Skip to main content
SpringerLink
Log in

Variational Foundations and Generalized Unified Theory of RVE-Based Multiscale Models

  • Original Paper
  • Published:
Archives of Computational Methods in Engineering Aims and scope Submit manuscript

Abstract

A unified variational theory is proposed for a general class of multiscale models based on the concept of Representative Volume Element. The entire theory lies on three fundamental principles: (1) kinematical admissibility, whereby the macro- and micro-scale kinematics are defined and linked in a physically meaningful way; (2) duality, through which the natures of the force- and stress-like quantities are uniquely identified as the duals (power-conjugates) of the adopted kinematical variables; and (3) the Principle of Multiscale Virtual Power, a generalization of the well-known Hill-Mandel Principle of Macrohomogeneity, from which equilibrium equations and homogenization relations for the force- and stress-like quantities are unequivocally obtained by straightforward variational arguments. The proposed theory provides a clear, logically-structured framework within which existing formulations can be rationally justified and new, more general multiscale models can be rigorously derived in well-defined steps. Its generality allows the treatment of problems involving phenomena as diverse as dynamics, higher order strain effects, material failure with kinematical discontinuities, fluid mechanics and coupled multi-physics. This is illustrated in a number of examples where a range of models is systematically derived by following the same steps. Due to the variational basis of the theory, the format in which derived models are presented is naturally well suited for discretization by finite element-based or related methods of numerical approximation. Numerical examples illustrate the use of resulting models, including a non-conventional failure-oriented model with discontinuous kinematics, in practical computations.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Notes

  1. Within the generalized setting of the present paper, the term kinematics (and corresponding kinematical variables etc) should be understood, in a broader sense, as relating to the primal variables of a given formulation. That is, we refer to kinematical variables as those whose rates produce power with the corresponding fluxes (stress- or force-like variables). In mechanical problems—the main motivation of our work—it has obviously the conventional meaning of generalized displacements and strains and their rates. In thermal problems, it refers to temperature, temperature gradient, and so on.

  2. The term equilibrium here is not limited to static equilibrium. If the force system \(f\) includes generalized inertia forces associated to the physical problem at hand, then dynamic equilibrium is automatically accounted for by the Principle of Virtual Power.

  3. This treatment simplifies the algorithmic procedure used for detecting the localization sub-domain \(\varOmega _{\mu }^{L}\), where the strain field localizes in the RVE, and thus the boundary \(\varGamma _{\mu }^{L}\) of \(\varOmega _{\mu }^{L}\), where new kinematical restrictions must be prescribed after the cohesive crack nucleation.

  4. The idea of forcing an elastic unloading behavior in those integration points located outside the cohesive crack in a strong discontinuity finite element, is a standard technique widely used in the phenomenological approach to fracture. We have adapted this procedure to the multiscale modeling context.

References

  1. Allaire G (1991) Homogenization of the Navier–Stokes equations in open sets perforated with tiny holes II: non-critical sizes of the holes for a volume distribution and a surface distribution of holes. Arch Ration Mech Anal 113:261–298

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Amstutz S, Giusti S, Novotny A, de Souza Neto E (2010) Topological derivative for multi-scale linear elasticity models applied to the synthesis of microstructures. Int J Numer Methods Eng 84:733–756

    Article  MathSciNet  MATH  Google Scholar 

  4. Andia P, Costanzo F, Gray G (2005) A Lagrangian-based continuum homogenization approach applicable to molecular dynamics simulations. Int J Solids Struct 42:6409–6432

    Article  MathSciNet  MATH  Google Scholar 

  5. Andia P, Costanzo F, Gray G (2006) A classical mechanics approach to the determination of the stress–strain response of particle systems. Model Simul Mater Sci Eng 14:741–757

    Article  Google Scholar 

  6. Bazant Z, Planas J (1998) Fracture and size efect in concrete and other quasibrittle materials. CRC Press, Boca Raton

    Google Scholar 

  7. Bazilevs Y, Calo V, Cottrell J, Hughes T, Reali A, Scovazzi G (2007) Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Comput Methods Appl Mech Eng 197:173–201

    Article  MathSciNet  MATH  Google Scholar 

  8. Belytschko T, Loehnert S, Song JH (2008) Multiscale aggregating discontinuities: a method for circumventing loss of material stability. Int J Numer Methods Eng 73:869–894

