A. R. Srinivasa, On the inelastic behavior of solids - Part 1: Twinning, Int. J. Plast., 11 (1995), 653-678. doi: 10.1016/S0749-6419(95)00027-5.

[23]

K. R. Rajagopal and A. R. Srinivasa, Inelastic behavior of materials - part II: Energetics associated with discontinuous twinning, Int. J. Plast., 13 (1997), 1-35. doi: 10.1016/S0749-6419(96)00049-6.

[24]

K. R. Rajagopal and A. R. Srinivasa, Mechanics of the inelastic behavior of materials - part I: Theoretical underpinnings, Int. J. Plast., 14) (1998), 945-967.

[25]

K. R. Rajagopal and A. R. Srinivasa, Mechanics of the inelastic behavior of materials - part II: Inelastic response, Int. J. Plast., 14) (1998), 969-995.

[26]

K. R. Rajagopal and A. R. Srinivasa, On the thermomechanics of shape memory wires, Zeitschrift für Angewandte Mathematik und Physik, 50 (1999), 459-496.

[27]

K. R. Rajagopal and A. R. Srinivasa, Thermodynamics of Rate type fluid model, Journal of Non-Newtonian Fluid Mechanics, 88 (2000), 207-227. doi: 10.1016/S0377-0257(99)00023-3.

[28]

K. R. Rajagopal and A. R. Srinivasa, Modeling anisotropic fluids within the framework of bodies with multiple natural configurations, Journal of Non-Newtonian Fluid Mechanics, 99 (2001), 109-124. doi: 10.1016/S0377-0257(01)00116-1.

[29]

K. R. Rajagopal and A. R. Srinivasa, On the thermomechanics of materials that have multiple natural configurations - part I: Viscoelasticity and classical plasticity, Zeitschrift für Angewandte Mathematik und Physik, 55 (2004), 861-893.

[30]

K. R. Rajagopal and A. R. Srinivasa, On the thermomechanics of materials that have multiple natural configurations - part II: Twinning and solid to solid phase transformation, Zeitschrift für Angewandte Mathematik und Physik, 55 (2004), 1074-1093.

[31]

K. R. Rajagopal and A. R. Srinivasa, On the response of non-dissipative solids, Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci., 463 (2007), 357-367.

[32]

K. R. Rajagopal and A. R. Srinivasa, On a class of non-dissipative materials that are not hyperelastic, Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci., 465 (2009), 495-500.

[33]

K. R. Rajagopal and A. R. Srinivasa, A Gibbs-potential-based formulation for obtaining the response functions for a class of viscoelastic materials, Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci., 467 (2011), 39-58.

[34]

K. R. Rajagopal and A. R. Srinivasa, "Restrictions Placed on Constitutive Relations by Angular Momentum Balance and Galilean Invariance," In Press, Zeitschrift für Angewandte Mathematik und Mechanik, 2012.

[35]

K. R. Rajagopal and A. S. Wineman, "Mechanical Response of Polymers," Cambridge University Press, Cambridge, 2001.

[36]

I. J. Rao and K. R. Rajagopal, A thermodynamic framework for the study of crystallization in polymers, Zeitschrift für Angewandte Mathematik und Physik, 53 (2002), 365-406.

[37]

C. Truesdell, Mechanical Foundations of Elasticity and Fluid Dynamics, Mechanics I, Gordon and Breach, New York (1966) (Reprinted from Journal of Rational Mechanics, 1, 125-300 (1952) as corrected in 2, 595-616 (1953), and 3, 801 (1954)).

[38]

C. Truesdell and W. Noll, "Non-Linear Field Theories of Mechanics," 2nd edition, Springer, Berlin, 1992.

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