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Impact response of single-lap composite joints
Composites Engineering, Vol. 5, No. 8, pp. 1011-1027, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0961-9526/95 $9.50+ .00
Pergamon
0961-9526(95) 00003-8
IMPACT RESPONSE OF SINGLE-LAP COMPOSITE JOINTS Su-Seng Pang t, Chihdar Yang and Yi Zhao Department of Mechanical Engineering, Louisiana State University, Baton Rouge, LA 70803, U.S.A. (Received 12 July 1994;final version accepted 9 December 1994)
Abstract--The low-velocity impact of adhesive-bonded single-lap composite joints has been studied using a spring-mass model. In this quasi-static model, the impact response is represented by a time-dependent force, and the target joint is represented by an equivalent mass with equivalent stiffness. An analytical model has been developed to determine the equivalent mass and stiffness of the joint. The laminated anisotropic plate theory was used in the derivation of the governing equations of the two bonded laminates. The entire coupled system, as well as the assumed peel stress, were solvedusing both the joint kinematics and suitable boundary conditions. With the combination of a spring-mass equilibrium system and the developed joint model, a relationship between the impact force and the duration has been established. Adhesive stresses, which are believed to be the cause of failure, were predicted from the impact force. Impact tests of single-lap composite joints with different sample thicknesses and overlay lengths have been conducted to verify the proposed model. INTRODUCTION With the introduction of high performance fibers such as carbon, boron and Kevlar, along with some new and improved matrix materials, advanced composites have established themselves as engineering structural materials. Adhesive-bonded joints have been widely used for composite materials as an alternative to conventional mechanical joint designs. The primary limitation of such designs arises from machining difficulties and subsequent damage to the laminate following these operations. In order to fully utilize composite materials, more research needs to be conducted in the field of joint design and analysis. One can find review papers regarding adhesive-bonded composite lap joints in the literature. Kutscha (1964) reviewed the early analytical work on isotropic adherends prior to 1961, and the analyses from 1961 to 1969 were reviewed by Kutscha and Hofer (1969). In 1982, Matthews et al. reviewed the theoretical work, including classical and finite element methods related to all aspects of adhesive-bonded joints in composite materials. Vinson (1989) summarized the published work dealing with the adhesive bonding of polymer matrix composite structures. When studying adhesive-bonded lap joints, the effects due to the rotation of the adherends were first taken into account by Goland and Reissner (1944). They introduced an equation to relate the bending moment of the adherend at the end of the overlay to the in-plane loading. The basic approach of the Goland and Reissner theory was based on beam theory, or rather, on cylindrically bentplate theory which treated the overlay section as a beam of twice the thickness of the adherend. Hart-Smith (1973a, b, c, 1974) published a series of papers regarding singlelap, double-lap, scarf, and stepped-lap joints involving a continuum model in which the adherends were isotropic or anisotropic elastic, and the adhesive was modeled as elastic, elastic-plastic, or bielastic. From these ample review papers, it is found that a considerable effort has been expended to determine the structural behavior of the most commonly used joints for composite laminates--single-lap joints. Most of the previous research on composite joints was based on the strength of materials theory and the classical plate theory, which neglect deflections and stresses attributed to the transverse shearing effects. In general, the *To whom correspondence should be addressed. 1011 C0E 5-8-0
1012
s.-s. Pang et al.
strength of materials approach can achieve acceptable accuracy when a plate has a large span-to-depth ratio (Reissner, 1945; Reddy, 1984). However, when the overlay length is small, the transverse shearing effects become significant. Neglecting the transverse shearing effects could result in considerable error in the model. Yang et al. (1992) studied double-lap composite joints under cantilevered bending and proposed a strain gap model to describe the stress-strain behavior. They also proposed (1993) an analytical model for adhesive-bonded single-lap composite joints under cylindrical bending based on the laminated anisotropic plate theory. The asymmetry of the adherend laminates and the transverse shear deformation were included in their models. Yang and Pang (1993) also developed an analytical model to predict the stress-strain distributions within a single-lap joint under tensile loading. In the current study, the dynamic behavior of the single-lap composite joint under impact load has been investigated. The analytical model has been developed based on Yang and Pang's previous work on single-lap joints which has been extended to specific loading and boundary conditions in the current paper. The spring-mass model (Shivakum a r et al., 1983) has also been applied to the impactor-target system to establish the relationship between impact duration and impact force. Impact tests have been conducted on single-lap joint samples with different lay-ups to verify the developed analytical model. IMPACT MODELOF SINGLE-LAPCOMPOSITEJOINT The system of an isotropic hemispherical impactor and a single-lap composite joint can be represented by a spring-mass model. The impact force can be related to the indentation through a modified Hertzian contact law. The equivalent stiffness keq of the singlelap composite joint can be theoretically determined by the model developed in a later section. By applying Newton's second law of motion, the equations of motion of the impactor-target system can be established. Using appropriate boundary conditions, a relationship between impact duration and maximum impact force can finally be obtained. C o n t a c t law f o r c o m p o s i t e s
The well known Hertzian contact law was originally proposed for contact of two elastic isotropic spheres. The contact between an elastic sphere and an elastic flat plate is a special case in which the radius of one sphere is infinite. In this case, the relationship between the contact force, F, and indentation, z, can be expressed as and
F K = 4R,/Z(1
3
(1)
= KZ ~/2
- v2
\
E1
1 - v22~ -1
+---~2 ]
(2)
where the subscripts 1 and 2 denote indenter and target, respectively, R is the radius of the indenter, v is Poisson's ratio, and E is Young's modulus. For contact between isotropic and composite materials, K can be determined from the modified contact law (Sun and Yang, 1980): /,"
= 4R, 3
(1 \
E
1) -1 +
(3)
where E is the Young's modulus of the isotropic indenter, and E c is the Young's modulus of the composite target in the direction normal to the contact plane. Spring-mass m o d e l
A spring-mass model was applied for the impact of laminated composite plates with a single-lap joint. The load-deflection behavior of the beam was represented by an equivalent stiffness keq. As shown in Fig. 1, z l ( t ) and z2(t) are the deflection of the impactor with a mass m~ and the central deflection of the single-lap joint with an equivalent m a s s m e q . Both the equivalent stiffness k,q and the equivalent m a s s meq for the single-lap joint are related to the geometry and material properties of the joint, and the contact and
Impact response of single-lapcompositejoints
1013
I-'--7
Fig. 1. Impactor displacement,indentation, and sample deflection. supporting conditions, Both will be determined through a detailed analytical model in the later sections. During the contact, the indentation at time t is
z(t) = zl(t) -
z2(t).
(4)
By applying Newton's second law of motion to the spring-mass system, the equations of motion are d2Zl ml--~ + d2z2 meq " - ~ +
KZa/2 =
0
(5)
keqz2 - KZ3/2 =
0
(6)
where keq is the equivalent stiffness of the single-lap joint. Here, the modified Hertzian contact law was applied to the single-lap joint with finite width and thickness. Such an application is valid only if the width of the beam is considerably larger than the contact area and if the thickness of the beam is considerably larger than the indentation. The impact force wave consists of several single waves, and each of them can be modeled as F = Fmsin
t
(7)
where Fm is the peak value of impact force within the single wave period, and tm is the time corresponding to Fm. A similar sinusoidal expression has been used by several researchers (Sun and Yang, 1980; Pang et al., 1991, etc.). The initial conditions for z~(t) can be expressed as Zl(O) = 0 dzl(0) dt = v0 where Vo is the impact velocity. From eqns (5)-(8), the solution of
zl(t)
z~(t) can
(8)
be obtained as
. r t ) ( + Vo 2Fmtm~t 4Fmt2msln(--t \2tm nml /
(9)
= rt2ml
and the indentation can be expressed as
z(t) = Zm sin 2/3(
rt t'~ \2tin / "
(lO)
The plate deflection can then be obtained from eqn (4):
z2(t )
=
4Fmt2m" ( rt ) REml ~ t ~
s i n
+
(
VO
2Fmtm~t /tml /
-
Zm
sin2/3// rt t~ \2tin /
.
(ll)
S.-S. Pang et al.
1014
To determine the relation between tm and Fro, the condition at tm was applied:
Z(tm) = Zl(tm) -- z2(tm)
(12)
where
2/3
Z(tm) = Zm =
/~m/
(13)
Zl(tm) = Votm - ( 1 - 2) 2t2m m
(14)
and z2(tm) can be determined from eqn (6):
d2z2 = KZ 3/2 -
keq --~-
d2z2
(15)
meq dt 2
where the second derivative of z2(t) can be obtained from eqn (11):
d2z2
Fm sm ( It t) R2Zmsin-4/3f --~-~t~ cos ( t )It ml ~ - ~ \2tm / 2~m •
=
+ It2Zmsin2/3~ It t) ~ \2tin
(16)
then z2(tm) can be written as
l[(
)
meq _3/2 _ It2m2
Z2(tm)=~q K 1 + ml ~m
]
6t 2 Zm •
(17)
6keq It2meq~z t2J m - Votm = 0
(18)
Thus eqn (12) becomes
[ ( 1 + meqx) K m/ l keq + (l-2)
2Kt2m13/2+( ~-m~JZm
1
or [ 1 (m_~) ~eq 1 +
+
( 2 ) 1-
2t2m]E + 1 ( It2m2~F2/3_ ItmlJ m ~7~ l - 6t2mkeqj m rot m=O.
(19)
Both eqns (18) and (19) are algebraic equations in which the maximum indentation (Zm)or the maximum impact force (Fro)is related to the impact duration. Once the impact duration is determined, the maximum indentation and maximum impact force can be solved iteratively. To determine the equivalent stiffness keq and the equivalent mass meq, an analytical model for the single-lap joint under transverse loading has been developed based on the laminated anisotropic plate theory.
