On implicit constitutive theories

Kumbakonam R. Rajagopal

Applications of Mathematics (2003)

  • Volume: 48, Issue: 4, page 279-319
  • ISSN: 0862-7940

Abstract

top
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).

How to cite

top

Rajagopal, Kumbakonam R.. "On implicit constitutive theories." Applications of Mathematics 48.4 (2003): 279-319. <http://eudml.org/doc/33150>.

@article{Rajagopal2003,
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).},
author = {Rajagopal, Kumbakonam R.},
journal = {Applications of Mathematics},
keywords = {constitutive relations; constraint; Lagrange multiplier; Helmholtz potential; rate of dissipation; elasticity; inelasticity; viscoelasticity; constraint; Lagrange multiplier; Helmholtz potential; rate of dissipation},
language = {eng},
number = {4},
pages = {279-319},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On implicit constitutive theories},
url = {http://eudml.org/doc/33150},
volume = {48},
year = {2003},
}

TY - JOUR
AU - Rajagopal, Kumbakonam R.
TI - On implicit constitutive theories
JO - Applications of Mathematics
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 48
IS - 4
SP - 279
EP - 319
AB - 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).
LA - eng
KW - constitutive relations; constraint; Lagrange multiplier; Helmholtz potential; rate of dissipation; elasticity; inelasticity; viscoelasticity; constraint; Lagrange multiplier; Helmholtz potential; rate of dissipation
UR - http://eudml.org/doc/33150
ER -

