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epoxy composite sandwich structures with three-dimensional corrugated cores
Composites Science and Technology 156 (2018) 296e304
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Composites Science and Technology journal homepage: http://www.elsevier.com/locate/compscitech
Mechanical response of carbon/epoxy composite sandwich structures with three-dimensional corrugated cores Guo-dong Xu a, Zhi-hai Wang a, Tao Zeng a, *, Su Cheng a, Dai-ning Fang b a b
Department of Engineering Mechanics, Harbin University of Science and Technology, Harbin, 150080, PR China School of Aerospace Engineering, Beijing Institute of Technology, 100081, Beijing, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 October 2017 Received in revised form 5 January 2018 Accepted 7 January 2018 Available online 10 January 2018
A novel three-dimensional (3D) corrugated core sandwich structure was designed and fabricated by auto-cutting process. Mechanical behaviors and failure mechanism of 3D corrugated core sandwich structures were investigated. Analytical models were developed to estimate the strength, stiffness and dominant failure modes. In order to demonstrate sensitivity of the graded parameters on mechanical behaviors of 3D corrugated core sandwich structures, the specimens with different graded parameters were fabricated and tested under compression and bending loads. Results showed that the graded parameters have obviously influences on the mechanical properties and failure modes of 3D corrugated core sandwich structures. The predictions were also compared with the experiments and the results showed good agreements. Failure maps were constructed to illustrate the controlling failure mechanisms in various regions with different parameters. © 2018 Elsevier Ltd. All rights reserved.
Keywords: Composites Sandwich structures Mechanical properties Corrugated cores
1. Introduction Sandwich structures consisted of solid face sheets and lowdensity core are widely used in aerospace, high-speed train and civil engineering due to their high stiffness/strength-to-weight ratio. Sandwich panels were traditionally composed of stochastic cores such as foams [1] or periodic cores such as honeycomb core [2], lattice core [3e5], corrugated core [6,7] and orthogrid core [8,9]. Currently, corrugated cores are preferred in sandwich panels as the light-weight cores due to its excellent mechanical properties and simple manufacturing process. There are significant amount of literatures about the static and dynamic behavior of sandwich panels with corrugated cores. Mohammadi et al [10] presented an analytical equivalent model for predicting the mechanical properties of the trapezoidal corrugated core. Cheon and Kim [11] suggested an equivalent plate model to analyze the mechanical behavior of sinusoidally corrugated-core sandwich panels under tensile and bending loads. Rubino et al [12] investigated the quasistatic three-point bending response of Y-frame and triangular corrugated sandwich structures with simply supported and
* Corresponding author. E-mail address: [email protected] (T. Zeng). https://doi.org/10.1016/j.compscitech.2018.01.015 0266-3538/© 2018 Elsevier Ltd. All rights reserved.
clamped boundary conditions. Zhang et al [13] investigated the bending strength, stiffness and energy absorption of trapezoid corrugated sandwich composite structure. Bartolozzi et al [14] developed the analytical model to predict the static and dynamic behavior of the sandwich structure with corrugated core and assess the precision of this model. Yang et al [15] investigate the modal response of all-composite corrugated sandwich cylindrical shells including axial and circular corrugated cores. Recently, novel corrugated sandwich structures attract more and more attention, including hybrid corrugation, multilayers corrugation and hierarchical corrugation. Paczos et al [16] studied the bending performance of corrugated sandwich beams that consist of five layers. Kooistra et al [17] analyzed collapse mechanisms of a second order, hierarchical corrugated truss structure. Han et al [18] presented a novel lightweight sandwich construction with honeycombcorrugation hybrid core and investigated the performance of the hybrid structure under quasi-static out-of-plane uniform compression. However, in most existing studies as mentioned above, the corrugated cores that have been considered for cellular materials are two-dimensional structures. In our previous work, a lengthdirection graded truss core sandwich structure was presented [19]. In this work, a new type of 3D corrugated core sandwich structure, which can implement the graded function in the
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thickness-direction, was designed and fabricated. The analytical models were presented to predict the compression and bending performance of 3D corrugated core sandwich structures. The corresponding experiments were performed to prove the validity of the analytical models. The relationships between geometry parameters and failure modes of 3D corrugated core sandwich structures were also studied by constructing the failure maps. 2. Material design and fabrication Fig. 1a shows a 3D corrugated core sandwich structure, which comprises upper and bottom face sheets and inclined trapezoid webs. The trapezoid webs have a constant thickness and viable width. Fig. 1b is a unit cell of a 3D corrugated core with geometrical quantities. In this work, the geometrical parameters are fixed as tw¼1 mm, lw¼11.6 mm, a¼9.5 mm, lu¼31.6 mm, hu¼10 mm, q¼60 . The graded parameter of the trapezoid web is defined as
n¼
bu bd
(1)
The auto-cutting and hot pressing method [19] was used to fabricate 3D corrugated core composite sandwich structures. Face sheets and corrugated cores were made from carbon fiber/epoxy laminates with the stack sequence [0/90]n. The detailed fabricate process can be described: first, the unidirectional carbon fiber/ epoxy prepreg (T700/E44, Zhongtan Co., China) were laminated and cut to the patterns (as shown in Fig. 2). Then, the patterns were manufactured to the trapezoid web cores and contacted to the face sheets by a metal model. Finally, the sample was vacuum bagged and cured in an autoclave at 135 C under pressure of 0.5 MPa for 120 min.
Fig. 2. Schematic of the manufacturing process of the 3D corrugated core composite sandwich panel.
the tested properties of the face sheets and trapezoid web cores are different although they have the same stack sequences. In this work, a once-curing method [19], which the lattice core and face sheets were simultaneous cured and formed the sandwich structures by hot press, was presented. Although this method can facilitate the manufacturing process, it indeed weakens the mechanical properties of the corrugated lattice core because it is difficult to apply the same pressure on the lattice core as that on the face sheets. 3.2. Compression tests Three kinds of 3D corrugated core sandwich structure (specimen C1, C2 and C3) were tested in this work. All the specimens have 33 unit cells and 1 mm face sheets thickness. The geometrical parameters of each unit cell are wu¼30 mm and bu¼5mm. The graded parameter n of the specimens varied from 0.17 to 0.50. The tests were conducted at a constant speed of 0.2 mm/min.