SINGLE-LAPCOMPOSITEJOINTUNDERTRANSVERSELOADING Analytical model Figure 2 shows the configuration of a single-lap joint under transverse loading, P, which is represented by force per unit width. The derivation of the displacement of the point under loading was based on the first-order laminated plate theory:
u = u°(x) + z~/(x)
(20)
w = w(x)
(21)
where u and w represent the displacement field of the two laminates in the x- and zdirections. The superscript " o " denotes the middle-plane dement, and ~, is its corresponding bending slope. Substitution of the displacement equations into kinematics relations and stress-strain equations yields the following relations regarding N x, the stress resultant in the x-direction, My, the unit width moment in the y-direction, and Qx, the
Impact response of single-lap composite joints
1015
transverse shear stress resultant, for orthotropic laminates under plane-strain conditions (Whitney, 1987) du °U N c = A ~ , - - ~ + Bx% dq/vdx
(22)
du oL N5 = ALl ~ + BL1 d~Ldx
(23)
du °U M y = BU ~ + D h dqzv dx
(24)
du °L M E = BE1 ~ + D~x d~'L dx
(25)
Q, = kU A~s(~uu + ~--~
(26) (27)
where k is the shear correction factor and [A], [B] and [D] are the matrices of the equivalent modulus for the laminate per unit width which are defined as hU/2
(AU,, B~,, Duo
(i) = I u Qt'(I'zI'z~I) dz" d - h /2
(Ah, Bh, oh) = I hL/2 ~(i)tl Z Z~2)dz2,
h°/2
(i)
A~s = .l_hu/2Qssdzl ALs =
(28)
hL/2
/| L rlti),4. ~ 5 5 u'~'2 " J-h /2
(29)
The ~l/3(i)land ~ssr~(i)represent the stiffness in x-direction and the shear stiffness of the ith ply, respectively. The superscripts U and L denote the upper and lower laminate, respectively, h is the thickness, and zl and z2 are measured from the middle-plane of the upper and lower laminate as shown in Fig. 2. Because of the varying laminate loading conditions, it is conventient to break the joint into four sections as shown in Fig. 2. The system of governing equations of each joint section is discussed separately in the following sections.
Joint section (1). In this joint section, the upper laminate sustains the force from the support and applied loading. The normal force resultant N ~ , moment resultant MyU~,and transverse shear stress resultant Q~ can be shown as: ¢
U
Nxl
= 0
P MyUi = - - ' 2 X l
,
ll
©
4-
P 2
-i= t ~ - - ~
l, =I~
:I: @-+-® +
(30)
l2
®
Fig. 2. Single-lapjoint under transverse load.
. T_ h L
1016
Pang et al.
S.-S.
~-- dz -'~
Q~
ov~dnU
+ Mya
: "r q
~M~+ d:
I
Fig. 3. Free body diagram and sign convention. where the sign conventions can be seen from Fig. 3, which shows a segment of the upper laminate from the joint section (3) as a free body diagram. Combining relations in eqns (22), (24) and (26) with the above, the system of governing equations regarding the three variables, u~U, ¢,~ and w~ becomes du~ U d¢/~ AIuI-~1 +BIU,~-T=0 au°U nu " I + •-'It dx---~-
DIU1
kUaU. u
de/u dx !
P Xl 2
-
dwU_
(31)
P
,-,55 ~1 + kUAU5 d x I
2"
Because the three eigenvalues of the above system are all zero, together with the particular solution, the solutions are three fourth-degree polynomials with three independent coefficients. These undertermined coefficients need to be solved from the boundary conditions. Joint section (2). The governing equations of the lower laminate in the joint section (2) are almost the same as those of the upper adherend in the joint section (1). The only difference is the direction of the transverse shear stress resultant.
ALI-d/,/~ L + B L d~/,/L = o P g du~'g O g d~2L B l l - ~ 2 + 11 dx 2 = - - - -2 (/2 - x2) kL . L
L
dwL
A55g/2 + kLAL5 dx 2 = --
(32)
P -2"
the solutions are three fourth-degree polynomials with three undetermined coefficients. Again,
Joint section (3). Joint section (3) together with joint section (4) are the overlay region. The governing equations can be described by (Yang and Pang, 1993)
d2u~ U
d2¢/~
r/
~ d2u~ L
AL
~,L + T ¢/3u + 2
+
(33)
d2¢/L
+ ah dx
= _G(u~L _ U~U) +
¢/L +
¢/~
--
+
(34)
Impact response of single-lapcompositejoints dq/3 U,--~-
BU , ~ + D
-
u u u k A55 qt3 +
Gr/ hU a G h° (h-~2 hU ) 2 (u~L - u°U) + ~ - ~ ~ + -~-- ~
B~1 ~
+
1017
m r d I],/3 r L XJll "-~ -- k A55
2 hU
G
+
22
+
~--~)
(35)
r
q/3 +
r/ 2 (u]L - u~u) + --r/-2-
~
kUAU5 - - + kLA~g5
+ -2- ~
(36)
dx 2It = q
(37)
+ ----~/] = - q
(38)
where r and q are the shear stress and peel stress (transverse normal stress) within the adhesive as shown in Fig. 3, respectively. Equations (33)-(38) are six coupled second-order ordinary differential equations with six variables: u~U, u°L3, q?U3,qJL3, W3 U, W3 r and with one unknown function, q, which is the peel stress of the adhesive between the two laminates. Because every function which is piece-wise smooth for a finite range can be expanded as a Fourier cosine series, the peel stress of the adhesive q can be represented by an (n + l)-term Fourier cosine series with (n + l) coefficients, Co-Cn, as
inx ci cos - -
q = i=0
(39)
13
where 13 stands for the length of this section and is half of the overlay length. By defining vi as the variables which need to be solved, U 1 = U 3OU ,
U2 = U~ L'
V3 = I]'/3U'
V4 = ~bcL,
05 = W 3U,
U6 =
wL
and by combining the homogeneous and particular solutions, vi can be written as 9
VJ=
i2=0 djixi Vj =
9 2
n IRX + ejlc°sh°taX + ejEsinh°tax + i~=i gj'" sin --,13 -
-
dji xi + ejl cosh a3x + ej2 sinh ot3x +
i=0
.
~
j=
inx gji c o s - - ,
i= 1
13
1,2,3,4
(40)
j = 5, 6
(41)
where ds and es are undetermined coefficients of the homogeneous solutions (Yang and Pang, 1993). Because eqns (33)-(38) are six second-order ordinary differential equations, the homogeneous solutions, shown as the first three terms in eqns (35) and (36) of the system, are so related that there are only 12 independent coefficients with which all dji, ej~, and ej2 can be determined. Also, gj~ can be obtained as functions of ci through the governing equations, eqns (33)-(38). The (n + 1) undetermined coefficients Co-% together with the 12 independent coefficients in ds and es result in a total of (n + 13) unknowns. The (n + 1) extra unknown coefficients from eqn (39) for the peel stress distribution can be covered by the (n + 1) extra equations from the joint kinematics and the orthogonal properties of Fourier series as
c, = 2EI.I/39 om= ~ o (dsm - d6m)X" COs ~,3rnx -dx + (e52 - %2)
f 13sinh(a3x) cos~-7\t3//rnx\ j dx 0
,.~//rnx\ dx + (es, - e6l) l] cosh(ot3x) cos/--~-)
w
+ (gs, - g6r) ~1 ,
r = 1, 2, 3 ..... n. (42)
S.-S. Pang et al.
1018
Joint section (4). The system of governing equations of the joint section (4) is identical to that of the joint section (3). Because the force is applied at the center of the overlay, the transverse shear stress resultant of the adhesive, peel and shear stresses of the adhesive will have discontinuity at the point under loading. It is for this reason that the overlay region has been split into sections (3) and (4). Combining the aggregate governing equations of the two adherends in all the four sections leads to a total of 12 second-order and six first-order ordinary differential equations including a total of 18 variables. The general solutions contain 30 undetermined coefficients. This is to say, 30 boundary conditions (or equations) are needed to solve the whole problem. These boundary conditions are obtained from either the continuity or the force equilibrium at the junctions between joint sections. Equivalent stiffness The equivalent stiffness keq for the spring-mass model can be obtained by solving the relation between the transverse deflection and the applied force at the center of the joint overlay. After solving the above systems of equations, the equivalent stiffness kCq can be written as P
(43)
keq = - - Wm
where P and wm( = wU(0)) are the transverse loading and deflection at the center of the overlay, respectively.
Equivalent mass In general, the solution of the displacement can be expressed as a summation of sinusoidal functions with various modes. For each specific mode, the magnitude of the acceleration of each cross-section of the laminate is proportional to the magnitude of its transverse deflection. After superposition of different modes, the resultant displacement and acceleration will not be directly proportional due to their phase shifts. However, in many applications, the first mode is dominant. In calculating the equivalent mass of this study, the acceleration of each cross-section is assumed to be proportional to its transverse displacement. Therefore, the equivalent mass for the developed spring-mass model as represented by the middle cross-section of the overlay can be determined from
meq
--
' (f
w4U(0)
+
0
p w U ( x I ) d x I -+-
i
o
PW2L(X2) d x 2 -I-
p[wU4(x4)+ w4L(x4)ldx,
fl
-I- wL(x3)] d X 3
(44)
0
where p is the mass of the laminate per unit length. E X P E R I M E N T A L STUDY
To verify the proposed model, low-velocity impact tests of single-lap composite joint samples were performed. A hemispherical-tipped impactor, with 25.4 mm diameter and a mass of 0.75 kg, was used in the tests. The sample material was Scotchply with cross-ply layout. Table 1 lists the material properties of the sample material which were provided by the manufacturer. The samples were 229 mm (9 in) long and 38 mm (1.5 in) wide with four different joint configurations: (a) 6.35 mm (1/4 in) thick adherends with 50.8 mm (2in) overlay; (b) 6.35mm (1/4in) thick adherends with 63.5 mm (2.5in) overlay; (c) 3.18 mm (1/8 in) thick adherends with 50.8 mm (2 in) overlay; and (d) 3.18 mm (1/8 in) thick adherends with 63.5 mm (2.5 in) overlay. The span between the two end supports was set at 178 mm (7 in) for the impact test. A total of 25 samples were tested.
