Các khía cạnh của việc thực hiện các điều kiện biên kéo không đổi trong quá trình đồng nhất hóa tính toán thông qua các điều kiện biên bán-Dirichlet

Hill R (1972) On constitutive macro-variables for heterogeneous solids at finite strain. Proc R Soc Lond A 326:131–147 Ogden RW (1974) On the overall moduli of non-linear elastic composite materials. J Mech Phys Solids 22:541–553 Oden JT, Vemaganti K, Moës N (1999) Hierarchical modeling of heterogeneous solids. Comput Methods Appl Mech Eng 172:3–25 Markovic D, Ibrahimbegovic A (2004) On micro-macro interface conditions for micro scale based FEM for inelastic behavior of heterogeneous materials. Comput Methods Appl Mech Eng 193:5503–5523 Lloberas-Valls O, Rixen DJ, Simone A, Sluys LJ (2012) On micro-to-macro connections in domain decomposition multiscale methods. Comput Methods Appl Mech Eng 225–228:177–196 Eshelby JD (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc Lond A 241:376–396 Hashin Z, Shtrikman S (1963) A variational approach to the theory of the elastic behaviour of multiphase materials. J Mech Phys Solids 11:127–140 Walpole LJ (1966) On bounds for the overall elastic moduli of inhomogeneous systems-II. J Mech Phys Solids 14:289–301 Mori T, Tanaka K (1973) Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall 21:571–574 Halpin JC, Kardos JL (1976) The Halpin-Tsai equations:The Halpin-Tsai equations: a review. Polym Eng Sci 16:344–352 Hori M, Nemat-Nasser S (1993) Double-inclusion model and overall moduli of multi-phase composites. Mech Mater 14:189–206 Willis JR (1986) Variational estimates for the overall response of an inhomogeneous nonlinear dielectric. In: Ericksen JL, Kinderlehrer D, Kohn R, Lions J-L (eds) Homogenization and effective moduli of materials and media, volume 1 of the IMA volumes in mathematics and its applications. Springer, New York, pp 247–263 Talbot DRS, Willis JR (1992) Some simple explicit bounds for the overall behaviour of nonlinear composites. Int J Solids Struct 29:1981–1987 Ponte Castañeda P, deBotton G, Li G (1992) Effective properties of nonlinear inhomogeneous dielectrics. Phys Rev B 46:4387–4394 Ponte P (1996) Castañeda. Exact second-order estimates for the effective mechanical properties of nonlinear composite materials. J Mech Phys Solids 44:827–862 Mura T, Shodja HM, Hirose Y (1996) Inclusion problems. Appl Mech Rev 49:118–127 Böhm HJ (1998) A short introduction to basic aspects of continuum mechanics. CDL-FMD Report 3. Technical report, Institute of Lightweight Design and Structural Biomechanics (ILSB), Vienna University of Technology Nemat-Nasser S, Hori M (1999) Micromechanics: overall properties of heterogeneous materials. Elsevier, Amsterdam Klusemann B, Böhm HJ, Svendsen B (2012) Homogenization methods for multi-phase elastic composites: comparisons and benchmarks. Eur J Mech A Solids 34:21–37 Miehe C, Schröder J, Schotte J (1999) Computational homogenization analysis in finite plasticity simulation of texture development in polycrystalline materials. Comput Methods Appl Mech Eng 171:387–418 Feyel F, Chaboche J-L (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 Terada K, Hori M, Kyoya T, Kikuchi N (2000) Simulation of the multi-scale convergence in computational homogenization approaches. Int J Solids Struct 37:2285–2311 Kouznetsova VG, Brekelmans WAM, Baaijens FPT (2001) An approach to micro-macro modeling of heterogeneous materials. Comput Mech 27:37–48 Jiang M, Alzebdeh K, Jasiuk I, Ostoja-Starzewski M (2001) Scale and boundary conditions effects in elastic properties of random composites. Acta Mech 148:63–78 Miehe C, Schröder J, Becker M (2002) Computational homogenization analysis in finite elasticity: material and structural instabilities on the micro- and macro-scales of periodic composites and their interaction. Comput Methods Appl Mech Eng 191:4971–5005 Moës N, Cloirec M, Cartraud P, Remacle J-F (2003) A computational approach to handle complex microstructure geometries. Comput Methods Appl Mech Eng 192:3163–3177 Zohdi TI, Wriggers P (2005) Introduction to computational micromechanics. Springer, Berlin Segurado J, Llorca J (2006) Computational micromechanics of composites: the effect of particle spatial distribution. Mech Mater 38:873–883 