- Email: [email protected]
Response of hybrid laminated composite plates under low-velocity impact
Computers 0
Pergamon PII: 0045-7949(95)00298-7
RESPONSE
OF HYBRID LAMINATED UNDER LOW-VELOCITY
& Sm,rru,~s Vol. 65, No 6, pp. 965-914, 1997 1997 Elswicr Science Ltd. All rights resfxwd Pnnted in Great Bntm 0045s7949/97 $17 00 + 0 on
COMPOSITE IMPACT
PLATES
Young-Shin Lee,? Kun-Hee Kang$ and Oung Park$ tDepartment
of Mechanical
Design Engineering, Chung-Nam National University, 305-764, Korea fADD, Tae-Jon P.O.Box 35, Tae-Jon 305600, Korea
You-soung,
Tae-Jon
(Received 9 August 1994) Abstract-The
response of hybrid laminated composite plates subjected to low velocity impact was investigated using shear deformation theory. As a result, the fractional energy loss of two hybrid composite plates with the same component ratio has different values according to the stacking sequence. A Graphite-Kevlar-Graphite plate has low energy loss and a Kevlar-Graphite-Kevlar plate much higher energy loss. Contact forces between the impactor and plates, center deflections of the plates and velocity changes of the impactor to time have different values according to the material properties of the impacted surface. Various composite plates with the same material in the impacted surface behaved with a similar response. 0 1997 Elsevier Science Ltd
1. INTRODUCTION
produces damage and consequently reduces the strength of composite materials, even though it has low energy. The damage modes usually include local permanent deformations, breakage of fibers, delaminations, etc. While the cause of these damages is still unknown and may not be simple in nature, in general, the dynamic response of the whole structure is of importance. A number of investigators have made the analytical and experimental studies of low velocity impact response of composites [14]. Particularly, several approaches have been taken to improve the impact resistance of composite materials and to enhance the residual strength of it. A part of these include the experimental studies of hybrid laminates. Adams and Zimmerman [5] demonstrated that the impact performance of a graphite/epoxy laminate can be increased by hybridizing with discrete plies of polyethylene/epoxy. Jang et al. [6], showed that nylon/epoxy, polyester/epoxy and polyethylene/ epoxy enhanced the impact resistance of graphite/ epoxy laminate according to stacking sequence. As mentioned above, many investigators are concerned about the hybrid laminated composites, and these have been studied experimentally, not analytically. The present investigation predicted numerically low velocity impact response of hybrid laminated composite plates (graphite/epoxy-glass/epoxy and graphite/epoxy-kevlar/epoxy) using the finite element method, and compared the results with those of single laminated composite plate (graphite/epoxy). Impact
965
Modified contact law suggested by Yang and Sun (71 was used for calculating contact force. Fractional energy loss was computed by velocity change of the impactor according to stacking sequences of hybrid laminates and this result was compared with that of single graphite/epoxy laminate which was susceptible to damage caused by foreign body impact. 2. ANALYSIS
2.1. Analytical
model
A hybrid laminated plate of constant thickness h consists of thin laminas of unidirectionally fiberreinforced composites perfectly bonded together (Fig. 1). Each lamina, whose fiber may orient in any arbitrary direction, can be regarded as a homogeneous orthotropic solid. A coordinate system (x,, x2) of a typical kth lamina is chosen in such a way that the x, - x2 plane coincides with the midplane of the lamina, and the xl and x2 axes are parallel and perpendicular to the fiber direction, respectively. 0 is the angle between the x-axis and xi-axis measured from x to xi counterclockwise, as shown in Fig. 1. The boundary condition of the hybrid laminates is simply supported. Impactor, mass m and radius r, is impacted on the center of the plate with velocity v. 2.2. Constitutive
equations of laminate
In the theory developed by Whitney and Pagan0 [8], we can obtain the following constitutive equations of a symmetrically laminated plate in the
966
Young-Shin Lee et al ?j=l
case of choosing the x-axis of the laminate reference system to coincide with the 0” fiber direction and neglecting the displacement components of the midplane in the x and y direction. Dll awax
+ 2Dla awax
ay + D66ayx/ay2
+D,6(a2$J/ax2 + av+iay) - h(aw/ax
Fig. 2. Nine-node isoparametric plate element.
+ (D,~ + o,)a2+y/axay
+ 4x) = I&
Using the shape functions, the plate displacements w, +X and 4y are approximated by
D16(a2+,iaX2 + a29.yiay2) + (D,~ +
066)a5px/axay (2)
+D6aa5p,/ax2 + 2Dlaa24,1axay + Dz2a24,iay2 -ffu(aw/ay
+ 44 = 4h
(1) where
Hsi(a2w/ax2 + ahlay +
ab,jax + a4?jay) + q =
PC.
2.3. Finite element analysis 1. Plate finite element. A nine-node isoparametric plate finite element (see Fig. 2) was used to model the dynamic motion of the hybrid laminated plate. At each node there are three degrees of freedom, namely the displacement component of z-direction, w and the rotations of the cross-sections perpendicular to the x- and y-axis, & and &,
The stiffness and mass matrices are obtained by numerical integration using Gauss quadrature. Following standard finite element procedures, the system stiffness matrix [I$] and mass matrix [MPl are assembled from the element matrices. The equations of motion are expressed in matrix form as
where (a)
{Pp} = (O,---,
N-layered laminate
F,---,O}’
is the force vector in which F is the contact force associated with the degree of freedom corresponding to the w-displacement at the impact point. 2. Impactor modeling. A higher order rod finite element was used to model the impactor. This element has two degrees of freedom at each node, namely the axial displacement u and its first derivative au/ax. The displacement function is taken as (b)
Coordinate system u = al + a2x + a3x2 + a4x3,
x
where a, are constant coefficients. Solving these coefficients in terms of the nodal degrees of freedom and substituting into eqn (4), we obtain
where Fig. 1. Geometry of a hybrid laminate.
