In this work, the two-scale asymptotic homogenization method (AHM) is developed to describe the effective behavior of multi-laminated elastic micropolar composites with Fibonacci and random structure under perfect contact conditions at the interfaces. The local problem statements over the periodic cell are presented, and the corresponding effective stiffness and torque properties are reported. The transversal cross-section of the periodic cell is characterized by a laminated structure where the pattern for the layers follows two distinct configurations: (a) a Fibonacci arrangement, and (b) a random sequence focused on the probabilistic binomial function. The non-null effective properties of multi-laminated Cosserat elastic composites with isotropic centro-symmetric constituents are listed. Numerical results for multi-laminated elastic micropolar composites with both types of structures and centro-symmetric isotropic constituents are illustrated and discussed. The overall effective behavior for both cases converges to specific effective values of periodic structures as the number of layers increases.
AddessiDSaccoE.A multi-scale enriched model for the analysis of masonry panels. Int J Solids Struct2012; 49: 865–880.
2.
MollonGZhaoJ.3D generation of realistic granular samples based on random fields theory and Fourier shape descriptors. Comput Methods Appl Mech Eng2014; 279: 46–65.
3.
VladMDFernández AguadoEGómez GonzálezS, et al. Novel titanium-apatite hybrid scaffolds with spongy bone-like micro architecture intended for spinal application: in vitro and in vivo study. Mater Sci Eng C2020; 110: 110658.
4.
DhariRSPatelNPWangH, et al. Numerical investigation of Fibonacci series based bio-inspired laminates under impact loading. Compos Struct2021; 255: 112985.
5.
ČernýMPetrušJPavliňákováV.A new approach to the structure-properties relationships determination for porous filled reinforced materials. Polymers2022; 14: 4390.
6.
XiaHZhangXKHuA, et al. Effective elastic constants and phonon spectrum in metallic ta/al quasiperiodic superlattices. Phys Rev B1993; 47(7): 3890–3895.
Guinovart-SanjuánDRodríguez-RamosRGuinovart-DíazR, et al. Effective properties of regular elastic laminated shell composite. Compos Part B Eng2016; 87: 12–20.
9.
RobertsAJ.Embed to rigorously and accurately homogenize quasi-periodic multi-scale heterogeneous PDEs, with computer álgebra. arXiv [preprint]2022; 2209.02822.
10.
PatraMMaitiS.Anomalous magnetic response of a quasi-periodic mesoscopic ring in presence of rashba and dresselhaus spin-orbit interactions. Eur Phys J B2016; 89: 88.
11.
PatraMMaitiS.Characteristics of persistent spin current components in a quasi-periodic fibonacci ring with spin-orbit interactions: prediction of spin-orbit coupling and on-site energy. Ann Phys2016; 375: 337–350.
12.
LiCZhangXCaoZ.Triangular and Fibonacci number patterns driven by stress on core/shell microstructures. Science2005; 309: 909–911.
DaviesP.The thermodynamic theory of black holes. Proc R Soc Lond A1977; 353: 499–521.
15.
DaviesP.Thermodynamics of black holes. Rep Prog Phys1978; 41: 1313–1355.
16.
CruzNOlivaresMVillanuevaJ.The golden ratio in Schwarzschild–Kottler black holes. Eur Phys J C2017; 77: 123.
17.
YuDXueDRatajczakH.Golden ratio and bond-valence parameters of hydrogen bonds of hydrated borates. J Mol Struct2006; 783: 210–214.
18.
ArneodoAArgoulFBacryE, et al. Fibonacci sequences in diffusion-limited aggregation. In: Garcia-RuizJMLouisEMeakinP, et al. (eds) Growth patterns in physical sciences and biology (NATO ASI series B), vol. 304. New York: Plenum Press, 1993, pp. 191–202.
19.
DouadySCouderY.Phyllotaxis as a dynamical self organizing process part I: the spiral modes resulting from time-periodic iterations. J Theor Biol1996; 178(3): 255–273.
20.
RivierNSadocJCharvolinJ.Phyllotaxis: a framework for foam topological evolution. Eur Phys J E2016; 39(1): 7.
21.
NégadiT.A mathematical model for the genetic code(s) based on Fibonacci numbers and their q-analogues. NeuroQuantology2015; 13(3): 259–272.
22.
KimJLeeJHamadaS, et al. Self-replication of DNA rings. Nat Nanotechnol2015; 10: 528–533.
LeeJJooKKimS, et al. Re-examination of structure optimization of off-lattice protein ab models by conformational space annealing. J Comput Chem2008; 29: 2479–2484.
25.
VaeziABarkeshliM.Fibonacci anyons from Abelian bilayer quantum hall states. Phys Rev Lett2014; 113: 236804.
26.
VaeziA.Superconducting analogue of the parafermion fractional quantum hall states. Phys Rev X2014; 4: 031009.
BajemaKMerlinR.Raman scattering by acoustic phonons in Fibonacci GaAs-AlAs superlattices. Phys Rev B1987; 36(8): 4555–4557.
29.
ToddJMerlinRClarkeR.Synchroton X-ray study of a Fibonacci superlattice. Phys Rev B1986; 57(9): 1157–1160.
30.
AlbuquerqueECottamM.Theory of elementary excitations in quasiperiodic structures. Phys Rep2003; 376: 225–337.
31.
AlbuquerqueECottamM.Polaritons in periodic and quasiperiodic structures. 1st ed. Amsterdam: Elsevier, 2004.
