Using the tool of two-scale convergence, we provide a rigorous mathematical setting for the homogenization result obtained by Fleck and Willis [J. Mech. Phys. Solids 52 (2004) 1855-1888] concerning the effective plastic behaviour of a strain gradient composite material. Moreover, moving from deformation theory to flow theory, we prove a convergence result for the homogenization of quasistatic evolutions in the presence of isotropic linear hardening.

Classification: 74C05, 74G65, 74Q05, 35B27, 49J45

Keywords: strain gradient plasticity, periodic homogenization, two-scale convergence, quasistatic evolutions

@article{COCV_2011__17_4_1035_0, author = {Giacomini, Alessandro and Musesti, Alessandro}, title = {Two-scale homogenization for a model in strain gradient plasticity}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, publisher = {EDP-Sciences}, volume = {17}, number = {4}, year = {2011}, pages = {1035-1065}, doi = {10.1051/cocv/2010036}, zbl = {1300.74008}, mrnumber = {2859864}, language = {en}, url = {http://www.numdam.org/item/COCV_2011__17_4_1035_0} }

Giacomini, Alessandro; Musesti, Alessandro. Two-scale homogenization for a model in strain gradient plasticity. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 4, pp. 1035-1065. doi : 10.1051/cocv/2010036. http://www.numdam.org/item/COCV_2011__17_4_1035_0/

[1] Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992) 1482-1518. | MR 1185639 | Zbl 0770.35005

,[2] The deformation of plastically non-homogeneous alloys. Philos. Mag. 21 (1970) 399-424.

,[3] Periodic unfolding and homogenization. C. R. Math. Acad. Sci. Paris 335 (2002) 99-104. | MR 1921004 | Zbl 1001.49016

, and ,[4] The periodic unfolding method in homogenization. SIAM J. Math. Anal. 40 (2008) 1585-1620. | MR 2466168 | Zbl 1167.49013

, and ,[5] Quasistatic evolution problems for linearly elastic-perfectly plastic materials. Arch. Ration. Mech. Anal. 180 (2006) 237-291. | MR 2210910 | Zbl 1093.74007

, and ,[6] Mathematical analysis and numerical methods for science and technology 2, Functional and variational methods. Springer-Verlag, Berlin (1988). | MR 969367 | Zbl 0755.35001

and ,[7] Strain gradient plasticity. Adv. Appl. Mech. 33 (1997) 295-361. | Zbl 0894.73031

and ,[8] A reformulation of strain gradient plasticity. J. Mech. Phys. Solids. 49 (2001) 2245-2271. | Zbl 1033.74006

and ,[9] Bounds and estimates for the effect of strain gradients upon the effective plastic properties of an isotropic two-phase composite. J. Mech. Phys. Solids 52 (2004) 1855-1888. | MR 2070929 | Zbl 1122.74486

and ,[10] Homogenization and mechanical dissipation in thermoviscoelasticity. Arch. Ration. Mech. Anal. 96 (1986) 265-293. | MR 855306 | Zbl 0621.73044

and ,[11] Quasi-static evolution for a model in strain gradient plasticity. SIAM J. Math. Anal. 40 (2008) 1201-1245. | MR 2452885 | Zbl 1162.74009

and ,[12] A unified treatment of strain gradient plasticity. J. Mech. Phys. Solids 52 (2004) 1379-1406. | MR 2049012 | Zbl 1114.74366

,[13] A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. I. Small deformations. J. Mech. Phys. Solids 53 (2005) 1624-1649. | MR 2146591 | Zbl 1120.74353

and ,[14] Two-scale convergence. Int. J. Pure Appl. Math. 2 (2002) 35-86. | MR 1912819 | Zbl 1061.35015

, and ,[15] Existence results for energetic models for rate-independent systems. Calc. Var. Partial Differential Equations 22 (2005) 73-99. | MR 2105969 | Zbl 1161.74387

and ,[16] Evolution of rate-independent systems, in Handb. Differ. Equ., Evolutionary equations II, Elsevier/North-Holland, Amsterdam (2005) 461-559. | MR 2182832 | Zbl 1120.47062

,[17] A mathematical model for rate independent phase transformations with hysteresis, in Proceedings of the Workshop on Models of Continuum Mechanics in Analysis and Engineering, H.-D. Alber, R. Balean and R. Farwig Eds., Shaker-Verlag, Aachen (1999) 117-129.

and ,[18] Two-scale homogenization for evolutionary variational inequalities via the energetic formulation. SIAM J. Math. Anal. 39 (2007) 642-668. | MR 2338425 | Zbl 1185.35282

and ,[19] A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608-623. | MR 990867 | Zbl 0688.35007

,[20] Nonlocal effects induced by homogenization, in Partial differential equations and the calculus of variations II, Progr. Nonlinear Differential Equations Appl. 2, BirkhĂ¤user Boston, Boston (1989) 925-938. | MR 1034036 | Zbl 0682.35028

,[21] Memory effects and homogenization. Arch. Ration. Mech. Anal. 111 (1990) 121-133. | MR 1057651 | Zbl 0725.45012

,[22] Homogenization of the nonlinear Kelvin-Voigt model of viscoelasticity and of the Prager model of plasticity. Contin. Mech. Thermodyn. 18 (2006) 223-252. | MR 2245987 | Zbl 1160.74331

,[23] Homogenization of the nonlinear Maxwell model of viscoelasticity and of the Prandtl-Reuss model of elastoplasticity. Proc. Roy. Soc. Edinburgh Sect. A 138 (2008) 1363-1401. | MR 2488064 | Zbl 1170.35016

,[24] Homogenization of nonlinear visco-elastic composites. J. Math. Pures Appl. 89 (2008) 477-504. | MR 2416672 | Zbl 1166.35004

,[25] Bounds and self-consistent estimates for the overall moduli of anisotropic composites. J. Mech. Phys. Solids 25 (1977) 182-202. | Zbl 0363.73014

,