Comptes Rendus
no. 6
Quasi-static response, implicit scheme and incremental problem in gradient plasticity
Quoc-Son Nguyen1
1 Laboratoire de Mécanique des Solides, École Polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau, France
Comptes Rendus. Mécanique, Volume 344 (2016) no. 6, pp. 439-447.

This paper is devoted to the study of gradient plasticity at small strains. Some time-independent dissipative processes such as brittle damage can also be considered in the same framework. Our attention is focussed on the description of the constitutive equations, on the formulation of the governing equations in terms of the energy potential and the dissipation potential of the solid. A time-discretization by the implicit scheme of the evolution equation leads to the study of the incremental problem which is different from the rate problem. The increment of the response under an increment of the loads must satisfy a variational inequality and, if the energy potential is convex, an incremental minimum principle. In particular, a local minimum of the incremental minimum principle is a stable solution to the variational inequality.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crme.2016.01.004
Keywords: Gradient plasticity, Brittle damage, Standard models, Evolution equation, Implicit scheme, Incremental problem, Variational principles

Quoc-Son Nguyen 1

1 Laboratoire de Mécanique des Solides, École Polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau, France 
@article{CRMECA_2016__344_6_439_0,
     author = {Quoc-Son Nguyen},
     title = {Quasi-static response, implicit scheme and incremental problem in gradient plasticity},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {439--447},
     publisher = {Elsevier},
     volume = {344},
     number = {6},
     year = {2016},
     doi = {10.1016/j.crme.2016.01.004},
     language = {en},
}
TY  - JOUR
AU  - Quoc-Son Nguyen
TI  - Quasi-static response, implicit scheme and incremental problem in gradient plasticity
JO  - Comptes Rendus. Mécanique
PY  - 2016
SP  - 439
EP  - 447
VL  - 344
IS  - 6
PB  - Elsevier
DO  - 10.1016/j.crme.2016.01.004
LA  - en
ID  - CRMECA_2016__344_6_439_0
ER  - 
%0 Journal Article
%A Quoc-Son Nguyen
%T Quasi-static response, implicit scheme and incremental problem in gradient plasticity
%J Comptes Rendus. Mécanique
%D 2016
%P 439-447
%V 344
%N 6
%I Elsevier
%R 10.1016/j.crme.2016.01.004
%G en
%F CRMECA_2016__344_6_439_0
Quoc-Son Nguyen. Quasi-static response, implicit scheme and incremental problem in gradient plasticity. Comptes Rendus. Mécanique, Volume 344 (2016) no. 6, pp. 439-447. doi : 10.1016/j.crme.2016.01.004. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2016.01.004/

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

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

[3] B. Bourdin; G.A. Francfort; J.J. Marigo Numerical experiments in revisited brittle fracture, J. Mech. Phys. Solids, Volume 48 (2000), pp. 797-826

[4] N.A. Fleck; J.R. Willis A mathematical basis for strain-gradient plasticity theory, part II, J. Mech. Phys. Solids, Volume 57 (2009), pp. 1045-1057

[5] A. Giacomini; L. Lussardi Quasi-static evolution for a model in strain gradient plasticity, SIAM J. Math. Anal., Volume 40 (2008), pp. 1201-1245

[6] P. Neff; A. Sydov; C. Wieners Numerical approximation of incremental infinitesimal gradient plasticity, Int. J. Numer. Methods Eng., Volume 77 (2009) no. 3, pp. 414-436

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

[8] M. Frémond Phase Change in Mechanics, Lecture Notes UMI, Springer-Verlag, Berlin, 2012

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

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

[11] G. Francfort; J.J. Marigo Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, Volume 46 (1998), pp. 1319-1342

[12] G. Duvaut; J.L. Lions Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972

[13] G. Dal Maso; A. De Simone; M.G. Mora Quasistatic evolution problems for linearly elastic-perfectly plastic materials, J. Ration. Mech. Anal., Volume 180 (2006), pp. 237-291

[14] A. Giacomini On the energetic formulation of the Gurtin and Anand model in strain gradient plasticity, Discrete Contin. Dyn. Syst., Volume 17 (2012), pp. 527-552

[15] Q.S. Nguyen; D. Radenkovic Stability of an equilibrium in elastic–plastic solids, Marseille (1975)

[16] N. Lahellec; P. Suquet Effective response and field statistics in elasto-plastic and elasto-viscoplastic composites under radial and non-radial loading, Int. J. Plast., Volume 42 (2013), pp. 1-30

Cited by Sources:

Comments - Policy