Mechanics of Composite Materials, Vol. 53, No. 2, May, 2017 (Russian Original Vol. 53, No. 2, March-April, 2017)
EXPERIMENTAL VALIDATION OF THE TRANSVERSE SHEAR BEHAVIOR OF A NOMEX CORE FOR SANDWICH PANELS
M. I. Farooqi,1 M. A. Nasir,1* H. M. Ali,1 and Y. Ali2
Keywords: Nomex honeycomb, transverse shear moduli, finite-element analysis This work deals with determination of the transverse shear moduli of a Nomex® honeycomb core of sandwich panels. Their out-of-plane shear characteristics depend on the transverse shear moduli of the honeycomb core. These moduli were determined experimentally, numerically, and analytically. Numerical simulations were performed by using a unit cell model and three analytical approaches. Analytical calculations showed that two of the approaches provided reasonable predictions for the transverse shear modulus as compared with experimental results. However, the approach based upon the classical lamination theory showed large deviations from experimental data. Numerical simulations also showed a trend similar to that resulting from the analytical models.
1. Introduction Structural sandwich panels are made from two thin face sheets and lightweight core between them. The core is usually made of some cellular material, such as foam or honeycomb. Structural sandwich panels are becoming more and more popular in industry because of their light weight and high stiffness. The core is responsible for their light weight, and the stiffness is provided by the face sheets, or skins [1-3]. In order to achieve the optimum benefits of structural sandwich panels, the core should have a sufficient stiffness so that the two face sheets should not slide over each other when a bending load is applied to the sandwich panel [4]. The core shows a high resistance to deformation due to the transverse shear moduli. Since honeycomb cores have a periodic structure, their analysis can be performed by using the equivalent elastic moduli [5]. It is important to note Department of Mechanical and Aeronautical Engineering, University of Engineering and Technology, Taxila, Pakistan University of Sargodha, Sargodha, Pakistan * Corresponding author; e-mail:
[email protected] 1 2
Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 53, No. 2, pp. 279-290 , March-April, 2017. Original article submitted February 11, 2015; revision submitted April 5, 2016. 0191-5665/17/5302-0193 © 2017 Springer Science+Business Media New York
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that the core exhibits an in-plane and skins out-of-plane mechanical behavior [6]. Therefore, it is very important to determine the equivalent transverse shear moduli of honeycomb. Gibson and Ashby supposed that [7] the out-of-plane transverse shear properties of honeycomb core are determined by two dissimilar transverse shear moduli. Various analytical methods have been presented by many authors for predicting the properties of honeycomb core as functions of its geometry and material characteristics. In all those investigations, the base material of the honeycomb core was isotropic, e.g., aluminum [3, 7, 8]. In the present research, the base material considered is Nomex®, which is not isotropic. Honeycomb cores are manufactured from a Nomex® paper, which is orthotropic and whose mechanical properties are not reported in the literature. Foo et al. [9] determined these properties experimentally and used them in numerical simulations to determine the elastic moduli of honeycomb core. Zie et al., [10] carried out a study on the out-of-plane shear properties of superalloys. Reyes et al. [11] carried out FEM analyses to determine the natural frequencies of sandwich panels and compared their results with published data. Mirko et al., [12] looked into the possibility of using 3D random fiber composites to repair damaged sandwich structures. This research extends the work done on Nomex® honeycomb cores by Nasir et al., [13]. In their work, the authors carried out various numerical and theoretical calculations. In the present work, a similar approach is adopted. In all analyses, the mechanical characteristics determined by Foo et al. [9] are used. A comparison experimental data and calculation results found using the approaches proposed by Meraghni et al., [6], Kelsey et al., [8], and Gibson and Ashby [7] is made.
