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Stress analysis of laminated E-glass epoxy composite plates subject to impact dynamic loading
Pergamon PII: SO0457949(%)00020-X
Compurers & Structures Vol. 61. No. I. pp. I-1 I. 1996 Copyright 0 1996 Ekvicr Science Ltd Printed-in &at Britain. All rights reserved 0045-7949/96 $15.00 + 0.00
STRESS ANALYSIS OF LAMINATED E-GLASS EPOXY COMPOSITE PLATES SUBJECT TO IMPACT DYNAMIC LOADING W. J. Liou, C. I. Tseng and L. P. Chao Department of Mechanical Engineering, Feng-Chia University, Taichung, Taiwan, Republic of China (Received
1I September 1995)
AbstracG-The transient response of an E-glass epoxy laminated composite plate impacted by a steel circular cylinder is analyzed. The analysis is performed by using a three-dimensional hybrid stress finite element program. The contact force between the projectile and the laminated plate is modeled by the Hertzian impact law. The predicted transverse deflection at the center of the laminated plate agrees with the deflection determined experimentally. The effects of projectile velocity and thickness of the laminated nlate on the imoacted stress distributions and the central transverse deflection are also investigated. copyright 0 1696 Elsevier Science Ltd.
using Hertzian contact theory. Sun et al. [7] established a contact law for a composite plate and a steel ball. The contact behavior was incorporated into a finite element program to analyze the wave propagation phenomenon. Concerning the plate impact experiments, a systematic experimental study of failure mechanisms on impact of laminated plates was conducted by Takeda [8]. In the experiments, a small steel cylinder was fired from a gas gun at subperforation velocities. It indicated that the flexural wave propagation was the principal response mode of moderately thin and thick plates subject to central local impact, and it led to delamination failure propagation. The effect of impactor mass and velocity, and ply orientation on laminated plate delamination was investigated by Takeda et al. [9, lo]. In the present investigation, a hybrid stress finite element program is applied to predict the displacement and stress distributions in an E-glass epoxy laminated composite plate subject to a local impact loading. The accuracy and versatility of the hybrid stress finite element method have been validated by many investigators [ 1I-141, they have proven that the hybrid stress finite element model is a useful tool in attacking complicated structure analysis problems. The detailed formulation of this model can be found in these references. The analysis herein is concerned only with the time that the laminated plate remains elastic, and that delamination has not yet occurred. The damping in the laminated plate is assumed to be negligible. The stresses and central transverse deflections are presented to show the transient response of the laminated plate when impacted by a projectile. Knowledge of where and when the stresses are maximum should be helpful in determining where,
INTRODUCTION
During the past three decades, the applications of composite materials have grown rapidly and are now very commonplace around the world. Industries such as aircraft, automobiles, sporting goods, electronics, and appliances are quite dependent on fiberreinforced composites. In the contemporary technology of structural composite materials, major deficiencies still exist with respect to the ability to determine the stress field within a multilayered composite laminate, especially for the impact loading. The transient response of laminated plates subject to local impact loadings has been of significant concern in many advanced engineering structures and components. For example, the leading edge of an aircraft wing or blades in a jet engine may be hit by foreign objects such as stones, nuts, bolts or birds. In these cases the impact resistance of laminated plates must be known in order to design the laminated plates which meet aircraft safety requirements. Impact problems with energies far below the penetration levels can cause significant damage to components made of composite materials, since delamination often occurs at energy levels which are an order of magnitude below those causing penetration [l]. The analysis of the plate impact problems have been concerned for a long time history. Goldsmith [2] provided a detailed discussion for impact on isotropic beams and plates. Chow [3] investigated the transient response of a rectangular laminated plate subject to a concentrated impact loading by using shear deformation theory [4]. Chen [S] applied a Hertzian contact law to analyze the response of impact on anisotropic materials. The dynamic response of laminates impacted by spherical impactors was discussed by Greszczuk and Chao [6] 1
2
W. J. Liou et al.
Y*V
I-
&=t: 7omm
---i
T
u=v=w=o
U=O
(aTe - aTSo/2 - pB2/2)dv
n
U=W=W=O
+
?v,,
T+(i - ufds-
s
7omm
w=o
Fig. 1.Clamped boundary conditions for a quadrant of the laminate.
when and how the delamination failure occurs. With the information of the laminated plate behavior, the structure members can be constructed to meet the specific design requirements.
FORMULATION
Based on the development of Tong et aE. (111, the hybrid functional for multilayered elements in dynamic problems without body force and initial stress can be defined in the matrix form as
T%ds dt,
s
(1)
where superscript T denotes the matrix transpose, u is the stress vector, e is the strain vector computed from the displacement vector u, S is the transformed compliance matrix, T is the vector of surface tractions, t is the vector of prescribed surface tractions, rl is the vector of boundary displacements and p is the density of the laminate. un is the volume of nth element having boundary au,, and su,, is the surface of nth element over which traction is prescribed. The stress, strain, and displacement fields are independently assumed as functions of unknown parameters a, and /3 and nodal displacements q, respectively,
Pfh T = Ru,
TV=
ii=Lq, HYBRID STRESS FINITE ELEMENT
3%I
u =Nq,
u= Ng,
e=Du=DiVq=Bq,
(2)
where the stress vector, a, satisfies the homogeneous equilibrium equations and the detailed development of the stress field can be found in Ref. [13]. P, R and L are matrices consisting of polynomial terms of coordinates (x, y, z). N is the shape function in terms of the natural coordinates (5, n, <).
2-l-----
*
Experimental
-2
TIME (microseconds) Fig. 2. Transverse deflections of the center vs time for a three-layer laminate determined e~perimentaily and by using hybrid stress finite element method, h = 4.29 mm, u = 22.6 m/see.
3
Laminated E-glass epoxy composite plates
TIME
(microseconds)
Fig. 3. Transverse deflection of the center vs time for a three-layer laminate under Hertzian impact, h = 3.81 mm, t’ = 22.6 m/set.