Impact response of single-lap compositejoints
1019
Table 1. Material properties of Scotchplylaminates Longitudinal modulus EI~ Transverse modulus E22 Shear modulus G12 In-plane Poisson's ratio v~2 Resin content (by weight) Fiber content (by weight)
39.30 GPa (5.7 Mpsi) 9.70 GPa (1.4 Mpsi) 4.14 GPa (0.6 Mpsi) 0.26 38%
62O/o
Preliminary tests Conducting the preliminary static test serves two purposes: to provide a calibration curve which converts the voltage readings into the impact force, and to determine the outof-plane Young's modulus of the sample (Ec) which is needed when applying the modified Hertzian contact law. These tests were conducted in an Instron servohydraulic machine.
Weight drop tower The impact tests were conducted using a specially designed weight drop tower. The schematic diagram is shown in Fig. 4. A 2.4 m tall frame was constructed on a concrete base to increase the rigidity of the entire system. Two steel rods were attached to the frame to guide the " H " shape steel beam that holds the impactor. A timer unit was used to measure the time taken by the impactor to travel through a fixed distance. All the samples were simply-supported at two opposite sides and were free at the other two sides. Two semiconductor strain gages were bonded to the impactor. The impactor stroke was at the center of the overlap region.
Impact velocity In order to reach different values of impact velocity, the impactor was dropped from different heights. By measuring the time taken by the impactor to travel through a fixed distance, the accelerations, and consequently the velocities, could be calculated. Here the acceleration for each case was assumed to be constant.
Impact force To obtain the force history during impact, two semiconductor strain gages with large gage factor (approximately 50) were attached to the impactor at the near-tip locations. A strain gage conditioner and amplifier system (model 2100) was used to convert the signal from the gage to the oscilloscope, and the signal was filtered at 100 kHz. A Nicolet digital oscilloscope (model 4090C) was used to record the force signal, which was then stored on a floppy disk for later analysis. The digital sampling rate was set at a 1/~s interval. Figures 5(a)-(d) demonstrate the impact force history of four of the samples, No. 3, No. 1 l, No. 14 and No. 24; each represents one of the four sample configurations.
Wheatstonef_L Bridge l__. Amplifier~
Digital OscilloscopeI~-~
V--I
Frame ,. ~ - Guide Rod ~ Support Plate ~ - Impactor
~ - Strain Gage Composite Plate Clamp ,/- Concrete Base ~'///, 1//11//////11/ ////~ Fig. 4. Weight drop tower.
S.-S. Pang et
1020
2.5 z
,
.
,
.
,
5.0
Sample No. 3 Thickness = I/8" Overlay ffi 2"
2.0
al. ,
4.0
o~ a.o
1.5
0
l
1.0
~" 2.0
~u
1.0
0.5 0.0
,
•
,
Sample No. 11 Thickness = 1/4" Overlay : 2"
j
v
o r~
•
0.0 i
0.5
0.0
1,0
1.5
f
2.0
0.0
0.5
Time ( m s )
,
I
i
1.5
2.0
Time ( m s )
(b)
(a)
2.5
i
1.0
•
,
•
,
.
,
5.0
.-''
,
-
,
•
,
Thickness= 1/8" Overlay = 2.5"
1.5
4.0
3
,
•
,
Sample No. 24 Thickness = I/4" Overlay = 2.5"
Sample No. 14
2,0
•
3.0
I. o
t~ 2.0
1.0
e~
E 0.5
,~
1
0.0
1.0
0.0 i
i
0.0
0.5
i
i
I
1.0 1.5 Time (ms)
2.0
i
i
i
i
i
0.0
0.5
1.0
1.5
2.0
(e)
Time (ms)
(d) Fig. 5. Impact force history.
RESULTS AND DISCUSSION
The spring-mass model was used for the analysis of a single-lap joint under lowvelocity impact. The effect of impact is represented by a time-dependent force which can be solved from a quasi-static equilibrium condition. For the case in which a wave can be reflected many times within the sample during the impact period, a quasi-static equilibrium condition can be assumed. In the current study, the wave speed within the laminated plate is approximately 2300 m/s (VEV~, where E = 9.7 GPa and p = 1829 kg/m3). Based on experimental data, the impact period corresponding to the first wave is about 0.2 ms, which represents at least 18 round trips of the wave during the impact period for the 6.35 mm (1/4 in) laminate or 36 round trips for the 3.18 mm (1/8 in) laminate. Therefore, the quasi-static equilibrium condition can be approximately established (Pang and Kailasam, 1991). The spring-mass model of a single-lap composite joint consists of two main parameters, equivalent mass and stiffness constants. Both of them are directly related to w, which is the transverse displacement of the structure. As can be seen from eqn (44), the deformed shape of the single-lap joint is utilized to determine the equivalent mass which is included in the spring-mass model. On the other hand, the equivalent stiffness represents the relation between the applied loading and the center deflection of the joint. Therefore, the development of the joint model is essential in order to obtain the equivalent stiffness and equivalent mass of the joint for the impact analysis. Tables 2 and 3 give the equivalent stiffness and mass for the four different configurations of impact test samples. In order to obtain different impact velocities, the drop height was varied for each individual case during the experimental study. When calculating the impact velocity, the acceleration was assumed to be constant for each individual case but different from case
1021
Impact response of single-lap composite joints Table 2. Equivalent stiffness of impact samples
3.18 mm (1/8 in) thick adherends 6.35 mm (I/4 in) thick adherends
50.8 mm (2 in) overlay
63.5 mm (2.5 in) overlay
97.60 kN/m 694.27 kN/m
114.43 kN/m 845.42 kN/m
Table 3. Ratio of equivalent mass to actual mass of impact samples
3.18 mm (1/8 in) thick adherends 6.35 mm (1/4 in) thick adherends
50.8 mm (2 in) overlay
63.5 mm (2.5 in) overlay
0.799 0.796
0.758 0.789
to case. It can be seen from the impact force history, Figs 5(a)-(d), that during the impact a compressive wave propagates to the top of the impactor and reflects as a tensile wave. Compressive and tensile waves cancel each other resulting in a decrease in the force. This process repeats until the separation of the impactor and the sample target occurs. It can also be noticed from Figs 5(a)-(d) that the maximum force occurs at the first wave for all cases. To predict the maximum impact force, the spring-mass system equations, eqns (5) and (6), were simply applied to the first wave period. Once the time duration is obtained from the impact test, the maximum impact force can be solved from the relation given by eqn (19). The analytical and the experimental results for all the test samples are tabulated in Tables 4-7, where the duration corresponds to the first wave
Table 4. Impact force of samples with 50.8 mm (2 in) overlay and 3.18 mm (1/8 in) thick adherends Sample no.
Velocity (m/s)
Duration (p.s)
Analytical force (kN)
Experimental force (kN)
Deviation (070)
1 2 3 4 5 6 7 8
4,476 4.449 4.035 4.040 3.856 3.830 3.787 3.635
164 168 170 171 164 172 175 171
1.919 1.674 1.557 1.506 1.553 1.453 1.318 1.497
1.983 1.890 1.816 1.584 1.812 1.495 1.437 1.486
3.23 11.43 14.26 4.92 14.29 2.81 8.28 0.74
Table 5. Impact force of samples with 50.8 mm (2 in) overlay and 6.35 mm (1/4 in) thick adherends Sample no.
Velocity (m/s)
Duration (~)
Analytical force (kN)
Experimental force (kN)
Deviation (%)
9 I0 11 12 13
4.578 4.342 4.147 4.086 3.837
181 188 184 188 191
4.652 4.017 4.213 3.768 3.445
4.443 4.142 4.011 3.266 3.441
4.70 3.02 5.04 15.37 0.12
Table 6. Impact force of samples with 63.5 mm (2.5 in) overlay and 3.18 mm 0 / 8 in) thick adherends Sample no.
Velocity (m/s)
Duration (/~s)
Analytical force (kN)
Experimental force (kN)
Deviation (070)
14 15 16 17 18 19
4.271 4.242 4.195 4.126 3.784 3.740
232 238 231 240 237 238
2.113 1.587 1.958 1.513 1.609 !.571
1.910 1.833 1.881 1.702 1.706 1.592
10.63 13.42 4.09 11.10 5.69 1.32
1022
S.-S. Pang et al.
Table 7. Impact force of samples with 63.5 mm (2.5 in) overlay and 6.35 mm (1/4 in) thick adherends Sample no.
Velocity (m/s)
Duration (p.s)
Analytical force (kN)
Experimental force (kN)
Deviation (%)
20 21 22 23 24 25
4.373 4.344 4.269 4.164 4.120 4.031
217 216 214 224 222 229
4.022 4.106 4.278 3.434 3.570 3.086
4.243 4.170 3.930 4.142 3.629 2.875
7.39 1.53 8.85 17.09 3.30 7.34
period. Figures 6(a)-(d) show the results o f the maximum impact force for four of the samples, each representing one type of sample configuration. The analytical curve for each case is represented by a dotted line which simulates the first wave of the impact curve. It is known that the maximum impact force depends upon many factors such as impact velocity, impactor weight and geometry, target-sample geometry (thickness, adhesive overlay) and support condition, as well as impactor and target material properties. To understand the effects of adhesive overlay length and adherend thickness on the impact force, four types of sample configurations have been used in the impact test, including two adherend thicknesses, 3.18 mm (1/8 in) and 6.35 mm (1/4in), and two overlay lengths, 50.8 mm (2 in) and 63.5 mm (2.5 in). It is shown from Tables 4-7 that the impact velocities directly affect the maximum impact forces. The higher the impact velocity, the greater the maximum impact force. 3.0
2.5
5.0
Sample No. 3 Thickness = I/8" Overlay = 2"
4.0
2.0
3.0
0= ..o
1.s
k
1.0
2.0
0.5
i:u ~ 1.0
~
Sample No. 11 Thickness : 1/4" O v e r l a y . 2"
r.
