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Dynamic response of fiber–metal laminates (FMLs) subjected to low-velocity impact
ARTICLE IN PRESS Thin-Walled Structures 48 (2010) 62–70
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Dynamic response of fiber–metal laminates (FMLs) subjected to low-velocity impact G.H. Payeganeh a, F. Ashenai Ghasemi a,, K. Malekzadeh b a b
Department of Mechanical Engineering, Shahid Rajaee Teacher Training University (SRTTU), Lavizan, Postal Code 16788-15811, Tehran, Iran Department of Mechanical Engineering, Malek Ashtar University of Technology, Tehran, Iran
a r t i c l e in f o
a b s t r a c t
Article history: Received 4 September 2008 Accepted 16 July 2009 Available online 18 August 2009
Fiber–metal laminates (FMLs) are high-performance hybrid structures based on alternating stacked arrangements of fiber-reinforced plastic (FRP) plies and metal alloy layers. The response of FMLs subjected to low-velocity impact is studied in this paper. The aluminum (Al) sheets are placed instead of some of layers of FRP plies. The effect of the Al layers on contact force history, deflection, in-plane strains and stresses of the structure is studied. The first-order shear deformation theory as well as the Fourier series method is used to solve the governing equations of the composite plate analytically. The interaction between the impactor and the plate is modeled with the use of a two degrees-of-freedom system, consisting of springs-masses. The Choi’s linearized Hertzian contact model is used in the impact analysis of the hybrid composite plate. The results indicated that some of the parameters like the layer sequence, mass and velocity of the impactor in a constant impact energy level, and the aspect ratio (a/b) of the plate are important factors affecting the dynamic response of the FMLs. Interaction among the mentioned geometrical parameters and material parameters like the aluminum 2024-T3 alloy layers is studied. The numerical results that are presented in this paper hitherto not reported in the published literature. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Impact Composite Fiber–metal laminate (FML) Fiber-reinforced plastic (FRP) Springs-masses model
1. Introduction Fiber–metal laminates (FMLs) are hybrid structures consisting of different metal sheets and fiber-reinforced plastic (FRP) layers. One of the most important objects of their production is to combine the good impact resistance of the metals with the better lightweight characteristic of FRP laminates [1–5]. Therefore, there will be an excellent candidate material to be used for aerospace structures [6–8]. Abrate [9–11] studied the impact behavior of composite structures extensively. Olsson [12–14] classified low-velocity impacts in two categories, the small mass and the large mass impact, which is based on the ratio of the impactor mass to the target mass. When the mass of the impactor is small in comparison with the weight of the target, there will be a small mass response dominated by shear and flexural waves in which the deflection, load, and flexural stresses are out of phase. When the mass of the impactor is much larger than the target mass, there will happen a ‘quasi-static’ large mass response, in which the deflection, peak load, and stresses are more or less in phase. Vlot [1] showed that the damaged zone of FMLs after the impact is
Corresponding author. Tel.: +98 21 22970052; fax: +1 678 8685363.
E-mail address: [email protected] (F. Ashenai Ghasemi). 0263-8231/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2009.07.005
smaller than the FRPs. Caprino et al. [2] demonstrated that the overall impact force–displacement curve of FMLs under lowvelocity impact only depends on the impact energy, rather than mass and speed of the impactor separately. Caprino et al. [3] presented a mechanistic model with neglecting the macroscopic behavior of the structure (i.e. neglecting local deformation due to indentation, overall deflection, damage initiation and development) to study the low-velocity impact. Atas [4] has done an experimental investigation to carry out the damage process of FMLs under low-velocity impact. Abdullah et al. [5] showed the positive effect of the FMLs in comparison with the FRP laminates in high-velocity impact too. They investigated that the stacking sequence is an important parameter on the perforation resistance of these structures. In this research, a complete model is developed so that the effect of low-velocity impact upon the FMLs demonstrated. With this model, the researchers will be enabled to investigate the complete response of FMLs subjected to low-velocity impact, considering of the shear deformation effect. They can also determine the contact force history independently and regardless of solving the motion equations of the plate. This also saves lot of the computational time efforts. The effect of using Al layers as well as some of the parameters such as the stacking sequence, the mass and the velocity of the impactor in a constant energy level and the aspect ratio of the structure on the impact response of
ARTICLE IN PRESS G.H. Payeganeh et al. / Thin-Walled Structures 48 (2010) 62–70
FMLs is studied in details. The present study of these parameters is useful to the designers too, which have not been presented in similar literatures.
63
boundary conditions are as follows: w ¼ cx;x ¼ cy ¼ 0; at x ¼ 0; a w ¼ cy;y ¼ cx ¼ 0; at y ¼ 0; b
ð4Þ
2. Governing equations Fig. 1 shows a FML, which is in contact with an impactor mass. Here, the plate equations developed by Whitney and Pagano [15] are used. They included the effect of transverse shear deformations, which the displacement field is u ¼ u0 ðx; y; tÞ þ zcx ðx; y; tÞ v ¼ v0 ðx; y; tÞ þ zcy ðx; y; tÞ 0
w ¼ w ðx; y; tÞ
ð1Þ
u0, v0 and w0 are the plate displacements in x, y and z directions at the plate mid-plane and cx and cy are the shear rotations in the x and y directions. For the specially orthotropic form (Bij ¼ 0, A16 ¼ A26 ¼ D16 ¼ D26 ¼ 0) results in
3. Constitutive equations Constitutive equations of stress–strain relationship for a FML are as follows [17]: 8 9 8 98 9 e = > > < s1 > = < Q11 Q12 0 > => < 1 > e2 s2 ¼ Q12 Q22 0 > > >> > :t > ; :0 0 Q ;: g ; 12
(
t13 t23
66
)
( ¼
Q55
0
0
Q44
)(
g13 g23
12
) ð5Þ
€ D11 cx;xx þ D66 cx;yy þ ðD12 þ D66 Þcy;xy ksh A55 cx ksh A55 w;x ¼ Ic x € ðD12 þ D66 Þcx;xy þ D66 cy;xx þ D22 cy;yy ksh A44 cy ksh A44 w;y ¼ Ic y € ksh A55 cx;x þ ksh A55 w;xx þ ksh A44 cy;y þ ksh A44 w;yy þ q ¼ rw
ð2Þ
ksh is the shear correction factor introduced by Mindlin [16], usually equals to p2/12 and q the dynamic normal load over the plate and also Z h=2 Z h=2 Qijk ð1; z; z2 Þdz; ðr; IÞ ¼ r0 ð1; z2 Þdz ð3Þ ðAij ; Bij ; Dij Þ ¼ h=2
h=2
In the above equation, {s} represents the stresses in the principle directions. In addition, the matrix {e} represents the strains in the principle directions. Qij represents the reduced stiffness matrices for the FML structure. Because of discontinuity function of stresses through the thickness in each layer, it is possible to determine the constitutive equation by considering the force-couple resultants in terms of stresses, using integration of Eq. (5) through the plate thickness, which yields #( ) "A Bij N ij e0 ; i; j ¼ 1; 2; 6 ¼ Bij Dij M k
In the above relation, r0 represents the density of each layer and r the total density of the plate. In addition, I is the moment of inertia and h the thickness of the plate. (Qij)k (i, j ¼ 1,2,6) are the reduced in-plane stiffness components and (Qij)k (i, j ¼ 4,5) are the reduced transverse shear stiffness components[15]. In the present research, a simply supported rectangular plate is chosen to study with the dimensions of a and b, which its
where N and S are vectors of forces and M the vector of moments, respectively. Aij, Bij, and Dij are the components of extensional and shear, coupling, and bending stiffness matrices, respectively. Also e0 and g are the mid-plane and shear strains, respectively and k is the curvatures. In addition, ksh is the shear correction factor.
Fig. 1. Schematic view of the FML impacted by a spherical mass on its center.