Substituting eqn (2) into eqn (l), transforms hybrid functional into the form
I& = 1
n
the H =
P%Pda, I ‘),
M =
pNTNda. s 1”
G, =
RTLds, s I+,
[BTGq - p’H/I/2 - 4’A44/2
G=
+ a’(Giq - Gq) - q’Qldt,
(3)
PTBdv, I v1,
n Gz = 1RTNds,
c Q = 1LTTds.
J
where
il
DISTANCE
FROM
IMPACT
I,
(mm)
0.00
G-l.00 d 5 3
-1.50
40 60 )cI*y 80 Rc*Ma?OO hb&.H120 140 160
Fig. 4. Transverse deflection along the I-axis under Hertzian impact for time from 40 to ZOOpsec, r = 22.6 misec.
(4)
W. J. Liou et DISTANCE
al.
FROM
IMPACT
(mm)
-
40
60 80
v ~100 ~120
-1.80
Fig. 5. Transverse deflection along
The variation p yields
of functional
G,q -
the y-axis under Hertzian L’= 22.6 m/set.
fIR with respect to GIand
G>q = 0,
Gq - HP = 0
impact
for time from
40 to 200 psec,
where
(5)
K = G’H-‘G,
M =
pwNdc.
(8)
s
1I,
and G, = Gz,
/I = H-IGq.
(6)
A substitution of eqn (6) into eqn (3) reduces functional fI, to the form
TIR = C [qTKq/2 - dTMh/2 - q’Q]dt, n $
DlSfANCE
Taking the variation to q yields
the
(7)
of fIR of eqn (7) with respect
Kq+Mij=Q.
Equation (9) is the equation of motion for the finite element method, where K, M and Q are the element stiffness matrix, mass matrix, and the load vector. respectively. FROM
IMPACT
(mm)
1
1001
0
4I
1t \
-100 .x
-200
20
.; 40 ~***y 60 A.& 80 zLU._G 100
-300
-600
(9)
’
Fig. 6. Normal stress along the x-axis under Hertzian impact for time from 20 to 100 psec, v = 22.6 mjsec.
Laminated E-glass epoxy composite plates DISTANCE
FROM
IMPACT
(mm)
‘Oc
20
.D
I X :
., AY~I**
-20
C&_&U 100
d 5 b”
40 80 fjo
-30
Fig. 7. Transverse normal stress along the x-axis under Hertzian impact for time from 20 to 100 psec, c = 22.6 mjsec.
IMPACT
MECHANISM AND LAMINATE CHARACTERISTICS
Scotchply 1002 E-glass epoxy with material properties for each lamina,
In the present investigation, a blunt-ended steel circular cylinder with 2.54 cm (1 in) in length and 0.9525 cm (3/8 in) in diameter is used as a projectile. The mass of the projectile is 14.175 gm (l/2 ounce). The target plate is a square E-glass epoxy laminated composite plate with a side length of 14 cm (5.5 in). The target plate is clamped around the four edges. The projectile dimensions and mass, and plate dimensions and boundary conditions are the same as those used in the experimental works conducted by Takeda [8]. The material used in the present investigation is the
DISTANCE
the following
& = 40.0 GPa ET = 8.27 GPa GLT= 4.13 GPa Grr = 3.03 GPa VLT= vm = 0.26 p = 1.90 g cm-‘,
FROM
IMPACT
(mm)
Fig. 8. Shear stress along the x-axis under Hertzian impact for time from 20 to 100 psec, v = 22.6 m/set.
(10)
W. J. Liou et al. DISTANCE
FROM
IMPACT
(mm)
0
7
x2
-50
3 d>; -100 Y
-
20 40 60 w 80 $+-&-es 100
b
-150
Fig. 9. Normal stress along the .r-axis under Hertzian impact for time from 20 to 100 psec, v = 22.6 mjsec where p is the density of the lamina, with units of mass per unit volume. Due to the laminate geometry, boundary conditions, and loading being symmetric with respect to the axes through the center of the laminate, it is sufficient to analyze only one quarter of the laminate. The clamped boundary conditions for the present model are shown in Fig. 1, which is shown for a quadrant of the laminate and the shaded area is the impact area.
projectile, only the velocity of the projectile can be observed. The actual force distribution between the projectile and the laminate during impact is unknown. An elastic impact, including indentation of the laminate surface by the projectile, is used to simulate the force distributions of the projectile on the laminated plate. The force between the projectile and the laminate during impact is assumed to be governed by the Hertzian theory in the form of F=H(r-w)p,
HERTZIAN
IMPACT
LAW AND FORCE DISTRIBUTION IMPACTED PLATE
In the present hybrid stress finite element model, equivalent nodal forces are required in the computation. However, when a laminate is impacted by a DISTANCE
(11)
ON
where F is the applied force of the projectile on the laminate during impact, His a material constant, and is set equal to 100,000,000 N m-‘,5, r is the displacement of the projectile, w is the displacement
FROM
IMPACT
(mm)
Fig. 10. Shear stress along the y-axis under Hertzian impact for time from 20 to 100 ,usec, ZJ= 22.6 m/set.
Laminated E-glass epoxy composite plates DlsTANCE
FROM
IMPACT
(mm)
200 g
100 0
r d s -100 b” -200 -300
.!!A
400
Fig. Il. Normal stress along the x-axis under Hertzian impact for time from 100 to 400 psec, v = 22.6 m/set.
at the point of impact, i.e. the transverse displacement of the center of the laminate, and p is an exponent and is equal to 1.5. Both displacements are measured from the surface of the undeformed laminate. The Hertzian theory of impact was discussed by Goldsmith [2]. Sun et al. [7] proposed a simple static indentation test to establish the contact law for the round steel ball and SMC panel. They found that a Hertzian contact law with p = 1.5 was a good approximation for the loading-indentation relationship. The Hertzian contact law was used successfully by Petersen [15] in the investigation of transient response of laminated plates subject to dynamic impact loading. In his study, p was set equal to 1.5, and the corresponding value of H was set equal to DISTANCE
100,000,000 N m-Is, which were found to be appropriate. The contact behavior is mathematically modeled and incorporated into the hybrid stress finite element program to perform the dynamic analysis of a square laminated plate subject to impact of a steel circular cylinder. To solve the dynamic impact problem, a three-dimensional hybrid stress isoparametric element is used in conjunction with the Newmark direct integration method [ 161. The procedures of the Newmark direct integration method have been documented in Ref. [ 161.After the establishments of the hybrid stress finite element model and the Hertzian impact law, the transient response of the E-glass epoxy laminated composite plate impacted by a steel circular cylinder projectile is investigated. The central transverse
FROM
IMPACT
(mm)
50
40 1
Fig. 12. Shear stress along the x-axis under Hertzian impact for time from 100 to 400 psec, cl = 22.6 m/set.
