Acta Materialia 53 (2005) 303–313 www.actamat-journals.com
Cellular metal lattices with hollow trusses Douglas T. Queheillalt *, Haydn N.G. Wadley Department of Materials Science and Engineering, University of Virginia, 116 Engineers Way, P.O. Box 400745, Charlottesville, Virginia 22904-4745, USA Received 14 June 2004; received in revised form 17 September 2004; accepted 21 September 2004 Available online 22 October 2004
Abstract Cellular metal lattice truss structures are being investigated for use as multifunctional load supporting structures where their other functionalities include thermal management, dynamic load protection and acoustic damping. A simple method for making lattice structures with either solid or hollow trusses is reported. The approach involves laying up collinear arrays of either solid wires or hollow cylinders and then alternating the direction of successive layers. The alternating collinear assembly is metallically bonded by a brazing process. The dimensions of the cylinders, the wall thickness of hollow truss structures and the spacing between the trusses enable independent control of the cell size and the relative density of the structure. The process has been used to create stainless steel lattices with either square or diamond topologies with relative densities from 0.03 to 0.23. The through thickness elastic modulus of these lattice truss structures is found to be proportional to relative density. The square topology has twice the stiffness of the diamond oriented trusses. The peak compressive strengths of both topologies are similar and is controlled by plastic buckling. The structural efficiency of hollow truss structures with a fixed cell size is approximately proportional to the relative density unlike equivalent structures made from solid trusses whose peak strength has been predicted to scale with the cube of the relative density. The experimental data for hollow trusses lie between predictions for trusses with built-in and pin-jointed nodes consistent with experimental observations of constrained node rotation. The use of hollow trusses increases the resistance to buckling offsetting the usually rapid drop in strength as the relative density decreases in cellular systems where truss buckling controls failure. The low relative density hollow truss structures reported here have the highest reported specific peak strength of any cellular metal reported to date. 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Cellular material; Porous material; Stainless steel; Mechanical properties
1. Introduction Numerous techniques have been proposed for making cellular metals [1,2]. Many initially exploited foaming techniques to create metal foams with stochastically distributed cell sizes and shapes with either or open or closed porosity [1]. The closed-cell variants of these metal foams have shown promise for impact energy absorption [3–6] and perhaps acoustic *
Corresponding author. Tel.: +1 434 982 5678; fax: +1 434 982 5677. E-mail address:
[email protected] (D.T. Queheillalt).
damping [7–9]. Open cell foams have been used for cross flow heat exchange [10–20]. Load supporting (structural) applications of these materials have been limited by their low elastic moduli and strengths which have power law dependencies upon density and are greatly inferior to those of honeycombs of the same density [21–23]. The low stiffness and strength of open cell metal foams has generated significant interest in alternative cell topologies which might offer strengths comparable to honeycombs while simultaneously facilitating the other functionalities of open cell metal foams [24–27]. Several open cell systems have been proposed based
1359-6454/$30.00 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2004.09.024
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on truss lattices with tetrahedral [28–35], pyramidal [36– 38], 3D-Kagome [39,40] and woven (metal textile) [41– 44] topologies, Fig. 1. The structural benefits of cellular materials are often realized when configured as the cores of sandwich panel structures. When appropriately oriented, the trusses are subject to either tension or compression during panel bending [28–30]. The strengths of these cellular core sandwich structures are then governed by the stress at which the various panel failure mechanisms (face sheet stretching/wrinkling, core plastic yielding, and either plastic or elastic buckling of core truss elements) are initiated [45–48].
