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Effects of solid distribution on the stiffness and strength of metallic foams
PII:
Acta mater. Vol. 46, No. 6, pp. 2139±2150, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 1359-6454/98 $19.00 + 0.00 S1359-6454(97)00421-7
EFFECTS OF SOLID DISTRIBUTION ON THE STIFFNESS AND STRENGTH OF METALLIC FOAMS A. E. SIMONE1 and L. J. GIBSON2 Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 and 2Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. 1
(Received 12 August 1997; accepted 3 October 1997) AbstractÐLightweight metallic cellular materials can be used in the construction of composite plates, shells and tubes with high structural eciency. Previous models for the mechanical performance of cellular materials have focused on their dependence on relative density, cell geometry and the properties of the solid material of which the cell faces and edges are composed. In this study, we consider the eect of the distribution of solid between the cell faces and edges on mechanical properties using ®nite element analysis of idealized 2D (hexagonal honeycomb) and 3D (closed-cell tetrakaidecahedral foam) cellular materials. The eects of the distribution of the solid on the stiness and strength of these materials are presented and discussed. # 1998 Acta Metallurgica Inc.
1. INTRODUCTION
Metallic foams have the potential for use in weightecient composite structures for a variety of applications. There are currently several liquid state processes used to make closed-cell metallic foams [1±4]. The processing of the liquid metal foam before solidi®cation can have a signi®cant eect on both the microstructure of the solid metallic foam and its mechanical properties [5]. A large body of literature on the behavior of liquid foams already exists; reviews of liquid foam formation, stability and behavior are given by Kraynik [6] and Weaire and Fortes [7]. In this study we are particularly interested in the eect of the ®nal distribution of solid in the cell edges and faces on the stiness and strength of metallic foams. Most relevant to this study are the mechanisms which determine the drainage pattern present in a liquid foam during its evolution from a wet, or undrained, foam to a dry, or drained, foam. At the junction of the cell faces in a cell edge, surface tension causes the liquid±gas interface to be curved in an arc of constant radius called a Plateau border [8]. The surface tension forces acting along the Plateau border cause the ¯uid pressure in the cell edge to be lower than that in the cell face. The drainage pattern in a wet foam, therefore, is from the cell face into the adjacent cell edges, where the Plateau border regions form channels through which gravitational forces drain the liquid over time. In a liquid foam with no impurities, the cell faces will drain until they burst due to the strong surface tension forces. In order for a liquid foam to become stable, the liquid±gas interface must be altered either by introducing a surfactant that
reduces surface tension or by increasing the viscosity of the surface layer [9±11]. A cell face in a liquid foam that is stabilized in this way will drain down to some critical thickness and then stop further drainage. A micrograph of an aluminum foam produced by a liquid state process (Alporas, Shinko Wire, Amagasaki) with Plateau borders is shown in Fig. 1. The drainage pattern of a liquid metal foam can aect the structure of the resulting solid foam in two ways. If the excess liquid is not allowed to drain away suciently before solidi®cation, a density gradient results causing a high degree of material heterogeneity and anisotropy. This has been previously noted in aluminum foam panels produced by the Alcan process [12, 5]. Of more direct relevance to this study is the in¯uence of drainage on the distribution of liquid between the cell faces and the Plateau borders at solidi®cation. As a stable liquid foam drains, the radii of curvature of the Plateau borders and the fraction of total liquid contained within them decrease. Here we study the eect of the ®nal distribution of solid in the cell edges and faces on the stiness and strength of metallic foams. The Young's modulus and plastic collapse stress of a foam can be described by the following equations, derived using dimensional analysis [13]: 2 E* r* r* f 1 ÿ f 1 Es rs rs and
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3=2 spl * r* r* 0:3f 1 ÿ f sYS rs rs
2
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tion made in this analysis is that all cell faces were made planar in order to eliminate the non-linearities caused by wall curvature and isolate the eects of the solid distribution. An aggregate of the tetrakaidecahedral cells which we use to represent a closed cell foam is shown in Fig. 2(b). These cells pack in a body-centered cubic (b.c.c.) lattice with the crystallographic axes h100i, h010i and h001i being parallel to the three normal directions of the square faces [directions X1, X2 and X3 in Fig. 2(b)]. Each cell edge in the tetrakaidecahedral foam consists of the intersection of two hexagonal and one square face. The angle between the planes of the two hexagonal faces is the tetrahedral angle, 109.478, while that between the planes of a hexagonal and square face is 125.268. In three dimensions, the parameter F3 is de®ned as the volume fraction of solid material bounded by the Plateau borders in the cell edges and vertices relative to the total volume of solid material. Fig. 1. Micrograph of an aluminum foam with Plateau borders (Alporas, Shinko Wire, Amagasaki, r* = 200 kg/m3).
2. ANALYSIS
where E*, r* and spl* are the modulus, density and plastic collapse stress of the foam, Es, rs and sYS are those of the solid cell wall material, and f is the volume fraction of solid contained in the cell edges. The dimensional arguments used to obtain these equations assume that the cell edges and faces are of uniform thickness and do not account for the Plateau borders observed in some foams. Here, we use ®nite element analysis to estimate the relative elastic modulus and relative plastic collapse strength of a 2D honeycomb and a 3D closedcell foam as a function of relative density, r*/rs, and the distribution of the solid material. For the 2D analysis, a regular hexagonal honeycomb structure, as shown in Fig. 2(a), is used for simplicity. In two dimensional analysis, the parameter F2 is de®ned as the area of solid material bounded by the Plateau borders at the vertices relative to the total area of solid. Warren and Kraynik [14] and Kraynik and Warren [15] ®nd analytically that the elastic modulus of a low density (r*/rs<0.15) honeycomb is increased when material is shifted away from the midspan of a member towards the nodes. Using ®nite continuum elements, this analysis can be extended to honeycombs of a much higher relative density. Several space-®lling unit cells are available to model 3D foams. While the Weaire-Phelan unit cell [16], an eight cell aggregate of two polyhedra, is the lowest surface energy unit cell yet discovered, the Kelvin tetrakaidecahedron is the lowest surface energy unit cell known consisting of a single polyhedron. The Kelvin cell is de®ned by six planar square faces and eight hexagonal faces that are non-planar, but have zero mean curvature. The one simpli®ca-
All ®nite element analyses were performed using the ABAQUS Analysis Package (Hibbit, Karlsson and Sorensen, Pawtucket, RI), running on a DEC Alphastation 500 (Digital Equipment, Maynard, MA). For the 2D hexagonal honeycomb modeling, the following elements from the ABAQUS element library were used: B22 (3-node quadratic beams) and CPS8 (8-node biquadratic plane stress elements). For the 3D closed-cell tetrakaidecahedral foam modeling, the following elements were used: S9R5 (9-node doubly curved thin shell elements with reduced integration), C3D20 (20-node quadratic bricks) and C3D15 (15-node quadratic triangular prisms). The material model for all ®nite element analyses was isotropic with elastic±perfectly plastic stress±strain behavior. This model was used to eliminate any dependencies of the measured results on a chosen strain hardening exponent. As-cast aluminum alloys typically have an elastic modulus of 69 to 73 GPa and a yield strength of 100 to 300 MPa [17]. The isotropic elastic±perfectly plastic material model used here was given material properties consistent with those of an as-cast aluminum alloy: Es=70 GPa, n = 0.33 and sYS=150 MPa. Input ®les for ABAQUS were produced using the MATLAB (Mathworks, Natick, MA) package. Using these scripts, the re®nement of each model could be explicitly de®ned. In order to balance solution accuracy and solution eciency, re®nement tests were performed for each series of models. In these tests, a model was analyzed at successively higher levels of re®nement. A re®nement level was chosen such that the solution was within 0.02% of the solution obtained with the maximum re®nement. Models with up to 95 000 degrees of freedom could be run.
