- Email: [email protected]
Determining effective contact stiffness between striker and composite shell
COMPOSITE STRUCTURES
ELSEVIER
Composite Structures 43 (1998) 137-145
Determining
effective contact stiffness between striker and composite shell S.W. Gong*, V.P.W. Shim, S.L. Toh
Department of Mechanical and Production Engineering, National University of Singapore, Singapore 119260, Singapore
Abstract An approach to determining the effective contact stiffness between a striker and composite shell is presented. The effective contact stiffness is obtained via impact tests in conjunction with an analytic impact function recently presented by the authors. The relationship between the effective contact stiffness and the contact coefficients defined by Hertzian contact laws is described.
Impact tests are conducted on simply-supported glass/epoxy cylindrical and ogival shells. The effective contact stiffnesses for these shells impacted by a solid striker are obtained using the present method. In addition, effects of impact mass and impact velocity on the contact force are examined and discussed. 0 1998 Elsevier Science Ltd. All rights reserved.
1. Introduction
Fibre-reinforced laminated composites are becoming widely utilized as primary structural components in aerospace applications. In view of designing for resistance to foreign object impact damage, prediction of impact force between a striker and the target composite structure is necessary. The impact force is a result of contact deformation between a striker and the target structure and should therefore be evaluated on this basis. To estimate the contact force between a striker and the target structure during impact, Hertz’s contact law has been used andi modified by many researchers. Timoshenko [l] initiated the basic approach in this area by combining the Hertz contact force law with Bernoulli-Euler beam theory. Karas [2] extended this approach to the study of central impact on a rectangular plate and obtained the contact force. Lee [3] presented a modified Hertz method, which includes flexural deflection of the target structure in addition to contact deformation, to predict the contact force on a beam. Preston and Cook [4] and Greszczuk [5] employed the modified Hertz method to evaluate the contact force for a cantilever and circular plates, respectively. Sun [6] and Dobyns [7] did a similar analysis for a rectangular plate. *Corresponding author. Tel.: 00 65 7761 647; Fax: 00 65 7761 647 962; E-mail: [email protected]
The contact stiffness is a key parameter in prediction of impact force. Shivakumar, Elber and Illg [8] proposed a spring-mass model, which was an extension of Lee’s spring-mass model for impact on a beam. In their model, the impactor and plate were represented by two rigid masses and the associated deformation characteristics modelled by springs. They evaluated the contact stiffness following Conway and Angew’s [9] procedure which was restricted to simulation of a composite plate impacted by a spherical isotropic striker. Sun et al. [lO,ll] established empirical contact laws for glass/epoxy and graphite/epoxy laminates in contact with steel balls. In their study, static indentation tests were conducted on glass/epoxy and graphite/ epoxy composite laminates using a steel sphere as the indentor. This yielded the contact stiffness between the composite laminate and the steel ball. Lin and Lee [12] following Sun et al. conducted a similar study on composite laminated shells. All these contact stiffness estimations are based on static indentation tests. In this study, an approach to determination of the effective contact stiffness is proposed, whereby it is obtained via impact tests in conjunction with an analytic impact function recently presented by Gong et al. [13,14]. An impact test rig was designed and built to determine the effective contact stiffness and study the contact force. Impact tests were conducted on simplysupported glass/epoxy cylindrical and ogival shells. The effective contact stiffnesses between the striker and the
0263-8223/98/$ - see front matter 0 1998 Elsevier Science Ltd. AI1 rights reserved. PII: SO263-8223(98)OOlO2-0
S. W Gong et al.lComposite Structures 43 (1998) 137-145
138
shells were obtained, from which the procedure for this is demonstrated.
The contact force therefore can be expressed terms of the five parameters as follow [13]: K%z,(c, - l)sin(o,r)+a,(c,
in
- l)sin(w2t)]
F(t)= 0
2. Impact force model and effective contact stiffness O
Consider a simply-supported composite shell of arbitrary shape, with a constant thickness h, mass ml and density p. The shell is impacted by a solid striker of mass m2 at a velocity V. To determine the contact force between the striker and the shell during impact, a simple spring-mass model is used, as shown in Fig. 1. This is essentially a modification of Shivakumar’s [8] model. The present S-M model contains five parameters; they are the effective shell mass m$, the striker mass m2, the equivalent stiffness K1 of the simplysupported shell, the effective contact stiffness K3 and the initial striker velocity V. Let w(t) and w2(t) represent respectively the normal displacement of the load point on the shell and that of the striker at any time t during impact. The effective contact stiffness K$ is defined by the relation of contact force to the contact deformation:
t>T
In the force function, T is the contact duration and
0 Cl =
Kq-wfrn,
K3 c2 =
Kq--wirn;, V al= wdc2-cJ
(1)
F(t)= K%6(t)
V
or
a2=
(4) W2(CI
F(t) = K%[w~(Q
-w(t)1
-
c2)
(2) The effective shell mass Rayleigh’s method [ 151: mi=ph
rnt
is determined
by
W2(x, YW
where dS is the differential surface area of the shell while S is the total surface area and W(x, y) is a function of position that defines the shape of the natural mode of shell vibration. The equivalent stiffness of the simply-supported shell is determined from: Boundaries: Simply supported
Fig. 1. Spring-mass model.
K,=w:,rnT
(6)
where wll is the fundamental frequency of the target structure. Equation (6) is based on stimulation of only the fundamental mode of vibration of the shell, because early studies [3,16] have shown that impact durations are generally many times longer than the time taken for stress waves generated to travel to opposite boundaries of the plate and return. Furthermore, the effects of higher modes, especially when the plate undergoes large deflection, are small and can be neglected [8]. The parameters m2 and V can be determined directly from the mass and initial velocity of the striker. Except for the effective contact stiffness K$, the parameters rni, K,, m2 and V can be easily determined.
