In general, asymptotic homogenization methods (AHMs) are based on the hypothesis of perfect scale separation. In practice, this is not always the case. The problem arises of improving the solution in such a way that it becomes applicable if the inhomogeneity parameter is not small. Our study focuses on the higher order asymptotic homogenization for dynamical problems. Systems with continuous and piecewise continuous parameters, discrete systems, and also continuous systems with discrete elements are considered. Both low-frequency and high-frequency vibrations are analyzed. For low-frequency vibrations, several approximations of the AHM are constructed. The influence of the boundary conditions and the system parameters is investigated. The accuracy of the resulting approximation is assessed in numerical simulations.
BoutinC. Microstructural influence on heat conduction. Int J Heat Mass Transfer1995; 38(17): 3181–3195.
2.
AmeenMMPeerlingsRHJGeersMGD. Higher-order asymptotic homogenization of periodic materials with low scale separation. In: ECCOMAS Congress 2016, VII European Congress on Computational Methods in Applied Science Engineering. PapadrakakisMPapadopoulosVStefanouGPlevrisV (eds.). Crete Island, Greece, 5–10.06 2016: 2346–2352.
3.
AmeenMMPeerlingsRHJGeersMGD. A quantitative assessment of the scale separation limits of classical and higher-order asymptotic homogenization. Europ J Mech.-A/Solids2018; 71: 89–100.
4.
AmeenMM. Scale-dependent homogenization of heterogeneous microstructures: application to mechanical metamaterials. PhD Thesis, Technische Univ. Eindhoven, 2019.
BakhvalovNPanasenkoG. Homogenisation: Averaging processes in periodic media. mathematical problems in the mechanics of composite materials. Dordrecht: Kluwer, 1989.
8.
BensoussanALionsJ-LPapanicolaouG. Asymptotic analysis for periodic structures. Amsterdam: North-Holland, 1978.
9.
AndrianovIVLesnichayaVAManevitchLI. Homogenization methods in statics and dynamics of ribbed shells. Moscow, Nauka, 1985 (in Russian).
10.
KesavanS. Homogenization of elliptic eigenvalue problems. I. Appl Math Optim1979; 5(2): 153–167.
11.
KesavanS. Homogenization of elliptic eigenvalue problems. II. Appl Math Optim1979; 5(3): 197–216.
12.
SantosaFVogeliusM. First-order corrections to the homogenized eigenvalues of periodic composite medium. SIAM J Appl Math1993; 53(6): 1636–1668; Erratum: SIAM J App. Math1995; 55(3): 684.
13.
MoskowShVogeliusM. First order corrections to the homogenized eigenvalues of a periodic composite medium. The case of Neumann boundary conditions. Preprint, 1996.
14.
MoskowShVogeliusM. First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof. Proc R Soc Edinb A Math1997; 127(6): 1263–1299.
15.
ChenWFishJ. A dispersive model for wave propagation in periodic heterogeneous media based on homogenization with multiple spatial and temporal scales. J Appl Mech2001; 68: 153–161.
16.
SunXZhouChIchchouMLainéJ-PZineA-M. Multi-scale homogenization of transversal waves in periodic composite beams. Int J Appl Mech2017; 9(3): 1750039.
17.
WangQVaradanVKQuekST. Small scale effect on elastic buckling of carbon nanotubes with nonlocal continuum models. Phys Lett A2006; 357(2): 130–135.
18.
AndrianovIVAwrejcewiczJDanishevskyyVV. Linear and nonlinear waves in microstructured solids: homogenization and asymptotic approaches. Boca Raton: CRC Press, Taylor & Francis, 2021.
19.
ManevitchLIOshmyanVG. An asymptotic study of the linear vibrations of a stretched beam with concentrated masses and discrete elastic supports. J Sound Vibr1999; 223(5): 679−691.
20.
ManevitchLIAndrianovIVOshmyanVO. Mechanics of periodically heterogeneous structures. Berlin: Springer-Verlag, 2002.
21.
CrasterRVJosephLMKaplunovJ. Long-wave asymptotic theories: the connection between functionally graded waveguides and periodic media. Wave Motion2014; 51(4): 581–588.
22.
GoldenveizerAL.KaplunovJDNoldeEV. On Timoshenko-Reissner type theories of plates and shells. Int J Solids Struct1993; 30(5): 675–694.
23.
KaplunovJDNoldeEVShorrBF. A perturbation approach for evaluating natural frequencies of moderately thick elliptic plates. J Sound Vib2005; 281(3-5): 905–919.
24.
PichuginAVAskesHTyasA. Asymptotic equivalence of homogenisation procedures and fine-tuning of continuum theories. J Sound Vib2008; 313(3-5): 858–874.
25.
CastroCZuazuaE. Low frequency asymptotic analysis of a string with rapidly oscillating density. SIAM J Appl Math2000; 60: 1205–1233.
26.
CastroCZuazuaE. High frequency asymptotic analysis of a string with rapidly oscillating density. Eur J Appl Math2000; 11(6): 595–622.
27.
ZevinAA. Estimates of eigenvalue of self-adjoint boundary-value problems with periodic coefficients. Ukr Math J1998; 50: 719–722.
28.
MovchanABMovchanNVPoultonCG. Asymptotic models of fields in dilute and densely packed composites. London: Imperial College Press, 2002.
29.
ColquittDJDanishevskyyVVKaplunovJ. Composite dynamic models for periodically heterogeneous media. Math Mech Solids2019; 24(9): 2663–2693.
30.
GoupillaudPL. An approach to inverse filtering of near-surface layer effects from seismic records. Geophysics1961; 26(6): 754–760.