0.0
0.0 i
i
0.10
0.00
Time
i
i
0.20
0.30
i
0.00
(ms)
i
i
i
0.10
0.20
0.30
Time
(a)
(ms)
(b)
3.0
5.0
S a m p l e No. 14 Thickness = 1/8"
2.5
Overlay : 2.5"
Z
2.0
= o o
1.s
/~"
\\
#\
"~ 1.0 a. E 0.5
~
Sample No. 24 Thickness = i/4" Overlay = 2.5"
4.0
.= 3.o #i.
',,
\\
2.0 o. E
0.0
t.0
0.0 i
i
i
i
i
i
,
0.00
0.10
0.20
0.30
0.00
0.10
0.20
Time
(ms)
'Time
(c)
(ms)
(d) Fig. 6. Maximum impact force.
0.30
1023
Impact response of single-lap composite joints
:
150
1200
125
iooo
z 800
g 100 e-
600
03
400 50
I
0.04
i
i
I
0.05
0.06
0.07
I
L
0.04
i
i
0.05 0,06 0.07 Overlay Length (m)
Overlay Length (m) ~)
(a)
Fig. 7. Effect of overlay length on equivalent stiffness.
Due to the limitation of the weight drop tower used in this test, only limited velocity variations have been achieved. As a result, no experimental relation between impact velocity and maximum impact force has been obtained. The effect of adherend thickness on the maximum impact force can also be identified from the analytical and experimental results. The maximum impact forces of 6.35 mm (1/4 in) samples are greater than those of 3.18 mm (1/8 in) samples. For samples with 50.8 mm (2 in) adhesive overlay, the average experimental maximum impact force of the 6.35 mm (1/4 in) samples is about 2.35 (or 2.58 for analytical value) times as large as that of the 3.18mm (1/8in) samples, while for samples with 63.5mm (2.5in) adhesive overlay, the average experimental maximum impact force of the 6.35 mm (1/4 in) samples is 2.16 (or 2.17 for analytical value) times as large as that of the 3.18 mm (1/8 in) ones. One of the reasons for such a difference is that the equivalent stiffness has been changed significantly when the thickness of the adherends changes from 3.18mm (1/8 in) to 6.35 mm (1/4 in). The length of adhesive overlay also affects the equivalent stiffness. An analytical relation between the overlay length and the equivalent stiffness is shown in Figs 7(a) and 7(b), for 3.18mm (1/8in) and 6.35 mm (1/4in) samples, respectively, with length of overlay varying from 38.1 mm (1.5 in) to 76.2 mm (3 in). By integrating the force history with respect to time, the impulse can be obtained. Tables 8-11 list both analytical and experimental results. Again, the impulse here corresponds to the impact force history within the first wave period. Figures 8(a)-(d) demonstrate the results for four of the samples; each represents one different sample configuration. The deviation between the analytical and the experimental results has been provided for all of the cases. The maximum deviation for impact force prediction is 17.09% (sample No. 23); while for impulse, the maximum deviation is 34.64% (sample No. 14).
Table 8. Impulse of samples with 50.8 mm (2 in) overlay and 3.18 mm (1/8 in) thick adherends Sample no.
Velocity (m/s)
Duration (ps)
Analytical impulse (N s)
Experimental impulse (N s)
Deviation (%)
1 2 3 4 5 6 7 8
4,476 4.449 4.035 4,040 3.856 3.830 3,787 3.635
164 168 170 171 164 172 175 171
0.1967 0.1790 0.1685 0.1639 0.1621 0.1591 0.1468 0,1630
0.1891 0.1885 0.1771 0.1756 0.1635 0.1871 0.1324 0.1843
4.02 5.04 4.86 6.66 0.86 14.97 10.88 11.56
1024
S.-S. Pang et al. Table 9. Impulse of samples with 50.8 mm (2 in) overlay and 6.35 mm (1/4 in) thick adherends Sample no.
Velocity (m/s)
Duration (p.s)
impulse(Ns)
Analytical
Experimental impulse(Ns)
Deviation (%)
9 10 11 12 13
4.578 4.342 4.147 4.086 3.837
181 188 184 188 191
0.5360 0.4808 0.4935 0.4510 0.4189
0.4761 0.4414 0.4606 0.3918 0.3862
12.58 8.93 7.14 15.11 8.47
Tabk 10. Impulse of samples with 63.5 mm (2.5 in) overlay and 3.18 mm (1/8 in) thick adherends Sample no.
Velocity
Duration
(m/s)
(~)
Analytical impulse (N s)
Experimental impulse (N s)
Deviation (%)
14 15 16 17 18 19
4.271 4.242 4.195 4.126 3.784 3.740
232 238 231 240 237 238
0.3121 0.2404 0.2879 0.2312 0.2428 0.2380
0.2318 0.2090 0.2157 0.2187 0.1929 0.1838
34.64 15.02 33.47 5.72 25.87 29.49
Table 11. Impulse of samples with 63.5 mm (2.5 in) overlay and 6.35 mm (1/4 in) thick adherends Sample no.
Velocity (m/s)
Duration (p.s)
Analytical impulse (N s)
Experimental impulse (N s)
Deviation (%)
20 21 22 23 24 25
4.373 4.344 4.269
217 216 214 224 222 229
0.5556 0.5646 0.5964 0.4897 0.5045 0.4499
0.4875 0.5819 0.4727 0.4516 0.4769 0.4162
13.97 2.97 26.17 8.44 5.79 8.10
4.164
4.120 4.031
•
0.4 "W 0.3 z 0.2
,
-
,
•
,
.
,
-
,
•
-
r
Sample No. 3 Thickness = 1/8" Overlay = 2"
,
•
,
.
=
0.6
.
,
.
.
,
,
I
,
,
z
Z
f
0.3 ,~ 0.2
0.0
0.0 I
i
i
0.05
0.10
0.15
i
0.00
i
i
.
0.20 0.25
=
.
i
.
i
i
0.00 0.05 0.10 0.15 0.20 0.25 Time (ms)
Time (ms) (a)
0.3
,
"G" 0.5
.~ 0.1
0.4
•
Sample No. 11 Thickness = 1/4" Overlay ffi2"
0a) "
Sample No. 14 Thickness= I/8" Overlay = 2.5"
0.6 ~ --"
0.5
,
.
,
-
i
-
,
Sample No. ?.4 Thickness = 1/4" Overlay = 2.5"
Z
0.2
0.3
.~0.1
~ 0.2
0.0
0.0 i
i
i
0.00
0.05
0.10
~
0.15
I
I
0.20
0.25
Time (ms)
0.00
0.05
0.10
0.15
Time (ms)
(c)
(d) Fig. 8. Impulse during impact.
0.20
0.2
Impact response of single-lap composite joints
1025
When measured by average values, a 7.60o70 deviation has been noticed for the 3.18 mm (1/8 in) thick samples with 50.8 mm (2 in) adhesive overlay, a 4.1007o deviation for the 6.35 mm (1/4 in) thick samples with 50.8 mm (2 in) overlay, a 2.57o7o deviation for the 3.18 mm (1/8 in) thick sample with 63.5 mm (2.5 in) overlay, and a 2.15070 deviation for the 6.35 mm (1/4 in) thick samples with 63.5 mm (2.5 in) overlay. There are many factors which may cause the analytical results to deviate from the experimental results. For example, the uneven friction between guiding rods and the bearings in the " H " shape beam which holds the impactor will cause the changes in acceleration from point to point, whereas the acceleration is assumed to be constant throughout the impact process in the calculation of the impact velocity. Damping is another cause for the deviation between the analytical and experimental results. In the analytical model, the damping effect has not been considered; while in the actual impact test, damping does exist due to the supporting conditions, adhesive layer, etc. Nethertheless, for the purpose of engineering design, the developed analytical model still provides a reasonably good prediction for the maximum impact force for such a complicated composite structure. From the impact force, a feedback calculation can be performed to predict the adhesive stresses, such as shear and peel. In order to accomplish these, the impact force is first used to calculate the P in the governing equations (eqns (30)-(32)) and the boundary conditions, then the six variables of the displacement field (u °U, u °L, WU, WE, wU, wE) in joint sections (3) and (4) can be calculated. Subsequently with eqns (34) and (37), the adhesive shear stress z and peel stress q can be calculated, respectively. CONCLUSIONS
The impact response of adhesive-bonded single-lap composite joints was investigated. A simplified spring-mass model has been applied to represent the impactor-target system. During the impact period of a single wave of the structure vibration, the transverse shock wave can travel at least 18 round trips for 6.35 mm (1/4 in) laminate or 36 round trips for 3.18 mm (l/8 in) laminate. The quasi-static equilibrium can then be assumed to be established during the impact process. An equivalent mass with equivalent stiffness has been used to represent the complicated single-lap composite joint in the model. An analytical model has been developed to determine the equivalent mass and the equivalent stiffness based on the laminated anisotropic plate theory in which the governing equations of the two bonded laminates have been derived, and the entire coupled system was solved in closed-form using Fourier series and appropriate boundary conditions. Combining the developed model with the quasi-static spring-mass model, a relationship between the impact duration and the impact force has been obtained. Low-velocity impact tests of single-lap joints with two different adherend thicknesses and two overlay lengths have been conducted to verify the proposed model. Analytical and experimental results of maximum impact forces, as well as impulses, have been provided. The relationship between the impact duration and the impact force obtained from the developed analytical model can be used to predict the maximum impact force during the low-velocity impact. The results show that the average deviations between the predicted values and test data range from 2.15o7o to 4.10070, depending upon the sample configurations. The maximum deviation for an individual case is around 17°70. The maximum impact force is affected by many factors such as the impact velocity and the adherend thickness. The impact velocity directly affects the maximum impact force. The higher the impact velocity, the greater the maximum impact force. The thickness of the adherends has a notable influence on the value of the maximum impact force because the thickness change will cause a significant change in the equivalent stiffness of the single-lap joint sample. On the other hand, the overlay length of the joint shows no obvious effect on the maximum impact force. Based on the current study, the spring-mass model can provide a reasonably good prediction for cases where impact failure occurs after a long enough time to allow wave reflections to establish quasi-static equilibrium. But it may not be adequate for situations that result from localized stress wave action near the point of impact.
1026
S.-S. Pang et al.