W. J. Liou
TIME
et al.
-
Y
=
20
-
” v
= =
40 60
m/xc m/set m/set
(microseconds)
Fig. 13. Transverse deflections of the center vs time for a three-layer laminate under Hertzian impact with three projectile velocities, h = 3.81 mm.
and stress distributions as functions of time are presented in the next section.
deflection
EGLASS
EPOXY LAMINATES SUBJECT LOCAL IMPACT
TO A
In this section, the central transverse deflection and stress distributions of an E-glass epoxy cross-ply laminate subject to impact loading are examined. The material properties of E-glass epoxy are given in eqn (10). Equal thickness is assumed in each layer. The laminate and projectile dimensions are as described in the previous section. The boundary conditions of the finite element model are given in Fig. 1. The force distribution between the projectile DISTANCE
Od
-650
and the laminated plate during impact is governed by the Hertzian impact law, which allows the projectile to indent the laminate and rebound. Figure 2 contains plots of the central transverse deflection of a three-layer cross-ply (O/90/0) E-glass epoxy laminate impacted by a circular cylinder projectile, as a function of time determined experimentally and by the hybrid stress finite element method. The laminate thickness is 4.29 mm and the projectile velocity is equal to 22.6 m s-l. The deflection predicted by the hybrid stress finite element method agrees with the deflection determined experimentally for time interval less than 200 ps. As time increases, the computed and experimental deflections do not agree closely, because the FROM
IMPACT
(mm)
*x
20
.z ***y* AU.
40 60 80
J
Fig. 14. Normal stress along the x-axis under Hertzian impact for time from 20 to 100 psec, v = 40 m/set.
Laminated
E-glass epoxy composite
DISTANCE
-30
,FROM
IMPACT
plates (mm)
’
Fig. 15. Shear stress along the -y-axis under Hertzian
delamination failure is initiated at the interface of the laminate and the material stiffness changes after the delamination failure occurs. The other reason is that the boundary conditions for the experimental work are not truly clamped around the four edges. The metal frame which holds the impacted laminate will allow the laminate to move in the horizontal direction and rotate a small amount. These effects cause the effective side length of the laminated plate to be larger than the distance between the supports. In the following investigations, effort will be concentrated on the numerical analysis. A perfect clamped boundary condition is assumed around the edges, and the side length will be set equal to 140 mm in the computation. The central transverse deflection as a function of time of a three-layer cross-ply (O/90/0) E-glass epoxy laminate impacted by a projectile is shown in Fig. 3, where the laminate thickness is equal to 3.81 mm and the projectile velocity is equal to 22.6 m SK’. After impact, the center point deflects from zero to a maximum about 1.7 mm at 300 ps. As time increases, the plate starts to rebound. The period of the transient response is longer than Fig. 2, where the thickness is equal to 4.29 mm, it indicated that the thickness of the laminate has an effect on the period of the transient response. The applied force between the projectile and the laminate computed from the Hertzian impact law is also included in Fig. 3. The first impact occurs from 0 to approximately 260 pts, at which time, the projectile loses contact with the laminate. The projectile recontacts the laminate at approximately 530 ps. These observations agree with the results of experimental works. It is also noticed that for a higher velocity of the projectile, a longer contact time is observed. The transverse deflections along the x-axis and the y-axis of the same laminate under Hertzian impact
impact
for time from 20 to 100 psec. L’= 40 mjsec
for time from 40-200 ps are shown in Figs 4 and 5. The results show the flexural wave motion along the x-axis and the v-axis from the center to the edge of the laminate. Since EL is greater than ET, the bending stiffness Dll is greater than Dzz for a three-layer cross-ply (O/90/0) laminate. This difference in bending stiffness causes the flexural wave to move faster in the x-direction than in the y-direction. For example, observe the deflection at time equal to 80~s the distance from the center to the nearest point of zero deflection is 36 mm in the x-direction and 28 mm in the y-direction. At a particular time, the transverse deflection is larger at a specified point along the r-axis than at a point at the same distance along the y-axis. The difference between them is due to the anisotropy exhibited by a three-layer cross-ply laminate. The normal stress, CT,,transverse normal stress, u, and transverse shear stress, r.,:, along the .u-axis in the three-layer cross-ply laminate are shown in Figs 68 for time from 20 to 100 ps. The normal stress, 0, , which is evaluated at the upper surface of the 90’ layer (2/3 laminate thickness above the bottom of the laminate) and the transverse shear stress, T,,, along the y-axis for time from 20 to 100 ps are shown in Figs 9 and 10. As the observation for the transverse deflection, in which the flexural wave moves faster in the .u-direction than in the y-direction, the stress wave is also seen to be moving in a similar fashion. The maximum value of the normal stress, or, along the x-axis is greater than the maximum value of the normal stress, o,, along the y-axis. The maximum value of the transverse shear stress, t,:, at the outer edge of the impact zone on the x-axis is slightly higher than the maximum value of the transverse shear stress, T,, , on the y-axis. The transverse normal stress (Fig. 7), al, which is evaluated at the top surface of the laminate is dependent on the contact