32.
GellermannWKohmotoMSutherlandB, et al. Theory of elementary excitations in quasiperiodic structures. Phys Rev Lett1994; 72(5): 633–636.
33.
CosseratECosseratF.Théorie des corps déformables. Paris: A. Hermann et Fils, 1909.
34.
AeroEKuvshinskiiE.Fundamental equations of the theory of elastic media with rotationally interacting particles. Sov Phys Solid State1961; 2: 1272–1281.
35.
EringenAC.Linear theory of micropolar elasticity. J Math Mech1966; 15(6): 909–923.
36.
EringenAC.Microcontinuum field theories I: foundations and solids. 1st ed. New York: Springer, 1999.
37.
StojanovićR.Mechanics of polar continua: theory and applications (CISM courses and lectures), vol. 2. Vienna: Springer-Verlag, 1969.
38.
StojanovićR.Recent developments in the theory of polar continua (CISM courses and lectures), vol. 27. Vienna: Springer-Verlag, 1969.
39.
StojanovićR.Nonlinear micropolar elasticity. In: NowackiWOlszakW (eds) Micropolar elasticity (International Centre for Mechanical Sciences), vol. 151. Vienna: Springer, 1974, pp. 73–103.
40.
NowackiW.The linear theory of micropolar elasticity. In: NowackiWOlszakW (eds) Micropolar elasticity (International Centre for Mechanical Sciences), 1st ed. Vienna: Springer, 1974, pp. 1–43.
41.
NowackiW.Theory of asymmetric elasticity. 1st ed. Moscow: Pergamon-Press, 1986.
42.
ForestS.Mechanics of generalized continua: construction by homogenization. J Phys IV1998; 8: 39–484.
43.
ForestS.Homogenization methods and mechanics of generalized continua - part 2. Theor Appl2002; 28–29: 113–144.
44.
ForestS.Micromorphic media. In: AltenbachHEremeyevV (eds) Generalized continua from the theory to engineering applications (CISM International Centre for Mechanical Sciences), vol. 541. Vienna: Springer, 2013, pp. 249–300.
45.
Ostoja-StarzewskiR.Microstructural randomness and scaling in mechanics of materials (Modern mechanics and mathematics series). Boca Raton, FL: Chapman & Hall/CRC, 2008.
46.
EremeyevVLeonidPAltenbachL.Foundations of micropolar mechanics. Berlin: Springer, 2013.
47.
AltenbachHEremeyevV.Generalized continua from the theory to engineering applications. Vienna: Springer, 2013.
48.
EremeyevVPietraszkiewiczW.Material symmetry group and constitutive equations of micropolar anisotropic elastic solids. Math Mech Solids2016; 21(2): 210–221.
49.
Dell’IsolaFEremeyevV.Some introductory and historical remarks on mechanics of microstructured materials. In: dell’IsolaFEremeyevVPorubovA (eds) Advances in mechanics of microstructured media and structures. Cham: Springer, 2018, pp. 1–20.
50.
TrovalusciPOstoja-StarzewskiMDe BellisM, et al. Scale-dependent homogenization of random composites as micropolar continua. Eur J Mech A Solids2015; 49: 396–407.
51.
TrovalusciPDe BellisMOstoja-StarzewskiM, et al. Particulate random composites homogenized as micropolar materials. Meccanica2014; 49: 2719–2727.
52.
Rodríguez-RamosRYanesVEspinosa-AlmeydaY, et al. Micro-macro asymptotic approach applied to heterogeneous elastic micropolar media. Analysis of some examples. Int J Solids Struct2022; 239–240: 111444.
53.
YanesVSabinaFJEspinosa-AlmeydaY, et al. Asymptotic homogenization approach applied to cosserat heterogeneous media. In: AndrianovIGluzmanSMityushevV (eds) Mechanics and physics of structured media. Cambridge, MA: Academic Press/Elsevier, 2022, pp. 459–491.
54.
ZhengQSSpencerAJM. On the canonical representation for Kronecker powers of orthogonal tensors with application to material symmetry problems. Int J Eng Sci1993; 31(4): 617–635.
BakhvalovNPanasenkoG.Homogenisation: averaging processes in periodic media (Mathematics and its applications Soviet series). 1st ed. Amsterdam: Kluwer Academic, 1989.
57.
EremeyevVPietraszkiewiczW.Material symmetry group of the non-linear polar-elastic continuum. Int J Solids Struct2012; 49(14): 1993–2005.
58.
HassanpourSHepplerGR.Micropolar elasticity theory: a survey of linear isotropic equations, representative notations, and experimental investigations. Math Mech Solids2015; 22: 224–242.
59.
LakesR.Experimental microelasticity of two porous solids pages. Int J Solids Struct1986; 22(1): 55–63.
60.
Rodríguez-RamosRSabinaFJGuinovart-DíazR, et al. Closed-form expressions for the effective coefficients of a fiber-reinforced composite with transversely isotropic constituents I. Elastic and square symmetry. Mech Mater2001; 33: 223–235.
61.
Guinovart-SanjuánDVajraveluKRodríguez-RamosR, et al. Effective predictions of heterogeneous flexoelectric multilayered composite with generalized periodicity. Int J Mech Sci2020; 181: 105755.
62.
Guinovart-SanjuánDVajraveluKRodríguez-RamosR, et al. Simple closed-form expressions for the effective properties of multilaminated flexoelectric composites. J Eng Math2021; 127: 4.