3. Experiments
3.3. Three-point bending tests
3.1. Material properties
Three-point bending tests in x and y-direction were performed for 3D corrugated core composite sandwich beams in this section due to their anisotropic properties, which are similar to Ref. [20]. Four kinds of 3D corrugated core composite sandwich beams in ydirection (specimen B1, B2, B3 and B4) were tested in three-point bending. All the specimens have 73 unit cells and they have the same span L¼190 mm, width B¼90 mm and overhang distance S¼15.6mm. Width of unit cell is wu¼30 mm. The face sheets thickness tf, upper width of the web bu, and graded parameter n are listed in Table 1. Among all the specimens, specimen B1 and specimen B2 have stronger face sheets (tf¼0.75 mm) and weaker trapezoid webs (bu¼3 mm). The graded parameters of the specimens are 0.60 and 0.17, respectively. Specimen B3 and specimen B4
The compression tests were performed to determine the mechanical properties of the trapezoid web core and face sheets. The compression tests were performed on a100 kN screw driven test machine at a constant speed of 0.5 mm/min. The INSTRON Advanced Video Extensometer was used to measure the strain. The straight strips made of carbon fiber/epoxy laminates were detached from the specimens. In order to keep the tested specimen vertical, a special fixture which have been mentioned in Ref. [19] was used in compression tests. The average Young's modulus and peak strength are 15.12 GPa and 254.45 MPa for the trapezoid web core, and 30.89 GPa and 571.39 MPa for the face sheets. It must be pointed out that
Fig. 1. Configuration of a 3D corrugated core composite sandwich panel.
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Table 1 Dimensions of four kinds of 3D corrugated core sandwich beams (in y-direction) under bending load. Specimen
tf (mm)
bu (mm)
n
B1 B2 B3 B4
0.75 0.75 0.50 0.50
3 3 5 5
0.60 0.17 0.50 0.25
have weaker face sheets (tf¼0.5 mm) and stronger trapezoid webs (bu¼5 mm), and the graded parameters of the specimens are 0.50 and 0.25. The tests were conducted at a constant speed of 0.5 mm/min. A central block was used to suppress local indentations at the face sheets. The 3D corrugated core composite sandwich beams in x-direction were tested in three-point bending. The specimens have 72 unit cells. The geometrical parameters are L¼147.0 mm, S¼ 23.5 mm, wu¼21.0 mm, bu¼5 mm, tf¼0.75 mm and n¼0.5. In order to determine the bending failure loads of the 3D corrugated lattice core sandwich beams, the shear experiments were also performed. The shear specimens have 63 unit cells and face sheets thickness is 1.5 mm. The geometrical parameters of each unit cell are wu¼30 mm, bu¼15mm and n¼0.5. A single-lap shear fixture was used to perform shear test. The samples were bonded to steel shear plates using an epoxy adhesive and the shear plates were attached to a hydraulic load frame. The displacement rate for the tests was 0.5 mm/min, and the measured load cell force was used to calculate the shear strength. The shear strength is 0.88MPa and failure mode is debonding of the face sheets and the cores, as shown in Fig. 3. 4. Analytical formulations The analytical models were presented for predicting the compression and bending responses of 3D corrugated core composite sandwich structures.
Fig. 4. Deformation and free-body diagram of sandwich structure with 3D corrugated core under compression.
subjected to a compression load F. The deformation of the 3D corrugated core panel along z-direction is given by
4.1. Compression performance 4.1.1. Compression stiffness The analytical model for predicting the effective compression stiffness of 3D corrugated core sandwich structures was presented by analyzing the elastic deformation of a trapezoid web. In this work, the trapezoid web is considered as a nonprismatic beam along z1-direction. For 3D corrugated core with a large slenderness ratio, the trapezoid web is assumed to be pin jointed to the face sheets [17,21]. Fig. 4 shows a unit cell of the 3D corrugated core
Zhu
Dz ¼ 0
FN sin2 qbðzÞtw Ew
dz
(2)
where bðzÞ is the width of the trapezoid web. Ew is Young's modulus of the web. The effective Young's modulus of the 3D corrugated core is
Ezz ¼
s ε
¼
2 sin3 qtw Ew hu Z h lu wu 1=bðzÞdz
(3)
0
4.1.2. Compression strength Two competing mechanisms including fracture and Euler buckling of the trapezoid web will ultimately determine the compression strength of the sandwich structure. The failure strength is (1) Fracture of the trapezoid web Fig. 3. Shear stress-strain curve and failure photograph of 3D corrugated core composite sandwich panel.
