Skip to main content
SpringerLink
Log in

Mathematical homogenization of inelastic dissipative materials: a survey and recent progress

  • Review Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

In this paper, a review of papers on mathematical homogenization of dissipative composites under small strains and on the interplay between homogenization procedure and dissipation due to mechanical work is presented. Moreover, a critical survey on the links between mathematical homogenization and computational homogenization is attempted.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aboudi, J.: Micromechanics-based thermoviscoelastic constitutive equations for rubber-like matrix composites at finite strains. Int. J. Solids Struct. 41, 5611–5629 (2004)

    Article  MATH  Google Scholar 

  2. Aboudi, J., Pindera, M.-J., Arnold, S.M.: Higher-order theory for periodic multiphase materials with inelastic phases. Int. J. Plast 19(6), 805–847 (2003)

    Article  MATH  Google Scholar 

  3. Alber, H.-D.: Materials with memory. In: Lecture Notes in Mathematics, vol. 1682. Springer-Verlag, Berlin (1998)

  4. Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23, 1482–1518 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Allaire, G.: Shape Optimization by the Homogenization Method. Springer, New York (2002)

    Book  MATH  Google Scholar 

  6. Allaire, G., Bonnetier, E., Francfort, G., Jouve, F.: Shape optimization by the homogenization method. Numer. Math. 76, 27–68 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Allaire, G., Briane, M.: Multiscale convergence and reiterated homogenization. Proc. R. Soc. Edinb. A 126, 297–342 (1996)

    Article  MATH  Google Scholar 

  8. Allaire, G., Brizzi, R.: A multiscale FEM for numerical homogenization. Multiscale Model. Simul. 4(3), 790–812 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Allaire, G., Kohn, R.V.: Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions. Q. Appl. Math. 51(4), 675–699 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. Allaire, G., Kohn, R.V.: Optimal bounds on the effective behavior of a mixture of two well-ordered elastic materials. Q. Appl. Math. 51(4), 643–674 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Allaire, G., Kohn, R.V.: Topology optimization and optimal shape design using homogenization. In: Topology Design of Structures, Springer, Dordrecht (1993)

  12. Amar, M.: Two-scale convergence and homogenization in BD. Asymptot. Anal. 16, 65–84 (1998)

    MathSciNet  MATH  Google Scholar 

  13. Anzellotti, G., Giaquinta, M.: Existence of the displacements field for an elasto–plastic body subject to Hencky’s law and von Mises yield condition. Manuscr. Math. 32, 101–136 (1980)

    Article  MATH  Google Scholar 

  14. Anzellotti, G., Luckhaus, S.: Dynamical evolution of elasto–plastic bodies. Appl. Math. Optim. 15(1), 121–140 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  15. Armero, F., Garikipati, K.: Recent advances in the analysis and numerical simulation of strain localization in inelastic solids. In: Owen, D.R.J., Oñate, E. (eds.) Proceedings of COMPLAS IV, 4th international conference on computational plasticity, pp. 547–561. Pineridge Press, Swansea (1995)

  16. Asada, T., Ohno, N.: Fully implicit formulation of elastoplastic homogenization problem for two-scale analysis. Int. J. Solids Struct. 44(22–23), 7261–7275 (2007)

    Article  MATH  Google Scholar 

  17. Asada, T., Tanaka, Y., Ohno, N.: Two-scale analysis of honeycombs indented by flat punch. In: Huh, H., Park, C., Lee, C., Keum, Y. (eds) Engineering Plasticity and its Applications. From Nanoscale to Macroscale. Proceedings of AEPA 2008, pp. 819–824. World Scientific Publishing, Singapore (2009)

  18. Babadjian, J.-F.: Méthodes variationnelles pour l’étude de milieux dissipatifs: applications en rupture, endommagement et plasticité. Université Pierre et Marie Curie, Habilitation (2013)

    Google Scholar 

  19. Benaarbia, A., Chrysochoos, A., Robert, G.: Thermomechanical behavior of PA6.6 composites subjected to low cycle fatigue. Compos. B 76, 52–64 (2015)

    Article  Google Scholar 

  20. Brassart, L., Stainier, L.: On convergence properties of variational constitutive updates for elasto-visco-plasticity. GAMM-Mitteilungen 35(1), 26–42 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Brassart, L., Stainier, L., Doghri, I., Delannay, L.: Homogenization of elasto-(visco) plastic composites based on an incremental variational principle. Int. J. Plast 36, 86–112 (2012)

    Article  MATH  Google Scholar 

  22. Bravo-Castillero, J., Rodríguez-Ramos, R., Mechkour, H., Otero, J.A., Cabanas, J.H., Sixto, L.M., Guinovart-Díaz, R., Sabina, F.J.: Homogenization and effective properties of periodic thermomagnetoelectroelastic composites. J. Mech. Mater. Struct. 4(5), 819–836 (2009)

    Article  Google Scholar 

  23. Bridgman, P.W.: The thermodynamics of plastic deformation and generalized entropy. Rev. Mod. Phys. 22(1), 56–63 (1950)

    Article  ADS  MATH  Google Scholar 

  24. Bruhns, O.: Some remarks on the history of plasticity. Heinrich Hencky, a pioneer of the early years. In: Stein, E. (ed.) The History of Theoretical, Material and Computational Mechanics-Mathematics Meets Mechanics and Engineering, vol. 1, Lecture Notes in Applied Mathematics and Mechanics, pp. 133–152. Springer, Berlin, Heidelberg (2014)

  25. Carstensen, C., Hackl, K., Mielke, A.: Non-convex potentials and microstructures in finite-strain plasticity. Proc. R. Soc. A 458, 299–317 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Cavalcante, M.A.A., Khatam, H., Pindera, M.J.: Homogenization of elastic–plastic periodic materials by FVDAM and FEM approaches—an assessment. Compos. Part B Eng. 42, 1713–1730 (2011)

