Two-scale homogenization for a model in strain gradient plasticity

Alessandro Giacomini; Alessandro Musesti

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 4, page 1035-1065
  • ISSN: 1292-8119

Abstract

top
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. Solids52 (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.

How to cite

top

Giacomini, Alessandro, and Musesti, Alessandro. "Two-scale homogenization for a model in strain gradient plasticity." ESAIM: Control, Optimisation and Calculus of Variations 17.4 (2011): 1035-1065. <http://eudml.org/doc/221925>.

@article{Giacomini2011,
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. Solids52 (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. },
author = {Giacomini, Alessandro, Musesti, Alessandro},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Strain gradient plasticity; periodic homogenization; two-scale convergence; quasistatic evolutions},
language = {eng},
month = {11},
number = {4},
pages = {1035-1065},
publisher = {EDP Sciences},
title = {Two-scale homogenization for a model in strain gradient plasticity},
url = {http://eudml.org/doc/221925},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Giacomini, Alessandro
AU - Musesti, Alessandro
TI - Two-scale homogenization for a model in strain gradient plasticity
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2011/11//
PB - EDP Sciences
VL - 17
IS - 4
SP - 1035
EP - 1065
AB - 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. Solids52 (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.
LA - eng
KW - Strain gradient plasticity; periodic homogenization; two-scale convergence; quasistatic evolutions
UR - http://eudml.org/doc/221925
ER -

References

top
  1. G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal.23 (1992) 1482–1518.  
  2. M.F. Ashby, The deformation of plastically non-homogeneous alloys. Philos. Mag.21 (1970) 399–424.  
  3. D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization. C. R. Math. Acad. Sci. Paris335 (2002) 99–104.  
  4. D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization. SIAM J. Math. Anal.40 (2008) 1585–1620.  
  5. G. Dal Maso, A. DeSimone and M.G. Mora, Quasistatic evolution problems for linearly elastic-perfectly plastic materials. Arch. Ration. Mech. Anal.180 (2006) 237–291.  
  6. R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science and technology 2, Functional and variational methods. Springer-Verlag, Berlin (1988).  
  7. N.A. Fleck and J.W. Hutchinson, Strain gradient plasticity. Adv. Appl. Mech.33 (1997) 295–361.  
  8. N.A. Fleck and J.W. Hutchinson, A reformulation of strain gradient plasticity. J. Mech. Phys. Solids.49 (2001) 2245–2271.  
  9. 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. Solids52 (2004) 1855–1888.  
  10. G. Francfort and P.-M. Suquet, Homogenization and mechanical dissipation in thermoviscoelasticity. Arch. Ration. Mech. Anal.96 (1986) 265–293.  
  11. A. Giacomini and L. Lussardi, Quasi-static evolution for a model in strain gradient plasticity. SIAM J. Math. Anal.40 (2008) 1201–1245.  
  12. P. Gudmundson, A unified treatment of strain gradient plasticity. J. Mech. Phys. Solids52 (2004) 1379–1406.  
  13. M.E. Gurtin and L. Anand, A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. I. Small deformations. J. Mech. Phys. Solids53 (2005) 1624–1649.  
  14. D. Lukkassen, G. Nguetseng and P. Wall, Two-scale convergence. Int. J. Pure Appl. Math.2 (2002) 35–86.  
  15. A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems. Calc. Var. Partial Differential Equations22 (2005) 73–99.  
  16. A. Mielke, Evolution of rate-independent systems, in Handb. Differ. Equ., Evolutionary equationsII, Elsevier/North-Holland, Amsterdam (2005) 461–559.  
  17. 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.  
  18. 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.  
  19. G. Nguetseng, A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal.20 (1989) 608–623.  
  20. L. Tartar, Nonlocal effects induced by homogenization, in Partial differential equations and the calculus of variationsII, Progr. Nonlinear Differential Equations Appl.2, Birkhäuser Boston, Boston (1989) 925–938.  
  21. L. Tartar, Memory effects and homogenization. Arch. Ration. Mech. Anal.111 (1990) 121–133.  
  22. A. Visintin, Homogenization of the nonlinear Kelvin-Voigt model of viscoelasticity and of the Prager model of plasticity. Contin. Mech. Thermodyn.18 (2006) 223–252.  
  23. A. Visintin, Homogenization of the nonlinear Maxwell model of viscoelasticity and of the Prandtl-Reuss model of elastoplasticity. Proc. Roy. Soc. Edinburgh Sect. A138 (2008) 1363–1401.  
  24. A. Visintin, Homogenization of nonlinear visco-elastic composites. J. Math. Pures Appl.89 (2008) 477–504.  
  25. J.R. Willis, Bounds and self-consistent estimates for the overall moduli of anisotropic composites. J. Mech. Phys. Solids25 (1977) 182–202.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.