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Mechanical property of lattice truss material in sandwich panel including strut flexural deformation
Composite Structures 94 (2012) 3448–3456
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Mechanical property of lattice truss material in sandwich panel including strut flexural deformation Hailong Chen b, Qing Zheng b,c, Long Zhao b,c, Yu Zhang b, Hualin Fan a,d,⇑ a
Laboratory of Structural Analysis for Defense Engineering and Equipment, College of Mechanics and Materials, Hohai University, Nanjing 210098, China PLA University of Science and Technology, Nanjing 210007, China c State Key Laboratory of Automotive Safety and Energy, Tsinghua University, Beijing 100084, China d State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200030, China b
a r t i c l e
i n f o
Article history: Available online 13 June 2012 Keywords: Lattice composites Sandwich Mechanical properties Strength Deformation
a b s t r a c t Lattice truss materials are usually assumed to be stretching dominated neglecting the bending resistance of struts. In this paper, bending resistance of struts is considered for lattice truss sandwich panels. The mechanical behaviors are not only decided by the relative density of the lattice and the strut inclination, but also the slenderness ratio of the strut. For stout and hierarchical struts, the slenderness ratio turns to smaller, and the shear force and the bending moment are comparable to the strut axial force. Compared with the stretching dominated theory, the stiffness of the lattice material should be improved while the strength reduced, which has been proved to be more consistent with experimental results. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Stretching dominated principle is adopted to design weight efficient lightweight lattice truss materials, which appears to be mechanically competitive alternatives to foam and honeycomb structures when configured as the core of a sandwich panel [1,2]. Numerous studies have addressed sandwich panel design for various lattice truss sandwich structures [2–4]. In analytical predictions, lattice structures are always simplified as pin-jointed trusses, where struts are only axially stretched or compressed. The flexural deformation and bending resistance of struts are neglected. The predicted equivalent stiffness and yield strength of the lattice truss material is proportional to its relative density [1–3]. Analytical predictions for strengths are always greater than experimental data, while predicted stiffness is a little smaller [5–9]. The discrepancy between predicted and measured yield strengths at low relative density appears to be a manifestation of manufacturing defects introduced during fabrication, as pointed out by Kooistra et al. [5]. They also concluded that analytical model using a built-in node assumption should predict the higher relative density response of lattices reasonably well, while the lower relative density data should be better approximated by a pin-jointed truss-facesheet connection condition consistent with less end constraint of slender trusses. Recently lattice structures with hollow
struts [6] and hierarchical walls [10] have been constructed and found to be significantly stronger than their solid truss counterparts. In experiments no evidence of node rotation was observed, which reveals that the pin-joint assumption is not applicable for stout struts with higher radius of gyration. Finnegan [7] and Xiong et al. [8] adopted built-in ends to predict mechanical properties and failure modes of low density carbon fiber composite pyramidal truss structures. In this paper, the strut is treated as a beam to deduce the equivalent mechanical property of lattice truss structures in the sandwich panel. The error of the stretching dominated principle would be discussed. 2. Topology of sandwich panel with pyramidal lattice core As shown in Fig. 1, the sandwich panel comprises of two solid skins and a pyramidal lattice truss core. Area of a representative unit cell in plane xy, S, is given by
S ¼ 2ðh tan h þ dÞ2 ;
where h is the height of the panel in z direction, h is the angle between a strut and axis z and d denotes the distance between neighboring struts as shown in Fig. 1. Volume of a unit cell, V, is
V ¼ 2hðh tan h þ dÞ2 : ⇑ Corresponding author at: Laboratory of Structural Analysis for Defense Engineering and Equipment, College of Mechanics and Materials, Hohai University, Nanjing 210098, China. E-mail address: [email protected] (H. Fan). 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.06.004
ð1Þ
ð2Þ
The relative density of the lattice material, q⁄, defined by the ratio of the volume of struts to the volume of the representative unit cell, is given by
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H. Chen et al. / Composite Structures 94 (2012) 3448–3456
mechanical property of the lattice truss core is proportional to its relative density.
z skin
3. Properties of the lattice including bending resistance
y d 3 x
1
h θ
2 3.1. Out-of-plane compression Under out-of-plane compression along z direction, the four struts in a unit cell sustain identical force as shown in Fig. 2. In compression, the deformation of the strut is along axis z and denoted by dz. Deformations in plane xy are assumed to be zero ideally. In the theoretical analysis, only one strut was included in the model. The axial deformation of the strut, dl, is
4
dl ¼ dz cos h;
ð8Þ
and the transverse deformation, dt, is
skin
dt ¼ dz sin h:
ð9Þ
According to the deformation at the strut end, strut forces are given by Fig. 1. Topology of the sandwcih panel with pyramidal lattice core.
EA dz cos2 h; h 12EI Q ¼ 3 dz sin h cos3 h; h
N¼ 4Ah 4A 2A ¼ ¼ q ¼ ; V cos h S cos h cos hðh tan h þ dÞ2
ð3Þ
M¼
rcs ¼ q rs cos x;
6EI h
2
dz sin h cos2 h
ð5Þ
1 2 Gcs ¼ q E sin 2x; 8 pffiffiffi 2 scs ¼ q rs sin 2x; 4
P ¼ N cos h þ Q sin h ¼
ð6Þ
EA 12EI 2 dz cos3 h þ 3 dz sin h cos3 h; h h
re ¼
where Ecs, rcs, Gcs and scs denote the compressive stiffness, the compressive strength, the shear stiffness and the shear strength of the lattice, respectively. Symbols E and rs denote the modulus and initial yield strength of the strut material, respectively. The
4P EA 48EI 2 ¼4 cos3 hee þ sin h cos3 hee ; 2 S S Sh
z
P y
M
δt δz δ l
x
N
Q
θ
z y
δxy
z
y
Ty
δly
δty
M
δy
x
δx
ϕ
ð14Þ
where the equivalent compression strain ee in z direction is equal to dz/h. Then the equivalent compression stiffness of the lattice, Ee, is given by
δz
(b)
ð13Þ
and the equivalent compression stress of the lattice, re, is defined as
ð7Þ
(a)
ð12Þ
for axial force N, shear force Q, and bending moment M, respectively. Symbol I denotes the moment of inertia of the strut. The equilibrium equation along axis z is given by
ð4Þ
2
ð11Þ
and
where A denotes the cross section area of the strut. Under strecthing dominated assumption, analytical expressions for the stiffness and strength of a pyramidal lattice truss core sandwich structure with elastic–ideally plastic struts are given by [9]
Ecs ¼ q E cos4 x;
ð10Þ
δxy
δy
Q
θ
x Fig. 2. Deformation and forces of (a) compressed and (b) sheared struts.
N
3450
re 4EA 3 48EI 2 ¼ sin h cos3 h; cos h þ 2 S ee Sh
ð15Þ
Combining I ¼ Ar 2 and Eq. (3), Eq. (15) is simplified as
Ee ¼ q E cos4 hð1 þ gÞ
ð16Þ
with
g ¼ 12
2 sin h : k
r1;3
ð17Þ
Coefficient g denotes the enhancement of the newly suggested equivalent compression stiffness compared with previous model where bending effects have been neglected. The effective slenderness ratio k of the strut is defined by
k¼
Coupling the axial compression force and the shear force, the maximum/minimum axial stress at the neutral axis of the beam, r1,3 is given by
ð18Þ
Symbol r is the radius of gyration. For circular cross sections with radius pffiffiffi r, r ¼ r=2. For rectangular cross sectionswith height of t, r ¼ 3t=6. Including the strut bending resistance, the compression stiffness of the lattice core should be enlarged, as shown in Fig. 3. The analytical prediction was also checked by commercial ANSYS code, where beam elements, BEAM3, were adopted for struts to compare with the calulated modulus using bar elements. In numerical calcultion, the material is linear elastic isotropic and each strut was divided into 100 elements. When the slenderness ratio of the stout strut gets smaller, the enhancement of the equivalent compression stiffness turns to greater. The strut inclination also greatly influences the mudulus of the lattice. Larger inclination (larger h) leads to greater stiffness enhancement. The enhancement can be negletced only when the slenderness ratio is large enough, for example, k > 20. It could be concluded that previous model should be applied to lattice truss materials with rather small relative densities. Ratios of the shear force and the bending moment to the axial force are given by
Q 6 sin 2h ¼ ; N k2
ð19Þ
2rs ¼ 1þ E cos2 h
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!1 256 2 : 1 þ 4 sin 2h k
ð22Þ
According to Eq. (16), the equivalent initial yield strength of the lattice, res, is
res ¼ q rs cos2 hg1
ð23Þ
with
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!1 2 ! sin h 256 2 1 þ 1 þ 4 sin 2h g1 ¼ 2 1 þ 12 k k
(a)
ð24Þ
0.4
0.3
0.2 0
θ =45
and
0.1
M 6 sin h ¼ ; Nh k2
ð21Þ
Under the maximum stress principle, the corresponding equivalent strain of the lattice at yielding, eezs, is given by
eezs
h : r
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 N2 4Q þ 3A 4A2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! E 256 2 2 ¼ eez cos h 1 1 þ 4 sin 2h : 2 k
N ¼ 2A
Q /N
Ee ¼
H. Chen et al. / Composite Structures 94 (2012) 3448–3456
0
30 ,60
0
ð20Þ
respectively. As shown in Fig. 4, the ratio cannot be neglected when the strut is stout enough, which means the shear force and the bending moment will obviously influence the lattice strength.