1.1. Analytical calculations The analytical approach presented by Meraghni, Desrumaux, and Benzeggagh [6] is based upon the classical lamination theory. Their model is for a regular hexagonal honeycomb, therefore, the resulting equations give single values rather than bounds. Meraghni et al. [14] suggested the following relations for the out-of-plane equivalent transverse shear moduli:
∗ Gxz =
eG (1 + 2 cos θ ) , 2a sin θ (1 + cos θ )
G ∗yz =
eG (e + 2a sin θ ) , 2a (1 + cos θ )(e + a sin θ )
where G is the elastic modulus of the base material, and e is the thickness of cell wall. Gibson and Ashby proposed an analytical model for calculating the equivalent transverse shear moduli in the rib∗ bon and transverse directions, Gxz and G ∗yz , respectively. The material was assumed isotropic. The authors also provided bounds for theses moduli to consider the nonuniform stress field in sheared honeycombs and showed that, for loading in the x direction (ribbon direction), these bounds were equal for any cell geometry, while for the y direction (transverse direction), the lower and upper bounds converged to a single value for a regular hexagon. Their analytical model is given by the equations [15]
∗ Gxz =
(1 + sin 2 θ )eG (1 + 2 sin 2 θ )eG eG cos θ , £ G ∗yz £ . a (1 + sin θ ) 3 cos θ 2(1 + sin θ ) cos θ
Kelsey, Gellaty, and Clark [8] proposed analytical relations to predict the equivalent stiffness of a honeycomb core made from metallic foils. They assumed that the base material is isotropic and also resolved the difficulty connected with the nonuniform stress field in the sheared honeycomb by giving bounds for the upper and lower bounds for the shear moduli. The bounds in the y direction (transverse direction) were assumed the same for all cell shapes. Further, for a regular hexagonal cell, they were also taken equal in the x direction (ribbon direction). The relation proposed by Kelsey et al. [8] for G ∗yz is
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G ∗yz =
eG sin θ . a (1 + cos θ )
b
a
Fig. 1. Angle θ according to [6, 8] (a) and [7] (b).
h
a
b
Fig. 2. Honeycomb core. Fig. 3. Unit cell of a hexagonal honeycomb (a quarter cell is highlighted).
In all previous relations, it is assumed that cell walls are of the same length. The angle q is defined differently by different authors, as shown in Fig. 1.
1.2. Finite-element analysis ∗ To determine the equivalent transverse shear moduli Gxz and G ∗yz of the Nomex® honeycomb core, finite-element analyses were carried out using the ANSYS software. The basic methodology adopted for the finite-element modeling used the unit cell of honeycomb instead of the whole hexagonal structure. In Fig. 2, the honeycomb used as a core is shown. In Fig. 3, a single cell of the honeycomb is illustrated. ∗ The boundary conditions used to find Gxz and G ∗yz are detailed in Table 1 (also see Fig. 4). The material used for the present work was Nomex® honeycomb, therefore, the material properties of Nomex® paper as the base material were used. They were taken from [9], where they had been determined experimentally. The
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∗ TABLE 1. Boundary Conditions Used for Calculating Gxz and G ∗yz
Boundary I II III IV Top face I II III IV Top face
Displacement uy
ux Constrained Free Free Free u/2 Constrained Free Constrained Constrained
Shear modulus
∗ Gxz
uz
Constrained Constrained Free Constrained Constrained
Constrained Constrained Constrained Free Constrained
Shear modulus G ∗yz Constrained Free Free Free u/2
Free Constrained Constrained Free Constrained
1 1 II
IV III
I I