    Article  MathSciNet  MATH  Google Scholar 

  9. Belytschko T, Song JH (2010) Coarse-graining of multiscale crack propagation. Int J Numer Methods Eng 81:537–563

    MathSciNet  MATH  Google Scholar 

  10. Bensoussan A, Lions J, Papanicolaou G (1978) Asymptotic analysis for periodic structures. Elsevier, North-Holland

    MATH  Google Scholar 

  11. Berezovski A, Engelbrecht J, Maugin G (2011) Thermoelasticity with dual internal variables. J Therm Stress 34:413–430

    Article  MATH  Google Scholar 

  12. Berezovski A, Engelbrecht J, Peets T (2010) Multiscale modeling of microstructured solids. Mech Res Commun 37:531–534

    Article  MATH  Google Scholar 

  13. Berezovski A, Maugin G (2005) Stress-induced phase-transition front propagation in thermoelastic solids. Eur J Mech A Solids 24:1–21

    Article  MathSciNet  MATH  Google Scholar 

  14. Bhadeshia H, Kundu S, Abreu H (2009) Mathematics of crystallographic texture in martensitic and related transformations. In: Haldar A, Suwas S, Bhattacharjee D (eds) Microstructure and texture in steels. Springer, New York, pp 19–32

    Chapter  Google Scholar 

  15. Blanco P, Giusti S (2014) Thermomechanical multiscale constitutive modeling: accounting for microstructural thermal effects. J Elast 115:27–46

    Article  MathSciNet  MATH  Google Scholar 

  16. Bowles J, Mackenzie J (1954) The crystallography of martensite transformation. III. Face centred cubic to body centred tetragonal transformations. Acta Metall 2:224–234

    Article  Google Scholar 

  17. Budiansky B (1965) On the elastic moduli of some heterogeneous materials. J Mech Phys Solids 13:223–227

    Article  Google Scholar 

  18. Christensen J, de Abajo F (2012) Anisotropic metamaterials for full control of acoustic waves. Phys Rev Lett 108:124,301

    Article  Google Scholar 

  19. Coenen E, Kouznetsova VG, Geers MGD (2012) Multi-scale continuous–discontinuous framework for computational-homogenization–localization. J Mech Phys Solids 60:1486–1507

    Article  MathSciNet  Google Scholar 

  20. Coenen E, Kouznetsova VG, Geers MGD (2012) Novel boundary conditions for strain localization analyses in microstructural volume elements. Int J Numer Methods Eng 90:1–21

    Article  MathSciNet  MATH  Google Scholar 

  21. Daher N, Maugin G (1986) The method of virtual power in continuum mechanics. Application to media presenting singular surfaces and interfaces. Acta Mech 60:217–240

    Article  MathSciNet  MATH  Google Scholar 

  22. d’Alembert J (1743) Traité de dynamique. J. B. Coignard, Paris

  23. Del Vescovo D, Giorgio I (2014) Dynamic problems for metamaterials: review of existing models and ideas for further research. Int J Eng Sci 80:153–172

    Article  MathSciNet  Google Scholar 

  24. Dorfmann L, Ogden R (2014) Nonlinear theory of electroelastic and magnetoelastic interactions. Springer, New York

    Book  MATH  Google Scholar 

  25. Ene H (1983) On linear thermoelasticity of composite materials. Int J Eng Sci 21:443–448

    Article  MATH  Google Scholar 

  26. Engheta N, Ziolkowski R (2006) Metamaterials: physics and engineering explorations. Wiley, New York

    Book  Google Scholar 

  27. Eringen A (1999) Microcontinuum field theories. I: foundations and solids. Springer, New York

    Book  MATH  Google Scholar 

  28. Eringen A (1999) Microcontinuum field theories. II. Fluent media. Springer, New York

    Book  MATH  Google Scholar 

  29. Fang N, Xi D, Xu J, Ambati M, Srituravanich W, Sun C, Zhang X (2006) Ultrasonic metamaterials with negative modulus. Nat Mater 5:452–456

    Article  Google Scholar 

  30. Feyel F, Chaboche J (2000) \(\text{ FE }^{2}\) multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials. Comput Methods Appl Mech Eng 183:309–330