(4)
961
Response of hybrid laminated composite plates is the vector of element nodal degrees of freedom, WI’ =
and
Ifi(x>Lh(xMxLf4~N
Following the usual manner, the system stiffness and mass matrices are assembled from the element stiffness and mass matrices, and the system equations of motion are expressed as
in which
(6) J(x)
= (1 - x/L)2(1 + 2x/L)
fi(x)
= x(1 - x/L)2
where {Pr} = {F,O,---,O}T
.fi(x) = (x*/L%3
- 2x/L)
J+(x) = W/L)(X/L
- 1)
in which impacting
F is the contact force end of the impactor.
applied
at the
2.4. Contact force analysis Yang and Sun [7] have conducted static indentation tests on glass/epoxy and graphite/epoxy com-
are shape functions.
v
START
Input data: geometry, boundary conditions, material properties, contact coefficients etc. Determine the mass matricies of the plate and impactor, and the stiffness matricies of the plate and impactor
I Modify the mass matrix of the plate, and form the effective stiffness matrix of the impactor
I
Calculate
the displacement
Calculate
at time t+0At
the impactor acceleration at time t+At
I
1
G
Calculate
the impactor
velocity
1
Fig. 3. Flow chart for impact analysis.
968
Young-Shin Lee et al 8X8
1.52.4mm ---_____)
Fig. 4. Finite element mesh for laminated plate and impactor.
using steel balls. From the results, they suggested the relations of contact force and indentation depth for loading, unloading and reloading curve.
where k, is reloading experimental value.
1. Loading curve. The contact force F and the indentation depth tl have the relation
The equations of motion, (3) and (6), are to be analyzed with the eqns (7)-(9). In order to analyze eqn (3), the mass matrix [MP] is diagonalized with the lumped mass method, and the central difference method is used for time integration. The Wilson-8 method is used for the time integration of eqn (6). A flow chart for impact analysis is shown in Fig. 3.
posite laminates
F = ku.“,
(7)
where k is contact coefficient calculated by the modified Hertzian contact law and n = 1.5 is the experimental value. 2. Unloading curve. In order to account for the permanent deformation, Crook suggested the power law F = Fm[(a - ~Q/(G - ao)14,
(8)
where F,,, is the maximum contact force just before unloading, x0 is the permanent indentation in an unloading cycle, urnis the indentation corresponding to F,,, and q = 2.5 is the experimental value. 3. Reloading curve. The assumed to be in the form
reloading
F = k,(cc - ao)P,
law
was
(9)
rigidity and p = 1.5 is the
2.5. Numerical analysis
3.
NUMERICAL RESULTS AND DISCUSSION
This study deals with the fractional energy loss and the impact response of hybrid laminated composite plates which are composed of graphite/epoxy laminates having high stiffness, but relatively poor resistance to the impact loadings, and glass/epoxy laminates and kevlar/epoxy laminates having high impact resistance. A five-ply sublaminate whose stacking sequence is arranged by [0”/45”/0”/ - 45”/0”] fiber angles is simply expressed as a single character, for instance, graphite/epoxy as C, glass/epoxy as G and kevlar/ epoxy as K. And there are seven types of hybrid
Table 1. Mechanical properties of composite plates and impactor [7, 91 Material Graphite/epoxy Glass/epoxy Kevlar/eDoxv Impacio; _
B @paI 120.7 39.3 35.4 207.0
EZ@pa) 7.93 8.27 22.0 -
Gu = GI, = G23 @pa) 5.52 4.14 7.36 -
VI2
P (Kg m-‘1
0.30 0.26 0.18 0.30
1600 2040 1469 7860
969
Response of hybrid laminated composite plates Table 2. Contact coefficients of specimen [7] [O/45/0/-4S/Obs Specimen
k (N mm-‘.‘)
4
a, (mm)
s,
P
25,700
2.5
0.0167
0.094
1.5
26,900
2.5
0.0211
0.094
1.5
67,400
2.5
0.0211
0.094
1.5
Graphite/epoxy (CCCC) Glass/epoxy (GGGG) Kevlar/epoxy (KKKK)
Table 3. Experimental data for the energy loss of various plates by Jang [6] Impact energy (J)
Absorbed
Energy
Specimen
energy (J)
loss (%)
cc
110
2.8
2.5
CP,
110
11.5
10.5
CP,
110
11.5
10.5
P,C
110
16.0
14.5
P, c P, P, P, P,
110 110 110
26.7 30.7 93.0
24.3 27.9 84.5
laminates such as CCCC, CGGC, GCCG, GGGG, CKKC, KCCK and KKKK which are symmetric to the midplane. For example, a CGGC laminated plate represents the hybrid laminated plate having its stacking arrangement of ([oO/45°/00/- 45”/O”]c [0”/45”/0”/ -45°/O”]o}S which is stacked by 20 composite laminas. The shapes and the finite element meshes of the laminated plate and impactor are shown in Fig. 4 and the thickness of the plate is 2.69 mm. The mechanical properties for three types of composite
----Sun
plates and the impactor are summarized in Table 1 and their contact coefficients in Table 2. A contact coefficient k for the composite plates, shown in Table 2, was computed by a modified Hertzian contact law and it was assumed that the values of cc, and S, for glass/epoxy laminates and all of coefficients for kevlar/epoxy laminates are equal to those for graphite/epoxy laminates. All of coefficients for the hybrid laminated composite plates which are not described in Table 2 should be obtained by experimental tests, however, it
[3]
25 04 0
.
