# American Institute of Mathematical Sciences

February  2013, 6(1): 43-62. doi: 10.3934/dcdss.2013.6.43

## On the Fleck and Willis homogenization procedure in strain gradient plasticity

 1 LAGA, Université Paris-Nord, Avenue J.-B. Clément 93430, Villetaneuse, France 2 Dipartimento di Matematica, Facoltà di Ingegneria, Università degli Studi di Brescia, Via Valotti 9, 25133 Brescia, Italy 3 Dipartimento di Matematica e Fisica “Niccolò Tartaglia", Università Cattolica del Sacro Cuore, Via dei Musei 41, 25121 Brescia, Italy

Received  May 2011 Revised  July 2011 Published  October 2012

We revisit the homogenization process for a heterogeneous small strain gradient plasticity model considered in [5]. We derive a precise homogenized behavior, independently of any kind of periodicity assumption and demonstrate that it reduces to a model studied in [8] when periodicity is re-introduced.
Citation: Gilles A. Francfort, Alessandro Giacomini, Alessandro Musesti. On the Fleck and Willis homogenization procedure in strain gradient plasticity. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 43-62. doi: 10.3934/dcdss.2013.6.43
##### References:
 [1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.  Google Scholar [2] M. F. Ashby, The deformation of plastically non-homogeneous alloys, Philos. Mag., 21 (1970), 399-424. doi: 10.1080/14786437008238426.  Google Scholar [3] N. A. Fleck and J. W. Hutchinson, Strain gradient plasticity, Adv. Appl. Mech., 33 (1997), 295-361. doi: 10.1016/S0065-2156(08)70388-0.  Google Scholar [4] N. A. Fleck and J. W. Hutchinson, A reformulation of strain gradient plasticity, J. Mech. Phys. Solids, 49 (2001), 2245-2271. doi: 10.1016/S0022-5096(01)00049-7.  Google Scholar [5] 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. doi: 10.1016/j.jmps.2004.02.001.  Google Scholar [6] G. A. Francfort and S. M\"uller, Combined effects of homogenization and singular perturbations in elasticity, J. Reine Angew. Math., 454 (1994), 1-35. doi: 10.1515/crll.1994.454.1.  Google Scholar [7] A. Garroni, G. Leoni and M. Ponsiglione, Gradient theory for plasticity via homogenization of discrete dislocations, J. Eur. Math. Soc., 12 (2010), 1231-1266. doi: 10.4171/JEMS/228.  Google Scholar [8] A. Giacomini and A. Musesti, Two-scale homogenization for a model in strain gradient plasticity, ESAIM Control Optim. Calc. Var, 17 (2011), 1035-1065. doi: 10.1051/cocv/2010036.  Google Scholar [9] P. Gudmundson, A unified treatment of strain gradient plasticity, J. Mech. Phys. Solids, 52 (2004), 1379-1406. doi: 10.1016/j.jmps.2003.11.002.  Google Scholar [10] 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.  Google Scholar [11] A. Mielke, Evolution of rate-independent systems, in "Evolutionary Equations. Vol. II'' (Edited by C. M. Dafermos and E. Feireisl), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam (2005), 461-559.  Google Scholar [12] 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. doi: 10.1137/060672790.  Google Scholar [13] F. Murat and L. Tartar, $H$-convergence, in "Topics in the Mathematical Modelling of Composite Materials,'' Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, 31 (1997), 21-43.  Google Scholar [14] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043.  Google Scholar

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##### References:
 [1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.  Google Scholar [2] M. F. Ashby, The deformation of plastically non-homogeneous alloys, Philos. Mag., 21 (1970), 399-424. doi: 10.1080/14786437008238426.  Google Scholar [3] N. A. Fleck and J. W. Hutchinson, Strain gradient plasticity, Adv. Appl. Mech., 33 (1997), 295-361. doi: 10.1016/S0065-2156(08)70388-0.  Google Scholar [4] N. A. Fleck and J. W. Hutchinson, A reformulation of strain gradient plasticity, J. Mech. Phys. Solids, 49 (2001), 2245-2271. doi: 10.1016/S0022-5096(01)00049-7.  Google Scholar [5] 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. doi: 10.1016/j.jmps.2004.02.001.  Google Scholar [6] G. A. Francfort and S. M\"uller, Combined effects of homogenization and singular perturbations in elasticity, J. Reine Angew. Math., 454 (1994), 1-35. doi: 10.1515/crll.1994.454.1.  Google Scholar [7] A. Garroni, G. Leoni and M. Ponsiglione, Gradient theory for plasticity via homogenization of discrete dislocations, J. Eur. Math. Soc., 12 (2010), 1231-1266. doi: 10.4171/JEMS/228.  Google Scholar [8] A. Giacomini and A. Musesti, Two-scale homogenization for a model in strain gradient plasticity, ESAIM Control Optim. Calc. Var, 17 (2011), 1035-1065. doi: 10.1051/cocv/2010036.  Google Scholar [9] P. Gudmundson, A unified treatment of strain gradient plasticity, J. Mech. Phys. Solids, 52 (2004), 1379-1406. doi: 10.1016/j.jmps.2003.11.002.  Google Scholar [10] 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.  Google Scholar [11] A. Mielke, Evolution of rate-independent systems, in "Evolutionary Equations. Vol. II'' (Edited by C. M. Dafermos and E. Feireisl), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam (2005), 461-559.  Google Scholar [12] 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. doi: 10.1137/060672790.  Google Scholar [13] F. Murat and L. Tartar, $H$-convergence, in "Topics in the Mathematical Modelling of Composite Materials,'' Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, 31 (1997), 21-43.  Google Scholar [14] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043.  Google Scholar
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