Applications of Mathematics

, Volume 48, Issue 4, pp 279–319 | Cite as

On Implicit Constitutive Theories

  • K. R. Rajagopal
Article

Abstract

In classical constitutive models such as the Navier-Stokes fluid model, and the Hookean or neo-Hookean solid models, the stress is given explicitly in terms of kinematical quantities. Models for viscoelastic and inelastic responses on the other hand are usually implicit relationships between the stress and the kinematical quantities. Another class of problems wherein it would be natural to develop implicit constitutive theories, though seldom resorted to, are models for bodies that are constrained. In general, for such materials the material moduli that characterize the extra stress could depend on the constraint reaction. (E.g., in an incompressible fluid, the viscosity could depend on the constraint reaction associated with the constraint of incompressibility. In the linear case, this would be the pressure.) Here we discuss such implicit constitutive theories. We also discuss a class of bodies described by an implicit constitutive relation for the specific Helmholtz potential that depends on both the stress and strain, and which does not dissipate in any admissible process. The stress in such a material is not derivable from a potential, i.e., the body is not hyperelastic (Green elastic).

constitutive relations constraint Lagrange multiplier Helmholtz potential rate of dissipation elasticity inelasticity viscoelasticity 

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References

  1. [1]
    G. Amontons: De la résistance causée dans les machines. Mémoires de l' Académie Royale A (1699), 257–282.Google Scholar
  2. [2]
    E. C. Andrade: Viscosity of liquids. Nature 125 (1930), 309–310.Google Scholar
  3. [3]
    B. Bernstein: A unified theory of elasticity and plasticity. Internat. J. Engrg. Sci. 15 (1977), 645–660.Google Scholar
  4. [4]
    B. Bernstein: Hypo-elasticity and elasticity. Arch. Rational Mech. Anal. 6 (1960), 89–104.Google Scholar
  5. [5]
    P. Bridgman: The Physics of High Pressure. The MacMillan Company, New York, 1931.Google Scholar
  6. [6]
    C. A. Coulomb: Theories des machines simples, en ayant égard au frottement de leurs parties, et a la roider des cordages. Mém. Math. Phys. X (1785), 161–332.Google Scholar
  7. [7]
    J. d'Alembert: Traité de Dynamique. Paris, Chez David, Libraire, 1758.Google Scholar
  8. [8]
    C. Eckart: The thermodynamics of irreversible processes IV. The theory of elasticity and inelasticity. Physical Rev. 73 (1948), 373–382.Google Scholar
  9. [9]
    M. Franta, J. Málek and K. R. Rajagopal: On steady ows of uids and shear dependent viscosities. Proc. Roy. Soc. London Ser. A: Mathematical, Physical and Engineering Sciences (2003). Submitted.Google Scholar
  10. [10]
    C.F. Gauss: On a new general principle of mechanics; Translation of Ñber ein neues allgemeines Grundgesetz der Mechanik. J. Reine Angew. Math. 4 (1829), 232–235.Google Scholar
  11. [11]
    H. Goldstein: Classical Mechanics. Addison-Wesley, Boston, 1980.Google Scholar
  12. [12]
    J. Hron, J. Málek and K. R. Rajagopal: Simple flows of fluids with pressure dependent viscosities. Proc. Roy. Soc. London Ser. A: Mathematical, Physical and Engineering Sciences 457 (2001), 1603–1622.Google Scholar
  13. [13]
    K. Kannan, K. R. Rajagopal: A thermomechanical framework for the transition of a viscoelastic liquid to a viscoelastic solid. Mathematics and Mechanics of Solids. To appear.Google Scholar
  14. [14]