Fig. 2. A two degrees-of-freedom springs-masses model [3,18].
fSg ¼ ½ksh Aij fgg; i; j ¼ 4; 5
ð6Þ
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after rearrangements [18] m1 z€ 1 ¼ K1 z1 K2 ðz1 z2 Þ
4. Solution of dynamic response of the plate 4.1. Analysis of contact force
m2 z€ 2 ¼ K2 ðz2 z1 Þ
Here, two degrees-of-freedom springs-masses model [3,18] is used to determine the impact force (Fig. 2). The motion equation is as follows:
ð10Þ
The above system of ordinary equations can be solved by the Runge–Kutta method, using of the MATLAB ode solver.
m2 z€ 2 þ F ¼ 0 4.2. Analysis of deflection and stress–strain m1 z€ 1 þ K1 z2 þ K2 z32 F ¼ 0
ð7Þ
The index 1 belongs to the hybrid plate and the index 2 indicates the impactor. F is the contact force, m1 ¼ mp and m2 ¼ mi represents, respectively the mass of the impactor and the FML plate, z1 and z2 are, respectively, the relative displacements of the impactor and the structure masses. K1 ¼ Kbs is the bendingshear stiffness the plate. Using of the Choi’s linearized model [19], instead of non-linear Hertzian contact law, the contact force can be obtained as FðtÞ ¼ Kl a ¼ Kl ðz2 z1 Þ 1=3
2=3
Kl ¼ F m Kc
ð8Þ
In the above equations, K2 ¼ Kl represents the linearized contact coefficient in Choi’s linearized contact law, Fm is the maximum predicted contact force, and Kc represents the contact stiffness in the improved Hertzian contact law, which can be calculated based on the following Eq. [19]: Kc ¼
1=2 R2
4 3 ð1 u22 =EÞ þ ð1=E22 Þ
ð9Þ
Neglecting the effects of rotary inertia [16], the current problem could be converted to a system of ordinary differential equations of second order in time for the Fourier coefficients of the transverse deflection [20]. The equation of the motion of a FML plate subjected to a point load q(x, y, t) is equal to [20,21] 39 2 38 2 39 82 0 L11 L12 L13 > > = = > < < Amn ðtÞ > 6 7 6L 7 6 7 ð11Þ 5 4 12 L22 L23 5 4 Bmn ðtÞ 5 ¼ 4 0 > > > ; € mn ðtÞ > L13 L23 L33 : Wmn ðtÞ ; : Pmn ðtÞ rhW where the Lij coefficients are introduced in [21] and mp np XX Pmn ðtÞSin qðx; y; tÞ ¼ xSin y a b m n
Also, for a concentrated load located at the point (xc, yc), Pmn(t) are the terms of the Fourier series as mp np 4FðtÞ ð13Þ Pmn ðtÞ ¼ Sin xc Sin yc a b ab where F(t) is the impact load (see Eq. (8)). The impact solution for a rectangular plate with simply supported boundary conditions is assumed to be in the following form [22,23]:
cx ðx; y; tÞ ¼ where R2 is radius of the curvature, n2 the Poisson’s ratio, E and E22 the elastic modulus of the isotropic impactor and the transverse elastic modulus of the top layer of the FML structure, respectively. Considering the Fig. 2 and replacing the value of F in Eq. (7) with the values presented by Choi’s differential equations
1 X
Amn ðtÞCos
mp a
m;n¼1 1 X
xSin
np y b
mp np xCos y cy ðx; y; tÞ ¼ Bmn ðtÞSin a b m;n¼1 wðx; y; tÞ ¼
mp np Wmn ðtÞSin xSin y a b m;n¼1 1 X
10000 9000
Present Delfosse [26]
8000
Pierson [25]
Contact Force (N)
7000 6000 5000 4000 3000 2000 1000 0
0
1
2
ð12Þ
3 Time (s)
4
5
6 x 10-3
Fig. 3. Comparison of the contact force from the present model with the Pierson [25] analytical model and with the Delfosse [26] experimental results.
ð14Þ
ARTICLE IN PRESS G.H. Payeganeh et al. / Thin-Walled Structures 48 (2010) 62–70
where Amn(t), Bmn(t) and Wmn(t) are the time-dependent coefficients. Using the changes of variables method [22], Eq. (11) simplifies as follows:
gyz ¼ cy þ @w=@y
€ mn ðtÞ þ o2 Wmn ðtÞ ¼ Pmn ðtÞ W mn rh
gxz ¼ cx þ @w=@x
ð15Þ
¼
¼ where
o2mn ¼
L13 KA þ L23 KB þ L33 rh
ð16Þ
omn2 is the square of the fundamental frequencies of the plate. For m ¼ n ¼ 1, the value of K1 in Eq. (7) is K1 ¼ m1 o211
ð17Þ
65
1 n mp np o X np Wmn ðtÞ þ Bmn ðtÞ Sin x Cos y b a b m;n¼1
1 n mp np o X mp :Wmn ðtÞ þ Amn ðtÞ Cos x Sin y a a b m;n¼1
ð21Þ
ð22Þ
where z is the thickness coordinate of the structure. In addition, the stress–strain expressions of the traditional composite structures can be used to calculate the stress distribution of the FML too as follows: 2 3 2 3 2 ex 3 sxx Q 11 Q 12 Q 16 6s 7 6Q 7 6 7 4 yy 5 ¼ 4 12 Q 22 Q 26 5 4 ey 5; g sxy Q 16 Q 26 Q 66 xy k " # " # " # gxz sxz Q 55 Q 54 ð23Þ gyz syz ¼ Q 54 Q 44 k
The value of Wmn(t), would be easily calculated based on the Runge–Kutta method of 4th and 5th ranks and using a software like MATLAB and its ode 45 solver. Substituting the results in Eq. (14), the values of w, cx and cy could be calculated. The values of ex, ey and gxy could be easily determined, as discussed earlier [24]
ex ¼ z cx;x ¼ z
1 X
Amn
m;n¼1
ey ¼ z cy;y ¼ z
1 X
Bmn
m;n¼1
mp mp np Sin x Sin y a a b
ð18Þ
np mp np Sin x Sin y b a b
ð19Þ
Table 1 Geometrical and material properties of the FML plate and the impactor [23,27,28]. Geometrical properties of FML plate Simply supported : Boundary: boundary conditions Length ¼ Widthwidth ¼ 200 mm Lay-up: [0/90/0/90/0]s Ply thickness ¼ 0.269 mm Material properties of FRP layer (glass–polyster) E11 ¼ 24.51 GPa; E22 ¼ E33 ¼ 7.77 GPa G13 ¼ G12 ¼ 3.34 GPa; G23 ¼ 1.34 GPa n12 ¼ n13 ¼ n23 ¼ 0.078 r ¼ 1800 kg/m3 Material properties of Al layer (2024-T3) E ¼ 72.4 GPa, G ¼ 27.6 GPa, n ¼ 0.33, r ¼ 2780 kg/m3
gxy ¼ zðcx;y þ cy;x Þ
m;n¼1
Amn
np b
mp mp np þ Bmn Cos x Cos y a a b ð20Þ
Properties of impactor (Steel) E ¼ 207 GPa, n ¼ 0.3, r ¼ 7800 kg/m3 Tip diameter ¼ 12.7 mm Mass ¼ 2.0 kg Velocity ¼ 1.0 m/s
1
1 9 25 100 400
0.8
w/h
¼ z
1 X
Term Terms Terms Terms Terms
0.6
0.4
0.2
0
0
0.5
1
1.5
2 Time (s)
2.5
3
3.5
Fig. 4. Effect of number of terms in Fourier series on deflection ratio of the plate.
4 10-3
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5. Results and discussions
symmetric and cross ply. The impactor is a heavy spherical object (impactor mass/plate mass 42). Material and geometrical properties of the FML plate as well as the impactor are presented in Table 1. The structure consists of 10 layers and they are numbered from top to bottom.
