Comptes Rendus
Variational principles in the theory of gradient plasticity
Comptes Rendus. Mécanique, Volume 339 (2011) no. 12, pp. 743-750.

Gradient models have been intensively discussed in the literature for the study of time-dependent or time-independent processes such as visco-plasticity, plasticity and damage. This Note is devoted to the theory of Gradient Plasticity. A general and consistent mathematical description available for common time-independent behavior is presented. Our attention is focused on the derivation of general results such as the description of the governing equations for the global response, for the rate response, the expression of the associated variational principles and the question of uniqueness in terms of the energy potential and the dissipation potential.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crme.2011.08.002
Keywords: Continuum mechanics, Gradient plasticity, Standard model

Quoc-Son Nguyen 1

1 Laboratoire de mécanique des solides CNRS (umr7649), École polytechnique, 91128 Palaiseau cedex, France
@article{CRMECA_2011__339_12_743_0,
     author = {Quoc-Son Nguyen},
     title = {Variational principles in the theory of gradient plasticity},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {743--750},
     publisher = {Elsevier},
     volume = {339},
     number = {12},
     year = {2011},
     doi = {10.1016/j.crme.2011.08.002},
     language = {en},
}
TY  - JOUR
AU  - Quoc-Son Nguyen
TI  - Variational principles in the theory of gradient plasticity
JO  - Comptes Rendus. Mécanique
PY  - 2011
SP  - 743
EP  - 750
VL  - 339
IS  - 12
PB  - Elsevier
DO  - 10.1016/j.crme.2011.08.002
LA  - en
ID  - CRMECA_2011__339_12_743_0
ER  - 
%0 Journal Article
%A Quoc-Son Nguyen
%T Variational principles in the theory of gradient plasticity
%J Comptes Rendus. Mécanique
%D 2011
%P 743-750
%V 339
%N 12
%I Elsevier
%R 10.1016/j.crme.2011.08.002
%G en
%F CRMECA_2011__339_12_743_0
Quoc-Son Nguyen. Variational principles in the theory of gradient plasticity. Comptes Rendus. Mécanique, Volume 339 (2011) no. 12, pp. 743-750. doi : 10.1016/j.crme.2011.08.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2011.08.002/

[1] R. de Borst; H.B. Muhlhaus Gradient-dependent plasticity: formulation and algorithmic aspects, Int. J. Num. Meth. Engrg., Volume 35 (1992), pp. 521-539

[2] M. Frémond Contact unilatéral avec adhérence une théorie du premier gradient (G. Del Piero; F. Maceri, eds.), Unilateral Problems in Structural Analysis, CISM Course, vol. 304, Springer-Verlag, Wien, 1985, pp. 117-137

[3] M. Frémond; B. Nedjar Damage, gradient of damage and principle of virtual power, Int. J. Solids and Structures, Volume 33 (1996), pp. 1083-1103

[4] G.A. Maugin; W. Muschik Thermodynamics with internal variables, part 1: general concepts, J. Non-equilibrium Thermodynamics, Volume 19 (1994), pp. 217-249

[5] M.E. Gurtin Generalized Ginzburg–Landau and Cahn–Hilliard equations based on a microforce balance, Physica D, Volume 92 (1996), pp. 178-192

[6] N.A. Fleck; J.W. Hutchinson A reformulation of strain gradient plasticity, JMPS, Volume 49 (2001), pp. 2245-2271

[7] C. Polizzotto Unified thermodynamic framework of nonlocal/gradient continuum theories, Eur. J. Mech. A/Solids, Volume 22 (2003), pp. 651-668

[8] S. Forest; J.M. Cardona; R. Sievert Thermoelasticity of second-grade media (Maugin; Drouot; Sidoroff, eds.), Continuum Thermodynamics, Kluwer, Dordrecht, 2000

[9] N.A. Fleck; J.R. Willis A mathematical basis for strain-gradient plasticity theory, part ii, JMPS, Volume 57 (2009), pp. 1045-1057

[10] E. Lorentz; S. Andrieux A variational formulation for nonlocal damage models, Int. J. Plasticity, Volume 15 (2003), pp. 119-138

[11] J.J. Moreau Sur les lois de frottement, de plasticité et de viscosité, C. R. Acad. Sciences, Volume 271 (1970), pp. 608-611

[12] Q.S. Nguyen Stability and Nonlinear Solid Mechanics, Wiley, Chichester, 2000

[13] G. Duvaut; J.L. Lions Les inéquations en mécanique et en physique, Dunod, Paris, 1972

[14] G. Francfort; A. Mielke Existence results for a class of rate-independent material models with nonconvex elastic energies, J. R. A. Math. Mech., Volume 595 (2006), pp. 55-91

[15] J. Mandel Cours de mécanique des milieux continus, Gauthier-Villars, Paris, 1966

[16] R. Hill A general theory of uniqueness and stability in elastic/plastic solids, J. Mech. Phys. Solids, Volume 6 (1958), pp. 236-249

[17] A. Giacomini; A. Musesti Two-scale homogenization for a model in strain gradient plasticity, ESAIM: Control, Optimisation and Calculus of Variations (2011) | DOI

[18] M.E. Gurtin; L. Anand A theory of strain-gradient plasticity for isotropic, plastically irrotational materials, JMPS, Volume 53 (2005), pp. 1624-1649

[19] A. Mainik; A. Mielke Global existence for rate-independent gradient plasticity at finite strain, J. Nonlinear Sci., Volume 19 (2009), pp. 221-248

Cited by Sources:

Comments - Policy