Fig. 4. Description of different faces in formulating the symmetry of one fourth of model of the unit cell. average results for the longitudinal and transverse elastic moduli were found to be 3.4 and 2.46 GPa, respectively. So, we ∗ took Gxz in the fiber direction G ∗yz in the transverse one. A displacement u/2 was applied to the upper face of the quarter model of unit cell to determine the reaction force F numerically. To obtain the equivalent strain ε, the displacement u was divided by the cell height h. To find the equivalent stress σ, the reaction force F was multiplied by four and then divided by the area S of the unit cell. The equivalent modulus was calculated as the ratio of the equivalent stress to the equivalent strain. The corresponding formulas are
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ε=
u 4F σ ∗ , σ= , S = 2a 2 cos θ (1 + sin θ ), Gxz or G ∗yz = . h S ε
TABLE 2. Analytical Results Cell wall thickness e, mm
Loading direction
0.10 0.16 0.26 0.10 0.16 0.26
x axis x axis x axis y axis y axis y axis
[8] 26.76 42.82 69.59 17.86 28.57 46.42
G ∗yz , МPa [7] 26.76 42.82 69.59 17.86 28.57 46.42
[6] 35.69 57.09 92.78 20.21 31.96 51.01
b
a
Fig. 5. Deformed shape of quarter model as result of application of a displacement to the top face in the x direction (a) and the relation forces obtained as the result of simulation (b). 1.3 Experiments Experiments were carried out on three types of Nomex® honeycomb cores with regular hexagonal cells to find the equivalent transverse shear moduli in the ribbon and transverse directions. The experiments were performed according to ASTM standard C-273, which allows one to carry out the test on a bare core, without face sheets, and sandwich face sheets attached to the test fixture. The outcome of the experiment was load--deflection data in the form of a curve. Postprocessing of the experimental data gave the shear stress at any point of loading, from which the effective shear modulus Gx∞ could be computed [16] by the formula
Gx• =
Rh , Lb
where R = ΔP/ΔL is the slope of the initial, linear Portion of the load–deflection curve, h is the thickness of core, and b is specimen width.
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a 35 30 25 20 15 10 5
P, kN
0
0.5
b
2
3
5
1.5
2.5
0
1
4 1.0
u, mm
18 14 10 6 2
2.0
P, kN 3 1
5 4
u, mm
2 1
2
3
4
Fig. 6. Load–deflection curves P–u in loading samples 1-5 in the ribbon (a) and transverse (b) directions. e = 0.1 mm.
TABLE 3. Results of Numerical Simulation e, mm
Loading direction
F, Н
e, %
s, МPa
Shear modulus, MPa
0.10 0.16 0.26 0.10 0.16 0.26
x axis x axis x axis y axis y axis y axis
3.487 5.579 9.066 1.803 2.885 4.688
0.025 0.025 0.025 0.025 0.025 0.025
0.684 1.095 1.780 0.354 0.566 0.921
27.36 43.80 71.20 14.16 22.64 36.84
2. Results and Discussion 2.1. Analytical Results Analytical calculations were performed by using the approaches developed by Kelsey et al., [8], Gibson and Ashby [7], and Meraghni et al., [6], and the results obtained are shown in Table 2. 2.2. Results of numerical simulation The numerical simulations were performed using the FEM-based ANSYS software for the quarter model of the unit cell. In each case small displacement (u/2 = 0.1 mm) were applied to the top face of the finite-element model. The resulting deformed shape is shown in Fig. 5a. The reaction forces at the bottom face are shown in Fig. 5b. Results of the numerical simulation are presented in Table 3. In these calculations, the area S was found from the formula
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S = 2a 2 cos θ (1 + sin θ ).
∗ TABLE 4. Postprocessing Results for Gxz (Cell Wall Thickness e = 0.1mm)
Sample
Pmax, kN
Shear stress, MPa
ΔL, mm
ΔP, kN
R = ΔP/ΔL, kN/mm
∗ Gxz , MPa
1 2 3 4 5
8.75 10.59 10.34 8.05 9
0.94 1.14 1.11 0.86 0.96
0.21 0.3 0.19 0.16 0.27
3.72 6.06 3.8 3.0 4.2
17.7 20.2 20.0 18.75 15.55
15.24 17.37 17.20 16.13 13.38
b
|
+
l, mm
+
+
+
+
55 Gyz, МPa 50 45 40 35 30 25 20 | 15 | 10 5 0.05 0.10 0.15
+
a 100 Gxz, MPa 90 80 70 60 | 50 40 | 30 20 l, mm | 10 0.05 0.10 0.15 0.20 0.25 0.30
0.20
0.25
0.30
∗ Fig. 7. Comparison of simulation, analytical and experimental results for Gxz (a) and G ∗yz (b): simulation (♦), [6] (x), [7] (▲), [8] (■), and experiment (*).