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Fig. 2. (a) Idealized regular hexagonal honeycomb. (b) Idealized tetrakaidecadedral foam structure.
Two solution methods were used for each model to be analyzed. An applied loading and small deformation linear elastic analysis were ®rst used to measure the elastic modulus and Poisson's ratio. Large displacement theory and an incremental, iterative solution method were then used to produce a stress±strain curve. The solution method used was either the Newton method (higher relative density models) or the Riks method (lower relative density
models) [18]. Displacement boundary conditions were imposed on the appropriate nodes in order to simulate a uniaxial compression test. The reaction forces on the nodes of the displaced boundary were recorded and summed in order to compute the stress at each increment of displacement. A spline curve was ®t to the stress±stain data points and the peak stress of this curve was taken as the measured plastic collapse stress.
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2.1. Hexagonal honeycomb A schematic of an idealized, regular, isotropic hexagonal honeycomb is shown in Fig. 2(a). The unit cell that was used in all analyses is shown in Fig. 3(a). No symmetry assumptions were made in the 2D analysis, so the required boundary conditions of this model are simply that compatibility is maintained between the corresponding elements in the periodically repeating unit cell. The mechanical properties of a low density regular hexagonal honeycomb can be accurately modeled using standard beam theory: 3 E1 * E2 * r* 1:5 3 Es Es rs 12 * 21 * 1 spl * 1 sYS 2
r* rs
2
:
4 5
For higher relative density honeycombs with less slender cell edges, the predictions of beam theory become less accurate due to signi®cant axial and shear deformation of the cell edges and deformation of the solid material at the vertices. Honeycombs with no Plateau borders and straight edges of uniform thickness were modeled through a range of relative densities using either beam or continuum elements. The ®nite beam elements used for the analysis allow for axial and shear deformation in the cell edges, but do not account for any deformation at the vertices. Continuum elements with a sucient mesh density can account for any signi®cant deformation in either the cell edges or the vertices. The relationship between the relative density, the cell edge thickness, t, and the edge length, l, is de®ned by r* 2 t 1 t 2 p ÿ : 6 rs 3l 3 l After running an initial series of tests to con®rm that equal properties were measured in the X1 and X2 directions, all further tests were performed with uniaxial loading in the X2 direction only. The geometry of an idealized hexagonal honeycomb with Plateau borders of radius of curvature Rp and width wp is shown in Fig. 3(b). Considering a representative vertex, the relative density of this honeycomb can be de®ned as r* Ap Ae , At rs
7
where Ap is the area of the solid contained in the Plateau borders at the vertex, p 1 3 Ap 2Rp t2 ÿ pR2p , 4 2 Ae is the area of the solid in the edges outside the Plateau borders, l Rp t=2 Ae 3 ÿ p t, 2 3 and At is the area represented by the joint in the space-®lling honeycomb, p 1 p 3 3 3 2 l : l At 3l 4 2 2 From equation (7), then, the relationship between the relative density, the cell edge thickness, edge length and the Plateau border radius of curvature is 2 r* 2 t 1 t 2 4 2p Rp p ÿ : 8 ÿ p l rs 3 3 3 3l 3 l The fraction of solid in the vertex and Plateau border region, F2, is Fig. 3. (a) Unit cell for idealized hexagonal honeycomb. (b) Geometry of a typical vertex with Plateau borders.
F2
Ap Ap rs , Ap Ae At r*
SIMONE and GIBSON:
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Fig. 4. Sample ®nite element meshes of a regular hexagonal honeycomb with r*/rs=0.09 and F2 equal to 0.02, 0.4, and 0.98.
or F2
2 rs 1 t 2 4 Rp t 4 2p Rp p ÿ : r* 3 l l 3 l2 3 3 3 9
For each value of relative density, 0.015, 0.03, 0.06 and 0.09, a series of models were generated and analyzed with the parameter, F2, being varied between close to zero (cell edges of uniform thickness) and unity (all of the solid contained in the Plateau borders). Sample meshes of a hexagonal honeycomb with a relative density of 0.09 and F2 equal to 0.02, 0.4, and 0.98 are shown in Fig. 4. 2.2. Closed-cell tetrakaidecahedral foam The idealized tetrakaidecahedral foam is shown in Fig. 2(b). The unit cell for this structure is shown in Fig. 5. When modeling with ®nite shell elements, no symmetry assumptions were made, so the required boundary conditions of this model were simply that compatibility was maintained between the corresponding elements in the periodically repeating unit cell. The ®rst objective was to evaluate the degree of anisotropy of the tetrakaidecahe-
Fig. 5. (a) Unit cell of a tetrakaidecahedral foam.
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dral foam. Shell element models representing various relative densities were tested in three directions corresponding to the following crystallographic directions of the b.c.c. lattice: h100i (normal to a square face), h111i (normal to a hexagonal face) and h110i (in the plane of a square face and perpendicular to one edge of the square). For relative densities less than or equal to 0.2, the relationship between the relative density, cell face thickness and edge length of a tetrakaidecahedral foam with uniform face thickness is approximated by 2 r* t t 1:185 ÿ 0:4622 : 10 rs l l A tetrakaidecahedral foam unit cell was then modeled using 20-node quadratic bricks. Because of the many degrees of freedom required to produce continuum element models with adequate mesh re®nement, these models were loaded only in the h100i direction, with the cubic symmetry of the tetrakaidecahedral cell being used to reduce the model size. The relationship of the reduced unit cell to the full unit cell is shown in Fig. 6(a, b), while the unit cell itself is shown in Fig. 6(c). Four of the planes which bound the model are simple symmetric boundaries. The normals of the planes of these boundary surfaces are either parallel to or perpendicular to the loading direction. The ®fth boundary is an antisymmetric boundary which requires that the unit cell be compatible with an identical unit cell rotated 1808 about the axis indicated as z±z in Fig. 6. The cross-sectional geometry of the cell edge of a tetrakaidecahedral foam with Plateau borders is shown in Fig. 7. The compromises made in using the tetrakaidecahedral unit cell impose limits on the values of F3 which can be reasonably modeled. Because the angle between the two hexagonal faces is smaller than the angle between a hexagonal face and a square face, the radius of curvature of two of the Plateau borders (Rp,1) on a given cell edge is larger than that of the third (Rp,2). Using simple geometry, it can be seen that Rp,1
t wp 1:932wp 2 tan 54:748=2
and Rp,2
p t wp wp 2, 2 tan 70:528=2
where wp is the width of the Plateau border as de®ned in Fig. 7(b). The relationship between the two Plateau border radii can then be de®ned as 1:932 t 1:932 p ÿ 1 : 11 Rp,1 p Rp,2 2 2 2 As the Plateau borders get smaller, the ratio Rp,1/ Rp,2 increases, causing the elements in the smaller Plateau border to become distorted. Highly dis-
torted brick elements reduce the accuracy of the analysis. For each relative density, a lower bound (excluding the foam with no Plateau borders) was placed on the range of F3 which could be modeled, such that Rp,1/Rp,2<2. Because the square cell faces are much smaller than the hexagonal cell faces, but the width of the Plateau border is the same on both, the upper bound of F3 is reached when the Plateau borders cover the entire square cell face. The shape of the oblate ellipsoid curve that is formed at a vertex, where the Plateau borders meet, was approximated by a series of spline curves. The relationship between the relative density, normalized cell face thickness and the normalized Plateau border width was determined by numerically measuring the unit cell volume of a large number of models over a range of relative densities and F3, and then ®tting a two variable polynomial curve ®t to this data. The relationship for the ranges (0.025 < t/l < 0.1) and (0.05 < r*/rs<0.2) is 2 r* t t 10ÿ3 ÿ 0:0125 1190 ÿ 520 r l l 2 t t wp 0:41 ÿ 339 464 l l l 2 2 t t w 551 ÿ 70 ÿ 1200 l l l 2 3 t t w : 12 ÿ 122 ÿ 198 ÿ 2000 l l l The fraction of solid in the Plateau borders, F3, can then be found using the volume of the cell faces not contained within the Plateau borders, the computed relative density and the volume of the tetrakaidecahedral unit cell: pÿ p 2 3t l ÿ 2wp 2 6t 3 l ÿ 2= 3wp F3 1 ÿ : 13 11:31l 3 r*=rs For each of the relative densities 0.05, 0.1, 0.15 and 0.2, models were generated and analyzed over the available range of F3. Sample meshes of a tetrakaidecahedral foam with a relative density of 0.15 and F3 equal to 0.075, 0.51, and 0.73 are shown in Fig. 8. 3. RESULTS
3.1. Hexagonal honeycomb The ®nite element analysis results for the relative elastic modulus and the relative peak stress of a hexagonal honeycomb with cell edges of uniform thickness are within 10% of the analytical results of Gibson and Ashby [13] for relative densities up to 0.12. The continuum elements give a modulus and strength slightly higher than their model while the beam elements give almost identical modulus and
SIMONE and GIBSON:
EFFECTS ON STIFFNESS AND STRENGTH OF METALLIC FOAMS
Fig. 6. (a) Relationship of reduced continuum element model to the tetrakaidecahedral foam unit cell. (b) Cell wall elements of the reduced unit cell. (c) Perspective of reduced continuum element model.