IO
W. J. Liou et al.
force computed from the Hertzian impact law (Fig. 3). Outside the impact region, the transverse normal stress approaches to zero. Figures 11 and 12 contain the normal stress, CJ,, and the transverse shear stress, r,,, along the x-axis for time from 100 to 400 ps. It is observed that the transverse shear stress near the impact region is much greater for time equal to 100 ps than for time equal to 300 ps. This fact is because at 300 ps the projectile has lost contact with the laminate and no force occurs between them, as shown in Fig. 3. It indicated that the transverse shear stress is dependent on the history of the contact force. Figure 13 contains plots of the central transverse deflection vs time for a three-layer cross-ply (O/90/0) laminate under three different projectile velocities, 20, 40 and 60 m ss’, respectively. The central transverse deflection increases with increasing projectile velocity and is almost proportional to the projectile velocity, but the period of the dynamic response remains the same. It is seen that the projectile velocity does not have an effect on the period of laminate response. The normal stress, rr, , and the transverse shear stress, TV:, along the x-axis in the three-layer laminate with projectile velocity equal to 40 m s-’ are shown in Figs 14 and 15 for time from 20 to 100 ps. As expected, stress increases with increasing projectile velocity. At a particular time, the distance from the center to the nearest point of the laminate, where the stress is zero, is the same as in the case for projectile velocity equal to 22.6 m ss’ (Figs 6 and 8). It is clear that flexural wave velocities are only slightly affected by the projectile velocity for time less than 100 ~LSafter impact. CONCLUSIONS
The deflection and stress distributions of a laminated E-glass epoxy composite plate subject to the impact of a steel circular cylinder projectile have been investigated and presented graphically. The contact force between the projectile and the laminated plate during impact was modeled by the Hertzian impact law. The contact force distribution was incorporated into the hybrid stress finite element program to perform the analysis. The central transverse deflection of the laminated plate predicted by the finite element method agreed with the deflection determined experimentally. The force distribution between the projectile and the laminated plate, computed from the Hertzian impact law, showed that the projectile impacted the laminated plate, lost contact, and then recontacted, which agreed with the experimental observations. The stress distributions along the x-axis and the y-axis of a laminated plate when impacted by a projectile showed evidence of flexural wave motion. The flexural wave started from the center of impact and propagated outwardly. For an anisotropic laminated plate, the flexural wave propagated faster
in the direction in which the bending stiffness is the greatest. Higher stresses also existed in the direction having higher stiffness. The transverse normal stress and shear stresses were dependent on the history of the contact force. After the projectile separated from the laminated plate these stresses decreased significantly. For a thicker laminated plate, the central transverse deflection and the period of the dynamic response were smaller. Laminate thickness also had an effect on the flexural wave velocity. The flexural wave propagated faster in the thicker laminate than in the thinner laminate. The deflections and stresses were proportional to the projectile velocity. It was also noticed that the projectile velocity had little effect on the velocity of the flexural wave for a time less than 100 ps. The deflection and stress analysis for the present impact problems was based on the assumption of linear elasticity and the damping in the laminated plate being neglected. It was assumed that no failures will occur within the structure. However, experimental studies of impact by Takeda et al. [lo] showed that delamination usually existed at the interface between layers. They observed a delamination crack beginning near the point of impact and propagating outwardly. It is seen that the flexural wave, which causes the stresses to propagate, may be the major reason for the delamination propagation. The present hybrid stress finite element model is capable of providing predictions of the interlaminar stresses which actually cause delamination cracks at the interfaces [17]. It is hoped that the interlaminar stresses can be incorporated into a suitable failure criterion to examine the failure envelope and investigate the initiation of the delamination crack and the characteristics of the delamination mechanism at a particular interface.
REFERENCES On the solution I. E. T. Moyer Jr and E. Ghashghai-Abdi, of problems involving impact type loading. In Advances in Aerospace Science and Engineering; ASME Winter Annual Meeting, New Orleans, LA, pp. 39-48 (1984). Impact, the Theory and Physical 2. W. Goldsmith, Behavior of, Colliding Solids. Edward Arnold, London (1960). of flexural waves in an 3. T. S. Chow, On the propagation orthotropic laminated -plate and its response to an impulsive load. J. Comput. Mater. 5, 306319 (1971). in 4. J. M. Whitney and N. J. Pagano, Shear deformation heterogeneous anisotropic plates. J. appl. Mech. Trans. ASME 92, 1031-1036 (1970). materials due 5. W. T. Chen, Stresses in some anisotropic to indentation and sliding. Int. J. Solids Struct. 5, 191-214 (1969). and H. Chao, Impact damage in 6. L. B. Greszczuk Composite composites. graphite-fiber-reinforced materials: testing and design. ASTM STP 617, 389408 (1977). 1. C. T. Sun, B. V. Sankar and T. M. Tan, Dynamic response of SMC to impact of a steel ball. In: Advances
Laminated E-glass epoxy composite plates in Aerospace Structures and Materials; ASME Winter Annual Meeting, Washington, DC, pp. 99-106 (1981). 8. N. Takeda, Experimental studies of the delamination mechanisms in impacted fiber-reinforced composite plates. Ph.D. dissertation, Department of Engineering Sciences, University of Florida (1980). 9. N. Takeda, R. L: Sierakowski and L. E. Malvern, Studies of impacted glass fiber reinforced composite laminates. SAMPE Q. 12, 9-12 (1981). 10. N. Takeda, R. L. Sierakowski, C. A. Ross and L. E. Malvern, Delamination-crack propagation in ballistically impacted glass/epoxy composite laminates. Exp. Mech.
22, 19-25 (1982).
II. P. Tong, S. T. Mau and T. H. H. Pian, Derivation of geometric stiffness and mass matrices for finite element hybrid models. Int. J. Solids Struct. 10,919-932 (1974). 12. R. L. Spilker, A hybrid-stress finite element formulation for thick multilayer laminates. Comput. Struct. 11, 507-514 (1980).
II
13. W. J. Liou and C. T. Sun, A three-dimensional hybrid stress isoparametric element for the analysis of laminated composite plates. Comput. Struct 25, 241-249
(1987).
14. C. T. Sun and W. J. Liou, A three-dimensional hybrid stress finite element formulation for free vibrations of laminated composite plates. J. Sound Vibr. 119, l-14 (1987).
15. B. R. Petersen, Finite element analysis of composite plate impacted by a projectile. Ph.D. dissertation, Department of Engineering Sciences, University of Florida ( 1985). 16. N. M. Newmark, A method of computation for structural dynamics. J. Engng Mech. Div. ASCE 85, 67-94 (1959).
17. N. J. Pagan0 and R. B. Pipes, Some observations on the interlaminar strength of composite laminates. ht. J. mech. Sci. 15, 679-688 (1973).
