Foam mechanics: the linear elastic response of two-dimensional spatially periodic cellular materials

Warren, William E.; Kraynik, Andrew M.

doi:10.1016/0167-6636(87)90020-2

Cited by 209 publications

(83 citation statements)

References 14 publications

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“…Specific models have been developed for the particular mechanical properties exhibited by some of these cocontinuous materials as the cellular materials or some open cell foams, in order to account for bending and torsion characteristics, due to strut-like elements [Gent and Thomas 1959;Ko 1965;Menges and Knipschild 1975;Warren and Kraynik 1987;1997]. These effects are partly handled in numerical modeling, they are not in rheological ones and they would not be at all in a homogenization approach.…”

Section: Patrick Franciosi Renald Brenner and Abderrahim El Omrimentioning

confidence: 99%

Effective property estimates for heterogeneous materials with cocontinuous phases

Franciosi

Brenner

Omri

2011

J. Mech. Mater. Struct.

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This work concerns heterogeneous multiphase materials which may exhibit omnidirectional full or partial cocontinuity of several or all phases. The estimate of their effective (mechanical or physical) properties is not yet well handled as compared to those for well-defined aggregate or reinforced-matrix structures, especially in the context of homogenization methods. We propose in this framework a modeling scheme which aims at accounting for such phase cocontinuity features. In the mechanical application field, the modeling validity restricts to elastic properties of unloaded materials or in load situations as far as bending and torsion effects of possibly strut-like phase parts are not essential. For other physical properties (dielectric, magnetic, etc.), the modeling applications concern those for which homogenization approaches are relevant. The modeling is based on a material's morphology description in terms of a generalization of so-called "fiber systems" that were introduced in early literature reports. Using parameters that describe the clustering characteristics of the individual phases and of their assemblage, we have considered these fiber systems both within a layer-based approach of the material structure and within an aggregate-like one. By these two routes, we have obtained two estimate forms that differ only slightly in definition. The presentation uses the elasticity formalism, in simple cases of isotropic mixtures of two-phase materials with isotropic phase behavior but the modeling extends to n-phase anisotropic materials as established separately. Our estimates are compared with basic variational bounds and homogenization estimates, with some literature data and with homogenization results obtained with the fast Fourier transform approach on numerical structures. All data are matched with different parameter sets corresponding to different types of phase organization. The two estimates remain nearly equal for all of the examined structures regardless of phase contrast and with only slight differences consistent with their definition difference.

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Section: Patrick Franciosi Renald Brenner and Abderrahim El Omrimentioning

confidence: 99%

Effective property estimates for heterogeneous materials with cocontinuous phases

Franciosi

Brenner

Omri

2011

J. Mech. Mater. Struct.

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“…In the previous section, we presented a method for finding the three equivalent elastic moduli for the directions parallel to a cell wall constituting the hexagon cell; however, these are only three of the nine components of the elastic modulus described in (2). To express the elastic characteristics of a hexagonal honeycomb, it is necessary to know all nine components.…”

Section: Calculation Of Elastic Modulus Matrixmentioning

confidence: 99%

“…To date, honeycomb materials consisting of regular hexagonal cells or symmetrical hexagonal cells [1][2][3][4] have been the subject of extensive research. In the present study, a general method is proposed for finding the equivalent elastic moduli for the two-dimensional (2D) problem of honeycomb consisting of an array of hexagonal cells, including asymmetrical hexagonal cells.…”

Section: Introductionmentioning

confidence: 99%

Equivalent Elastic Modulus of Asymmetrical Honeycomb

Chen

Masuda

2011

ISRN Mechanical Engineering

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The equivalent elastic moduli of asymmetrical hexagonal honeycomb are studied by using a theoretical approach. The deformation of honeycomb consists of two types of deformations. The first is deformation inside the unit, which is caused by bending, stretching, and shearing of cell walls and rigid rotation of the unit; the second is relative displacement between units. The equivalent elastic modulus related to a direction parallel to one cell wall of the honeycomb is determined from the relative deformation between units. In addition, a method for calculating other elastic moduli by coordinate transformation is described, and the elastic moduli for various shapes of hexagon, which are obtained by systematically altering the regular hexagon, are investigated. It is found that the maximum compliance C y y | max and the minimum compliance C y y | min of elastic modulus C y y in one rotation of the (x, y) coordinate system vary as the shape of the hexagon is changed. However, C y y | max takes a minimum and C y y | min takes a maximum when the honeycomb cell is a regular hexagon, for which the equivalent elastic moduli are unrelated to the selected coordinate system, and are constant with C 11 = C 22 .

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“…In the element design formulation all the elements of the unit cell are identical and the shape of a typical prismatic element of variable crosssection is optimized. Note, elements with variable thickness were considered in [Warren and Kraynik 1987], where the linear elastic response of an undamaged hexagonal honeycomb was studied for three different element shapes, and in [Warren and Kraynik 1988] for a tetrahedral unit cell, but the shape of the elements were not optimized. The present paper deals with two basic topologies: regular triangular and regular hexagonal lattices.…”

Section: Introductionmentioning

confidence: 99%

Design of crack-resistant two-dimensional periodic cellular materials

Lipperman

Ryvkin

Fuchs

2009

J. Mech. Mater. Struct.

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The resistance to macrocrack propagation in two-dimensional periodic cellular materials subjected to uniaxial remote stresses is improved by redistributing the material of the solid phase. The materials are represented by beam lattices with regular triangular or hexagonal patterns. The purpose of the design is to minimize the maximum tensile stress for all possible crack locations allowed by the material microstructure. Two design cases are considered. In the cell design case material is redistributed between the otherwise uniform elements of the repetitive cell. In the element design case the shape of identical elements is optimized. The analysis of such infinite trellis with an arbitrary macroscopic crack is enabled by an efficient exact structural analysis approach. It is shown that the fracture toughness of the triangular layout can be significantly increased by redistribution of the material between the elements with uniform cross sections while for the case of hexagonal lattice the effect is achieved mainly by using identical elements with variable thickness distribution.

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Foam mechanics: the linear elastic response of two-dimensional spatially periodic cellular materials

Cited by 209 publications

References 14 publications

Effective property estimates for heterogeneous materials with cocontinuous phases

Effective property estimates for heterogeneous materials with cocontinuous phases

Equivalent Elastic Modulus of Asymmetrical Honeycomb

Design of crack-resistant two-dimensional periodic cellular materials

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