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Fig. 7. (a) Geometry of the cross-section of a tetrakaidecahedral cell edge. (b) Geometry of the cross-section of a tetrakaidecahedral cell edge with Plateau borders.
strength. The Gibson and Ashby models correlate well with material properties predicted using ®nite beam elements at low relative densities, but predict a slightly higher modulus as relative density increases due to their exclusion of axial and transverse shear deformation. The mechanical properties predicted using continuum elements are somewhat higher than those predicted by beam theory. With increasing relative density and t/l, the size of the vertex increases, decreasing the eective edge length in bending and stiening the honeycomb. The increased vertex size also causes the plastic hinge to form slightly closer to the in¯ection point at the mid-span of the cell edge, increasing the stress required to initiate plastic collapse. The ®nite element analysis results for the mechanical properties of honeycombs with Plateau borders are contained in Fig. 9. The modulus and peak stress values shown are normalized by the modulus and peak stress of a honeycomb with no Plateau borders (as modeled with continuum elements). As F2 increases, the modulus and peak stress at ®rst increase above the values for honeycombs with uni-
form edge thickness and then eventually fall below them. The value of F2 at which the modulus and peak stress fall below those for honeycombs with uniform edge thickness increases with increasing relative density. This behavior can be understood in terms of the bending moment distribution along a cell edge and the redistribution of material from the edge into the Plateau borders with increasing F2. The moment varies linearly from one end to the other and has a value of zero at the beam in¯ection point at mid-span. Moving a fraction of the solid into the Plateau borders increases the moment of inertia of the edge cross-section in the Plateau border region, where the moment is the largest, and decreases the moment of inertia of the edge crosssection outside the Plateau border region where the moment is smallest. For a low density honeycomb, moving material from the cell edge into the Plateau border region causes a small increase in the Plateau border width, wp, but a large relative decrease in the thickness of the cell edge outside the Plateau border, t. The moment of inertia of the cell edge cross-section in the Plateau border region is increased, but because the width of the Plateau border is small, the cell wall outside the Plateau border, which has a signi®cantly reduced moment of inertia, is still subjected to a high bending moment. The ¯exural stiness of the honeycomb is therefore reduced. This is supported by the ®ndings of Warren and Kraynik [14] and Kraynik and Warren [15]. The plastic hinge which initiates plastic collapse forms at a distance roughly equal to wp from the vertex. Because wp is small and the plastic moment capacity of the cell edge cross section is diminished by the reduction in thickness, the stress required to develop plastic hinges, which is equivalent to the peak stress, also decreases. For a high density honeycomb with material moved from the cell edge into the Plateau border region, the increase in the Plateau border width is much larger relative to the decrease in the thickness of the cell edge outside the Plateau border. The increase in moment of inertia near the vertex and the resulting reduction in eective beam length are therefore more signi®cant than the decrease in moment of inertia of the cell edge outside the Plateau border. This results in the modulus increases reported in Fig. 9(a) and in Ref. [14]. Similarly, although the plastic moment capacity of the cell edge cross-section where the plastic hinge forms is slightly reduced, the stress required to initiate hinge formation is larger because of the reduced eective length of the cell edge in bending. This results in the peak stress increases reported in Fig. 9(b). For each relative density, however, there is a critical value of F2, beyond which the transfer of any additional solid material into the Plateau borders leads to a reduction in the mechanical properties. This critical value of F2 increases with
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Fig. 8. Sample ®nite element meshes of a tetrakaidecahedral foam with r*/rs=0.15 and F3 equal to 0.075, 0.51, and 0.73.
relative density, and, for each relative density, is somewhat lower with respect to peak stress than with respect to modulus. 3.2. Closed-cell tetrakaidecahedral foam The ®nite element analysis using shell elements gives Young's moduli which are almost equal for loading in the h100i, h111i, and h110i directions. The results are well described by: h100i: E*=Es 0:3152 r*=rs 0:2089 r*=rs 2 h111i: E*=Es 0:3026 r*=rs 0:3141 r*=rs 2 h110i: E*=Es 0:3325 r*=rs 0:3116 r*=rs 2 for relative densities less than 0.2. The tetrakaidecahedral cell is relatively isotropic with respect to elastic modulus, with the maximum stiness being less than 10% greater than the minimum stiness for relative densities less than 0.2. The highest stiness is measured in what corresponds to the h110i direction in the b.c.c. lattice. The stinesses in the h100i and h111i directions are very close to that in the h110i direction. The h100i direction was chosen for further modeling using continuum elements because the symmetry assumptions that could be made for loading in this direction allowed the model size to be signi®cantly reduced. The relative Young's modulus and peak stress for loading in the h100i direction were estimated using both shell and continuum elements. The results are well described by:
Continuum: E*=Es 0:3163 r*=rs 0:3188 r* =rs 2 Shell: E*=Es 0:3152 r*=rs 0:2089 r*=rs 2 Continuum: spl *=sYS 0:4445 r*=rs 0:3346 r*=rs 2 Shell: spl *=sYS 0:4459 r*=rs 0:3211 r*=rs 2 for relative densities less than 0.2. The modulus predicted by continuum elements is slightly higher than that predicted by shell elements. The peak stresses predicted by the two element types are approximately the same. As previously noted in the hexagonal honeycomb results, the higher modulus measured using continuum elements is due to the increased constraint at the cell edges as the face thickness increases. The ®nite element results suggest that Gibson and Ashby's model (equations (1) and (2)) correctly predicts both a linear and quadratic dependence on relative density, but the constants of proportionality are dierent from those suggested by Gibson and Ashby. The parameter f in these models represents the volume fraction of solid contained in the cell edges. In a tetrakaidecahedral foam cell, the area contained in an edge cross-section, computed using simple geometry, can be expressed in terms of the face thickness, t, as r t2 1 3 Ae : 14 4 1:932 2
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to elastic modulus [Fig. 10(a)], the behavior of the tetrakaidecahedral foams is qualitatively similar to that measured in the hexagonal honeycomb, although the magnitude of the measured modulus increase is much lower. The mechanisms of modulus increase are similar to those previously discussed in the honeycomb analysis, but the in¯uence of these mechanisms on the modulus are reduced because the deformation behavior of the idealized foam is dominated by the axial deformation of cell faces and edges, rather than bending. The peak stress of the tetrakaidecahedral foam model decreases with the formation of a Plateau border for all values of relative density considered [Fig. 10(b)]. The plastic collapse of the model foam is also dominated by axial deformation in the cell faces, rather than bending. The mechanism which increased the peak stress in the honeycomb model with Plateau borders has a lesser in¯uence on the foam model behavior because the cell faces yield axially before the fully plastic moment can develop along the cell edge. The transfer of solid material
Fig. 9. (a) Normalized modulus vs F2, (b) normalized peak stress vs F2 for regular hexagonal honeycombs with Plateau borders.