A methodology of predicting adhesive stresses during impact was proposed. The maximum adhesive stresses can be utilized to predict the joint strength if the adhesive failure criterion is available. DEDICATION
This paper is specially dedicated to Professor Werner Goldsmith's 70th birthday on May 23, 1994. Professor Goldsmith is the Professor Emeritus of the Mechanical Engineering Department at the University of California, Berkeley, and served as the PhD advisor of Su-Seng Pang, the first author of this paper, during the period of 1983-86. REFERENCES Adams, R. D. and Wake, W. C. (1984). Structural Adhesive Joints in Engineering. Elsevier Applied Science Publishers, London and New York. Bert, C. W. (1973). Simplified analysis of static shear factors for beams of nonhomogeneous cross section. J. Comp. Mater. 7, 525-529. Chow, J. S. (1971). On the propagation of flexural waves in an orthotropic plate and its response to an impulsive load. J. Comp. Mater. 5, 306-319. Dharmarajan, S. and McCutchen, H. Jr (1973). Shear coefficients for orthotropic beams. J. Comp. Mater. 7, 530-535. Goland, M. and Reissner, E. (1944). The stresses in cemented joints. J. Appl. Mech. 11, AI7-A27. Goldsmith, W. (1960). Impact: The Theory and Physical Behavior of Colliding Solids. Edward Arnold, London. Griffin, S. A., Pang, S. S. and Yang, C. (1991). Strength model of adhesive bonded composite pipe joints under tension. Polym. Engng Sci. 31(7), 533-538. Hart-Smith, L. J. (1973a). Adhesive-bonded single-lap joints. Douglas Aircraft Co., NASA Langley Report CR-112236. Hart-Smith, L. J. (1973b). Adhesive-bonded double-lap joints. Douglas Aircraft Co., NASA Langley Report CR-112235. Hart-Smith, L. J. (1973c). Adhesive-bonded scarf and stepped-lap joints. Douglas Aircraft Co., NASA Langley Report CR- 112237. Hart-Smith, L. J. (1974). Analysis and design of advanced composite bonded joints. Douglas Aircraft Co., NASA Langley Report CR-2218. Kutscha, D. (1964). Mechanics of adhesive-bonded lap-type joints: survey and review. Technical Report AFML-TDR-64-298. Kutscha, D. and Hofer, K. E. Jr (1969). Feasibility of joining advanced composite flight vehicles. Technical Report AFML-TR-68-391. Matthews, F. L., Kiity, P. F. and Goodwin, E. W. (1982). A review of the strength of joints in fibre-reinforced plastics: Part 2. Adhesively bonded joints. Composites 13, 29-37. Mindlin, R. D. (1951). Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J. AppL Mech. 18, 336-343. Pang, S. S. and Kailasam, A. A. (1991). A study of impact response of composite pipe. J. Energy Resource TechnoL 113, 182-188. Pang, S. S., Zhao, Y., Yang, C. and Griffin, S. A. (1991). Impact response of composite laminates with a hemispherical indenter. Polym. Engng Sci. 31(20), 1461-1466. Reddy, J. N. (1984). A refined nonlinear theory of plates with transverse shear deformation. Int. J. Solids Struct. 20(9/10), 881-896. Reissner, E. (1945). The effect of transverse shear deformation on the bending of elastic plates. J. AppL Mech. 12, A69-A77. Shivakumar, K. N., Elber, W. and lllg, W. (1983). Prediction of impact force and duration during low velocity impact on circular composite laminates. NASA-TM-85703. Sun, C. T. and Yang, S. H. (1980). Contact law and impact response of laminated composites. NASACR-159884. Tsai, S. W. and Hahn, H. T. (1980). Introduction to Composite Materials. Technomic Publishing Company, Inc., Lancaster, PA. Vinson, J. R. (1989). Adhesive bonding of polymer composites. Polym. Engng Sci. 29(19), 1325-1331. Whitney, J. M. (1972). Stress analysis of thick laminated composite and sandwich plates. J. Comp. Mater. 6, July, 426-440. Whitney, J. M. (1973). Shear correction factors for orthotropic laminates under static load. J. Appl. Mech. 40, 302-304. Whitney, J. M. (1987). Structural Analysis of Laminated Anisotropic Plates. Technomic Publishing Company, Inc., Lancaster, PA. Whitney, J. M. and Pagano, N. J. (1970). Shear deformation in heterogeneous anisotropic plates. J. Appl. Mech. 37, 1031-1036. Yang, C. and Pang, S. S. (1993). Stress-strain analysis of single-lap composite joints under tension. Proc. 1993 A S M E Energy-Sources Technology Conference & Exhibition, Composite Material Technology (Edited by D. Hui, T. J. Kozik and O. O. Ochoa). PD-Vol. 53, pp. 85-94.
Impact response of single-lap composite joints
1027
Yang, C. and Pang, S. S. (1993). Stress-strain analysis of adhesive-bonded single-lap composite joints under cylindrical bending. Comp. Engng 3(1 l), 1051-1063. Yang, C., Pang, S. S. and Griffin, S. A. (1992). Failure analysis of adhesive-bonded double-lap joints under cantilevered bending. Polym. Engng Sci. 32(9), 632-640.
COE 5-8-E
Pergamon
0961-9526(95) 00003-8
IMPACT RESPONSE OF SINGLE-LAP COMPOSITE JOINTS Su-Seng Pang t, Chihdar Yang and Yi Zhao Department of Mechanical Engineering, Louisiana State University, Baton Rouge, LA 70803, U.S.A. (Received 12 July 1994;final version accepted 9 December 1994)
Abstract--The low-velocity impact of adhesive-bonded single-lap composite joints has been studied using a spring-mass model. In this quasi-static model, the impact response is represented by a time-dependent force, and the target joint is represented by an equivalent mass with equivalent stiffness. An analytical model has been developed to determine the equivalent mass and stiffness of the joint. The laminated anisotropic plate theory was used in the derivation of the governing equations of the two bonded laminates. The entire coupled system, as well as the assumed peel stress, were solvedusing both the joint kinematics and suitable boundary conditions. With the combination of a spring-mass equilibrium system and the developed joint model, a relationship between the impact force and the duration has been established. Adhesive stresses, which are believed to be the cause of failure, were predicted from the impact force. Impact tests of single-lap composite joints with different sample thicknesses and overlay lengths have been conducted to verify the proposed model. INTRODUCTION With the introduction of high performance fibers such as carbon, boron and Kevlar, along with some new and improved matrix materials, advanced composites have established themselves as engineering structural materials. Adhesive-bonded joints have been widely used for composite materials as an alternative to conventional mechanical joint designs. The primary limitation of such designs arises from machining difficulties and subsequent damage to the laminate following these operations. In order to fully utilize composite materials, more research needs to be conducted in the field of joint design and analysis. One can find review papers regarding adhesive-bonded composite lap joints in the literature. Kutscha (1964) reviewed the early analytical work on isotropic adherends prior to 1961, and the analyses from 1961 to 1969 were reviewed by Kutscha and Hofer (1969). In 1982, Matthews et al. reviewed the theoretical work, including classical and finite element methods related to all aspects of adhesive-bonded joints in composite materials. Vinson (1989) summarized the published work dealing with the adhesive bonding of polymer matrix composite structures. When studying adhesive-bonded lap joints, the effects due to the rotation of the adherends were first taken into account by Goland and Reissner (1944). They introduced an equation to relate the bending moment of the adherend at the end of the overlay to the in-plane loading. The basic approach of the Goland and Reissner theory was based on beam theory, or rather, on cylindrically bentplate theory which treated the overlay section as a beam of twice the thickness of the adherend. Hart-Smith (1973a, b, c, 1974) published a series of papers regarding singlelap, double-lap, scarf, and stepped-lap joints involving a continuum model in which the adherends were isotropic or anisotropic elastic, and the adhesive was modeled as elastic, elastic-plastic, or bielastic. From these ample review papers, it is found that a considerable effort has been expended to determine the structural behavior of the most commonly used joints for composite laminates--single-lap joints. Most of the previous research on composite joints was based on the strength of materials theory and the classical plate theory, which neglect deflections and stresses attributed to the transverse shearing effects. In general, the *To whom correspondence should be addressed. 1011 C0E 5-8-0
1012
s.-s. Pang et al.
strength of materials approach can achieve acceptable accuracy when a plate has a large span-to-depth ratio (Reissner, 1945; Reddy, 1984). However, when the overlay length is small, the transverse shearing effects become significant. Neglecting the transverse shearing effects could result in considerable error in the model. Yang et al. (1992) studied double-lap composite joints under cantilevered bending and proposed a strain gap model to describe the stress-strain behavior. They also proposed (1993) an analytical model for adhesive-bonded single-lap composite joints under cylindrical bending based on the laminated anisotropic plate theory. The asymmetry of the adherend laminates and the transverse shear deformation were included in their models. Yang and Pang (1993) also developed an analytical model to predict the stress-strain distributions within a single-lap joint under tensile loading. In the current study, the dynamic behavior of the single-lap composite joint under impact load has been investigated. The analytical model has been developed based on Yang and Pang's previous work on single-lap joints which has been extended to specific loading and boundary conditions in the current paper. The spring-mass model (Shivakum a r et al., 1983) has also been applied to the impactor-target system to establish the relationship between impact duration and impact force. Impact tests have been conducted on single-lap joint samples with different lay-ups to verify the developed analytical model. IMPACT MODELOF SINGLE-LAPCOMPOSITEJOINT The system of an isotropic hemispherical impactor and a single-lap composite joint can be represented by a spring-mass model. The impact force can be related to the indentation through a modified Hertzian contact law. The equivalent stiffness keq of the singlelap composite joint can be theoretically determined by the model developed in a later section. By applying Newton's second law of motion, the equations of motion of the impactor-target system can be established. Using appropriate boundary conditions, a relationship between impact duration and maximum impact force can finally be obtained. C o n t a c t law f o r c o m p o s i t e s
The well known Hertzian contact law was originally proposed for contact of two elastic isotropic spheres. The contact between an elastic sphere and an elastic flat plate is a special case in which the radius of one sphere is infinite. In this case, the relationship between the contact force, F, and indentation, z, can be expressed as and
F K = 4R,/Z(1
3
(1)
= KZ ~/2
- v2
\
E1
1 - v22~ -1
+---~2 ]
(2)
where the subscripts 1 and 2 denote indenter and target, respectively, R is the radius of the indenter, v is Poisson's ratio, and E is Young's modulus. For contact between isotropic and composite materials, K can be determined from the modified contact law (Sun and Yang, 1980): /,"
= 4R, 3
(1 \
E
1) -1 +
(3)
where E is the Young's modulus of the isotropic indenter, and E c is the Young's modulus of the composite target in the direction normal to the contact plane. Spring-mass m o d e l
A spring-mass model was applied for the impact of laminated composite plates with a single-lap joint. The load-deflection behavior of the beam was represented by an equivalent stiffness keq. As shown in Fig. 1, z l ( t ) and z2(t) are the deflection of the impactor with a mass m~ and the central deflection of the single-lap joint with an equivalent m a s s m e q . Both the equivalent stiffness k,q and the equivalent m a s s meq for the single-lap joint are related to the geometry and material properties of the joint, and the contact and
Impact response of single-lapcompositejoints
1013
I-'--7
Fig. 1. Impactor displacement,indentation, and sample deflection. supporting conditions, Both will be determined through a detailed analytical model in the later sections. During the contact, the indentation at time t is
z(t) = zl(t) -
z2(t).