0.0 4
8
12
16
λ
(b)
0.6
Analytical prediction Numerical simulation
0.3
η
M /(N⋅h)
0.4
θ =60
0
0.2
0.2 0
θ =60 0
45
0.1
0
45
0
30
0
30 0.0
0.0 0
4
8
12
16
20
λ Fig. 3. Compression stiffness enhancement with respect to effective slenderness ratio.
4
8
12
16
λ Fig. 4. Ratios of (a) shear force to axial force and (b) bending moment to axial force under compression with respect to effective slenderness ratio.
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H. Chen et al. / Composite Structures 94 (2012) 3448–3456
(a)
0.8
1.5
Analytical prediction Numerical simulation
0
θ =60
0.7 0
η3
η1
θ = 30 1.2
0
45
0.6
0
45
0
60 0.5 0
30 0.9
0.4 0
4
8
12
16
20
0
4
0
θ =60
0
η2
45 1.0
E h
rbar ¼ dz cos2 h þ
0
30
0
4
8
12
16
20
dz ¼
λ Fig. 5. Strength reduction for (a) the maximum stress principle and (b) the Mises rule with respect to effective slenderness ratio.
h
2
dz sin h cos2 h;
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r21 þ r22 þ ðr1 r2 Þ2 ¼ ¼ rs : 2
ð29Þ
ð25Þ
ð26Þ
According to Eq. (16), the equivalent initial yield strength of the lattice is
res ¼ q rs cos2 hg2
ð27Þ
with
2 ! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!1 sin h 192 2 1 þ 12 : 1 þ 4 sin 2h k k
ð30Þ
and the equivalent compression yield strength of the lattice material is
ð31Þ
with
when yielded, the corresponding equivalent strain of the lattice is given by
1=2 2r s 256 2 sin 2h : 1 þ 3 1 þ E cos2 h k4
1 6dE cos2 h þ ; sin h cos2 h E h
rs h
res ¼ q rs cos2 hg3
with r2 = 0, the Mises yield stress, rM, is given by
g2 ¼
6EdE
where dE is the maximum distance between the edge points to the neural axis. For circular cross sections with radius r, dE = r. For rectangular cross sectionswith height of t, dE = t/2. When rbar = rs, the equivalent stress of the lattice material is equal to its initial yield strength res with
0.8
eez ¼
20
At the edge of the cross section of the beam, the coupling effect between the axial force and the bending moment must be discussed seriously. The maximum axial stress in the beam, rbar, is given by
1.2
rM
16
Fig. 6. Compression strength reduction with respect to effective slenderness ratio.
1.4
0.6
12
λ
λ
(b)
8
ð28Þ
Variations of g1 and g2 with k and h are depicted in Fig. 5, which reveals that the shear force will not reduce the strength of the lattice. So the coupling between the axial force and the shear force should not control the load capacity of the lattice.
g3 ¼
1þg pffiffiffiffiffiffi : 1 þ 2 3g
ð32Þ
Controlled by coupled axial compression and bending, the strength of the lattice should be reduced, as shown in Fig. 6. When k < 8, the suggested strength is less than a half of the strength predicted by previous models. Strength reduction gets larger for dense lattices. With smaller inclination (smaller h), the lattice would have better anti-compression performance with greater strength. Analytical predictions were also checked by commercial ANSYS code where beam elements (BEAM3) were adopted to simulate the strut to compare with the calulated strength using bar elements, as shown in Fig. 6. In numerical calcultion, the material is linear elastic isotropic and each strut was divided into 100 elements. Previous models overestimated compression strengths of lattices with stouter struts [11]. Tested strengths were always smaller than predicted data, which ordinarily be explained by manufacturing defaults. According to the improved model, the bending deformation of the strut is an important factor for lattice strength reduction, as shown in Fig. 7, which shows that the suggested model captures the strength dependence upon relative density. The analytical prediction of the compressive yield strengths are also plotted for elastic buckling. For the buckling cases it was assumed that the struts were either built-in at the facesheets, whereupon k = 2, or pin-jointed for which k = 1. The factor k depends on the rotational stiffness of the constraints at the nodes. Fig. 7 shows
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H. Chen et al. / Composite Structures 94 (2012) 3448–3456
0.6
Summation of shear forces along axis y of a unit cell is
0.5
Ty ¼ Plastic yield
σes/( ρ*σs )
0.3
Tx ¼ Buckling
0.0 0.00
0.01
0.02
0.03
2EA 24EI 24EI 2 dx cos h sin h þ 3 dx cos5 h þ 3 dx cos3 h; h h h
ð46Þ
where Tx is the shear force along axis x. According to Fig. 2b, to a shear displacement dxy, shear displacements along x and y directions are given by
Experimental data [11] Previous model Suggested model
0.1
ð45Þ
Accordingly, the summation of shear forces along axis x of a unit cell under shear displacment dx is
0.4
0.2
2EA 24EI 24EI 2 dy cos h sin h þ 3 dy cos5 h þ 3 dy cos3 h: h h h
dy ¼ dxy sin u and dx ¼ dxy cos u;
0.04
ρ*
ð47Þ
respectively. The shear force along the shear displacement direction, Txy, is
Fig. 7. Compression strength compared with experiments [11].
T xy ¼ T x cos u þ T y sin u;
that the elastic buckling model captures the strength dependence upon relative density for samples smaller than 1.0% by the k = 2 approximation in compression [11].
ð48Þ
and extended to be
2EA 24EI 24EI 2 dxy cos h sin h þ 3 dxy cos5 h þ 3 dxy cos3 h: h h h
T xy ¼
ð49Þ
Combining with Eq. (34), the shear force is given by
3.2. Shear As shown in Fig. 2, the axial deformation, dly, of struts 1 and 2 under shear deformation dy along y direction is
dly ¼ dy sin h;
where the shear strain, cxy, is defined by the ratio of dxy to h. The shear force is irrelavant to angle u. The equivalent shear stress of the lattice, sxy, is defined as
ð34Þ
sxy ¼ q E cos2 h sin2 h þ
According to the deformation at the strut end, strut forces are given by
EA dy cos h sin h; h 12EI Q ¼ 3 dy cos4 h; h 6EI M ¼ 2 dy cos3 h; h
N¼
EA 12EI 2 dy cos h sin h þ 3 dy cos5 h: h h
ð36Þ
with
ð37Þ
g4 ¼
where Ty is the shear force along axis y. The axial deformation of struts 3 and 4 is
dly ¼ 0;
12 k2
cos4 h þ
k2
cos2 h cxy :
ð51Þ
sxy 1 2 ¼ q E sin 2h½1 þ g4 cxy 8
12 k2
ð52Þ
ð1 þ cos2 hÞcot2 h:
ð53Þ
Including the strut bending resistance, the shear stiffness of the lattice core should be enlarged, as shown in Fig. 8. Slenderness ratios get smaller for stout or hierarchical struts, and the stiffness enhancement turns to greater. The enhancement can be negletced when the slenderness ratio is large enough, for example, k > 20.