    K. Kannan, I. Rao and K. R. Rajagopal: A thermomechanical framework for the glass transition phenomenon in certain polymers and its application to the fiber spinning problems. J. of Rheology 46 (2002), 977–999.Google Scholar
  15. [15]
    J. L. Lagrange: Mécanique Analytique. Mme Ve Courcier, Paris, 1787, Translation (by A. Boissonnade, V. N. Vagliente), Kluwer Academic Publishers, Dordrecht, 1997.Google Scholar
  16. [16]
    M. Levy: Mémoire sur les équations générales des mouvements intérieurs des corps ductiles au delà des limites en élasticité pourrait les ramener à leur premier état. C. R. Acad. Sci. 70 (1870), 1323–1325.Google Scholar
  17. [17]
    J. Málek, J. Nečas and K. R. Rajagopal: Global analysis of the flows of fluids with pressure dependent viscosities. Arch. Rational Mech. Anal. 165 (2002), 243–269.Google Scholar
  18. [18]
    J. C. Maxwell: On the Dynamical Theory of Gases. Philosophical Transactions of the Royal Society of London, Series A (1866), 26–78.Google Scholar
  19. [19]
    A. J. A. Morgan: Some properties of media by constitutive equations in implicit form. Internat. J. Engrg. Sci. 4 (1966), 155–178.Google Scholar
  20. [20]
    J. Murali Krishnan, K. R. Rajagopal: A thermodynamic framework for the constitutive modeling of asphalt concrete: theory and aplications. ASCE Journal of Materials. Accepted for publication.Google Scholar
  21. [21]
    C. L. M. H. Navier: Mémoire sur les lois du mouvement des fluides. Mém. Acad. Re. Sci., Paris 6 (1823), 389–416.Google Scholar
  22. [22]
    W. Noll: A mathematical theory of the mechanical behavior of continuous media. Arch. Rational Mech. Anal. 2 (1958), 197–226.Google Scholar
  23. [23]
    W. Noll: A new mathematical theory of simple materials. Arch. Rational Mech. Anal. 48 (1972), 1–50.Google Scholar
  24. [24]
    S. D. Poisson: Mémoire sur les équations générales de l'équilibre et du mouvement des corps solides élastiques et des fluides. Journal de l'Ecole Polytechnique 13 (1831), 1–174.Google Scholar
  25. [25]
    L. Prandtl: Spannungsverteilung in plastischen Körpern. In: Proceeding of the 1st International Congress in Applied Mechanics. Delft, 1924.Google Scholar
  26. [26]
    K. R. Rajagopal, A. S. Wineman: On constitutive equations for branching of response with selectivity. Internat. J. Non-Linear Mech. 15 (1980), 83–91.Google Scholar
  27. [27]
    K. R. Rajagopal, A. S. Wineman: A constitutive equation for nonlinear solids which undergo deformation induced micro-structural changes. International Journal of Plasticity 8 (1992), 385–395.Google Scholar
  28. [28]
    K. R. Rajagopal: Multiple configurations in Continuum Mechanics. Report 6. Institute of Computational and Applied Mechanics, University of Pittsburgh, 1995.Google Scholar
  29. [29]
    K. R. Rajagopal, A. R. Srinivasa: On the inelastic behavior of solids—Part I: Twinning. International Journal of Plasticity 11 (1995), 653–678.Google Scholar
  30. [30]
    K. R. Rajagopal, A. R. Srinivasa: On the inelastic behavior of solids—Part II: Energetics associated with discontinuous deformation twinning. International Journal of Plasticity 13 (1997), 1–35.Google Scholar
  31. [31]
    K. R. Rajagopal, A. S. Wineman: A linearized theory for materials undergoing microstructural change. ARI 51 (1998), 160–168.Google Scholar
  32. [32]
    K. R. Rajagopal, A. R. Srinivasa: Mechanics of the inelastic behavior of materials—Part I: Theoretical underpinnings. International Journal of Plasticity 14 (1998), 945–967.Google Scholar
  33. [33]
    K. R. Rajagopal, A. R. Srinivasa: Mechanics of the inelastic behavior of materials—Part II: Inelastic response. International Journal of Plasticity 14 (1998), 969–995.Google Scholar
  34. [34]