Compurers & Structures Vol. 61. No. I. pp. I-1 I. 1996 Copyright 0 1996 Ekvicr Science Ltd Printed-in &at Britain. All rights reserved 0045-7949/96 $15.00 + 0.00
STRESS ANALYSIS OF LAMINATED E-GLASS EPOXY COMPOSITE PLATES SUBJECT TO IMPACT DYNAMIC LOADING W. J. Liou, C. I. Tseng and L. P. Chao Department of Mechanical Engineering, Feng-Chia University, Taichung, Taiwan, Republic of China (Received
1I September 1995)
AbstracG-The transient response of an E-glass epoxy laminated composite plate impacted by a steel circular cylinder is analyzed. The analysis is performed by using a three-dimensional hybrid stress finite element program. The contact force between the projectile and the laminated plate is modeled by the Hertzian impact law. The predicted transverse deflection at the center of the laminated plate agrees with the deflection determined experimentally. The effects of projectile velocity and thickness of the laminated nlate on the imoacted stress distributions and the central transverse deflection are also investigated. copyright 0 1696 Elsevier Science Ltd.
using Hertzian contact theory. Sun et al. [7] established a contact law for a composite plate and a steel ball. The contact behavior was incorporated into a finite element program to analyze the wave propagation phenomenon. Concerning the plate impact experiments, a systematic experimental study of failure mechanisms on impact of laminated plates was conducted by Takeda [8]. In the experiments, a small steel cylinder was fired from a gas gun at subperforation velocities. It indicated that the flexural wave propagation was the principal response mode of moderately thin and thick plates subject to central local impact, and it led to delamination failure propagation. The effect of impactor mass and velocity, and ply orientation on laminated plate delamination was investigated by Takeda et al. [9, lo]. In the present investigation, a hybrid stress finite element program is applied to predict the displacement and stress distributions in an E-glass epoxy laminated composite plate subject to a local impact loading. The accuracy and versatility of the hybrid stress finite element method have been validated by many investigators [ 1I-141, they have proven that the hybrid stress finite element model is a useful tool in attacking complicated structure analysis problems. The detailed formulation of this model can be found in these references. The analysis herein is concerned only with the time that the laminated plate remains elastic, and that delamination has not yet occurred. The damping in the laminated plate is assumed to be negligible. The stresses and central transverse deflections are presented to show the transient response of the laminated plate when impacted by a projectile. Knowledge of where and when the stresses are maximum should be helpful in determining where,
INTRODUCTION
During the past three decades, the applications of composite materials have grown rapidly and are now very commonplace around the world. Industries such as aircraft, automobiles, sporting goods, electronics, and appliances are quite dependent on fiberreinforced composites. In the contemporary technology of structural composite materials, major deficiencies still exist with respect to the ability to determine the stress field within a multilayered composite laminate, especially for the impact loading. The transient response of laminated plates subject to local impact loadings has been of significant concern in many advanced engineering structures and components. For example, the leading edge of an aircraft wing or blades in a jet engine may be hit by foreign objects such as stones, nuts, bolts or birds. In these cases the impact resistance of laminated plates must be known in order to design the laminated plates which meet aircraft safety requirements. Impact problems with energies far below the penetration levels can cause significant damage to components made of composite materials, since delamination often occurs at energy levels which are an order of magnitude below those causing penetration [l]. The analysis of the plate impact problems have been concerned for a long time history. Goldsmith [2] provided a detailed discussion for impact on isotropic beams and plates. Chow [3] investigated the transient response of a rectangular laminated plate subject to a concentrated impact loading by using shear deformation theory [4]. Chen [S] applied a Hertzian contact law to analyze the response of impact on anisotropic materials. The dynamic response of laminates impacted by spherical impactors was discussed by Greszczuk and Chao [6] 1
2
W. J. Liou et al.
Y*V
I-
&=t: 7omm
---i
T
u=v=w=o
U=O
(aTe - aTSo/2 - pB2/2)dv
n
U=W=W=O
+
?v,,
T+(i - ufds-
s
7omm
w=o
Fig. 1.Clamped boundary conditions for a quadrant of the laminate.
when and how the delamination failure occurs. With the information of the laminated plate behavior, the structure members can be constructed to meet the specific design requirements.
FORMULATION
Based on the development of Tong et aE. (111, the hybrid functional for multilayered elements in dynamic problems without body force and initial stress can be defined in the matrix form as
T%ds dt,
s
(1)
where superscript T denotes the matrix transpose, u is the stress vector, e is the strain vector computed from the displacement vector u, S is the transformed compliance matrix, T is the vector of surface tractions, t is the vector of prescribed surface tractions, rl is the vector of boundary displacements and p is the density of the laminate. un is the volume of nth element having boundary au,, and su,, is the surface of nth element over which traction is prescribed. The stress, strain, and displacement fields are independently assumed as functions of unknown parameters a, and /3 and nodal displacements q, respectively,
Pfh T = Ru,
TV=
ii=Lq, HYBRID STRESS FINITE ELEMENT
3%I
u =Nq,
u= Ng,
e=Du=DiVq=Bq,
(2)
where the stress vector, a, satisfies the homogeneous equilibrium equations and the detailed development of the stress field can be found in Ref. [13]. P, R and L are matrices consisting of polynomial terms of coordinates (x, y, z). N is the shape function in terms of the natural coordinates (5, n, <).
2-l-----
*
Experimental
-2
TIME (microseconds) Fig. 2. Transverse deflections of the center vs time for a three-layer laminate determined e~perimentaily and by using hybrid stress finite element method, h = 4.29 mm, u = 22.6 m/see.
3
Laminated E-glass epoxy composite plates
TIME
(microseconds)
Fig. 3. Transverse deflection of the center vs time for a three-layer laminate under Hertzian impact, h = 3.81 mm, t’ = 22.6 m/set.
Substituting eqn (2) into eqn (l), transforms hybrid functional into the form
I& = 1
n
the H =
P%Pda, I ‘),
M =
pNTNda. s 1”
G, =
RTLds, s I+,
[BTGq - p’H/I/2 - 4’A44/2
G=
+ a’(Giq - Gq) - q’Qldt,
(3)
PTBdv, I v1,
n Gz = 1RTNds,
c Q = 1LTTds.