    Article  Google Scholar 

  27. Cavalcante, M.A.A., Marques, S.P.C., Pindera, M.J.: Computational aspects of the parametric finite-volume theory for functionally graded materials. Comput. Mater. Sci. 44, 422–438 (2008)

    Article  Google Scholar 

  28. Cavalcante, M.A.A., Marques, S.P.C., Pindera, M.J.: Transient thermomechanical analysis of a layered cylinder by the parametric finite-volume theory. J. Therm. Stresses 32, 112–134 (2009)

    Article  Google Scholar 

  29. Cavalcante, M.A.A., Pindera, M.-J.: Finite-volume enabled transformation field analysis of periodic materials. Int. J. Mech. Mater. Des. 9, 153–179 (2013)

    Article  Google Scholar 

  30. Cavalcante, M.A.A., Pindera, M.J.: Generalized FVDAM theory for elastic–plastic periodic materials. Int. J. Plast 77, 90–117 (2016)

    Article  Google Scholar 

  31. Chaboche, J., Kanoute, P., Ross, A.: On the capabilities of mean field approaches for the description of plasticity in metal matrix composites. Int. J. Plast. 21, 1409–1434 (2005)

    Article  MATH  Google Scholar 

  32. Chaboche, J., Kruch, S., Maire, J., Pottier, T.: Towards a micromechanics based inelastic and damage modeling of composites. Int. J. Plast. 17, 411–439 (2001)

    Article  MATH  Google Scholar 

  33. Charalambakis, N., Murat, F.: Weak solutions to initial boundary value problems for the shearing of nonhomogeneous thermoviscoplastic materials. Proc. R. Soc. Edinb. 113A, 257–265 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  34. Charalambakis, N., Murat, F.: Approximation by finite elements, existence and uniqueness for a model of stratified thermoviscoplastic materials. Ricerche Mat. 55, 171–218 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  35. Charalambakis, N., Murat, F.: Homogenization of stratified thermoviscoplastic materials. Q. Appl. Math. 64, 359–399 (2006)

  36. Charalambakis, N., Murat, F.: Stability by homogenization of thermoviscoplastic problems. Math. Models Methods Appl. Sci. 20, 1591–1616 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. Charalambakis, N., Murat, F.: Two stable by homogenization models in simple shearing of rate-dependent non-homogeneous materials. Q. Appl. Math. 68, 395–419 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Chatzigeorgiou, G., Charalambakis, N., Chemisky, Y., Meraghni, F.: Periodic homogenization for fully coupled thermomechanical modeling of dissipative generalized standard materials. Int. J. Plast. 81, 18–39 (2016)

    Article  Google Scholar 

  39. Chatzigeorgiou, G., Chemisky, Y., Meraghni, F.: Computational micro to macro transitions for shape memory alloy composites using periodic homogenization. Smart Mater. Struct. 24, 035009 (2015)

    Article  ADS  Google Scholar 

  40. Chatzigeorgiou, G., Javili, A., Steinmann, P.: Unified magnetomechanical homogenization framework with application to magnetorheological elastomers. Math. Mech. Solids 19(2), 194–212 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  41. Chatzigeorgiou, G., Meraghni, F., Javili, A.: Generalized interfacial energy and size effects in composites. J. Mech. Phys. Solids 106, 257–282 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  42. Chen, W.F., Han, D.J.: Plasticity for Structural Engineers. Springer, Berlin (1988)

    Book  MATH  Google Scholar 

  43. Cherkaev, A., Kohn, R.: Topics in the Mathematical Modelling of Composite Materials. Birkhäuser, Boston (1997)

    Book  MATH  Google Scholar 

  44. Christensen, R.M., Lo, K.H.: Solutions for effective shear properties in three phase sphere and cylinder models. J. Mech. Phys. Solids 27, 315–330 (1979)

    Article  ADS  MATH  Google Scholar 

  45. Ciarlet, P.: Mathematical Elasticity: Volume I, Three dimensional Elasticity, Studies in Mathematics and its Applications. North-Holland Elsevier Science Pub., Amsterdam (1988)

    MATH  Google Scholar 

  46. Dal Maso, G.: Intoduction to \(\Gamma \)-convergence. In: Brezis, H. (ed.) Progress in Non-linear Differential Equations and their Application, vol. 8. Springer Science+Business Media, LLC, Boston (1993)

    Google Scholar 

  47. Dal Maso, G., DeSimone, A., Mora, M.G.: Quasistatic evolution problems for linearly elastic-perfectly plastic materials. Arch. Ration. Mech. Anal. 180, 237–291 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  48. De Giorgi, E.: Sulla convergenza di alcune successioni di integrali del tipo dell’area. Rendiconti Matematici 8, 277–294 (1975)

    MathSciNet  MATH  Google Scholar 

  49. De Giorgi, E.: G-operators and \(\Gamma \)-convergence. PWN Polish Scientific Publishers and North-Holland, Amsterdam (1984)

    Google Scholar 

  50. Debordes, O.: Dualité des théorèmes statique et cinématique dans la théorie de l’adaptation des milieux continus élastoplastiques. Comptes Rendus de l’Académie des Sciences de Paris 282(Serie A), 535–537 (1976)

    MathSciNet  MATH  Google Scholar 

  51. Debordes, O., Nayrolles, B.: Sur la théorie et le calcul a l’adaptation des structures élastoplastiques. J. Méc. 15, 1–53 (1976)

    MATH  Google Scholar 

  52. Defranceschi, A.: An introduction to homogenization and G-convergence. School on Homogenization and G-Convergence, ICTP, Trieste, September 6–17 (1993)