    Article  MATH  Google Scholar 

  31. Forest S, Amestoy M (2008) Hypertemperature in thermoelastic solids. C R Méc 336:347–353

    Article  MATH  Google Scholar 

  32. Francescato P, Pastor J, Riveill-Reydet B (2004) Ductile failure of cylindrically porous materials. Part I: plane stress problem and experimental results. Eur J Mech A Solids 23:181–190

    Article  MATH  Google Scholar 

  33. Francfort G (1983) Homogenization and linear thermoelasticity. SIAM J Math Anal 14:696–708

    Article  MathSciNet  MATH  Google Scholar 

  34. Geijselaers H, Perdahcioğlu E (2009) Mechanically induced martensitic transformation as a stress-driven process. Scr Mater 60:29–31

    Article  Google Scholar 

  35. Germain P (1973) La méthode des puissances virtuelles en mécanique des milieux continus. Premiére partie. Théorie du second gradient. J Méc 12:235–274

    MathSciNet  MATH  Google Scholar 

  36. Germain P (1973) The method of virtual power in continuum mechanics. part 2: microstructure. SIAM J Appl Math 25:556–575

    Article  MathSciNet  MATH  Google Scholar 

  37. Gitman I, Askes H, Sluys L (2007) Representative volume: existence and size determination. Eng Fract Mech 74:2518–2534

    Article  Google Scholar 

  38. Giusti S, Blanco P, de Souza Neto E, Feijóo R (2009) An assessment of the Gurson yield criterion by a computational multi-scale approach. Eng Comput 26(3):281–301

    Article  MATH  Google Scholar 

  39. Guest J, Prevost J (2006) Optimizing multifunctional materials: design of microstructures for maximized stiffness and fluid permeability. Int J Solids Struct 43(22–23):7028–7047

    Article  MATH  Google Scholar 

  40. Gurson A (1977) Continuum theory of ductile rupture by void nucleation and growth—part I: yield criteria and flow rule for porous media. J Eng Mater Technol 99:2–15

    Article  Google Scholar 

  41. Hashin Z, Shtrikman S (1963) A variational approach to the theory of elastic behaviour of multiphase materials. J Mech Phys Solids 11:127–140

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  43. Hill R (1965) Continuum micro-mechanics of elastoplastic polycrystals. J Mech Phys Solids 13:89–101

    Article  MATH  Google Scholar 

  44. Hill R (1965) A self-consistent mechanics of composite materials. J Mech Phys Solids 13:213–222

    Article  Google Scholar 

  45. Hill R (1972) On constitutive macro-variables for heterogeneous solids at finite strain. Proc R Soc Lond A 326:131–147

    Article  MATH  Google Scholar 

  46. Hou T, Hua X, Hussain F (2013) Multiscale modeling of incompressible turbulent flows. J Comp Phys 232:383–396

    Article  MathSciNet  MATH  Google Scholar 

  47. Hughes T, Feijóo G, Mazzei L, Quincy JB (1998) The variational multiscale method—a paradigm for computational mechanics. Comput Methods Appl Mech Eng 166:3–24

    Article  MathSciNet  MATH  Google Scholar 

  48. Hughes T, Mazzei L, Jansen K (2000) Large eddy simulation and the variational multiscale method. Comput Vis Sci 3:47–59

    Article  MATH  Google Scholar 

  49. Hughes T, Oberai A, Mazzei L (2001) Large eddy simulation of turbulent channel flows by the variational multiscale method. Phys Fluids 13:1784–1799

    Article  MATH  Google Scholar 

  50. Ieşan D, Nappa L (2005) On the theory of heat for micromorphic bodies. Int J Eng Sci 43:17–32

    Article  MathSciNet  MATH  Google Scholar 

  51. Ieşan D, Quintanilla R (2009) On thermoelastic bodies with inner structure and microtemperatures. J Math Anal Appl 354:12–23

  52. Irving J, Kirkwood J (1950) The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics. J Chem Phys 18:817–829

    Article  MathSciNet  Google Scholar 

  53. Kanouté P, Boso DP, Chaboche J, Schrefler B (2009) Multiscale methods for composites: a review. Arch Comput Methods Eng 16:31–75 P.