The accuracy is verified by comparison of force–time relationships obtained by the present solution with those generated using an analytical method of Pierson [25] and those obtained from the experimental results of Delfosse [26]. Fig. 3 shows a good agreement in the results. The effect of number of terms of the Fourier series in the solution for the transverse deflection ratio (deflection to thickness) of the plate is illustrated in Fig. 4. Fig. 4 shows that the convergence of the present series solution is achieved by using of only nine terms, but the full convergence is demonstrated with 100 terms. The effect of using Al layers and some of the parameters such as their stacking sequence, mass and velocity of the impactor in a constant energy level and aspect ratio (a/b ratio) of the structure on the impact response of FMLs is studied. The FML plate used is
5.1. Effect of the layer sequence of the FMLs In this research, only 2 layers of the Al 2027-T3 alloy layers are placed instead of the glass–polyster ones of the FRP structures symmetrically to see their position effect on impact resistance of the structure. It means the Al layers are placed instead of these glass–polyster layers of the structure separately: (1, 10), (2, 9), (3, 8), (4, 7), and (5, 6). Fig. 5(a) shows the time history of non-dimensional transverse deflection w/h ratio (the ratio of composite plate deflection to its
1.5 1) No
0.9945 0.9539
1
Al Layers
2) (1, 10) Al Layers
0.8490
3) (2, 9) 4) (3, 8)
Al Layers Al Layers
5) (4, 7)
Al Layers
6) (5, 6)
Al Layers
w/h
0.6830 0.5243 0.5222
0.5
1
6
5 3
2
0
0
0.5
1
1.5
4
2 Time (s)
3
2.5
3.5
4 10-3
1000 900
1) No
800
2 (Max. 730.0 N)
700 Contact Force (N)
Al Layers
2) (1, 10) Al Layers 3) (2, 9)
Al Layers
4) (3, 8)
Al Layers
5) (4, 7)
Al Layers
6) (5, 6)
Al Layers
600 6 (Max. 329.5 N) 5 (Max. 329.5 N) 4 (Max. 330.0 N) 3 (Max. 330.5 N) 1 (Max. 310.0 N)
500 400 300 200 100 0
228 µs
0
1
485 µs
2
3
4 Time (s)
521 µs
5
6
7 10-4
Fig. 5. Effect of the layer sequence of the Al layers on: (a) non-dimensional deflection (w/h) history and (b) contact force history.
ARTICLE IN PRESS G.H. Payeganeh et al. / Thin-Walled Structures 48 (2010) 62–70
xx (MPa)
5
67
x 10-3
4
1) No Al Layers 2) (1, 10) Al Layers 3) (2, 9) Al Layers
3
4) (3, 8) 5) (4, 7) 6) (5, 6)
Al Layers Al Layers Al Layers
2 1
1
2
3
4
5
6
0 -1 -2 0
1 Time (s)
2 x 10-4
Fig. 6. Effect of the layer sequence of the Al layers on sxx.
thickness) of the plate. It is seen that this value changes from 0.9539 for the FRP plate (the composite without the Al layers, curve 1) to: 0.5222 (curve 2), 0.5243 (curve 3), 0.6830 (curve 4), 0.8490 (curve 5), and 0.9945 (curve 6) that corresponds to composites when the Al sheets are in these layers of the structure: (1, 10), (2, 9), (3, 8), (4, 7), and (5, 6), respectively. One can see, except the last case that the Al layers are replaced with the middle ones of the structure (here, layers (5, 6)), using of the Al sheets results in a decrease in the value of w/h ratio. This reduction in w/h ratio of the structure results in a decrease in the fibers breakage probability. Therefore, the impact resistance of the structure increases. There are two other valuable results too. First, the most reduction of the max. w/h ratio belongs to the time, which the Al sheets are placed in layers (1, 10). This is maybe why most of the researchers are studied on the FMLs in which their Al layers are placed in the outer ones [1–5]. However, the second result is more interesting and it is not reported up to now. As is evident in Fig. 5(b), by replacing the glass–polyster layers with the Al ones, the maximum contact force (hereinafter-called MCF) increases from 310 N in the case of the structure without the Al layers to: 730, 330.5, 330, 329.5, and 329 N when the Al sheets are replaced with these layers of the structure: (1, 10), (2, 9), (3, 8), (4, 7), and (5, 6), respectively. Except the curve 2, the MCF is increased about 6 percent, while the maximum contact force time (MCFT) tends to move to the right side of the diagram by the distance l in Fig. 5(b), and the contact time (CT) is increased from 485 ms up to at last 521 ms. Thus, the shocking effect of the impact force that transfers to the plate decreases and a weaker impact is inflicted upon the structure. But, for the curve 2 (the Al sheets are placed in layers (1,10)), this shocking effect is increased considerably that is not a good result. This increase in contact force is about 155 percent, which reduces the damage resistance of the structure. This is happened since the impact is a local phenomenon. If the Al sheets are placed in the outer layers of the structure, the impactor hits a stiffer layer. Therefore, the MCF increases (from 310 to 730 N) but the MCFT reduces (from 240 to 113 ms) and the CT decreases (from 485 to
228 ms). Accordingly, the impact produces a greater shock and the structure experiences a larger impact load (in comparison with the plate having the Al sheets at the other positions). So, replacing of only the outer layers (here, layers (1, 10)) of a FRP structure with Al sheets cannot always lead to the best result. Because of this, in all of the other parts of the present research, we consider the FML sheets to be placed in layers (2, 9) of the structure. The time history of the xx in-plane stresses (sxx) during the impact is shown in Fig. 6. Fig. 6 demonstrates the variation of the sxx with time, when the Al sheets are replaced with the glass–polyster ones. It is seen that except the case that the Al layers are placed in the middle layers (here, layers (5, 6)) of the structure (curve 6), Al sheets has a good effect on reducing of the in-plane stresses either. This result is exactly coincided with the results taken from the Fig. 5(a) too. 5.2. Effect of mass and velocity of the impactor in a constant impact energy level Here, the effect of different masses and velocities of the impactor in a constant impact energy level are studied. The mass and the velocity of the impactor are chosen as (1) m ¼ 1.0 kg, v ¼ 1.4 m/s (2) m ¼ 2.0 kg, v ¼ 1.0 m/s (3) m ¼ 4.0 kg, v ¼ 0.7 m/s which all of the above cases results in an impact energy level equal to 1.0 J. The Al sheets are used in layers of (2, 9) of the structure. Other properties are presented in Table 1. Fig. 7 leads to the following results in a constant impact energy level but with different mass and velocity of the impactor: (1) No matter if the Al layers are placed in the structure (curves 4–6) or not (curves 1–3), the max. of w/h ratio and the MCF are increased by increasing of the velocity of the impactor (from curve 3 to curve 2 and from curve 2 to curve 1 when the Al
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2 1.8 1.6
1) m=1.0, v=1.4, No
Al Layer
2) m=2.0, v=1.0, No 3) m=4.0, v=0.7, No
Al Layer Al Layer
4) m=1.0, v=1.4, (2,9) Al Layers 5) m=2.0, v=1.0, (2,9) Al Layers 6) m=4.0, v=0.7, (2,9) Al Layers
1.4
w/h
1.2 1 1
0.8 2
0.6 4
0.4 0.2 0
3
5
3333 µs 2061 µs
6
0
0.5
1
1.5
2 Time (s)
2.5
3
3.5 10-3
800 700
1) m=1.0, v=1.4, No 2) m=2.0, v=1.0, No
3) m=4.0, v=0.7, No Al Layer 4) m=1.0, v=1.4, (2,9) Al Layers
600 Contact Force (N)
Al Layer Al Layer
500
5) m=2.0, v=1.0, (2,9) Al Layers 6) m=4.0, v=0.7, (2,9) Al Layers
6
400
1 5
300
2 4
200
3
100 0
0
1
2
3
4 Time (s)
5
6
7
8 10-4
Fig. 7. Effect of various mass and velocity of impactor in constant energy level on: (a) w/h ratio and (b) contact force history of the plate.