References

top
  1. De la résistance causée dans les machines, Mémoires de l’ Académie Royale  A  (1699), 257–282. (1699) 
  2. 10.1038/125309b0, Nature 125 (1930), 309–310. (1930) DOI10.1038/125309b0
  3. 10.1016/0020-7225(77)90016-7, Internat. J. Engrg. Sci. 15 (1977), 645–660. (1977) Zbl0364.73017MR0667196DOI10.1016/0020-7225(77)90016-7
  4. 10.1007/BF00276156, Arch. Rational Mech. Anal. 6 (1960), 89–104. (1960) Zbl0094.36702MR0118016DOI10.1007/BF00276156
  5. The Physics of High Pressure, The MacMillan Company, New York, 1931. (1931) 
  6. 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. (1785) 
  7. Traité de Dynamique, Paris, Chez David, Libraire, 1758. (1758) 
  8. 10.1103/PhysRev.73.373, Physical Rev. 73 (1948), 373–382. (1948) MR0023698DOI10.1103/PhysRev.73.373
  9. On steady flows of fluids and shear dependent viscosities, Proc. Roy. Soc. London Ser. A: Mathematical, Physical and Engineering Sciences (2003), Submitted. (2003) MR2121929
  10. On a new general principle of mechanics, 8 (1830), 137–140;. (1830) 
  11. Classical Mechanics, Addison-Wesley, Boston, 1980. (1980) Zbl0491.70001MR0575343
  12. Simple flows of fluids with pressure dependent viscosities, Proc. Roy. Soc. London Ser.  A: Mathematical, Physical and Engineering Sciences 457 (2001), 1603–1622. (2001) 
  13. A thermomechanical framework for the transition of a viscoelastic liquid to a viscoelastic solid, Mathematics and Mechanics of Solids (to appear). (to appear) MR2053023
  14. 10.1122/1.1485281, J.  of Rheology 46 (2002), 977–999. (2002) DOI10.1122/1.1485281
  15. Mécanique Analytique, Mme Ve Courcier, Paris, 1787, Translation (by A. Boissonnade, V. N.  Vagliente), Kluwer Academic Publishers, Dordrecht, 1997. (1787) MR1784335
  16. 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. (1870) 
  17. 10.1007/s00205-002-0219-4, Arch. Rational Mech. Anal. 165 (2002), 243–269. (2002) MR1941479DOI10.1007/s00205-002-0219-4
  18. On the Dynamical Theory of Gases, Philosophical Transactions of the Royal Society of London, Series A  (1866), 26–78. (1866) 
  19. 10.1016/0020-7225(66)90021-8, Internat. J. Engrg. Sci. 4 (1966), 155–178. (1966) MR0195307DOI10.1016/0020-7225(66)90021-8
  20. A thermodynamic framework for the constitutive modeling of asphalt concrete: theory and aplications, ASCE Journal of Materials, Accepted for publication. 
  21. Mémoire sur les lois du mouvement des fluides, Mém. Acad. Re. Sci., Paris 6 (1823), 389–416. (1823) 
  22. 10.1007/BF00277929, Arch. Rational Mech. Anal. 2 (1958), 197–226. (1958) Zbl0083.39303MR0105862DOI10.1007/BF00277929
  23. 10.1007/BF00253367, Arch. Rational Mech. Anal. 48 (1972), 1–50. (1972) Zbl0271.73006MR0445985DOI10.1007/BF00253367
  24. 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. (1831) 
  25. Spannungsverteilung in plastischen Körpern, In: Proceeding of the 1st International Congress in Applied Mechanics, Delft, 1924. (1924) 
  26. 10.1016/0020-7462(80)90002-5, Internat. J. Non-Linear Mech. 15 (1980), 83–91. (1980) MR0580724DOI10.1016/0020-7462(80)90002-5
  27. 10.1016/0749-6419(92)90056-I, International Journal of Plasticity 8 (1992), 385–395. (1992) DOI10.1016/0749-6419(92)90056-I
  28. Multiple configurations in Continuum Mechanics,, Report  6, Institute of Computational and Applied Mechanics, University of Pittsburgh, 1995. (1995) 
  29. On the inelastic behavior of solids—Part  I: Twinning, International Journal of Plasticity 11 (1995), 653–678. (1995) 
  30. 10.1016/S0749-6419(96)00049-6, International Journal of Plasticity 13 (1997), 1–35. (1997) DOI10.1016/S0749-6419(96)00049-6
  31. 10.1007/s007770050049, ARI 51 (1998), 160–168. (1998) DOI10.1007/s007770050049
  32. 10.1016/S0749-6419(98)00037-0, International Journal of Plasticity 14 (1998), 945–967. (1998) DOI10.1016/S0749-6419(98)00037-0
  33. 10.1016/S0749-6419(98)00041-2, International Journal of Plasticity 14 (1998), 969–995. (1998) DOI10.1016/S0749-6419(98)00041-2
  34. 10.1007/s000330050028, ZAMP 50 (1999), 459–496. (1999) MR1697717DOI10.1007/s000330050028
  35. 10.1016/S0377-0257(99)00023-3, Journal of Non-Newtonian Fluid Mechanics 88 (2000), 207–227. (2000) DOI10.1016/S0377-0257(99)00023-3
  36. Modeling anistropic fluids within the framework of bodies with multiple natural configurations, Journal of Non-Newtonian Fluid Mechanics (2001), . (2001) 
  37. 10.1007/PL00012620, ZAMP 53 (2002), 923–948. (2002) MR1963544DOI10.1007/PL00012620
  38. 10.1016/S0020-7683(00)00079-2, International Journal of Solids and Structures 38 (2001), 1149–1167. (2001) DOI10.1016/S0020-7683(00)00079-2
  39. Growth and remodeling in a dynamically loaded axial tissue, Computational Method in Engineering Science, In press. 
  40. Phenomenological modeling of polymer crystallization using the notion of multiple natural configurations, Interfaces and free boundaries 2 (2000), 73–94. (2000) MR1759500
  41. 10.1007/s00033-002-8161-8, Z.  Angew. Math. Phys. 53 (2002), 365–406. (2002) MR1910337DOI10.1007/s00033-002-8161-8
  42. 10.1002/zamm.19300100308, Z. Angew. Math. Mech. 10 (1939), 266–274. (1939) DOI10.1002/zamm.19300100308
  43. Note à joindre au Mémoire sur la dynamique des fluides, C.  R.  Acad. Sci. 17 (1843), 1240–1243. (1843) 
  44. Continuum Physics, Vol.  3, A. C. Eringen (ed.), Academic Press, New York, 1975. (1975) MR0468444
  45. A phenomenological model of twinning based on dual reference structures, Acta. Metall. (1998), 1–14. (1998) 
  46. 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. (1845) 
  47. 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. (1867) 
  48. A First Course in Rational Continuum Mechanics, Academic Press, Boston-San Diego, 1991. (1991) Zbl0866.73001MR1162744
  49. Hypo-elasticity, J.  Rational Mechanics and Analysis 4 (1955), 323–425. (1955) Zbl0064.42002MR0068412
  50. The Non-Linear Field Theories of Mechanics. Handbuch der Physik, I I I 3 , Springer-Verlag, Berlin-Heidelberg-New York, 1965. (1965) MR0193816
  51. Mechanik der festen Körpern im plastisch-deformablen Zustand. Nachrichten von der königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1913), 582–592. (1913) 
  52. On a constitutive theory for materials undergoing microstructural changes, Archives of Mechanics 42 (1990), 53–74. (1990) MR1108245
  53. Some extremum principles in irreversible thermodynamics, In: Progress in Solid Mechanics, I. Sneddon, R.  Hill (eds.), North-Holland Publishing Company, Amsterdam, 1963. (1963) MR0163470

Citations in EuDML Documents

top
  1. M. Bulíček, Josef Málek, Kumbakonam R. Rajagopal, Analysis of the flows of incompressible fluids with pressure dependent viscosity fulfilling ν ( p , · ) → + ∞ as p → + ∞
  2. Satish Karra, Kumbakonam R. Rajagopal, Development of three dimensional constitutive theories based on lower dimensional experimental data
  3. Pavel Krejčí, Josef Málek, Vít Průša, Special issue dedicated to professor K. R. Rajagopal
  4. Andreas Almqvist, Evgeniya Burtseva, Kumbakonam R. Rajagopal, Peter Wall, On modeling flow between adjacent surfaces where the fluid is governed by implicit algebraic constitutive relations
  5. Christiaan Le Roux, Kumbakonam R. Rajagopal, Shear flows of a new class of power-law fluids
  6. Tomáš Bárta, One-dimensional model describing the non-linear viscoelastic response of materials
  7. Kumbakonam R. Rajagopal, A generalization of the classical Euler and Korteweg fluids

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.