ð39Þ
2.0
Analytical prediction Numerical simulation
and the transverse deformation is
dty ¼ dy :
12
The shear modulus, Gxy, is given by
Gxy ¼
ð38Þ
1 2
ð35Þ
respectively. The equilibrium equation in shear direction is given by
T y ¼ N sin h þ Q cos h ¼
ð50Þ
ð33Þ
and the transverse deformation, dty, is
dty ¼ dy cos h:
12 12 2 T xy ¼ 2EA cos h sin h þ 2 cos4 h þ 2 cos2 h cxy ; k k
ð40Þ 1.5
N ¼ 0; 12EI Q ¼ 3 dy cos3 h; h
ð41Þ
0
θ =30
1.0
ð42Þ
0
45
and
M¼
η4
According to the deformation at the strut end, strut forces are given by,
0.5
6EI h
2
dy cos2 h:
0.0
The equilibrium equation in the shear direction is given by
Ty ¼ Q ¼
12EI h
3
dy cos3 h:
0
60
ð43Þ 0
4
8
12
16
20
λ
ð44Þ Fig. 8. Shear stiffness enhancement with respect to effective slenderness ratio.
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H. Chen et al. / Composite Structures 94 (2012) 3448–3456
Inclination of the strut also greatly influences the shear mudulus of the lattice. With smaller inclination (smaller h), the stiffness enhancement is much more obvious. The analytical prediction was also checked by commercial ANSYS code, where beam elements (BEAM3) were adopted for struts to compare with the calulated shear modulus using bar elements. In numerical calcultion, the material is linear elastic isotropic and each strut was divided into 100 elements. The forces under shear deformation dxy are given by
EA dxy cos h sin h sin u; h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12EI 2 Q ¼ 3 dxy cos3 h cos2 h sin u þ cos2 u; h
N¼
ð54Þ ð55Þ
and
M¼
6EI h
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 dxy cos2 h cos2 h sin u þ cos2 u;
ð56Þ
for struts 1 and 2, respectively, and
EA dxy cos h sin h cos u; h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12EI 2 Q ¼ 3 dxy cos3 h cos2 h cos2 u þ sin u; h
N¼
ð57Þ ð58Þ
for struts 3 and 4, respectively. Ratio of the shear force to the axial force is given by
Q 12 cos2 h ¼ N k2 sin h
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M 6 ¼ 2 cot h cos2 h þ tan2 u: Nh k
M¼
h
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 dxy cos h cos2 h cos2 u þ sin u;
ð61Þ
for struts 1 and 2, respectively. As shown in Fig. 9, the ratio cannot be neglected when the strut is stout enough, which means the shear force and the bending moment will obviously influence the strength of the lattice. With smaller inclination (smaller h), the shear force and bending moment increase rapidly when struts turn to stouter, which should greatly influence the lattice strength. Coupling the axial compression force with the shear force, the maximum/ minimum axial stress at the central line of the beam is given by E 2
r1;3 ¼ cxy 0
cos h sin h sin u@1
2
ð60Þ
and the ratio of the bending moment to the axial force is given by
and
6EI
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos2 h þ tan2 u;
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 64 12 cos2 h 1þ cos2 h þ tan2 u A: 2 9 k sin h
ð62Þ
ð59Þ According to Eq. (52), the equivalent initial yield strength of the lattice, ses, is
(a)
0.35
ses ¼ ϕ =45 ϕ =00
0.30
θ =30
0.25
0
g4 ¼2 1 þ
12
cot2 h
0
30
ð64Þ
0
60
0.10
cos2 hcot2 h þ
k2 k2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 64 12 cos2 h @1 þ 1 þ cos2 h þ tan2 u A : 9 k2 sin h
0.15 0
12
0
0
45
ð63Þ
with
0
0.20
Q /N
1 q rs sin 2hg4 4 sin u
45
with r2 = 0, the Mises yield stress, rM, is given by 0.05
0
rM ¼ Ecxy
60
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 16 12 cos2 h cos2 h þ tan2 u cos h sin h sin u 1 þ 3 k2 sin h
0.00 4
8
12
16
λ
(b)
¼ rs :
0.35
30
ð65Þ
ϕ =450 ϕ =00
0.30 0
when yielded, the corresponding equivalent initial yield stress of the lattice is
M /(N⋅h)
0.25 0.20
θ =30
0
45
ses ¼
0
1 q rs sin 2hg5 4 sin u
ð66Þ
with
0.15
0
45 0.10
g5 ¼ 1 þ 0
"
60
0.05 0.00 4
8
12
16
λ Fig. 9. Ratios of (a) shear force to axial force and (b) bending moment to axial force under shearing with respect to effective slenderness ratio.
k2
cos2 h cot2 h þ
12
cot2 h
k2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 #1=2 16 12 cos2 h cos2 h þ tan2 u : 1þ 3 k2 sin h
0
60
12
ð67Þ
Variations of g4 and g5 with k and h are close to or even greater than 1, which reveals that the shear force will not reduce the lattice strength. The coupling between the axial force and the shear force should not control the load capacity of the lattice.
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H. Chen et al. / Composite Structures 94 (2012) 3448–3456
At the edge of the cross section of the beam, the coupling effect between the axial force and the bending moment must be considered. The maximum stress in struts 1 and 2 are determined by
E h
rbar ¼ dxy cos h sin h sin / þ
6EdE
dxy h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cos2 h cos2 h sin / þ cos2 /: 2
ð68Þ
when rbar = rs, the corresponding equivalent strain cxy of the lattice material is
rs E cos h
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 12 2 cos h cos2 h sin u þ cos2 u : k
sin h sin u þ
ð69Þ According to Eq. (52), the equivalent initial yield stength of the lattice is
ses ¼
1 q rs sin 2hg6 4 sin u
ð70Þ
with
g6 ¼ 1 þ
12 2
k
cos2 hcot2 h þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 12 cot h cos2 h þ cot2 u cot2 h 1 þ k k
12
ð71Þ
2
for struts 1 and 2 and
1 q rs cos h sin hg7 2 cos u
(a)
ð72Þ
1.0
k2
cos2 hcot2 h þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 12 2 cot h 1 þ cot h cos2 h þ tan2 u k k2
12
1.2
Analytical prediction Numerical simulation 1.0
0
θ = 60 0.8
η7
η7
0
0
45
ϕ =0
30
0
0
ϕ =15
0
0.4 0
4
8
12
16
20
0
4
8
12
16
20
λ
λ
(c)
0
0.6
0
30
0.6
0.5
θ = 60
0.8
45
0.7
1.0
Analytical prediction Numerical simulation
(d)
0.9
0.6
Analytical prediction Numerical simulation
0.9
0.8
0
θ = 60
0
θ = 60 0.7
0
45 0
45 0.6
0
30
0
ϕ = 45
0
30
0.5 ϕ =300 4
8
12
λ
16
20
ð73Þ
for struts 3 and 4, respectively. Controlled by coupled axial compression and bending, the initial yield strength of the lattice should be greatly reduced, even for slender struts with k > 20, as shown in Fig. 10. The reduction would be greater for dense lattices. With smaller inclination (smaller h), the shear strength reduces much more rapidly. The analytical prediction was also checked by commercial ANSYS code, where beam elements (BEAM3) were adopted for struts to compare with the calulated shear strength using bar elements. In numerical calcultion, the material is linear elastic isotropic and each strut was divided into 100 elements. For lattices with stouter struts, previous models overestimated their shear strengths [11]. In experiments, tested strengths were always smaller than predicted data, which ordinarily be explained by manufacturing defaults. In fact, the flexural deformation of the strut is an important factor for strength reduction, especially for lattices with stout struts and small inclinations. As shown in Fig. 11, compared with previous model, the suggested model combining with elastic buckling captures the shear strength dependence upon relative density for samples and leads to a smaller deviations. Fig. 11 shows that the elastic buckling model captures the strength dependence upon relative density for samples smaller than 2.5% by the k = 1 approximation in shear, where the node may be sheared and rotation constraints would be weakened [11].
(b)
Analytical prediction Numerical simulation
12
0.9
η7
ses ¼
g7 ¼ 1 þ
η7
cxy ¼
with
0.3
0
0
4
8
12
16
λ
Fig. 10. Shear strength reduction for (a) u = 00; (b) u = 150; (c) u = 300 and (d) u = 450 with respect to effective slenderness ratio.
