American Institute of Mathematical Sciences

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

On the Fleck and Willis homogenization procedure in strain gradient plasticity

Gilles A. Francfort 1, , Alessandro Giacomini 2, and  Alessandro Musesti 3,

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.
Keywords: Homogenization, H-convergence., gradient plasticity.
Mathematics Subject Classification: Primary: 74Q05, 74C05; Secondary: 35B2.
Citation: Gilles A. Francfort, Alessandro Giacomini, Alessandro Musesti. On the Fleck and Willis homogenization procedure in strain gradient plasticity. Discrete and 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.

[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.

[1]

Alessandro Giacomini. On the energetic formulation of the Gurtin and Anand model in strain gradient plasticity. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 527-552. doi: 10.3934/dcdsb.2012.17.527
[2]

Michiel Bertsch, Roberta Dal Passo, Lorenzo Giacomelli, Giuseppe Tomassetti. A nonlocal and fully nonlinear degenerate parabolic system from strain-gradient plasticity. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 15-43. doi: 10.3934/dcdsb.2011.15.15
[3]

Jean Louis Woukeng. $\sum $-convergence and reiterated homogenization of nonlinear parabolic operators. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1753-1789. doi: 10.3934/cpaa.2010.9.1753
[4]

Jie Zhao. Convergence rates for elliptic reiterated homogenization problems. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2787-2795. doi: 10.3934/cpaa.2013.12.2787
[5]

Andriy Bondarenko, Guy Bouchitté, Luísa Mascarenhas, Rajesh Mahadevan. Rate of convergence for correctors in almost periodic homogenization. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 503-514. doi: 10.3934/dcds.2005.13.503
[6]

Stefan Kindermann. Convergence of the gradient method for ill-posed problems. Inverse Problems and Imaging, 2017, 11 (4) : 703-720. doi: 10.3934/ipi.2017033
[7]

Martin Heida, Stefan Neukamm, Mario Varga. Stochastic homogenization of $ \Lambda $-convex gradient flows. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 427-453. doi: 10.3934/dcdss.2020328
[8]

Craig Cowan, Abdolrahman Razani. Singular solutions of a Hénon equation involving a nonlinear gradient term. Communications on Pure and Applied Analysis, 2022, 21 (1) : 141-158. doi: 10.3934/cpaa.2021172
[9]

Weisheng Niu, Yao Xu. Convergence rates in homogenization of higher-order parabolic systems. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4203-4229. doi: 10.3934/dcds.2018183
[10]

Fabio Camilli, Claudio Marchi. On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems. Networks and Heterogeneous Media, 2011, 6 (1) : 61-75. doi: 10.3934/nhm.2011.6.61
[11]

Patrick Henning. Convergence of MsFEM approximations for elliptic, non-periodic homogenization problems. Networks and Heterogeneous Media, 2012, 7 (3) : 503-524. doi: 10.3934/nhm.2012.7.503
[12]

Arianna Giunti. Convergence rates for the homogenization of the Poisson problem in randomly perforated domains. Networks and Heterogeneous Media, 2021, 16 (3) : 341-375. doi: 10.3934/nhm.2021009
[13]

Erik Kropat. Homogenization of optimal control problems on curvilinear networks with a periodic microstructure --Results on $\boldsymbol{S}$-homogenization and $\boldsymbol{Γ}$-convergence. Numerical Algebra, Control and Optimization, 2017, 7 (1) : 51-76. doi: 10.3934/naco.2017003
[14]

Wenqing Hu, Chris Junchi Li. A convergence analysis of the perturbed compositional gradient flow: Averaging principle and normal deviations. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 4951-4977. doi: 10.3934/dcds.2018216
[15]

C.Y. Wang, M.X. Li. Convergence property of the Fletcher-Reeves conjugate gradient method with errors. Journal of Industrial and Management Optimization, 2005, 1 (2) : 193-200. doi: 10.3934/jimo.2005.1.193
[16]

Yu-Ning Yang, Su Zhang. On linear convergence of projected gradient method for a class of affine rank minimization problems. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1507-1519. doi: 10.3934/jimo.2016.12.1507
[17]

Sylvia Serfaty. Gamma-convergence of gradient flows on Hilbert and metric spaces and applications. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1427-1451. doi: 10.3934/dcds.2011.31.1427
[18]

Sanming Liu, Zhijie Wang, Chongyang Liu. On convergence analysis of dual proximal-gradient methods with approximate gradient for a class of nonsmooth convex minimization problems. Journal of Industrial and Management Optimization, 2016, 12 (1) : 389-402. doi: 10.3934/jimo.2016.12.389
[19]

Tomáš Roubíček. Thermodynamics of perfect plasticity. Discrete and Continuous Dynamical Systems - S, 2013, 6 (1) : 193-214. doi: 10.3934/dcdss.2013.6.193
[20]

Luis Silvestre. Hölder continuity for integro-differential parabolic equations with polynomial growth respect to the gradient. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1069-1081. doi: 10.3934/dcds.2010.28.1069

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (63)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy