In this paper, the analytical solution of the multi-step homogenization problem for multi-rank composites with generalized periodicity made of elastic materials is presented. The proposed homogenization scheme may be combined with computational homogenization for solving more complex microstructures. Three numerical examples are presented, concerning locally periodic stratified materials, matrices with wavy layers and wavy fiber-reinforced composites.
LakesR.Materials with structural hierarchy. Nature1993; 361: 511–515.
2.
JabinPTzavarasA.Kinetic decomposition for periodic homogenization problems. SIAM J Math Anal2009; 41: 360–390.
3.
MorenoRBorgaMShedbyO. Techniques for computing fabric tensors in visualisation and processing of tensors and higher order description for multi-valued data. In: Mathematics and Visualization. New York: Springer, 2014; pp. 271–292.
4.
MiltonG.The Theory of Composites. Cambridge: Cambridge University Press, 2002.
5.
LebeeASabK.Homogenization of thick periodic plate: application of the bending-gradient plate theory to a folded core sandwich panel. Int J Solids Struct2012; 49: 2778–2792.
6.
BensoussanALionsJLPapanicolaouG.Asymptotic methods for periodic structures. Amsterdam: North-Holland, 1978.
7.
BrianeM.Homogénéisation de matériaux fibrés. PhD Thesis, Université Pierre et Marie Curie (Paris VI), 1990.
8.
BrianeM.Three models of non periodic fibrous materials obtained by homogenization. RAIRO Mod Math Anal Numér1993; 27: 759–775.
9.
TsalisDChatzigeorgiouGCharalambakisN.Homogenization of structures with generalized periodicity. Composites B2012; 43: 2495–2512.
10.
ChatzigeorgiouGEfendievYCharalambakisN. Effective thermoelastic properties of composites with periodicity in cylindrical coordinates. Int J Solids Struct2012; 49: 2590–2603.
11.
ChanBLeongK.Scaffolding in tissue engineering: general approaches and tissue specific considerations. Eur Spine J2008; 17: 467–479.
12.
NilakantanGKeefeMGillespieJ. Novel multi-scale modeling of woven fabric composites for use in impact studies. In: 10th international LS-DYNA users conference, 2008.
13.
RajanSMobasherBVaildyaiB. Ls-dyna implemented multi-layer fabric material model development for engine fragment mitigation. In: 11th international LS-DYNA users conference, 2010.
14.
ZhuDVaidyaAMobasherB. Finite element modeling of ballistic impact on multi-layer kevlar 49 fabrics. Composites B2014; 56: 254–262.
15.
GuedesJKikuchiN.Preprocessing and posprocessing for materials based on the homogenization method with adaptive finite element methods. Comput Meth Appl Mech Eng1990; 83: 143–198.
16.
HollisterSKikuchiN.A comparison of homogenization and standard mechanics analyses for periodic porous composites. Computat Mech1992; 10: 73–95.
17.
HashinZ.Analysis of composite materials: A survey. J Appl Mech1983; 50: 481–505.
18.
KalamkarovALKolpakovAG.Analysis, design and optimization of composite structures. New York: Wiley, 1997.
19.
Nemat-NasserSHoriS.Micromechanics: Overall Properties of Heterogeneous Materials. Amsterdam: Elsevier, 1999.
20.
AboudiJPinderaMArnoldS.Higher-order theory for functionally graded materials. Composites B1999; 30: 777–832.
21.
Nemat-NasserS.Averaging theorems in finite deformation plasticity. Mech Mater1999; 31: 493–523.
22.
Guinovart-DiazRRodriguez-RamosRBravo-CastilleroJ. A recursive asymptotic homogenization scheme for multi-phase fiber-reinforced composites. Mech Mater2005; 37: 1119–1131.
23.
ChatzigeorgiouGCharalambakisN.Instability analysis of non-homogeneous materials under biaxial loading. Int J Plasticity2005; 21: 1970–1999.
24.
LoveBBatraR.Determination of effective thermomechanical parameters of a mixture of two elastothermoviscoplastic constituents. Int J Plasticity2006; 22: 1026–1061.
KalamkarovAAndrianovIDanishevs’kyyV.Asymptotic homogenization of composite materials and structures. Appl Mech Rev2009; 62: 030802.
27.
PinderaMKhatamHDragoS. Micromechanics of spatially uniform heterogeneous media: A critical review and emerging approaches. Composites B2009; 40: 349–378.
28.
NieGZhongZBatraR.Material tailoring for functionally graded hollow cylinders and spheres. Compos Sci Technol2011; 71: 666–673.
29.
WuYNieYYangZ.Prediction of effective properties for random heterogeneous materials with extrapolation. Arch Appl Mech2014; 84: 247–261.
30.
