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Rate constitutive theories for ordered thermoviscoelastic fluids: polymers

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Abstract

This paper presents development of rate constitutive theories for compressible as well as in incompressible ordered thermoviscoelastic fluids, i.e., polymeric fluids in Eulerian description. The polymeric fluids in this paper are considered as ordered thermoviscoelastic fluids in which the stress rate of a desired order, i.e., the convected time derivative of a desired order ‘m’ of the chosen deviatoric Cauchy stress tensor, and the heat vector are functions of density, temperature, temperature gradient, convected time derivatives of the chosen strain tensor up to any desired order ‘n’ and the convected time derivative of up to orders ‘m−1’ of the chosen deviatoric Cauchy stress tensor. The development of the constitutive theories is presented in contravariant and covariant bases, as well as using Jaumann rates. The polymeric fluids described by these constitutive theories will be referred to as ordered thermoviscoelastic fluids due to the fact that the constitutive theories are dependent on the orders ‘m’ and ‘n’ of the convected time derivatives of the deviatoric Cauchy stress and conjugate strain tensors. The highest orders of the convected time derivative of the deviatoric Cauchy stress and strain tensors define the orders of the polymeric fluid. The admissibility requirement necessitates that the constitutive theories for the stress tensor and heat vector satisfy conservation laws, hence, in addition to conservation of mass, balance of momenta, and conservation of energy, the second law of thermodynamics, i.e., Clausius–Duhem inequality must also be satisfied by the constitutive theories or be used in their derivations. If we decompose the total Cauchy stress tensor into equilibrium and deviatoric components, then Clausius–Duhem inequality and Helmholtz free-energy density can be used to determine the equilibrium stress in terms of thermodynamic pressure for compressible fluids and in terms of mechanical pressure for incompressible fluids, but the second law of thermodynamics provides no mechanism for deriving the constitutive theories for the deviatoric Cauchy stress tensor. In the development of the constitutive theories in Eulerian description, the covariant and contravariant convected coordinate systems and Jaumann measures are natural choices. Furthermore, the mathematical models for fluids require Eulerian description in which material point displacements are not measurable. This precludes the use of displacement gradients, i.e., strain measures, in the development of the constitutive theories. It is shown that compatible conjugate pairs of convected time derivatives of the deviatoric Cauchy stress and strain measures in co-, contravariant and Jaumann bases in conjunction with the theory of generators and invariants provide a general mathematical framework for the development of constitutive theories for ordered thermofluids in Eulerian description. This framework has a foundation based on the basic principles and axioms of continuum mechanics, but the resulting constitutive theories for the deviatoric Cauchy stress tensor must satisfy the condition of positive work expanded, a requirement resulting from the entropy inequality. The paper presents a general theory of constitutive equations for ordered thermoviscoelastic fluids which is then specialized to obtain commonly used constitutive equations for Maxwell, Giesekus and Oldroyd-B constitutive models in contra- and covariant bases and using Jaumann rates.

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References

  1. Brown R.: A brief account of microscopical observations made in the months of June, July, and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. Edinb. New Philos. J. 5, 358–371 (1828)

    Google Scholar 

  2. Brown R.: Additional remarks on active molecules. Edinb. J. Sci. 1, 314–319 (1829)

    Google Scholar 

  3. Maxwell J.C.: On the dynamical theory of gases. Philos. Trans. R. Soc. Lond. A 157, 49–88 (1867)

    Article  Google Scholar 

  4. Bird R.B., Armstrong R.C., Hassager O.: Dynamics of Polymeric Liquids, vol. 1, Fluid Mechanics, 2nd edn. Wiley, London (1987)

    Google Scholar 

  5. Jeffreys H.: The Earth. Cambridge University Press, Cambridge (1929)

    MATH  Google Scholar 

  6. Oldroyd J.G.: On the formulation of rheological equations of state. Proc. R. Soc. Lond. A 200, 523–541 (1950)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Giesekus H.: A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility. J. Non-Newton. Fluid Mech. 11, 69–110 (1982)

    Article  MATH  Google Scholar 

  8. Phan-Thien N., Tanner R.I.: A new constitutive equation derived from network theory. J. Non-Newton. Fluid Mech. 2, 353–365 (1977)