    Article  MATH  Google Scholar 

  54. Kato J, Yachi D, Terada K, Kyoya T (2014) Topology optimization of micro-structure for composites applying a decoupling multi-scale analysis. Struct Multidiscip Optim 49:595–608

    Article  MathSciNet  Google Scholar 

  55. Kirkwood J (1946) The statistical mechanical theory of transport processes. I. General theory. J Chem Phys 14:180–201

    Article  Google Scholar 

  56. Kirkwood J (1947) The statistical mechanical theory of transport processes. II. Transport in gases. J Chem Phys 15:72–76

    Article  Google Scholar 

  57. Kirkwood J, Buff F, Greenn M (1949) The statistical mechanical theory of transport processes. III. The coefficients of shear and bulk viscosity of liquids. J Chem Phys 17:988–994

    Article  Google Scholar 

  58. Kouznetsova V, Brekelmans W, Baaijens F (2001) An approach to micro–macro modeling of heterogeneous materials. Comput Mech 27:37–48

    Article  MATH  Google Scholar 

  59. Kouznetsova V, Geers M, Brekelmans W (2002) Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. Int J Numer Methods Eng 54:1235–1260

    Article  MATH  Google Scholar 

  60. Kouznetsova V, Geers M, Brekelmans W (2004) Multi-scale second order computational homogenization of multi-phase materials: a nested finite element solution strategy. Comput Methods Appl Mech Eng 193:5525–5550

    Article  MATH  Google Scholar 

  61. Kundu S, Bhadeshia H (2006) Transformation texture in deformed stainless steel. Scr Mater 55:779–781

    Article  Google Scholar 

  62. Kundu S, Bhadeshia H (2007) Crystallographic texture and intervening transformations. Scr Mater 57:869–872

    Article  Google Scholar 

  63. Lakes R (1987) Foam structures with negative Poisson’s ratio. Science AAAS 235(4792):1038–1040

    Article  Google Scholar 

  64. Lakes R (1987) Negative Poisson’s ratio materials. Science AAAS 238(4826):551

    Article  Google Scholar 

  65. Larsson F, Runesson K, Su F (2010) Variationally consistent computational homogenization of transient heat flow. Int J Numer Methods Eng 81:1659–1686

    MathSciNet  MATH  Google Scholar 

  66. Larsson R, Diebels S (2007) A second-order homogenization procedure for multi-scale analysis based on micropolar kinematics. Int J Numer Methods Eng 69:2485–2512

    Article  MathSciNet  MATH  Google Scholar 

  67. Larsson R, Zhang Y (2007) Homogenization of microsystem interconnects based on micropolar theory and discontinuous kinematics. J Mech Phys Solids 55:819–841

    Article  MathSciNet  MATH  Google Scholar 

  68. Lee S, Park C, Seo Y, Wang Z, Kim C (2010) Composite acoustic medium with simultaneously negative density and modulus. Phys Rev Lett 104:054,301

    Article  Google Scholar 

  69. Li J, Chan C (2004) Double-negative acoustic metamaterial. Phys Rev E 70:055,602(R)

    Article  Google Scholar 

  70. Lidström P (2011) On the volume average of energy and net power. Contin Mech Thermodyn 23:275–304

    Article  MathSciNet  MATH  Google Scholar 

  71. Liu Z, Zhang X, Mao Y, Zhu Y, Yang Z, Chan C, Sheng P (2000) Locally resonant sonic materials. Science 289:1734–1736

    Article  Google Scholar 

  72. Luscher D, McDowell D, Bronkhorst C (2010) A second gradient theoretical framework for hierarchical multiscale modeling of materials. Int J Plast 26:1248–1275

    Article  MATH  Google Scholar 

  73. Mandel J (1971) Plasticité Classique et Viscoplasticité. CISM Lecture Notes No. 97. Springer, Udine

    Google Scholar 

  74. Marsden J, Hughes T (1983) Mathematical foundations of elasticity. Dover, New York

    MATH  Google Scholar 

  75. Maugin G (1980) The method of virtual power in continuum mechanics: application to coupled fields. Acta Mech 35:1–70

    Article  MathSciNet  MATH  Google Scholar 

  76. McDowell D (2007) Simulation-assisted materials design for the concurrent design of materials and products. JOM 59:21–25

    Article  Google Scholar 

  77. McDowell D (2010) A perspective on trends in multiscale plasticity. Int J Plast 26:1280–1309

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  79. Miehe C, Koch A (2002) Computational micro-to-macro transition of discretized microstructures undergoing small strain. Arch Appl Mech 72:300–317