  53. Desrumaux, F., Meraghni, F., Benzeggagh, M.L.: Generalised Mori–Tanaka scheme to model anisotropic damage using numerical Eshelby tensor. J. Compos. Mater. 35(7), 603–624 (2001)

    Article  ADS  Google Scholar 

  54. Dodd, B., Bai, Y.: Adiabatic Shear Localization. Elsevier, London (2012)

    Google Scholar 

  55. Doghri, I., Ouaar, A.: Homogenization of two-phase elasto–plastic composite materials and structures: study of tangent operators, cyclic plasticity and numerical algorithms. Int. J. Solids Struct. 40, 1681–1712 (2003)

    Article  MATH  Google Scholar 

  56. Drucker, D.C.: Variational principles in the mathematical theory of plasticity. Technical Report Nonr 562(10), Division of Applied Mathematics, Brown University (1956)

  57. Drucker, D.C.: A definition of stable inelastic material. Technical Report Nonr 562(20)/2, Division of Applied Mathematics, Brown University (1957)

  58. Duvaut, G., Lions, J.-L.: Les inéquations en Mécanique et en Physique. Dunod, Paris (1972)

    MATH  Google Scholar 

  59. Dvorak, G.: Transformation field analysis of inelastic composite materials. Proc. R. Soc. Lond. A 437, 311–327 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  60. Dvorak, G., Bahel-El-Din, Y., Wafa, A.: Implementation of the transformation field analysis for inelastic composite materials. Comput. Mech. 14, 201–228 (1994)

    Article  MATH  Google Scholar 

  61. Ebobisse, F., Reddy, B.D.: Some mathematical problems in perfect plasticity. Comput. Methods Appl. Mech. Eng. 193, 5071–5094 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  62. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)

    MATH  Google Scholar 

  63. Eshelby, J .D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. Lond. A 241(1226), 376–396 (1957)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  64. Fenchel, W.: Convex Cones, Sets, and Functions. Princeton University Press, Princeton (1953)

    MATH  Google Scholar 

  65. Fish, J., Shek, K., Pandheeradi, M., Shephard, M.: Computational plasticity for composite structures based on mathematical homogenization: theory and practice. Comput. Methods Appl. Mech. Eng. 148, 53–73 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  66. Francfort, G., Giacomini, A.: Small strain heterogeneous elasto-plasticity revisited. Commun. Pure Appl. Math. 65, 1185–1241 (2012)

    Article  MATH  Google Scholar 

  67. Francfort, G., Giacomini, A.: Heterogeneous Elasto-plasticity. In: Seminar in Laboratory J.-L. Lions. University Paris 6, Paris (2013)

  68. Francfort, G., Giacomini, A.: On periodic homogenization in perfect elasto-plasticity. J. Eur. Math. Soc. 16, 409–461 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  69. Francfort, G., Giacomini, A.: The role of a vanishing interfacial layer in perfect elasto-plasticity. Chin. Ann. Math. Ser. B 36B(5), 813–828 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  70. Francfort, G., Suquet, P.: Homogenization and mechanical dissipation in thermoviscoelasticity. Arch. Ration. Mech. Anal. 96, 268–293 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  71. Francfort, G.A., Giacomini, A., Marigo, J.J.: The taming of plastic slips in Von Mises elasto-plasticity. Interfaces Free Bound. 17(4), 497–516 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  72. Francfort, G.A., Giacomini, A., Marigo, J.J.: A case study for uniqueness of elasto-plastic evolutions: the bi-axial test. J. Math. Pures. Appl. 105(2), 198–227 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  73. Francfort, G.A., Murat, F.: Homogenization and optimal bounds in linear elasticity. Arch. Ration. Mech. Anal. 94(4), 307–334 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  74. Germain, P.: Sur certaines définitions liées à l’énergie en mécanique des solides. Int. J. Eng. Sci. 20(2), 245–259 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  75. Greenberg, H.J.: Complementary minimum principles for an elastic–plastic material. Q. Appl. Math. 7, 85 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  76. Guedes, J.M., Kikuchi, N.: Preprocessing and posprocessing for materials based on the homogenization method with adaptive finite element methods. Comput. Methods Appl. Mech. Eng. 83, 143–198 (1990)

    Article  ADS  MATH  Google Scholar 

  77. Gurtin, M.E., Fried, E., Anand, L.: The Mechanics and Thermodynamics of Continua. Cambridge University Press, Cambridge (2009)

    Google Scholar 

  78. Halphen, B.: Sur les discontinuités de vitesse en elastoplasticité. Comptes Rendus de l’Académie des Sciences de Paris, Ser. A-B 287(7), A569–A572 (1978)

    MathSciNet  MATH  Google Scholar 

  79. Halphen, B., Nguyen, Q.S.: Sur les matériaux standards généralisés. J. Méc. 14(1), 39–63 (1975)

    MATH  Google Scholar 

  80. Hartl, D.J., Chatzigeorgiou, G., Lagoudas, D.C.: Three-dimensional modeling and numerical analysis of rate-dependent irrecoverable deformation in shape memory alloys. Int. J. Plast. 26(10), 1485–1507 (2010)

    Article  MATH  Google Scholar 

  81. Hashin, Z.: Theory of mechanical behavior of heterogeneous media. Appl. Mech. Rev. 17, 1–9 (1963)

    Google Scholar 

  82. Hashin, Z.: Analysis of composite materials: a survey. J. Appl. Mech. 50, 481–505 (1983)

    Article  ADS  MATH  Google Scholar 

  83. Hashin, Z., Rosen, B.W.: The elastic moduli of fiber-reinforced materials. J. Appl. Mech. 31, 223–232 (1964)

    Article  ADS  Google Scholar 

  84. Hashin, Z., Shtrikman, S.: A variational approach to the theory of the effective magnetic permeability of multiphase materials. J. Appl. Phys. 33, 3125–3131 (1962)

