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IMA Journal of Applied Mathematics Advance Access originally published online on March 11, 2009
IMA Journal of Applied Mathematics 2009 74(4):507-532; doi:10.1093/imamat/hxp001
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© The Author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

On the homogenization of orthotropic elastic composites by the strong-property-fluctuation theory

Andrew J. Duncan{dagger} and Tom G. Mackay{ddagger}

School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh EH9 3JZ, UK

Akhlesh Lakhtakia§

NanoMM—Nanoengineered Metamaterials Group, Department of Engineering Science and Mechanics, Pennsylvania State University, University Park, PA 16802-6812, USA

{dagger} Email: andrew.duncan{at}ed.ac.uk

{ddagger} Corresponding author. Email: t.mackay{at}ed.ac.uk

§ Email: akhlesh{at}psu.edu

Received on March 27, 2008; Revision received November 26, 2008. Accepted on January 14, 2009

The strong-property-fluctuation theory (SPFT) provides a general framework for estimating the constitutive parameters of a homogenized composite material (HCM). We developed the elastodynamic SPFT for orthotropic HCMs in order to undertake numerical studies. A specific choice of two-point covariance function—which characterizes the distributional statistics of the generally ellipsoidal particles that constitute the component materials—was implemented. Representative numerical examples revealed that the lowest-order SPFT estimate of the HCM stiffness tensor is qualitatively similar to the estimate provided by the Mori–Tanaka mean-field formalism, but the differences between the two estimates vary as the orthotropic nature of the HCM is accentuated. The second-order SPFT provides a correction to the lowest-order estimate of the HCM stiffness tensor and density. The correction, indicating effective dissipation due to scattering loss, increases as the HCM becomes less orthotropic but decreases as the correlation length becomes smaller.

Keywords: homogenization; strong-property-fluctuation theory; metamaterials; Mori–Tanaka mean-field formalism.


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