    Article  MATH  Google Scholar 

  80. Miehe C, Schotte J, Lambrecht J (2002) Homogenization of inelastic solid materials at finite strains based on incremental minimization principles. Application to the texture analysis of polycrystals. J Mech Phys Solids 50:2123–2167

    Article  MathSciNet  MATH  Google Scholar 

  81. Miehe C, Schotte J, Schröder J (1999) Computational micro-macro transitions and overall moduli in the analysis of polycrystals at large strains. Comput Mater Sci 6:372–382

    Article  Google Scholar 

  82. Miehe C, Schroder J, Becker M (2002) Computational homogenization analysis in finite elasticity: material and structural instabilities on the micro- and macro-scales of periodics composites and their interaction. Comput Methods Appl Mech Eng 191:4971–5005

    Article  MathSciNet  MATH  Google Scholar 

  83. Nemat-Nasser S (1999) Averaging theorems in finite deformation plasticity. Mech Mater 31:493–523

    Article  Google Scholar 

  84. Nguyen QS (2010) On standard dissipative gradient models. Ann Solid Struct Mech 1:79–86

    Article  Google Scholar 

  85. Nguyen QS, Andrieux S (2005) The non-local generalized standard approach: a consistent gradient theory. C R Méc 333:139–145

    Article  MATH  Google Scholar 

  86. Nguyen V, Lloberas-Valls O, Stroeven M, Sluys L (2011) Homogenization-based multiscale crack modelling: from micro-diffusive damage to macro-cracks. Comput Methods Appl Mech Eng 200:1220–1236

    Article  MATH  Google Scholar 

  87. Nguyen V, Stroeven M, Sluys L (2012) Multiscale failure modeling of concrete: micromechanical modeling, discontinuous homogenization and parallel computations. Comput Methods Appl Mech Eng 201–204:139–156

    Article  MathSciNet  MATH  Google Scholar 

  88. Nguyen V, Valls O, Stroeven M, Sluys L (2010) On the existence of representative volumes for softening quasi-brittle materials—a failure zone averaging scheme. Comput Methods Appl Mech Eng 199:3028–3038

    Article  MATH  Google Scholar 

  89. Nguyen V, Valls O, Stroeven M, Sluys L (2012) Computational homogenization for multiscale crack modeling. Implementational and computational aspects. Int J Numer Methods Eng 89:192–226

    Article  MathSciNet  MATH  Google Scholar 

  90. Oden J (1979) Applied functional analysis. Prentice-Hall, New Jersey

    MATH  Google Scholar 

  91. Oliver J (1989) A consistent characteristic length for smeared craking models. Int J Numer Methods Eng 28:461–474

    Article  MATH  Google Scholar 

  92. Özdemir I, Brekelmans W, Geers M (2008) Computational homogenization for heat conduction in heterogeneous solids. Int J Numer Methods Eng 73(2):185–204

    Article  MathSciNet  MATH  Google Scholar 

  93. Özdemir I, Brekelmans W, Geers M (2008) \(\text{ FE }^{2}\) computational homogenization for the thermo-mechanical analysis of heterogeneous solids. Comput Methods Appl Mech Eng 198:602–613

    Article  MathSciNet  MATH  Google Scholar 

  94. Pastor J, Francescato P, Trillat M, Loute E, Rousselier G (2004) Ductile failure of cylindrically porous materials. Part II: other cases of symmetry. Eur J Mech A Solids 23:190–201

    Article  MATH  Google Scholar 

  95. Patel J, Cohen M (1953) Criterion for the action of applied stress in the martensitic transformation. Acta Metall 1:531–538

    Article  Google Scholar 

  96. Perdahcioğlu E, Geijselaers H (2012) A macroscopic model to simulate the mechanically indyced martensitic transformation in metastable austenitic stainless steel. Acta Mater 60:4409–4419

    Article  Google Scholar 

  97. Perić D, de Souza Neto E, Feijóo R, Partovi M, Molina AC (2011) On micro-to-macro transitions for multi-scale analysis of non-linear heterogeneous materials: unified variational basis and finite element implementation. Int J Numer Methods Eng 87:149–170

    Article  MATH  Google Scholar 

  98. Pham K, Kouznetsova V, Geers M (2013) Transient computational homogenization for heterogeneous materials under dynamic excitation. J Mech Phys Solids 61:2125–2146

