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Applicable Analysis
An International Journal
Volume 94, 2015 - Issue 2
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Articles

Homogenization of plasticity equations with two-scale convergence methods

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Pages 375-398 | Received 03 Dec 2013, Accepted 17 Feb 2014, Published online: 19 Mar 2014

References

  • Alber H-D. Evolving microstructure and homogenization. Contin. Mech. Thermodyn. 2000;12:235–286.
  • Visintin A. On homogenization of elasto-plasticity. J. Phys: Conf. Ser. 2005;22:222–234.
  • Visintin A. Homogenization of the nonlinear Kelvin–Voigt model of viscoelasticity and of the Prager model of plasticity. Contin. Mech. Thermodyn. 2006;18:223–252.
  • Nesenenko S. Homogenization in viscoplasticity. SIAM J. Math. Anal. 2007;39:236–262.
  • Francfort GA, Giacomini A. On periodic homogenization in perfect elasto-plasticity. J. Eur. Math. Soc. 2014;16:409–461.
  • Mielke A. Evolution of rate-independent systems. In: Dafermos C, Feireisl E, editors. Handbook of differential equations, evolutionary equations. Vol. II. Amsterdam: Elsevier; 2005. p. 461–559.
  • Mielke A, Theil F. On rate-independent hysteresis models. NoDEA Nonlinear Differ. Equ. Appl. 2004;11:151–189.
  • Alber H-D, Nesenenko S. Justification of homogenization in viscoplasticity: from convergence on two scales to an asymptotic solution in L2(Ω). J. Multiscale Model. 2009;1:223–244.
  • Cioranescu D, Damlamian A, Griso G. The periodic unfolding method in homogenization. SIAM J. Math. Anal. 2008;40:1585–1620.
  • Alber H-D. Materials with memory. Vol. 1682, Lecture notes in mathematics. Berlin: Springer-Verlag; 1998. Initial-boundary value problems for constitutive equations with internal variables.
  • Han W, Reddy BD. Plasticity. Vol. 9, Interdisciplinary applied mathematics. New York (NY): Springer-Verlag; 1999. Mathematical theory and numerical analysis.
  • Visintin A. Homogenization of the nonlinear Maxwell model of viscoelasticity and of the Prandtl-Reuss model of elastoplasticity. Proc. Roy. Soc. Edinb. Sect. A. 2008;138:1363–1401.
  • Schweizer B, Veneroni M. Periodic homogenization of the Prandtl-Reuss model with hardening. J. Multiscale Modelling. 2010;2:69–106.
  • Braides A, Chiadò Piat V, Defranceschi A. Homogenization of almost periodic monotone operators. Ann. Inst. H. Poincaré Anal. Non Linéaire. 1992;9:399–432.
  • Castaing C, Valadier M. Convex analysis and measurable multifunctions. Vol. 580, Lecture notes in mathematics. Berlin: Springer-Verlag; 1977.
  • Chiadò Piat V, Defranceschi A. Homogenization of monotone operators. Nonlinear Anal. 1990;14:717–732.
  • Damlamian A, Meunier N, Van Schaftingen J. Periodic homogenization for convex functionals using Mosco convergence. Ric. Mat. 2008;57:209–249.
  • Francfort GA, Murat F, Tartar L. Homogenization of monotone operators in divergence form with x-dependent multivalued graphs. Ann. Mat. Pura Appl. (4). 2009;188:631–652.
  • Schweizer B. Homogenization of the Prager model in one-dimensional plasticity. Contin. Mech. Thermodyn. 2009;20:459–477.
  • Veneroni M. Stochastic homogenization of subdifferential inclusions via scale integration. Int. J. Struct. Changes Solids. 2011;3:83–98.
  • Visintin A. Homogenization of nonlinear visco-elastic composites. J. Math. Pures Appl. (9). 2008;89:477–504.
  • Nesenenko S, Neff P. Homogenization for dislocation based gradient visco-plasticity. Technical Report arXiv:1301.2911, Arxiv; 2013.
  • Mielke A, Roubíček T, Stefanelli U. Г-limits and relaxations for rate-independent evolutionary problems. Calc. Var. Partial Differ. Equ. 2008;31:387–416.
  • Mielke A, Timofte AM. Two-scale homogenization for evolutionary variational inequalities via the energetic formulation. SIAM J. Math. Anal. 2007;39:642–668 (electronic).
  • Schweizer B. Averaging of flows with capillary hysteresis in stochastic porous media. Eur. J. Appl. Math. 2007;18:389–415.
  • Brezis H. Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert [Maximal monotone operators and contraction semigroups in Hilbert spaces]. Amsterdam: North Holland/American Elsevier; 1973.
  • Allaire G. Homogenization and two-scale convergence. SIAM J. Math. Anal. 1992;23:1482–1518.
  • Nguetseng G. A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 1989;20:608–623.
  • Visintin A. Some properties of two-scale convergence. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 2004;15:93–107.
  • Visintin A. Two-scale convergence of some integral functionals. Calc. Var. Partial Differ. Equ. 2007;29:239–265.
  • Braides A, Defranceschi A. 1998. Homogenization of multiple integrals. Vol. 12, Oxford lecture series in mathematics and its applications. New York (NY): The Clarendon Press Oxford University Press.
  • Braess D 2001. Finite elements. 2nd ed. Cambridge: Cambridge University Press. Theory, fast solvers, and applications in solid mechanics, Translated from the 1992 German edition by Larry L. Schumaker.

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