American Institute of Mathematical Sciences

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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 & 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.  Google Scholar

[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.  Google Scholar

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[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. Google Scholar

[6]

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

[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. Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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). Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[35]

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

[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.  Google Scholar

[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)). Google Scholar

[38]

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

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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. Google Scholar

[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. Google Scholar

[6]

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

[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. Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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). Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[35]

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

[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.  Google Scholar

[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)). Google Scholar

[38]

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

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