    Article  MathSciNet  MATH  Google Scholar 

  99. Podio-Guidugli P (2009) A virtual power format for thermomechanics. Contin Mech Thermodyn 20:479–487

    Article  MathSciNet  MATH  Google Scholar 

  100. Ricker S, Mergheim J, Steinmann P (2009) On the multiscale computation of defect driving forces. Int J Multiscale Comput Eng 7:457–474

    Article  Google Scholar 

  101. Romano G, Diaco M, Barretta R (2010) Variational formulation of the first principle of continuum thermodynamics. Contin Mech Thermodyn 22:177–187

    Article  MathSciNet  MATH  Google Scholar 

  102. Rots J, Nauta P, Kusters G, Blaauwendraad T (1985) Smeared crack approach and fracture localization in concrete. HERON 30:1–48

    Google Scholar 

  103. Sánchez P, Blanco P, Huespe A, Feijóo R (2011) Failure-oriented multi-scale variational formulation for softening materials. Tech. Rep. P&D No. 6, LNCC-MCTI Laboratório Nacional de Computação Científica

  104. Sánchez P, Blanco P, Huespe A, Feijóo R (2013) Failure-oriented multi-scale variational formulation: micro-structures with nucleation and evolution of softening bands. Comput Methods Appl Mech Eng 257:221–247

    Article  MathSciNet  MATH  Google Scholar 

  105. Sanchez-Palencia E (1980) Non-homogeneous media and vibration theory. Lecture notes in physics, vol 127. Springer, Berlin

    Google Scholar 

  106. Sanchez-Palencia E (1983) Homogenization method for the study of composite media. In: Verhulst F (ed) Asymptotic analysis II. Surveys and new trends. Lecture notes in mathematics, vol 985. Springer, Berlin, pp 192–214

    Chapter  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  108. Silva E, Fonseca J, Kikuchi N (1997) Optimal design of periodic microstructures. Comput Mech 19(5):397–410

    Article  MATH  Google Scholar 

  109. Song JH, Belytschko T (2009) Multiscale aggregating discontinuities method for micro–macro failure of composites. Compos Part B 40:417–426

    Article  Google Scholar 

  110. de Souza Neto E, Feijóo R (2006) Variational foundations of multi-scale constitutive models of solid: Small and large strain kinematical formulation. Tech Rep. P&D No. 16, LNCC-MCTI Laboratório Nacional de Computação Científica

  111. de Souza Neto E, Feijóo R (2008) On the equivalence between spatial and material volume averaging of stress in large strain multi-scale constitutive models. Mech Mater 40:803–811

    Article  Google Scholar 

  112. de Souza Neto E, Feijóo R (2010) Variational foundations of large strain multiscale solid constitutive models: kinematical formulation. In: Vaz M Jr, de Souza Neto E, Muñoz Rojas P (eds) Computational materials modelling: from classical to multi-scale techniques. Wiley, Chichester, pp 341–378

  113. Speirs D, de Souza Neto E, Perić D (2008) An approach to the mechanical constitutive modelling of arterial tissue based on homogenization and optimization. J Biomech 41:2673–2680

  114. Sunyk R, Steinmann P (2003) On higher gradients in continuum-atomistic modelling. Int J Solids Struct 40:6877–6896

    Article  MATH  Google Scholar 

  115. Swan C (1994) Techniques for stress- and strain-controlled homogenization of inelastic periodic composites. Comput Methods Appl Mech Eng 117:249–267

    Article  MathSciNet  MATH  Google Scholar 

  116. Tamura I (1982) Deformation-induced martensitic transformation and transformation-induced plasticity in steels. Met Sci 16:245–253

    Article  Google Scholar 

  117. Tartar L (1980) Incompressible fluid flow in a porous medium-convergence of the homogenization process. Volume 127 on Lecture notes in physics. Springer, Berlin

    Google Scholar 

  118. Temizer I, Wriggers P (2011) Homogenization in finite thermoelasticity. J Mech Phys Solids 59:344–372

    Article  MathSciNet  MATH  Google Scholar 

  119. Terada K, Inugai T, Hamana Y, Miyori A, Hirayama N (2008) Parameter identification for anisotropic hyperelastic materials by numerical material testing. Trans Jpn Soc Comput Eng Sci 23:190–201