The volume fraction of the solid contained in the cell edge can be estimated using the edge length per unit volume and the relative density of the tetrakaidecahedral unit cell: 12Ae 1 : 15 f 11:31l 2 r*=rs The magnitude of f becomes negligible at low relative density. The ®nite element analysis results for the mechanical properties of a tetrakaidecahedral closed-cell foam with Plateau borders are summarized in Fig. 10. The modulus and peak stress values shown are normalized by the modulus and peak stress of a tetrakaidecahedral foam with no Plateau borders (i.e. uniformly thick cell faces). The values are interpolated over the range of F3 which could not be modeled due to the nonuniformity of the cell face incidence angles in the tetrakaidecahedral cell. This range was between the value of F3=0, for the foam with no Plateau borders and F3=0.047, 0.094, 0.143 and 0.192, respectively, for the models with relative density 0.05, 0.1, 0.15 and 0.2. With respect
Fig. 10. (a) Normalized modulus vs F3, (b) normalized peak stress vs F3 for tetrakaidecahedral foams with Plateau borders.
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from the cell face into the Plateau border reduces the net area of the cell face with respect to in-plane axial deformation, resulting in a lower measured peak stress. 4. DISCUSSION
Honeycombs deform primarily by bending of the cell edges, with the maximum bending moment occurring at the ends of the member. Shifting material away from the cell edges into Plateau borders at the vertices initially increases the modulus and peak stress of the honeycomb by increasing the moment of inertia at the ends of the members where the bending moments are the highest. As more material is moved, the increase in moment of inertia does not compensate for the reduction of moment of inertia in the cell edge, causing a reduction in the mechanical properties. Similar behavior would be expected in open cell foam structures with respect to the shifting of solid material from the cell edges into the vertices. In contrast, closed cell foams deform primarily by the in-plane stretching of the cell faces. As a result, shifting material from the cell faces of a closed cell foam into Plateau borders along the edges has little eect on the modulus and causes a reduction in the peak stress. As the volume fraction of solid shifted to the edges increases to the point where almost all of the solid material is in the edges rather than the faces (F3 approaches 1) the foam behaves more like an open cell foam. Open cell foams with Plateau borders along the edges have been analyzed by Warren and Kraynik [19]. Changing the shape of the edge cross-section from circular or square to a Plateau border con®guration increases the stiness of these foams by 60±70%. Metallic foams with high relative density (r*/ rs>0.1) have been produced that have a modulus comparable to that of the idealized tetrakaidecahedral foam with no Plateau borders ( [3]; mfr. data). The peak stress values of these foams, however, are signi®cantly below the idealized values. Most metallic foams with lower relative density have both modulus and peak stress values far below the idealized behavior [5]. All of these foams exhibit the microstructure of reasonably dry foams. Cell edges generally have small Plateau borders (0.05 < F < 0.2), although localized regions of higher relative density and large Plateau borders can be observed in many foam samples [5]. The results presented in Fig. 10 suggest that the material distribution probably does not have a signi®cant eect on the modulus of these foams, but may contribute to the lower measured peak stress. The poor mechanical properties of the low relative density foams are more likely attributable to the presence of defects (curvature, corrugations) in the cell faces [5, 20]. If lower density metallic foams can approach the idealized behavior presented here,
Fig. 11. Weight eciency of sandwich panels with idealized foam and honeycomb cores.
their potential for use in structural applications would increase signi®cantly. Structural sandwich panels using conventional low density metallic foam cores are inferior to skin-stringer construction or aluminum honeycomb sandwich construction in terms of weight eciency [21]. A structural sandwich panel with an idealized tetrakaidecahedral closed-cell foam core, however, nearly matches the weight eciency of a honeycomb sandwich. Figure 11 shows the normalized weight eciency of optimized sandwich panels over a range of normalized bending stiness (analysis after Simone and Gibson [21]). The three curves in Fig. 11 represent panels with cores consisting of an idealized open cell foam (with mechanical properties comparable to conventional low density metallic foams), an idealized hexagonal honeycomb and an idealized tetrakaidecahedral closed-cell foam. The faces of the sandwich panels are composed of the same material as the solid material in the cellular cores. Improved weight eciency, such as that illustrated in Fig. 11, in addition to the various secondary bene®ts (damping, ®re resistance, energy absorption, the ability to make 3D sandwich structures) would make sandwich panels with metallic foam cores a cost eective alternative for many structural applications. AcknowledgementsÐWe wish to acknowledge the ®nancial support of the Advanced Research Projects Agency (Contract # PNO 39141). REFERENCES 1. Akiyama, S., Imagawa, K., Kitahara, A., Nagata, S., Morimoto, K., Nishikawa, T. and Itoh, M., Foamed Metal and Method of Producing Same. U.S. Patent No. 4,713,277, 1987. 2. Jin, I., Kenny, L. and Sang, H., Method of Producing Lightweight Foamed Metal. U.S. Patent No. 4,973,358, 1990. 3. Kunze, H. D., Baumeister, J., Banhart, J. and Weber, M., Powder Metall. Int., 1993, 25(4), 182±185.
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4. Prakash, O., Sang, H. and Embury, J. D., Mater. Sci. Engng A, 1995, 199, 195±203. 5. Simone, A. E. and Gibson, L. J., Acta metall. mater., 1997, submitted. 6. Kraynik, A. M., Ann. Rev. Fluid Mech., 1988, 20, 325±357. 7. Weaire, D. and Fortes, M. A., Adv. Phys., 1994, 43(6), 685±738. 8. Plateau, J. A. F., Statique Experimentale et Teorique des Liquides Soumis aux Seules Forces Moleculaires. Gauthier-Villiard, Paris, 1873. 9. Bikerman, J. J., Foams. Springer-Verlag, New York, 1973. 10. Sebba, F., Foams and Biliquid Aprons. John Wiley and Sons, Chichester, U.K., 1987. 11. Walstra, P., in Foams: Physics, Chemistry and Structure, ed. A. J. Wilson. Springer-Verlag, London, 1989. 12. Beals, J. T. and Thompson, M. S., J. Mater. Sci., 1997, 32, 3595±3600. 13. Gibson, L. J. and Ashby, M. F., Cellular Solids: Structure and Properties, 2nd edn. Cambridge University Press, 1997.
14. Warren, W. E. and Kraynik, A. M., Mech. Mater., 1987, 6, 27±37. 15. Kraynik, A. M. and Warren, W. E., in Low Density Cellular Plastics: Physical Basis of Behavior, ed. N. C. Hilyard and A. Cunningham. Chapman and Hall, London, 1994. 16. Weaire, D. and Phelan, R. Phil. Mag. Lett., 1994, 69, 107±110. 17. American Society of Metals, ASM Handbook, Vol. 2 Properties and Selection: Nonferrous Alloys and Special-Purpose Materials. ASM International, Metals Park, OH, 1990. 18. ABAQUS Theory Manual. Hibbet, Karlsson and Sorenson, Inc., Pawtucket, RI, 1995. 19. Warren, W. E. and Kraynik, A. M., J. appl. Mech., 1988, 55, 341. 20. Simone, A. E. and Gibson, L. J., Acta metall. mater., 1997, submitted. 21. Simone, A. E. and Gibson, L. J., J. Mater. Sci., 1997, in print.