(4)
By applying Newton's second law of motion to the spring-mass system, the equations of motion are d2Zl ml--~ + d2z2 meq " - ~ +
KZa/2 =
0
(5)
keqz2 - KZ3/2 =
0
(6)
where keq is the equivalent stiffness of the single-lap joint. Here, the modified Hertzian contact law was applied to the single-lap joint with finite width and thickness. Such an application is valid only if the width of the beam is considerably larger than the contact area and if the thickness of the beam is considerably larger than the indentation. The impact force wave consists of several single waves, and each of them can be modeled as F = Fmsin
t
(7)
where Fm is the peak value of impact force within the single wave period, and tm is the time corresponding to Fm. A similar sinusoidal expression has been used by several researchers (Sun and Yang, 1980; Pang et al., 1991, etc.). The initial conditions for z~(t) can be expressed as Zl(O) = 0 dzl(0) dt = v0 where Vo is the impact velocity. From eqns (5)-(8), the solution of
zl(t)
z~(t) can
(8)
be obtained as
. r t ) ( + Vo 2Fmtm~t 4Fmt2msln(--t \2tm nml /
(9)
= rt2ml
and the indentation can be expressed as
z(t) = Zm sin 2/3(
rt t'~ \2tin / "
(lO)
The plate deflection can then be obtained from eqn (4):
z2(t )
=
4Fmt2m" ( rt ) REml ~ t ~
s i n
+
(
VO
2Fmtm~t /tml /
-
Zm
sin2/3// rt t~ \2tin /
.
(ll)
S.-S. Pang et al.
1014
To determine the relation between tm and Fro, the condition at tm was applied:
Z(tm) = Zl(tm) -- z2(tm)
(12)
where
2/3
Z(tm) = Zm =
/~m/
(13)
Zl(tm) = Votm - ( 1 - 2) 2t2m m
(14)
and z2(tm) can be determined from eqn (6):
d2z2 = KZ 3/2 -
keq --~-
d2z2
(15)
meq dt 2
where the second derivative of z2(t) can be obtained from eqn (11):
d2z2
Fm sm ( It t) R2Zmsin-4/3f --~-~t~ cos ( t )It ml ~ - ~ \2tm / 2~m •
=
+ It2Zmsin2/3~ It t) ~ \2tin
(16)
then z2(tm) can be written as
l[(
)
meq _3/2 _ It2m2
Z2(tm)=~q K 1 + ml ~m
]
6t 2 Zm •
(17)
6keq It2meq~z t2J m - Votm = 0
(18)
Thus eqn (12) becomes
[ ( 1 + meqx) K m/ l keq + (l-2)
2Kt2m13/2+( ~-m~JZm
1
or [ 1 (m_~) ~eq 1 +
+
( 2 ) 1-
2t2m]E + 1 ( It2m2~F2/3_ ItmlJ m ~7~ l - 6t2mkeqj m rot m=O.
(19)
Both eqns (18) and (19) are algebraic equations in which the maximum indentation (Zm)or the maximum impact force (Fro)is related to the impact duration. Once the impact duration is determined, the maximum indentation and maximum impact force can be solved iteratively. To determine the equivalent stiffness keq and the equivalent mass meq, an analytical model for the single-lap joint under transverse loading has been developed based on the laminated anisotropic plate theory.
SINGLE-LAPCOMPOSITEJOINTUNDERTRANSVERSELOADING Analytical model Figure 2 shows the configuration of a single-lap joint under transverse loading, P, which is represented by force per unit width. The derivation of the displacement of the point under loading was based on the first-order laminated plate theory:
u = u°(x) + z~/(x)
(20)
w = w(x)
(21)
where u and w represent the displacement field of the two laminates in the x- and zdirections. The superscript " o " denotes the middle-plane dement, and ~, is its corresponding bending slope. Substitution of the displacement equations into kinematics relations and stress-strain equations yields the following relations regarding N x, the stress resultant in the x-direction, My, the unit width moment in the y-direction, and Qx, the
Impact response of single-lap composite joints
1015
transverse shear stress resultant, for orthotropic laminates under plane-strain conditions (Whitney, 1987) du °U N c = A ~ , - - ~ + Bx% dq/vdx
(22)
du oL N5 = ALl ~ + BL1 d~Ldx
(23)
du °U M y = BU ~ + D h dqzv dx
(24)
du °L M E = BE1 ~ + D~x d~'L dx
(25)
Q, = kU A~s(~uu + ~--~
(26) (27)
where k is the shear correction factor and [A], [B] and [D] are the matrices of the equivalent modulus for the laminate per unit width which are defined as hU/2
(AU,, B~,, Duo
(i) = I u Qt'(I'zI'z~I) dz" d - h /2
(Ah, Bh, oh) = I hL/2 ~(i)tl Z Z~2)dz2,
h°/2
(i)
A~s = .l_hu/2Qssdzl ALs =
(28)
hL/2
/| L rlti),4. ~ 5 5 u'~'2 " J-h /2
(29)
The ~l/3(i)land ~ssr~(i)represent the stiffness in x-direction and the shear stiffness of the ith ply, respectively. The superscripts U and L denote the upper and lower laminate, respectively, h is the thickness, and zl and z2 are measured from the middle-plane of the upper and lower laminate as shown in Fig. 2. Because of the varying laminate loading conditions, it is conventient to break the joint into four sections as shown in Fig. 2. The system of governing equations of each joint section is discussed separately in the following sections.
Joint section (1). In this joint section, the upper laminate sustains the force from the support and applied loading. The normal force resultant N ~ , moment resultant MyU~,and transverse shear stress resultant Q~ can be shown as: ¢
U
Nxl
= 0
P MyUi = - - ' 2 X l
,
ll
©
4-
P 2
-i= t ~ - - ~
l, =I~
:I: @-+-® +
(30)
l2
®
Fig. 2. Single-lapjoint under transverse load.
. T_ h L
1016
Pang et al.
S.-S.
~-- dz -'~
Q~
ov~dnU
+ Mya
: "r q
~M~+ d:
I
Fig. 3. Free body diagram and sign convention. where the sign conventions can be seen from Fig. 3, which shows a segment of the upper laminate from the joint section (3) as a free body diagram. Combining relations in eqns (22), (24) and (26) with the above, the system of governing equations regarding the three variables, u~U, ¢,~ and w~ becomes du~ U d¢/~ AIuI-~1 +BIU,~-T=0 au°U nu " I + •-'It dx---~-
DIU1
kUaU. u
de/u dx !
P Xl 2
-
dwU_
(31)
P
,-,55 ~1 + kUAU5 d x I
2"
Because the three eigenvalues of the above system are all zero, together with the particular solution, the solutions are three fourth-degree polynomials with three independent coefficients. These undertermined coefficients need to be solved from the boundary conditions. Joint section (2). The governing equations of the lower laminate in the joint section (2) are almost the same as those of the upper adherend in the joint section (1). The only difference is the direction of the transverse shear stress resultant.
ALI-d/,/~ L + B L d~/,/L = o P g du~'g O g d~2L B l l - ~ 2 + 11 dx 2 = - - - -2 (/2 - x2) kL . L
L
dwL
A55g/2 + kLAL5 dx 2 = --
(32)
P -2"
the solutions are three fourth-degree polynomials with three undetermined coefficients. Again,
Joint section (3). Joint section (3) together with joint section (4) are the overlay region. The governing equations can be described by (Yang and Pang, 1993)
d2u~ U
d2¢/~
r/
~ d2u~ L
AL
~,L + T ¢/3u + 2
+
(33)
d2¢/L
+ ah dx
= _G(u~L _ U~U) +
¢/L +
¢/~
--
+
(34)
Impact response of single-lapcompositejoints dq/3 U,--~-
BU , ~ + D
-
u u u k A55 qt3 +
Gr/ hU a G h° (h-~2 hU ) 2 (u~L - u°U) + ~ - ~ ~ + -~-- ~
B~1 ~
+
1017
m r d I],/3 r L XJll "-~ -- k A55
2 hU
G
+
22
+
~--~)
(35)
r
q/3 +
r/ 2 (u]L - u~u) + --r/-2-
~
kUAU5 - - + kLA~g5
+ -2- ~
(36)
dx 2It = q
(37)
+ ----~/] = - q
(38)
where r and q are the shear stress and peel stress (transverse normal stress) within the adhesive as shown in Fig. 3, respectively. Equations (33)-(38) are six coupled second-order ordinary differential equations with six variables: u~U, u°L3, q?U3,qJL3, W3 U, W3 r and with one unknown function, q, which is the peel stress of the adhesive between the two laminates. Because every function which is piece-wise smooth for a finite range can be expanded as a Fourier cosine series, the peel stress of the adhesive q can be represented by an (n + l)-term Fourier cosine series with (n + l) coefficients, Co-Cn, as
inx ci cos - -
q = i=0
(39)
13
where 13 stands for the length of this section and is half of the overlay length. By defining vi as the variables which need to be solved, U 1 = U 3OU ,
U2 = U~ L'
V3 = I]'/3U'
V4 = ~bcL,
05 = W 3U,
U6 =
wL
and by combining the homogeneous and particular solutions, vi can be written as 9
VJ=
i2=0 djixi Vj =
9 2
n IRX + ejlc°sh°taX + ejEsinh°tax + i~=i gj'" sin --,13 -
-
dji xi + ejl cosh a3x + ej2 sinh ot3x +
i=0
.