loo
200
300
Remark C: Carbon/epoxy P,: Polyester/epoxy P,: Polyethylene/epoxy Specimen size: 475 mm Mass of impactor: 1 kg Impact velocity: 4.5 m s - ’ Boundary condition: clamped
400
SOO
600
i I 700
\ I 800
900
‘l 1OOO
Time (msec) Fig. 5. Contact force between the impactor and plate for CCCC plate ([O/45/O/-45/O]2s) with simply supported edges. Impacted by 12.7 mm diameter and, 8.54 gm steel ball at 3.0 m s-l.
970
Young-Shin Lee et al Table 4. Calculated data for the energy loss of various plates in the present analysis Impact energy (J)
Absorbed energy (J)
Energy loss (%)
cccc
0.0675
0.0121
18
CKKP
0.0675
0.0128
19
CGGC
0.0675
0.0189
28
GCCG
0.0675
0.0282
42
KCCX GGGG KKKK
0.0675 0.0675 0.0675
0.042 1 0.0372 0.0426
62 55 63
Specimen
is assumed that their coefficients are the same as those of a laminated plate which consists of composite material of the impact surface. To observe the effectiveness of this study, the fractional energy loss is compared with the experimental results in Ref. [6] and the impact response compared with the numerical results of a CCCC laminated plate in Ref. [3]. Table 3 represents the experimental data of the absorbed energy and the energy loss for various hybrid laminates when the impactor penetrates the plates in Reference [6]. Table 4 represents the numerical results of the absorbed energy {Eu = f m(W - I+*)} and the energy loss (lOOEu/Ei) computed by the rebouncing velocity of an impactor in the present analysis, where m, Vi, Vr and Ei (=f mVi*) are the mass of an impactor, the impact velocity, the rebouncing velocity and the impact energy, respectively.
Remark C: Graphite/epoxy G: Glass/epoxy K: Kevlar/epoxy Mass of impactor: 15 gm Impact velocity: 3 m s - I Boundary condition: simply supported
It can be seen in Table 3 that the carbon/epoxy laminated plate having poor resistance to the impact loading has low energy loss, but for hybrid laminated plates adding polyester/epoxy or polyethylene/epoxy which have good impact characteristics the absorbing rate of impact energy has increased more than four times. The energy loss varies with the material properties of the impact surfaces. It can be found that hybrid laminated plates whose impact surfaces consist of the material having good impact characteristics absorb much impact energy. The numerical results computed by the present analysis, which are summarized in Table 4, also have a similar tendency. The hybrid laminated plates which are composed of graphite/epoxy and glass/ and kevlar/epoxy) epoxy, (or graphite/epoxy absorb higher impact energy than the single
0.4 T
----Sun 0.3 ..
-
[3] Present
0.2 .. 0.1 ..
I:;: 1 0
100
200
300
400
500
600
700
800
900
1000
Time (msec) Fig. 6. Displacement of the plate center for CCCC plate ([O/45/0/-45/0]& with simply supported edges. Impacted by 12.7 mm diameter and 8.54 gm steel ball at 3.0 m s-i.
971
Response of hybrid laminated composite plates 400 350 300 250 zc 2 200 z 8 B I50 z U 100
50
0
Time (msec) Fig. 7. Contact forces between the impactor and various pIates-CCCC, CGGC, GCCG and GGGG-with simply supported edges. Impacted by 12.7mm diameter and 15 gm steel rod at 3.0 m s-l.
graphite/epoxy laminated plate (CCCC). Even though the hybrid laminated plates consist of the same types of sublaminates the energy loss may vary with their stacking sequence. For instance, the energy loss for KCCK plate which is stacked on impact surface by plates with a good impact resistance is better than that of CKKC by three times. Therefore it can be seen that the sublaminate with a good impact resistance should be
arranged on the impact surface in the case of hybrid laminated plates which consist of the same types of sublaminates. When a hybrid laminated plate is collided by an impactor at a low impact velocity of 3 m s-l, impact responses such as the contact forces to time, the displacements at the midpoint of the plate and the velocity changes of an impactor will be evaluated in the following.
1. 0.8 0.
h
0.4
3y
0.2 !
-1.0. 0
100
200
300
400
500 600 Time (msec)
700
800
900
4
1000
Fig. 8, Displacements of the plate center for various plates-CCCC, CGGC, GCCG and GGGG-with simply supported edges. Impacted by 12.7mm diameter and 15 gm steel rod at 3.0 m s-l. CAS 65/6-G
912
Young-Shin Lee et al 3.0, 2.5..
-
cccc
zz ----
E% GGGG
400
500
600
Time (msec)
Fig. 9. Impactor velocities on various plates--CCCC,
CGGC, GCCG and GGGG-with supported edges. Impacted by 12.7 mm diameter and 15 gm steel rod at 3.0 m s-‘.