Acta mater. Vol. 46, No. 6, pp. 2139±2150, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 1359-6454/98 $19.00 + 0.00 S1359-6454(97)00421-7
EFFECTS OF SOLID DISTRIBUTION ON THE STIFFNESS AND STRENGTH OF METALLIC FOAMS A. E. SIMONE1 and L. J. GIBSON2 Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 and 2Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. 1
(Received 12 August 1997; accepted 3 October 1997) AbstractÐLightweight metallic cellular materials can be used in the construction of composite plates, shells and tubes with high structural eciency. Previous models for the mechanical performance of cellular materials have focused on their dependence on relative density, cell geometry and the properties of the solid material of which the cell faces and edges are composed. In this study, we consider the eect of the distribution of solid between the cell faces and edges on mechanical properties using ®nite element analysis of idealized 2D (hexagonal honeycomb) and 3D (closed-cell tetrakaidecahedral foam) cellular materials. The eects of the distribution of the solid on the stiness and strength of these materials are presented and discussed. # 1998 Acta Metallurgica Inc.
1. INTRODUCTION
Metallic foams have the potential for use in weightecient composite structures for a variety of applications. There are currently several liquid state processes used to make closed-cell metallic foams [1±4]. The processing of the liquid metal foam before solidi®cation can have a signi®cant eect on both the microstructure of the solid metallic foam and its mechanical properties [5]. A large body of literature on the behavior of liquid foams already exists; reviews of liquid foam formation, stability and behavior are given by Kraynik [6] and Weaire and Fortes [7]. In this study we are particularly interested in the eect of the ®nal distribution of solid in the cell edges and faces on the stiness and strength of metallic foams. Most relevant to this study are the mechanisms which determine the drainage pattern present in a liquid foam during its evolution from a wet, or undrained, foam to a dry, or drained, foam. At the junction of the cell faces in a cell edge, surface tension causes the liquid±gas interface to be curved in an arc of constant radius called a Plateau border [8]. The surface tension forces acting along the Plateau border cause the ¯uid pressure in the cell edge to be lower than that in the cell face. The drainage pattern in a wet foam, therefore, is from the cell face into the adjacent cell edges, where the Plateau border regions form channels through which gravitational forces drain the liquid over time. In a liquid foam with no impurities, the cell faces will drain until they burst due to the strong surface tension forces. In order for a liquid foam to become stable, the liquid±gas interface must be altered either by introducing a surfactant that
reduces surface tension or by increasing the viscosity of the surface layer [9±11]. A cell face in a liquid foam that is stabilized in this way will drain down to some critical thickness and then stop further drainage. A micrograph of an aluminum foam produced by a liquid state process (Alporas, Shinko Wire, Amagasaki) with Plateau borders is shown in Fig. 1. The drainage pattern of a liquid metal foam can aect the structure of the resulting solid foam in two ways. If the excess liquid is not allowed to drain away suciently before solidi®cation, a density gradient results causing a high degree of material heterogeneity and anisotropy. This has been previously noted in aluminum foam panels produced by the Alcan process [12, 5]. Of more direct relevance to this study is the in¯uence of drainage on the distribution of liquid between the cell faces and the Plateau borders at solidi®cation. As a stable liquid foam drains, the radii of curvature of the Plateau borders and the fraction of total liquid contained within them decrease. Here we study the eect of the ®nal distribution of solid in the cell edges and faces on the stiness and strength of metallic foams. The Young's modulus and plastic collapse stress of a foam can be described by the following equations, derived using dimensional analysis [13]: 2 E* r* r* f 1 ÿ f 1 Es rs rs and
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3=2 spl * r* r* 0:3f 1 ÿ f sYS rs rs
2
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tion made in this analysis is that all cell faces were made planar in order to eliminate the non-linearities caused by wall curvature and isolate the eects of the solid distribution. An aggregate of the tetrakaidecahedral cells which we use to represent a closed cell foam is shown in Fig. 2(b). These cells pack in a body-centered cubic (b.c.c.) lattice with the crystallographic axes h100i, h010i and h001i being parallel to the three normal directions of the square faces [directions X1, X2 and X3 in Fig. 2(b)]. Each cell edge in the tetrakaidecahedral foam consists of the intersection of two hexagonal and one square face. The angle between the planes of the two hexagonal faces is the tetrahedral angle, 109.478, while that between the planes of a hexagonal and square face is 125.268. In three dimensions, the parameter F3 is de®ned as the volume fraction of solid material bounded by the Plateau borders in the cell edges and vertices relative to the total volume of solid material. Fig. 1. Micrograph of an aluminum foam with Plateau borders (Alporas, Shinko Wire, Amagasaki, r* = 200 kg/m3).
2. ANALYSIS
where E*, r* and spl* are the modulus, density and plastic collapse stress of the foam, Es, rs and sYS are those of the solid cell wall material, and f is the volume fraction of solid contained in the cell edges. The dimensional arguments used to obtain these equations assume that the cell edges and faces are of uniform thickness and do not account for the Plateau borders observed in some foams. Here, we use ®nite element analysis to estimate the relative elastic modulus and relative plastic collapse strength of a 2D honeycomb and a 3D closedcell foam as a function of relative density, r*/rs, and the distribution of the solid material. For the 2D analysis, a regular hexagonal honeycomb structure, as shown in Fig. 2(a), is used for simplicity. In two dimensional analysis, the parameter F2 is de®ned as the area of solid material bounded by the Plateau borders at the vertices relative to the total area of solid. Warren and Kraynik [14] and Kraynik and Warren [15] ®nd analytically that the elastic modulus of a low density (r*/rs<0.15) honeycomb is increased when material is shifted away from the midspan of a member towards the nodes. Using ®nite continuum elements, this analysis can be extended to honeycombs of a much higher relative density. Several space-®lling unit cells are available to model 3D foams. While the Weaire-Phelan unit cell [16], an eight cell aggregate of two polyhedra, is the lowest surface energy unit cell yet discovered, the Kelvin tetrakaidecahedron is the lowest surface energy unit cell known consisting of a single polyhedron. The Kelvin cell is de®ned by six planar square faces and eight hexagonal faces that are non-planar, but have zero mean curvature. The one simpli®ca-
All ®nite element analyses were performed using the ABAQUS Analysis Package (Hibbit, Karlsson and Sorensen, Pawtucket, RI), running on a DEC Alphastation 500 (Digital Equipment, Maynard, MA). For the 2D hexagonal honeycomb modeling, the following elements from the ABAQUS element library were used: B22 (3-node quadratic beams) and CPS8 (8-node biquadratic plane stress elements). For the 3D closed-cell tetrakaidecahedral foam modeling, the following elements were used: S9R5 (9-node doubly curved thin shell elements with reduced integration), C3D20 (20-node quadratic bricks) and C3D15 (15-node quadratic triangular prisms). The material model for all ®nite element analyses was isotropic with elastic±perfectly plastic stress±strain behavior. This model was used to eliminate any dependencies of the measured results on a chosen strain hardening exponent. As-cast aluminum alloys typically have an elastic modulus of 69 to 73 GPa and a yield strength of 100 to 300 MPa [17]. The isotropic elastic±perfectly plastic material model used here was given material properties consistent with those of an as-cast aluminum alloy: Es=70 GPa, n = 0.33 and sYS=150 MPa. Input ®les for ABAQUS were produced using the MATLAB (Mathworks, Natick, MA) package. Using these scripts, the re®nement of each model could be explicitly de®ned. In order to balance solution accuracy and solution eciency, re®nement tests were performed for each series of models. In these tests, a model was analyzed at successively higher levels of re®nement. A re®nement level was chosen such that the solution was within 0.02% of the solution obtained with the maximum re®nement. Models with up to 95 000 degrees of freedom could be run.