During the compressive loading of cellular trusses only core deformation mechanisms are active. Trusses aligned with the loading direction are the most efficient at supporting stress, while those that are inclined are limited by force resolution considerations [45–48]. The truss failure mechanism (elastic or plastic buckling or plastic yield) depends upon the slenderness ratio of the trusses. Since the slenderness ratio and relative density of a lattice made from solid trusses are interdependent, both the truss strength and responsible failure mecha (defined nism depend upon the lattice relative density, q as the density of the cellular structure, qc, divided by that of the solid material from which it is made, qs), the lattice topology and the material used to make the trusses [27]. At high lattice truss relative densities, elastic-perfectly plastic trusses fail by yielding of the truss columns even when they are inclined. The compressive collapse strength, rpk, scales linearly with the relative density [27] ; rpk ¼ Rry q
ð1Þ
where R is a lattice topology dependent scaling factor and ry is the yield stress of the solid material. Hexagonal honeycomb cores have all of their webs aligned in the loading direction and R = 1.0. For tetrahedral and pyramidal lattices, Fig. 1(a) and (b), R = sin2 x, where x is the angle of inclination between the truss elements and the loading direction. For these two topologies the optimal angle for resisting both through thickness compression and shear is about 55 [37] and in this case R = 0.67. For a diamond lattice truss structure, Fig. 1(d), R = sin2 x but in this case x = 45 and R = 0.50. Zupan et al. [44] have shown that edge effects are important during compressive loading of low aspect ratio diamond truss samples. The load supporting area is reduced because some trusses are not connected to both face sheets. The reduction in load bearing area depends on the truss angle and sample aspect ratio. They suggest that for low aspect ratio samples R = sin2 x[1 (1/Atan x)], where A is the length to height (aspect) ratio of the sample [44]. For square truss structures (made by rotating the core in Fig. 1(d) by 45) half the trusses have x = 0 with the remainder having x = 90. In this case the effective value of R is again 0.5. The compressive strength of lattice structures can exceed these predictions when they are made from made from high work hardening rate materials. Low relative density lattice structures have long slender trusses that fail by elastic buckling. In this case the compressive strength is found by replacing ry in Eq. (1) by the elastic bifurcation stress of a compressively loaded cylindrical column. The stress is given by Fig. 1. Schematic illustrations of microtruss lattice structures with tetrahedral, pyramidal, Kagome and woven textile truss topologies.
rcr ¼
p2 k 2 Es a2 ; l 4
ð2Þ
D.T. Queheillalt, H.N.G. Wadley / Acta Materialia 53 (2005) 303–313
where Es is the elastic modulus, a the radius and l the length of the column [49]. The factor k depends on the rotational stiffness of the nodes; k = 1 for a pin-joint that can freely rotate while k = 2 for fixed-joints which cannot rotate [50]. We note that the relative density of a square or diamond truss lattice made from solid circular trusses depends upon the a/l ratio p a ¼ q : ð3Þ 2 l The elastic buckling strength obtained by substituting Eqs. (2) and (3) into Eq. (1) therefore depends upon 3 . By substituting for a/l in Eq. (2) and equating the q plastic yield and elastic buckling strengths the square and diamond lattice structures can be shown to collapse by elastic buckling when rffiffiffiffiffiffiffiffiffi ry < q : ð4Þ k 2 Es At intermediate relative densities the compressive strength of a truss lattice is controlled by inelastic buckling at an inelastic bifurcation stress given by Shanley– Engesser tangent modulus theory [49]. In this case the compressive strength is found by replacing ry in Eq. (1) by the inelastic bifurcation stress of a compressively loaded column. The peak compressive strength for inelastic buckling of a square or diamond lattice truss structure is then given by rpk ¼ k 2 Et sin2 x q3 ;
ð5Þ
where Et is the tangent modulus. For both a square and diamond topology the sin2 x term is again 0.5 and the predicted through thickness compressive strength of the two topologies is predicted to be the same. Note that for solid circular trusses where either elastic or plastic 3 , buckling controls the peak strength scales with q whereas when the strength is controlled by plastic yielding it scales linearly with the relative density. Hollow truss structures provide a means for increasing the second moment of inertia of the trusses and thus their resistance to elastic or plastic buckling. They also provide a means for varying the cellular structures relative density without changing the cell size (while maintaining a constant truss slenderness ratio); a feature that could be used in the optimization of multifunctional systems. Here, a method for making truss lattice structures with either solid or hollow circular cross-section trusses is described. The method enables fabrication of sandwich structures with controlled relative density, cell orientation and size with a cell topology analogous to that of metal textile structures. However, unlike the metal textile process, this approach does not require weaving (so the trusses remain straight) and it is more amenable to the fabrication of hollow trusses with potentially
305
superior buckling resistances to their solid counterparts. The processing method and compressive mechanical properties for solid and hollow trusses in both a 0/90 (i.e., square) and ±45 (i.e., diamond) orientation with identical cell sizes are presented and explored.