Figures 5 and 6 show the profile of the contact forces and the displacements at the midpoint of the plate, respectively, comparing the numerical results computed by the present method with the results obtained by the input data from Ref. [3] in the case of a steel ball impactor with the mass of 8.54 gm collides on a CCCC graphite/ epoxy plate; and the numerical results in the case of an impactor with a mass of 15 gm colliding with
various hybrid laminated plates are shown in Figs 7-12. Figure 5 indicates that the initial peak and trend of the contact forces have an excellent agreement with the results of Ref. [3] (and the ending peak is slightly lower than the results of Ref. [3]). However, the displacement variation at a plate center during the impact shown in Fig. 6 agrees very well with the results of Ref. [3].
400 1
0
0
100
200
300
400
500
600
700
800
900
1000
Time (msec) Fig. 10. Contact forces between the impactor and various plates-CCCC, CKKC, KCCK and KKKK-with simply supported edges. Impacted by 12.7 mm diameter and 15 gm steel rod at 3.0 m s-‘.
Response of hybrid laminated composite plates I.01
973
El -CCCC +cKKc
aoKCCK ----KKKK
0.8” 0.6”
-0.6..
::
t
0
100
200
300
400 500 600 Time (msec)
700
800
900
rOO0
Fig. 11. Displacements of the plate center for various plates--CCCC, CKKC, KCCK and KKKK-with simply supported edges. Impacted by 12.7 mm diameter and 15 gm steel rod at 3.0 m s-l.
Figure 7 shows the profile of contact forces to time between hybrid laminated plates, such as CGGC and GCCG, and laminated plates, such as CCCC and GGGG, when four edges of the plates are simply supported. It can be seen here that contact forces vary with the stacking sequence of hybrid laminated plates and the impact responses of the laminated plates whose impact surfaces consist of the same types of,sublaminates are fairly similar.
Figures 8 and 9 show the plate center displacements to time and the velocity changes of impactor for the same hybrid laminated plates as in Fig. 7. It can be seen in Fig. 8 that the displacements of the plate center vary with the stacking sequence of the hybrid laminated plates and the displacement characteristics of the plates whose impact surfaces consist of the same types of sublaminates are similar. Figure 9 shows that the velocity changes of the impactor vary with the
3.0 2.5 -
KCCK
2.0
-3.0
0
100
200
300
400 500 600 Time (msec)
700
800
900
I
1000
Fig. 12. Impactor velocities on various plates--CCCC, CKKC, KCCK and KKKK-with supported edges. Impacted by 12.7 mm diameter and 15 gm steel rod at 3.0 m s-l.
simply
Young-Shin1 Lee et al
974
stacking sequences of the hybrid laminated plates and have a similar profile in the case where it has the same types of sublaminates on the impact surfaces. Figure 10 shows the profile of contact forces to time between CGGC and GCCG hybrid laminated plates and CCCC and GGGG plates when four edges of the plates are simply supported. It can also be seen that contact forces vary with the stacking sequence of hybrid laminated plates and the impact responses of the laminated plates whose impact surfaces consist of the same types of sublaminates are quite similar. Figures 11 and 12 show the plate center displacements to time and the velocity changes of impactor for the same hybrid laminated plates as in Fig. 10. We can see here also that the impact responses of the laminated plates whose impact surfaces consist of the same types of sublaminates are quite similar. From the above results it may be concluded that the bending behavior of the hybrid laminated plates dominantly relies upon the material properties of the impact surfaces. 4. CONCLUSION
A numerical analysis using the finite element method was employed in the prediction of the response of hybrid laminated composite plates subjected to low velocity impact. As a result, the fractional energy loss of two hybrid composite plates with the same component ratio have different values according to the stacking sequence. A Graphite-KevlarGraphite plate has low energy loss and a Kevlar-Graphite-Kevlar plate has over three times as much. It can be seen that the material having good impact resistance should be laid on an
impacted surface in the hybrid laminated composite plates. Contact forces between the impactor and plates, center deflections of the plates and velocity change of the impactor with time have different values according to material properties of the impacted surface. Various composite plates with the same material in the impacted surface behaved in a similar manner. REFERENCES
1. Sun, T. C. and Chattopadhyay, S., Dynamic response of anisotropic laminated plates under initial stress to impact of a mass. Trans. ASME J. appl. Mech., 1975, 42, 693698. 2. Greszcuk, L. B., Damage in composite materials due to low velocity impact. In: Zmpact Dynamics, pp. 55-94. Wiley, New York 1982. 3. Sun. C. T. and Chen. J. K.. On the imuact of initiallv stressed composite laminates. J. Compod. Mater., 198<, 19, 490-504. 4. Lee, Young-Shin and Park, Oung, Low-velocity impact
response of laminated composite plates using a higher order shear deformation theory. Trans. KSME, 1990, 14, 1365-1381 (in Korean). 5. Adams, D. F. and Zimmerman, R. S., Static and impact performance of polyethylene fiber/graphite fiber hybrid composites. SAkpE J:, 1986, 22,-l&16. 6. Jane. B. Z.. Chen. L. C.. Wane. C. Z.. Lin. H. T. and Zeey’R. H.; Impact resistanceand energy absorption mechanisms in hybrid composites. Compos. Sci. Technol., 1989, 34, 305-335. 7. Yang, S. H. and Sun, C. T., Indentation law for composite laminates. ASTM STP, 1982, 786, 425449. 8. Whitney, J. M. and Pagano, N. J., Shear deformation in heterogeneous anisotropic plates. Trans. ASME J. appl. Meih., 1970, 37, 1031-1036.
9. Meyer, P. I., Low-velocity hard-object impact of filament-wound kevlar/epoxy composite. Compos. Sci. Technol., 1988, 33, 279-293.