    Article  ADS  MATH  Google Scholar 

  85. Hill, R.: A variational principle of maximum plastic work in classical plasticity. Q. J. Mech. Appl. Mech. 1, 18–28 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  86. Hill, R.: The Mathematical Theory of Plasticity. Oxford University Press, New York (1950)

    MATH  Google Scholar 

  87. Hill, R.: On the problem of uniqueness in the theory of a rigid-plastic solid-1. J. Mech. Phys. Solids 4, 247–255 (1956)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  88. Hill, R.: Acceleration waves in solids. J. Mech. Phys. Solids 10, 1–16 (1963)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  89. Hill, R.: Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11, 357–372 (1963)

    Article  ADS  MATH  Google Scholar 

  90. Hill, R.: The essential structure of constitutive laws for metal composites and polycrystals. J. Mech. Phys. Solids 15(2), 79–95 (1967)

    Article  ADS  Google Scholar 

  91. Hill, R.: On macroscopic effects of heterogeneity in elastoplastic media at finite strain. Math. Proc. Camb. Philos. Soc. 95, 481–494 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  92. Hill, R., Rice, J.: Constitutive analysis of elastic–plastic crystals at arbitrary strain. J. Mech. Phys. Solids 20(6), 401–413 (1972)

    Article  ADS  MATH  Google Scholar 

  93. Hollister, S.J., Kikuchi, N.: A comparison of homogenization and standard mechanics analyses for periodic porous composites. Comput. Mech. 10, 73–95 (1992)

    Article  MATH  Google Scholar 

  94. Houi, P.M., Stroud, D.: Nonlinear susceptibilities of granular media. Phys. Rev. B 37, 8719–8724 (1988)

    Article  ADS  Google Scholar 

  95. Javili, A., Chatzigeorgiou, G., Steinmann, P.: Computational homogenization in magneto-mechanics. Int. J. Solids Struct. 50, 4197–4216 (2013)

    Article  Google Scholar 

  96. Jendli, Z., Meraghni, F., Fitoussi, J., Baptist, D.: Multi-scales modelling of dynamic behaviour for discontinuous fibre SMC composites. Compos. Sci. Technol. 69(1), 97–103 (2009)

    Article  Google Scholar 

  97. Jirásek, M.: Objective modeling of strain localization. Revue Française de Génie Civil 6(6), 1119–1132 (2002)

    Article  Google Scholar 

  98. Johnson, C.: Existence theorems for plasticity problems. J. Math. Pures Appl. 55, 431–444 (1976)

    MathSciNet  MATH  Google Scholar 

  99. Johnson, C.: On finite element methods for plasticity problems. Numer. Math. 26, 79–84 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  100. Johnson, C.: A mixed finite element method for plasticity problems with hardening. SIAM J. Numer. Anal. 14, 575–583 (1977)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  101. Johnson, C.: On plasticity with hardening. J. Math. Anal. Appl. 62, 325–335 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  102. Kalamkarov, A.L., Andrianov, I.V., Danishevs’kyy, V.V.: Asymptotic homogenization of composite materials and structures. Appl. Mech. Rev. 62, 030802 (2009)

    Article  ADS  Google Scholar 

  103. Kestin, J., Rice, J.R.: Paradoxes in the application of thermodynamics to strained solids. In: Stuart, E.G., Gal-Or, B., Brainard, A.J. (eds.) A Critical Review of Thermodynamics, pp. 275–298. Mono Book Corp, Baltimore (1970)

    Google Scholar 

  104. Khon, R.V., Milton, G.W.: On bounding the effective conductivity of anisotropic composites. In: Ericksen, J.L., Kinderlehrer, D., Kohn, R., Lions, J.-L. (eds.) Homogenization and Effective Moduli of Materials and Media, pp. 97–125. Springer-Verlag, New York (1986)

    Chapter  Google Scholar 

  105. Kohn, R.V., Milton, G.W.: Variational bounds on the effective moduli of anisotropic composites. J. Mech. Phys. Solids 36, 597–629 (1988)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  106. Kohn, R.V., Temam, R.: Dual spaces of stresses and strains, with applications to Hencky plasticity. Appl. Math. Optim. 10, 1–35 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  107. Koiter, W.: General theorems for elasto–plastic solids. In: Sneddon, I.N., Hill, R. (eds.) Progress in Solid Mechanics, vol. 1, pp. 164–221. North-Holland, Amsterdam (1960)

    Google Scholar 

  108. Kruch, S., Chaboche, J.L.: Multi-scale analysis in elasto-viscoplasticity coupled with damage. Int. J. Plast. 27, 2026–2039 (2011)

    Article  MATH  Google Scholar 

  109. Lagoudas, D., Hartl, D., Chemisky, Y., Machado, L., Popov, P.: Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys. Int. J. Plast. 32–33, 155–183 (2012)

    Article  Google Scholar 

  110. Lagoudas, D.C.: Shape Memory Alloys: Modeling and Engineering Applications. Springer, New York (2008)

    MATH  Google Scholar 

  111. Lagoudas, D.C., Gavazzi, A.C., Nigam, H.: Elastoplastic behavior of metal matrix composittes based on incremental plasticity and the Mori–Tanaka averaging scheme. Comput. Mech. 8, 193–203 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  112. Lahellec, N., Suquet, P.: Nonlinear composites: a linearization procedure, exact to second-order in contrast and for which the strain-energy and the affine formulations coincide. Comptes Rendus Mécanique 332, 693–700 (2004)