layers are not placed in the structure and from curve 6 to curve 5 and from curve 5 to curve 4 when the Al sheets are placed in the laminate, respectively). This result is the same as what the Sun and Chen [29] are reported in variable impact energy levels for the FRP plates. It is also seen that in a constant impact energy level and in comparison with the impactor mass, the impactor velocity is a more effective parameter on maximum w/h ratio and the MCF of the structure (Fig. 7(a) and (b)). (2) When the Al layers are placed instead of glass–polyster ones, the max. of w/h ratio is decreased (from curves 1 to 4, 2 to 5, and 3 to 6, respectively, in Fig. 7(a)). In addition, the maximum contact force is increased too (from curves 1 to 4, 2 to 5, and 3 to 6, respectively, in Fig. 7(b)). The same as what the Sun and Chen [29] reported in variable impact energy
levels for the FRP plates, one could see that in a constant impact energy level and in comparison with the impactor mass, the impactor velocity is a more effective parameter on max. w/h ratio and the MCF of the FMLs too (Fig. 7(a) and (b)). (3) No matter whether the Al layers placed instead of glass– polyster ones in the structure or they are not, in a constant energy level, changing the mass and velocity of the impactor nearly has no effect on CT. However, it must be noted that when the metal sheets are placed in the structure, this period reduces nearly to half (from about 3333 ms for non-Al layered structures to about 2061 ms for the Al layered ones). This means that in FMLs the oscillation motion of the structure could be damped quicker. (4) The results obtained from Fig. 7 are classified in Table 2. As it is visible, the same reduction in max. w/h ratio (and the same
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Table 2 Effect of mass and velocity of the impactor in a constant impact energy level. Constant energy Level ¼ 1.0 J, FML sheets are placed in (2, 9) layers of structure Studied parameters Different m and v
Max. w/h reduction percent
MCF increase percent
(1) m ¼ 1.0 kg, v ¼ 1.4 m/s (2) m ¼ 2.0 kg, v ¼ 1.0 m/s (3) m ¼ 4.0 kg, v ¼ 0.7 m/s
46 45 46
7 6 6
Table 3 Effect of a/b ratio of the plate. a/b Ratio Composite plate
1
1.25
1.5
Max. w/h of FRP plate Max. w/h of FML plate Percent of reduction
0.9535 0.5243 45
0.6939 0.3798 45
0.5404 0.3736 30
increase in MCF) is happened in all cases. This means that in a constant impact energy level, the amount of increase in the impact resistance of the structure obtained by using the Al sheets is nearly independent of the value of mass or velocity of the impactor. This result and the one obtained by Caprino et al. [2] could somehow verify each other. Because, Caprino et al. [2] showed that the overall impact force–displacement curve of FMLs under low-velocity impact depends on the impact energy, rather than mass and speed of the impactor separately.
5.3. Effect of the aspect ratio (a/b ratio) of the plate The effect of aspect ratio (a/b ratio) is also studied. Again, the Al layers are placed instead of the (2, 9) ones of the structure and the other properties are the same as in Table 1. The results of this section are presented in Table 3. Table 3 leads to First, no matter if the Al layers are placed in the structure (FML) or not (FRP), increasing of the aspect ratio of the structure leads to decrease in the max. w/h ratio of the structure. The reason is that with increasing of the a/b ratio, the supports of the plate become close to each other. Therefore, the plate tends to become a strip, which is hinged simply supported. This results in the max. w/h ratio of the structure decreases and the impact resistance of the structure increases. Second, it must be noted that as much as the a/b ratio of the plate increases, the positive effect of the Al layers decreases (from 45 percent decrease for the a/b ¼ 1 to 30 percent decrease for the a/b ¼ 1.5). This result is exactly in coinciding with the first result of this section too.
6. Conclusions In the present research, the dynamic response of low-velocity impact upon FMLs is studied using the first-order shear deformation theory and Fourier series method to solve the system of governing differential equations of the plate analytically. To model the interaction between the impactor and the plate, a system having two degrees-of-freedom consisting of springs-masses is used. The results of the above research demonstrated that the use of the Al sheets inside the FRP plates improve their global
69
behavior against the impact. The plate with the Al layers damps the impact phenomenon more uniformly and rapidly. The results indicated that some of the parameters like the layer sequence, mass and velocity in a constant impact energy, and aspect ratio of the plate are important factors affecting the dynamic response of the FMLs. These results have not been presented in the published literature yet. The Al layers affect the MCF, the MCFT and the CT. Accordingly, the shocking effect of the impact will be changed. The changes occurred in the deflection and in-plane stresses too. The location of the Al layers is an important factor to change the overall and the local behavior of the structure. Placing of the Al layers near of the impact zone layer can improve the impact resistance of the structure the most. If the Al sheets are placed further from the impact zone layer, this positive effect tends to zero or even negative. It is seen that in a constant impact energy level, the amount of increase in the impact resistance of the structure obtained by using the Al layers is nearly independent of the value of mass or velocity of the impactor. Although, placing of the Al sheets in the traditional composite structure improves the impact resistance of the structure, this positive effect reduces considerably by increasing of the a/b ratio of the plate.
References [1] Vlot A. Impact loading on fiber metal laminates. International Journal of Impact Engineering 1996;18(3):291–307. [2] Caprino G, Spataro G, Del Luongo S. Low-velocity impact behavior of fiberglass-aluminium laminates. Composites: Part A 2004;35:605–16. [3] Caprino G, Lopresto V, Iaccarino P. A simple mechanical model to predict the macroscopic response of fiberglass-aluminium laminates under low-velocity impact. Composites Part A 2007;38:290–300. [4] Atas C. An experimental investigation on the impact response of fiberglass/ aluminum composites. Journal of Reinforced Plastics and Composites 2007;26(14):1479–91. [5] Abdullah MR, Cantwell WJ. The impact resistance of polypropylene-based fiber metal laminates. Composites Science and Technology 2006;66:1682–93. [6] Asundi A, Choi AYN. Fiber metal laminates: an advanced material for future aircraft. Journal of Materials Processing Technology 1997;63(1-3):384–94. [7] Vlot A, Vogelesang LB, De Vries. Towards application of fiber metal laminates in large aircraft. Aircraft Engineering and Aerospace Technology 1999;71(6):558–70. [8] Marsh G. Airframes exploit composites in battle for supremacy. Reinforced Plastics 2005;49(3):26–32. [9] Abrate S. Impact on laminated composite materials. Applied Mechanics Review 1991;44:155–90. [10] Abrate S. Impact on laminated composites. Applied Mechanics Review 1994;47:517–44. [11] Abrate S. Modeling of impacts on composite structures. Composite Structures 2001;51:129–38. [12] Olsson R. Mass criterion for wave controlled impact response of composite plates. Composites: Part A 2000;31:879–87. [13] Olsson R. Analytical prediction of large mass impact damage in composite laminates. Composites: Part A 2001;32:1207–15. [14] Olsson R. Closed form prediction of peak load and delamination onset under small mass impact. Composite Structures 2003;59:341–9. [15] Whitney JM, Pagano NJ. Shear deformation in heterogeneous anisotropic plates. Journal of Applied Mechanics 1970;37:1031–6. [16] Mindlin RD. Influence of rotary inertia and shear on flexural motions of isotropic elastic plates. Journal of Applied Mechanics 1951;18:31–8. [17] Reddy JN. Mechanics of laminated composite plates and shells, 2nd ed. United States of America: CRC Press; 1997. [18] Shokuhfar A, Khalili SMR, Ashenai Ghasemi F, Malekzadeh K, Raissi S. Analysis and optimization of smart hybrid composite plates subjected to low-velocity impact using the response surface methodology (RSM). Thin-Walled Structures 2008;46:1204–12. [19] Choi IH, Lim CH. Low-velocity impact analysis of composite laminates using linearized contact law. Composite Structures 2004;66:125–32. [20] Carvalho A, Soares CG. Dynamic response of rectangular plates of composite materials subjected to impact loads. Composite Structures 1996;34:55–63. [21] Khalili SMR, Shokuhfar A, Malekzadeh K, Ashenai Ghasemi F. Low-velocity impact response of active thin-walled hybrid composite structures embedded with SMA wires. Thin-Walled Structures 2007;45:799–808. [22] Christoforou AP, Swanson SR. Analysis of Impact response in composite plates. International Journal of Solids Structures 1991;27(2):161–70. [23] Khalili SMR, Shokuhfar A, Ashenai Ghasemi F. Effect of smart stiffening procedure on low-velocity impact response of smart structures. Journal of Materials Processing Technology 2007;190(1–3):142–52.
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[24] Khalili SMR, Shokuhfar A, Ashenai Ghasemi F, Malekzadeh K. Dynamic response of smart hybrid plate subjected to low-velocity. Journal of Composite Materials 2007;41(19):2347–70. [25] Pierson MO, Vaziri R. Analytical solution for low-velocity impact response of composite plates. AIAA Journal 1996;34(8):1633–40. [26] Delfosse D, Vaziri R, Pierson MO, Poursartip A. Analysis of the non-penetrating impact behavior of CFRP laminates. In: Proceeding of the 9th International Conference on Composite Materials (Madrid, Spain), Vol. 5. Cambridge, England, UK: Woodhead Publishing; 1993. p. 366–73.
[27] Malekzadeh K, Khalili MR, Jafari A, Mittal RK. Dynamic response of in-plane pre-stressed sandwich panels with a viscoelastic flexible core and different boundary conditions. Journal of Composite Materials 2006;40: 1449–69. [28] Krimbalis PP, Poon C, Fawaz Z, Behdinan K. Prediction of bearing strength in fiber metal laminates. Journal of Composite Materials 2007;41: 1137–57. [29] Sun CT, Chen JK. On the impact of initially stressed composite laminates. Journal of Composite Materials 1985;19:490–503.