sfc ¼
2sw tw bu sin q lu wu
(4)
G.-d. Xu et al. / Composites Science and Technology 156 (2018) 296e304
h i M C ¼0
(2) Euler buckling of the trapezoid web Considering an elastic trapezoid web as shown in Fig. 5, the differential equation of the deflection curve can be expressed as [22]
Ew Iðz1 Þ
d2 x1 þ FN x1 ¼ 0 dz21
(5)
where Iðz1 Þ is principal moment of inertia of trapezoid web at the level z1, which can be expressed as
Iðz1 Þ ¼ Imin z1 =b
(6)
3 =12 is the minimum principal moment of inertia Here, Imin ¼ bu tw about y1 axis for the trapezoid web (z1 ¼ b). b is the distance from the origin (O) in the coordinate (x1-y1-z1) to the top of the web, as shown in Fig. 5a. Letting T ¼ EwbImin , Eq. (5) can be rewritten as
d2 x1 FN þ x1 z1 1 ¼0 T dz21
(7)
Introducing two variables z1 ¼ t 2 T=ð4FN Þ and pffiffiffiffiffi x1 ðz1 Þ ¼ Vðz1 Þ z1 , the differential equation of the deflection curve can be simplified as
d2 V 1 dV 1 þ 1 2 V ¼0 þ t dt dt 2 t
(8)
Eq. (8) is a Bessel's differential equation of the first order. The possible expression of analytical solution is given by
VðtÞ ¼ C1 J1 ðtÞ þ C2 N1 ðtÞ
(9)
where C1 and C2 are the integral constant. J1 ðtÞ and N1 ðtÞ are the Bessel function of the first and second kinds. pffiffiffiffiffi Considering x1 ðz1 Þ ¼ Vðz1 Þ z1 , the expression of x1 ðz1 Þ is
" rffiffiffiffiffiffiffiffiffiffi ! rffiffiffiffiffiffiffiffiffiffi !# pffiffiffiffiffi FN z1 FN z1 þ C2 N1 2 x1 ðz1 Þ ¼ z1 C1 J1 2 T T
(10)
For a simple web with pinned end, the corresponding boundary conditions are
x1 ðz1 Þjz1 ¼b ¼ 0;
x1 ðz1 Þjz1 ¼bþlw ¼ 0
299
(11)
Substituting Eq. (10) into Eq. (11) yields a system of linear equations
(12)
A non-trivial solution to Eq. (12) exists when the determinant of matrix ½M equal to zero. For a particular value of n, the critical load FNcr can be determined by solving Eq. (12) numerically and then we can obtain the failure strength. 2 Letting FNcr ¼ m p Elw2 Imin , the buckling parameter m can be w obtained. 4.2. Bending performance The analytical models [23] were used for predicting the bending stiffness and collapse strength of the 3D corrugated core sandwich beam. It was assumed that the core alone carries the shear load, while the face sheets carry the applied moment. 4.2.1. Bending deflection The total deflection d at the mid-point of a sandwich beam loaded by P is the sum of the deflections due to bending of the face sheets and shear of the core:
d ¼ dB þ dS ¼
PL3 PL þ 48ðEIÞeq 4ðAGÞeq
(13)
where (EI)eq is the equivalent flexural rigidity and (AG)eq is the equivalent shear rigidity
2 1 ðEIÞeq ¼ Ef Btf hu þ 2tw þ tf 2
ðAGÞeq ¼
2 B hu þ 2tw þ tf hu þ 2tw
Gcxz
(14)
(15)
Here Ef is Young's modulus of the face sheet and Gcxz is the shear stiffness of the 3D corrugated core, which can be obtained by equivalence theory of the unit cell [19]. 4.2.2. Bending failure load There are four possible failure modes in three-point bending for 3D corrugated core sandwich beams. According to Ref. [24], the failure load is (1) Face crushing (FC)
PFC ¼
4Btf hu þ 2tw þ tf L
sfc
(16)
where sfc denotes the yielding strength of the face sheet. (2) Face wrinkling (FW)
PFW ¼
4p2 Ef hc þ 2tw þ tf Btf3 3Lðlu aÞ2
(17)
(3) Core shear failure
PCF ¼ 2Bðhu þ 2tw Þtfc ¼ Fig. 5. Buckling deformation of two ends hinged trapezoidal web with non-uniform cross-section.
4sw tw bu cos qBðhu þ 2tw Þ ðweb fracture ðCFÞÞ lu wu
(18)
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4.3. Failure mechanism maps
PCB ¼ 2Bðhu þ 2tw Þtbc ¼m
3 b cos qðh þ 2t Þ BEw p2 tw u u w ðweb buckling ðCBÞÞ 3lu wu l2w
PCD ¼ 2Bðhu þ 2tw ÞtD c ðcore debonding ðCDÞÞ
(19)
(20)
f
where tc and tbc are the shear strength of the core including fracture and buckling. tD c is interfacial shear strength which can be obtained through shear test. (4) Indentation
PIF ¼
PIB
The failure modes of 3D corrugated core sandwich panels under three-point bending can be illustrated in collapse mechanism maps based on analytical parameters. Face wrinkling, Face crushing, core web crushing and core web buckling were studied in this work. The failure maps were developed as a function of the non-dimensional geometrical parameters tf/l, tw/l (l is the ratio of the maximum moment to the maximum transverse force) and graded parameter n, and the boundaries of failure modes were obtained by equating the critical loads for different failure modes. 5. Results and discussions
2sw tw bu B sin q ðweb fracture ðIFÞÞ lu
3 b B sin q Ew p2 tw u ¼m ðweb buckling ðIBÞÞ 6lu l2w
(21)
5.1. Compression tests
(22)
Fig. 6a shows the measured response of specimen C1, which has the smallest graded parameter. The stress firstly increases until the first peak. Then the stress decreased due to the fracture of the trapezoid web. With the increasing of strain, the stress increases again and reaches the second peak. Fig. 6b shows the measured
Fig. 6. Compression stress-strain curves and failure photographs of 3D corrugated core composite sandwich panels.