    Article  ADS  Google Scholar 

  113. Lahellec, N., Suquet, P.: Effective behavior of linear viscoelastic composites: a time-integration approach. Int. J. Solids Struct. 44, 507–529 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  114. Lahellec, N., Suquet, P.: On the effective behavior of nonlinear inelastic composites: I. Incremental variational principles. J. Mech. Phys. Solids 55(9), 1932–1963 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  115. Lahellec, N., Suquet, P.: On the effective behavior of nonlinear inelastic composites: II. A second order procedure. J. Mech. Phys. Solids 55(9), 1964–1992 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  116. Lee, C.Y., Yu, W.: Variational asymptotic modeling of composite beams with spanwise heterogeneity. Comput. Struct. 89, 1503–1511 (2011)

    Article  Google Scholar 

  117. Love, B., Batra, R.C.: Determination of effective thermomechanical parameters of a mixture of two elastothermoviscoplastic constituents. Int. J. Plast. 22, 1026–1061 (2006)

    Article  MATH  Google Scholar 

  118. Lukkassen, D., Nguetseng, G., Wall, P.: Two-scale convergence. Int. J. Pure Appl. Math. 2, 33–81 (2002)

    MathSciNet  MATH  Google Scholar 

  119. Lurie, K., Cherkaev, A.: Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportion. Proc. R. Soc. Edinb. A 99, 71–87 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  120. Lurie, K., Cherkaev, A.: Exact estimates of conductivity of a binary mixture of isotropic components. Proc. R. Soc. Edinb. A 104, 21–38 (1986)

    Article  MATH  Google Scholar 

  121. Maghous, S., Creus, G.J.: Periodic homogenization in thermoviscoelasticity: case of multilayered media with ageing. Int. J. Solids Struct. 40, 851–870 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  122. Mainik, A., Mielke, A.: Existence results for energetic models for rate-independent systems. Calc. Var. Partial. Differ. Equ. 22, 73–99 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  123. Matthies, H., Strang, G., Christiansen, E.: The saddle point of a differential programm in energy methods in FEA. In: Glowinski, R., Robin, E., Zienckiewicz, O.C. (eds.) Energy Methods in Finite Element Analysis. Wiley and Sons, New York (1979)

    Google Scholar 

  124. McPhedran, R.C., McKenzie, D.R.: The conductivity of lattices and spheres: 1. The simple cubic lattices. Proc. R. Soc. Lond. A 362, 45–63 (1978)

    Article  ADS  Google Scholar 

  125. Meraghni, F., Benzeggagh, M.L.: Micromechanical modelling of matrix degradation in randomly oriented discontinuous-fibre composites. Compos. Sci. Technol. 55(2), 171–186 (1995)

    Article  Google Scholar 

  126. Mercier, B.: Sur la théorie et l’analyse numérique de problèmes de plasticité. Ph.D. thesis, Université Pierre et Marie Curie (Paris VI) (1977)

  127. Mercier, S., Molinari, A.: Homogenization of elastic–viscoplastic heterogeneous materials: self-consistent and Mori–Tanaka schemes. Int. J. Plast. 25, 1024–1048 (2009)

    Article  MATH  Google Scholar 

  128. Mercier, S., Molinari, A., Berbenni, S., Berveiller, M.: Comparison of different homogenization approaches for elastic–viscoplastic materials. Modell. Simul. Mater. Sci. Eng. 20(2), 024004 (2012)

    Article  ADS  Google Scholar 

  129. Michel, J.C., Galvanetto, U., Suquet, P.: Constitutive relations involving internal variables based on a micromechanical analysis. In: Maugin, G., Drouot, R., Sidoroff, F. (eds.) Continuum Thermomechanics: The Art and Science of Modelling Material Behaviour, pp. 301–312. Kluwer Academic Publishers, New York (2000)

    Google Scholar 

  130. Michel, J.C., Moulinec, H., Suquet, P.: Effective properties of composite materials with periodic microstructure: a computational approach. Comput. Methods Appl. Mech. Eng. 172, 109–143 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  131. Michel, J.C., Suquet, P.: Nonuniform transformation field analysis. Int. J. Solids Struct. 40, 6937–6955 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  132. Michel, J.C., Suquet, P.: Computational analysis of nonlinear composites structures using the nonuniform transformation field analysis. Comput. Methods Appl. Mech. Eng. 193, 5477–5502 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  133. Michel, J.C., Suquet, P.: Non-uniform transformation field analysis: a reduced model for multiscale non-linear problems in solid mechanics. In: Galvanetto, U., Aliabadi, F. (eds.) Multiscale Modelling in Solid Mechanics: Computational Approaches, pp. 159–206. Imperial College Press, Singapore (2010)

    Google Scholar 

  134. Miehe, C.: Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation. Int. J. Numer. Methods Eng. 55, 1285–1322 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  135. Miehe, C., Koch, A.: Computational micro-to-macro transitions of discretized microstructures undergoing small strains. Arch. Appl. Mech. 72(4–5), 300–317 (2002)

    Article  ADS  MATH  Google Scholar 

  136. Mielke, A.: Private Communication

  137. Mielke, A.: Existence of minimizers in incremental elastoplasticity with finite strains. SIAM J. Math. Anal. 36, 384–404 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  138. Mielke, A.: Evolution of rate-independent systems. In: Dafermos, C.M., Feireisl, E. (eds.) Handbook of Differential Equations: Evolutionary Equations, vol. 2, pp. 461–559. North-Holland, Amsterdam (2005)

    Chapter  Google Scholar 

  139. Mielke, A., Timofte, A.: Two-scale homogenization for evolutionary variational inequalities via the energetic formulation. SIAM J. Math. Anal. 39, 642–668 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  140. Miksis, M.J.: Effective dielectric constant of a nonlinear composite material. SIAM J. Appl. Math. 43, 1140–1155 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  141. Milton, G.: Bounds on the electromagnetic, elastic, and other properties of two-component composites. Phys. Rev. Lett. 46, 542–545 (1981)