J
where
il
DISTANCE
FROM
IMPACT
I,
(mm)
0.00
G-l.00 d 5 3
-1.50
40 60 )cI*y 80 Rc*Ma?OO hb&.H120 140 160
Fig. 4. Transverse deflection along the I-axis under Hertzian impact for time from 40 to ZOOpsec, r = 22.6 misec.
(4)
W. J. Liou et DISTANCE
al.
FROM
IMPACT
(mm)
-
40
60 80
v ~100 ~120
-1.80
Fig. 5. Transverse deflection along
The variation p yields
of functional
G,q -
the y-axis under Hertzian L’= 22.6 m/set.
fIR with respect to GIand
G>q = 0,
Gq - HP = 0
impact
for time from
40 to 200 psec,
where
(5)
K = G’H-‘G,
M =
pwNdc.
(8)
s
1I,
and G, = Gz,
/I = H-IGq.
(6)
A substitution of eqn (6) into eqn (3) reduces functional fI, to the form
TIR = C [qTKq/2 - dTMh/2 - q’Q]dt, n $
DlSfANCE
Taking the variation to q yields
the
(7)
of fIR of eqn (7) with respect
Kq+Mij=Q.
Equation (9) is the equation of motion for the finite element method, where K, M and Q are the element stiffness matrix, mass matrix, and the load vector. respectively. FROM
IMPACT
(mm)
1
1001
0
4I
1t \
-100 .x
-200
20
.; 40 ~***y 60 A.& 80 zLU._G 100
-300
-600
(9)
’
Fig. 6. Normal stress along the x-axis under Hertzian impact for time from 20 to 100 psec, v = 22.6 mjsec.
Laminated E-glass epoxy composite plates DISTANCE
FROM
IMPACT
(mm)
‘Oc
20
.D
I X :
., AY~I**
-20
C&_&U 100
d 5 b”
40 80 fjo
-30
Fig. 7. Transverse normal stress along the x-axis under Hertzian impact for time from 20 to 100 psec, c = 22.6 mjsec.
IMPACT
MECHANISM AND LAMINATE CHARACTERISTICS
Scotchply 1002 E-glass epoxy with material properties for each lamina,
In the present investigation, a blunt-ended steel circular cylinder with 2.54 cm (1 in) in length and 0.9525 cm (3/8 in) in diameter is used as a projectile. The mass of the projectile is 14.175 gm (l/2 ounce). The target plate is a square E-glass epoxy laminated composite plate with a side length of 14 cm (5.5 in). The target plate is clamped around the four edges. The projectile dimensions and mass, and plate dimensions and boundary conditions are the same as those used in the experimental works conducted by Takeda [8]. The material used in the present investigation is the
DISTANCE
the following
& = 40.0 GPa ET = 8.27 GPa GLT= 4.13 GPa Grr = 3.03 GPa VLT= vm = 0.26 p = 1.90 g cm-‘,
FROM
IMPACT
(mm)
Fig. 8. Shear stress along the x-axis under Hertzian impact for time from 20 to 100 psec, v = 22.6 m/set.
(10)
W. J. Liou et al. DISTANCE
FROM
IMPACT
(mm)
0
7
x2
-50
3 d>; -100 Y
-
20 40 60 w 80 $+-&-es 100
b
-150
Fig. 9. Normal stress along the .r-axis under Hertzian impact for time from 20 to 100 psec, v = 22.6 mjsec where p is the density of the lamina, with units of mass per unit volume. Due to the laminate geometry, boundary conditions, and loading being symmetric with respect to the axes through the center of the laminate, it is sufficient to analyze only one quarter of the laminate. The clamped boundary conditions for the present model are shown in Fig. 1, which is shown for a quadrant of the laminate and the shaded area is the impact area.
projectile, only the velocity of the projectile can be observed. The actual force distribution between the projectile and the laminate during impact is unknown. An elastic impact, including indentation of the laminate surface by the projectile, is used to simulate the force distributions of the projectile on the laminated plate. The force between the projectile and the laminate during impact is assumed to be governed by the Hertzian theory in the form of F=H(r-w)p,
HERTZIAN
IMPACT
LAW AND FORCE DISTRIBUTION IMPACTED PLATE
In the present hybrid stress finite element model, equivalent nodal forces are required in the computation. However, when a laminate is impacted by a DISTANCE
(11)
ON
where F is the applied force of the projectile on the laminate during impact, His a material constant, and is set equal to 100,000,000 N m-‘,5, r is the displacement of the projectile, w is the displacement
FROM
IMPACT
(mm)
Fig. 10. Shear stress along the y-axis under Hertzian impact for time from 20 to 100 ,usec, ZJ= 22.6 m/set.
Laminated E-glass epoxy composite plates DlsTANCE
FROM
IMPACT
(mm)
200 g
100 0
r d s -100 b” -200 -300
.!!A
400
Fig. Il. Normal stress along the x-axis under Hertzian impact for time from 100 to 400 psec, v = 22.6 m/set.
at the point of impact, i.e. the transverse displacement of the center of the laminate, and p is an exponent and is equal to 1.5. Both displacements are measured from the surface of the undeformed laminate. The Hertzian theory of impact was discussed by Goldsmith [2]. Sun et al. [7] proposed a simple static indentation test to establish the contact law for the round steel ball and SMC panel. They found that a Hertzian contact law with p = 1.5 was a good approximation for the loading-indentation relationship. The Hertzian contact law was used successfully by Petersen [15] in the investigation of transient response of laminated plates subject to dynamic impact loading. In his study, p was set equal to 1.5, and the corresponding value of H was set equal to DISTANCE
100,000,000 N m-Is, which were found to be appropriate. The contact behavior is mathematically modeled and incorporated into the hybrid stress finite element program to perform the dynamic analysis of a square laminated plate subject to impact of a steel circular cylinder. To solve the dynamic impact problem, a three-dimensional hybrid stress isoparametric element is used in conjunction with the Newmark direct integration method [ 161. The procedures of the Newmark direct integration method have been documented in Ref. [ 161.After the establishments of the hybrid stress finite element model and the Hertzian impact law, the transient response of the E-glass epoxy laminated composite plate impacted by a steel circular cylinder projectile is investigated. The central transverse
FROM
IMPACT
(mm)
50
40 1
Fig. 12. Shear stress along the x-axis under Hertzian impact for time from 100 to 400 psec, cl = 22.6 m/set.