20
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H. Chen et al. / Composite Structures 94 (2012) 3448–3456
0.5
τes/( ρ*σs )
0.3
Plastic yield
0.2
0.5
0.4
σb13/(ρ*σs)
0.4
(a)
Experimental data [11] Previous model Suggested model
Buckling
Experimental data [5] Previous prediction Suggested prediction
0.3
0.2
Yield strength
0.1 0.1
Buckling 0.0 0.00
0.01
0.02
0.03
0.04
0.0 0.00
ρ*
0.05
Fig. 11. Predicted shear strength compared with experiments [11].
(b)
4. Sandwich panel with tetrahedral lattices Kooistra et al. [5] tested the shear behavior of age hardenable 6061 aluminum sandwich panels with tetrahedral lattice truss cores made of struts of rectangular cross sections. According to the suggested method, behaviors of tetrahedral lattice truss sandwich panel structures were predicted, where bending resistance of struts were included in the modified equations. The modified shear modulus, Gb13, is
Gb13 ¼
1 q E sin2 2hg8 8
0.10
0.15
ρ* 0.5
σb13/(ρ*σs)
0.4
Experimental data [5] Previous prediction Suggested prediction
0.3
0.2
Yield strength
ð74Þ
0.1
Buckling
with
g8 ¼ 1 þ
pffiffiffi 3 q sin xð1 þ sin2 hÞ: 2
0.0 0.00
ð75Þ
0.05
The modified initial yield shear strength, rb13, is
1 4
rb13 ¼ q rs sin 2hg9
ð76Þ
with
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11 pffiffiffi 9 3 3 g9 ¼ g8 @1 þ q sin hA : 2 0
ð77Þ
Experimental data were compared with analytical predictions. As shown in Figs. 12 and 13, ratios of Gb13/(q⁄E) and rb13/(q⁄rs) are constant in previous analytical predictions. In experiments, ratios of Gb13/(q⁄E) are smaller than previous predictions and get 0.16
Gb13/( ρ*E )
0.12
0.08
Experimental data [5] Previous prediction Suggested prediction
0.04
0.00 0.00
0.02
0.04
0.06
0.08
0.10
0.15
ρ*
0.10
Fig. 13. Comparisons between experimental data [5] and analytical predictions for (a) initial shear strength and (b) ultimate shear strength in the aged hardened condition.
larger when the relative density increases. The modified model suggested in this paper reveals this regualtion. The initial yield strength should be overestimated in previous models, where all the cross section is assumed to yield simultaneously. In fact, when the maximum stress equals to the yield strength, the lattice truss structures woud be in plastic yield state. The modified model is much more consistent, as shown in Fig. 13. Fig. 13 shows that the elastic buckling model captures the strength dependence upon relative density by the k = 2 approximation for aged hardened samples with higher yield stresses, while by the k = 1 approximation for annealed samples with rather smaller yield stresses [5]. After the initial yield, stresses in the cross section will continously get larger if the strut material is plastic hardening. Tested data were determined by 0.2% offset yield strength, which will be larger than the initial yield strength predicted by elastic–ideally plastic model of struts, especially for lattices with large relative density. It must be pointed that Timoshenko beam models must be more suitable for stout struts, although they were not adopted here. To predict of the peak load of tetrahedral lattices, the ultimate shear strength, rp13, is suggested as
1 4
rp13 ¼ g10 g11 q rs sin 2hg9 ; 0.12
ρ* Fig. 12. Comparisons between experimental data [5] and analytical predictions for shear stiffness.
ð78Þ
with
g10 ¼ rp =rs ;
ð79Þ
>where rp is the ultimate strength of the lattice material, and g11 denotes the ratio of the ultimate plastic strength to the initial yield
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H. Chen et al. / Composite Structures 94 (2012) 3448–3456
(a)
0.5
σp13/(ρ*σs )
0.4
Experimental data [5] Suggested prediction
Ultimate yield strength
0.3
0.2
0.1
0.0 0.00
Buckling
0.05
0.10
0.15
ρ*
(b)
1.2
1.0
Experimental data [5] Suggested prediction Ultimate yield strength
σp13/(ρ*σs )
0.8
Acknowledgements
0.6
0.4
0.2
0.0 0.00
(2) Including the strut flexural deformation, the shear force and the bending moment of the strut play an important role in the mechanical behavior of the lattice. The coupling between the axial force and the bending moment would reduce the lattice strength, especially for lattices with stout struts. The influence of the bending deformation can be neglected when k > 20. (3) The analytical model suggested in this paper reveals that the stiffness should be a little greater while the strength should be smaller than those predicted by the stretching domainated principle. The discrepancy would be enlarged for dense lattices. The suggested model would make a prediction more consistent with experiments. (4) Mechanical properties are not only decided by the relative density and the strut incliantion, but also depend on the slenderness ratio of struts. With identical relative density and strut incliantion, stouter lattices have greater stiffness but smaller strength. Strut inclinations greatly affect lattice properties. In compression, the strength reduction would be enlarged with increasing inclination (increasing h), while in shearing struts with smaller inclination (smaller h) are apt to be bended failure.
Buckling
0.05
0.10
0.15
ρ* Fig. 14. Comparisons between experimental data [5] and analytical predictions for (a) initial shear strength and (b) ultimate shear strength in the annealed condition.
strength load of elastic-perfectly plastic beams under bending. For rectangular cross sections, g11 = 1.5. In the aged hardened condition and in the annealed condition, value of g10 is 1.14 and 2.97 respectively [5]. As shown in Fig. 14, predictions are well consistent with experimental results, much better than the prediction of the initial yield strength, which indicates that ultimate plastic strength of bent beams can be used to predict the ultimate strength of the lattice truss material. 5. Conclusions The equivalent mechanical property of the lattice truss material was discussed including the flexural deformation and bending resistance of the strut in lattice structures. According to the research, it can be concluded that: (1) Analytical predictions based on the stretching dominated principle overestimated the strength of lattice truss materials, while underestimated their stiffness. For stout lattices, the discrepancy gets larger, which has been revealed by experiments.
Supports from National Natural Science Foundation of China (11172089), Program for New Century Excellent Talents in University, Science Fund for Creative Research Groups of the National Natural Science Foundations of China (51021001), State Key Laboratory of Automotive Safety and Energy of Tsinghua University (KF11031) and State Key Laboratory of Ocean Engineering of Shanghai Jiao Tong University (Grant No. 1001) are gratefully acknowledged. References [1] Deshpande VS, Ashby MF, Fleck N A. Foam topology bending versus stretching dominated architectures. Acta Mater 2001;49:1035–40. [2] Fan HL, Zeng T, Fang DN, et al. Mechanics of advanced fiber reinforced lattice composites. Acta Mech Sinica 2010;26(6):825–35. [3] Deshpande VS, Fleck NA, Ashby MF. Yield of truss core sandwich beams in 3-point bending. Int J Solids Struct 2001;38:6275–305. [4] Leekitwattana M, Boyd SW, Shenoi RA. Evaluation of the transverse shear stiffness of a steel bi-directional corrugated-strip-core sandwich beam. J Construct Steel Res 2011;67:248–54. [5] Kooistra GW, Queheillalt DT, Wadley HNG. Shear behavior of aluminum lattice truss sandwich panel structures. Mater Sci Eng A 2008;472:242–50. [6] Queheillalt DT, Wadley HNG. Pyramidal lattice truss structures with hollow trusses. Mater Sci Eng A 2005;397:132–7. [7] Finnegan K, Kooistra G, Wadley HNG, et al. The compressive response of carbon fiber composite pyramidal truss sandwich cores. Int J Mater Res 2007;98:1264–72. [8] Xiong J, Ma L, Wu LZ, et al. Fabrication and crushing behavior of low density carbon fiber composite pyramidal truss structures. Compos Struct 2010;92:2695–702. [9] Queheillalt DT, Murty Y, Wadley HNG. Mechanical properties of an extruded pyramidal lattice truss sandwich structure. Scripta Mater 2008;58:76–9. [10] Kooistra GW, Deshpande VS, Wadley HNG. Hierarchical corrugated core sandwich panel concepts. J Appl Mech 2007;4:259–68. [11] Queheillalt DT, Wadley HNG. Titanium alloy lattice truss structures. Mater Des 2009;30:1966–75.