Contents lists available at ScienceDirect
Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Dynamic response of fiber–metal laminates (FMLs) subjected to low-velocity impact G.H. Payeganeh a, F. Ashenai Ghasemi a,, K. Malekzadeh b a b
Department of Mechanical Engineering, Shahid Rajaee Teacher Training University (SRTTU), Lavizan, Postal Code 16788-15811, Tehran, Iran Department of Mechanical Engineering, Malek Ashtar University of Technology, Tehran, Iran
a r t i c l e in f o
a b s t r a c t
Article history: Received 4 September 2008 Accepted 16 July 2009 Available online 18 August 2009
Fiber–metal laminates (FMLs) are high-performance hybrid structures based on alternating stacked arrangements of fiber-reinforced plastic (FRP) plies and metal alloy layers. The response of FMLs subjected to low-velocity impact is studied in this paper. The aluminum (Al) sheets are placed instead of some of layers of FRP plies. The effect of the Al layers on contact force history, deflection, in-plane strains and stresses of the structure is studied. The first-order shear deformation theory as well as the Fourier series method is used to solve the governing equations of the composite plate analytically. The interaction between the impactor and the plate is modeled with the use of a two degrees-of-freedom system, consisting of springs-masses. The Choi’s linearized Hertzian contact model is used in the impact analysis of the hybrid composite plate. The results indicated that some of the parameters like the layer sequence, mass and velocity of the impactor in a constant impact energy level, and the aspect ratio (a/b) of the plate are important factors affecting the dynamic response of the FMLs. Interaction among the mentioned geometrical parameters and material parameters like the aluminum 2024-T3 alloy layers is studied. The numerical results that are presented in this paper hitherto not reported in the published literature. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Impact Composite Fiber–metal laminate (FML) Fiber-reinforced plastic (FRP) Springs-masses model
1. Introduction Fiber–metal laminates (FMLs) are hybrid structures consisting of different metal sheets and fiber-reinforced plastic (FRP) layers. One of the most important objects of their production is to combine the good impact resistance of the metals with the better lightweight characteristic of FRP laminates [1–5]. Therefore, there will be an excellent candidate material to be used for aerospace structures [6–8]. Abrate [9–11] studied the impact behavior of composite structures extensively. Olsson [12–14] classified low-velocity impacts in two categories, the small mass and the large mass impact, which is based on the ratio of the impactor mass to the target mass. When the mass of the impactor is small in comparison with the weight of the target, there will be a small mass response dominated by shear and flexural waves in which the deflection, load, and flexural stresses are out of phase. When the mass of the impactor is much larger than the target mass, there will happen a ‘quasi-static’ large mass response, in which the deflection, peak load, and stresses are more or less in phase. Vlot [1] showed that the damaged zone of FMLs after the impact is
Corresponding author. Tel.: +98 21 22970052; fax: +1 678 8685363.
E-mail address: [email protected] (F. Ashenai Ghasemi). 0263-8231/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2009.07.005
smaller than the FRPs. Caprino et al. [2] demonstrated that the overall impact force–displacement curve of FMLs under lowvelocity impact only depends on the impact energy, rather than mass and speed of the impactor separately. Caprino et al. [3] presented a mechanistic model with neglecting the macroscopic behavior of the structure (i.e. neglecting local deformation due to indentation, overall deflection, damage initiation and development) to study the low-velocity impact. Atas [4] has done an experimental investigation to carry out the damage process of FMLs under low-velocity impact. Abdullah et al. [5] showed the positive effect of the FMLs in comparison with the FRP laminates in high-velocity impact too. They investigated that the stacking sequence is an important parameter on the perforation resistance of these structures. In this research, a complete model is developed so that the effect of low-velocity impact upon the FMLs demonstrated. With this model, the researchers will be enabled to investigate the complete response of FMLs subjected to low-velocity impact, considering of the shear deformation effect. They can also determine the contact force history independently and regardless of solving the motion equations of the plate. This also saves lot of the computational time efforts. The effect of using Al layers as well as some of the parameters such as the stacking sequence, the mass and the velocity of the impactor in a constant energy level and the aspect ratio of the structure on the impact response of
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FMLs is studied in details. The present study of these parameters is useful to the designers too, which have not been presented in similar literatures.
63
boundary conditions are as follows: w ¼ cx;x ¼ cy ¼ 0; at x ¼ 0; a w ¼ cy;y ¼ cx ¼ 0; at y ¼ 0; b
ð4Þ
2. Governing equations Fig. 1 shows a FML, which is in contact with an impactor mass. Here, the plate equations developed by Whitney and Pagano [15] are used. They included the effect of transverse shear deformations, which the displacement field is u ¼ u0 ðx; y; tÞ þ zcx ðx; y; tÞ v ¼ v0 ðx; y; tÞ þ zcy ðx; y; tÞ 0
w ¼ w ðx; y; tÞ
ð1Þ
u0, v0 and w0 are the plate displacements in x, y and z directions at the plate mid-plane and cx and cy are the shear rotations in the x and y directions. For the specially orthotropic form (Bij ¼ 0, A16 ¼ A26 ¼ D16 ¼ D26 ¼ 0) results in
3. Constitutive equations Constitutive equations of stress–strain relationship for a FML are as follows [17]: 8 9 8 98 9 e = > > < s1 > = < Q11 Q12 0 > => < 1 > e2 s2 ¼ Q12 Q22 0 > > >> > :t > ; :0 0 Q ;: g ; 12
(
t13 t23
66
)
( ¼
Q55
0
0
Q44
)(
g13 g23
12
) ð5Þ
€ D11 cx;xx þ D66 cx;yy þ ðD12 þ D66 Þcy;xy ksh A55 cx ksh A55 w;x ¼ Ic x € ðD12 þ D66 Þcx;xy þ D66 cy;xx þ D22 cy;yy ksh A44 cy ksh A44 w;y ¼ Ic y € ksh A55 cx;x þ ksh A55 w;xx þ ksh A44 cy;y þ ksh A44 w;yy þ q ¼ rw
ð2Þ
ksh is the shear correction factor introduced by Mindlin [16], usually equals to p2/12 and q the dynamic normal load over the plate and also Z h=2 Z h=2 Qijk ð1; z; z2 Þdz; ðr; IÞ ¼ r0 ð1; z2 Þdz ð3Þ ðAij ; Bij ; Dij Þ ¼ h=2
h=2
In the above equation, {s} represents the stresses in the principle directions. In addition, the matrix {e} represents the strains in the principle directions. Qij represents the reduced stiffness matrices for the FML structure. Because of discontinuity function of stresses through the thickness in each layer, it is possible to determine the constitutive equation by considering the force-couple resultants in terms of stresses, using integration of Eq. (5) through the plate thickness, which yields #( ) "A Bij N ij e0 ; i; j ¼ 1; 2; 6 ¼ Bij Dij M k
In the above relation, r0 represents the density of each layer and r the total density of the plate. In addition, I is the moment of inertia and h the thickness of the plate. (Qij)k (i, j ¼ 1,2,6) are the reduced in-plane stiffness components and (Qij)k (i, j ¼ 4,5) are the reduced transverse shear stiffness components[15]. In the present research, a simply supported rectangular plate is chosen to study with the dimensions of a and b, which its
where N and S are vectors of forces and M the vector of moments, respectively. Aij, Bij, and Dij are the components of extensional and shear, coupling, and bending stiffness matrices, respectively. Also e0 and g are the mid-plane and shear strains, respectively and k is the curvatures. In addition, ksh is the shear correction factor.
Fig. 1. Schematic view of the FML impacted by a spherical mass on its center.