Pergamon PII: 0045-7949(95)00298-7
RESPONSE
OF HYBRID LAMINATED UNDER LOW-VELOCITY
& Sm,rru,~s Vol. 65, No 6, pp. 965-914, 1997 1997 Elswicr Science Ltd. All rights resfxwd Pnnted in Great Bntm 0045s7949/97 $17 00 + 0 on
COMPOSITE IMPACT
PLATES
Young-Shin Lee,? Kun-Hee Kang$ and Oung Park$ tDepartment
of Mechanical
Design Engineering, Chung-Nam National University, 305-764, Korea fADD, Tae-Jon P.O.Box 35, Tae-Jon 305600, Korea
You-soung,
Tae-Jon
(Received 9 August 1994) Abstract-The
response of hybrid laminated composite plates subjected to low velocity impact was investigated using shear deformation theory. As a result, the fractional energy loss of two hybrid composite plates with the same component ratio has different values according to the stacking sequence. A Graphite-Kevlar-Graphite plate has low energy loss and a Kevlar-Graphite-Kevlar plate much higher energy loss. Contact forces between the impactor and plates, center deflections of the plates and velocity changes of the impactor to time have different values according to the material properties of the impacted surface. Various composite plates with the same material in the impacted surface behaved with a similar response. 0 1997 Elsevier Science Ltd
1. INTRODUCTION
produces damage and consequently reduces the strength of composite materials, even though it has low energy. The damage modes usually include local permanent deformations, breakage of fibers, delaminations, etc. While the cause of these damages is still unknown and may not be simple in nature, in general, the dynamic response of the whole structure is of importance. A number of investigators have made the analytical and experimental studies of low velocity impact response of composites [14]. Particularly, several approaches have been taken to improve the impact resistance of composite materials and to enhance the residual strength of it. A part of these include the experimental studies of hybrid laminates. Adams and Zimmerman [5] demonstrated that the impact performance of a graphite/epoxy laminate can be increased by hybridizing with discrete plies of polyethylene/epoxy. Jang et al. [6], showed that nylon/epoxy, polyester/epoxy and polyethylene/ epoxy enhanced the impact resistance of graphite/ epoxy laminate according to stacking sequence. As mentioned above, many investigators are concerned about the hybrid laminated composites, and these have been studied experimentally, not analytically. The present investigation predicted numerically low velocity impact response of hybrid laminated composite plates (graphite/epoxy-glass/epoxy and graphite/epoxy-kevlar/epoxy) using the finite element method, and compared the results with those of single laminated composite plate (graphite/epoxy). Impact
965
Modified contact law suggested by Yang and Sun (71 was used for calculating contact force. Fractional energy loss was computed by velocity change of the impactor according to stacking sequences of hybrid laminates and this result was compared with that of single graphite/epoxy laminate which was susceptible to damage caused by foreign body impact. 2. ANALYSIS
2.1. Analytical
model
A hybrid laminated plate of constant thickness h consists of thin laminas of unidirectionally fiberreinforced composites perfectly bonded together (Fig. 1). Each lamina, whose fiber may orient in any arbitrary direction, can be regarded as a homogeneous orthotropic solid. A coordinate system (x,, x2) of a typical kth lamina is chosen in such a way that the x, - x2 plane coincides with the midplane of the lamina, and the xl and x2 axes are parallel and perpendicular to the fiber direction, respectively. 0 is the angle between the x-axis and xi-axis measured from x to xi counterclockwise, as shown in Fig. 1. The boundary condition of the hybrid laminates is simply supported. Impactor, mass m and radius r, is impacted on the center of the plate with velocity v. 2.2. Constitutive
equations of laminate
In the theory developed by Whitney and Pagan0 [8], we can obtain the following constitutive equations of a symmetrically laminated plate in the
966
Young-Shin Lee et al ?j=l
case of choosing the x-axis of the laminate reference system to coincide with the 0” fiber direction and neglecting the displacement components of the midplane in the x and y direction. Dll awax
+ 2Dla awax
ay + D66ayx/ay2
+D,6(a2$J/ax2 + av+iay) - h(aw/ax
Fig. 2. Nine-node isoparametric plate element.
+ (D,~ + o,)a2+y/axay
+ 4x) = I&
Using the shape functions, the plate displacements w, +X and 4y are approximated by
D16(a2+,iaX2 + a29.yiay2) + (D,~ +
066)a5px/axay (2)
+D6aa5p,/ax2 + 2Dlaa24,1axay + Dz2a24,iay2 -ffu(aw/ay
+ 44 = 4h
(1) where
Hsi(a2w/ax2 + ahlay +
ab,jax + a4?jay) + q =
PC.
2.3. Finite element analysis 1. Plate finite element. A nine-node isoparametric plate finite element (see Fig. 2) was used to model the dynamic motion of the hybrid laminated plate. At each node there are three degrees of freedom, namely the displacement component of z-direction, w and the rotations of the cross-sections perpendicular to the x- and y-axis, & and &,
The stiffness and mass matrices are obtained by numerical integration using Gauss quadrature. Following standard finite element procedures, the system stiffness matrix [I$] and mass matrix [MPl are assembled from the element matrices. The equations of motion are expressed in matrix form as
where (a)
{Pp} = (O,---,
N-layered laminate
F,---,O}’
is the force vector in which F is the contact force associated with the degree of freedom corresponding to the w-displacement at the impact point. 2. Impactor modeling. A higher order rod finite element was used to model the impactor. This element has two degrees of freedom at each node, namely the axial displacement u and its first derivative au/ax. The displacement function is taken as (b)
Coordinate system u = al + a2x + a3x2 + a4x3,
x
where a, are constant coefficients. Solving these coefficients in terms of the nodal degrees of freedom and substituting into eqn (4), we obtain
where Fig. 1. Geometry of a hybrid laminate.
