On implicit constitutive theories

Kumbakonam R. Rajagopal

Applications of Mathematics (2003)

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

Abstract

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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

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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 -

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Citations in EuDML Documents

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  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. Pavel Krejčí, Josef Málek, Vít Průša, Special issue dedicated to professor K. R. Rajagopal
  3. Satish Karra, Kumbakonam R. Rajagopal, Development of three dimensional constitutive theories based on lower dimensional experimental data
  4. Tomáš Bárta, One-dimensional model describing the non-linear viscoelastic response of materials
  5. Christiaan Le Roux, Kumbakonam R. Rajagopal, Shear flows of a new class of power-law fluids

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