Fig. 2. A two degrees-of-freedom springs-masses model [3,18].
fSg ¼ ½ksh Aij fgg; i; j ¼ 4; 5
ð6Þ
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after rearrangements [18] m1 z€ 1 ¼ K1 z1 K2 ðz1 z2 Þ
4. Solution of dynamic response of the plate 4.1. Analysis of contact force
m2 z€ 2 ¼ K2 ðz2 z1 Þ
Here, two degrees-of-freedom springs-masses model [3,18] is used to determine the impact force (Fig. 2). The motion equation is as follows:
ð10Þ
The above system of ordinary equations can be solved by the Runge–Kutta method, using of the MATLAB ode solver.
m2 z€ 2 þ F ¼ 0 4.2. Analysis of deflection and stress–strain m1 z€ 1 þ K1 z2 þ K2 z32 F ¼ 0
ð7Þ
The index 1 belongs to the hybrid plate and the index 2 indicates the impactor. F is the contact force, m1 ¼ mp and m2 ¼ mi represents, respectively the mass of the impactor and the FML plate, z1 and z2 are, respectively, the relative displacements of the impactor and the structure masses. K1 ¼ Kbs is the bendingshear stiffness the plate. Using of the Choi’s linearized model [19], instead of non-linear Hertzian contact law, the contact force can be obtained as FðtÞ ¼ Kl a ¼ Kl ðz2 z1 Þ 1=3
2=3
Kl ¼ F m Kc
ð8Þ
In the above equations, K2 ¼ Kl represents the linearized contact coefficient in Choi’s linearized contact law, Fm is the maximum predicted contact force, and Kc represents the contact stiffness in the improved Hertzian contact law, which can be calculated based on the following Eq. [19]: Kc ¼
1=2 R2
4 3 ð1 u22 =EÞ þ ð1=E22 Þ
ð9Þ
Neglecting the effects of rotary inertia [16], the current problem could be converted to a system of ordinary differential equations of second order in time for the Fourier coefficients of the transverse deflection [20]. The equation of the motion of a FML plate subjected to a point load q(x, y, t) is equal to [20,21] 39 2 38 2 39 82 0 L11 L12 L13 > > = = > < < Amn ðtÞ > 6 7 6L 7 6 7 ð11Þ 5 4 12 L22 L23 5 4 Bmn ðtÞ 5 ¼ 4 0 > > > ; € mn ðtÞ > L13 L23 L33 : Wmn ðtÞ ; : Pmn ðtÞ rhW where the Lij coefficients are introduced in [21] and mp np XX Pmn ðtÞSin qðx; y; tÞ ¼ xSin y a b m n
Also, for a concentrated load located at the point (xc, yc), Pmn(t) are the terms of the Fourier series as mp np 4FðtÞ ð13Þ Pmn ðtÞ ¼ Sin xc Sin yc a b ab where F(t) is the impact load (see Eq. (8)). The impact solution for a rectangular plate with simply supported boundary conditions is assumed to be in the following form [22,23]:
cx ðx; y; tÞ ¼ where R2 is radius of the curvature, n2 the Poisson’s ratio, E and E22 the elastic modulus of the isotropic impactor and the transverse elastic modulus of the top layer of the FML structure, respectively. Considering the Fig. 2 and replacing the value of F in Eq. (7) with the values presented by Choi’s differential equations
1 X
Amn ðtÞCos
mp a
m;n¼1 1 X
xSin
np y b
mp np xCos y cy ðx; y; tÞ ¼ Bmn ðtÞSin a b m;n¼1 wðx; y; tÞ ¼
mp np Wmn ðtÞSin xSin y a b m;n¼1 1 X
10000 9000
Present Delfosse [26]
8000
Pierson [25]
Contact Force (N)
7000 6000 5000 4000 3000 2000 1000 0
0
1
2
ð12Þ
3 Time (s)
4
5
6 x 10-3
Fig. 3. Comparison of the contact force from the present model with the Pierson [25] analytical model and with the Delfosse [26] experimental results.
ð14Þ
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where Amn(t), Bmn(t) and Wmn(t) are the time-dependent coefficients. Using the changes of variables method [22], Eq. (11) simplifies as follows:
gyz ¼ cy þ @w=@y
€ mn ðtÞ þ o2 Wmn ðtÞ ¼ Pmn ðtÞ W mn rh
gxz ¼ cx þ @w=@x
ð15Þ
¼
¼ where
o2mn ¼
L13 KA þ L23 KB þ L33 rh
ð16Þ
omn2 is the square of the fundamental frequencies of the plate. For m ¼ n ¼ 1, the value of K1 in Eq. (7) is K1 ¼ m1 o211
ð17Þ
65
1 n mp np o X np Wmn ðtÞ þ Bmn ðtÞ Sin x Cos y b a b m;n¼1
1 n mp np o X mp :Wmn ðtÞ þ Amn ðtÞ Cos x Sin y a a b m;n¼1
ð21Þ
ð22Þ
where z is the thickness coordinate of the structure. In addition, the stress–strain expressions of the traditional composite structures can be used to calculate the stress distribution of the FML too as follows: 2 3 2 3 2 ex 3 sxx Q 11 Q 12 Q 16 6s 7 6Q 7 6 7 4 yy 5 ¼ 4 12 Q 22 Q 26 5 4 ey 5; g sxy Q 16 Q 26 Q 66 xy k " # " # " # gxz sxz Q 55 Q 54 ð23Þ gyz syz ¼ Q 54 Q 44 k
The value of Wmn(t), would be easily calculated based on the Runge–Kutta method of 4th and 5th ranks and using a software like MATLAB and its ode 45 solver. Substituting the results in Eq. (14), the values of w, cx and cy could be calculated. The values of ex, ey and gxy could be easily determined, as discussed earlier [24]
ex ¼ z cx;x ¼ z
1 X
Amn
m;n¼1
ey ¼ z cy;y ¼ z
1 X
Bmn
m;n¼1
mp mp np Sin x Sin y a a b
ð18Þ
np mp np Sin x Sin y b a b
ð19Þ
Table 1 Geometrical and material properties of the FML plate and the impactor [23,27,28]. Geometrical properties of FML plate Simply supported : Boundary: boundary conditions Length ¼ Widthwidth ¼ 200 mm Lay-up: [0/90/0/90/0]s Ply thickness ¼ 0.269 mm Material properties of FRP layer (glass–polyster) E11 ¼ 24.51 GPa; E22 ¼ E33 ¼ 7.77 GPa G13 ¼ G12 ¼ 3.34 GPa; G23 ¼ 1.34 GPa n12 ¼ n13 ¼ n23 ¼ 0.078 r ¼ 1800 kg/m3 Material properties of Al layer (2024-T3) E ¼ 72.4 GPa, G ¼ 27.6 GPa, n ¼ 0.33, r ¼ 2780 kg/m3
gxy ¼ zðcx;y þ cy;x Þ
m;n¼1
Amn
np b
mp mp np þ Bmn Cos x Cos y a a b ð20Þ
Properties of impactor (Steel) E ¼ 207 GPa, n ¼ 0.3, r ¼ 7800 kg/m3 Tip diameter ¼ 12.7 mm Mass ¼ 2.0 kg Velocity ¼ 1.0 m/s
1
1 9 25 100 400
0.8
w/h
¼ z
1 X
Term Terms Terms Terms Terms
0.6
0.4
0.2
0
0
0.5
1
1.5
2 Time (s)
2.5
3
3.5
Fig. 4. Effect of number of terms in Fourier series on deflection ratio of the plate.
4 10-3
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5. Results and discussions
symmetric and cross ply. The impactor is a heavy spherical object (impactor mass/plate mass 42). Material and geometrical properties of the FML plate as well as the impactor are presented in Table 1. The structure consists of 10 layers and they are numbered from top to bottom.