    K. R. Rajagopal, A. R. Srinivasa: Thermomechanical modeling of shape memory alloys. ZAMP 50 (1999), 459–496.Google Scholar
  35. [35]
    K. R. Rajagopal, A. R. Srinivasa: A thermodynamic framework for rate type fluid models. Journal of Non-Newtonian Fluid Mechanics 88 (2000), 207–227.Google Scholar
  36. [36]
    K. R. Rajagopal, A. R. Srinivasa: Modeling anistropic fluids within the framework of bodies with multiple natural configurations. Journal of Non-Newtonian Fluid Mechanics (2001).Google Scholar
  37. [37]
    K. R. Rajagopal, L. Tao: Modeling of the microwave drying process of aqueous dielectrics. ZAMP 53 (2002), 923–948.Google Scholar
  38. [38]
    I. J. Rao, K. R. Rajagopal: A study of strain-induced crystallization of polymers. International Journal of Solids and Structures 38 (2001), 1149–1167.Google Scholar
  39. [39]
    I. J. Rao, J. D. Humphrey and K. R. Rajagopal: Growth and remodeling in a dynamically loaded axial tissue. Computational Method in Engineering Science. In press.Google Scholar
  40. [40]
    I. J. Rao, K. R. Rajagopal: Phenomenological modeling of polymer crystallization using the notion of multiple natural configurations. Interfaces and free boundaries 2 (2000), 73–94.Google Scholar
  41. [41]
    I. J. Rao, K. R. Rajagopal: A thermomechanical framework for crystallization of polymers. Z. Angew. Math. Phys. 53 (2002), 365–406.Google Scholar
  42. [42]
    E. Reuss: Berücksichtigung der elastischen Formänderung in der Plastizitätstheorie. Z. Angew. Math. Mech. 10 (1939), 266–274.Google Scholar
  43. [43]
    A. J. C. B. Saint-Venant: Note à joindre au Mémoire sur la dynamique des fluides. C. R. Acad. Sci. 17 (1843), 1240–1243.Google Scholar
  44. [44]
    A. J. M. Spencer: Continuum Physics, Vol. 3 (A. C. Eringen, ed.). Academic Press, New York, 1975.Google Scholar
  45. [45]
    A. R. Srinivasa, K. R. Rajagopal and R. Armstrong: A phenomenological model of twinning based on dual reference structures. Acta. Metall. (1998), 1–14.Google Scholar
  46. [46]
    G. G. Stokes: On the theories of internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. Transactions of the Cambridge Philosophical Society 8 (1845), 287–305.Google Scholar
  47. [47]
    H. E. Tresca: On the `flow of solids' with the practical application in some forgings. In: Proceedings of the Institution of Mechanical Engineers. London, 1867, pp. 114–150.Google Scholar
  48. [48]
    C. Truesdell: A First Course in Rational Continuum Mechanics. Academic Press, Boston-San Diego, 1991.Google Scholar
  49. [49]
    C. A. Truesdell: Hypo-elasticity. J. Rational Mechanics and Analysis 4 (1955), 323–425.Google Scholar
  50. [50]
    C. Truesdell, W. Noll: The Non-Linear Field Theories of Mechanics. Handbuch der Physik, III 3. Springer-Verlag, Berlin-Heidelberg-New York, 1965.Google Scholar
  51. [51]
    R. von Mises: Mechanik der festen Körpern im plastisch-deformablen Zustand. Nachrichten von der königlichen Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse (1913), 582–592.Google Scholar
  52. [52]
    A. S. Wineman, K. R. Rajagopal: On a constitutive theory for materials undergoing microstructural changes. Archives of Mechanics 42 (1990), 53–74.Google Scholar
  53. [53]
    H. Ziegler: Some extremum principles in irreversible thermodynamics. In: Progress in Solid Mechanics (I. Sneddon, R. Hill, eds.). North-Holland Publishing Company, Amsterdam, 1963.Google Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2003

Authors and Affiliations

  • K. R. Rajagopal
    • 1
  1. 1.Department of Mechanical Engineering Texas A&MUniversity College StationUSA

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