TABLE 5. Postprocessing Results for G ∗yz (Cell Wall Thickness e = 0.1 mm) Sample
Pmax, kN
Shear stress, MPa
ΔL, mm
ΔP, kN
R = ΔP/ΔL, kN/mm
Gyz, MPa
1
7.25
0.78
0.4
4.42
11.05
9.50
2
7.7
0.83
0.4
5.49
13.72
11.80
3
8.75
0.94
0.4
4.25
10.62
9.14
4
7.95
0.85
0.4
4.56
11.4
9.80
5
7.6
0.82
0.4
4.67
11.67
10.04
2.3. Experimental results Separate tests were performed for the ribbon and transverse directions, and the found results are presented in the foregoing Sects. ∗ 2.3.1. Tests for determining Gxz . In Fig. 6a, experimental results for five samples of honeycomb loaded in the ribbon direction are shown. As is seen, they all follow almost the same pattern. The elastic and plastic regions are separated from each other very clearly. The initial, linear portions of the curves were used for determining the equivalent shear moduli, and the peak loads were employed for calculating the shear strengths. The plastic region were about three times as large as the elastic one.
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TABLE 6. Averaged Experimental Results Sample
e, mm
Loading direction
Average shear strength, MPa
Average modulus, MPa
1 2 3 4 5 6
0.10 0.16 0.26 0.10 0.16 0.26
x axis x axis x axis y axis y axis y axis
1 2.65 3.07 0.84 1.07 1.78
15.86 28.94 52.83 10.06 17.31 24.32
TABLE 7. Comparison of Simulation, Analytical, and Experimental Results e, mm
Loading direction
0.10 0.16 0.26 0.10 0.16 0.26
x axis x axis x axis y axis y axis y axis
Simulation 27.36 43.80 71.20 14.16 22.64 36.84
[8] 26.76 42.82 69.59 17.86 28.57 46.42
Shear modulus, MPa Analytical [7] [6] 26.76 35.69 42.82 57.09 69.59 92.78 17.86 20.21 28.57 31.96 46.42 51.01
Experiment 15.86 28.94 52.83 10.06 17.31 24.32
The experimental results found, after postprocessing, are given in Table 4. Values of the parameter R were obtained from experimental curves. From the initial, linear portions of the curves, two arbitrary points were selected on the displacement axis, and the corresponding value of forces were obtained on the load axis. These values are tabulated as ΔL and ΔP, whose ratio gives R. Similar tests is loading in the ribbon direction were carried out on samples with wall thicknesses e = 0.16 and 0.26 mm. 2.3.2. Tests for G ∗yz . Figure 6b shows the actual experimental results found on honeycomb samples with e = 0.1 mm. Table 5 presents the results obtained by post processing experimental data for a cell with e = 0.1 mm. Table 7 presents the average values obtained as a result of the above-mentioned tests. The results of finite-element simulations, analytical calculations, and experiments are presented in Table 7. Figure 7 compares the same results. Discussion In all experimental curves, two regions can be clearly identified — before and after the peak load. The first one can be regarded as the linear elastic region, but the second one — as the plastic region. These regions can be divided into four categories on the basis of load–deflection trends. The first one is the elastic region. After the peak load, a rapid drop in load can be observed, and this is the region where the deformation of inclined walls takes place. Also, the wrinkling of cell walls occurs here. At the third stage, the load becomes almost constant with increasing deformation. Here, the fracture of inclined cell walls and debonding between cell walls proceed. In the last stage, load first decreases rapidly and then becomes constant. This is the stage of gross failure. The initial, linear portions of the experimental curves were used to determine the equivalent transverse shear moduli in the transverse and ribbon directions. The methodology used agrees with the guidelines of ASTM C-273. The loading rates for the ribbon and transverse directions were different — according to ASTM C-273, they to be such that the peak load is reached between 3 to 6 min. 200