S.U! Gong et al.lComposite
The effective contact stiffness Kq depends on the material and geometrica. properties of the striker and target structure and is used to predict the contact force between a striker and shell during impact. Therefore, it is preferable that this stiffness be determined from impact tests. The procedure to determine effective contact stiffness is relatively simple. First, a shell is impacted by a pendulum striker at a given velocity. The contact force between the striker and shell during impact is captured by a force transducer attached to the striker. An impact force-time curve is obtained from the impact test, and the impact force function eqn (3) is used to fit the: experimental data. Except for the effective contact stiffness K% the other parameters in the impact force function eqn (3) are known; rnt and Kl are determined from eqn (5) and (6) and m2 and V are the mass and initial velocity of the striker in the impact test. A trial stiffness (KQ,, is assumed for an initial calculation of the impact force. By comparing calculated values of the maximum impact force F,, and contact duration 1” (i.e. when the striker is in contact with the shell) with those from the impact test, (KQ,, is appropriately adjusted to a second trial value (KQ2 and substituted back into the impact force function eqn (3). This process of comparison and modification is repeated, until the value of (Kt),, converges such that the difference between calculated and experimental values of maximum impact force F,,,, and contact duration T -fall within specified tolerances; i.e. the effective contact stiffness is: Kq = (KV,,
(7)
3. Relating effective contact stiffness to contact
Structures 43 (1998) 137-145
139
velocity of the striker. For a stationary target structure:
(11) For a target structure with simply-supported the following relationship exists: (&J, < (&Z),< (&J,,
(12)
where the subscripts fi, ss and st refer respectively to free, simply-supported and stationary structures. For p = 1.5 and ml > 10 nz2, the difference between (c?,,,),~ and (6,),, is less than 4%. Therefore, for a striker of small mass, the maximum contact deformation of a target structure with simply-supported edges can be estimated by eqn (11). Substitution of eqn (9) into eqn (8) and (1) yields respectively: P
(13) and F(t) = KqG,sin f
(14)
The respective impulses experienced during the impact duration T, corresponding to eqn (13) and (14) are:
(15)
10*K,(&sin G )”
coefficients defined by Hertzian contact laws
The contact coefficients K2 and p are defined by Hertzian theory of the form: F=(t) = KdW)lP
(8)
They can be obtained by static indentation tests [lo, 111. Greszczuk [5] has shown that contact deformation 6 can be approximated fairly accurately by:
edges,
and T
K@,sin F dt
(16)
s0
For impacts with a common impulse I and impact duration T:
s T
s(t) = &sin F
I=
K$&,,sin~dt=
[OTK,(&,,sin~~t
(17)
0
where S, is the maximum contact deformation. For a target structure with free edges:
Hence
&-‘K2 = q&-‘K2
where ml and m2 are the masses of the target structure and the striker, respectively, and V is the impact
(18)
140
S.W Gong et al.lComposite Structures 43 (1998) 137-145
where T(x) is the gamma function and
(19)
However, equality in the peak force requires that:
By substituting eqn (26) into (14) and eqn (16) with p = 1.5, the difference between peak forces estimated by eqn (13) and (14) and the difference between impulses predicted by eqn (15) and (16) are both approx. 5%. Equation (26) implies the effective contact stiffness Kq can also be determined when the contact coefficients K2 andp are known. These coefficients can be measured from static indentation tests. The available data on K2 and p for a steel striker in contact with graphite/epoxy laminated plates and shells can be found in the Ref. [lo-121.
from which K%= #-‘K, m
= r3m#-‘K
4.Experimental arrangement for impact tests 2
(21)
where y/3 =
(22)
To resolve the two different requirements defined value of by eqn (18) and (21), an appropriate ~(~1<~2<~3) must be found. The effective contact stiffness K% can be written as:
K~=rj&-'K2q,
(23)
of q ( = Q) which minimizes the differthe peak forces estimated by eqn (13) the impulses predicted by eqn (15) and by the extreme point of the following
An impact test rig was designed and built to determine the effective contact stiffness and study the contact force between the striker and target structure. The rig consists of three separate units: two holding fixtures - one for a cylindrical shell and the other for an ogival shell, an impact pendulum device and a data measurement and recording system. Figure 2 shows the glass/epoxy composite cylindrical and ogival shell specimens and their hxtures. The cylindrical shell was attached to a wooden fixture by thin
where the first term is the relative difference between q and Q while the second describes the relative difference between q and q3. This extreme point of eqn (24) occurs at:
2r(~+,),,rw
y+1>
v12=&r-
2
(25)
4P(
~+l)+sQ( $)
Substituting q2 into eqn (23) yields the equation relating the effective contact stiffness to the contact coefficients defined by Hertzian contact laws:
(,,l
K*,=k
yj-
)
2r(f+1)+Lr(Y) 6P_,K
4P(~+l)+nT($)
m
’
(26)
Fig. 2. Cylindrical and ogival specimens and their fixtures.