    Article  Google Scholar 

  9. Phan-Thien N., Tanner R.I.: A nonlinear network viscoelastic model. J. Rheol. 22, 259–283 (1978)

    Article  ADS  MATH  Google Scholar 

  10. Bird R.B., Armstrong R.C., Hassager O.: Dynamics of Polymeric Liquids, vol. 2, Kinetic Theory, 2nd edn. Wiley, London (1987)

    Google Scholar 

  11. Grmela M., Ottinger H.C.: Dynamic and thermodynamics of complex fluids I—development of a general formalism. Phys. Rev. E 56(6), 6620–6632 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  12. Dunn J.E., Rajagopal K.R.: Fluids of differential type: critical reveiw and thermodynamic analysis. Int. J. Eng. Sci. 33(5), 689–729 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  13. Rajagopal K.R., Srinivasa A.R.: On the development of fluid models of the differential type within a new thermodynamic framework. Mech. Res. Commun. 35, 483–489 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Leonov A.I.: Analysis of simple constitutive equations for viscoelastic liquids. J. Non-Newton. Fluid Mech. 42, 323–350 (1992)

    Article  MATH  Google Scholar 

  15. Rajagopal, K.R., Srinivasa, A.R.: A thermodynamic framework for rate type fluid models. J. Non-Newton. Fluid Mech. 88, 207–227

  16. Rajagopal K.R., Srinivasa A.R.: A Gibbs potential based formulation for obtaining the response functions for a class of viscoelastic materials. Proc. R. Soc. A 467, 39–58 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  17. Surana, K.S., Nunez, D., Reddy, J.N., Romkes, A.: Rate constitutive theory for ordered thermofluids. J. Continuum Mech. Thermodyn. 25 (pp. N/A yet) (2013)

  18. Surana, K.S., Reddy, J.N., Nunez, D.: Ordered rate constitutive theories for thermoviscoelastic solids with memory in Lagrangian description using Gibbs potential (under review) (2013)

  19. Eringen A.C.: Mechanics of Continua. Wiley, London (1967)

    MATH  Google Scholar 

  20. Eringen A.C.: Nonlinear Theory of Continuous Media. McGraw-Hill, New York (1962)

    Google Scholar 

  21. Surana, K.S., Reddy, J.N.: Continuum Mechanics (manuscript of the textbook in preparation) (2013)

  22. Prager W.: Strain hardening under combined stresses. J. Appl. Phys. 16, 837–840 (1945)

    Article  ADS  MathSciNet  Google Scholar 

  23. Reiner M.: A mathematical theory of dilatancy. Am. J. Math. 67, 350–362 (1945)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  24. Rivlin R.S., Ericksen J.L.: Stress-deformation relations for isotropic materials. J. Ration. Mech. Anal. 4, 323–425 (1955)

    MATH  MathSciNet  Google Scholar 

  25. Rivlin R.S.: Further remarks on the stress-deformation relations for isotropic materials. J. Ration. Mech. Anal. 4, 681–702 (1955)

    MATH  MathSciNet  Google Scholar 

  26. Todd J.A.: Ternary quadratic types. Philos. Trans. R. Soc. Lond. Ser A Math. Phys. Sci. 241, 399–456 (1948)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  27. Wang C.C.: On representations for isotropic functions, part I. Arch. Ration. Mech. Anal. 33, 249 (1969)

    Article  MATH  Google Scholar 

  28. Wang C.C.: On representations for isotropic functions, part II. Arch. Ration. Mech. Anal. 33, 268 (1969)

    Article  Google Scholar 

  29. Wang C.C.: A new representation theorem for isotropic functions, part I and part II. Arch. Ration. Mech. Anal. 36, 166–223 (1970)

    Article  MATH  Google Scholar 

  30. Wang C.C.: Corrigendum to ‘Representations for isotropic functions’. Arch. Ration. Mech. Anal. 43, 392–395 (1971)

    Article  Google Scholar 

  31. Smith G.F.: On a fundamental error in two papers of C.C. Wang, ‘On representations for isotropic functions, part I and part II’. Arch. Ration. Mech. Anal. 36, 161–165 (1970)