Contents lists available at SciVerse ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Mechanical property of lattice truss material in sandwich panel including strut flexural deformation Hailong Chen b, Qing Zheng b,c, Long Zhao b,c, Yu Zhang b, Hualin Fan a,d,⇑ a
Laboratory of Structural Analysis for Defense Engineering and Equipment, College of Mechanics and Materials, Hohai University, Nanjing 210098, China PLA University of Science and Technology, Nanjing 210007, China c State Key Laboratory of Automotive Safety and Energy, Tsinghua University, Beijing 100084, China d State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200030, China b
a r t i c l e
i n f o
Article history: Available online 13 June 2012 Keywords: Lattice composites Sandwich Mechanical properties Strength Deformation
a b s t r a c t Lattice truss materials are usually assumed to be stretching dominated neglecting the bending resistance of struts. In this paper, bending resistance of struts is considered for lattice truss sandwich panels. The mechanical behaviors are not only decided by the relative density of the lattice and the strut inclination, but also the slenderness ratio of the strut. For stout and hierarchical struts, the slenderness ratio turns to smaller, and the shear force and the bending moment are comparable to the strut axial force. Compared with the stretching dominated theory, the stiffness of the lattice material should be improved while the strength reduced, which has been proved to be more consistent with experimental results. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Stretching dominated principle is adopted to design weight efficient lightweight lattice truss materials, which appears to be mechanically competitive alternatives to foam and honeycomb structures when configured as the core of a sandwich panel [1,2]. Numerous studies have addressed sandwich panel design for various lattice truss sandwich structures [2–4]. In analytical predictions, lattice structures are always simplified as pin-jointed trusses, where struts are only axially stretched or compressed. The flexural deformation and bending resistance of struts are neglected. The predicted equivalent stiffness and yield strength of the lattice truss material is proportional to its relative density [1–3]. Analytical predictions for strengths are always greater than experimental data, while predicted stiffness is a little smaller [5–9]. The discrepancy between predicted and measured yield strengths at low relative density appears to be a manifestation of manufacturing defects introduced during fabrication, as pointed out by Kooistra et al. [5]. They also concluded that analytical model using a built-in node assumption should predict the higher relative density response of lattices reasonably well, while the lower relative density data should be better approximated by a pin-jointed truss-facesheet connection condition consistent with less end constraint of slender trusses. Recently lattice structures with hollow
struts [6] and hierarchical walls [10] have been constructed and found to be significantly stronger than their solid truss counterparts. In experiments no evidence of node rotation was observed, which reveals that the pin-joint assumption is not applicable for stout struts with higher radius of gyration. Finnegan [7] and Xiong et al. [8] adopted built-in ends to predict mechanical properties and failure modes of low density carbon fiber composite pyramidal truss structures. In this paper, the strut is treated as a beam to deduce the equivalent mechanical property of lattice truss structures in the sandwich panel. The error of the stretching dominated principle would be discussed. 2. Topology of sandwich panel with pyramidal lattice core As shown in Fig. 1, the sandwich panel comprises of two solid skins and a pyramidal lattice truss core. Area of a representative unit cell in plane xy, S, is given by
S ¼ 2ðh tan h þ dÞ2 ;
where h is the height of the panel in z direction, h is the angle between a strut and axis z and d denotes the distance between neighboring struts as shown in Fig. 1. Volume of a unit cell, V, is
V ¼ 2hðh tan h þ dÞ2 : ⇑ Corresponding author at: Laboratory of Structural Analysis for Defense Engineering and Equipment, College of Mechanics and Materials, Hohai University, Nanjing 210098, China. E-mail address: [email protected] (H. Fan). 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.06.004
ð1Þ
ð2Þ
The relative density of the lattice material, q⁄, defined by the ratio of the volume of struts to the volume of the representative unit cell, is given by
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H. Chen et al. / Composite Structures 94 (2012) 3448–3456
mechanical property of the lattice truss core is proportional to its relative density.
z skin
3. Properties of the lattice including bending resistance
y d 3 x
1
h θ
2 3.1. Out-of-plane compression Under out-of-plane compression along z direction, the four struts in a unit cell sustain identical force as shown in Fig. 2. In compression, the deformation of the strut is along axis z and denoted by dz. Deformations in plane xy are assumed to be zero ideally. In the theoretical analysis, only one strut was included in the model. The axial deformation of the strut, dl, is
4
dl ¼ dz cos h;
ð8Þ
and the transverse deformation, dt, is
skin
dt ¼ dz sin h:
ð9Þ
According to the deformation at the strut end, strut forces are given by Fig. 1. Topology of the sandwcih panel with pyramidal lattice core.
EA dz cos2 h; h 12EI Q ¼ 3 dz sin h cos3 h; h
N¼ 4Ah 4A 2A ¼ ¼ q ¼ ; V cos h S cos h cos hðh tan h þ dÞ2
ð3Þ
M¼
rcs ¼ q rs cos x;
6EI h
2
dz sin h cos2 h
ð5Þ
1 2 Gcs ¼ q E sin 2x; 8 pffiffiffi 2 scs ¼ q rs sin 2x; 4
P ¼ N cos h þ Q sin h ¼
ð6Þ
EA 12EI 2 dz cos3 h þ 3 dz sin h cos3 h; h h
re ¼
where Ecs, rcs, Gcs and scs denote the compressive stiffness, the compressive strength, the shear stiffness and the shear strength of the lattice, respectively. Symbols E and rs denote the modulus and initial yield strength of the strut material, respectively. The
4P EA 48EI 2 ¼4 cos3 hee þ sin h cos3 hee ; 2 S S Sh
z
P y
M
δt δz δ l
x
N
Q
θ
z y
δxy
z
y
Ty
δly
δty
M
δy
x
δx
ϕ
ð14Þ
where the equivalent compression strain ee in z direction is equal to dz/h. Then the equivalent compression stiffness of the lattice, Ee, is given by
δz
(b)
ð13Þ
and the equivalent compression stress of the lattice, re, is defined as
ð7Þ
(a)
ð12Þ
for axial force N, shear force Q, and bending moment M, respectively. Symbol I denotes the moment of inertia of the strut. The equilibrium equation along axis z is given by
ð4Þ
2
ð11Þ
and
where A denotes the cross section area of the strut. Under strecthing dominated assumption, analytical expressions for the stiffness and strength of a pyramidal lattice truss core sandwich structure with elastic–ideally plastic struts are given by [9]
Ecs ¼ q E cos4 x;
ð10Þ
δxy
δy
Q
θ
x Fig. 2. Deformation and forces of (a) compressed and (b) sheared struts.
N
3450
re 4EA 3 48EI 2 ¼ sin h cos3 h; cos h þ 2 S ee Sh
ð15Þ
Combining I ¼ Ar 2 and Eq. (3), Eq. (15) is simplified as
Ee ¼ q E cos4 hð1 þ gÞ
ð16Þ
with
g ¼ 12
2 sin h : k
r1;3
ð17Þ
Coefficient g denotes the enhancement of the newly suggested equivalent compression stiffness compared with previous model where bending effects have been neglected. The effective slenderness ratio k of the strut is defined by
k¼
Coupling the axial compression force and the shear force, the maximum/minimum axial stress at the neutral axis of the beam, r1,3 is given by
ð18Þ
Symbol r is the radius of gyration. For circular cross sections with radius pffiffiffi r, r ¼ r=2. For rectangular cross sectionswith height of t, r ¼ 3t=6. Including the strut bending resistance, the compression stiffness of the lattice core should be enlarged, as shown in Fig. 3. The analytical prediction was also checked by commercial ANSYS code, where beam elements, BEAM3, were adopted for struts to compare with the calulated modulus using bar elements. In numerical calcultion, the material is linear elastic isotropic and each strut was divided into 100 elements. When the slenderness ratio of the stout strut gets smaller, the enhancement of the equivalent compression stiffness turns to greater. The strut inclination also greatly influences the mudulus of the lattice. Larger inclination (larger h) leads to greater stiffness enhancement. The enhancement can be negletced only when the slenderness ratio is large enough, for example, k > 20. It could be concluded that previous model should be applied to lattice truss materials with rather small relative densities. Ratios of the shear force and the bending moment to the axial force are given by
Q 6 sin 2h ¼ ; N k2
ð19Þ
2rs ¼ 1þ E cos2 h
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!1 256 2 : 1 þ 4 sin 2h k
ð22Þ
According to Eq. (16), the equivalent initial yield strength of the lattice, res, is
res ¼ q rs cos2 hg1
ð23Þ
with
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!1 2 ! sin h 256 2 1 þ 1 þ 4 sin 2h g1 ¼ 2 1 þ 12 k k
(a)
ð24Þ
0.4
0.3
0.2 0
θ =45
and
0.1
M 6 sin h ¼ ; Nh k2
ð21Þ
Under the maximum stress principle, the corresponding equivalent strain of the lattice at yielding, eezs, is given by
eezs
h : r
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 N2 4Q þ 3A 4A2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! E 256 2 2 ¼ eez cos h 1 1 þ 4 sin 2h : 2 k
N ¼ 2A
Q /N
Ee ¼
H. Chen et al. / Composite Structures 94 (2012) 3448–3456
0
30 ,60
0
ð20Þ
respectively. As shown in Fig. 4, the ratio cannot be neglected when the strut is stout enough, which means the shear force and the bending moment will obviously influence the lattice strength.