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Fig. 2. (a) Idealized regular hexagonal honeycomb. (b) Idealized tetrakaidecadedral foam structure.
Two solution methods were used for each model to be analyzed. An applied loading and small deformation linear elastic analysis were ®rst used to measure the elastic modulus and Poisson's ratio. Large displacement theory and an incremental, iterative solution method were then used to produce a stress±strain curve. The solution method used was either the Newton method (higher relative density models) or the Riks method (lower relative density
models) [18]. Displacement boundary conditions were imposed on the appropriate nodes in order to simulate a uniaxial compression test. The reaction forces on the nodes of the displaced boundary were recorded and summed in order to compute the stress at each increment of displacement. A spline curve was ®t to the stress±stain data points and the peak stress of this curve was taken as the measured plastic collapse stress.
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2.1. Hexagonal honeycomb A schematic of an idealized, regular, isotropic hexagonal honeycomb is shown in Fig. 2(a). The unit cell that was used in all analyses is shown in Fig. 3(a). No symmetry assumptions were made in the 2D analysis, so the required boundary conditions of this model are simply that compatibility is maintained between the corresponding elements in the periodically repeating unit cell. The mechanical properties of a low density regular hexagonal honeycomb can be accurately modeled using standard beam theory: 3 E1 * E2 * r* 1:5 3 Es Es rs 12 * 21 * 1 spl * 1 sYS 2
r* rs
2
:
4 5
For higher relative density honeycombs with less slender cell edges, the predictions of beam theory become less accurate due to signi®cant axial and shear deformation of the cell edges and deformation of the solid material at the vertices. Honeycombs with no Plateau borders and straight edges of uniform thickness were modeled through a range of relative densities using either beam or continuum elements. The ®nite beam elements used for the analysis allow for axial and shear deformation in the cell edges, but do not account for any deformation at the vertices. Continuum elements with a sucient mesh density can account for any signi®cant deformation in either the cell edges or the vertices. The relationship between the relative density, the cell edge thickness, t, and the edge length, l, is de®ned by r* 2 t 1 t 2 p ÿ : 6 rs 3l 3 l After running an initial series of tests to con®rm that equal properties were measured in the X1 and X2 directions, all further tests were performed with uniaxial loading in the X2 direction only. The geometry of an idealized hexagonal honeycomb with Plateau borders of radius of curvature Rp and width wp is shown in Fig. 3(b). Considering a representative vertex, the relative density of this honeycomb can be de®ned as r* Ap Ae , At rs
7
where Ap is the area of the solid contained in the Plateau borders at the vertex, p 1 3 Ap 2Rp t2 ÿ pR2p , 4 2 Ae is the area of the solid in the edges outside the Plateau borders, l Rp t=2 Ae 3 ÿ p t, 2 3 and At is the area represented by the joint in the space-®lling honeycomb, p 1 p 3 3 3 2 l : l At 3l 4 2 2 From equation (7), then, the relationship between the relative density, the cell edge thickness, edge length and the Plateau border radius of curvature is 2 r* 2 t 1 t 2 4 2p Rp p ÿ : 8 ÿ p l rs 3 3 3 3l 3 l The fraction of solid in the vertex and Plateau border region, F2, is Fig. 3. (a) Unit cell for idealized hexagonal honeycomb. (b) Geometry of a typical vertex with Plateau borders.
F2
Ap Ap rs , Ap Ae At r*
SIMONE and GIBSON:
EFFECTS ON STIFFNESS AND STRENGTH OF METALLIC FOAMS
Fig. 4. Sample ®nite element meshes of a regular hexagonal honeycomb with r*/rs=0.09 and F2 equal to 0.02, 0.4, and 0.98.
or F2
2 rs 1 t 2 4 Rp t 4 2p Rp p ÿ : r* 3 l l 3 l2 3 3 3 9
For each value of relative density, 0.015, 0.03, 0.06 and 0.09, a series of models were generated and analyzed with the parameter, F2, being varied between close to zero (cell edges of uniform thickness) and unity (all of the solid contained in the Plateau borders). Sample meshes of a hexagonal honeycomb with a relative density of 0.09 and F2 equal to 0.02, 0.4, and 0.98 are shown in Fig. 4. 2.2. Closed-cell tetrakaidecahedral foam The idealized tetrakaidecahedral foam is shown in Fig. 2(b). The unit cell for this structure is shown in Fig. 5. When modeling with ®nite shell elements, no symmetry assumptions were made, so the required boundary conditions of this model were simply that compatibility was maintained between the corresponding elements in the periodically repeating unit cell. The ®rst objective was to evaluate the degree of anisotropy of the tetrakaidecahe-
Fig. 5. (a) Unit cell of a tetrakaidecahedral foam.
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dral foam. Shell element models representing various relative densities were tested in three directions corresponding to the following crystallographic directions of the b.c.c. lattice: h100i (normal to a square face), h111i (normal to a hexagonal face) and h110i (in the plane of a square face and perpendicular to one edge of the square). For relative densities less than or equal to 0.2, the relationship between the relative density, cell face thickness and edge length of a tetrakaidecahedral foam with uniform face thickness is approximated by 2 r* t t 1:185 ÿ 0:4622 : 10 rs l l A tetrakaidecahedral foam unit cell was then modeled using 20-node quadratic bricks. Because of the many degrees of freedom required to produce continuum element models with adequate mesh re®nement, these models were loaded only in the h100i direction, with the cubic symmetry of the tetrakaidecahedral cell being used to reduce the model size. The relationship of the reduced unit cell to the full unit cell is shown in Fig. 6(a, b), while the unit cell itself is shown in Fig. 6(c). Four of the planes which bound the model are simple symmetric boundaries. The normals of the planes of these boundary surfaces are either parallel to or perpendicular to the loading direction. The ®fth boundary is an antisymmetric boundary which requires that the unit cell be compatible with an identical unit cell rotated 1808 about the axis indicated as z±z in Fig. 6. The cross-sectional geometry of the cell edge of a tetrakaidecahedral foam with Plateau borders is shown in Fig. 7. The compromises made in using the tetrakaidecahedral unit cell impose limits on the values of F3 which can be reasonably modeled. Because the angle between the two hexagonal faces is smaller than the angle between a hexagonal face and a square face, the radius of curvature of two of the Plateau borders (Rp,1) on a given cell edge is larger than that of the third (Rp,2). Using simple geometry, it can be seen that Rp,1
t wp 1:932wp 2 tan 54:748=2
and Rp,2
p t wp wp 2, 2 tan 70:528=2
where wp is the width of the Plateau border as de®ned in Fig. 7(b). The relationship between the two Plateau border radii can then be de®ned as 1:932 t 1:932 p ÿ 1 : 11 Rp,1 p Rp,2 2 2 2 As the Plateau borders get smaller, the ratio Rp,1/ Rp,2 increases, causing the elements in the smaller Plateau border to become distorted. Highly dis-
torted brick elements reduce the accuracy of the analysis. For each relative density, a lower bound (excluding the foam with no Plateau borders) was placed on the range of F3 which could be modeled, such that Rp,1/Rp,2<2. Because the square cell faces are much smaller than the hexagonal cell faces, but the width of the Plateau border is the same on both, the upper bound of F3 is reached when the Plateau borders cover the entire square cell face. The shape of the oblate ellipsoid curve that is formed at a vertex, where the Plateau borders meet, was approximated by a series of spline curves. The relationship between the relative density, normalized cell face thickness and the normalized Plateau border width was determined by numerically measuring the unit cell volume of a large number of models over a range of relative densities and F3, and then ®tting a two variable polynomial curve ®t to this data. The relationship for the ranges (0.025 < t/l < 0.1) and (0.05 < r*/rs<0.2) is 2 r* t t 10ÿ3 ÿ 0:0125 1190 ÿ 520 r l l 2 t t wp 0:41 ÿ 339 464 l l l 2 2 t t w 551 ÿ 70 ÿ 1200 l l l 2 3 t t w : 12 ÿ 122 ÿ 198 ÿ 2000 l l l The fraction of solid in the Plateau borders, F3, can then be found using the volume of the cell faces not contained within the Plateau borders, the computed relative density and the volume of the tetrakaidecahedral unit cell: pÿ p 2 3t l ÿ 2wp 2 6t 3 l ÿ 2= 3wp F3 1 ÿ : 13 11:31l 3 r*=rs For each of the relative densities 0.05, 0.1, 0.15 and 0.2, models were generated and analyzed over the available range of F3. Sample meshes of a tetrakaidecahedral foam with a relative density of 0.15 and F3 equal to 0.075, 0.51, and 0.73 are shown in Fig. 8. 3. RESULTS
3.1. Hexagonal honeycomb The ®nite element analysis results for the relative elastic modulus and the relative peak stress of a hexagonal honeycomb with cell edges of uniform thickness are within 10% of the analytical results of Gibson and Ashby [13] for relative densities up to 0.12. The continuum elements give a modulus and strength slightly higher than their model while the beam elements give almost identical modulus and
SIMONE and GIBSON:
EFFECTS ON STIFFNESS AND STRENGTH OF METALLIC FOAMS
Fig. 6. (a) Relationship of reduced continuum element model to the tetrakaidecahedral foam unit cell. (b) Cell wall elements of the reduced unit cell. (c) Perspective of reduced continuum element model.