141
SW! Gong et aLlComposite Structures 43 (1998) 137-145
wires, while the ogival shell was mounted onto the boss of a circular plate. These impose approximately simplysupported boundaries for the cylindrical and ogival shells, respectively. In impact tests, the fixture holding the shells was mounted on a movable fixture base, as shown in Fig. 3. This Gxture base was not part of the pendulum frame, so that it could be moved in the X-, Y- and Z-directions and rotated about the X-axis, as illustrated in Fig. 3. This allowed the striker attached to the pendulum to impact any point on the shell specimens in a direction normal to the shell surface. The pendulum impact device illustrated in Fig. 3 was designed for a striker to impact the composite shells at specified velocities. A IPCB 208AO3 force transducer was attached to a removable steel block at the tip of a slender rod in the impa’ct pendulum device. The block could be changed to vary the impacting mass. The transducer had a removable cap which made direct contact with the target during impact and this could be changed to vary the contact stiffness. Since contact duration is governed by contact stiffness and impact mass, it can be easily controlled using this device. Impact velocity is calculated based on the mass and dimensions of the pendulum shown in Fig. 4, where Lp is the pendulum length, m, the mass of the rod and m,
angle scale
slender rod
the mass of the force transducer and steel block. Assuming that friction is negligible and employing conservation of energy, the impact velocity can be determined from:
=
1 I m,+ ;,s 112, \/2gL,(cos/L?- cosa) 1
J
(27)
m,+ - m,
3
where g is gravitational acceleration; LYand B are the angles shown in Fig. 4 and can be measured during impact tests by the angle scale fixed on the pendulum frame. The effective impact mass m, of the pendulum which includes the effect of the mass of the rod, can be determined from [ 151: 33 m,=m,+-m,
140
where the subscripts e, s, r refer respectively to the effective impact mass, mass of the striker and mass of the rod.
.
1111
force .transducer
,
pendulumframe
Fixture base
Steel block
Fig. 3. Pendulum impact device.
142
S. W Gong et al./Composite Structures 43 (1998) 137-145
A schematic diagram of the data measurement and recording system is shown in Fig. 5. In impact tests, the force transducer constituted part of the striker and made direct contact with the shell specimens. Thus, the contact force between the striker and shell was measured directly by the force transducer. The output signal was fed to a PCB 480D09 power unit which acted as a combined power supply and signal amplifier. Amplified force output was captured on a Yokogawa DL 1200A digital storage oscilloscope. The data were then transferred to a personal computer via an RS232 interface card and cable for subsequent processing and analysis.
5. Experimental results and discussion Impact tests were conducted on a [O,] glass/epoxy cylindrical shell and a [04] glass/epoxy ogival shell. Material properties of the shells are listed in Table 1. A pendulum striker with an effective mass of 0.075 kg and an impact velocity of 1 m/s was used for both cylindrical and ogival shells. Early studies [17] on
Fig. 4. Impact pendulum model.
impact of spheres have shown that contact stiffness is governed by the material properties (Young’s modulus) of the spheres and their radii. Since the cylindrical shell has a constant radius, impact at its centre was deemed to be sufficient to examine its contact stiffness. In contrast with the cylindrical shell, the radius r of the ogival shell varies with angle JI as shown in Fig. 6; therefore, the ogival shell was impacted at points A, B, C, D and E along a meridian (see Table 3). The contact forces were measured and the procedure described above was used to determine the effective contact stiffnesses between the force transducer and the shell specimens. Tables 2 and 3 show values of the effective contact stiffnesses obtained from impact tests. From Table 3, it is seen that contact stiffness for the ogival shell decreases slightly with $ because the radius r decreases with +. This result may seem counter-intuitive, i.e. that the shell is stiffer for a larger radius r, but it should be noted that contact stiffness is the ‘local stiffness’ concentrated at the contact area; the experimental results imply that deformation at the contact area is slightly less for a larger radius r, which is consistent with earlier studies [17] on the impact of spheres. However, the variation of contact stiffness with radius r of the ogival shell is relatively small and does not exceed 6%. This indicates that the effect of the ogival shell curvature on effective contact stiffness is negligible. Figures 7 and 8 illustrate the calculated and measured contact forces for glass/epoxy cylindrical and ogival shells, respectively. The cylindrical shell was impacted at its centre while the ogival shell was impacted at point C (I,+= 1joJ2, 8 = 0). It is observed that both the calculated and measured contact force for the cylindrical shell exhibit longer durations than the contact force between the ogival shell and the same striker. This is due to the fact that the Young’s moduli of the cylindrical shell are smaller than that of the ogival shell, as shown in Table 1. These results indicate that the stiffness (Young’s modulus) of the impacted shell has a significant effect on contact duration. Comparisons between the calculated and measured impact forces in Figs 7 and 8 show that the calculated
PCB 480D09 Force transducer
_
Power uint
Steel block / Transducer,
cap Fig. 5. Measurement system for impact test.
S. W Gong et al.lComposite Structures 43 (1998) 137-145 Table 1 Material properties of shells Properties
Err @Pa) Ezz @Pa) GU @Pa) cl2 P (kg/m?
143
Table 3 contact stiffness between [O,] glass/epoxy ogival shell and striker derived from impact tests (see Fig. 6)
[Or]glass/epoxy cylinder
[O,] glass/epoxy ogival shell
14.506 5.362 2.509 0.231 1526
15.063 14.924 3.566 0.283 1637
Striker
Impact point
Effective contact stiffness
Steel cylinder with diameter of 12.7 mm
contact duration and peak force differ from measured values by 5% at most; also, the calculated force-time curves match the measured curves well, especially in the case of impact on the ogival shell. This indicates that the values of effective contact stiffness obtained from impact tests are reasonable. The effects of impact mass and velocity on contact force were examined experimentally. Tests were first
587 kN/m 593 kN/m 601 kN/m 609 kN/m 618 kN/m
performed on the [04] glass/epoxy ogival shell, using three steel blocks of 0.025, 0.050 and 0.075 kg mass. The rod and force transducer have masses of 0.108 and 0.025 kg, respectively. By combining the mass of each block with that of the transducer and the pendulum
200 1 L
I
43 ,.i
..”
!
,..’
,.i
/
,.:
l
,I
,,‘_/
. . . . . . . . -..-‘.I--‘.
.-...
_..__
,,,.,,
r / ._____._._..._!_._._._._____ ..._
/_/ ,__.... $-‘“ul, /_,_ ,__.. /..Cs’ i ,,,.. I /. ....-...-....r-.........._.. _.____(,._,.