ChatzigeorgiouGJaviliASteinmannP.Unified magnetomechanical homogenization framework with application to magnetorheological elastomers. Math Mech Solids2014; 19: 193–211.
31.
BerrehiliY.Lognitudinal traction of a fiber-reinforced composite beam with debonded fibers. Int J Theoret Math Phys2014; 4(2): 53–57.
32.
Abd-AllaAEdfawyEMahmoudS.On problem of the non-homogeneity of an infinite orthotropic elastic cylinder. J Computat Theoret Nanosci2014; 11: 945–952.
33.
TuWPinderaM.Cohesive zone-based damage evolution in periodic materials via finite-volume homogenization. J Appl Mech2014; 81: 100–115.
34.
SavvasDStefanouGPapadrakakisM. Homogenization of random heterogeneous media with inclusions of arbitrary shape modeled by XFEM. Computat Mech2014; 54: 1221–1235.
35.
MahmoudSAbd-AllaATounsiA. The problem of wave propagation in magnetorotating orthotropic non-homogeneous medium. J Vibr Control2014; in press.
36.
BabuskaI.Homogenization and its application, Mathematical and computational problems. In: Proceedings of the third symposium on numerical solution of partial differential equations, III (SYNSPADE), University of Maryland, 1975. New York: Academic Press, 1976, pp. 89–116.
37.
BabuskaI.Homogenization approach in engineering. In: Computing methods in applied sciences and engineering (second international symposium, Versailles, 1975), Part 1 (Lecture Notes in Economics and Mathematical Systems, Vol. 134). New York: Springer, 1976, pp. 137–153.
38.
Sanchez-PalenciaE.Non-homogeneous media and vibration theory (Lecture Notes in Physics, Vol. 127). Berlin: Springer Verlag, 1978.
39.
FrancfortG.Homogenization and linear thermoelasticity. SIAM J Math Anal1983; 14: 696–708.
40.
FrancfortGMuratF.Homogenization and optimal bounds in linear elasticity. Arch Rat Mech Anal1986; 94: 307–334.
41.
AllaireG.Homogenization and two-scale convergence. SIAM J Math Anal1992; 23: 1482–1518.
42.
MuratFTartarL. H-convergence. In: CherkaevAKohnRVTopics in the mathematical modelling of composite materials (Progress in Nonlinear Differential Equations and their Applications, Vol. 31). Basel: Birkhäuser, 1997, pp. 21–43.
43.
ZhigangSChaoxianZXiguangG. A quadrilateral element-based method for calculation of multi-scale temperature field. Chin J Aeronaut2010; 23: 529–536.
44.
GelebartL.Periodic boundary conditions for the numerical homogenization of composite tubes. C R Mecanique2011; 339: 12–19.
45.
LeeCYuW.Variational asymptotic modeling of composite beams with spanwise heterogeneity. Comput Struct2011; 89: 1503–1511.
46.
NackenhorstUKardasDHelmichT. Computational techniques for multiscale analysis of materials and interfaces (Lecture Notes in Applied and Computational Mechanics, Vol. 57). New York: Springer, 2011, pp. 133–167.
47.
PinderaMCharalambakisNLagoudasD. Special issue of composites part B: Homogenization and micromechanics of smart and multifunctional materials. Composites B2012; 43: 2493–2494.
48.
BirmanVChandrashekharaKHopkinsM. Strength analysis of particulate polymers. Composites B2013; 54: 278–288.
49.
BalokhonovRRomanovaVSchmauderS. A mesomechanical analysis of plastic strain and fracture localization in a material with a bilayer coating. Composites B2014; 66: 276–286.
50.
KatzATrinhCWrightJ. Plastic strain localization in periodic materials with wavy brick-and-mortar architectures and its effect on the homogenized response. Composites B2015; 68: 270–280.
51.
FreundJKarakocASjolundJ.Computational homogenization of regular cellular material according to classical elasticity. Mech Mater2014; 78: 56–65.
52.
TsalisDBaxevanisTChatzigeorgiouG. Homogenization of elastoplastic composites with generalized periodicity in the microstructure. Int J Plasticity2013; 51: 161–187.
TartarL.Nonlinear constitutive relations and homogenization. In: Contemporary Developments in Continuum Mechanics and Partial Differential Equations. Amsterdam: North-Holand, 1978.
55.
SuquetPM.Elements of homogenization for inelastic solid mechanics. Technical Report 85-2, Université des Sciences et Techniques du Languedoc, Montpellier, 1985.
56.
ZesersAKruminsJ.Surface properties of a hooked steel fiber and their effects on the fiber pullout and composite cracking 1. experimental study. Mech Compos Mater2014; 50: 437–446.
57.
KhatamHPinderaM.Parametric finite-volume micromechanics of periodic materials with elastoplastic phases. Int J Plasticity2009; 25: 1386–1411.