2. Lattice truss structure fabrication Cellular structures were assembled from 304 stainless steel solid wires and hollow tubes using a tool to align the cylinders in collinear layers, Fig. 2(a). The orientation of successive layers was alternated to create the lattice truss architecture. The cylinders were metallurgically bonded using a brazing technique [51,52]. A Wall Colmonoy NICROBRAZ alloy 51 with a nominal composition of Ni–25Cr–10P–0.03C (wt%) was used for bonding. These brazing alloys contain (a) Lattice truss layup in tool tool
hollow tubes or solid wires
(b) Lattice truss machined
(c) Face sheets attached
H
W L
Fig. 2. (a) The stacking arrangement for the hollow truss structures, (b) removing the lattice truss structure from the brazing assembly and cutting to size and (c) attaching the face sheets to form a sandwich panel with a diamond orientation. An analogous process is used for the fabrication of square oriented samples.
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melting point depressants such as boron, phosphorous or silicon to achieve desirable liquid flow and wetting behavior. The alloy was applied to each truss as a powder contained in a polymer binder. The alternating collinear assembly was placed in a vacuum furnace for high-temperature brazing and heated at 10 C/min up to 550 C, held for 20 min (to volatilize and remove the polymer binder), then heated to 1020 C, for 60 min at 104 Torr before furnace cooling to ambient temperature at 25 C/min. After removing the bonded cellular structures from their tooling, they were cut to size (length L = 50 mm, width W = 30 mm, height H = 15 mm) using wire electro-discharge machining, Fig. 2(b). A second brazing treatment was then used to attach 304 stainless steel (0.1000 thick) face sheets to the cellular lattice truss structure, Fig. 2(c). Photographs of the lattice truss structures with solid and hollow trusses and cells oriented in both the square (0/90) and diamond (±45) orientations are shown in Fig. 3. Fig. 4 shows micrographs of truss–truss nodes and the truss–face sheet interface for diamond oriented lattice structures. Fig. 5 shows cross-sectional micrographs of a truss–truss node and a truss–face sheet interface for square oriented lattice truss structures. It can be seen that during transient liquid phase bonding, capillary action draws the molten braze material into the region where the cylinders make contact and forms a thick fillet at the contact point. The molten braze material is also drawn to the truss face sheet contacts forming a contact that is several times that of the tube
Fig. 4. Micrographs of nodes at: (a) truss–truss nodes; (b) truss–face sheet connections for diamond lattice truss structures.
Fig. 3. Photographs showing: (a) square orientation; (b) diamond orientation solid and hollow microtruss lattice structures.
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True tensile stress (MPa)
1000
307
(a)
304 stainless steel
800
600
400
200
Es = 203 GPa 0
0
0.1
0.2
0.3
0.4
0.5
0.4
0.5
True tensile strain
Tangent modulus, Et (MPa)
4000
Fig. 5. Cross-sectional micrographs showing: (a) truss–truss node; (b) truss–face sheet bond. Dashed lines show the approximate edge positions of the original tube and face sheet boundary.
(b)
3000
2000
1000
0
0
0.1
0.2
0.3
True tensile strain
cross-sectional area. It can be seen from Fig. 5 that there is good braze metal penetration of the base metal providing a strong bond between the trusses and face sheet and at truss to truss connections (nodes). The square and diamond oriented lattice structures with both solid and hollow trusses were tested in compression at a nominal strain rate of 4 · 102 s1. The measured load cell force was used to calculate the nominal stress applied to the structure. The nominal through thickness strain was obtained from a laser extensometer on the sample centerline.
Fig. 6. (a) Average uniaxial tensile response (true stress–true strain) for 304 stainless steel and (b) Shanley–Engesser tangent modulus.
shown in Fig. 6(b). Note its rapid change with plastic strain. The density, qs, of the steel was 8.0 gm/cm3 [53].
3. Relative density calculation Fig. 7 shows a unit cell of the lattice structure created by the alternating collinear cylinder array process. If the Lattice Truss Structure
2.1. Parent material response In order to compare measured and predicted values of strength, the uniaxial tensile response of 304 stainless steel subjected to the same thermal history as the lattice truss structures was determined. The tensile response measured at a strain rate of 104 s1 is shown in Fig. 6(a). The elastic modulus and 0.2% offset yield strength for 304 stainless steel were 203 GPa and 176 MPa, respectively. Significant work hardening occurred in the plastic region. The tangent modulus given by the slope dr/de of the true stress–true strain response is
Unit cell
2ao
2ai
l l Fig. 7. Truss stacking sequence and a unit cell of the hollow microtruss lattice structure.