_._.._. ._f ._._._._._ _._._._._._._._._._ ._
100
Experiment Theory
-.-.-
150 -
!
-
50 -
\ I
._.-..
Or -5ot* -2
_._._
’ * ’ * ’ ’ ’ * ’ * ’ * * ’ ’ * * *
0
2 Time
4 (ms)
6
8
Fig. 7. Contact force between striker and cylindrical shell during
3oo/
-.-.-
Experiment
Fig. 6. Location of j.mpact point for ogival shell.
Table 2 Effective contact stiffness betsveen [O,] glass/epoxy cylindrical shell and striker derived from impact tests Striker
Impact point
Effective contact stiffness
Steel cylinder with diameter of 12.17 mm
cylindrical shell centre
253 kN/m
-2
0
2 4 Time (ms)
6
Fig. 8. Contact force between striker and ogival shell during impact.
S.W Gong et al.lComposite Structures 43 (1998) 137-145
144
;_*. ,,:. *.,
duration T remains unchanged. The reason is that a higher impact velocity induces a larger deformation and thereby a larger contact force. However, impact duration is governed by the material properties of the striker and the ogival shell; hence, it remains unchanged.
me=0.075kg -.-.m =O.lOOkg . . . .. . . . . . m;=O. 125kg
6. Conclusions
-1
0
1 Time
2
3
4
(ms)
Fig. 9. Effect of impact mass on contact force.
rod using eqn (6), three different effective impact masses are obtained; i.e. 0.075, 0.100 and 0.125 kg. In the tests, the impact velocity was maintained at 1 m/s. Figure 9 shows the measured contact force for the three impact masses. It is seen that the contact force magnitude and contact duration T (when the impact force reduces to zero) increase with impact mass, because a larger mass carries a higher incident kinetic energy to be dissipated via larger shell deflection. The [O,] glass/epoxy ogival shell was then impacted respectively at 1, 3 and 5 m/s. These different velocities were achieved by releasing the impact pendulum with a length of 0.6 m at a = 22”, a = 69” and a = 140”, respectively, and setting p = 0” when the striker contacts the ogival shell. The effective impact mass was 0.075 kg (the 0.025 kg steel block was used). Figure 10 illustrates measured contact forces for the three impact velocities. It is seen that the magnitude of the contact force increases with impact velocity, but the contact
1200 I 1000
I.
2 g
800
8
600
b 0 2 2 S
V=l m/s -.-.V=3m/s .. ......_ V=Sm/s
; ‘; :: : : ; : j ; i *\ ; :, a :. \i :, .: :. I:1: :, :. I:
400 200 0 -200
h -2
0
2 Time
4 (ms)
6
Fig. 10. Effect of impact velocity on contact force.
8
The proposed approach facilitates the determination of the effective contact stiffness between a striker and a target composite shell. The method is easy to use and has been proved effective. Results show that the structural stiffness (Young’s modulus) of the impacted shell has a significant effect on contact duration. A larger mass carries a higher incident kinetic energy to be dissipated and thus both contact force and contact duration increase with impact mass. Impact velocity dominates impact energy but has little effect on contact duration. These results suggest that in impact tests, a given peak impact force can be obtained by varying impact velocity or impact mass, while a required contact duration can be obtained by adjusting the mass or material of the striker.
References PI Timoshenko S. Zur frage nach der wirkung eines stosse auf einer blken. Z. Math. physik. 1913;62:198-209.
PI Karas K. Platten unter seitlichem stoss. Ing.-Arch. 1939;lO: 237-50. [31 Lee EH. Impact of a mass striking a beam. ASME Trans. J Appl. Mech. 1941;A-129-38. 141 Preston JL Jr, Cook TS. Foreign object impact damage to composites. ASTM STP 568, Philadelphia PA, 1975;49. PI Greszczuk LB. Impact dynamics. New York John Wiley, 1982:55. Fl Sun CT, Chattopadhyay S. Dynamic response of anisotropic laminated plates under initial stress due to impact of a mass. ASME Trans. J Appl. Mech. 1975;42:693-8. [71 Dobyns AL. Analysis of simply-supported orthotropic plates subject to static and dynamic loads. AIAA Journal 1981;19:642-50. PI Shivakumar KN, Elber W, Illg W. Prediction of impact force and duration due to low-velocity on circular composite laminates. ASME Trans. J Appl. Mech. 1985;52:674-80. [91 Conway HD, Angew Z. The pressure distribution between two elastic bodies in contact. J. Math. Phys. 1996;7:460-5. PO1 Yang SH, Sun CT. Composite material: testing and design. ASTM STP 787, Philadelphia PA, 1982;425. WI Tan TM, Sun CT. Use of statical indentation laws in the impact analysis of laminated composite plates. ASME Trans. J Appl. Mech. 1985;52:6-12. WI Lin HJ, Lee YJ. Use of statical indentation laws in the impact analysis of composite laminated plates and shells. ASME Trans. J Appl. Mech. 1985;57:787-9. P31 Gong SW, Shim VPW, Toh SL. Impact response of laminated orthogonal shells with curvatures. Comp. Engng 1995;5:257-275.
S.lK Gong et al./Composite Structures 43 (1998) 137-145 [14] Gong SW, Shim VPW, Toh SL. Central and noncentral impact on orthotropic composite cylindrical shells. AIAA Journal 1996;34:1619-26. [15] Weaver W Jr, Timoshenko SP, Young DH. Vibration problems in engineering. New York.: John Wiley, 1990~24.
14.5
[16] Lord Rayleigh. On the prediction of vibration by forces of relatively long duration, with application to the theory of collisions. Philosophical magazine, Series 6, 1906;11:283-92. [17] Timoshenko SP, Goodier JN. Theory of elasticity. New York: McGraw-Hill, 1970.
