Overall dynamic constitutive relations of layered elastic composites

Abstract

A method for homogenization of a heterogeneous (finite or periodic) elastic composite is presented. It allows direct, consistent, and accurate evaluation of the averaged overall frequency-dependent dynamic material constitutive relations. It is shown that when the spatial variation of the field variables is restricted by a Bloch-form (Floquet-form) periodicity, then these relations together with the overall conservation and kinematical equations accurately yield the displacement or stress modeshapes and, necessarily, the dispersion relations. It also gives as a matter of course point-wise solution of the elasto-dynamic field equations, to any desired degree of accuracy. The resulting overall dynamic constitutive relations however, are general and need not be restricted by the Bloch-form periodicity. The formulation is based on micro-mechanical modeling of a representative unit cell of the composite proposed by Nemat-Nasser and coworkers; see, e.g., [1] and [2].We show that, for a micro-structured elastic composite, the overall effective massdensity and compliance (stiffness) are always real-valued and positive, whether or not the corresponding unit cell (representative volume element used as a unit cell) is geometrically and/or materially symmetric. The average strain and linear momentum are however couple and the coupling constitutive parameters are always each others complex conjugates for any heterogeneous elastic unit cell, such that the overall energy-density is always real and positive. In this paper, we have sought to separate the overall constitutive relations which should depend only on the composition and structure of the unit cell, from the overall field equations which should hold for any elastic composite; i.e., we use only the local field equations and material properties to deduce the overall constitutive relations.It is shown, by way of an example of a bi-layered composite, that dispersion curves obtained by our method accurately produce the exact results of Rytov [3]. The method is also used to calculate the effective parameters for a 2-layered composite and the results are compared with those of homogenization based on the field integration of the exact solution (Willis [4], and Nemat-Nasser et al. [5]), and certain relevant issues are clarified. Finally the method is used to homogenize both a symmetric and a non-symmetric 4-layered composite and the results for the symmetric case are compared with those reported by Nemat-Nasser et al. as the exact solution. Thus, this method provides a powerful solution and homogenization tool to use in many cases where the unit cell contains inclusions of complex geometry.

Elastic metamaterials and dynamic homogenization: a review

“…The above relations hold generally for Bloch waves in three-dimensional linear periodic composites and are in congruence with effective dynamic properties for electromagnetic Bloch/Floquet waves [95]. In one dimension these relations simplify further [96]:…”

“…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.…”