Higher‐Order Terms in the Homogenized Stress‐Strain Relation of Periodic Elastic Media

Abstract

Applying the method of two-scale asymptotic expansions to the displacement field in periodic elastic structures the completed macroscopic stress-strain relation of such media is derived. This relation contains beside the usually considered local part also weakly nonlocal contributions in the form of strain gradients up to infinite order. Formulae for the determination of the pertaining homogenized material tensors are derived.Unter Benntzung der Methode der asymptotischen Entwicklung fur die Verschiebungsfelder in periodischen elastischen Strukturen wird die vollstandige makroskopische Spannungs-Dehnungs-Beziehung solcher Medien abgeleitet. Diese Beziehung enthalt neben dem iiblicherweise beriicksichtigten lokalen Anteil auch schwach nichtlokale Beitrage in Form von Spannungsgradienten bis zur unendlichen Ordnung. Formeln fur die Bestimmung der zugehorig homogenisierten Materialtensoren werden abgeleitet. Ba.sic EquationsI n the theory of homogenization of microheterogeneous media with periodically distributed elasticity [l to 41 the linear operator L"(rC) = -vz * c"(5)v, Pfaffenwaldring 57, D-7000 Stuttgart 80, FRG. 33*

Non‐local dispersive model for wave propagation in heterogeneous media: one‐dimensional case

“…} is applied; a 0 = 1= 4 and h(x) denote the Heaviside step function; f 0 , x 0 and are the magnitude, the location of the maximum value and the half-width of the initial pulse, respectively. The pulse is similar in shape to the Gaussian distribution function.…”

“…The material properties of the two constituents are E 1 = 200 GPa, E 2 = 5 GPa, 1 = 2 = 8000 kg=m 3 , and = 0:5. The calculated homogenized material properties are: E 0 = 9:76 GPa, The bar is subjected to an impact load q(t) = q 0 a 0 t 4 (t − T ) 4 [1 − h(t − T )] at l = 1 m, where T is the duration of the impact pulse, q 0 = −50 K N and a 0 is scaled in such a way that 06q(t)6q 0 . The time-varying displacements at x = 0:5 m are plotted in Figures 4 and 5, which correspond to pulse duration T = 62:83 and 15:71 s, respectively.…”

“…The role of higher-order terms in the asymptotic expansion has been investigated in statics by Gambin and Kroner [4], and Boutin [5]. In elastodynamics, Boutin and Auriault [6] demonstrated the terms of a higher-order successively introduced e ects of polarization, dispersion and attenuation.…”

Thermo‐mechanical analysis of periodic multiphase materials by a multiscale asymptotic homogenization approach

“…References [2][3][4][5]. Boutin [6]; Schrefler et al [7] and Gambin and Kroner [8] studied the role of higher-order terms in the multiscale asymptotic expansion in statics. Boutin and Auriault [9] investigated the effects of dispersion and attenuation introduced by higher-order terms in elastokinetics.…”

Higher order asymptotic homogenization and wave propagation in periodic composite materials

“…The higher order terms arising in static problems were considered by Gambin & Kröner (1989) and Boutin (1996). They have shown that the heterogeneity of the medium results in the induction of an infinite series of displacement fields with successively lower amplitudes.…”