The accuracy is verified by comparison of force–time relationships obtained by the present solution with those generated using an analytical method of Pierson [25] and those obtained from the experimental results of Delfosse [26]. Fig. 3 shows a good agreement in the results. The effect of number of terms of the Fourier series in the solution for the transverse deflection ratio (deflection to thickness) of the plate is illustrated in Fig. 4. Fig. 4 shows that the convergence of the present series solution is achieved by using of only nine terms, but the full convergence is demonstrated with 100 terms. The effect of using Al layers and some of the parameters such as their stacking sequence, mass and velocity of the impactor in a constant energy level and aspect ratio (a/b ratio) of the structure on the impact response of FMLs is studied. The FML plate used is
5.1. Effect of the layer sequence of the FMLs In this research, only 2 layers of the Al 2027-T3 alloy layers are placed instead of the glass–polyster ones of the FRP structures symmetrically to see their position effect on impact resistance of the structure. It means the Al layers are placed instead of these glass–polyster layers of the structure separately: (1, 10), (2, 9), (3, 8), (4, 7), and (5, 6). Fig. 5(a) shows the time history of non-dimensional transverse deflection w/h ratio (the ratio of composite plate deflection to its
1.5 1) No
0.9945 0.9539
1
Al Layers
2) (1, 10) Al Layers
0.8490
3) (2, 9) 4) (3, 8)
Al Layers Al Layers
5) (4, 7)
Al Layers
6) (5, 6)
Al Layers
w/h
0.6830 0.5243 0.5222
0.5
1
6
5 3
2
0
0
0.5
1
1.5
4
2 Time (s)
3
2.5
3.5
4 10-3
1000 900
1) No
800
2 (Max. 730.0 N)
700 Contact Force (N)
Al Layers
2) (1, 10) Al Layers 3) (2, 9)
Al Layers
4) (3, 8)
Al Layers
5) (4, 7)
Al Layers
6) (5, 6)
Al Layers
600 6 (Max. 329.5 N) 5 (Max. 329.5 N) 4 (Max. 330.0 N) 3 (Max. 330.5 N) 1 (Max. 310.0 N)
500 400 300 200 100 0
228 µs
0
1
485 µs
2
3
4 Time (s)
521 µs
5
6
7 10-4
Fig. 5. Effect of the layer sequence of the Al layers on: (a) non-dimensional deflection (w/h) history and (b) contact force history.
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xx (MPa)
5
67
x 10-3
4
1) No Al Layers 2) (1, 10) Al Layers 3) (2, 9) Al Layers
3
4) (3, 8) 5) (4, 7) 6) (5, 6)
Al Layers Al Layers Al Layers
2 1
1
2
3
4
5
6
0 -1 -2 0
1 Time (s)
2 x 10-4
Fig. 6. Effect of the layer sequence of the Al layers on sxx.
thickness) of the plate. It is seen that this value changes from 0.9539 for the FRP plate (the composite without the Al layers, curve 1) to: 0.5222 (curve 2), 0.5243 (curve 3), 0.6830 (curve 4), 0.8490 (curve 5), and 0.9945 (curve 6) that corresponds to composites when the Al sheets are in these layers of the structure: (1, 10), (2, 9), (3, 8), (4, 7), and (5, 6), respectively. One can see, except the last case that the Al layers are replaced with the middle ones of the structure (here, layers (5, 6)), using of the Al sheets results in a decrease in the value of w/h ratio. This reduction in w/h ratio of the structure results in a decrease in the fibers breakage probability. Therefore, the impact resistance of the structure increases. There are two other valuable results too. First, the most reduction of the max. w/h ratio belongs to the time, which the Al sheets are placed in layers (1, 10). This is maybe why most of the researchers are studied on the FMLs in which their Al layers are placed in the outer ones [1–5]. However, the second result is more interesting and it is not reported up to now. As is evident in Fig. 5(b), by replacing the glass–polyster layers with the Al ones, the maximum contact force (hereinafter-called MCF) increases from 310 N in the case of the structure without the Al layers to: 730, 330.5, 330, 329.5, and 329 N when the Al sheets are replaced with these layers of the structure: (1, 10), (2, 9), (3, 8), (4, 7), and (5, 6), respectively. Except the curve 2, the MCF is increased about 6 percent, while the maximum contact force time (MCFT) tends to move to the right side of the diagram by the distance l in Fig. 5(b), and the contact time (CT) is increased from 485 ms up to at last 521 ms. Thus, the shocking effect of the impact force that transfers to the plate decreases and a weaker impact is inflicted upon the structure. But, for the curve 2 (the Al sheets are placed in layers (1,10)), this shocking effect is increased considerably that is not a good result. This increase in contact force is about 155 percent, which reduces the damage resistance of the structure. This is happened since the impact is a local phenomenon. If the Al sheets are placed in the outer layers of the structure, the impactor hits a stiffer layer. Therefore, the MCF increases (from 310 to 730 N) but the MCFT reduces (from 240 to 113 ms) and the CT decreases (from 485 to
228 ms). Accordingly, the impact produces a greater shock and the structure experiences a larger impact load (in comparison with the plate having the Al sheets at the other positions). So, replacing of only the outer layers (here, layers (1, 10)) of a FRP structure with Al sheets cannot always lead to the best result. Because of this, in all of the other parts of the present research, we consider the FML sheets to be placed in layers (2, 9) of the structure. The time history of the xx in-plane stresses (sxx) during the impact is shown in Fig. 6. Fig. 6 demonstrates the variation of the sxx with time, when the Al sheets are replaced with the glass–polyster ones. It is seen that except the case that the Al layers are placed in the middle layers (here, layers (5, 6)) of the structure (curve 6), Al sheets has a good effect on reducing of the in-plane stresses either. This result is exactly coincided with the results taken from the Fig. 5(a) too. 5.2. Effect of mass and velocity of the impactor in a constant impact energy level Here, the effect of different masses and velocities of the impactor in a constant impact energy level are studied. The mass and the velocity of the impactor are chosen as (1) m ¼ 1.0 kg, v ¼ 1.4 m/s (2) m ¼ 2.0 kg, v ¼ 1.0 m/s (3) m ¼ 4.0 kg, v ¼ 0.7 m/s which all of the above cases results in an impact energy level equal to 1.0 J. The Al sheets are used in layers of (2, 9) of the structure. Other properties are presented in Table 1. Fig. 7 leads to the following results in a constant impact energy level but with different mass and velocity of the impactor: (1) No matter if the Al layers are placed in the structure (curves 4–6) or not (curves 1–3), the max. of w/h ratio and the MCF are increased by increasing of the velocity of the impactor (from curve 3 to curve 2 and from curve 2 to curve 1 when the Al
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2 1.8 1.6
1) m=1.0, v=1.4, No
Al Layer
2) m=2.0, v=1.0, No 3) m=4.0, v=0.7, No
Al Layer Al Layer
4) m=1.0, v=1.4, (2,9) Al Layers 5) m=2.0, v=1.0, (2,9) Al Layers 6) m=4.0, v=0.7, (2,9) Al Layers
1.4
w/h
1.2 1 1
0.8 2
0.6 4
0.4 0.2 0
3
5
3333 µs 2061 µs
6
0
0.5
1
1.5
2 Time (s)
2.5
3
3.5 10-3
800 700
1) m=1.0, v=1.4, No 2) m=2.0, v=1.0, No
3) m=4.0, v=0.7, No Al Layer 4) m=1.0, v=1.4, (2,9) Al Layers
600 Contact Force (N)
Al Layer Al Layer
500
5) m=2.0, v=1.0, (2,9) Al Layers 6) m=4.0, v=0.7, (2,9) Al Layers
6
400
1 5
300
2 4
200
3
100 0
0
1
2
3
4 Time (s)
5
6
7
8 10-4
Fig. 7. Effect of various mass and velocity of impactor in constant energy level on: (a) w/h ratio and (b) contact force history of the plate.
layers are not placed in the structure and from curve 6 to curve 5 and from curve 5 to curve 4 when the Al sheets are placed in the laminate, respectively). This result is the same as what the Sun and Chen [29] are reported in variable impact energy levels for the FRP plates. It is also seen that in a constant impact energy level and in comparison with the impactor mass, the impactor velocity is a more effective parameter on maximum w/h ratio and the MCF of the structure (Fig. 7(a) and (b)). (2) When the Al layers are placed instead of glass–polyster ones, the max. of w/h ratio is decreased (from curves 1 to 4, 2 to 5, and 3 to 6, respectively, in Fig. 7(a)). In addition, the maximum contact force is increased too (from curves 1 to 4, 2 to 5, and 3 to 6, respectively, in Fig. 7(b)). The same as what the Sun and Chen [29] reported in variable impact energy
levels for the FRP plates, one could see that in a constant impact energy level and in comparison with the impactor mass, the impactor velocity is a more effective parameter on max. w/h ratio and the MCF of the FMLs too (Fig. 7(a) and (b)). (3) No matter whether the Al layers placed instead of glass– polyster ones in the structure or they are not, in a constant energy level, changing the mass and velocity of the impactor nearly has no effect on CT. However, it must be noted that when the metal sheets are placed in the structure, this period reduces nearly to half (from about 3333 ms for non-Al layered structures to about 2061 ms for the Al layered ones). This means that in FMLs the oscillation motion of the structure could be damped quicker. (4) The results obtained from Fig. 7 are classified in Table 2. As it is visible, the same reduction in max. w/h ratio (and the same
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Table 2 Effect of mass and velocity of the impactor in a constant impact energy level. Constant energy Level ¼ 1.0 J, FML sheets are placed in (2, 9) layers of structure Studied parameters Different m and v
Max. w/h reduction percent
MCF increase percent
(1) m ¼ 1.0 kg, v ¼ 1.4 m/s (2) m ¼ 2.0 kg, v ¼ 1.0 m/s (3) m ¼ 4.0 kg, v ¼ 0.7 m/s
46 45 46
7 6 6
Table 3 Effect of a/b ratio of the plate. a/b Ratio Composite plate
1
1.25
1.5
Max. w/h of FRP plate Max. w/h of FML plate Percent of reduction
0.9535 0.5243 45
0.6939 0.3798 45
0.5404 0.3736 30
increase in MCF) is happened in all cases. This means that in a constant impact energy level, the amount of increase in the impact resistance of the structure obtained by using the Al sheets is nearly independent of the value of mass or velocity of the impactor. This result and the one obtained by Caprino et al. [2] could somehow verify each other. Because, Caprino et al. [2] showed that the overall impact force–displacement curve of FMLs under low-velocity impact depends on the impact energy, rather than mass and speed of the impactor separately.