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301
Table 2 Theoretical and experimental value of compression strength and modulus of 3D corrugated core sandwich structures with different graded parameter n. Specimen
n
C1
0.17
C2
0.25
C3
0.50
Analytical
Experiment
Failure mode
Strength/stiffness (MPa)
Failure mode
Strength/stiffness (MPa)
CF CB CF CB CF CB
2.42/286.79 2.79/286.79 2.42/222.40 2.07/222.40 2.42/148.27 1.14/148.27
CF
2.37/252.13
CB
1.88/198.72
CB
1.01/124.88
response of specimen C2, which has an intermediate graded parameter. The stress firstly increases until the failure occurs at 1.88 MPa. This collapse is caused by the Euler buckling of the trapezoid webs. Then, the stress begins to decrease and followed by a plateau region along with the fluctuation. This is due to the gradual delamination of the trapezoid webs as the load increased. The compressive stress-strain curves measured for specimen C3 is shown in Fig. 6c. Comparing with specimens C1 and C2, specimen C3 has the largest graded parameter. That means the trapezoid web is closed to a rectangle and the failure mode is similar to conventional corrugated core sandwich structure. When the stress reaches the maximum, Euler buckling failure occurs. With increasing the strain, the stress drops rapidly. The predicted and tested results of the 3D corrugated core sandwich panels in compression are summarized in Table 2. Fig. 7 shows a comparison of the compression strength between 3D corrugated lattice sandwich structures with other lattice truss structures. Although the strength is relative low, 3D corrugated sandwich structures have potential applications because of their variable cross-section cores, which can facilitate the graded function in thicknessdirection. 5.2. Bending tests Four kinds of bending specimens (in y-direction) with different graded parameter and face sheets thickness (listed in Table 1) were tested. Fig. 8a shows the measured response of specimen B1, which have the strong face sheets, weak trapezoid webs and large graded parameter. The load firstly increases until the shear buckling occurs at a load of ~584.10 N. Then, the load slowly increases again
combining with short undulation plateaus until the maximum peak load reaches. Fig. 8b shows the measured response of specimen B2. The failure load is ~1262.73 N and the failure mode is the fracture of trapezoid webs. It is found that the failure load of specimen B2 is much higher than that specimen B1. This indicates that the graded parameter has obvious influences on the bending failure load of 3D corrugated core sandwich beams when the face sheets are strong. Fig. 8c and d shows the measured responses of specimen B3 and B4. Comparing specimen B3 with specimen B4, it is found that the graded parameter has little influences on the failure mode and the failure load of the 3D corrugated core sandwich beam when the face sheets are weak. The predicted and tested bending deformation, stiffness of 3D corrugated core sandwich beams (in y-direction) are listed in Table 3 as well as the failure loads and collapse modes are summarized in Table 4. The tested values are in good agreement with the theoretical predictions. Fig. 9 shows the load-displacement curve and failure photograph of 3D corrugated core composite sandwich beam (in x-direction) under bending loads. It is found that the load firstly increases until the delamination occurs at a load of ~879.10 N. The mechanism maps for 3D corrugated core sandwich panels were plotted in Figs. 10 and 11. Fig. 10a illustrates the failure map as functions of tf/l and tw/l for n¼0.25. With high ratio tf/l and tw/l, core shear fracture and face sheets crushing are the mainly collapse mechanism due to the strong trapezoid webs. With low ratio tf/l and tw/l, core shear buckling and face sheets wrinkling are the mainly collapse mechanism as observed in specimen B4. Fig. 10b illustrates the failure maps for 3D corrugated core sandwich structures for n¼0.50. In comparison with the counterpart for n¼0.25, the maps are slightly shifted to the right hand side, and the domains of core shear buckling is slightly expanded, due to the buckling parameter m decreases as the graded parameter n increase and the core becomes weaker against the shear load. Fig. 11 plotted the failure modes in terms of normalized parameters tf/l and n. At low ratio tf/l, face sheets wrinkling is the mainly collapse mechanism. At high ratio tf/l and small graded parameter n, core shear fracture is the mainly collapse mechanism as observed in specimens B2. At high ratio tf/l and large graded parameter n, core shear buckling is the mainly collapse mechanism as observed in specimens B1. 6. Conclusions
Fig. 7. Compression strengths of typical lattice core composite structures [8].
A novel 3D corrugated core sandwich structure has been presented, which is different from the conventional corrugated core sandwich structures. Analytical models were developed to predict the mechanical behaviors of 3D corrugated core sandwich structures. Compression and bending carbon/epoxy composite sandwich specimens with 3D corrugated cores were designed and tested to investigate the influence of the geometry and graded parameters on the mechanical response and failure modes. The results showed that mechanical properties and the failure modes of
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G.-d. Xu et al. / Composites Science and Technology 156 (2018) 296e304
Fig. 8. Load-displacement curves and failure photographs of 3D corrugated core composite sandwich beams in y-direction under bending loads.
Table 3 Three-point bending deformation of 3D corrugated core sandwich beams (in y-direction) along with predicted and measured P/d. -1
Specimen
ds and db (mm)
Total d (mm)
Analytical P/d (N mm
B1
1.156 0.806 1.154 1.719 0.603 1.163 0.403 1.163
1.962
346.172
313.523
2.873
504.212
456.725
1.766
353.014
325.177
1.565
399.634
365.156
B2 B3 B4
)
Test P/d (N mm
-1
)
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Table 4 Predicted and measured failure loads and collapse modes of 3D corrugated core sandwich beams (in y-direction) under three-point bending tests. Specimen
B1
B2
B3
B4
Analytical results
Experimental results
Failure mode
Failure force (N)
Failure mode
Failure force (N)
FC FW CF CB CD IF IB FC FW CF CB CD IF IB FC FW CF CB CD IF IB FC FW CF CB CD IF IB
10345.60 2146.14 1739.28 679.18 1900.80 3966.48 1884.12 10345.60 2146.14 1449.40 1809.11 1900.80 4945.81 3966.48 6761.84 623.42 2898.80 1272.22 1900.80 6610.80 3502.72 6761.84 623.42 2898.80 2153.71 1900.80 6610.80 5887.86
CB
584.10
CF
1262.73
FW
629.45
Fig. 11. Failure map illustrated as functions of tf/l and n.
FW
707.98
3D corrugated core composite sandwich structures mainly depend on the geometry and graded parameters. Under compression, the failure mode of the sandwich structure with small graded parameter is fracture of the core. But that with large graded parameter tends to core buckling. The graded parameter has little influence on the bending strength and failure mode for 3D corrugated core composite sandwich structures with thin face sheets. But it has obvious influence when face sheets are thick. The measured peak loads and failure modes were in good agreement with the analytical predictions. Failure maps were constructed to predict the bending response of 3D corrugated core sandwich structure.
Fig. 9. Load-displacement curve and failure photographs of 3D corrugated core composite sandwich beams in x-direction under bending loads.