ELSEVIER
Composite Structures 43 (1998) 137-145
Determining
effective contact stiffness between striker and composite shell S.W. Gong*, V.P.W. Shim, S.L. Toh
Department of Mechanical and Production Engineering, National University of Singapore, Singapore 119260, Singapore
Abstract An approach to determining the effective contact stiffness between a striker and composite shell is presented. The effective contact stiffness is obtained via impact tests in conjunction with an analytic impact function recently presented by the authors. The relationship between the effective contact stiffness and the contact coefficients defined by Hertzian contact laws is described.
Impact tests are conducted on simply-supported glass/epoxy cylindrical and ogival shells. The effective contact stiffnesses for these shells impacted by a solid striker are obtained using the present method. In addition, effects of impact mass and impact velocity on the contact force are examined and discussed. 0 1998 Elsevier Science Ltd. All rights reserved.
1. Introduction
Fibre-reinforced laminated composites are becoming widely utilized as primary structural components in aerospace applications. In view of designing for resistance to foreign object impact damage, prediction of impact force between a striker and the target composite structure is necessary. The impact force is a result of contact deformation between a striker and the target structure and should therefore be evaluated on this basis. To estimate the contact force between a striker and the target structure during impact, Hertz’s contact law has been used andi modified by many researchers. Timoshenko [l] initiated the basic approach in this area by combining the Hertz contact force law with Bernoulli-Euler beam theory. Karas [2] extended this approach to the study of central impact on a rectangular plate and obtained the contact force. Lee [3] presented a modified Hertz method, which includes flexural deflection of the target structure in addition to contact deformation, to predict the contact force on a beam. Preston and Cook [4] and Greszczuk [5] employed the modified Hertz method to evaluate the contact force for a cantilever and circular plates, respectively. Sun [6] and Dobyns [7] did a similar analysis for a rectangular plate. *Corresponding author. Tel.: 00 65 7761 647; Fax: 00 65 7761 647 962; E-mail: [email protected]
The contact stiffness is a key parameter in prediction of impact force. Shivakumar, Elber and Illg [8] proposed a spring-mass model, which was an extension of Lee’s spring-mass model for impact on a beam. In their model, the impactor and plate were represented by two rigid masses and the associated deformation characteristics modelled by springs. They evaluated the contact stiffness following Conway and Angew’s [9] procedure which was restricted to simulation of a composite plate impacted by a spherical isotropic striker. Sun et al. [lO,ll] established empirical contact laws for glass/epoxy and graphite/epoxy laminates in contact with steel balls. In their study, static indentation tests were conducted on glass/epoxy and graphite/ epoxy composite laminates using a steel sphere as the indentor. This yielded the contact stiffness between the composite laminate and the steel ball. Lin and Lee [12] following Sun et al. conducted a similar study on composite laminated shells. All these contact stiffness estimations are based on static indentation tests. In this study, an approach to determination of the effective contact stiffness is proposed, whereby it is obtained via impact tests in conjunction with an analytic impact function recently presented by Gong et al. [13,14]. An impact test rig was designed and built to determine the effective contact stiffness and study the contact force. Impact tests were conducted on simplysupported glass/epoxy cylindrical and ogival shells. The effective contact stiffnesses between the striker and the
0263-8223/98/$ - see front matter 0 1998 Elsevier Science Ltd. AI1 rights reserved. PII: SO263-8223(98)OOlO2-0
S. W Gong et al.lComposite Structures 43 (1998) 137-145
138
shells were obtained, from which the procedure for this is demonstrated.
The contact force therefore can be expressed terms of the five parameters as follow [13]: K%z,(c, - l)sin(o,r)+a,(c,
in
- l)sin(w2t)]
F(t)= 0
2. Impact force model and effective contact stiffness O
Consider a simply-supported composite shell of arbitrary shape, with a constant thickness h, mass ml and density p. The shell is impacted by a solid striker of mass m2 at a velocity V. To determine the contact force between the striker and the shell during impact, a simple spring-mass model is used, as shown in Fig. 1. This is essentially a modification of Shivakumar’s [8] model. The present S-M model contains five parameters; they are the effective shell mass m$, the striker mass m2, the equivalent stiffness K1 of the simplysupported shell, the effective contact stiffness K3 and the initial striker velocity V. Let w(t) and w2(t) represent respectively the normal displacement of the load point on the shell and that of the striker at any time t during impact. The effective contact stiffness K$ is defined by the relation of contact force to the contact deformation:
t>T
In the force function, T is the contact duration and
0 Cl =
Kq-wfrn,
K3 c2 =
Kq--wirn;, V al= wdc2-cJ
(1)
F(t)= K%6(t)
V
or
a2=
(4) W2(CI
F(t) = K%[w~(Q
-w(t)1
-
c2)
(2) The effective shell mass Rayleigh’s method [ 151: mi=ph
rnt
is determined
by
W2(x, YW
where dS is the differential surface area of the shell while S is the total surface area and W(x, y) is a function of position that defines the shape of the natural mode of shell vibration. The equivalent stiffness of the simply-supported shell is determined from: Boundaries: Simply supported
Fig. 1. Spring-mass model.
K,=w:,rnT
(6)
where wll is the fundamental frequency of the target structure. Equation (6) is based on stimulation of only the fundamental mode of vibration of the shell, because early studies [3,16] have shown that impact durations are generally many times longer than the time taken for stress waves generated to travel to opposite boundaries of the plate and return. Furthermore, the effects of higher modes, especially when the plate undergoes large deflection, are small and can be neglected [8]. The parameters m2 and V can be determined directly from the mass and initial velocity of the striker. Except for the effective contact stiffness K$, the parameters rni, K,, m2 and V can be easily determined.