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Table 1 Tube parameters and corresponding relative densities (calculated and measured)
Outside (2ao, mm) 1.481 1.473 1.473 1.473 1.473
Inside (2ai, mm) 0.000 1.067 1.219 1.270 1.372
Calculated 0.233 0.110 0.073 0.060 0.031
Measured 0.23 ± 0.005 0.11 ± 0.003 0.08 ± 0.003 0.06 ± 0.003 0.03 ± 0.003
Stress (MPa)
Relative density ð qÞ
Tube diameter
80
(a) Square Orientation
60 0.23
40 0.11
20 0.08 0.06 0.03
0
The center-to-center cell size (l) was 5 mm for all samples.
0
0.2
0.4
0.6
0.4
0.6
Strain
2
V c ¼ 4ao l :
ð7Þ
, is the ratio of the metal volume The relative density, q to that of the unit cell ¼ q
V s pða2o a2i Þ : ¼ 2ao l Vc
(b) Diamond Orientation
40 0.23
20 0.11
ð6Þ
If the cylinders have an inner diameter, 2ai, the volume occupied by solid metal in the unit cell is V s ¼ 2plða2o a2i Þ:
Stress (MPa)
60
added weight of the braze alloy is ignored; only the diameter of the wires, the wall thickness and diameter of the hollow cylinders, and the cell size determine the relative density of the cellular structure. For a stacking of cylinders of outer diameter, 2ao, and a center-to-center cylinder (cell) spacing of l; the unit cell volume
ð8Þ
¼ pao =2l which is identical to the relative As ai ! 0, q density expression for a plain woven textile structure, Eq. (3) [41]. The relative density given by Eq. (8) is independent of the orientation of the cellular truss structure. Table 1 compares the measure and predicted relative densities using Eq. (8). It can be seen that the measured relative densities are well predicted by Eq. (8).
4. Results and analysis 4.1. Compressive response The through thickness compressive nominal stress– strain responses for square and diamond oriented samples are shown in Fig. 8(a) and (b). They exhibit characteristics typical of many cellular metal structures including a region of elastic response, plastic yielding, a post yield peak stress followed by a plateau region and finally hardening associated with densification that began at a densification strain (eD) of 50–60%. Core ‘‘softening’’, especially for the square trusses, was observed once the peak compressive strength of the structure was surpassed. ¼ 0:11 Photographs of hollow truss structures with q undergoing compressive loading are shown in Fig. 9 at
0.08 0.06 0.03
0 0
0.2
Strain Fig. 8. Compressive stress–strain response for sandwich structures with: (a) square oriented; (b) diamond oriented lattice truss structures.
nominal strains of 0%, 10%, 20%, 30% and 40%. It can be clearly seen in Fig. 9(a) that as the square orientation lattice truss structure was compressed the trusses parallel to the loading direction cooperatively buckled near their midpoints. This buckling coincided with the peak in strength, Fig. 8. The buckling half wavelength corresponded to the face sheet separation distance. As deformation progressed beyond peak load, the trusses continued to cooperatively buckle causing the core to extend laterally beyond the face sheets. This was accompanied by a significant decrease in flow stress. Some node rotation can be observed in Fig. 8. The horizontal trusses provided an efficient means of coupling the buckling behavior of the individual vertical trusses. The diamond orientation lattice truss structure buckled differently. While the onset of truss buckling again coincided with the peak in strength, Fig. 9(b), shows that as the diamond lattice truss structure was compressed from 0% to 10% the truss members buckled with a half wavelength approximately that of the cell size. The nodes acted as plastic hinges and significant rotation again occurred. As deformation progressed, the truss members continued to buckle and the core again began to barrel outwards. The higher order deformation buckling mode of the trusses was accompanied by a smaller post peak drop in flow stress. The solid truss structure buckling behaved similarly to that of the hollow trusses for both orientations.