    Article  ADS  Google Scholar 

  142. Milton, G.: The coherent potential approximation is a realizable effective medium scheme. Commun. Math. Phys. 99, 463–500 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  143. Milton, G.: Modelling the properties of composites by laminates. In: Ericksen, J.L., Kinderlehrer, D., Kohn, R., Lions, J.-L. (eds.) Homogenization and effective moduli of materials and media, pp. 150–174. Springer-Verlag, New York (1986)

    Chapter  Google Scholar 

  144. Milton, G.: On characterizing the set of possible effective tensors: the variational method and the translation method. Commun. Pure Appl. Math. 43, 63–125 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  145. Moreau, J.J.: Application of convex analysis to the treatment of elastoplastic systems. In: Germain, P., Nayrolles, B. (eds.) Applications of Methods of Functional Analysis to Problems in Mechanics, pp. 56–89. Springer, Berlin (1976)

    Chapter  Google Scholar 

  146. Mori, T., Tanaka, K.: Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 21(5), 571–574 (1973)

    Article  Google Scholar 

  147. Mura, T.: Micromechanics of Defects in Solids. In: Mechanics of elastic and inelastic solids. Second, Revised edition, Kluwer Academic Publishers, Dordrecht (1987)

  148. Murat, F.: H-convergence. Séminaire d’analyse fonctionnelle et numérique de l’Université d’Alger. Multicopied (1977)

  149. Murat, F.: Compacité par compensation. Annali della Scuola Normale Superiore di Pisa 5, 489–507 (1978)

    MathSciNet  MATH  Google Scholar 

  150. Murat, F., Tartar, L.: Calcul des Variations et Homogeneisation, Les methodes de l’homogénéisation: Théorie et Applications en Physique. In: Coll. Dir. Etudes et Recherches EDF, pp. 319–369. Eyrolles, Paris (1985)

  151. Murat, F., Tartar, L.: H-convergence, in topics in the mathematical modelling of composite materials. In: Cherkaev, A., Kohn, R.V. (eds.) Progress in Nonlinear Differential Equations and Their Applications, vol. 31, pp. 21–43. Birkhäuser, Boston (1997)

    Google Scholar 

  152. Needleman, A., Tvergaard, V.: Analyses of plastic localization in metals. Appl. Mech. Rev. 45(3, part 2), S3–S18 (1992)

    Article  ADS  Google Scholar 

  153. Neff, P., Pauly, D., Witsch, K.J.: Poincare meets Korn via Maxwell: extending Korn’s first inequality to incompatible tensor fields. J. Differ. Equ. 258(4), 1267–1302 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  154. Nemat-Nasser, S., Hori, M.: Micromechanics: Overall Properties of Heterogeneous Materials, 2nd edn. North-Holland, Amsterdam (1999)

    MATH  Google Scholar 

  155. Nesenenko, S., Neff, P.: Homogenization for dislocation based gradient viscoplasticity. J. Math. Anal. Appl. 425(1), 133–159 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  156. Nguetseng, G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20, 608–623 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  157. Nguyen, Q.S.: Mechanical modelling of anelasticity. Rev. Phys. Appl. 23, 325–330 (1988)

    Article  Google Scholar 

  158. Oliver, J.: Continuum modeling of strong discontinuities in Solid Mechanics. In: Owen, D.R.J., Oñate, E. (eds.) Proceedings of COMPLAS IV, 4th International Conference on Computational Plasticity, pp. 455–479. Pineridge Press, Swansea (1995)

  159. Oliver, J.: Modeling strong discontinuities in solid mechanics via strain softening constitutive equations. Part 1: fundamentals. Int. J. Numer. Methods Eng. 39, 3575–3600 (1996)

    Article  MATH  Google Scholar 

  160. Oliver, J., Simo, J.: Modeling strong discontinuities by means of strain softening constitutive equations. In: Computer Modelling of Concrete Structures, Proceedings of EURO-C 1994, pp. 363–372. Pineridge Press, Swansea, UK (1994)

  161. Ortiz, M., Simo, J.C.: An analysis of a new class of integration algorithms for elastoplastic constitutive relations. Int. J. Numer. Methods Eng. 23, 353–366 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  162. Pindera, M.J., Khatam, H., Drago, A.S., Bansal, Y.: Micromechanics of spatially uniform heterogeneous media: a critical review and emerging approaches. Compos. B Eng. 40(5), 349–378 (2009)

    Article  Google Scholar 

  163. Ponte-Castañeda, P.: A new variational principle and its applications to nonlinear heterogeneous systems. SIAM J. Appl. Math. 52, 1321–1341 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  164. Ponte-Castañeda, P.: Bounds and estimates for the properties of nonlinear heterogeneous systems. Philos. Trans. A 340, 532–566 (1992)

    MathSciNet  MATH  Google Scholar 

  165. Ponte-Castañeda, P.: New variational principles in plasticity and their application to composite materials. J. Mech. Phys. Solids 40, 1757–1788 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  166. Ponte-Castañeda, P.: Exact second order estimates for the effective mechanical properties of nonlinear composite materials. J. Mech. Phys. Solids 44, 827–862 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  167. Ponte-Castañeda, P.: Second-order homogenization estimates for nonlinear composites incorporating field fluctuations. I—theory. J. Mech. Phys. Solids 50, 737–757 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  168. Ponte-Castañeda, P., Suquet, P.: Nonlinear composites. Adv. Appl. Mech. 34, 171–302 (1998)

    Article  MATH  Google Scholar 

  169. Ponte-Castañeda, P., Willis, J.R.: Variational second-order estimates for nonlinear composites. Proc. R. Soc. Lond. A 455, 1799–1811 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  170. Ponte-Castañeda, P.: The effective mechanical properties of nonlinear isotropic composites. J. Mech. Phys. Solids 39, 45–71 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  171. Prager, W., Hodge, P.: Theory of Perfectly Plastic Solids. Wiley, New York (1951)

