Statistical Continuum Theories

Beran, Mark J.

doi:10.1122/1.548991

Cited by 153 publications

(221 citation statements)

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“…[1][2][3][4][5] One of the most important morphological descriptors is the volume fraction of the phases, which, in the case of porous media, is just the porosity ͑i.e., the volume fraction of the fluid phase͒. The volume fraction of twophase random media fluctuates on a spatially local level, even for statistically homogeneous media.…”

Section: Introductionmentioning

confidence: 99%

Local volume fraction fluctuations in random media

Quintanilla

Torquato

1997

The Journal of Chemical Physics

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Although the volume fraction is a constant for a statistically homogeneous random medium, on a spatially local level it fluctuates. We study the full distribution of volume fraction within an observation window of finite size for models of random media. A formula due to Lu and Torquato for the standard deviation or ''coarseness'' associated with the local volume fraction is extended for the nth moment of for any n. The distribution function F L of the local volume fraction of five different model microstructures is evaluated using analytical and computer-simulation methods for a wide range of window sizes and overall volume fractions. On the line, we examine a system of fully penetrable rods and a system of totally impenetrable rods formed by random sequential addition ͑RSA͒. In the plane, we study RSA totally impenetrable disks and fully penetrable aligned squares. In three dimensions, we study fully penetrable aligned cubes. In the case of fully penetrable rods, we will also simplify and numerically invert a prior analytical result for the Laplace transform of F L . In all of these models, we show that, for sufficiently large window sizes, F L can be reasonably approximated by the normal distribution.

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Section: Introductionmentioning

confidence: 99%

Local volume fraction fluctuations in random media

Quintanilla

Torquato

1997

The Journal of Chemical Physics

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“…8 -11, while perturbation techniques have been used in Refs. [12][13][14][15][16][17][18][19] to treat the random polycrystals. The mathematical methods using translations with null Lagrangians 20 and Padé approximants [21][22][23] appear also effective in constructing bounds for macroscopically isotropic aggregates.…”

Section: Introductionmentioning

confidence: 99%

Uncertainty ranges for the macroscopic resistivities and permeabilities of random polycrystalline aggregates

Chinh

2001

Phys. Rev. B

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The completely random polycrystals considered here are aggregates of the anisotropic grains, the crystalline and shape orientations of which are uncorrelated. Because of irregular shapes of the grains ͑uncertainty of the microgeometry͒, one expects some uncertainty in observed macroscopic behavior of the aggregates. Our recently derived bounds on the possible ranges for polycrystal transport properties, which are proportional coefficients relating solenoidal and irrotational vector fields, are applied to evaluate electrical resistivities, permeabilities, and thermal conductivities of a number of practical aggregates, and considered for possible improvements by an interpolation method. We argue that the statistical homogeneity and isotropy hypotheses for the random aggregates are valid only within some uncertainty, and our bounds can still provide measures of this uncertainty asymptotically.

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“…[1][2][3][4][5] The analysis and evaluation of the distribution of the local field ͑i.e., fluctuations of the local field͒ has received far less attention. Nonetheless, the distribution of the local field is of great fundamental and practical importance in understanding many crucial material properties such as the breakdown phenomenon 6,7 and the nonlinear behavior of composites.…”

Section: Introductionmentioning

confidence: 99%

Electric-field fluctuations in random dielectric composites

Cheng

Torquato

1997

Phys. Rev. B

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When a composite is subjected to a constant applied electric, thermal, or stress field, the associated local fields exhibit strong spatial fluctuations. In this paper, we evaluate the distribution of the local electric field ͑i.e., all moments of the field͒ for continuum ͑off-lattice͒ models of random dielectric composites. The local electric field in the composite is calculated by solving the governing partial differential equations using efficient and accurate integral equation techniques. We consider three different two-dimensional dispersions in which the inclusions are either ͑i͒ circular disks, ͑ii͒ squares, or ͑iii͒ needles. Our results show that in general the probability density function associated with the electric field for disks and squares exhibits a double-peak character. Therefore, the variance or second moment of the field is inadequate in characterizing the field fluctuations in the composite. Moreover, our results suggest that the variances for each phase are generally not equal to each other. In the case of a dilute concentration of needles, the probability density function is a singly peaked one, but the higher-order moments are appreciably larger for needles than for either disks or squares.

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Statistical Continuum Theories

Cited by 153 publications

References 0 publications

Local volume fraction fluctuations in random media

Local volume fraction fluctuations in random media

Uncertainty ranges for the macroscopic resistivities and permeabilities of random polycrystalline aggregates

Electric-field fluctuations in random dielectric composites

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