Optimizing transport in a homogeneous network

Durand, Marc; Weaire, D.

doi:10.1103/physreve.70.046125

Cited by 28 publications

(25 citation statements)

References 13 publications

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“…Indeed, isotropy conditions (3.6) encompass the isotropy conditions e α ij e β ij = δ α,β /d for transport properties. Furthermore, using the identity d α=1 (e α ij ) 2 = 1, it is easy to show that condition (c) implies j s ij e ij = 0, which is the condition required at every junction of a network (together with conditions (a) and (b)) to have optimal transport properties [32,33].…”

Section: (B) Mechanical Conditionsmentioning

confidence: 99%

Stiffest elastic networks

Gurtner

Durand

2014

Proc. R. Soc. A.

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The rigidity of a network of elastic beams is closely related to its microstructure. We show both numerically and theoretically that there is a class of isotropic networks, which are stiffer than any other isotropic network of same density. The elastic moduli of these stiffest elastic networks are explicitly given. They constitute upper-bounds, which compete or improve the well-known Hashin-Shtrikman bounds. We provide a convenient set of criteria (necessary and sufficient conditions) to identify these networks and show that their displacement field under uniform loading conditions is affine down to the microscopic scale. Finally, examples of such networks with periodic arrangement are presented, in both two and three dimensions. In particular, we present an optimal and isotropic three-dimensional structure which, to our knowledge, is the first one to be presented as such.

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Section: (B) Mechanical Conditionsmentioning

confidence: 99%

Stiffest elastic networks

Gurtner

Durand

2014

Proc. R. Soc. A.

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“…Ultimately, we expect this growing database of nets to provide a substantial foundation for a range of investigations into the physical features of nets, extending current work on percolation, transport and elastic responses (Gibson & Ashby, 1997;Roberts & Garboczi, 2002;Durand, 2005;Durand & Weaire, 2004). This suite of examples will allow detailed exploration of possible correlations between physical, topological and geometric features of crystal nets.…”

Section: Figure 31mentioning

confidence: 99%

Three-dimensional Euclidean nets from two-dimensional hyperbolic tilings: kaleidoscopic examples

Ramsden

Robins

Hyde

2009

Acta Crystallogr A Found Crystallogr

121

106

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We present a method for geometric construction of periodic three-dimensional Euclidean nets by projecting two-dimensional hyperbolic tilings onto a family of triply periodic minimal surfaces (TPMSs). Our techniques extend the combinatorial tiling theory of Dress, Huson & Delgado-Friedrichs to enumerate simple reticulations of these TPMSs. We include a taxonomy of all networks arising from kaleidoscopic hyperbolic tilings with up to two distinct tile types (and their duals, with two distinct vertices), mapped to three related TPMSs, namely Schwarz's primitive (P) and diamond (D) surfaces, and Schoen's gyroid (G).

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“…Conditions a), b), c) are the necessary and sufficient conditions to maximize the average conductivity of a network of thin wires as well [12]. Why structures satisfying these conditions have both maximal bulk modulus and maximal conductivity ?…”

Section: Some Commentsmentioning

confidence: 99%

“…We shall answer to all these questions in the present study. Indeed, Durand & Weaire [11] [12] already established necessary and sufficient conditions on the structure of cellular networks having maximal average conductivity. A quite similar approach is used in this paper to show that the very same conditions are also necessary and sufficient to maximize the bulk modulus of an open-cell material.…”

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Optimizing the bulk modulus of low-density cellular networks

Durand

2005

Phys. Rev. E

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We present an alternative derivation of upper-bounds for the bulk modulus of both two-dimensional and three-dimensional cellular materials. For two-dimensional materials, we recover exactly the expression of the Hashin-Shtrikman (HS) upper-bound in the low-density limit, while for three-dimensional materials we even improve the HS bound. Furthermore, we establish necessary and sufficient conditions on the cellular structure for maximizing the bulk modulus, for a given solid volume fraction. These conditions are found to be exactly those under which the electrical (or thermal) conductivity of the material reaches its maximal value as well. These results provide a set of straightforward criteria allowing to address the design of optimized cellular materials, and shed light on recent studies of structures with both maximal bulk modulus and maximal conductivity. Finally, we discuss the compatibility of the criteria presented here with the geometrical constraints caused by minimization of surface energy in a real foam.Cellular solids appear widely in nature and are manufactured on a large scale by man. Examples include wood, cancellous bone, cork, foams for insulation and packaging, or sandwich panels in aircraft. Material density, or solid volume fraction, φ, is a predominant parameter for the mechanical properties of cellular materials. Various theoretical studies on the mechanical properties of both two-dimensional (2D) and three-dimensional (3D) structures have been attempted [1]. Unfortunately, exact calculations can be achieved for cellular materials with simple geometry only [2], and numerical simulations [2][3] or semi-empirical models [4][5][6] are required in order to study the mechanical properties of more complex structures. However, expression of bounds on the effective moduli can be established. Perhaps the most famous bounds are those given by Z. Hashin and S. Shtrikman for isotropic heterogeneous media [7][8]. In particular, the Hashin-Shtrikman bounds for the effective bulk modulus in the low-density asymptotic limit (φ ≪ 1) read:for 2D cellular structures [7][9], and:for 3D structures [8]. κ (2D) and κ (3D) are the actual bulk modulus respectively for 2D and 3D structures, and E, G, K are the Young modulus, shear modulus, and bulk modulus of the solid phase, respectively. These three elastic moduli are related by: E = 4KG K+G for 2D bodies and by: E = 9KG 3K+G for 3D bodies. The search for optimal structures maximizing some specific modulus (for a given value of solid volume fraction φ), is of evident practical importance. In a recent study, Torquato et al. [9][10] identified values of conductivity and elastic moduli of the two-dimensional square, hexagonal, kagomé and triangular cellular structures, and observed that the bulk modulus of these structures is equal to the HS upper-bound value. The authors did not attempt to explain this result, although they noticed that such structures under uniform compression deform without bend (affine compression). Are these structures the only structures...

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Optimizing transport in a homogeneous network

Cited by 28 publications

References 13 publications

Stiffest elastic networks

Stiffest elastic networks

Three-dimensional Euclidean nets from two-dimensional hyperbolic tilings: kaleidoscopic examples

Optimizing the bulk modulus of low-density cellular networks

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