Effective Dynamic Constitutive Relations for 3-D Periodic Elastic Composites
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Effective Dynamic Constitutive Relations for 3-D Periodic Elastic Composites Ankit Srivastava and Sia Nemat-Nasser Department of Mechanical and Aerospace Engineering University of California, San Diego La Jolla, CA, 92093-0416 USA
ABSTRACT Central to the idea of metamaterials is the concept of dynamic homogenization which seeks to define frequency dependent effective properties for Bloch wave propagation. Recent advances in the theory of dynamic homogenization have established the coupled form of the constitutive relation (Willis constitutive relation). This coupled form of the constitutive relation naturally emerges from ensemble averaging of the dynamic fields and automatically satisfies the dispersion relation in the case of periodic composites. Its importance is also notable due to its invariance under transformational acoustics. Here we discuss the explicit form of the effective dynamic constitutive equations. We elaborate upon the existence and emergence of coupling in the dynamic constitutive relation and further symmetries of the effective tensors.
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INTRODUCTION
Recent interest in the character of the overall dynamic properties of composites with tailored microstructure necessitates a systematic homogenization procedure to express the dynamic response of an elastic composite in terms of its average effective compliance and density. A thorough review of advances in the field of dynamic homogenization for the electromagnetic case can be found in [1]. The elastostatic response of composites has been long understood to be non-local in space [2-5] but in the context of inhomogeneous elastodynamics, the effective constitutive relations are non-local in both space and time [6,7]. Homogenization for calculating these overall dynamic properties of composites, based on the integration of the field variables, has been proposed by a number of researchers. For electromagnetic waves, see, for example [8-11]. For elastodynamic waves [12] has presented a homogenization method based on an ensemble averaging technique of the ’Bloch’ reduced form of the wave propagating in a periodic composite; see also [13,14]. In the present paper we discuss the explicit form of the effective dynamic constitutive equations. We elaborate upon the existence and emergence of coupling in the dynamic constitutive relation and its dependence upon the architectural symmetries of the unit cell. We show that the averaged dynamic constitutive parameters are tensorial in nature and that the average strain tensor is coupled with the average momentum tensor. Such a form of the averaged constitutive relation where the constitutive parameters (including mass density) are tensors and where the average strain (average Send correspondence to Ankit Srivastava: E-mail: [email protected]
stress) is coupled with average linear momentum has been predicted in the literature [12,14-16] and references cited therein). In general, the effective properties for the dynamic problem are not uniquely determined but they become unique in the presence of incompatible s
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