S.U! Gong et al.lComposite
The effective contact stiffness Kq depends on the material and geometrica. properties of the striker and target structure and is used to predict the contact force between a striker and shell during impact. Therefore, it is preferable that this stiffness be determined from impact tests. The procedure to determine effective contact stiffness is relatively simple. First, a shell is impacted by a pendulum striker at a given velocity. The contact force between the striker and shell during impact is captured by a force transducer attached to the striker. An impact force-time curve is obtained from the impact test, and the impact force function eqn (3) is used to fit the: experimental data. Except for the effective contact stiffness K% the other parameters in the impact force function eqn (3) are known; rnt and Kl are determined from eqn (5) and (6) and m2 and V are the mass and initial velocity of the striker in the impact test. A trial stiffness (KQ,, is assumed for an initial calculation of the impact force. By comparing calculated values of the maximum impact force F,, and contact duration 1” (i.e. when the striker is in contact with the shell) with those from the impact test, (KQ,, is appropriately adjusted to a second trial value (KQ2 and substituted back into the impact force function eqn (3). This process of comparison and modification is repeated, until the value of (Kt),, converges such that the difference between calculated and experimental values of maximum impact force F,,,, and contact duration T -fall within specified tolerances; i.e. the effective contact stiffness is: Kq = (KV,,
(7)
3. Relating effective contact stiffness to contact
Structures 43 (1998) 137-145
139
velocity of the striker. For a stationary target structure:
(11) For a target structure with simply-supported the following relationship exists: (&J, < (&Z),< (&J,,
(12)
where the subscripts fi, ss and st refer respectively to free, simply-supported and stationary structures. For p = 1.5 and ml > 10 nz2, the difference between (c?,,,),~ and (6,),, is less than 4%. Therefore, for a striker of small mass, the maximum contact deformation of a target structure with simply-supported edges can be estimated by eqn (11). Substitution of eqn (9) into eqn (8) and (1) yields respectively: P
(13) and F(t) = KqG,sin f
(14)
The respective impulses experienced during the impact duration T, corresponding to eqn (13) and (14) are:
(15)
10*K,(&sin G )”
coefficients defined by Hertzian contact laws
The contact coefficients K2 and p are defined by Hertzian theory of the form: F=(t) = KdW)lP
(8)
They can be obtained by static indentation tests [lo, 111. Greszczuk [5] has shown that contact deformation 6 can be approximated fairly accurately by:
edges,
and T
K@,sin F dt
(16)
s0
For impacts with a common impulse I and impact duration T:
s T
s(t) = &sin F
I=
K$&,,sin~dt=
[OTK,(&,,sin~~t
(17)
0
where S, is the maximum contact deformation. For a target structure with free edges:
Hence
&-‘K2 = q&-‘K2
where ml and m2 are the masses of the target structure and the striker, respectively, and V is the impact
(18)
140
S.W Gong et al.lComposite Structures 43 (1998) 137-145
where T(x) is the gamma function and
(19)
However, equality in the peak force requires that:
By substituting eqn (26) into (14) and eqn (16) with p = 1.5, the difference between peak forces estimated by eqn (13) and (14) and the difference between impulses predicted by eqn (15) and (16) are both approx. 5%. Equation (26) implies the effective contact stiffness Kq can also be determined when the contact coefficients K2 andp are known. These coefficients can be measured from static indentation tests. The available data on K2 and p for a steel striker in contact with graphite/epoxy laminated plates and shells can be found in the Ref. [lo-121.
from which K%= #-‘K, m
= r3m#-‘K
4.Experimental arrangement for impact tests 2
(21)
where y/3 =
(22)
To resolve the two different requirements defined value of by eqn (18) and (21), an appropriate ~(~1<~2<~3) must be found. The effective contact stiffness K% can be written as:
K~=rj&-'K2q,
(23)
of q ( = Q) which minimizes the differthe peak forces estimated by eqn (13) the impulses predicted by eqn (15) and by the extreme point of the following
An impact test rig was designed and built to determine the effective contact stiffness and study the contact force between the striker and target structure. The rig consists of three separate units: two holding fixtures - one for a cylindrical shell and the other for an ogival shell, an impact pendulum device and a data measurement and recording system. Figure 2 shows the glass/epoxy composite cylindrical and ogival shell specimens and their hxtures. The cylindrical shell was attached to a wooden fixture by thin
where the first term is the relative difference between q and Q while the second describes the relative difference between q and q3. This extreme point of eqn (24) occurs at:
2r(~+,),,rw
y+1>
v12=&r-
2
(25)
4P(
~+l)+sQ( $)
Substituting q2 into eqn (23) yields the equation relating the effective contact stiffness to the contact coefficients defined by Hertzian contact laws:
(,,l
K*,=k
yj-
)
2r(f+1)+Lr(Y) 6P_,K
4P(~+l)+nT($)
m
’
(26)
Fig. 2. Cylindrical and ogival specimens and their fixtures.