    Article  MATH  Google Scholar 

  32. Smith G.F.: On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors. Int. J. Eng. Sci. 9, 899–916 (1971)

    Article  MATH  Google Scholar 

  33. Spencer A.J.M., Rivlin R.S.: The theory of matrix polynomials and its application to the mechanics of isotropic continua. Arch. Ration. Mech. Anal. 2, 309–336 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  34. Spencer A.J.M., Rivlin R.S.: Further results in the theory of matrix polynomials. Arch. Ration. Mech. Anal. 4, 214–230 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  35. Spencer, A.J.M.: Theory of invariants, chapter 3. In: Eringen, A.C. (ed.) Treatise on Continuum Physics, I. Academic Press, London (1971)

  36. Boehler J.P.: On irreducible representations for isotropic scalar functions. J. Appl. Math. Mech. (Zeitschrift für Angewandte Mathematik und Mechanik) 57, 323–327 (1977)

    MATH  MathSciNet  Google Scholar 

  37. Zheng Q.S.: On the representations for isotropic vector-valued, symmetric tensor-valued and skew-symmetric tensor-valued functions. Int. J. Eng. Sci. 31, 1013–1024 (1993)

    Article  MATH  Google Scholar 

  38. Zheng Q.S.: On transversely isotropic, orthotropic and relatively isotropic functions of symmetric tensors, skew-symmetric tensors, and vectors. Int. J. Eng. Sci. 31, 1399–1453 (1993)

    Article  MATH  Google Scholar 

  39. Surana K.S., Nunez D., Reddy J.N., Romkes A.: Rate constitutive theory for ordered thermoelastic solids. Ann. Solid Struct. Mech. 3, 27–54 (2012)

    Article  Google Scholar 

  40. Udaykumar H.S., Tran L., Belk D.M., Vanden K.J.: An Eulerian method for computation of multimaterial impact with ENO shock-capturing and sharp interfaces. J. Comput. Phys. 186, 136–177 (2003)

    Article  ADS  MATH  Google Scholar 

  41. Tran L., Udaykumar H.S.: A particle-level set-based sharp interface cartesian grid method for impact, penetration, and void collapse. J. Comput. Phys. 193, 469–510 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  42. White F.M.: Fluid Mechanics, 7th edn. McGraw-Hill, New York (2010)

    Google Scholar 

  43. Panton R.L.: Incompressible Flow, 3rd edn. Wiley, London (2005)

    Google Scholar 

  44. Surana K.S., MA Y., Reddy J.N., Romkes A.: The rate constitutive equations and their validity for progressively increasing deformation. Mech. Adv. Mater. Struct. 17, 509–533 (2010)

    Article  Google Scholar 

  45. Giesekus H.: Die Rheologische Zustandsgleichung Elasto-Viskoser Flüssigkeiten—Insbesondere Von Weissenberg-Flüssigkeiten—für Allgemeine und Stationäre Fließvorgänge. J. Appl. Math. Mech. (Zeitschrift für Angewandte Mathematik und Mechanik) 42, 32–61 (1962)

    MATH  Google Scholar 

  46. Rajagopalan D., Armstrong R.C., Brown R.A.: Finite element methods for calculation of steady viscoelastic flow using constitutive equations with a Newtonian viscosity. J. Non-Newton. Fluid Mech. 36, 159–192 (1990)

    Article  MATH  Google Scholar 

  47. Rajagopalan D., Phillips R.J., Armstrong R.C., Brown R.A., Bose A.: The influence of viscoelasticity on the existence of steady solutions in two-dimensional rimming flow. J. Fluid Mech. Digit. Arch. 235, 611–642 (1992)

    ADS  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Kansas, Lawrence, KS, USA

    K. S. Surana, D. Nunez & A. Romkes

  2. Department of Mechanical Engineering, Texas A&M University, College Station, TX, USA

    J. N. Reddy

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  1. K. S. Surana
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  2. D. Nunez
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  3. J. N. Reddy
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  4. A. Romkes
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Correspondence to K. S. Surana.

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Communicated by Andreas Öchsner.

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Surana, K.S., Nunez, D., Reddy, J.N. et al. Rate constitutive theories for ordered thermoviscoelastic fluids: polymers. Continuum Mech. Thermodyn. 26, 143–181 (2014). https://doi.org/10.1007/s00161-013-0295-8

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