D.T. Queheillalt, H.N.G. Wadley / Acta Materialia 53 (2005) 303–313
309
sheets, x1 = 0 and x2 = 90, and Rel = 0.50. For a diamond lattice truss structure ignoring edge effects x = 45 and Rel = 0.25 [44]. Edge effects for low aspect ratio diamond truss samples can be important and in this case, Rel = sin4 x[1 (1/A tan x)]. For the samples tested here (A = 3.3) the predicted modulus drops from 0.25 to 0.18. The non-dimensional modulus coefficient Rel ¼ E= qEs for both the experimental data (collected from unload/reload data prior to yielding) and the predicted moduli for these lattice truss structures is shown in Fig. 10. Fig. 10 shows that the predictions and measurements are in very good agreement. 4.3. Compressive strength The compressive peak strength of a hollow lattice structure can be predicted from the inelastic buckling strength of a truss member. The inelastic buckling stress of a hollow cylinder is given by 2 ao þ a2i p2 k 2 IEt 2 2 rcr ¼ ¼ p k Et ; ð10Þ 4l2 A c l2 where I is the second moment of inertia and Ac the cross-sectional area of the column. Note that the stress for the onset of buckling in Eq. (10) is minimized when the columns are solid (i.e., ai = 0). Inserting Eq. (10) for ry in Eq. (1) gives the peak compressive strength for a lattice truss structure failing by inelastic buckling 2 ao þ a2i 2 2 rpk ¼ p k Et q: ð11Þ sin2 x 4l2 This result applies to both square and diamond oriented lattice truss structures. The elastic buckling behavior of the lattice truss structures can by found by replacing Et in Eq. (11) with 0.75
Fig. 9. Photographs showing the deformation characteristics of: (a) square orientation hollow truss structure; (b) diamond orientation hollow truss samples at plastic strains of 0%, 10%, 20%, 30% and 40%.
4.2. Elastic moduli
0.45
0.3
Zupan and Fleck have shown the out-of-plane modulus, E, of a metal textile structure scales linearly with the relative density [44] ; E ¼ Rel Es q
0.6
ð9Þ
where Rel = sin4 x. For a square lattice truss structure Rel = V1sin4 x1 + V2sin4 x2, where V1 is the volume fraction of trusses making contact with the face sheets and V2 is the volume fraction of trusses parallel to the face
0.15 Square (measured) Diamond (measured)
0 0
0.05
0.1
0.15
0.2
0.25
Fig. 10. Comparisons between the measured and predicted elastic moduli coefficient, Rel, of square and diamond oriented lattice truss structures as a function of relative density.
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Es. The transition from elastic buckling to plastic yielding or inelastic buckling can then be found by equating the resulting elastic buckling expression to the plastic strength relation (Eq. (1)) or that for plastic buckling (Eq. (11)). The relative densities of all samples investigated here were well above the elastic buckling–plastic and elastic buckling–inelastic buckling transitions, Table 2. The predicted and measured non-dimensional peak for strength coefficient R ¼ rpk = qry is plotted against q both the square and diamond lattice truss structures in Fig. 11(a) and (b), respectively. The measured value of the peak strength coefficient for the square lattice truss structure was approximately R = 1.2 and independent . That of the diamond lattice truss structure was also of q but had a lower value of R = 0.85. independent of q However, the sample aspect ratio was low (A = 3.3) and thus the strength was reduced by a factor of 0.7. When this sample size effect was accounted for, R = 1.2 for both the square and diamond lattices. The predicted and measured responses for the square and diamond topology lattices are shown on Fig. 11. The predictions for the diamond lattices were reduced by a factor of 0.7 to account for the samples low aspect ratio. Only the plastic yield mechanism with R = 0.5, Eq. (1) predicts a structural efficiency parameter that is independent of relative density. Strengths significantly exceeded this and were experimentally observed to be coincident with buckling. The experimental data are seen to lie between the k = 1 and k = 2 predictions for plastic buckling consistent with the observed constrained node rotation, Fig. 8. These results can be compared to recently reported measurements for square oriented textile (solid truss) structures where the strength coefficient varied from 0.50 to 0.65 for relative densities of 0.17 to 0.31 [41]. Recent measurements for diamond oriented textile structures had through thickness peak strength coefficients of 0.30–0.70 for relative densities ranging from 0.18 to 0.23 [44]. The solid truss structures fabricated by the alternating collinear cylinder lay-up process de-
Table 2 Relative densities for the transition of elastic buckling to plastic yielding for square and diamond lattice truss samples with fixed and pinned end conditions Sample relative density ð qÞ
0.23 0.11 0.08 0.06 0.03
Transition relative density Square orientation
Diamond orientation
k=2
k=1
k=2
k=1
0.015 0.006 0.004 0.003 0.001
0.030 0.011 0.007 0.006 0.003
0.018 0.007 0.004 0.003 0.002
0.011 0.014 0.009 0.007 0.003
2
(a) Square Inelastic buckling fixed nodes (k=2)
1.6
1.2 Experimental data
0.8
Elastic buckling (solid trusses) Plastic yielding
0.4
0
0
1.5
Inelastic buckling pinned nodes (k=1)
0.05
0.1
0.15
0.2
0.25
(b) Diamond Inelastic buckling fixed nodes (k=2)
1.2
0.9
0.6
0.3
0
0
Elastic buckling (solid trusses)
Experimental data Plastic yielding
Inelastic buckling pinned nodes (k=1)
0.05
0.1
0.15
0.2
0.25
Fig. 11. Comparisons between the measured and predicted peak strength coefficient, R, of (a) square and (b) diamond oriented lattice truss structures as a function of relative density. A sample aspect ratio of 3.3 was used for the strength predictions of the diamond lattice.