(4)
961
Response of hybrid laminated composite plates is the vector of element nodal degrees of freedom, WI’ =
and
Ifi(x>Lh(xMxLf4~N
Following the usual manner, the system stiffness and mass matrices are assembled from the element stiffness and mass matrices, and the system equations of motion are expressed as
in which
(6) J(x)
= (1 - x/L)2(1 + 2x/L)
fi(x)
= x(1 - x/L)2
where {Pr} = {F,O,---,O}T
.fi(x) = (x*/L%3
- 2x/L)
J+(x) = W/L)(X/L
- 1)
in which impacting
F is the contact force end of the impactor.
applied
at the
2.4. Contact force analysis Yang and Sun [7] have conducted static indentation tests on glass/epoxy and graphite/epoxy com-
are shape functions.
v
START
Input data: geometry, boundary conditions, material properties, contact coefficients etc. Determine the mass matricies of the plate and impactor, and the stiffness matricies of the plate and impactor
I Modify the mass matrix of the plate, and form the effective stiffness matrix of the impactor
I
Calculate
the displacement
Calculate
at time t+0At
the impactor acceleration at time t+At
I
1
G
Calculate
the impactor
velocity
1
Fig. 3. Flow chart for impact analysis.
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Young-Shin Lee et al 8X8
1.52.4mm ---_____)
Fig. 4. Finite element mesh for laminated plate and impactor.
using steel balls. From the results, they suggested the relations of contact force and indentation depth for loading, unloading and reloading curve.
where k, is reloading experimental value.
1. Loading curve. The contact force F and the indentation depth tl have the relation
The equations of motion, (3) and (6), are to be analyzed with the eqns (7)-(9). In order to analyze eqn (3), the mass matrix [MP] is diagonalized with the lumped mass method, and the central difference method is used for time integration. The Wilson-8 method is used for the time integration of eqn (6). A flow chart for impact analysis is shown in Fig. 3.
posite laminates
F = ku.“,
(7)
where k is contact coefficient calculated by the modified Hertzian contact law and n = 1.5 is the experimental value. 2. Unloading curve. In order to account for the permanent deformation, Crook suggested the power law F = Fm[(a - ~Q/(G - ao)14,
(8)
where F,,, is the maximum contact force just before unloading, x0 is the permanent indentation in an unloading cycle, urnis the indentation corresponding to F,,, and q = 2.5 is the experimental value. 3. Reloading curve. The assumed to be in the form
reloading
F = k,(cc - ao)P,
law
was
(9)
rigidity and p = 1.5 is the
2.5. Numerical analysis
3.
NUMERICAL RESULTS AND DISCUSSION
This study deals with the fractional energy loss and the impact response of hybrid laminated composite plates which are composed of graphite/epoxy laminates having high stiffness, but relatively poor resistance to the impact loadings, and glass/epoxy laminates and kevlar/epoxy laminates having high impact resistance. A five-ply sublaminate whose stacking sequence is arranged by [0”/45”/0”/ - 45”/0”] fiber angles is simply expressed as a single character, for instance, graphite/epoxy as C, glass/epoxy as G and kevlar/ epoxy as K. And there are seven types of hybrid
Table 1. Mechanical properties of composite plates and impactor [7, 91 Material Graphite/epoxy Glass/epoxy Kevlar/eDoxv Impacio; _
B @paI 120.7 39.3 35.4 207.0
EZ@pa) 7.93 8.27 22.0 -
Gu = GI, = G23 @pa) 5.52 4.14 7.36 -
VI2
P (Kg m-‘1
0.30 0.26 0.18 0.30
1600 2040 1469 7860
969
Response of hybrid laminated composite plates Table 2. Contact coefficients of specimen [7] [O/45/0/-4S/Obs Specimen
k (N mm-‘.‘)
4
a, (mm)
s,
P
25,700
2.5
0.0167
0.094
1.5
26,900
2.5
0.0211
0.094
1.5
67,400
2.5
0.0211
0.094
1.5
Graphite/epoxy (CCCC) Glass/epoxy (GGGG) Kevlar/epoxy (KKKK)
Table 3. Experimental data for the energy loss of various plates by Jang [6] Impact energy (J)
Absorbed
Energy
Specimen
energy (J)
loss (%)
cc
110
2.8
2.5
CP,
110
11.5
10.5
CP,
110
11.5
10.5
P,C
110
16.0
14.5
P, c P, P, P, P,
110 110 110
26.7 30.7 93.0
24.3 27.9 84.5
laminates such as CCCC, CGGC, GCCG, GGGG, CKKC, KCCK and KKKK which are symmetric to the midplane. For example, a CGGC laminated plate represents the hybrid laminated plate having its stacking arrangement of ([oO/45°/00/- 45”/O”]c [0”/45”/0”/ -45°/O”]o}S which is stacked by 20 composite laminas. The shapes and the finite element meshes of the laminated plate and impactor are shown in Fig. 4 and the thickness of the plate is 2.69 mm. The mechanical properties for three types of composite
----Sun
plates and the impactor are summarized in Table 1 and their contact coefficients in Table 2. A contact coefficient k for the composite plates, shown in Table 2, was computed by a modified Hertzian contact law and it was assumed that the values of cc, and S, for glass/epoxy laminates and all of coefficients for kevlar/epoxy laminates are equal to those for graphite/epoxy laminates. All of coefficients for the hybrid laminated composite plates which are not described in Table 2 should be obtained by experimental tests, however, it
[3]
25 04 0
.