~
j=
inx gji c o s - - ,
i= 1
13
1,2,3,4
(40)
j = 5, 6
(41)
where ds and es are undetermined coefficients of the homogeneous solutions (Yang and Pang, 1993). Because eqns (33)-(38) are six second-order ordinary differential equations, the homogeneous solutions, shown as the first three terms in eqns (35) and (36) of the system, are so related that there are only 12 independent coefficients with which all dji, ej~, and ej2 can be determined. Also, gj~ can be obtained as functions of ci through the governing equations, eqns (33)-(38). The (n + 1) undetermined coefficients Co-% together with the 12 independent coefficients in ds and es result in a total of (n + 13) unknowns. The (n + 1) extra unknown coefficients from eqn (39) for the peel stress distribution can be covered by the (n + 1) extra equations from the joint kinematics and the orthogonal properties of Fourier series as
c, = 2EI.I/39 om= ~ o (dsm - d6m)X" COs ~,3rnx -dx + (e52 - %2)
f 13sinh(a3x) cos~-7\t3//rnx\ j dx 0
,.~//rnx\ dx + (es, - e6l) l] cosh(ot3x) cos/--~-)
w
+ (gs, - g6r) ~1 ,
r = 1, 2, 3 ..... n. (42)
S.-S. Pang et al.
1018
Joint section (4). The system of governing equations of the joint section (4) is identical to that of the joint section (3). Because the force is applied at the center of the overlay, the transverse shear stress resultant of the adhesive, peel and shear stresses of the adhesive will have discontinuity at the point under loading. It is for this reason that the overlay region has been split into sections (3) and (4). Combining the aggregate governing equations of the two adherends in all the four sections leads to a total of 12 second-order and six first-order ordinary differential equations including a total of 18 variables. The general solutions contain 30 undetermined coefficients. This is to say, 30 boundary conditions (or equations) are needed to solve the whole problem. These boundary conditions are obtained from either the continuity or the force equilibrium at the junctions between joint sections. Equivalent stiffness The equivalent stiffness keq for the spring-mass model can be obtained by solving the relation between the transverse deflection and the applied force at the center of the joint overlay. After solving the above systems of equations, the equivalent stiffness kCq can be written as P
(43)
keq = - - Wm
where P and wm( = wU(0)) are the transverse loading and deflection at the center of the overlay, respectively.
Equivalent mass In general, the solution of the displacement can be expressed as a summation of sinusoidal functions with various modes. For each specific mode, the magnitude of the acceleration of each cross-section of the laminate is proportional to the magnitude of its transverse deflection. After superposition of different modes, the resultant displacement and acceleration will not be directly proportional due to their phase shifts. However, in many applications, the first mode is dominant. In calculating the equivalent mass of this study, the acceleration of each cross-section is assumed to be proportional to its transverse displacement. Therefore, the equivalent mass for the developed spring-mass model as represented by the middle cross-section of the overlay can be determined from
meq
--
' (f
w4U(0)
+
0
p w U ( x I ) d x I -+-
i
o
PW2L(X2) d x 2 -I-
p[wU4(x4)+ w4L(x4)ldx,
fl
-I- wL(x3)] d X 3
(44)
0
where p is the mass of the laminate per unit length. E X P E R I M E N T A L STUDY
To verify the proposed model, low-velocity impact tests of single-lap composite joint samples were performed. A hemispherical-tipped impactor, with 25.4 mm diameter and a mass of 0.75 kg, was used in the tests. The sample material was Scotchply with cross-ply layout. Table 1 lists the material properties of the sample material which were provided by the manufacturer. The samples were 229 mm (9 in) long and 38 mm (1.5 in) wide with four different joint configurations: (a) 6.35 mm (1/4 in) thick adherends with 50.8 mm (2in) overlay; (b) 6.35mm (1/4in) thick adherends with 63.5 mm (2.5in) overlay; (c) 3.18 mm (1/8 in) thick adherends with 50.8 mm (2 in) overlay; and (d) 3.18 mm (1/8 in) thick adherends with 63.5 mm (2.5 in) overlay. The span between the two end supports was set at 178 mm (7 in) for the impact test. A total of 25 samples were tested.
Impact response of single-lap compositejoints
1019
Table 1. Material properties of Scotchplylaminates Longitudinal modulus EI~ Transverse modulus E22 Shear modulus G12 In-plane Poisson's ratio v~2 Resin content (by weight) Fiber content (by weight)
39.30 GPa (5.7 Mpsi) 9.70 GPa (1.4 Mpsi) 4.14 GPa (0.6 Mpsi) 0.26 38%
62O/o
Preliminary tests Conducting the preliminary static test serves two purposes: to provide a calibration curve which converts the voltage readings into the impact force, and to determine the outof-plane Young's modulus of the sample (Ec) which is needed when applying the modified Hertzian contact law. These tests were conducted in an Instron servohydraulic machine.
Weight drop tower The impact tests were conducted using a specially designed weight drop tower. The schematic diagram is shown in Fig. 4. A 2.4 m tall frame was constructed on a concrete base to increase the rigidity of the entire system. Two steel rods were attached to the frame to guide the " H " shape steel beam that holds the impactor. A timer unit was used to measure the time taken by the impactor to travel through a fixed distance. All the samples were simply-supported at two opposite sides and were free at the other two sides. Two semiconductor strain gages were bonded to the impactor. The impactor stroke was at the center of the overlap region.
Impact velocity In order to reach different values of impact velocity, the impactor was dropped from different heights. By measuring the time taken by the impactor to travel through a fixed distance, the accelerations, and consequently the velocities, could be calculated. Here the acceleration for each case was assumed to be constant.
Impact force To obtain the force history during impact, two semiconductor strain gages with large gage factor (approximately 50) were attached to the impactor at the near-tip locations. A strain gage conditioner and amplifier system (model 2100) was used to convert the signal from the gage to the oscilloscope, and the signal was filtered at 100 kHz. A Nicolet digital oscilloscope (model 4090C) was used to record the force signal, which was then stored on a floppy disk for later analysis. The digital sampling rate was set at a 1/~s interval. Figures 5(a)-(d) demonstrate the impact force history of four of the samples, No. 3, No. 1 l, No. 14 and No. 24; each represents one of the four sample configurations.
Wheatstonef_L Bridge l__. Amplifier~
Digital OscilloscopeI~-~
V--I
Frame ,. ~ - Guide Rod ~ Support Plate ~ - Impactor
~ - Strain Gage Composite Plate Clamp ,/- Concrete Base ~'///, 1//11//////11/ ////~ Fig. 4. Weight drop tower.
S.-S. Pang et
1020
2.5 z
,
.
,
.
,
5.0
Sample No. 3 Thickness = I/8" Overlay ffi 2"
2.0
al. ,
4.0
o~ a.o
1.5
0
l
1.0
~" 2.0
~u
1.0
0.5 0.0
,
•
,
Sample No. 11 Thickness = 1/4" Overlay : 2"
j
v
o r~
•
0.0 i
0.5
0.0
1,0
1.5
f
2.0
0.0
0.5
Time ( m s )
,
I
i
1.5
2.0
Time ( m s )
(b)
(a)
2.5
i
1.0
•
,
•
,
.
,
5.0
.-''
,
-
,
•
,
Thickness= 1/8" Overlay = 2.5"
1.5
4.0
3
,
•
,
Sample No. 24 Thickness = I/4" Overlay = 2.5"
Sample No. 14
2,0
•
3.0
I. o
t~ 2.0
1.0
e~
E 0.5
,~
1
0.0
1.0
0.0 i
i
0.0
0.5
i
i
I
1.0 1.5 Time (ms)
2.0
i
i
i
i
i
0.0
0.5
1.0
1.5
2.0
(e)
Time (ms)
(d) Fig. 5. Impact force history.