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Fig. 7. (a) Geometry of the cross-section of a tetrakaidecahedral cell edge. (b) Geometry of the cross-section of a tetrakaidecahedral cell edge with Plateau borders.
strength. The Gibson and Ashby models correlate well with material properties predicted using ®nite beam elements at low relative densities, but predict a slightly higher modulus as relative density increases due to their exclusion of axial and transverse shear deformation. The mechanical properties predicted using continuum elements are somewhat higher than those predicted by beam theory. With increasing relative density and t/l, the size of the vertex increases, decreasing the eective edge length in bending and stiening the honeycomb. The increased vertex size also causes the plastic hinge to form slightly closer to the in¯ection point at the mid-span of the cell edge, increasing the stress required to initiate plastic collapse. The ®nite element analysis results for the mechanical properties of honeycombs with Plateau borders are contained in Fig. 9. The modulus and peak stress values shown are normalized by the modulus and peak stress of a honeycomb with no Plateau borders (as modeled with continuum elements). As F2 increases, the modulus and peak stress at ®rst increase above the values for honeycombs with uni-
form edge thickness and then eventually fall below them. The value of F2 at which the modulus and peak stress fall below those for honeycombs with uniform edge thickness increases with increasing relative density. This behavior can be understood in terms of the bending moment distribution along a cell edge and the redistribution of material from the edge into the Plateau borders with increasing F2. The moment varies linearly from one end to the other and has a value of zero at the beam in¯ection point at mid-span. Moving a fraction of the solid into the Plateau borders increases the moment of inertia of the edge cross-section in the Plateau border region, where the moment is the largest, and decreases the moment of inertia of the edge crosssection outside the Plateau border region where the moment is smallest. For a low density honeycomb, moving material from the cell edge into the Plateau border region causes a small increase in the Plateau border width, wp, but a large relative decrease in the thickness of the cell edge outside the Plateau border, t. The moment of inertia of the cell edge cross-section in the Plateau border region is increased, but because the width of the Plateau border is small, the cell wall outside the Plateau border, which has a signi®cantly reduced moment of inertia, is still subjected to a high bending moment. The ¯exural stiness of the honeycomb is therefore reduced. This is supported by the ®ndings of Warren and Kraynik [14] and Kraynik and Warren [15]. The plastic hinge which initiates plastic collapse forms at a distance roughly equal to wp from the vertex. Because wp is small and the plastic moment capacity of the cell edge cross section is diminished by the reduction in thickness, the stress required to develop plastic hinges, which is equivalent to the peak stress, also decreases. For a high density honeycomb with material moved from the cell edge into the Plateau border region, the increase in the Plateau border width is much larger relative to the decrease in the thickness of the cell edge outside the Plateau border. The increase in moment of inertia near the vertex and the resulting reduction in eective beam length are therefore more signi®cant than the decrease in moment of inertia of the cell edge outside the Plateau border. This results in the modulus increases reported in Fig. 9(a) and in Ref. [14]. Similarly, although the plastic moment capacity of the cell edge cross-section where the plastic hinge forms is slightly reduced, the stress required to initiate hinge formation is larger because of the reduced eective length of the cell edge in bending. This results in the peak stress increases reported in Fig. 9(b). For each relative density, however, there is a critical value of F2, beyond which the transfer of any additional solid material into the Plateau borders leads to a reduction in the mechanical properties. This critical value of F2 increases with
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Fig. 8. Sample ®nite element meshes of a tetrakaidecahedral foam with r*/rs=0.15 and F3 equal to 0.075, 0.51, and 0.73.
relative density, and, for each relative density, is somewhat lower with respect to peak stress than with respect to modulus. 3.2. Closed-cell tetrakaidecahedral foam The ®nite element analysis using shell elements gives Young's moduli which are almost equal for loading in the h100i, h111i, and h110i directions. The results are well described by: h100i: E*=Es 0:3152 r*=rs 0:2089 r*=rs 2 h111i: E*=Es 0:3026 r*=rs 0:3141 r*=rs 2 h110i: E*=Es 0:3325 r*=rs 0:3116 r*=rs 2 for relative densities less than 0.2. The tetrakaidecahedral cell is relatively isotropic with respect to elastic modulus, with the maximum stiness being less than 10% greater than the minimum stiness for relative densities less than 0.2. The highest stiness is measured in what corresponds to the h110i direction in the b.c.c. lattice. The stinesses in the h100i and h111i directions are very close to that in the h110i direction. The h100i direction was chosen for further modeling using continuum elements because the symmetry assumptions that could be made for loading in this direction allowed the model size to be signi®cantly reduced. The relative Young's modulus and peak stress for loading in the h100i direction were estimated using both shell and continuum elements. The results are well described by:
Continuum: E*=Es 0:3163 r*=rs 0:3188 r* =rs 2 Shell: E*=Es 0:3152 r*=rs 0:2089 r*=rs 2 Continuum: spl *=sYS 0:4445 r*=rs 0:3346 r*=rs 2 Shell: spl *=sYS 0:4459 r*=rs 0:3211 r*=rs 2 for relative densities less than 0.2. The modulus predicted by continuum elements is slightly higher than that predicted by shell elements. The peak stresses predicted by the two element types are approximately the same. As previously noted in the hexagonal honeycomb results, the higher modulus measured using continuum elements is due to the increased constraint at the cell edges as the face thickness increases. The ®nite element results suggest that Gibson and Ashby's model (equations (1) and (2)) correctly predicts both a linear and quadratic dependence on relative density, but the constants of proportionality are dierent from those suggested by Gibson and Ashby. The parameter f in these models represents the volume fraction of solid contained in the cell edges. In a tetrakaidecahedral foam cell, the area contained in an edge cross-section, computed using simple geometry, can be expressed in terms of the face thickness, t, as r t2 1 3 Ae : 14 4 1:932 2
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to elastic modulus [Fig. 10(a)], the behavior of the tetrakaidecahedral foams is qualitatively similar to that measured in the hexagonal honeycomb, although the magnitude of the measured modulus increase is much lower. The mechanisms of modulus increase are similar to those previously discussed in the honeycomb analysis, but the in¯uence of these mechanisms on the modulus are reduced because the deformation behavior of the idealized foam is dominated by the axial deformation of cell faces and edges, rather than bending. The peak stress of the tetrakaidecahedral foam model decreases with the formation of a Plateau border for all values of relative density considered [Fig. 10(b)]. The plastic collapse of the model foam is also dominated by axial deformation in the cell faces, rather than bending. The mechanism which increased the peak stress in the honeycomb model with Plateau borders has a lesser in¯uence on the foam model behavior because the cell faces yield axially before the fully plastic moment can develop along the cell edge. The transfer of solid material
Fig. 9. (a) Normalized modulus vs F2, (b) normalized peak stress vs F2 for regular hexagonal honeycombs with Plateau borders.