Fig. 10. Failure maps illustrated as functions of tf/l and tw/l for two given graded parameter n a) n¼0.25; b) n¼0.50.
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Contents lists available at ScienceDirect
Composites Science and Technology journal homepage: http://www.elsevier.com/locate/compscitech
Mechanical response of carbon/epoxy composite sandwich structures with three-dimensional corrugated cores Guo-dong Xu a, Zhi-hai Wang a, Tao Zeng a, *, Su Cheng a, Dai-ning Fang b a b
Department of Engineering Mechanics, Harbin University of Science and Technology, Harbin, 150080, PR China School of Aerospace Engineering, Beijing Institute of Technology, 100081, Beijing, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 October 2017 Received in revised form 5 January 2018 Accepted 7 January 2018 Available online 10 January 2018
A novel three-dimensional (3D) corrugated core sandwich structure was designed and fabricated by auto-cutting process. Mechanical behaviors and failure mechanism of 3D corrugated core sandwich structures were investigated. Analytical models were developed to estimate the strength, stiffness and dominant failure modes. In order to demonstrate sensitivity of the graded parameters on mechanical behaviors of 3D corrugated core sandwich structures, the specimens with different graded parameters were fabricated and tested under compression and bending loads. Results showed that the graded parameters have obviously influences on the mechanical properties and failure modes of 3D corrugated core sandwich structures. The predictions were also compared with the experiments and the results showed good agreements. Failure maps were constructed to illustrate the controlling failure mechanisms in various regions with different parameters. © 2018 Elsevier Ltd. All rights reserved.
Keywords: Composites Sandwich structures Mechanical properties Corrugated cores
1. Introduction Sandwich structures consisted of solid face sheets and lowdensity core are widely used in aerospace, high-speed train and civil engineering due to their high stiffness/strength-to-weight ratio. Sandwich panels were traditionally composed of stochastic cores such as foams [1] or periodic cores such as honeycomb core [2], lattice core [3e5], corrugated core [6,7] and orthogrid core [8,9]. Currently, corrugated cores are preferred in sandwich panels as the light-weight cores due to its excellent mechanical properties and simple manufacturing process. There are significant amount of literatures about the static and dynamic behavior of sandwich panels with corrugated cores. Mohammadi et al [10] presented an analytical equivalent model for predicting the mechanical properties of the trapezoidal corrugated core. Cheon and Kim [11] suggested an equivalent plate model to analyze the mechanical behavior of sinusoidally corrugated-core sandwich panels under tensile and bending loads. Rubino et al [12] investigated the quasistatic three-point bending response of Y-frame and triangular corrugated sandwich structures with simply supported and
* Corresponding author. E-mail address: [email protected] (T. Zeng). https://doi.org/10.1016/j.compscitech.2018.01.015 0266-3538/© 2018 Elsevier Ltd. All rights reserved.
clamped boundary conditions. Zhang et al [13] investigated the bending strength, stiffness and energy absorption of trapezoid corrugated sandwich composite structure. Bartolozzi et al [14] developed the analytical model to predict the static and dynamic behavior of the sandwich structure with corrugated core and assess the precision of this model. Yang et al [15] investigate the modal response of all-composite corrugated sandwich cylindrical shells including axial and circular corrugated cores. Recently, novel corrugated sandwich structures attract more and more attention, including hybrid corrugation, multilayers corrugation and hierarchical corrugation. Paczos et al [16] studied the bending performance of corrugated sandwich beams that consist of five layers. Kooistra et al [17] analyzed collapse mechanisms of a second order, hierarchical corrugated truss structure. Han et al [18] presented a novel lightweight sandwich construction with honeycombcorrugation hybrid core and investigated the performance of the hybrid structure under quasi-static out-of-plane uniform compression. However, in most existing studies as mentioned above, the corrugated cores that have been considered for cellular materials are two-dimensional structures. In our previous work, a lengthdirection graded truss core sandwich structure was presented [19]. In this work, a new type of 3D corrugated core sandwich structure, which can implement the graded function in the
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thickness-direction, was designed and fabricated. The analytical models were presented to predict the compression and bending performance of 3D corrugated core sandwich structures. The corresponding experiments were performed to prove the validity of the analytical models. The relationships between geometry parameters and failure modes of 3D corrugated core sandwich structures were also studied by constructing the failure maps. 2. Material design and fabrication Fig. 1a shows a 3D corrugated core sandwich structure, which comprises upper and bottom face sheets and inclined trapezoid webs. The trapezoid webs have a constant thickness and viable width. Fig. 1b is a unit cell of a 3D corrugated core with geometrical quantities. In this work, the geometrical parameters are fixed as tw¼1 mm, lw¼11.6 mm, a¼9.5 mm, lu¼31.6 mm, hu¼10 mm, q¼60 . The graded parameter of the trapezoid web is defined as
n¼
bu bd
(1)
The auto-cutting and hot pressing method [19] was used to fabricate 3D corrugated core composite sandwich structures. Face sheets and corrugated cores were made from carbon fiber/epoxy laminates with the stack sequence [0/90]n. The detailed fabricate process can be described: first, the unidirectional carbon fiber/ epoxy prepreg (T700/E44, Zhongtan Co., China) were laminated and cut to the patterns (as shown in Fig. 2). Then, the patterns were manufactured to the trapezoid web cores and contacted to the face sheets by a metal model. Finally, the sample was vacuum bagged and cured in an autoclave at 135 C under pressure of 0.5 MPa for 120 min.
Fig. 2. Schematic of the manufacturing process of the 3D corrugated core composite sandwich panel.
the tested properties of the face sheets and trapezoid web cores are different although they have the same stack sequences. In this work, a once-curing method [19], which the lattice core and face sheets were simultaneous cured and formed the sandwich structures by hot press, was presented. Although this method can facilitate the manufacturing process, it indeed weakens the mechanical properties of the corrugated lattice core because it is difficult to apply the same pressure on the lattice core as that on the face sheets. 3.2. Compression tests Three kinds of 3D corrugated core sandwich structure (specimen C1, C2 and C3) were tested in this work. All the specimens have 33 unit cells and 1 mm face sheets thickness. The geometrical parameters of each unit cell are wu¼30 mm and bu¼5mm. The graded parameter n of the specimens varied from 0.17 to 0.50. The tests were conducted at a constant speed of 0.2 mm/min.