141
SW! Gong et aLlComposite Structures 43 (1998) 137-145
wires, while the ogival shell was mounted onto the boss of a circular plate. These impose approximately simplysupported boundaries for the cylindrical and ogival shells, respectively. In impact tests, the fixture holding the shells was mounted on a movable fixture base, as shown in Fig. 3. This Gxture base was not part of the pendulum frame, so that it could be moved in the X-, Y- and Z-directions and rotated about the X-axis, as illustrated in Fig. 3. This allowed the striker attached to the pendulum to impact any point on the shell specimens in a direction normal to the shell surface. The pendulum impact device illustrated in Fig. 3 was designed for a striker to impact the composite shells at specified velocities. A IPCB 208AO3 force transducer was attached to a removable steel block at the tip of a slender rod in the impa’ct pendulum device. The block could be changed to vary the impacting mass. The transducer had a removable cap which made direct contact with the target during impact and this could be changed to vary the contact stiffness. Since contact duration is governed by contact stiffness and impact mass, it can be easily controlled using this device. Impact velocity is calculated based on the mass and dimensions of the pendulum shown in Fig. 4, where Lp is the pendulum length, m, the mass of the rod and m,
angle scale
slender rod
the mass of the force transducer and steel block. Assuming that friction is negligible and employing conservation of energy, the impact velocity can be determined from:
=
1 I m,+ ;,s 112, \/2gL,(cos/L?- cosa) 1
J
(27)
m,+ - m,
3
where g is gravitational acceleration; LYand B are the angles shown in Fig. 4 and can be measured during impact tests by the angle scale fixed on the pendulum frame. The effective impact mass m, of the pendulum which includes the effect of the mass of the rod, can be determined from [ 151: 33 m,=m,+-m,
140
where the subscripts e, s, r refer respectively to the effective impact mass, mass of the striker and mass of the rod.
.
1111
force .transducer
,
pendulumframe
Fixture base
Steel block
Fig. 3. Pendulum impact device.
142
S. W Gong et al./Composite Structures 43 (1998) 137-145
A schematic diagram of the data measurement and recording system is shown in Fig. 5. In impact tests, the force transducer constituted part of the striker and made direct contact with the shell specimens. Thus, the contact force between the striker and shell was measured directly by the force transducer. The output signal was fed to a PCB 480D09 power unit which acted as a combined power supply and signal amplifier. Amplified force output was captured on a Yokogawa DL 1200A digital storage oscilloscope. The data were then transferred to a personal computer via an RS232 interface card and cable for subsequent processing and analysis.
5. Experimental results and discussion Impact tests were conducted on a [O,] glass/epoxy cylindrical shell and a [04] glass/epoxy ogival shell. Material properties of the shells are listed in Table 1. A pendulum striker with an effective mass of 0.075 kg and an impact velocity of 1 m/s was used for both cylindrical and ogival shells. Early studies [17] on
Fig. 4. Impact pendulum model.
impact of spheres have shown that contact stiffness is governed by the material properties (Young’s modulus) of the spheres and their radii. Since the cylindrical shell has a constant radius, impact at its centre was deemed to be sufficient to examine its contact stiffness. In contrast with the cylindrical shell, the radius r of the ogival shell varies with angle JI as shown in Fig. 6; therefore, the ogival shell was impacted at points A, B, C, D and E along a meridian (see Table 3). The contact forces were measured and the procedure described above was used to determine the effective contact stiffnesses between the force transducer and the shell specimens. Tables 2 and 3 show values of the effective contact stiffnesses obtained from impact tests. From Table 3, it is seen that contact stiffness for the ogival shell decreases slightly with $ because the radius r decreases with +. This result may seem counter-intuitive, i.e. that the shell is stiffer for a larger radius r, but it should be noted that contact stiffness is the ‘local stiffness’ concentrated at the contact area; the experimental results imply that deformation at the contact area is slightly less for a larger radius r, which is consistent with earlier studies [17] on the impact of spheres. However, the variation of contact stiffness with radius r of the ogival shell is relatively small and does not exceed 6%. This indicates that the effect of the ogival shell curvature on effective contact stiffness is negligible. Figures 7 and 8 illustrate the calculated and measured contact forces for glass/epoxy cylindrical and ogival shells, respectively. The cylindrical shell was impacted at its centre while the ogival shell was impacted at point C (I,+= 1joJ2, 8 = 0). It is observed that both the calculated and measured contact force for the cylindrical shell exhibit longer durations than the contact force between the ogival shell and the same striker. This is due to the fact that the Young’s moduli of the cylindrical shell are smaller than that of the ogival shell, as shown in Table 1. These results indicate that the stiffness (Young’s modulus) of the impacted shell has a significant effect on contact duration. Comparisons between the calculated and measured impact forces in Figs 7 and 8 show that the calculated
PCB 480D09 Force transducer
_
Power uint
Steel block / Transducer,
cap Fig. 5. Measurement system for impact test.
S. W Gong et al.lComposite Structures 43 (1998) 137-145 Table 1 Material properties of shells Properties
Err @Pa) Ezz @Pa) GU @Pa) cl2 P (kg/m?
143
Table 3 contact stiffness between [O,] glass/epoxy ogival shell and striker derived from impact tests (see Fig. 6)
[Or]glass/epoxy cylinder
[O,] glass/epoxy ogival shell
14.506 5.362 2.509 0.231 1526
15.063 14.924 3.566 0.283 1637
Striker
Impact point
Effective contact stiffness
Steel cylinder with diameter of 12.7 mm
contact duration and peak force differ from measured values by 5% at most; also, the calculated force-time curves match the measured curves well, especially in the case of impact on the ogival shell. This indicates that the values of effective contact stiffness obtained from impact tests are reasonable. The effects of impact mass and velocity on contact force were examined experimentally. Tests were first
587 kN/m 593 kN/m 601 kN/m 609 kN/m 618 kN/m
performed on the [04] glass/epoxy ogival shell, using three steel blocks of 0.025, 0.050 and 0.075 kg mass. The rod and force transducer have masses of 0.108 and 0.025 kg, respectively. By combining the mass of each block with that of the transducer and the pendulum
200 1 L
I
43 ,.i
..”
!
,..’
,.i
/
,.:
l
,I
,,‘_/
. . . . . . . . -..-‘.I--‘.
.-...
_..__
,,,.,,
r / ._____._._..._!_._._._._____ ..._
/_/ ,__.... $-‘“ul, /_,_ ,__.. /..Cs’ i ,,,.. I /. ....-...-....r-.........._.. _.____(,._,.