The volume fraction of the solid contained in the cell edge can be estimated using the edge length per unit volume and the relative density of the tetrakaidecahedral unit cell: 12Ae 1 : 15 f 11:31l 2 r*=rs The magnitude of f becomes negligible at low relative density. The ®nite element analysis results for the mechanical properties of a tetrakaidecahedral closed-cell foam with Plateau borders are summarized in Fig. 10. The modulus and peak stress values shown are normalized by the modulus and peak stress of a tetrakaidecahedral foam with no Plateau borders (i.e. uniformly thick cell faces). The values are interpolated over the range of F3 which could not be modeled due to the nonuniformity of the cell face incidence angles in the tetrakaidecahedral cell. This range was between the value of F3=0, for the foam with no Plateau borders and F3=0.047, 0.094, 0.143 and 0.192, respectively, for the models with relative density 0.05, 0.1, 0.15 and 0.2. With respect
Fig. 10. (a) Normalized modulus vs F3, (b) normalized peak stress vs F3 for tetrakaidecahedral foams with Plateau borders.
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from the cell face into the Plateau border reduces the net area of the cell face with respect to in-plane axial deformation, resulting in a lower measured peak stress. 4. DISCUSSION
Honeycombs deform primarily by bending of the cell edges, with the maximum bending moment occurring at the ends of the member. Shifting material away from the cell edges into Plateau borders at the vertices initially increases the modulus and peak stress of the honeycomb by increasing the moment of inertia at the ends of the members where the bending moments are the highest. As more material is moved, the increase in moment of inertia does not compensate for the reduction of moment of inertia in the cell edge, causing a reduction in the mechanical properties. Similar behavior would be expected in open cell foam structures with respect to the shifting of solid material from the cell edges into the vertices. In contrast, closed cell foams deform primarily by the in-plane stretching of the cell faces. As a result, shifting material from the cell faces of a closed cell foam into Plateau borders along the edges has little eect on the modulus and causes a reduction in the peak stress. As the volume fraction of solid shifted to the edges increases to the point where almost all of the solid material is in the edges rather than the faces (F3 approaches 1) the foam behaves more like an open cell foam. Open cell foams with Plateau borders along the edges have been analyzed by Warren and Kraynik [19]. Changing the shape of the edge cross-section from circular or square to a Plateau border con®guration increases the stiness of these foams by 60±70%. Metallic foams with high relative density (r*/ rs>0.1) have been produced that have a modulus comparable to that of the idealized tetrakaidecahedral foam with no Plateau borders ( [3]; mfr. data). The peak stress values of these foams, however, are signi®cantly below the idealized values. Most metallic foams with lower relative density have both modulus and peak stress values far below the idealized behavior [5]. All of these foams exhibit the microstructure of reasonably dry foams. Cell edges generally have small Plateau borders (0.05 < F < 0.2), although localized regions of higher relative density and large Plateau borders can be observed in many foam samples [5]. The results presented in Fig. 10 suggest that the material distribution probably does not have a signi®cant eect on the modulus of these foams, but may contribute to the lower measured peak stress. The poor mechanical properties of the low relative density foams are more likely attributable to the presence of defects (curvature, corrugations) in the cell faces [5, 20]. If lower density metallic foams can approach the idealized behavior presented here,
Fig. 11. Weight eciency of sandwich panels with idealized foam and honeycomb cores.
their potential for use in structural applications would increase signi®cantly. Structural sandwich panels using conventional low density metallic foam cores are inferior to skin-stringer construction or aluminum honeycomb sandwich construction in terms of weight eciency [21]. A structural sandwich panel with an idealized tetrakaidecahedral closed-cell foam core, however, nearly matches the weight eciency of a honeycomb sandwich. Figure 11 shows the normalized weight eciency of optimized sandwich panels over a range of normalized bending stiness (analysis after Simone and Gibson [21]). The three curves in Fig. 11 represent panels with cores consisting of an idealized open cell foam (with mechanical properties comparable to conventional low density metallic foams), an idealized hexagonal honeycomb and an idealized tetrakaidecahedral closed-cell foam. The faces of the sandwich panels are composed of the same material as the solid material in the cellular cores. Improved weight eciency, such as that illustrated in Fig. 11, in addition to the various secondary bene®ts (damping, ®re resistance, energy absorption, the ability to make 3D sandwich structures) would make sandwich panels with metallic foam cores a cost eective alternative for many structural applications. AcknowledgementsÐWe wish to acknowledge the ®nancial support of the Advanced Research Projects Agency (Contract # PNO 39141). REFERENCES 1. Akiyama, S., Imagawa, K., Kitahara, A., Nagata, S., Morimoto, K., Nishikawa, T. and Itoh, M., Foamed Metal and Method of Producing Same. U.S. Patent No. 4,713,277, 1987. 2. Jin, I., Kenny, L. and Sang, H., Method of Producing Lightweight Foamed Metal. U.S. Patent No. 4,973,358, 1990. 3. Kunze, H. D., Baumeister, J., Banhart, J. and Weber, M., Powder Metall. Int., 1993, 25(4), 182±185.
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4. Prakash, O., Sang, H. and Embury, J. D., Mater. Sci. Engng A, 1995, 199, 195±203. 5. Simone, A. E. and Gibson, L. J., Acta metall. mater., 1997, submitted. 6. Kraynik, A. M., Ann. Rev. Fluid Mech., 1988, 20, 325±357. 7. Weaire, D. and Fortes, M. A., Adv. Phys., 1994, 43(6), 685±738. 8. Plateau, J. A. F., Statique Experimentale et Teorique des Liquides Soumis aux Seules Forces Moleculaires. Gauthier-Villiard, Paris, 1873. 9. Bikerman, J. J., Foams. Springer-Verlag, New York, 1973. 10. Sebba, F., Foams and Biliquid Aprons. John Wiley and Sons, Chichester, U.K., 1987. 11. Walstra, P., in Foams: Physics, Chemistry and Structure, ed. A. J. Wilson. Springer-Verlag, London, 1989. 12. Beals, J. T. and Thompson, M. S., J. Mater. Sci., 1997, 32, 3595±3600. 13. Gibson, L. J. and Ashby, M. F., Cellular Solids: Structure and Properties, 2nd edn. Cambridge University Press, 1997.
14. Warren, W. E. and Kraynik, A. M., Mech. Mater., 1987, 6, 27±37. 15. Kraynik, A. M. and Warren, W. E., in Low Density Cellular Plastics: Physical Basis of Behavior, ed. N. C. Hilyard and A. Cunningham. Chapman and Hall, London, 1994. 16. Weaire, D. and Phelan, R. Phil. Mag. Lett., 1994, 69, 107±110. 17. American Society of Metals, ASM Handbook, Vol. 2 Properties and Selection: Nonferrous Alloys and Special-Purpose Materials. ASM International, Metals Park, OH, 1990. 18. ABAQUS Theory Manual. Hibbet, Karlsson and Sorenson, Inc., Pawtucket, RI, 1995. 19. Warren, W. E. and Kraynik, A. M., J. appl. Mech., 1988, 55, 341. 20. Simone, A. E. and Gibson, L. J., Acta metall. mater., 1997, submitted. 21. Simone, A. E. and Gibson, L. J., J. Mater. Sci., 1997, in print.