3. Experiments
3.3. Three-point bending tests
3.1. Material properties
Three-point bending tests in x and y-direction were performed for 3D corrugated core composite sandwich beams in this section due to their anisotropic properties, which are similar to Ref. [20]. Four kinds of 3D corrugated core composite sandwich beams in ydirection (specimen B1, B2, B3 and B4) were tested in three-point bending. All the specimens have 73 unit cells and they have the same span L¼190 mm, width B¼90 mm and overhang distance S¼15.6mm. Width of unit cell is wu¼30 mm. The face sheets thickness tf, upper width of the web bu, and graded parameter n are listed in Table 1. Among all the specimens, specimen B1 and specimen B2 have stronger face sheets (tf¼0.75 mm) and weaker trapezoid webs (bu¼3 mm). The graded parameters of the specimens are 0.60 and 0.17, respectively. Specimen B3 and specimen B4
The compression tests were performed to determine the mechanical properties of the trapezoid web core and face sheets. The compression tests were performed on a100 kN screw driven test machine at a constant speed of 0.5 mm/min. The INSTRON Advanced Video Extensometer was used to measure the strain. The straight strips made of carbon fiber/epoxy laminates were detached from the specimens. In order to keep the tested specimen vertical, a special fixture which have been mentioned in Ref. [19] was used in compression tests. The average Young's modulus and peak strength are 15.12 GPa and 254.45 MPa for the trapezoid web core, and 30.89 GPa and 571.39 MPa for the face sheets. It must be pointed out that
Fig. 1. Configuration of a 3D corrugated core composite sandwich panel.
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Table 1 Dimensions of four kinds of 3D corrugated core sandwich beams (in y-direction) under bending load. Specimen
tf (mm)
bu (mm)
n
B1 B2 B3 B4
0.75 0.75 0.50 0.50
3 3 5 5
0.60 0.17 0.50 0.25
have weaker face sheets (tf¼0.5 mm) and stronger trapezoid webs (bu¼5 mm), and the graded parameters of the specimens are 0.50 and 0.25. The tests were conducted at a constant speed of 0.5 mm/min. A central block was used to suppress local indentations at the face sheets. The 3D corrugated core composite sandwich beams in x-direction were tested in three-point bending. The specimens have 72 unit cells. The geometrical parameters are L¼147.0 mm, S¼ 23.5 mm, wu¼21.0 mm, bu¼5 mm, tf¼0.75 mm and n¼0.5. In order to determine the bending failure loads of the 3D corrugated lattice core sandwich beams, the shear experiments were also performed. The shear specimens have 63 unit cells and face sheets thickness is 1.5 mm. The geometrical parameters of each unit cell are wu¼30 mm, bu¼15mm and n¼0.5. A single-lap shear fixture was used to perform shear test. The samples were bonded to steel shear plates using an epoxy adhesive and the shear plates were attached to a hydraulic load frame. The displacement rate for the tests was 0.5 mm/min, and the measured load cell force was used to calculate the shear strength. The shear strength is 0.88MPa and failure mode is debonding of the face sheets and the cores, as shown in Fig. 3. 4. Analytical formulations The analytical models were presented for predicting the compression and bending responses of 3D corrugated core composite sandwich structures.
Fig. 4. Deformation and free-body diagram of sandwich structure with 3D corrugated core under compression.
subjected to a compression load F. The deformation of the 3D corrugated core panel along z-direction is given by
4.1. Compression performance 4.1.1. Compression stiffness The analytical model for predicting the effective compression stiffness of 3D corrugated core sandwich structures was presented by analyzing the elastic deformation of a trapezoid web. In this work, the trapezoid web is considered as a nonprismatic beam along z1-direction. For 3D corrugated core with a large slenderness ratio, the trapezoid web is assumed to be pin jointed to the face sheets [17,21]. Fig. 4 shows a unit cell of the 3D corrugated core
Zhu
Dz ¼ 0
FN sin2 qbðzÞtw Ew
dz
(2)
where bðzÞ is the width of the trapezoid web. Ew is Young's modulus of the web. The effective Young's modulus of the 3D corrugated core is
Ezz ¼
s ε
¼
2 sin3 qtw Ew hu Z h lu wu 1=bðzÞdz
(3)
0
4.1.2. Compression strength Two competing mechanisms including fracture and Euler buckling of the trapezoid web will ultimately determine the compression strength of the sandwich structure. The failure strength is (1) Fracture of the trapezoid web Fig. 3. Shear stress-strain curve and failure photograph of 3D corrugated core composite sandwich panel.
