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. doi: 10.1137/0523084. Google Scholar

[2]

M. F. Ashby, The deformation of plastically non-homogeneous alloys,, Philos. Mag., 21 (1970), 399. doi: 10.1080/14786437008238426. Google Scholar

[3]

N. A. Fleck and J. W. Hutchinson, Strain gradient plasticity,, Adv. Appl. Mech., 33 (1997), 295. 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. 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. 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. 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. 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. doi: 10.1051/cocv/2010036. Google Scholar

[9]

P. Gudmundson, A unified treatment of strain gradient plasticity,, J. Mech. Phys. Solids, 52 (2004), 1379. 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), 53 (2005), 1624. Google Scholar

[11]

A. Mielke, Evolution of rate-independent systems,, in, (2005), 461. 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. doi: 10.1137/060672790. Google Scholar

[13]

F. Murat and L. Tartar, $H$-convergence,, in, 31 (1997), 21. 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. doi: 10.1137/0520043. Google Scholar

show all references

References:
[1]

G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482. doi: 10.1137/0523084. Google Scholar

[2]

M. F. Ashby, The deformation of plastically non-homogeneous alloys,, Philos. Mag., 21 (1970), 399. doi: 10.1080/14786437008238426. Google Scholar

[3]

N. A. Fleck and J. W. Hutchinson, Strain gradient plasticity,, Adv. Appl. Mech., 33 (1997), 295. 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. 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. 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. 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. 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. doi: 10.1051/cocv/2010036. Google Scholar

[9]

P. Gudmundson, A unified treatment of strain gradient plasticity,, J. Mech. Phys. Solids, 52 (2004), 1379. 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), 53 (2005), 1624. Google Scholar

[11]

A. Mielke, Evolution of rate-independent systems,, in, (2005), 461. 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. doi: 10.1137/060672790. Google Scholar

[13]

F. Murat and L. Tartar, $H$-convergence,, in, 31 (1997), 21. 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. doi: 10.1137/0520043. Google Scholar

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