    Google Scholar 

  120. Terada K, Kato J, Hirayama N, Inugai T, Yamamoto K (2013) A method of two-scale analysis with micro–macro decoupling scheme: application to hyperelastic composite materials. Comput Mech 52:1199–1219

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  122. Terada K, Watanabe I, Akiyama M (2006) Effects of shape and size of crystal grains on the strengths of polycrystalline metals. Int J Multiscale Comput Eng 4:445–460

    Article  Google Scholar 

  123. Toro S, Sánchez P, Huespe A, Giusti S, Blanco P, Feijóo R (2014) A two-scale failure model for heterogeneous materials: numerical implementation based on the finite element method. Int J Numer Methods Eng 97:313–351

    Article  MathSciNet  Google Scholar 

  124. Vatanabe S, Paulino G, Silva E (2013) Design of functionally graded piezocomposites using topology optimization and homogenization—toward effective energy harvesting materials. Comput Methods Appl Mech Eng 266:205–218

    Article  MathSciNet  MATH  Google Scholar 

  125. Verhoosel C, Remmers J, Gutiérrez M, de Borst R (2010) Computational homogenization for adhesive and cohesive failure in quasi-brittle solids. Int J Numer Methods Eng 83:1155–1179

  126. Watanabe I, Terada K, de Souza Neto E, Perić D (2006) Characterization of macroscopic tensile strength of polycrystalline metals with two-scale finite element analysis. J Mech Phys Solids 56:1105–1125

    Article  MathSciNet  MATH  Google Scholar 

  127. Wechsler M, Lieberman D, Read T (1953) On the theory of the formation of martensite. Trans AIME 197:1503–1515

    Google Scholar 

  128. Xu B, Arias F, Brittain S, Zhao XM, Grzybowski B, Torquato S (1999) Making negative Poisson’s ratio microstructures by soft lithography. Adv Mater 11:1186–1189

    Article  Google Scholar 

  129. Yvonnet J, He Q (2010) A non-concurrent multiscale method for computing the response of hyperelastic heterogeneous structures. Eur J Comput Mech 19:105–116

    MATH  Google Scholar 

  130. Zhu J, Christensen J, Jung J, Martin-Moreno L, Yin X, Fok L, Zhang X, Garcia-Vidal F (2011) A holey-structured metamaterial for acoustic deep-subwavelength imaging. Nat Phys 7:52–55

  131. Zimmerman J, Jones R, Templeton J (2010) A material frame approach for evaluating continuum variables in atomistic simulations. J Comput Phys 229:2364–2389

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work was partially supported by the Brazilian agencies CNPq and FAPERJ. The support of these agencies is gratefully acknowledged. P.J. Sánchez acknowledges the financial support from CONICET (grant PIP 2013-2015 631) and from the European Research Council under the European Unions Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement N. 320815 (ERC Advanced Grant Project Advanced tools for computational design of engineering materials COMP-DES-MAT).

Author information

Authors and Affiliations

  1. National Laboratory for Scientific Computing, LNCC/MCTI, Av. Getúlio Vargas, 333, Petrópolis, RJ, 25651-075, Brazil

    Pablo J. Blanco & Raúl A. Feijóo

  2. National Institute of Science and Technology in Medicine Assisted by Scientific Computing (INCT-MACC), Petrópolis, Brazil

    Pablo J. Blanco & Raúl A. Feijóo

  3. CIMEC-UNL-CONICET, Güemes 3450, 3000, Santa Fe, Argentina

    Pablo J. Sánchez

  4. GIMNI-UTN-FRSF, Lavaise 610, 3000, Santa Fe, Argentina

    Pablo J. Sánchez

  5. Zienkiewicz Centre for Computational Engineering, Swansea University, Swansea, SA2 8PP, UK

    Eduardo A. de Souza Neto

Authors
  1. Pablo J. Blanco
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pablo J. Sánchez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Eduardo A. de Souza Neto
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Raúl A. Feijóo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Raúl A. Feijóo.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Blanco, P.J., Sánchez, P.J., de Souza Neto, E.A. et al. Variational Foundations and Generalized Unified Theory of RVE-Based Multiscale Models. Arch Computat Methods Eng 23, 191–253 (2016). https://doi.org/10.1007/s11831-014-9137-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11831-014-9137-5

Search

Navigation