    MATH  Google Scholar 

  172. Puglisi, G., Truskinovsky, L.: Thermodynamics of rate-independent plasticity. J. Mech. Phys. Solids 53, 655–679 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  173. Qidwai, M.A., Lagoudas, D.C.: Numerical implementation of a shape memory alloy thermomechanical constitutive model using return mapping algorithms. Int. J. Numer. Methods Eng. 47, 1123–1168 (2000)

    Article  MATH  Google Scholar 

  174. Qu, J., Cherkaoui, M.: Fundamentals of Micromechanics of Solids. Wiley, New Jersey (2006)

    Book  Google Scholar 

  175. Schweizer, B., Veneroni, M.: Periodic homogenization of the Prandtl–Reuss model with hardening. J. Multiscale Model. 2, 69–106 (2010)

    Article  Google Scholar 

  176. Sengupta, A., Papadopoulos, P., Taylor, R.L.: A multiscale finite element method for modeling fully coupled thermomechanical problems in solids. Int. J. Numer. Methods Eng. 91, 1386–1405 (2012)

    Article  MathSciNet  Google Scholar 

  177. Simo, J., Oliver, J.: A new approach to the analysis and simulation of strong discontinuities. In: Bažant, Z.P., Bittnar, Z., Jirásek, M., Mazars, J. (eds.) Fracture and Damage in Quasibrittle Structures, pp. 25–39. E&FN Spon, London (1994)

    Google Scholar 

  178. Simo, J., Oliver, J., Armero, F.: An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids. Comput. Mech. 12, 277–296 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  179. Simo, J.C.: A new methodology for the numerical simulation of strain softening in inelastic solids. Technical Report CR 95.002, Naval Facilities Engineering Service Center, Port Hueneme, California (1994)

  180. Simo, J.C., Hughes, T.J.R.: Computational Inelasticity. Springer-Verlag, New York (1998)

    MATH  Google Scholar 

  181. Simo, J.C., Taylor, R.T.: A return mapping algorithm for plane stress elastoplasticity. Int. J. Numer. Meth. Eng. 22, 649–670 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  182. Solombrino, F.: Quasistatic evolution problems for nonhomogeneous elastic plastic materials. J. Convex Anal. 16, 89–119 (2009)

    MathSciNet  MATH  Google Scholar 

  183. Solombrino, F.: Quasistatic evolution in perfect plasticity for general heterogeneous materials. Arch. Ration. Mech. Anal. 212(1), 283–330 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  184. Spagnolo, S.: Convergence in energy for elliptic operators. In: Hubbart, B. (ed.) Numerical Solutions of Partial Differential Equations III, pp. 469–498. Academic Press, New York (1976)

    Chapter  Google Scholar 

  185. Suquet, P.: Existence et régularité des solutions des équations de la plasticité. Comptes Rendus de l’Académie des Sciences de Paris 286(Serie A), 1201–1204 (1978)

    MATH  Google Scholar 

  186. Suquet, P.: Un espace fonctionnel pour les équations de la plasticité. Annales de la Faculté des Sciences de Toulouse, 5e série 1(1), 77–87 (1979)

    Article  MATH  Google Scholar 

  187. Suquet, P.: Evolution problems for a class of dissipative materials. Q. Appl. Math. 38, 391–414 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  188. Suquet, P.: Sur les équations de la plasticité: existence et régulatité des solutions. J. Méc. 20(1), 1–39 (1981)

    Google Scholar 

  189. Suquet, P.: Plasticité et homogénéisation. Ph.D. thesis, Université Pierre et Marie Curie (Paris VI) (1982)

  190. Suquet, P.: Discontinuities and Plasticity. In: Panagiotopoulos, P., Moreau, J.-J. (eds.) Nonsmooth Mechanics and Applications, International Center for Mechanical Sciences, Courses and Lectures, vol. 302, pp. 280–340. Springer Vienna, Vienna (1988)

    Google Scholar 

  191. Suquet, P.: Overall properties of non-linear composites: a modified secant moduli theory and its link with Ponte Castañeda’s nonlinear variational procedure. Comptes Rendus de l’Académie des Sciences de Paris, Serie IIb 320, 563–571 (1995)

    MATH  Google Scholar 

  192. Suquet, P.: Effective properties of nonlinear composites. In: Suquet, P. (ed.) Continuum Micromechanics, pp. 197–264. Springer Vienna, Vienna (1997)

    Chapter  Google Scholar 

  193. Suquet, P.M.: Elements of homogenization for inelastic solid mechanics. Lecture Notes in Physics, vol. 272, pp. 193–278. Springer, Berlin (1987)

    Google Scholar 

  194. Talbot, D.R.S., Willis, J.R.: Variational principles for inhomogeneous non-linear media. IMA J. Appl. Math. 35, 39–54 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  195. Talbot, D.R.S., Willis, J.R.: Bounds and self-consistent estimates for the overall properties of nonlinear composites. IMA J. Appl. Math. 39, 215–240 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  196. Tartar, L.: Homogénéisation et compacité par compensation. Cours Peccot, Collège de France (1977)

    MATH  Google Scholar 

  197. Tartar, L.: Nonlinear Constitutive Relations and Homogenization. North-Holand, Amsterdam (1978)

    MATH  Google Scholar 

  198. Tartar, L.: Etude des oscillations dans les equations aux derivees partielles non-lineaires. Lect. Notes Phys. 195, 385–412 (1984)

