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Research Papers

Modeling the Elastic Properties of Reticulated Porous Ceramics

[+] Author and Article Information
Stephen J. Sedler

Department of Mechanical Engineering,
University of Minnesota—Twin Cities,
111 Church Street SE,
Minneapolis, MN 55455
e-mail: sedl0033@umn.edu

Thomas R. Chase

Department of Mechanical Engineering,
University of Minnesota—Twin Cities,
111 Church Street SE,
Minneapolis, MN 55455
e-mail: trchase@umn.edu

Jane H. Davidson

Department of Mechanical Engineering,
University of Minnesota—Twin Cities,
111 Church Street SE,
Minneapolis, MN 55455
e-mail: jhd@umn.edu

1Corresponding author.

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received September 16, 2016; final manuscript received October 19, 2016; published online November 16, 2016. Assoc. Editor: Peter W. Chung.

J. Eng. Mater. Technol 139(1), 011011 (Nov 16, 2016) (10 pages) Paper No: MATS-16-1267; doi: 10.1115/1.4035098 History: Received September 16, 2016; Revised October 19, 2016

A model to predict the elastic material properties of reticulated porous ceramics (RPCs) based on the microstructural geometry is presented. The RPC is represented by a repeating unit structure of truncated octahedrons (tetrakaidecahedrons) with the ligaments represented by the cell edges. The deformations of the ligaments in the cellular structure under applied loads are used to determine the effective moduli and Poisson's ratio of the bulk material. The ligament cross section is represented as having a Plateau border exterior surface with a cusp half-angle that is varied between 0 and 90 deg, and a Plateau border interior void with a cusp half-angle of zero, representative of the ranges seen in RPCs. The ligament cross-sectional area is permitted to vary along its length and the distance between internal and external cusps is assumed constant. The relative density of the foam, corresponding to the length, cusp distances, and external-cusp half-angle of the ligaments, is determined using solid geometry. The relative density has the dominant effect on the moduli, while normalized ligament length varies the moduli by 11–49% at a specified relative density. The impact of the external shape of a ligament on the relative moduli is insignificant. The model is validated through comparisons with the measured elastic properties of RPCs in the literature and new data. The model is the first to consider the effect of the microstructural features of ligaments of RPCs on the elastic moduli of the bulk material.

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Figures

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Fig. 1

A tetrakaidecahedron, representative of the cellular structure of RPCs

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Fig. 2

Macroscale photograph of an RPC with an inset scanning electron micrograph of a sectioned RPC ligament, exposing the Plateau border shaped void in the center

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Fig. 3

The double Plateau border cross section with an external cusp half-angle of (a) less than 30 deg and (b) greater than 30 deg. Arcs 1, 2, and 3 construct the internal Plateau border while arcs 4, 5, and 6 construct the external Plateau border.

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Fig. 4

The concentric Plateau borders ligament shape

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Fig. 5

Cross sections (a) at the midplane (ξ = 0) and (b) at an end-plane (ξ=(1/2)) of a ligament defined by ζ = 2 and θo=15 deg. Distance t is assumed to remain constant along the entire length of the ligament.

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Fig. 6

Two tetrakaidecahedron unit cells. The dotted lines form a rectangular repeating section having mirror symmetry across its faces. The ligaments in the repeating section are indicated with bold lines. The one-, two-, and three-directions are perpendicular to the square faces of the tetrakaidecahedron.

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Fig. 7

Rectangular repeating section of ligaments with mirror symmetry across the faces

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Fig. 8

Free-body diagram of one of the diagonal ligaments

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Fig. 9

Parameters used to define the statical moment of area

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Fig. 10

Eighth-order polynomial curves fit to the shear form factor data (generated using Eq. (26)) for example ligament geometries. Some example polynomials are shown.

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Fig. 11

Eighth-order polynomial curves fit to the nondimensionalized torsional constant for example ligament geometries. Some example polynomials are shown.

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Fig. 12

The graphical representation of (a) the ligament with the 45 deg end cuts, (b) the unit square assembled using four ligaments, and (c) two unit squares assembled with a common node, collinear diagonal lines, and rotated planes

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Fig. 13

Solid model of a unit cell geometry used to determine the relative density as it relates to ζ, θo, and L/Ai(0). Geometry is created by assembling an array of unit squares and removing all material outside of the unit cube volume. Visible irregularities in the geometry are caused by a slight mismatch between adjacent unit squares due to ligament shape. These irregularities have a negligible contribution to the total volume.

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Fig. 14

Relative density contours, generated using the unit cell geometry, for varying ζ and θo with L/Ai(0)=20.3. The marked locations are used in the generation of the curves in Fig. 18.

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Fig. 15

Relative density dependence of the relative elastic modulus. The different curves are for different ligament L/Ai(0) ratios, and plotting the entire range of ζ and θo investigated. The θo range examined spans from 0 to 90 deg. The curves nearly collapse on each other regardless of ζ and θo.

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Fig. 16

Relative density dependence of the relative shear modulus. The different curves are for different ligament L/Ai(0) ratios, and plotting the entire range of ζ and θo investigated. The “thickness” of the curves is attributable to varying ζ and θo.

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Fig. 17

Relative density dependence of Poisson's ratio. The different curves are for different ligament L/Ai(0) ratios, and plotting the entire range of ζ and θo investigated. The “thickness” of the curves is attributable to varying ζ and θo.

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Fig. 18

Second moment of area of the ligament along its nondimensionalized length. One set of curves is for relative densities of 0.08, and the other is for relative densities of 0.18. The ζ and θo pairs used to produce the individual curves are marked in Fig. 14.

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Fig. 19

Relative density dependence of the relative elastic modulus. The different curves are for different ligament L/Ai(0) ratios. The +, ×, △, •, and □ symbols are data reported in the open literature for RPCs manufactured using the replication method [13,14,19,20].

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Fig. 20

Relative density dependence of the relative shear modulus. The different curves are for different ligament L/Ai(0) ratios. The × and △ symbols are data reported in the open literature for RPCs manufactured using the replication method [14,19].

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