_._.._. ._f ._._._._._ _._._._._._._._._._ ._
100
Experiment Theory
-.-.-
150 -
!
-
50 -
\ I
._.-..
Or -5ot* -2
_._._
’ * ’ * ’ ’ ’ * ’ * ’ * * ’ ’ * * *
0
2 Time
4 (ms)
6
8
Fig. 7. Contact force between striker and cylindrical shell during
3oo/
-.-.-
Experiment
Fig. 6. Location of j.mpact point for ogival shell.
Table 2 Effective contact stiffness betsveen [O,] glass/epoxy cylindrical shell and striker derived from impact tests Striker
Impact point
Effective contact stiffness
Steel cylinder with diameter of 12.17 mm
cylindrical shell centre
253 kN/m
-2
0
2 4 Time (ms)
6
Fig. 8. Contact force between striker and ogival shell during impact.
S.W Gong et al.lComposite Structures 43 (1998) 137-145
144
;_*. ,,:. *.,
duration T remains unchanged. The reason is that a higher impact velocity induces a larger deformation and thereby a larger contact force. However, impact duration is governed by the material properties of the striker and the ogival shell; hence, it remains unchanged.
me=0.075kg -.-.m =O.lOOkg . . . .. . . . . . m;=O. 125kg
6. Conclusions
-1
0
1 Time
2
3
4
(ms)
Fig. 9. Effect of impact mass on contact force.
rod using eqn (6), three different effective impact masses are obtained; i.e. 0.075, 0.100 and 0.125 kg. In the tests, the impact velocity was maintained at 1 m/s. Figure 9 shows the measured contact force for the three impact masses. It is seen that the contact force magnitude and contact duration T (when the impact force reduces to zero) increase with impact mass, because a larger mass carries a higher incident kinetic energy to be dissipated via larger shell deflection. The [O,] glass/epoxy ogival shell was then impacted respectively at 1, 3 and 5 m/s. These different velocities were achieved by releasing the impact pendulum with a length of 0.6 m at a = 22”, a = 69” and a = 140”, respectively, and setting p = 0” when the striker contacts the ogival shell. The effective impact mass was 0.075 kg (the 0.025 kg steel block was used). Figure 10 illustrates measured contact forces for the three impact velocities. It is seen that the magnitude of the contact force increases with impact velocity, but the contact
1200 I 1000
I.
2 g
800
8
600
b 0 2 2 S
V=l m/s -.-.V=3m/s .. ......_ V=Sm/s
; ‘; :: : : ; : j ; i *\ ; :, a :. \i :, .: :. I:1: :, :. I:
400 200 0 -200
h -2
0
2 Time
4 (ms)
6
Fig. 10. Effect of impact velocity on contact force.
8
The proposed approach facilitates the determination of the effective contact stiffness between a striker and a target composite shell. The method is easy to use and has been proved effective. Results show that the structural stiffness (Young’s modulus) of the impacted shell has a significant effect on contact duration. A larger mass carries a higher incident kinetic energy to be dissipated and thus both contact force and contact duration increase with impact mass. Impact velocity dominates impact energy but has little effect on contact duration. These results suggest that in impact tests, a given peak impact force can be obtained by varying impact velocity or impact mass, while a required contact duration can be obtained by adjusting the mass or material of the striker.
References PI Timoshenko S. Zur frage nach der wirkung eines stosse auf einer blken. Z. Math. physik. 1913;62:198-209.
PI Karas K. Platten unter seitlichem stoss. Ing.-Arch. 1939;lO: 237-50. [31 Lee EH. Impact of a mass striking a beam. ASME Trans. J Appl. Mech. 1941;A-129-38. 141 Preston JL Jr, Cook TS. Foreign object impact damage to composites. ASTM STP 568, Philadelphia PA, 1975;49. PI Greszczuk LB. Impact dynamics. New York John Wiley, 1982:55. Fl Sun CT, Chattopadhyay S. Dynamic response of anisotropic laminated plates under initial stress due to impact of a mass. ASME Trans. J Appl. Mech. 1975;42:693-8. [71 Dobyns AL. Analysis of simply-supported orthotropic plates subject to static and dynamic loads. AIAA Journal 1981;19:642-50. PI Shivakumar KN, Elber W, Illg W. Prediction of impact force and duration due to low-velocity on circular composite laminates. ASME Trans. J Appl. Mech. 1985;52:674-80. [91 Conway HD, Angew Z. The pressure distribution between two elastic bodies in contact. J. Math. Phys. 1996;7:460-5. PO1 Yang SH, Sun CT. Composite material: testing and design. ASTM STP 787, Philadelphia PA, 1982;425. WI Tan TM, Sun CT. Use of statical indentation laws in the impact analysis of laminated composite plates. ASME Trans. J Appl. Mech. 1985;52:6-12. WI Lin HJ, Lee YJ. Use of statical indentation laws in the impact analysis of composite laminated plates and shells. ASME Trans. J Appl. Mech. 1985;57:787-9. P31 Gong SW, Shim VPW, Toh SL. Impact response of laminated orthogonal shells with curvatures. Comp. Engng 1995;5:257-275.
S.lK Gong et al./Composite Structures 43 (1998) 137-145 [14] Gong SW, Shim VPW, Toh SL. Central and noncentral impact on orthotropic composite cylindrical shells. AIAA Journal 1996;34:1619-26. [15] Weaver W Jr, Timoshenko SP, Young DH. Vibration problems in engineering. New York.: John Wiley, 1990~24.
14.5
[16] Lord Rayleigh. On the prediction of vibration by forces of relatively long duration, with application to the theory of collisions. Philosophical magazine, Series 6, 1906;11:283-92. [17] Timoshenko SP, Goodier JN. Theory of elasticity. New York: McGraw-Hill, 1970.