loo
200
300
Remark C: Carbon/epoxy P,: Polyester/epoxy P,: Polyethylene/epoxy Specimen size: 475 mm Mass of impactor: 1 kg Impact velocity: 4.5 m s - ’ Boundary condition: clamped
400
SOO
600
i I 700
\ I 800
900
‘l 1OOO
Time (msec) Fig. 5. Contact force between the impactor and plate for CCCC plate ([O/45/O/-45/O]2s) with simply supported edges. Impacted by 12.7 mm diameter and, 8.54 gm steel ball at 3.0 m s-l.
970
Young-Shin Lee et al Table 4. Calculated data for the energy loss of various plates in the present analysis Impact energy (J)
Absorbed energy (J)
Energy loss (%)
cccc
0.0675
0.0121
18
CKKP
0.0675
0.0128
19
CGGC
0.0675
0.0189
28
GCCG
0.0675
0.0282
42
KCCX GGGG KKKK
0.0675 0.0675 0.0675
0.042 1 0.0372 0.0426
62 55 63
Specimen
is assumed that their coefficients are the same as those of a laminated plate which consists of composite material of the impact surface. To observe the effectiveness of this study, the fractional energy loss is compared with the experimental results in Ref. [6] and the impact response compared with the numerical results of a CCCC laminated plate in Ref. [3]. Table 3 represents the experimental data of the absorbed energy and the energy loss for various hybrid laminates when the impactor penetrates the plates in Reference [6]. Table 4 represents the numerical results of the absorbed energy {Eu = f m(W - I+*)} and the energy loss (lOOEu/Ei) computed by the rebouncing velocity of an impactor in the present analysis, where m, Vi, Vr and Ei (=f mVi*) are the mass of an impactor, the impact velocity, the rebouncing velocity and the impact energy, respectively.
Remark C: Graphite/epoxy G: Glass/epoxy K: Kevlar/epoxy Mass of impactor: 15 gm Impact velocity: 3 m s - I Boundary condition: simply supported
It can be seen in Table 3 that the carbon/epoxy laminated plate having poor resistance to the impact loading has low energy loss, but for hybrid laminated plates adding polyester/epoxy or polyethylene/epoxy which have good impact characteristics the absorbing rate of impact energy has increased more than four times. The energy loss varies with the material properties of the impact surfaces. It can be found that hybrid laminated plates whose impact surfaces consist of the material having good impact characteristics absorb much impact energy. The numerical results computed by the present analysis, which are summarized in Table 4, also have a similar tendency. The hybrid laminated plates which are composed of graphite/epoxy and glass/ and kevlar/epoxy) epoxy, (or graphite/epoxy absorb higher impact energy than the single
0.4 T
----Sun 0.3 ..
-
[3] Present
0.2 .. 0.1 ..
I:;: 1 0
100
200
300
400
500
600
700
800
900
1000
Time (msec) Fig. 6. Displacement of the plate center for CCCC plate ([O/45/0/-45/0]& with simply supported edges. Impacted by 12.7 mm diameter and 8.54 gm steel ball at 3.0 m s-i.
971
Response of hybrid laminated composite plates 400 350 300 250 zc 2 200 z 8 B I50 z U 100
50
0
Time (msec) Fig. 7. Contact forces between the impactor and various pIates-CCCC, CGGC, GCCG and GGGG-with simply supported edges. Impacted by 12.7mm diameter and 15 gm steel rod at 3.0 m s-l.
graphite/epoxy laminated plate (CCCC). Even though the hybrid laminated plates consist of the same types of sublaminates the energy loss may vary with their stacking sequence. For instance, the energy loss for KCCK plate which is stacked on impact surface by plates with a good impact resistance is better than that of CKKC by three times. Therefore it can be seen that the sublaminate with a good impact resistance should be
arranged on the impact surface in the case of hybrid laminated plates which consist of the same types of sublaminates. When a hybrid laminated plate is collided by an impactor at a low impact velocity of 3 m s-l, impact responses such as the contact forces to time, the displacements at the midpoint of the plate and the velocity changes of an impactor will be evaluated in the following.
1. 0.8 0.
h
0.4
3y
0.2 !
-1.0. 0
100
200
300
400
500 600 Time (msec)
700
800
900
4
1000
Fig. 8, Displacements of the plate center for various plates-CCCC, CGGC, GCCG and GGGG-with simply supported edges. Impacted by 12.7mm diameter and 15 gm steel rod at 3.0 m s-l. CAS 65/6-G
912
Young-Shin Lee et al 3.0, 2.5..
-
cccc
zz ----
E% GGGG
400
500
600
Time (msec)
Fig. 9. Impactor velocities on various plates--CCCC,
CGGC, GCCG and GGGG-with supported edges. Impacted by 12.7 mm diameter and 15 gm steel rod at 3.0 m s-‘.