0.0 4
8
12
16
λ
(b)
0.6
Analytical prediction Numerical simulation
0.3
η
M /(N⋅h)
0.4
θ =60
0
0.2
0.2 0
θ =60 0
45
0.1
0
45
0
30
0
30 0.0
0.0 0
4
8
12
16
20
λ Fig. 3. Compression stiffness enhancement with respect to effective slenderness ratio.
4
8
12
16
λ Fig. 4. Ratios of (a) shear force to axial force and (b) bending moment to axial force under compression with respect to effective slenderness ratio.
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H. Chen et al. / Composite Structures 94 (2012) 3448–3456
(a)
0.8
1.5
Analytical prediction Numerical simulation
0
θ =60
0.7 0
η3
η1
θ = 30 1.2
0
45
0.6
0
45
0
60 0.5 0
30 0.9
0.4 0
4
8
12
16
20
0
4
0
θ =60
0
η2
45 1.0
E h
rbar ¼ dz cos2 h þ
0
30
0
4
8
12
16
20
dz ¼
λ Fig. 5. Strength reduction for (a) the maximum stress principle and (b) the Mises rule with respect to effective slenderness ratio.
h
2
dz sin h cos2 h;
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r21 þ r22 þ ðr1 r2 Þ2 ¼ ¼ rs : 2
ð29Þ
ð25Þ
ð26Þ
According to Eq. (16), the equivalent initial yield strength of the lattice is
res ¼ q rs cos2 hg2
ð27Þ
with
2 ! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!1 sin h 192 2 1 þ 12 : 1 þ 4 sin 2h k k
ð30Þ
and the equivalent compression yield strength of the lattice material is
ð31Þ
with
when yielded, the corresponding equivalent strain of the lattice is given by
1=2 2r s 256 2 sin 2h : 1 þ 3 1 þ E cos2 h k4
1 6dE cos2 h þ ; sin h cos2 h E h
rs h
res ¼ q rs cos2 hg3
with r2 = 0, the Mises yield stress, rM, is given by
g2 ¼
6EdE
where dE is the maximum distance between the edge points to the neural axis. For circular cross sections with radius r, dE = r. For rectangular cross sectionswith height of t, dE = t/2. When rbar = rs, the equivalent stress of the lattice material is equal to its initial yield strength res with
0.8
eez ¼
20
At the edge of the cross section of the beam, the coupling effect between the axial force and the bending moment must be discussed seriously. The maximum axial stress in the beam, rbar, is given by
1.2
rM
16
Fig. 6. Compression strength reduction with respect to effective slenderness ratio.
1.4
0.6
12
λ
λ
(b)
8
ð28Þ
Variations of g1 and g2 with k and h are depicted in Fig. 5, which reveals that the shear force will not reduce the strength of the lattice. So the coupling between the axial force and the shear force should not control the load capacity of the lattice.
g3 ¼
1þg pffiffiffiffiffiffi : 1 þ 2 3g
ð32Þ
Controlled by coupled axial compression and bending, the strength of the lattice should be reduced, as shown in Fig. 6. When k < 8, the suggested strength is less than a half of the strength predicted by previous models. Strength reduction gets larger for dense lattices. With smaller inclination (smaller h), the lattice would have better anti-compression performance with greater strength. Analytical predictions were also checked by commercial ANSYS code where beam elements (BEAM3) were adopted to simulate the strut to compare with the calulated strength using bar elements, as shown in Fig. 6. In numerical calcultion, the material is linear elastic isotropic and each strut was divided into 100 elements. Previous models overestimated compression strengths of lattices with stouter struts [11]. Tested strengths were always smaller than predicted data, which ordinarily be explained by manufacturing defaults. According to the improved model, the bending deformation of the strut is an important factor for lattice strength reduction, as shown in Fig. 7, which shows that the suggested model captures the strength dependence upon relative density. The analytical prediction of the compressive yield strengths are also plotted for elastic buckling. For the buckling cases it was assumed that the struts were either built-in at the facesheets, whereupon k = 2, or pin-jointed for which k = 1. The factor k depends on the rotational stiffness of the constraints at the nodes. Fig. 7 shows
3452
H. Chen et al. / Composite Structures 94 (2012) 3448–3456
0.6
Summation of shear forces along axis y of a unit cell is
0.5
Ty ¼ Plastic yield
σes/( ρ*σs )
0.3
Tx ¼ Buckling
0.0 0.00
0.01
0.02
0.03
2EA 24EI 24EI 2 dx cos h sin h þ 3 dx cos5 h þ 3 dx cos3 h; h h h
ð46Þ
where Tx is the shear force along axis x. According to Fig. 2b, to a shear displacement dxy, shear displacements along x and y directions are given by
Experimental data [11] Previous model Suggested model
0.1
ð45Þ
Accordingly, the summation of shear forces along axis x of a unit cell under shear displacment dx is
0.4
0.2
2EA 24EI 24EI 2 dy cos h sin h þ 3 dy cos5 h þ 3 dy cos3 h: h h h
dy ¼ dxy sin u and dx ¼ dxy cos u;
0.04
ρ*
ð47Þ
respectively. The shear force along the shear displacement direction, Txy, is
Fig. 7. Compression strength compared with experiments [11].
T xy ¼ T x cos u þ T y sin u;
that the elastic buckling model captures the strength dependence upon relative density for samples smaller than 1.0% by the k = 2 approximation in compression [11].
ð48Þ
and extended to be
2EA 24EI 24EI 2 dxy cos h sin h þ 3 dxy cos5 h þ 3 dxy cos3 h: h h h
T xy ¼
ð49Þ
Combining with Eq. (34), the shear force is given by
3.2. Shear As shown in Fig. 2, the axial deformation, dly, of struts 1 and 2 under shear deformation dy along y direction is
dly ¼ dy sin h;
where the shear strain, cxy, is defined by the ratio of dxy to h. The shear force is irrelavant to angle u. The equivalent shear stress of the lattice, sxy, is defined as
ð34Þ
sxy ¼ q E cos2 h sin2 h þ
According to the deformation at the strut end, strut forces are given by
EA dy cos h sin h; h 12EI Q ¼ 3 dy cos4 h; h 6EI M ¼ 2 dy cos3 h; h
N¼
EA 12EI 2 dy cos h sin h þ 3 dy cos5 h: h h
ð36Þ
with
ð37Þ
g4 ¼
where Ty is the shear force along axis y. The axial deformation of struts 3 and 4 is
dly ¼ 0;
12 k2
cos4 h þ
k2
cos2 h cxy :
ð51Þ
sxy 1 2 ¼ q E sin 2h½1 þ g4 cxy 8
12 k2
ð52Þ
ð1 þ cos2 hÞcot2 h:
ð53Þ
Including the strut bending resistance, the shear stiffness of the lattice core should be enlarged, as shown in Fig. 8. Slenderness ratios get smaller for stout or hierarchical struts, and the stiffness enhancement turns to greater. The enhancement can be negletced when the slenderness ratio is large enough, for example, k > 20.
