Two-scale homogenization for a model in strain gradient plasticity
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 4, p. 1035-1065

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.

DOI : https://doi.org/10.1051/cocv/2010036
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] G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992) 1482-1518. | MR 1185639 | Zbl 0770.35005

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

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

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

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

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

[7] N.A. Fleck and J.W. Hutchinson, Strain gradient plasticity. Adv. Appl. Mech. 33 (1997) 295-361. | Zbl 0894.73031

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

[9] N.A. Fleck and J.R. Willis, 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

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

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

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

[13] M.E. Gurtin and L. Anand, 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

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

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

[16] A. Mielke, 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. Mielke and F. Theil, 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.

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

[19] G. Nguetseng, 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] L. Tartar, Nonlocal effects induced by homogenization, in Partial differential equations and the calculus of variations II, Progr. Nonlinear Differential Equations Appl. 2, Birkhäuser Boston, Boston (1989) 925-938. | MR 1034036 | Zbl 0682.35028

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

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

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