Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-14T09:29:44.500Z Has data issue: false hasContentIssue false
  • English
  • Français

Two-scale homogenization for a modelin strain gradient plasticity

Published online by Cambridge University Press:  28 October 2010

Alessandro Giacomini  and
Alessandro Musesti
Alessandro Giacomini
Affiliation:
Dipartimento di Matematica, Facoltà di Ingegneria, Università degli Studi di Brescia, Via Valotti 9, 25133 Brescia, Italy. [email protected]
Alessandro Musesti
Affiliation:
Dipartimento di Matematica e Fisica “Niccolò Tartaglia”, Università Cattolica del Sacro Cuore, Via dei Musei 41, 25121 Brescia, Italy. [email protected]
Get access

Abstract

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.

Keywords

Strain gradient plasticityperiodic homogenizationtwo-scale convergencequasistatic evolutions
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 4 , October 2011 , pp. 1035 - 1065
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Allaire, G., Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992) 1482–1518. CrossRef
Ashby, M.F., The deformation of plastically non-homogeneous alloys. Philos. Mag. 21 (1970) 399–424. CrossRef
Cioranescu, D., Damlamian, A. and Griso, G., Periodic unfolding and homogenization. C. R. Math. Acad. Sci. Paris 335 (2002) 99–104. CrossRef
Cioranescu, D., Damlamian, A. and Griso, G., The periodic unfolding method in homogenization. SIAM J. Math. Anal. 40 (2008) 1585–1620. CrossRef
Dal Maso, G., DeSimone, A. and Mora, M.G., Quasistatic evolution problems for linearly elastic-perfectly plastic materials. Arch. Ration. Mech. Anal. 180 (2006) 237–291. CrossRef
R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science and technology 2, Functional and variational methods. Springer-Verlag, Berlin (1988).
Fleck, N.A. and Hutchinson, J.W., Strain gradient plasticity. Adv. Appl. Mech. 33 (1997) 295–361. CrossRef
Fleck, N.A. and Hutchinson, J.W., A reformulation of strain gradient plasticity. J. Mech. Phys. Solids. 49 (2001) 2245–2271. CrossRef
Fleck, N.A. and Willis, J.R., 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. CrossRef
Francfort, G. and Suquet, P.-M., Homogenization and mechanical dissipation in thermoviscoelasticity. Arch. Ration. Mech. Anal. 96 (1986) 265–293. CrossRef
Giacomini, A. and Lussardi, L., Quasi-static evolution for a model in strain gradient plasticity. SIAM J. Math. Anal. 40 (2008) 1201–1245. CrossRef
Gudmundson, P., A unified treatment of strain gradient plasticity. J. Mech. Phys. Solids 52 (2004) 1379–1406. CrossRef
Gurtin, M.E. and Anand, L., A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. I. Small deformations. J. Mech. Phys. Solids 53 (2005) 1624–1649. CrossRef
Lukkassen, D., Nguetseng, G. and Wall, P., Two-scale convergence. Int. J. Pure Appl. Math. 2 (2002) 35–86.
Mainik, A. and Mielke, A., Existence results for energetic models for rate-independent systems. Calc. Var. Partial Differential Equations 22 (2005) 73–99. CrossRef
A. Mielke, Evolution of rate-independent systems, in Handb. Differ. Equ., Evolutionary equations II, Elsevier/North-Holland, Amsterdam (2005) 461–559.
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.
Mielke, A. and Timofte, A.M., Two-scale homogenization for evolutionary variational inequalities via the energetic formulation. SIAM J. Math. Anal. 39 (2007) 642–668. CrossRef
Nguetseng, G., A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608–623. CrossRef
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.
Tartar, L., Memory effects and homogenization. Arch. Ration. Mech. Anal. 111 (1990) 121–133. CrossRef
Visintin, A., Homogenization of the nonlinear Kelvin-Voigt model of viscoelasticity and of the Prager model of plasticity. Contin. Mech. Thermodyn. 18 (2006) 223–252. CrossRef
Visintin, A., 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. CrossRef
Visintin, A., Homogenization of nonlinear visco-elastic composites. J. Math. Pures Appl. 89 (2008) 477–504. CrossRef
Willis, J.R., Bounds and self-consistent estimates for the overall moduli of anisotropic composites. J. Mech. Phys. Solids 25 (1977) 182–202. CrossRef