ð39Þ
2.0
Analytical prediction Numerical simulation
and the transverse deformation is
dty ¼ dy :
12
The shear modulus, Gxy, is given by
Gxy ¼
ð38Þ
1 2
ð35Þ
respectively. The equilibrium equation in shear direction is given by
T y ¼ N sin h þ Q cos h ¼
ð50Þ
ð33Þ
and the transverse deformation, dty, is
dty ¼ dy cos h:
12 12 2 T xy ¼ 2EA cos h sin h þ 2 cos4 h þ 2 cos2 h cxy ; k k
ð40Þ 1.5
N ¼ 0; 12EI Q ¼ 3 dy cos3 h; h
ð41Þ
0
θ =30
1.0
ð42Þ
0
45
and
M¼
η4
According to the deformation at the strut end, strut forces are given by,
0.5
6EI h
2
dy cos2 h:
0.0
The equilibrium equation in the shear direction is given by
Ty ¼ Q ¼
12EI h
3
dy cos3 h:
0
60
ð43Þ 0
4
8
12
16
20
λ
ð44Þ Fig. 8. Shear stiffness enhancement with respect to effective slenderness ratio.
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H. Chen et al. / Composite Structures 94 (2012) 3448–3456
Inclination of the strut also greatly influences the shear mudulus of the lattice. With smaller inclination (smaller h), the stiffness enhancement is much more obvious. The analytical prediction was also checked by commercial ANSYS code, where beam elements (BEAM3) were adopted for struts to compare with the calulated shear modulus using bar elements. In numerical calcultion, the material is linear elastic isotropic and each strut was divided into 100 elements. The forces under shear deformation dxy are given by
EA dxy cos h sin h sin u; h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12EI 2 Q ¼ 3 dxy cos3 h cos2 h sin u þ cos2 u; h
N¼
ð54Þ ð55Þ
and
M¼
6EI h
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 dxy cos2 h cos2 h sin u þ cos2 u;
ð56Þ
for struts 1 and 2, respectively, and
EA dxy cos h sin h cos u; h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12EI 2 Q ¼ 3 dxy cos3 h cos2 h cos2 u þ sin u; h
N¼
ð57Þ ð58Þ
for struts 3 and 4, respectively. Ratio of the shear force to the axial force is given by
Q 12 cos2 h ¼ N k2 sin h
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M 6 ¼ 2 cot h cos2 h þ tan2 u: Nh k
M¼
h
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 dxy cos h cos2 h cos2 u þ sin u;
ð61Þ
for struts 1 and 2, respectively. As shown in Fig. 9, the ratio cannot be neglected when the strut is stout enough, which means the shear force and the bending moment will obviously influence the strength of the lattice. With smaller inclination (smaller h), the shear force and bending moment increase rapidly when struts turn to stouter, which should greatly influence the lattice strength. Coupling the axial compression force with the shear force, the maximum/ minimum axial stress at the central line of the beam is given by E 2
r1;3 ¼ cxy 0
cos h sin h sin u@1
2
ð60Þ
and the ratio of the bending moment to the axial force is given by
and
6EI
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos2 h þ tan2 u;
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 64 12 cos2 h 1þ cos2 h þ tan2 u A: 2 9 k sin h
ð62Þ
ð59Þ According to Eq. (52), the equivalent initial yield strength of the lattice, ses, is
(a)
0.35
ses ¼ ϕ =45 ϕ =00
0.30
θ =30
0.25
0
g4 ¼2 1 þ
12
cot2 h
0
30
ð64Þ
0
60
0.10
cos2 hcot2 h þ
k2 k2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 64 12 cos2 h @1 þ 1 þ cos2 h þ tan2 u A : 9 k2 sin h
0.15 0
12
0
0
45
ð63Þ
with
0
0.20
Q /N
1 q rs sin 2hg4 4 sin u
45
with r2 = 0, the Mises yield stress, rM, is given by 0.05
0
rM ¼ Ecxy
60
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 16 12 cos2 h cos2 h þ tan2 u cos h sin h sin u 1 þ 3 k2 sin h
0.00 4
8
12
16
λ
(b)
¼ rs :
0.35
30
ð65Þ
ϕ =450 ϕ =00
0.30 0
when yielded, the corresponding equivalent initial yield stress of the lattice is
M /(N⋅h)
0.25 0.20
θ =30
0
45
ses ¼
0
1 q rs sin 2hg5 4 sin u
ð66Þ
with
0.15
0
45 0.10
g5 ¼ 1 þ 0
"
60
0.05 0.00 4
8
12
16
λ Fig. 9. Ratios of (a) shear force to axial force and (b) bending moment to axial force under shearing with respect to effective slenderness ratio.
k2
cos2 h cot2 h þ
12
cot2 h
k2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 #1=2 16 12 cos2 h cos2 h þ tan2 u : 1þ 3 k2 sin h
0
60
12
ð67Þ
Variations of g4 and g5 with k and h are close to or even greater than 1, which reveals that the shear force will not reduce the lattice strength. The coupling between the axial force and the shear force should not control the load capacity of the lattice.
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H. Chen et al. / Composite Structures 94 (2012) 3448–3456
At the edge of the cross section of the beam, the coupling effect between the axial force and the bending moment must be considered. The maximum stress in struts 1 and 2 are determined by
E h
rbar ¼ dxy cos h sin h sin / þ
6EdE
dxy h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cos2 h cos2 h sin / þ cos2 /: 2
ð68Þ
when rbar = rs, the corresponding equivalent strain cxy of the lattice material is
rs E cos h
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 12 2 cos h cos2 h sin u þ cos2 u : k
sin h sin u þ
ð69Þ According to Eq. (52), the equivalent initial yield stength of the lattice is
ses ¼
1 q rs sin 2hg6 4 sin u
ð70Þ
with
g6 ¼ 1 þ
12 2
k
cos2 hcot2 h þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 12 cot h cos2 h þ cot2 u cot2 h 1 þ k k
12
ð71Þ
2
for struts 1 and 2 and
1 q rs cos h sin hg7 2 cos u
(a)
ð72Þ
1.0
k2
cos2 hcot2 h þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 12 2 cot h 1 þ cot h cos2 h þ tan2 u k k2
12
1.2
Analytical prediction Numerical simulation 1.0
0
θ = 60 0.8
η7
η7
0
0
45
ϕ =0
30
0
0
ϕ =15
0
0.4 0
4
8
12
16
20
0
4
8
12
16
20
λ
λ
(c)
0
0.6
0
30
0.6
0.5
θ = 60
0.8
45
0.7
1.0
Analytical prediction Numerical simulation
(d)
0.9
0.6
Analytical prediction Numerical simulation
0.9
0.8
0
θ = 60
0
θ = 60 0.7
0
45 0
45 0.6
0
30
0
ϕ = 45
0
30
0.5 ϕ =300 4
8
12
λ
16
20
ð73Þ
for struts 3 and 4, respectively. Controlled by coupled axial compression and bending, the initial yield strength of the lattice should be greatly reduced, even for slender struts with k > 20, as shown in Fig. 10. The reduction would be greater for dense lattices. With smaller inclination (smaller h), the shear strength reduces much more rapidly. The analytical prediction was also checked by commercial ANSYS code, where beam elements (BEAM3) were adopted for struts to compare with the calulated shear strength using bar elements. In numerical calcultion, the material is linear elastic isotropic and each strut was divided into 100 elements. For lattices with stouter struts, previous models overestimated their shear strengths [11]. In experiments, tested strengths were always smaller than predicted data, which ordinarily be explained by manufacturing defaults. In fact, the flexural deformation of the strut is an important factor for strength reduction, especially for lattices with stout struts and small inclinations. As shown in Fig. 11, compared with previous model, the suggested model combining with elastic buckling captures the shear strength dependence upon relative density for samples and leads to a smaller deviations. Fig. 11 shows that the elastic buckling model captures the strength dependence upon relative density for samples smaller than 2.5% by the k = 1 approximation in shear, where the node may be sheared and rotation constraints would be weakened [11].
(b)
Analytical prediction Numerical simulation
12
0.9
η7
ses ¼
g7 ¼ 1 þ
η7
cxy ¼
with
0.3
0
0
4
8
12
16
λ
Fig. 10. Shear strength reduction for (a) u = 00; (b) u = 150; (c) u = 300 and (d) u = 450 with respect to effective slenderness ratio.