scribed here are more than 1.7 times more efficient at supporting compressive loads than their metal textile counterparts. The normalized strength coefficient of the square and diamond lattice truss structures is compared with competing stainless steel cores in Fig. 12 including, woven metal textile, pyramidal and square honeycomb cores [44]. At a relative density of 0.03, the hollow truss structures are 2.4 times stronger than pyramidal trusses about 2.6 times stronger than square honeycombs. The low relative density hollow lattice truss structures made by the alternating collinear lay up process appear to exhibit higher compressive strengths than any other cellular metal topology reported to date. 4.4. Impact energy absorption Preferred impact energy absorbing structures have stress strain curves that exhibit yielding followed by a
D.T. Queheillalt, H.N.G. Wadley / Acta Materialia 53 (2005) 303–313
Wv ¼
Z
1.000
(a)
0.100
Square orientation Diamond orientation
Alternating Colinear Layup
Wvol /
long plateau with no work hardening. Ideal structures collapse at a constant (plateau) stress which establishes a fixed load level transmitted to the supporting structure [54]. Numerous structures have been assessed for such behavior including circular tubes, square tubes, honeycombs, corrugated tubes, sandwich plates and metal foams. Cellular structures are of considerable interest for impact protection because of their high specific impact energy absorption and peak to plateau stress ratio near unity [2]. The energy absorption per unit volume, Wv, can be used as a merit index to compare different cellular structures. This specific energy is defined from the area under the nominal stress nominal strain curve as
311
Metal Textiles
0.010 Pyramidal lattice truss
0.001 0.001
0.01
0.1
1.0
/ 1.00
eD
(b)
ð12Þ
rðeÞ de; o
where r(e) is the flow stress of the structure and eD is the densification strain. The corresponding energy absorption per unit mass is calculated by dividing Eq. (12) by the samples density, i.e., product of the relative density and the parent alloyÕs density qs. q Fig. 13(a) and (b) shows the energy absorption per unit volume and mass, respectively, for the square and diamond oriented samples. It is compared with data recently reported for stainless steel woven metal textile lattice truss structure and pyramidal truss structures [44,55]. Fig. 13(a) shows the energy absorption per unit volume of the square and diamond hollow truss structures are comparable with woven textile lattice truss and pyramidal truss structures. However, when the energy absorption is normalized per unit mass, Fig. 13(b), the square and diamond hollow truss structures
10 Square orientation Diamond orientation Diamond woven textile Pyramidal truss Square honeycomb
1
0.1 0.001
0.01
0.1
1
Fig. 12. Comparisons between the peak strength coefficient, R, of the square and diamond oriented lattice truss structures and stainless steel square honeycomb, pyramidal, and woven metal textile cores as a function of relative density.
/
Alternating Colinear Layup
Metal Textiles Pyramidal lattice truss
0.10
Square orientation Diamond orientation
0.01 0.001
0.01
0.1
1.0
/ Fig. 13. Energy absorption (a) per unit volume and (b) per unit mass for the square and diamond orientation samples and various other core topologies.
are comparable with the pyramidal lattice truss structures and significantly higher than the woven textile lattice truss structures. The superior strength and energy absorption characteristics of the structures fabricated by the alternating collinear lay-up process is due to an absence of the plastic kinks created at crossing points in a woven textile structure. These imperfections reduce the truss buckling strength under axial loading which in turn reduces the macroscopic peak strength of the cellular structures. We also note that low relative density (
