On the effective conductivity of polycrystals and a three-dimensional phase-interchange inequality

Avellaneda, Marco; Cherkaev, Andrej; Lurie, Konstantin A.; Milton, Graeme W.

doi:10.1063/1.340445

Cited by 115 publications

(110 citation statements)

References 24 publications

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“…It will be shown here that polycrystals with statistically isotropic microstructures (which have isotropic overall behaviour even in the nonlinear case) form only a subclass of the set of all linear polycrystals with isotropic overall behaviour. This will help explain why the bound derived using the linear comparison method in conjunction with the linear bound of Avellaneda et al (1988) turns out to be less sharp than the bound derived directly using the translation method.…”

Section: Introductionmentioning

confidence: 99%

“…each crystal in the LCC can, in principle, have different properties and is not simply a rotated version of the same basic crystal). In this section, we recall the generalized HS bounds and SC estimates of Willis (1977) for the effective conductivity of linear composites and polycrystals and compare them with the Voigt UB and the LB of Avellaneda et al (1988) for isotropic polycrystals. Before proceeding, however, it is useful to emphasize that the Willis and the Avellaneda et al estimates make different assumptions about the polycrystal microstructures.…”

Section: Effective Conductivity Of Linear Polycrystalsmentioning

confidence: 99%

“…Note that there are alternative derivations of these results for 'cell polycrystals' (see Milton 2002, §23.10). Also, Helsing (1993) In a separate development, Avellaneda et al (1988) made use of the translation method to obtain an LB for the effective conductivity of polycrystals. In contrast with the Willis estimates, this bound involves no hypothesis whatsoever about the microstructures involved.…”

Section: Effective Conductivity Of Linear Polycrystalsmentioning

confidence: 99%

“…In their paper, Garroni & Kohn (2003) show that the UB for effective resistivity derived using the translation method gives a superior scaling law than a bound derived using the linear comparison method in conjunction with the bound by Avellaneda et al (1988), which is optimal for linear isotropic polycrystals. It will be shown here that polycrystals with statistically isotropic microstructures (which have isotropic overall behaviour even in the nonlinear case) form only a subclass of the set of all linear polycrystals with isotropic overall behaviour.…”

Section: Introductionmentioning

confidence: 99%

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Linear comparison estimates for the effective resistivity of three-dimensional nonlinear polycrystals

Racherla

Castañeda

2008

Proc. R. Soc. A.

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Estimates for the effective resistivity of nonlinear polycrystals are obtained using the 'linear comparison' homogenization scheme of DeBotton and Ponte Castañeda (DeBotton & Ponte Castañeda 1995 Proc. R. Soc. A 448, 121-142). Computing the effective properties of linear composites, with the same microstructure as the nonlinear composite, is an essential part of this scheme. The classical self-consistent method is employed for this purpose. An important characteristic of these estimates, for polycrystals with field thresholds, is that they satisfy the recent bound of Garroni and Kohn (Garroni & Kohn 2003 Proc. R. Soc. A 459, 2613-2625), which dramatically improves upon the classical Taylor upper bound at large crystal anisotropy. In addition, the estimates also satisfy the Hashin-Shtrikman bounds, which are more restrictive than the Garroni-Kohn bound at small crystal anisotropy. Interestingly, the scaling exponents for the linear comparison estimates are found to be independent of the constitutive nonlinearity. This last observation provides an explanation for the relative weakness of an earlier linear comparison bound obtained by Garroni and Kohn.

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Section: Introductionmentioning

confidence: 99%

Section: Effective Conductivity Of Linear Polycrystalsmentioning

confidence: 99%

Section: Effective Conductivity Of Linear Polycrystalsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Linear comparison estimates for the effective resistivity of three-dimensional nonlinear polycrystals

Racherla

Castañeda

2008

Proc. R. Soc. A.

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“…These bounds are known not to be the most general ones since they rely on an implicit assumption that the grains are equiaxed. A more general lower bound that is known to be optimal is due to Schulgasser [59] and Avellaneda et al [60]:…”

Section: Conductivity For Random Polycrystals Of Laminatesmentioning

confidence: 99%

Measures of microstructure to improve estimates and bounds on elastic constants and transport coefficients in heterogeneous media

Berryman

2006

Mechanics of Materials

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The most commonly discussed measures of microstructure in composite materials are the spatial correlation functions, which in a porous medium measure either the grain-to-grain correlations, or the pore-to-pore correlations in space. Improved bounds based on this information such as the Beran-Molyneux bounds for bulk modulus and the Beran bounds for conducticity are well-known.It is first shown here how to make direct use of this information to provide estimates that always lie between these upper and lower bounds for any microstructure whenever the microgeometry parameters are known. Then comparisons are made between these estimates, the bounds, and two new types of estimates. One new estimate for elastic constants makes use of the PeselnickMeister bounds (based on Hashin-Shtrikman methods) for random polycrystals of laminates to generate self-consistent values that always lie between the bounds. A second new type of estimate for conductivity assumes that measurements of formation factors (of which there are at least two distinct types in porous media, associated respectively with pores and grains) are available, and computes new bounds based on this information. The paper compares and contrasts these various methods in order to clarify just what microstructural information and how precisely that information needs to be known in order to be useful for estimating material constants in random and heterogeneous media. * berryman1@llnl.gov

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On characterizing the set of possible effective tensors of composites: The variational method and the translation method

Milton

1990

Comm Pure Appl Math

197

180

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AbstrsctA general algebraic framework is developed for characterizing the set of possible effective tensors of composites. A transformation to the polarization-problem simplifies the derivation of the HashinShtrikman variational principles and simplifies the calculation of the effective tensors of laminate materials. A general connection is established between two methods for bounding effective tensors of composites. The first method is based on the variational principles of Hashin and Shtrikman. The second method, due to Tartar, Murat, Lurie, and Cherkaev, uses translation operators or, equivalently. quadratic quasiconvex functions. A correspondence is established between these translation operators and bounding operators on the relevant non-local projection operator, r,. An important class of bounds, namely trace bounds on the effective tensors of two-component media, are given a geometrical interpretation: after a suitable fractional linear transformation of the tensor space each bound corresponds to a tangent plane to the set of possible tensors. A wide class of translation operators that generate these bounds is found. The extremal translation operators in this class incorporate projections onto spaces of antisymmetric tensors. These extremal translations generate attainable trace bounds even when the tensors of the two-components are not well ordered. In particular. they generate the bounds of Walpole on the erective bulk modulus. The variational principles of Gibiansky and Cherkaev for bounding complex effective tensors are reviewed and used to derive some rigorous bounds that generalize the bounds conjectured by Golden and Papanicolaou.An isomorphism is shown to underlie their variational principles. This isomorphism is used to obtain Dirichlet-type variational principles and bounds for the effective tensors of general non-selfadjoint problems. It is anticipated that these variational principles, which stem from the work of Gibiansky and Cherkaev, will have applications in many fields of science.

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On the effective conductivity of polycrystals and a three-dimensional phase-interchange inequality

Cited by 115 publications

References 24 publications

Linear comparison estimates for the effective resistivity of three-dimensional nonlinear polycrystals

Linear comparison estimates for the effective resistivity of three-dimensional nonlinear polycrystals

Measures of microstructure to improve estimates and bounds on elastic constants and transport coefficients in heterogeneous media

On characterizing the set of possible effective tensors of composites: The variational method and the translation method

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