20
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H. Chen et al. / Composite Structures 94 (2012) 3448–3456
0.5
τes/( ρ*σs )
0.3
Plastic yield
0.2
0.5
0.4
σb13/(ρ*σs)
0.4
(a)
Experimental data [11] Previous model Suggested model
Buckling
Experimental data [5] Previous prediction Suggested prediction
0.3
0.2
Yield strength
0.1 0.1
Buckling 0.0 0.00
0.01
0.02
0.03
0.04
0.0 0.00
ρ*
0.05
Fig. 11. Predicted shear strength compared with experiments [11].
(b)
4. Sandwich panel with tetrahedral lattices Kooistra et al. [5] tested the shear behavior of age hardenable 6061 aluminum sandwich panels with tetrahedral lattice truss cores made of struts of rectangular cross sections. According to the suggested method, behaviors of tetrahedral lattice truss sandwich panel structures were predicted, where bending resistance of struts were included in the modified equations. The modified shear modulus, Gb13, is
Gb13 ¼
1 q E sin2 2hg8 8
0.10
0.15
ρ* 0.5
σb13/(ρ*σs)
0.4
Experimental data [5] Previous prediction Suggested prediction
0.3
0.2
Yield strength
ð74Þ
0.1
Buckling
with
g8 ¼ 1 þ
pffiffiffi 3 q sin xð1 þ sin2 hÞ: 2
0.0 0.00
ð75Þ
0.05
The modified initial yield shear strength, rb13, is
1 4
rb13 ¼ q rs sin 2hg9
ð76Þ
with
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11 pffiffiffi 9 3 3 g9 ¼ g8 @1 þ q sin hA : 2 0
ð77Þ
Experimental data were compared with analytical predictions. As shown in Figs. 12 and 13, ratios of Gb13/(q⁄E) and rb13/(q⁄rs) are constant in previous analytical predictions. In experiments, ratios of Gb13/(q⁄E) are smaller than previous predictions and get 0.16
Gb13/( ρ*E )
0.12
0.08
Experimental data [5] Previous prediction Suggested prediction
0.04
0.00 0.00
0.02
0.04
0.06
0.08
0.10
0.15
ρ*
0.10
Fig. 13. Comparisons between experimental data [5] and analytical predictions for (a) initial shear strength and (b) ultimate shear strength in the aged hardened condition.
larger when the relative density increases. The modified model suggested in this paper reveals this regualtion. The initial yield strength should be overestimated in previous models, where all the cross section is assumed to yield simultaneously. In fact, when the maximum stress equals to the yield strength, the lattice truss structures woud be in plastic yield state. The modified model is much more consistent, as shown in Fig. 13. Fig. 13 shows that the elastic buckling model captures the strength dependence upon relative density by the k = 2 approximation for aged hardened samples with higher yield stresses, while by the k = 1 approximation for annealed samples with rather smaller yield stresses [5]. After the initial yield, stresses in the cross section will continously get larger if the strut material is plastic hardening. Tested data were determined by 0.2% offset yield strength, which will be larger than the initial yield strength predicted by elastic–ideally plastic model of struts, especially for lattices with large relative density. It must be pointed that Timoshenko beam models must be more suitable for stout struts, although they were not adopted here. To predict of the peak load of tetrahedral lattices, the ultimate shear strength, rp13, is suggested as
1 4
rp13 ¼ g10 g11 q rs sin 2hg9 ; 0.12
ρ* Fig. 12. Comparisons between experimental data [5] and analytical predictions for shear stiffness.
ð78Þ
with
g10 ¼ rp =rs ;
ð79Þ
>where rp is the ultimate strength of the lattice material, and g11 denotes the ratio of the ultimate plastic strength to the initial yield
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H. Chen et al. / Composite Structures 94 (2012) 3448–3456
(a)
0.5
σp13/(ρ*σs )
0.4
Experimental data [5] Suggested prediction
Ultimate yield strength
0.3
0.2
0.1
0.0 0.00
Buckling
0.05
0.10
0.15
ρ*
(b)
1.2
1.0
Experimental data [5] Suggested prediction Ultimate yield strength
σp13/(ρ*σs )
0.8
Acknowledgements
0.6
0.4
0.2
0.0 0.00
(2) Including the strut flexural deformation, the shear force and the bending moment of the strut play an important role in the mechanical behavior of the lattice. The coupling between the axial force and the bending moment would reduce the lattice strength, especially for lattices with stout struts. The influence of the bending deformation can be neglected when k > 20. (3) The analytical model suggested in this paper reveals that the stiffness should be a little greater while the strength should be smaller than those predicted by the stretching domainated principle. The discrepancy would be enlarged for dense lattices. The suggested model would make a prediction more consistent with experiments. (4) Mechanical properties are not only decided by the relative density and the strut incliantion, but also depend on the slenderness ratio of struts. With identical relative density and strut incliantion, stouter lattices have greater stiffness but smaller strength. Strut inclinations greatly affect lattice properties. In compression, the strength reduction would be enlarged with increasing inclination (increasing h), while in shearing struts with smaller inclination (smaller h) are apt to be bended failure.
Buckling
0.05
0.10
0.15
ρ* Fig. 14. Comparisons between experimental data [5] and analytical predictions for (a) initial shear strength and (b) ultimate shear strength in the annealed condition.
strength load of elastic-perfectly plastic beams under bending. For rectangular cross sections, g11 = 1.5. In the aged hardened condition and in the annealed condition, value of g10 is 1.14 and 2.97 respectively [5]. As shown in Fig. 14, predictions are well consistent with experimental results, much better than the prediction of the initial yield strength, which indicates that ultimate plastic strength of bent beams can be used to predict the ultimate strength of the lattice truss material. 5. Conclusions The equivalent mechanical property of the lattice truss material was discussed including the flexural deformation and bending resistance of the strut in lattice structures. According to the research, it can be concluded that: (1) Analytical predictions based on the stretching dominated principle overestimated the strength of lattice truss materials, while underestimated their stiffness. For stout lattices, the discrepancy gets larger, which has been revealed by experiments.
Supports from National Natural Science Foundation of China (11172089), Program for New Century Excellent Talents in University, Science Fund for Creative Research Groups of the National Natural Science Foundations of China (51021001), State Key Laboratory of Automotive Safety and Energy of Tsinghua University (KF11031) and State Key Laboratory of Ocean Engineering of Shanghai Jiao Tong University (Grant No. 1001) are gratefully acknowledged. References [1] Deshpande VS, Ashby MF, Fleck N A. Foam topology bending versus stretching dominated architectures. Acta Mater 2001;49:1035–40. [2] Fan HL, Zeng T, Fang DN, et al. Mechanics of advanced fiber reinforced lattice composites. Acta Mech Sinica 2010;26(6):825–35. [3] Deshpande VS, Fleck NA, Ashby MF. Yield of truss core sandwich beams in 3-point bending. Int J Solids Struct 2001;38:6275–305. [4] Leekitwattana M, Boyd SW, Shenoi RA. Evaluation of the transverse shear stiffness of a steel bi-directional corrugated-strip-core sandwich beam. J Construct Steel Res 2011;67:248–54. [5] Kooistra GW, Queheillalt DT, Wadley HNG. Shear behavior of aluminum lattice truss sandwich panel structures. Mater Sci Eng A 2008;472:242–50. [6] Queheillalt DT, Wadley HNG. Pyramidal lattice truss structures with hollow trusses. Mater Sci Eng A 2005;397:132–7. [7] Finnegan K, Kooistra G, Wadley HNG, et al. The compressive response of carbon fiber composite pyramidal truss sandwich cores. Int J Mater Res 2007;98:1264–72. [8] Xiong J, Ma L, Wu LZ, et al. Fabrication and crushing behavior of low density carbon fiber composite pyramidal truss structures. Compos Struct 2010;92:2695–702. [9] Queheillalt DT, Murty Y, Wadley HNG. Mechanical properties of an extruded pyramidal lattice truss sandwich structure. Scripta Mater 2008;58:76–9. [10] Kooistra GW, Deshpande VS, Wadley HNG. Hierarchical corrugated core sandwich panel concepts. J Appl Mech 2007;4:259–68. [11] Queheillalt DT, Wadley HNG. Titanium alloy lattice truss structures. Mater Des 2009;30:1966–75.