From the experimental curves found, it is seen that the peak load always increased and the full deformation decreased as the thickness of cell wall increased, i.e., the honeycomb stiffness increased. At the same cell wall thickness, the ribbondirection curves showed higher peak loads than the transverse-direction ones. Further investigation revealed smaller deformations at rupture for the ribbon-direction loading than for the transverse-direction one at the same cell wall thickness. These facts indicate that the ribbon-direction stiffnesses was higher than that of the transverse direction one. A comparison of experimental, numerical, and analytical results is illustrated in Fig. 7 and show that the simulation and analytical results are quite close. As is seen, the Kelsey et al., and Gibson and Ashby approaches gave exactly the same results The lower and upper bounds presented by these approaches also converged to the same values. But this kind of behav∗ ior was observed for regular hexagonal honeycombs with the same length of all cell walls. For Gxz , the Meraghni et al. approach (the classical lamination theory) showed larger deviations from experimental and numerical results than the other two analytical ones. A similar trend was observed for G ∗yz . Taking into account these facts, the approach presented by Meraghni et al., should be used carefully in predicting the equivalent transverse shear moduli for honeycomb cores. Both in the ribbon and transverse directions, the experimental values of elastic moduli were lower than the numerical and analytical results. The work of Meraghni et al., [6] also involves experimentation on honeycombs made from an isotropic base material (aluminum). In their work, the experimental results were underestimated by the analytical approach. The reason is quite obvious — the authors used an isotropic base material, but the honeycomb cores considered in the present study were made from a Nomex® paper, which is orthotropic in nature. Conclusions The numerical and analytical approaches originally used for isotropic materials have been successfully used to predict the shear moduli for a honeycomb based on a Nomex® paper, which is orthotropic in nature. The classical lamination theory showed comparatively large deviations from experimental results, because it is developed for rather thin laminates and not for thick cores. The Kelsey et al., & Gibson and Ashby approaches provided better estimates for the transverse shear moduli of Nomex® honeycomb cores. Numerical simulations followed the same trend as the analytical approaches. Also, the results obtained show that the design of sandwich panels with honeycomb cores should be based upon the transverse shear stiffness rather than the ribbon-direction ones. REFERENCES 1. S. D. Pan, L. Z. Wu, Y. G. Sun, Z. G. Zhou, J. L. Qu, “Longitudinal shear strength and failure process of honeycomb cores,” Composite Structures 72, 42-46 (2006). 2. C. W. Schwingshackl, G. S. Aglietti, and P. R. Cunningham, “Determination of honeycomb material properties: Existing theories and an alternative dynamic approach,” J. of Aerospace Engineering, 177-183 (2006). 3. S. Balawi and J. L. Abot, “A refined model for the effective in-plane elastic moduli of hexagonal honeycombs,” Compos. Struct., 84, 147-158 (2008). 4. T. Nordstrand and A. C. Leif, “Evaluation of transverse shear stiffness of structural core sandwich plates,” Compos. Struct., 37, 145-153 (1997). 5. A. Abbadi, Y. Koutsawa, A. Carmasol, S. Belouettar, and Z. Azari, “Experimental and numerical characterization of honeycomb sandwich composite panels,” Simulation Modeling Practice and Theory, 17, 1533-1547 (2009). 6. F. Meraghni, F. Desrumaux, and M. L. Benzeggagh, “Mechanical behavior of cellular core for structural sandwich panels,” Composites: Part A, 30, 767-779 (1999). 7. L. J. Gibson and M. F. Ashby, “Cellular Solid Structure and Properties,” Pergman Press England (1988). 8. S. Kelsey, R. A. Gellaty, and B. W. Clark, “The shear modulus of foil honeycomb core,” Aircraft Engn, 294-302 (1958).
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