Wriggers P, Moftah SO (2006) Mesoscale models for concrete: homogenisation and damage behaviour. Finite Elem Anal Des 42:623–636 Geers MGD, Coenen EWC, Kouznetsova VG (2007) Multi-scale computational homogenization of structured thin sheets. Modell Simul Mater Sci Eng 15:393–404 Gitman IM, Askes H, Sluys LJ (2007) Representative volume: existence and size determination. Eng Fract Mech 74:2518–2534 Yvonnet J, He Q-C (2007) The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains. J Comput Phys 223:341–368 Temizer İ, Zohdi TI (2007) A numerical method for homogenization in non-linear elasticity. Comput Mech 40:281–298 Temizer İ, Wriggers P (2007) An adaptive method for homogenization in orthotropic nonlinear elasticity. Comput Methods Appl Mech Eng 196:3409–3423 Temizer İ, Wriggers P (2008) On the computation of the macroscopic tangent for multiscale volumetric homogenization problems. Comput Methods Appl Mech Eng 198:495–510 Özdemir I, Brekelmans WAM, Geers MGD (2008) Computational homogenization for heat conduction in heterogeneous solids. Int J Numer Meth Eng 73:185–204 Nezamabadi S, Yvonnet J, Zahrouni H, Potier-Ferry M (2009) A multilevel computational strategy for handling microscopic and macroscopic instabilities. Comput Methods Appl Mech Eng 198:2099–2110 Larsson F, Runesson K (2011) On two-scale adaptive FE analysis of micro-heterogeneous media with seamless scale-bridging. Comput Methods Appl Mech Eng 200:2662–2674 Temizer İ, Wriggers P (2011) Homogenization in finite thermoelasticity. J Mech Phys Solids 59:344–372 Coenen EWC, Kouznetsova VG, Bosco E, Geers MGD (2012) A multi-scale approach to bridge microscale damage and macroscale failure: a nested computational homogenization-localization framework. Int J Fract 178:157–178 Nguyen VP, Stroeven M, Sluys LJ (2012) Multiscale failure modeling of concrete: Micromechanical modeling, discontinuous homogenization and parallel computations. Comput Methods Appl Mech Eng 201–204:139–156 Fritzen F, Leuschner M (2013) Reduced basis hybrid computational homogenization based on a mixed incremental formulation. Comput Methods Appl Mech Eng 260:143–154 Javili A, Chatzigeorgiou G, Steinmann P (2013) Computational homogenization in magneto-mechanics. Int J Solids Struct 50:4197–4216 Kochmann DM, Venturini GN (2013) Homogenized mechanical properties of auxetic composite materials in finite-strain elasticity. Smart Mater Struct 22:084004 Yvonnet J, Monteiro E, He Q-C (2013) Computational homogenization method and reduced database model for hyperelastic heterogeneous structures. Int J Multiscale Comput Eng 11:201–225 Balzani D, Scheunemann L, Brands D, Schröder J (2014) Construction of two- and three-dimensional statistically similar RVEs for coupled micro-macro simulations. Comput Mech 54:1269–1284 Keip M-A, Steinmann P, Schröder J (2014) Two-scale computational homogenization of electro-elasticity at finite strains. Comput Methods Appl Mech Eng 278:62–79 Fritzen F, Kochmann DM (2014) Material instability-induced extreme damping in composites: A computational study. Int J Solids Struct 51:4101–4112 Le BA, Yvonnet J, He Q-C (2015) Computational homogenization of nonlinear elastic materials using neural networks. Int J Numer Meth Eng 104:1061–1084 van Dijk NP (2016) Formulation and implementation of stress-driven and/or strain-driven computational homogenization for finite strain. Int J Numer Meth Eng 107:1009–1028 Kanouté P, Boso DP, Chaboche JL, Schrefler BA (2009) Multiscale methods for composites: a review. Arch Comput Methods Eng 16:31–75 Charalambakis N (2010) Homogenization techniques and micromechanics. A survey and perspectives. Appl Mech Rev 63:030803 Geers MGD, Kouznetsova VG, Brekelmans WAM (2010) Multi-scale computational homogenization: Trends and challenges. J Comput Appl Math 234:2175–2182 Nguyen VP, Stroeven M, Sluys LJ (2011) Multiscale continuous and discontinuous modelling of heterogeneous materials: A review on recent developments. J Multiscale Model 3:229–270 Saeb S, Steinmann P, Javili A (2016) Aspects of computational homogenization at finite deformations. A unifying review from Reuss’ to Voigt’s bound. Appl Mech Rev 68:050801 Linder C, Tkachuk M, Miehe C (2011) A micromechanically motivated