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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 |
References:
[1] |
G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.
doi: 10.1137/0523084. |
[2] |
M. F. Ashby, The deformation of plastically non-homogeneous alloys, Philos. Mag., 21 (1970), 399-424.
doi: 10.1080/14786437008238426. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
show all references
References:
[1] |
G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.
doi: 10.1137/0523084. |
[2] |
M. F. Ashby, The deformation of plastically non-homogeneous alloys, Philos. Mag., 21 (1970), 399-424.
doi: 10.1080/14786437008238426. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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