    ADS  MATH  Google Scholar 

  199. Tartar, L.: Estimations fines de coefficients homogénéisés. In: Kree, P. (ed.) Ennio De Giorgi Colloquium (Research Notes in Mathematics), pp. 168–187. Pitman, Boston (1985)

    Google Scholar 

  200. Tartar, L.: Memory effects and homogenization. Arch. Ration. Mech. Anal. 111, 121–133 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  201. Tartar, L.: General Theory of Homogenization. A Personalized Introduction. In: Lecture Notes of the Unione Matematica Italiana, vol. 7. Springer, Berlin (2009)

  202. Temam, R.: Existence theorems for the variational problems of plasticity. In: Atteia, M., Bancel, D., Gumowski, I. (eds.) Nonlinear Problems of Analysis in Geometry and Mechanics, pp. 57–70. Pitman, London (1981)

    Google Scholar 

  203. Temam, R.: On the continuity of the trace of vector functions with bounded deformation. Appl. Anal. Int. J. 11(4), 291–302 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  204. Temam, R.: Approximation de fonctions convexes sur un espace de mesures et applications. Can. Math. Bull. 25, 392–413 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  205. Temam, R., Strang, G.: Duality and relaxation in the variational problems of plasticity. J. Méc. 19, 493–527 (1980)

    MathSciNet  MATH  Google Scholar 

  206. Temam, R., Strang, G.: Functions of bounded deformation. Arch. Ration. Mech. Anal. 75, 7–21 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  207. Terada, K., Kikuchi, N.: A class of general algorithms for multi-scale analyses of heterogeneous media. Comput. Methods Appl. Mech. Eng. 190, 5427–5464 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  208. Tsalis, D., Baxevanis, T., Chatzigeorgiou, G., Charalambakis, N.: Homogenization of elastoplastic composites with generalized periodicity in the microstructure. Int. J. Plast. 51, 161–187 (2013)

    Article  Google Scholar 

  209. Tsalis, D., Chatzigeorgiou, G., Tsakmakis, C., Charalambakis, N.: Dissipation inequality-based periodic homogenization of wavy materials. Compos. B 76, 89–104 (2015)

    Article  Google Scholar 

  210. Tu, W., Pindera, M.J.: Targeting the finite-deformation response of wavy biological tissues with bio-inspired material architectures. J. Mech. Behav. Biomed. Mater. 28, 291–308 (2013)

    Article  Google Scholar 

  211. Tzavaras, A.: Effect of thermal softening in shearing of strain-rate dependent materials. Arch. Ration. Mech. Anal. 99(4), 349–374 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  212. Visintin, A.: Homogenization of the nonlinear Kelvin–Voigt model of viscoelasticity and of the Prager model of plasticity. Continuum Mech. Thermodyn. 18, 223–252 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  213. Visintin, A.: Homogenization of the nonlinear Maxwell model of viscoelasticity and of the Prandtl-Reuss model of elastoplasticity. Proc. R. Soc. Edinb. 138 A, 1363–1401 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  214. Walpole, L.J.: On bounds for the overall elastic moduli of inhomogeneous systems—I. J. Mech. Phys. Solids 14, 151–162 (1966)

    Article  ADS  MATH  Google Scholar 

  215. Walpole, L.J.: On the overall elastic moduli of composite materials. J. Mech. Phys. Solids 17, 235–251 (1969)

    Article  ADS  MATH  Google Scholar 

  216. Willis, J.: Bounds and self-consistent estimates for the overall properties of anisotropic composites. J. Mech. Phys. Solids 25, 185–202 (1977)

    Article  ADS  MATH  Google Scholar 

  217. Willis, J.R.: Variational estimates for the overall response of an inhomogeneous nonlinear dielectric. In: Ericksen, J.L., Kinderlehrer, D., Kohn, R., Lions, J.-L. (eds.) Homogenization and Effective Moduli of Materials and Media, pp. 247–263. Springer-Verlag, New York (1986)

    Chapter  Google Scholar 

  218. Wright, T.W.: The Physics and Mathematics of Shear Bands. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  219. Zeng, X.C., Bergman, D.J., Houi, P.M., Stroud, D.: Effective medium theory for weakly nonlinear composites. Phys. Rev. B 38, 10970–10973 (1988)

    Article  ADS  Google Scholar 

  220. Zhigang, S., Chaoxian, Z., Xiguang, G., Yingdong, S.: A quadrilateral element-based method for calculation of multi-scale temperature field. Chin. J. Aeronaut. 23, 529–536 (2010)

    Article  Google Scholar 

Download references

Acknowledgements

The authors wish to thank the anonymous reviewers for their constructive comments. The first author wishes to express his gratitude to the laboratory LEM3-UMR 7239 CNRS, Arts et Metiers ParisTech, Metz, for his worm hospitality during September–October 2016.

Author information

Authors and Affiliations

  1. Department of Civil Engineering, Aristotle University of Thessaloniki and Center for Research and Development of Advanced Materials AUTH-Texas A&M, GR 54124, Thessaloníki, Greece

    Nicolas Charalambakis

  2. Arts et Métiers ParisTech ENSAM, LEM3-UMR 7239 CNRS, Metz, 4 Rue Augustin Fresnel, 57078, Metz, France

    George Chatzigeorgiou, Yves Chemisky & Fodil Meraghni

Authors
  1. Nicolas Charalambakis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. George Chatzigeorgiou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yves Chemisky
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Fodil Meraghni
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicolas Charalambakis.

Additional information

Communicated by Andreas Öchsner.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Charalambakis, N., Chatzigeorgiou, G., Chemisky, Y. et al. Mathematical homogenization of inelastic dissipative materials: a survey and recent progress. Continuum Mech. Thermodyn. 30, 1–51 (2018). https://doi.org/10.1007/s00161-017-0587-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-017-0587-5

Keywords

Search

Navigation