diffusion-based transient network model and its incorporation into finite rubber viscoelasticity. J Mech Phys Solids 59:2134–2156 Tkachuk M, Linder C (2012) The maximal advance path constraint for the homogenization of materials with random network microstructure. Philos Mag 92:2779–2808 Raina A, Linder C (2014) A homogenization approach for nonwoven materials based on fiber undulations and reorientation. J Mech Phys Solids 65:12–34 Kanit T, Forest S, Galliet I, Mounoury V, Jeulin D (2003) Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int J Solids Struct 40:3647–3679 Kouznetsova VG, Geers MGD, Brekelmans WAM (2002) Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. Int J Numer Methods Eng 54:1235–1260 Khisaeva ZF, Ostoja-Starzewski M (2006) On the size of RVE in finite elasticity of random composites. J Elast 85:153–173 Javili A, Chatzigeorgiou G, McBride A, Steinmann P, Linder C (2015) Computational homogenization of nano-materials accounting for size effects via surface elasticity. GAMM Mitteilungen 38:285–312 Nemat-Nasser S, Hori M (1995) Universal bounds for overall properties of linear and nonlinear heterogeneous solids. J Eng Mater Technol 117:412–432 Miehe C (2002) Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation. Int J Numer Methods Eng 55:1285–1322 Kaczmarczyk L, Pearce CJ, Bićanić N (2008) Scale transition and enforcement of RVE boundary conditions in second-order computational homogenization. Int J Numer Methods Eng 74:506–522 Perić D, de Souza Neto EA, Feijóo RA, Partovi M, Carneiro Molina AJ (2011) On micro-to-macro transitions for multi-scale analysis of non-linear heterogeneous materials. Int J Numer Methods Eng 87:149–170 Mesarovic SD, Padbidri J (2005) Minimal kinematic boundary conditions for simulations of disordered microstructures. Philos Mag 85:65–78 Shen H, Brinson LC (2006) A numerical investigation of the effect of boundary conditions and representative volume element size for porous titanium. J Mech Mater Struct 1:1179–1204 Drago A, Pindera M-J (2007) Micro-macromechanical analysis of heterogeneous materials: macroscopically homogeneous vs periodic microstructures. Compos Sci Technol 67:1243–1263 Miehe C (2003) Computational micro-to-macro transitions for discretized micro-structures of heterogeneous materials at finite strains based on the minimization of averaged incremental energy. Comput Methods Appl Mech Eng 192:559–591 Felippa CA, Park KC (2002) The construction of free-free flexibility matrices for multilevel structural analysis. Comput Methods Appl Mech Eng 191:2139–2168 Podio-Guidugli Paolo (2000) Primer in elasticity. J Elast 58(1):1–104 Yuan Z, Fish J (2008) Toward realization of computational homogenization in practice. Int J Numer Methods Eng 73:361–380 Larsson F, Runesson K, Saroukhani S, Vafadari R (2011) Computational homogenization based on a weak format of micro-periodicity for RVE-problems. Comput Methods Appl Mech Eng 200:11–26 Nguyen V-D, Béchet E, Geuzaine C, Noels L (2012) Imposing periodic boundary condition on arbitrary meshes by polynomial interpolation. Comput Mater Sci 55:390–406 Hirschberger CB, Ricker S, Steinmann P, Sukumar N (2009) Computational multiscale modelling of heterogeneous material layers. Eng Fract Mech 76(6):793–812 McBride A, Mergheim J, Javili A, Steinmann P, Bargmann S (2012) Micro-to-macro transitions for heterogeneous material layers accounting for in-plane stretch. J Mech Phys Solids 60:1221–1239 Miehe C, Koch A (2002) Computational micro-to-macro transitions of discretized microstructures undergoing small strains. Arch Appl Mech 72:300–317 Larsson F, Runesson K (2007) RVE computations with error control and adaptivity: the power of duality. Comput Mech 39:647–661 Fish J, Fan R (2008) Mathematical homogenization of nonperiodic heterogeneous media subjected to large deformation transient loading. Int J Numer Methods Eng 76:1044–1064 Javili A, Dortdivanlioglu B, Kuhl E, Linder C (2015) Computational aspects of growth-induced instabilities through eigenvalue analysis. Comput Mech 56(3):405–420 Dortdivanlioglu B, Javili A, Linder C. Computational aspects of morphological instabilities using isogeometric analysis. Comput Methods Appl Mech Eng. doi:10.1016/j.cma.2016.06.028