5.3. Effect of the aspect ratio (a/b ratio) of the plate The effect of aspect ratio (a/b ratio) is also studied. Again, the Al layers are placed instead of the (2, 9) ones of the structure and the other properties are the same as in Table 1. The results of this section are presented in Table 3. Table 3 leads to First, no matter if the Al layers are placed in the structure (FML) or not (FRP), increasing of the aspect ratio of the structure leads to decrease in the max. w/h ratio of the structure. The reason is that with increasing of the a/b ratio, the supports of the plate become close to each other. Therefore, the plate tends to become a strip, which is hinged simply supported. This results in the max. w/h ratio of the structure decreases and the impact resistance of the structure increases. Second, it must be noted that as much as the a/b ratio of the plate increases, the positive effect of the Al layers decreases (from 45 percent decrease for the a/b ¼ 1 to 30 percent decrease for the a/b ¼ 1.5). This result is exactly in coinciding with the first result of this section too.
6. Conclusions In the present research, the dynamic response of low-velocity impact upon FMLs is studied using the first-order shear deformation theory and Fourier series method to solve the system of governing differential equations of the plate analytically. To model the interaction between the impactor and the plate, a system having two degrees-of-freedom consisting of springs-masses is used. The results of the above research demonstrated that the use of the Al sheets inside the FRP plates improve their global
69
behavior against the impact. The plate with the Al layers damps the impact phenomenon more uniformly and rapidly. The results indicated that some of the parameters like the layer sequence, mass and velocity in a constant impact energy, and aspect ratio of the plate are important factors affecting the dynamic response of the FMLs. These results have not been presented in the published literature yet. The Al layers affect the MCF, the MCFT and the CT. Accordingly, the shocking effect of the impact will be changed. The changes occurred in the deflection and in-plane stresses too. The location of the Al layers is an important factor to change the overall and the local behavior of the structure. Placing of the Al layers near of the impact zone layer can improve the impact resistance of the structure the most. If the Al sheets are placed further from the impact zone layer, this positive effect tends to zero or even negative. It is seen that in a constant impact energy level, the amount of increase in the impact resistance of the structure obtained by using the Al layers is nearly independent of the value of mass or velocity of the impactor. Although, placing of the Al sheets in the traditional composite structure improves the impact resistance of the structure, this positive effect reduces considerably by increasing of the a/b ratio of the plate.
References [1] Vlot A. Impact loading on fiber metal laminates. International Journal of Impact Engineering 1996;18(3):291–307. [2] Caprino G, Spataro G, Del Luongo S. Low-velocity impact behavior of fiberglass-aluminium laminates. Composites: Part A 2004;35:605–16. [3] Caprino G, Lopresto V, Iaccarino P. A simple mechanical model to predict the macroscopic response of fiberglass-aluminium laminates under low-velocity impact. Composites Part A 2007;38:290–300. [4] Atas C. An experimental investigation on the impact response of fiberglass/ aluminum composites. Journal of Reinforced Plastics and Composites 2007;26(14):1479–91. [5] Abdullah MR, Cantwell WJ. The impact resistance of polypropylene-based fiber metal laminates. Composites Science and Technology 2006;66:1682–93. [6] Asundi A, Choi AYN. Fiber metal laminates: an advanced material for future aircraft. Journal of Materials Processing Technology 1997;63(1-3):384–94. [7] Vlot A, Vogelesang LB, De Vries. Towards application of fiber metal laminates in large aircraft. Aircraft Engineering and Aerospace Technology 1999;71(6):558–70. [8] Marsh G. Airframes exploit composites in battle for supremacy. Reinforced Plastics 2005;49(3):26–32. [9] Abrate S. Impact on laminated composite materials. Applied Mechanics Review 1991;44:155–90. [10] Abrate S. Impact on laminated composites. Applied Mechanics Review 1994;47:517–44. [11] Abrate S. Modeling of impacts on composite structures. Composite Structures 2001;51:129–38. [12] Olsson R. Mass criterion for wave controlled impact response of composite plates. Composites: Part A 2000;31:879–87. [13] Olsson R. Analytical prediction of large mass impact damage in composite laminates. Composites: Part A 2001;32:1207–15. [14] Olsson R. Closed form prediction of peak load and delamination onset under small mass impact. Composite Structures 2003;59:341–9. [15] Whitney JM, Pagano NJ. Shear deformation in heterogeneous anisotropic plates. Journal of Applied Mechanics 1970;37:1031–6. [16] Mindlin RD. Influence of rotary inertia and shear on flexural motions of isotropic elastic plates. Journal of Applied Mechanics 1951;18:31–8. [17] Reddy JN. Mechanics of laminated composite plates and shells, 2nd ed. United States of America: CRC Press; 1997. [18] Shokuhfar A, Khalili SMR, Ashenai Ghasemi F, Malekzadeh K, Raissi S. Analysis and optimization of smart hybrid composite plates subjected to low-velocity impact using the response surface methodology (RSM). Thin-Walled Structures 2008;46:1204–12. [19] Choi IH, Lim CH. Low-velocity impact analysis of composite laminates using linearized contact law. Composite Structures 2004;66:125–32. [20] Carvalho A, Soares CG. Dynamic response of rectangular plates of composite materials subjected to impact loads. Composite Structures 1996;34:55–63. [21] Khalili SMR, Shokuhfar A, Malekzadeh K, Ashenai Ghasemi F. Low-velocity impact response of active thin-walled hybrid composite structures embedded with SMA wires. Thin-Walled Structures 2007;45:799–808. [22] Christoforou AP, Swanson SR. Analysis of Impact response in composite plates. International Journal of Solids Structures 1991;27(2):161–70. [23] Khalili SMR, Shokuhfar A, Ashenai Ghasemi F. Effect of smart stiffening procedure on low-velocity impact response of smart structures. Journal of Materials Processing Technology 2007;190(1–3):142–52.
ARTICLE IN PRESS 70
G.H. Payeganeh et al. / Thin-Walled Structures 48 (2010) 62–70
[24] Khalili SMR, Shokuhfar A, Ashenai Ghasemi F, Malekzadeh K. Dynamic response of smart hybrid plate subjected to low-velocity. Journal of Composite Materials 2007;41(19):2347–70. [25] Pierson MO, Vaziri R. Analytical solution for low-velocity impact response of composite plates. AIAA Journal 1996;34(8):1633–40. [26] Delfosse D, Vaziri R, Pierson MO, Poursartip A. Analysis of the non-penetrating impact behavior of CFRP laminates. In: Proceeding of the 9th International Conference on Composite Materials (Madrid, Spain), Vol. 5. Cambridge, England, UK: Woodhead Publishing; 1993. p. 366–73.
[27] Malekzadeh K, Khalili MR, Jafari A, Mittal RK. Dynamic response of in-plane pre-stressed sandwich panels with a viscoelastic flexible core and different boundary conditions. Journal of Composite Materials 2006;40: 1449–69. [28] Krimbalis PP, Poon C, Fawaz Z, Behdinan K. Prediction of bearing strength in fiber metal laminates. Journal of Composite Materials 2007;41: 1137–57. [29] Sun CT, Chen JK. On the impact of initially stressed composite laminates. Journal of Composite Materials 1985;19:490–503.