W. J. Liou
TIME
et al.
-
Y
=
20
-
” v
= =
40 60
m/xc m/set m/set
(microseconds)
Fig. 13. Transverse deflections of the center vs time for a three-layer laminate under Hertzian impact with three projectile velocities, h = 3.81 mm.
and stress distributions as functions of time are presented in the next section.
deflection
EGLASS
EPOXY LAMINATES SUBJECT LOCAL IMPACT
TO A
In this section, the central transverse deflection and stress distributions of an E-glass epoxy cross-ply laminate subject to impact loading are examined. The material properties of E-glass epoxy are given in eqn (10). Equal thickness is assumed in each layer. The laminate and projectile dimensions are as described in the previous section. The boundary conditions of the finite element model are given in Fig. 1. The force distribution between the projectile DISTANCE
Od
-650
and the laminated plate during impact is governed by the Hertzian impact law, which allows the projectile to indent the laminate and rebound. Figure 2 contains plots of the central transverse deflection of a three-layer cross-ply (O/90/0) E-glass epoxy laminate impacted by a circular cylinder projectile, as a function of time determined experimentally and by the hybrid stress finite element method. The laminate thickness is 4.29 mm and the projectile velocity is equal to 22.6 m s-l. The deflection predicted by the hybrid stress finite element method agrees with the deflection determined experimentally for time interval less than 200 ps. As time increases, the computed and experimental deflections do not agree closely, because the FROM
IMPACT
(mm)
*x
20
.z ***y* AU.
40 60 80
J
Fig. 14. Normal stress along the x-axis under Hertzian impact for time from 20 to 100 psec, v = 40 m/set.
Laminated
E-glass epoxy composite
DISTANCE
-30
,FROM
IMPACT
plates (mm)
’
Fig. 15. Shear stress along the -y-axis under Hertzian
delamination failure is initiated at the interface of the laminate and the material stiffness changes after the delamination failure occurs. The other reason is that the boundary conditions for the experimental work are not truly clamped around the four edges. The metal frame which holds the impacted laminate will allow the laminate to move in the horizontal direction and rotate a small amount. These effects cause the effective side length of the laminated plate to be larger than the distance between the supports. In the following investigations, effort will be concentrated on the numerical analysis. A perfect clamped boundary condition is assumed around the edges, and the side length will be set equal to 140 mm in the computation. The central transverse deflection as a function of time of a three-layer cross-ply (O/90/0) E-glass epoxy laminate impacted by a projectile is shown in Fig. 3, where the laminate thickness is equal to 3.81 mm and the projectile velocity is equal to 22.6 m SK’. After impact, the center point deflects from zero to a maximum about 1.7 mm at 300 ps. As time increases, the plate starts to rebound. The period of the transient response is longer than Fig. 2, where the thickness is equal to 4.29 mm, it indicated that the thickness of the laminate has an effect on the period of the transient response. The applied force between the projectile and the laminate computed from the Hertzian impact law is also included in Fig. 3. The first impact occurs from 0 to approximately 260 pts, at which time, the projectile loses contact with the laminate. The projectile recontacts the laminate at approximately 530 ps. These observations agree with the results of experimental works. It is also noticed that for a higher velocity of the projectile, a longer contact time is observed. The transverse deflections along the x-axis and the y-axis of the same laminate under Hertzian impact
impact
for time from 20 to 100 psec. L’= 40 mjsec
for time from 40-200 ps are shown in Figs 4 and 5. The results show the flexural wave motion along the x-axis and the v-axis from the center to the edge of the laminate. Since EL is greater than ET, the bending stiffness Dll is greater than Dzz for a three-layer cross-ply (O/90/0) laminate. This difference in bending stiffness causes the flexural wave to move faster in the x-direction than in the y-direction. For example, observe the deflection at time equal to 80~s the distance from the center to the nearest point of zero deflection is 36 mm in the x-direction and 28 mm in the y-direction. At a particular time, the transverse deflection is larger at a specified point along the r-axis than at a point at the same distance along the y-axis. The difference between them is due to the anisotropy exhibited by a three-layer cross-ply laminate. The normal stress, CT,,transverse normal stress, u, and transverse shear stress, r.,:, along the .u-axis in the three-layer cross-ply laminate are shown in Figs 68 for time from 20 to 100 ps. The normal stress, 0, , which is evaluated at the upper surface of the 90’ layer (2/3 laminate thickness above the bottom of the laminate) and the transverse shear stress, T,,, along the y-axis for time from 20 to 100 ps are shown in Figs 9 and 10. As the observation for the transverse deflection, in which the flexural wave moves faster in the .u-direction than in the y-direction, the stress wave is also seen to be moving in a similar fashion. The maximum value of the normal stress, or, along the x-axis is greater than the maximum value of the normal stress, o,, along the y-axis. The maximum value of the transverse shear stress, t,:, at the outer edge of the impact zone on the x-axis is slightly higher than the maximum value of the transverse shear stress, T,, , on the y-axis. The transverse normal stress (Fig. 7), al, which is evaluated at the top surface of the laminate is dependent on the contact