Figures 5 and 6 show the profile of the contact forces and the displacements at the midpoint of the plate, respectively, comparing the numerical results computed by the present method with the results obtained by the input data from Ref. [3] in the case of a steel ball impactor with the mass of 8.54 gm collides on a CCCC graphite/ epoxy plate; and the numerical results in the case of an impactor with a mass of 15 gm colliding with
various hybrid laminated plates are shown in Figs 7-12. Figure 5 indicates that the initial peak and trend of the contact forces have an excellent agreement with the results of Ref. [3] (and the ending peak is slightly lower than the results of Ref. [3]). However, the displacement variation at a plate center during the impact shown in Fig. 6 agrees very well with the results of Ref. [3].
400 1
0
0
100
200
300
400
500
600
700
800
900
1000
Time (msec) Fig. 10. Contact forces between the impactor and various plates-CCCC, CKKC, KCCK and KKKK-with simply supported edges. Impacted by 12.7 mm diameter and 15 gm steel rod at 3.0 m s-‘.
Response of hybrid laminated composite plates I.01
973
El -CCCC +cKKc
aoKCCK ----KKKK
0.8” 0.6”
-0.6..
::
t
0
100
200
300
400 500 600 Time (msec)
700
800
900
rOO0
Fig. 11. Displacements of the plate center for various plates--CCCC, CKKC, KCCK and KKKK-with simply supported edges. Impacted by 12.7 mm diameter and 15 gm steel rod at 3.0 m s-l.
Figure 7 shows the profile of contact forces to time between hybrid laminated plates, such as CGGC and GCCG, and laminated plates, such as CCCC and GGGG, when four edges of the plates are simply supported. It can be seen here that contact forces vary with the stacking sequence of hybrid laminated plates and the impact responses of the laminated plates whose impact surfaces consist of the same types of,sublaminates are fairly similar.
Figures 8 and 9 show the plate center displacements to time and the velocity changes of impactor for the same hybrid laminated plates as in Fig. 7. It can be seen in Fig. 8 that the displacements of the plate center vary with the stacking sequence of the hybrid laminated plates and the displacement characteristics of the plates whose impact surfaces consist of the same types of sublaminates are similar. Figure 9 shows that the velocity changes of the impactor vary with the
3.0 2.5 -
KCCK
2.0
-3.0
0
100
200
300
400 500 600 Time (msec)
700
800
900
I
1000
Fig. 12. Impactor velocities on various plates--CCCC, CKKC, KCCK and KKKK-with supported edges. Impacted by 12.7 mm diameter and 15 gm steel rod at 3.0 m s-l.
simply
Young-Shin1 Lee et al
974
stacking sequences of the hybrid laminated plates and have a similar profile in the case where it has the same types of sublaminates on the impact surfaces. Figure 10 shows the profile of contact forces to time between CGGC and GCCG hybrid laminated plates and CCCC and GGGG plates when four edges of the plates are simply supported. It can also be seen that contact forces vary with the stacking sequence of hybrid laminated plates and the impact responses of the laminated plates whose impact surfaces consist of the same types of sublaminates are quite similar. Figures 11 and 12 show the plate center displacements to time and the velocity changes of impactor for the same hybrid laminated plates as in Fig. 10. We can see here also that the impact responses of the laminated plates whose impact surfaces consist of the same types of sublaminates are quite similar. From the above results it may be concluded that the bending behavior of the hybrid laminated plates dominantly relies upon the material properties of the impact surfaces. 4. CONCLUSION
A numerical analysis using the finite element method was employed in the prediction of the response of hybrid laminated composite plates subjected to low velocity impact. As a result, the fractional energy loss of two hybrid composite plates with the same component ratio have different values according to the stacking sequence. A Graphite-KevlarGraphite plate has low energy loss and a Kevlar-Graphite-Kevlar plate has over three times as much. It can be seen that the material having good impact resistance should be laid on an
impacted surface in the hybrid laminated composite plates. Contact forces between the impactor and plates, center deflections of the plates and velocity change of the impactor with time have different values according to material properties of the impacted surface. Various composite plates with the same material in the impacted surface behaved in a similar manner. REFERENCES
1. Sun, T. C. and Chattopadhyay, S., Dynamic response of anisotropic laminated plates under initial stress to impact of a mass. Trans. ASME J. appl. Mech., 1975, 42, 693698. 2. Greszcuk, L. B., Damage in composite materials due to low velocity impact. In: Zmpact Dynamics, pp. 55-94. Wiley, New York 1982. 3. Sun. C. T. and Chen. J. K.. On the imuact of initiallv stressed composite laminates. J. Compod. Mater., 198<, 19, 490-504. 4. Lee, Young-Shin and Park, Oung, Low-velocity impact
response of laminated composite plates using a higher order shear deformation theory. Trans. KSME, 1990, 14, 1365-1381 (in Korean). 5. Adams, D. F. and Zimmerman, R. S., Static and impact performance of polyethylene fiber/graphite fiber hybrid composites. SAkpE J:, 1986, 22,-l&16. 6. Jane. B. Z.. Chen. L. C.. Wane. C. Z.. Lin. H. T. and Zeey’R. H.; Impact resistanceand energy absorption mechanisms in hybrid composites. Compos. Sci. Technol., 1989, 34, 305-335. 7. Yang, S. H. and Sun, C. T., Indentation law for composite laminates. ASTM STP, 1982, 786, 425449. 8. Whitney, J. M. and Pagano, N. J., Shear deformation in heterogeneous anisotropic plates. Trans. ASME J. appl. Meih., 1970, 37, 1031-1036.
9. Meyer, P. I., Low-velocity hard-object impact of filament-wound kevlar/epoxy composite. Compos. Sci. Technol., 1988, 33, 279-293.