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Research Papers

A Locally Exact Homogenization Theory for Periodic Microstructures With Isotropic Phases

[+] Author and Article Information
Anthony S. Drago, Marek-Jerzy Pindera

Civil Engineering Department, University of Virginia, Charlottesville, VA 22904-4742

J. Appl. Mech 75(5), 051010 (Jul 17, 2008) (14 pages) doi:10.1115/1.2913043 History: Received September 27, 2007; Revised February 05, 2008; Published July 17, 2008

Elements of the homogenization theory are utilized to develop a new micromechanics approach for unit cells of periodic heterogeneous materials based on locally exact elasticity solutions. The interior inclusion problem is exactly solved by using Fourier series representation of the local displacement field. The exterior unit cell periodic boundary-value problem is tackled by using a new variational principle for this class of nonseparable elasticity problems, which leads to exceptionally fast and well-behaved convergence of the Fourier series coefficients. Closed-form expressions for the homogenized moduli of unidirectionally reinforced heterogeneous materials are obtained in terms of Hill’s strain concentration matrices valid under arbitrary combined loading, which yield homogenized Hooke’s law. Homogenized engineering moduli and local displacement and stress fields of unit cells with offset fibers, which require the use of periodic boundary conditions, are compared to corresponding finite-element results demonstrating excellent correlation.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

(a) Periodically arranged inclusions in a square array and (b) RUC with an offset fiber that is the fundamental building block for the entire array

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Figure 2

Convergence of the predicted homogenized moduli with the number of harmonics used in the displacement field representation relative to the finite-element results for a unit cell with an off-center fiber for fiber volume fractions of 0.05, 0.35, and 0.60 with Ef∕Em=10 moduli ratio: (a) E22*, (b) G23*, (c) ν23*, and (d) G12*

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Figure 3

Comparison of the locally exact predictions for the effective moduli as a function of the fiber volume fraction with the finite-element calculations for unidirectional composites with Ef∕Em=10 moduli ratio: (a) E22*, (b) G23*, (c) ν23*, and (d) G12*

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Figure 4

Comparison of the locally exact predictions for the effective moduli as a function of the fiber volume fraction with the finite-element calculations for unidirectional composites with Ef∕Em=10−6 moduli ratio: (a) E22*, (b) G23*, (c) ν23*, and (d) G12*

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Figure 5

(a) Convergence of the coefficients Fnjf with the number of harmonics for a unit cell with an off-center fiber, vf=0.35, and Ef∕Em=10 moduli ratio subjected to loading by ε¯22 only and (b) amplitude (×10−3) of 20 coefficients (left) and the convergence of ε¯22 with the number of harmonics (right)

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Figure 6

Convergence of the predicted local displacement fields with the number of harmonic terms used in the displacement field representation relative to the finite-element results under loading by ε¯22 only for a unit cell with an off-center fiber and Ef∕Em=10 moduli ratio: (a) u2 and (b) u3

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Figure 7

Convergence of the predicted local stress fields with number of harmonics used in the displacement field representation relative to the finite-element results under loading by ε¯22 only for a unit cell with an off-center fiber and Ef∕Em=10 moduli ratio: (a) σ22, (b) σ23, and (c) σ33

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Figure 8

Average solution time as a function of the number of harmonic terms in the displacement field representation in a unit cell with an off-center fiber and Ef∕Em=10 moduli ratio loaded by ε¯22 only

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Figure 9

Comparison of the converged local stress fields predicted by the locally exact homogenization theory with the finite-element results for loading by ε¯23 only for a unit cell with an off-center fiber and Ef∕Em=10 moduli ratio: (a) σ23, (b) σ22, and (c) σ33

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Figure 10

Comparison of the converged local stress fields predicted by the locally exact homogenization theory with the finite-element results for loading by ε¯13 only for a unit cell with an off-center fiber and Ef∕Em=10 moduli ratio: (a) σ13 and (b) σ12

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Figure 11

Convergence of the predicted homogenized moduli with the number of harmonics used in the displacement field representation relative to the finite-element results for a unit cell with an off-center fiber, fiber volume fraction νf=0.35, and Ef∕Em=10 moduli ratio using collocation-based periodic boundary conditions

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