IO
W. J. Liou et al.
force computed from the Hertzian impact law (Fig. 3). Outside the impact region, the transverse normal stress approaches to zero. Figures 11 and 12 contain the normal stress, CJ,, and the transverse shear stress, r,,, along the x-axis for time from 100 to 400 ps. It is observed that the transverse shear stress near the impact region is much greater for time equal to 100 ps than for time equal to 300 ps. This fact is because at 300 ps the projectile has lost contact with the laminate and no force occurs between them, as shown in Fig. 3. It indicated that the transverse shear stress is dependent on the history of the contact force. Figure 13 contains plots of the central transverse deflection vs time for a three-layer cross-ply (O/90/0) laminate under three different projectile velocities, 20, 40 and 60 m ss’, respectively. The central transverse deflection increases with increasing projectile velocity and is almost proportional to the projectile velocity, but the period of the dynamic response remains the same. It is seen that the projectile velocity does not have an effect on the period of laminate response. The normal stress, rr, , and the transverse shear stress, TV:, along the x-axis in the three-layer laminate with projectile velocity equal to 40 m s-’ are shown in Figs 14 and 15 for time from 20 to 100 ps. As expected, stress increases with increasing projectile velocity. At a particular time, the distance from the center to the nearest point of the laminate, where the stress is zero, is the same as in the case for projectile velocity equal to 22.6 m ss’ (Figs 6 and 8). It is clear that flexural wave velocities are only slightly affected by the projectile velocity for time less than 100 ~LSafter impact. CONCLUSIONS
The deflection and stress distributions of a laminated E-glass epoxy composite plate subject to the impact of a steel circular cylinder projectile have been investigated and presented graphically. The contact force between the projectile and the laminated plate during impact was modeled by the Hertzian impact law. The contact force distribution was incorporated into the hybrid stress finite element program to perform the analysis. The central transverse deflection of the laminated plate predicted by the finite element method agreed with the deflection determined experimentally. The force distribution between the projectile and the laminated plate, computed from the Hertzian impact law, showed that the projectile impacted the laminated plate, lost contact, and then recontacted, which agreed with the experimental observations. The stress distributions along the x-axis and the y-axis of a laminated plate when impacted by a projectile showed evidence of flexural wave motion. The flexural wave started from the center of impact and propagated outwardly. For an anisotropic laminated plate, the flexural wave propagated faster
in the direction in which the bending stiffness is the greatest. Higher stresses also existed in the direction having higher stiffness. The transverse normal stress and shear stresses were dependent on the history of the contact force. After the projectile separated from the laminated plate these stresses decreased significantly. For a thicker laminated plate, the central transverse deflection and the period of the dynamic response were smaller. Laminate thickness also had an effect on the flexural wave velocity. The flexural wave propagated faster in the thicker laminate than in the thinner laminate. The deflections and stresses were proportional to the projectile velocity. It was also noticed that the projectile velocity had little effect on the velocity of the flexural wave for a time less than 100 ps. The deflection and stress analysis for the present impact problems was based on the assumption of linear elasticity and the damping in the laminated plate being neglected. It was assumed that no failures will occur within the structure. However, experimental studies of impact by Takeda et al. [lo] showed that delamination usually existed at the interface between layers. They observed a delamination crack beginning near the point of impact and propagating outwardly. It is seen that the flexural wave, which causes the stresses to propagate, may be the major reason for the delamination propagation. The present hybrid stress finite element model is capable of providing predictions of the interlaminar stresses which actually cause delamination cracks at the interfaces [17]. It is hoped that the interlaminar stresses can be incorporated into a suitable failure criterion to examine the failure envelope and investigate the initiation of the delamination crack and the characteristics of the delamination mechanism at a particular interface.
REFERENCES On the solution I. E. T. Moyer Jr and E. Ghashghai-Abdi, of problems involving impact type loading. In Advances in Aerospace Science and Engineering; ASME Winter Annual Meeting, New Orleans, LA, pp. 39-48 (1984). Impact, the Theory and Physical 2. W. Goldsmith, Behavior of, Colliding Solids. Edward Arnold, London (1960). of flexural waves in an 3. T. S. Chow, On the propagation orthotropic laminated -plate and its response to an impulsive load. J. Comput. Mater. 5, 306319 (1971). in 4. J. M. Whitney and N. J. Pagano, Shear deformation heterogeneous anisotropic plates. J. appl. Mech. Trans. ASME 92, 1031-1036 (1970). materials due 5. W. T. Chen, Stresses in some anisotropic to indentation and sliding. Int. J. Solids Struct. 5, 191-214 (1969). and H. Chao, Impact damage in 6. L. B. Greszczuk Composite composites. graphite-fiber-reinforced materials: testing and design. ASTM STP 617, 389408 (1977). 1. C. T. Sun, B. V. Sankar and T. M. Tan, Dynamic response of SMC to impact of a steel ball. In: Advances
Laminated E-glass epoxy composite plates in Aerospace Structures and Materials; ASME Winter Annual Meeting, Washington, DC, pp. 99-106 (1981). 8. N. Takeda, Experimental studies of the delamination mechanisms in impacted fiber-reinforced composite plates. Ph.D. dissertation, Department of Engineering Sciences, University of Florida (1980). 9. N. Takeda, R. L: Sierakowski and L. E. Malvern, Studies of impacted glass fiber reinforced composite laminates. SAMPE Q. 12, 9-12 (1981). 10. N. Takeda, R. L. Sierakowski, C. A. Ross and L. E. Malvern, Delamination-crack propagation in ballistically impacted glass/epoxy composite laminates. Exp. Mech.
22, 19-25 (1982).
II. P. Tong, S. T. Mau and T. H. H. Pian, Derivation of geometric stiffness and mass matrices for finite element hybrid models. Int. J. Solids Struct. 10,919-932 (1974). 12. R. L. Spilker, A hybrid-stress finite element formulation for thick multilayer laminates. Comput. Struct. 11, 507-514 (1980).
II
13. W. J. Liou and C. T. Sun, A three-dimensional hybrid stress isoparametric element for the analysis of laminated composite plates. Comput. Struct 25, 241-249
(1987).
14. C. T. Sun and W. J. Liou, A three-dimensional hybrid stress finite element formulation for free vibrations of laminated composite plates. J. Sound Vibr. 119, l-14 (1987).
15. B. R. Petersen, Finite element analysis of composite plate impacted by a projectile. Ph.D. dissertation, Department of Engineering Sciences, University of Florida ( 1985). 16. N. M. Newmark, A method of computation for structural dynamics. J. Engng Mech. Div. ASCE 85, 67-94 (1959).
17. N. J. Pagan0 and R. B. Pipes, Some observations on the interlaminar strength of composite laminates. ht. J. mech. Sci. 15, 679-688 (1973).