Overall dynamic properties of three-dimensional periodic elastic composites

Abstract

This article presents a method for the homogenization of three-dimensional periodic elastic composites. It allows for the evaluation of the averaged overall frequencydependent dynamic material constitutive tensors relating the averaged dynamic field variable tensors of velocity, strain, stress and linear momentum. Although the form of the dynamic constitutive relation for three-dimensional elastodynamic wave propagation has been known, this is the first time that explicit calculations of the effective parameters (for three dimensions) are presented. We show that for three-dimensional periodic composites, the overall compliance (stiffness) tensor, as produced directly by our formulation, is Hermitian, regardless of whether the corresponding unit cell is geometrically or materially symmetric. Overall, mass density is shown to be a tensor and, like the overall compliance tensor, always Hermitian. The average strain and linear momentum tensors are, however, coupled, and the coupling tensors are shown to be each others' Hermitian transpose. Finally, we present a numerical example of a three-dimensional periodic composite composed of elastic cubes periodically distributed in an elastic matrix. The presented results corroborate the predictions of the theoretical treatment illustrating the frequency dependence of the constitutive parameters. We also show that the effective properties calculated in this paper satisfy the dispersion relation of the composite.

Elastic metamaterials and dynamic homogenization: a review

“…It can be shown that when the averaging is performed over the Ω periodic parts of the field variables then the resulting effective properties automatically satisfy the dispersion relation of the periodic composite [93]. For periodic composites, the homogenized effective constitutive relation (Equation (15)) can be written in the tensorial form [94]:…”

“…0. Explicit calculations of effective properties in three-and one-dimensional periodic composites are provided in Refs [92,94,96].…”

“…As a consequence of the nonuniqueness of the effective dynamic constitutive relations, Equation (21), in one dimension, can be transformed so as to subsume the effects of S eff ; " S eff into a modified set of effective modulus and density parameters [93][94][95][96][97][98][99][100]:…”

Negative effective dynamic mass-density and stiffness: Micro-architecture and phononic transport in periodic composites

“…[31][32][33][34][35][36] Subsequent efforts have led to effective property definitions which satisfy both the averaged field equations and the dispersion relation of the composite. [37][38][39][40][41][42] In the present paper, we show that the effective properties thus defined become negative in the presence of local resonances, thereby capturing the dynamic effect first anticipated by Veselago 24 for the electromagnetic waves. Although the current treatment concerns 1-D composites, it is expected that the physical intuition gained will help to design 3-D composites with extreme material property profiles.…”

Evanescent wave boundary layers in metamaterials and sidestepping them through a variational approach

“…Currently, there are two main ways of doing this. The first is based upon asymptotic methods [8][9][10][11][12][13][14][15][16] and the second is based upon field averaging methods [17][18][19][20][21][22][23][24][25][26][27]. In addition to these, there are also scattering measurements based analytical and experimental techniques.…”