Statistical Continuum Theories

Beran, Mark J.; Pytte, Agnar

doi:10.1119/1.1974326

Cited by 338 publications

(93 citation statements)

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“…These functions have been introduced by Brown [6], and important applications with respect to effective elastic moduli are given in e.g. Beran [3]. Let D (i) (α) denote the domain occupied by the considered phase i in the particular sample α, then the indicator function reads…”

Section: N-point Probability Functionsmentioning

confidence: 99%

“…Then, the description of the microstructure is based on statistical considerations, see the basic literature in e.g. Beran [3] and Kröner [11]. Due to the fact that we focus on two-phase microstructures, we concentrate on measures characterizing the inclusion phase, if it is not specified otherwise, since the measures computed for the matrix will be dependent in general.…”

Section: Statistical Measures For Microstructure Characterizationmentioning

confidence: 99%

“…Then, ellipsoids with the semi-axis r x and r y of predefined aspect ratios, here r x /r y = 14.3, and random semi-axis r x ∈ [3,6]µm are randomly placed in the space of interest. This process is stopped if a certain phase fraction of P V = 0.1872 is reached.…”

Section: Setup Of Virtual Experimentsmentioning

confidence: 99%

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Approximation of random microstructures by periodic statistically similar representative volume elements based on lineal-path functions

2010

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For the direct incorporation of micromechanical information into macroscopic boundary value problems, the FE 2 -method provides a suitable numerical framework. Here, an additional microscopic boundary value problem, based on evaluations of representative volume elements (RVEs), is attached to each Gauss point of the discretized macrostructure. However, for real random heterogeneous microstructures the choice of a "large" RVE with a huge number of inclusions is much too time-consuming for the simulation of complex macroscopic boundary value problems, especially when history-dependent constitutive laws are adapted for the description of individual phases of the mircostructure. Therefore, we propose a method for the construction of statistically similar RVEs (SSRVEs), which have much less complexity but reflect the essential morphological attributes of the microscale. If this procedure is prosperous, we arrive at the conclusion that the SSRVEs can be discretized with significantly less degrees of freedom than the original microstructure. The basic idea for the design of such SSRVEs is to minimize a least-square functional taking into account suitable statistical measures, which characterize the inclusion morphology. It turns out that the combination of the volume fraction and the spectral density seems not to be sufficient. Therefore, a hybrid reconstruction method, which takes into account the lineal-path function additionally, is proposed that yields promising realizations of the SSRVEs. In order to demonstrate the performance of the proposed procedure, we analyze several representative numerical examples.

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Section: N-point Probability Functionsmentioning

confidence: 99%

Section: Statistical Measures For Microstructure Characterizationmentioning

confidence: 99%

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Approximation of random microstructures by periodic statistically similar representative volume elements based on lineal-path functions

2010

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“…In addition, one must sometimes deal with structural changes that occur during the diffusion process (Gavalas and Kim, 1981;Bhatia and Perlmutter, 1983). The theoretical approaches are as diverse as the problems and they include the 'dusty gas' (Mason and Malinauskas, 1983), random sphere models (Strieder and Aris, 1973), nonuniform capillary tube models (Foster and Butt, 1966), random capillary tube models (Johnson and Stewart, 1965;Gavalas and Kim, 1981), general statistical methods (Beran, 1968;Batchelor, 1974), effective medium theory (Webman, 1982) and multiscale methods (Bensoussan et al, 1978;Chang, 1982). The theory used in this study is based on the method of volume averaging (Anderson and Jackson, 1967;Marie, 1967;Slattery, 1967;Whitaker, 1967) and the closure scheme developed by Ryan et al (1980Ryan et al ( , 1981.…”

Section: Introductionmentioning

confidence: 99%

Diffusion in anisotropic porous media

1987

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An experimental system was constructed in order to measure the two distinct components of the effective diffusivity tensor in transversely isotropic, unconsolidated porous media. Measurements were made for porous media consisting of glass spheres, mica particles, and disks made from mylar sheets. Both the particle geometry and the void fraction of the porous media were determined experimentally, and theoretical calculations for the two components of the effective diffusivity tensor were carried out. The comparison between theory and experiment clearly indicates that the void fraction and particle geometry are insufficient to characterize the process of diffusion in anisotropic porous media. ~K Ave a b CA r y C De~ DyyDeft f h interfacial area between y-and r-phases for the macroscopic system, m 2 area of entrances and exits of the r-phase for the macroscopic system, m: interfacial area contained within the averaging volume, m 2 characteristic length of a particle, m average thickness of a particle, m concentration of species A, moles/m 3 reference concentration of species A, moles/m 3 intrinsic phase average concentration of species A, moles/m 3 ca-(CA) v, spatial deviation concentration of species A, moles/m 3 (CA)Y/Co, dimensionless concentration of species A binary molecular diffusion coefficient, m2/s effective diffusivity tensor, m2/s component of the effective diffusivity tensor associated with diffusion parallel to the bedding plane, m2/s component of the effective diffusivity tensor associated with diffusion perpendicular to the bedding plane, m2/s effective diffusivity for isotropic systems, m2/s vector field that maps V (CA) ~ on to ca, m depth of the mixing chamber, m "*Current address:

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“…The elastostatic response of composites has been long understood to be non-local in space (Beran 1968;Willis 1983;Diener et al 1984;Bakhvalov & Panasenko 1989), but in the context of inhomogeneous elastodynamics, the effective constitutive relations are non-local in both space and time (Willis 1981a,b). Homogenization for calculating these overall dynamic *Author for correspondence (ansrivas@ucsd.edu).…”

Section: Introductionmentioning

confidence: 99%

Overall dynamic properties of three-dimensional periodic elastic composites

Srivastava

Nemat‐Nasser

2011

Proc. R. Soc. A.

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This article presents a method for the homogenization of three-dimensional periodic elastic composites. It allows for the evaluation of the averaged overall frequencydependent dynamic material constitutive tensors relating the averaged dynamic field variable tensors of velocity, strain, stress and linear momentum. Although the form of the dynamic constitutive relation for three-dimensional elastodynamic wave propagation has been known, this is the first time that explicit calculations of the effective parameters (for three dimensions) are presented. We show that for three-dimensional periodic composites, the overall compliance (stiffness) tensor, as produced directly by our formulation, is Hermitian, regardless of whether the corresponding unit cell is geometrically or materially symmetric. Overall, mass density is shown to be a tensor and, like the overall compliance tensor, always Hermitian. The average strain and linear momentum tensors are, however, coupled, and the coupling tensors are shown to be each others' Hermitian transpose. Finally, we present a numerical example of a three-dimensional periodic composite composed of elastic cubes periodically distributed in an elastic matrix. The presented results corroborate the predictions of the theoretical treatment illustrating the frequency dependence of the constitutive parameters. We also show that the effective properties calculated in this paper satisfy the dispersion relation of the composite.

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Statistical Continuum Theories

Cited by 338 publications

References 0 publications

Approximation of random microstructures by periodic statistically similar representative volume elements based on lineal-path functions

Approximation of random microstructures by periodic statistically similar representative volume elements based on lineal-path functions

Diffusion in anisotropic porous media

Overall dynamic properties of three-dimensional periodic elastic composites

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