sfc ¼
2sw tw bu sin q lu wu
(4)
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h i M C ¼0
(2) Euler buckling of the trapezoid web Considering an elastic trapezoid web as shown in Fig. 5, the differential equation of the deflection curve can be expressed as [22]
Ew Iðz1 Þ
d2 x1 þ FN x1 ¼ 0 dz21
(5)
where Iðz1 Þ is principal moment of inertia of trapezoid web at the level z1, which can be expressed as
Iðz1 Þ ¼ Imin z1 =b
(6)
3 =12 is the minimum principal moment of inertia Here, Imin ¼ bu tw about y1 axis for the trapezoid web (z1 ¼ b). b is the distance from the origin (O) in the coordinate (x1-y1-z1) to the top of the web, as shown in Fig. 5a. Letting T ¼ EwbImin , Eq. (5) can be rewritten as
d2 x1 FN þ x1 z1 1 ¼0 T dz21
(7)
Introducing two variables z1 ¼ t 2 T=ð4FN Þ and pffiffiffiffiffi x1 ðz1 Þ ¼ Vðz1 Þ z1 , the differential equation of the deflection curve can be simplified as
d2 V 1 dV 1 þ 1 2 V ¼0 þ t dt dt 2 t
(8)
Eq. (8) is a Bessel's differential equation of the first order. The possible expression of analytical solution is given by
VðtÞ ¼ C1 J1 ðtÞ þ C2 N1 ðtÞ
(9)
where C1 and C2 are the integral constant. J1 ðtÞ and N1 ðtÞ are the Bessel function of the first and second kinds. pffiffiffiffiffi Considering x1 ðz1 Þ ¼ Vðz1 Þ z1 , the expression of x1 ðz1 Þ is
" rffiffiffiffiffiffiffiffiffiffi ! rffiffiffiffiffiffiffiffiffiffi !# pffiffiffiffiffi FN z1 FN z1 þ C2 N1 2 x1 ðz1 Þ ¼ z1 C1 J1 2 T T
(10)
For a simple web with pinned end, the corresponding boundary conditions are
x1 ðz1 Þjz1 ¼b ¼ 0;
x1 ðz1 Þjz1 ¼bþlw ¼ 0
299
(11)
Substituting Eq. (10) into Eq. (11) yields a system of linear equations
(12)
A non-trivial solution to Eq. (12) exists when the determinant of matrix ½M equal to zero. For a particular value of n, the critical load FNcr can be determined by solving Eq. (12) numerically and then we can obtain the failure strength. 2 Letting FNcr ¼ m p Elw2 Imin , the buckling parameter m can be w obtained. 4.2. Bending performance The analytical models [23] were used for predicting the bending stiffness and collapse strength of the 3D corrugated core sandwich beam. It was assumed that the core alone carries the shear load, while the face sheets carry the applied moment. 4.2.1. Bending deflection The total deflection d at the mid-point of a sandwich beam loaded by P is the sum of the deflections due to bending of the face sheets and shear of the core:
d ¼ dB þ dS ¼
PL3 PL þ 48ðEIÞeq 4ðAGÞeq
(13)
where (EI)eq is the equivalent flexural rigidity and (AG)eq is the equivalent shear rigidity
2 1 ðEIÞeq ¼ Ef Btf hu þ 2tw þ tf 2
ðAGÞeq ¼
2 B hu þ 2tw þ tf hu þ 2tw
Gcxz
(14)
(15)
Here Ef is Young's modulus of the face sheet and Gcxz is the shear stiffness of the 3D corrugated core, which can be obtained by equivalence theory of the unit cell [19]. 4.2.2. Bending failure load There are four possible failure modes in three-point bending for 3D corrugated core sandwich beams. According to Ref. [24], the failure load is (1) Face crushing (FC)
PFC ¼
4Btf hu þ 2tw þ tf L
sfc
(16)
where sfc denotes the yielding strength of the face sheet. (2) Face wrinkling (FW)
PFW ¼
4p2 Ef hc þ 2tw þ tf Btf3 3Lðlu aÞ2
(17)
(3) Core shear failure
PCF ¼ 2Bðhu þ 2tw Þtfc ¼ Fig. 5. Buckling deformation of two ends hinged trapezoidal web with non-uniform cross-section.
4sw tw bu cos qBðhu þ 2tw Þ ðweb fracture ðCFÞÞ lu wu
(18)
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4.3. Failure mechanism maps
PCB ¼ 2Bðhu þ 2tw Þtbc ¼m
3 b cos qðh þ 2t Þ BEw p2 tw u u w ðweb buckling ðCBÞÞ 3lu wu l2w
PCD ¼ 2Bðhu þ 2tw ÞtD c ðcore debonding ðCDÞÞ
(19)
(20)
f
where tc and tbc are the shear strength of the core including fracture and buckling. tD c is interfacial shear strength which can be obtained through shear test. (4) Indentation
PIF ¼
PIB
The failure modes of 3D corrugated core sandwich panels under three-point bending can be illustrated in collapse mechanism maps based on analytical parameters. Face wrinkling, Face crushing, core web crushing and core web buckling were studied in this work. The failure maps were developed as a function of the non-dimensional geometrical parameters tf/l, tw/l (l is the ratio of the maximum moment to the maximum transverse force) and graded parameter n, and the boundaries of failure modes were obtained by equating the critical loads for different failure modes. 5. Results and discussions
2sw tw bu B sin q ðweb fracture ðIFÞÞ lu
3 b B sin q Ew p2 tw u ¼m ðweb buckling ðIBÞÞ 6lu l2w
(21)
5.1. Compression tests
(22)
Fig. 6a shows the measured response of specimen C1, which has the smallest graded parameter. The stress firstly increases until the first peak. Then the stress decreased due to the fracture of the trapezoid web. With the increasing of strain, the stress increases again and reaches the second peak. Fig. 6b shows the measured
Fig. 6. Compression stress-strain curves and failure photographs of 3D corrugated core composite sandwich panels.
