Relation Between Microscopic and Macroscopic Mechanical Properties in Random Mixtures of Elastic Media

[+] Author and Article Information
Stefano Giordano

Department of Biophysical and Electronic Engineering (DIBE), University of Genoa, Via Opera Pia 11a, 16145 Genoa, Italy1stefgiord14@libero.it


New address to be used for all the correspondences: Department of Physics, University of Cagliari, Cittadella Universitaria, I-09042 Monserrato (Cagliari), Italy. E-mail: stefano.giordano@dsf.unica.it and stefgiord14@libero.it

J. Eng. Mater. Technol 129(3), 453-461 (Sep 25, 2006) (9 pages) doi:10.1115/1.2400282 History: Received June 28, 2006; Revised September 25, 2006

A material composed of a mixture of distinct homogeneous media can be considered as a homogeneous one at a sufficiently large observation scale. In this work, the problem of the elastic mixture characterization is solved in the case of linear random mixtures, that is, materials for which the various components are isotropic, linear, and mixed together as an ensemble of particles having completely random shapes and positions. The proposed solution of this problem has been obtained in terms of the elastic properties of each constituent and of the stoichiometric coefficients. In other words, we have explicitly given the features of the micro-macro transition for a random mixture of elastic material. This result, in a large number of limiting cases, reduces to various analytical expressions that appear in earlier literature. Moreover, some comparisons with the similar problem concerning the electric characterization of random mixtures have been drawn. The specific analysis of porous random materials has been performed and largely discussed. Such an analysis leads to the evaluation of the percolation threshold, to the determination of the convergence properties of Poisson’s ratio, and to good agreements with experimental data.

Copyright © 2007 by American Society of Mechanical Engineers
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Figure 4

Young’s modulus of different ceramic oxides. In (a) the properties of holmium oxide, Ho2O3 (holmia, (23), diamonds), and those of ytterbium oxide, Yb2O3 (ytterbia, (24), plus), are represented. Moreover, in (b) the properties of yttrium oxide, Y2O3 (yttria, (23), triangles), and those of samarium oxide, Sm2O3 (samaria, (25), circles), are reported.

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Figure 3

Bulk modulus k in GPa of sintered glass (circles), measured (see Ref. 22) for different values of the porosity c, compared with data obtained by Eq. 33 (solid line). Bulk and shear moduli of the pure glass were measured to be k1=46.06GPa and μ1=29.24GPa.

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Figure 2

Poisson’s ratio for a random porous material versus matrix Poisson’s ratio and porosity. Dashed lines correspond to the two-dimensional case, Eq. 30, continuous lines correspond to the three-dimensional case, Eq. 36. In both cases 15 different values of the matrix Poisson’s ratio are considered ranging from −1 to 1∕2. The intercepts of the lines at zero porosity (c=0) correspond to the solid matrix Poisson’s ratio. Note the convergence of the equivalent Poisson’s ratio for c=1∕3 to the value v=1∕3 (2D) and for c=1∕2 to the value v=1∕5 (3D).

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Figure 1

Young’s modulus for a random porous material versus the elastic moduli of the pure matrix and the porosity. Dashed line corresponds to the two-dimensional case, Eq. 29, continuous lines correspond to the three-dimensional case, Eq. 35. The two-dimensional result is independent on the matrix Poisson’s ratio while the three-dimensional result is parameterized by the matrix Poisson’s ratio (15 different values uniformly spaced ranging from −1 to 1∕2) but is practically independent on it in the range from 0 to 1∕2.



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