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Table 2 Theoretical and experimental value of compression strength and modulus of 3D corrugated core sandwich structures with different graded parameter n. Specimen
n
C1
0.17
C2
0.25
C3
0.50
Analytical
Experiment
Failure mode
Strength/stiffness (MPa)
Failure mode
Strength/stiffness (MPa)
CF CB CF CB CF CB
2.42/286.79 2.79/286.79 2.42/222.40 2.07/222.40 2.42/148.27 1.14/148.27
CF
2.37/252.13
CB
1.88/198.72
CB
1.01/124.88
response of specimen C2, which has an intermediate graded parameter. The stress firstly increases until the failure occurs at 1.88 MPa. This collapse is caused by the Euler buckling of the trapezoid webs. Then, the stress begins to decrease and followed by a plateau region along with the fluctuation. This is due to the gradual delamination of the trapezoid webs as the load increased. The compressive stress-strain curves measured for specimen C3 is shown in Fig. 6c. Comparing with specimens C1 and C2, specimen C3 has the largest graded parameter. That means the trapezoid web is closed to a rectangle and the failure mode is similar to conventional corrugated core sandwich structure. When the stress reaches the maximum, Euler buckling failure occurs. With increasing the strain, the stress drops rapidly. The predicted and tested results of the 3D corrugated core sandwich panels in compression are summarized in Table 2. Fig. 7 shows a comparison of the compression strength between 3D corrugated lattice sandwich structures with other lattice truss structures. Although the strength is relative low, 3D corrugated sandwich structures have potential applications because of their variable cross-section cores, which can facilitate the graded function in thicknessdirection. 5.2. Bending tests Four kinds of bending specimens (in y-direction) with different graded parameter and face sheets thickness (listed in Table 1) were tested. Fig. 8a shows the measured response of specimen B1, which have the strong face sheets, weak trapezoid webs and large graded parameter. The load firstly increases until the shear buckling occurs at a load of ~584.10 N. Then, the load slowly increases again
combining with short undulation plateaus until the maximum peak load reaches. Fig. 8b shows the measured response of specimen B2. The failure load is ~1262.73 N and the failure mode is the fracture of trapezoid webs. It is found that the failure load of specimen B2 is much higher than that specimen B1. This indicates that the graded parameter has obvious influences on the bending failure load of 3D corrugated core sandwich beams when the face sheets are strong. Fig. 8c and d shows the measured responses of specimen B3 and B4. Comparing specimen B3 with specimen B4, it is found that the graded parameter has little influences on the failure mode and the failure load of the 3D corrugated core sandwich beam when the face sheets are weak. The predicted and tested bending deformation, stiffness of 3D corrugated core sandwich beams (in y-direction) are listed in Table 3 as well as the failure loads and collapse modes are summarized in Table 4. The tested values are in good agreement with the theoretical predictions. Fig. 9 shows the load-displacement curve and failure photograph of 3D corrugated core composite sandwich beam (in x-direction) under bending loads. It is found that the load firstly increases until the delamination occurs at a load of ~879.10 N. The mechanism maps for 3D corrugated core sandwich panels were plotted in Figs. 10 and 11. Fig. 10a illustrates the failure map as functions of tf/l and tw/l for n¼0.25. With high ratio tf/l and tw/l, core shear fracture and face sheets crushing are the mainly collapse mechanism due to the strong trapezoid webs. With low ratio tf/l and tw/l, core shear buckling and face sheets wrinkling are the mainly collapse mechanism as observed in specimen B4. Fig. 10b illustrates the failure maps for 3D corrugated core sandwich structures for n¼0.50. In comparison with the counterpart for n¼0.25, the maps are slightly shifted to the right hand side, and the domains of core shear buckling is slightly expanded, due to the buckling parameter m decreases as the graded parameter n increase and the core becomes weaker against the shear load. Fig. 11 plotted the failure modes in terms of normalized parameters tf/l and n. At low ratio tf/l, face sheets wrinkling is the mainly collapse mechanism. At high ratio tf/l and small graded parameter n, core shear fracture is the mainly collapse mechanism as observed in specimens B2. At high ratio tf/l and large graded parameter n, core shear buckling is the mainly collapse mechanism as observed in specimens B1. 6. Conclusions
Fig. 7. Compression strengths of typical lattice core composite structures [8].
A novel 3D corrugated core sandwich structure has been presented, which is different from the conventional corrugated core sandwich structures. Analytical models were developed to predict the mechanical behaviors of 3D corrugated core sandwich structures. Compression and bending carbon/epoxy composite sandwich specimens with 3D corrugated cores were designed and tested to investigate the influence of the geometry and graded parameters on the mechanical response and failure modes. The results showed that mechanical properties and the failure modes of
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Fig. 8. Load-displacement curves and failure photographs of 3D corrugated core composite sandwich beams in y-direction under bending loads.
Table 3 Three-point bending deformation of 3D corrugated core sandwich beams (in y-direction) along with predicted and measured P/d. -1
Specimen
ds and db (mm)
Total d (mm)
Analytical P/d (N mm
B1
1.156 0.806 1.154 1.719 0.603 1.163 0.403 1.163
1.962
346.172
313.523
2.873
504.212
456.725
1.766
353.014
325.177
1.565
399.634
365.156
B2 B3 B4
)
Test P/d (N mm
-1
)
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Table 4 Predicted and measured failure loads and collapse modes of 3D corrugated core sandwich beams (in y-direction) under three-point bending tests. Specimen
B1
B2
B3
B4
Analytical results
Experimental results
Failure mode
Failure force (N)
Failure mode
Failure force (N)
FC FW CF CB CD IF IB FC FW CF CB CD IF IB FC FW CF CB CD IF IB FC FW CF CB CD IF IB
10345.60 2146.14 1739.28 679.18 1900.80 3966.48 1884.12 10345.60 2146.14 1449.40 1809.11 1900.80 4945.81 3966.48 6761.84 623.42 2898.80 1272.22 1900.80 6610.80 3502.72 6761.84 623.42 2898.80 2153.71 1900.80 6610.80 5887.86
CB
584.10
CF
1262.73
FW
629.45
Fig. 11. Failure map illustrated as functions of tf/l and n.
FW
707.98
3D corrugated core composite sandwich structures mainly depend on the geometry and graded parameters. Under compression, the failure mode of the sandwich structure with small graded parameter is fracture of the core. But that with large graded parameter tends to core buckling. The graded parameter has little influence on the bending strength and failure mode for 3D corrugated core composite sandwich structures with thin face sheets. But it has obvious influence when face sheets are thick. The measured peak loads and failure modes were in good agreement with the analytical predictions. Failure maps were constructed to predict the bending response of 3D corrugated core sandwich structure.
Fig. 9. Load-displacement curve and failure photographs of 3D corrugated core composite sandwich beams in x-direction under bending loads.
Fig. 10. Failure maps illustrated as functions of tf/l and tw/l for two given graded parameter n a) n¼0.25; b) n¼0.50.
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