RESULTS AND DISCUSSION
The spring-mass model was used for the analysis of a single-lap joint under lowvelocity impact. The effect of impact is represented by a time-dependent force which can be solved from a quasi-static equilibrium condition. For the case in which a wave can be reflected many times within the sample during the impact period, a quasi-static equilibrium condition can be assumed. In the current study, the wave speed within the laminated plate is approximately 2300 m/s (VEV~, where E = 9.7 GPa and p = 1829 kg/m3). Based on experimental data, the impact period corresponding to the first wave is about 0.2 ms, which represents at least 18 round trips of the wave during the impact period for the 6.35 mm (1/4 in) laminate or 36 round trips for the 3.18 mm (1/8 in) laminate. Therefore, the quasi-static equilibrium condition can be approximately established (Pang and Kailasam, 1991). The spring-mass model of a single-lap composite joint consists of two main parameters, equivalent mass and stiffness constants. Both of them are directly related to w, which is the transverse displacement of the structure. As can be seen from eqn (44), the deformed shape of the single-lap joint is utilized to determine the equivalent mass which is included in the spring-mass model. On the other hand, the equivalent stiffness represents the relation between the applied loading and the center deflection of the joint. Therefore, the development of the joint model is essential in order to obtain the equivalent stiffness and equivalent mass of the joint for the impact analysis. Tables 2 and 3 give the equivalent stiffness and mass for the four different configurations of impact test samples. In order to obtain different impact velocities, the drop height was varied for each individual case during the experimental study. When calculating the impact velocity, the acceleration was assumed to be constant for each individual case but different from case
1021
Impact response of single-lap composite joints Table 2. Equivalent stiffness of impact samples
3.18 mm (1/8 in) thick adherends 6.35 mm (I/4 in) thick adherends
50.8 mm (2 in) overlay
63.5 mm (2.5 in) overlay
97.60 kN/m 694.27 kN/m
114.43 kN/m 845.42 kN/m
Table 3. Ratio of equivalent mass to actual mass of impact samples
3.18 mm (1/8 in) thick adherends 6.35 mm (1/4 in) thick adherends
50.8 mm (2 in) overlay
63.5 mm (2.5 in) overlay
0.799 0.796
0.758 0.789
to case. It can be seen from the impact force history, Figs 5(a)-(d), that during the impact a compressive wave propagates to the top of the impactor and reflects as a tensile wave. Compressive and tensile waves cancel each other resulting in a decrease in the force. This process repeats until the separation of the impactor and the sample target occurs. It can also be noticed from Figs 5(a)-(d) that the maximum force occurs at the first wave for all cases. To predict the maximum impact force, the spring-mass system equations, eqns (5) and (6), were simply applied to the first wave period. Once the time duration is obtained from the impact test, the maximum impact force can be solved from the relation given by eqn (19). The analytical and the experimental results for all the test samples are tabulated in Tables 4-7, where the duration corresponds to the first wave
Table 4. Impact force of samples with 50.8 mm (2 in) overlay and 3.18 mm (1/8 in) thick adherends Sample no.
Velocity (m/s)
Duration (p.s)
Analytical force (kN)
Experimental force (kN)
Deviation (070)
1 2 3 4 5 6 7 8
4,476 4.449 4.035 4.040 3.856 3.830 3.787 3.635
164 168 170 171 164 172 175 171
1.919 1.674 1.557 1.506 1.553 1.453 1.318 1.497
1.983 1.890 1.816 1.584 1.812 1.495 1.437 1.486
3.23 11.43 14.26 4.92 14.29 2.81 8.28 0.74
Table 5. Impact force of samples with 50.8 mm (2 in) overlay and 6.35 mm (1/4 in) thick adherends Sample no.
Velocity (m/s)
Duration (~)
Analytical force (kN)
Experimental force (kN)
Deviation (%)
9 I0 11 12 13
4.578 4.342 4.147 4.086 3.837
181 188 184 188 191
4.652 4.017 4.213 3.768 3.445
4.443 4.142 4.011 3.266 3.441
4.70 3.02 5.04 15.37 0.12
Table 6. Impact force of samples with 63.5 mm (2.5 in) overlay and 3.18 mm 0 / 8 in) thick adherends Sample no.
Velocity (m/s)
Duration (/~s)
Analytical force (kN)
Experimental force (kN)
Deviation (070)
14 15 16 17 18 19
4.271 4.242 4.195 4.126 3.784 3.740
232 238 231 240 237 238
2.113 1.587 1.958 1.513 1.609 !.571
1.910 1.833 1.881 1.702 1.706 1.592
10.63 13.42 4.09 11.10 5.69 1.32
1022
S.-S. Pang et al.
Table 7. Impact force of samples with 63.5 mm (2.5 in) overlay and 6.35 mm (1/4 in) thick adherends Sample no.
Velocity (m/s)
Duration (p.s)
Analytical force (kN)
Experimental force (kN)
Deviation (%)
20 21 22 23 24 25
4.373 4.344 4.269 4.164 4.120 4.031
217 216 214 224 222 229
4.022 4.106 4.278 3.434 3.570 3.086
4.243 4.170 3.930 4.142 3.629 2.875
7.39 1.53 8.85 17.09 3.30 7.34
period. Figures 6(a)-(d) show the results o f the maximum impact force for four of the samples, each representing one type of sample configuration. The analytical curve for each case is represented by a dotted line which simulates the first wave of the impact curve. It is known that the maximum impact force depends upon many factors such as impact velocity, impactor weight and geometry, target-sample geometry (thickness, adhesive overlay) and support condition, as well as impactor and target material properties. To understand the effects of adhesive overlay length and adherend thickness on the impact force, four types of sample configurations have been used in the impact test, including two adherend thicknesses, 3.18 mm (1/8 in) and 6.35 mm (1/4in), and two overlay lengths, 50.8 mm (2 in) and 63.5 mm (2.5 in). It is shown from Tables 4-7 that the impact velocities directly affect the maximum impact forces. The higher the impact velocity, the greater the maximum impact force. 3.0
2.5
5.0
Sample No. 3 Thickness = I/8" Overlay = 2"
4.0
2.0
3.0
0= ..o
1.s
k
1.0
2.0
0.5
i:u ~ 1.0
~
Sample No. 11 Thickness : 1/4" O v e r l a y . 2"
r.
0.0
0.0 i
i
0.10
0.00
Time
i
i
0.20
0.30
i
0.00
(ms)
i
i
i
0.10
0.20
0.30
Time
(a)
(ms)
(b)
3.0
5.0
S a m p l e No. 14 Thickness = 1/8"
2.5
Overlay : 2.5"
Z
2.0
= o o
1.s
/~"
\\
#\
"~ 1.0 a. E 0.5
~
Sample No. 24 Thickness = i/4" Overlay = 2.5"
4.0
.= 3.o #i.
',,
\\
2.0 o. E
0.0
t.0
0.0 i
i
i
i
i
i
,
0.00
0.10
0.20
0.30
0.00
0.10
0.20
Time
(ms)
'Time
(c)
(ms)
(d) Fig. 6. Maximum impact force.
0.30
1023
Impact response of single-lap composite joints
:
150
1200
125
iooo
z 800
g 100 e-
600
03
400 50
I
0.04
i
i
I
0.05
0.06
0.07
I
L
0.04
i
i
0.05 0,06 0.07 Overlay Length (m)
Overlay Length (m) ~)
(a)
Fig. 7. Effect of overlay length on equivalent stiffness.
Due to the limitation of the weight drop tower used in this test, only limited velocity variations have been achieved. As a result, no experimental relation between impact velocity and maximum impact force has been obtained. The effect of adherend thickness on the maximum impact force can also be identified from the analytical and experimental results. The maximum impact forces of 6.35 mm (1/4 in) samples are greater than those of 3.18 mm (1/8 in) samples. For samples with 50.8 mm (2 in) adhesive overlay, the average experimental maximum impact force of the 6.35 mm (1/4 in) samples is about 2.35 (or 2.58 for analytical value) times as large as that of the 3.18mm (1/8in) samples, while for samples with 63.5mm (2.5in) adhesive overlay, the average experimental maximum impact force of the 6.35 mm (1/4 in) samples is 2.16 (or 2.17 for analytical value) times as large as that of the 3.18 mm (1/8 in) ones. One of the reasons for such a difference is that the equivalent stiffness has been changed significantly when the thickness of the adherends changes from 3.18mm (1/8 in) to 6.35 mm (1/4 in). The length of adhesive overlay also affects the equivalent stiffness. An analytical relation between the overlay length and the equivalent stiffness is shown in Figs 7(a) and 7(b), for 3.18mm (1/8in) and 6.35 mm (1/4in) samples, respectively, with length of overlay varying from 38.1 mm (1.5 in) to 76.2 mm (3 in). By integrating the force history with respect to time, the impulse can be obtained. Tables 8-11 list both analytical and experimental results. Again, the impulse here corresponds to the impact force history within the first wave period. Figures 8(a)-(d) demonstrate the results for four of the samples; each represents one different sample configuration. The deviation between the analytical and the experimental results has been provided for all of the cases. The maximum deviation for impact force prediction is 17.09% (sample No. 23); while for impulse, the maximum deviation is 34.64% (sample No. 14).
Table 8. Impulse of samples with 50.8 mm (2 in) overlay and 3.18 mm (1/8 in) thick adherends Sample no.
Velocity (m/s)
Duration (ps)
Analytical impulse (N s)
Experimental impulse (N s)
Deviation (%)
1 2 3 4 5 6 7 8
4,476 4.449 4.035 4,040 3.856 3.830 3,787 3.635
164 168 170 171 164 172 175 171
0.1967 0.1790 0.1685 0.1639 0.1621 0.1591 0.1468 0,1630
0.1891 0.1885 0.1771 0.1756 0.1635 0.1871 0.1324 0.1843
4.02 5.04 4.86 6.66 0.86 14.97 10.88 11.56
1024
S.-S. Pang et al. Table 9. Impulse of samples with 50.8 mm (2 in) overlay and 6.35 mm (1/4 in) thick adherends Sample no.
Velocity (m/s)
Duration (p.s)
impulse(Ns)
Analytical
Experimental impulse(Ns)
Deviation (%)
9 10 11 12 13
4.578 4.342 4.147 4.086 3.837
181 188 184 188 191
0.5360 0.4808 0.4935 0.4510 0.4189
0.4761 0.4414 0.4606 0.3918 0.3862
12.58 8.93 7.14 15.11 8.47
Tabk 10. Impulse of samples with 63.5 mm (2.5 in) overlay and 3.18 mm (1/8 in) thick adherends Sample no.
Velocity
Duration
(m/s)
(~)
Analytical impulse (N s)
Experimental impulse (N s)
Deviation (%)
14 15 16 17 18 19
4.271 4.242 4.195 4.126 3.784 3.740
232 238 231 240 237 238
0.3121 0.2404 0.2879 0.2312 0.2428 0.2380
0.2318 0.2090 0.2157 0.2187 0.1929 0.1838
34.64 15.02 33.47 5.72 25.87 29.49
Table 11. Impulse of samples with 63.5 mm (2.5 in) overlay and 6.35 mm (1/4 in) thick adherends Sample no.
Velocity (m/s)
Duration (p.s)
Analytical impulse (N s)
Experimental impulse (N s)
Deviation (%)
20 21 22 23 24 25
4.373 4.344 4.269
217 216 214 224 222 229
0.5556 0.5646 0.5964 0.4897 0.5045 0.4499
0.4875 0.5819 0.4727 0.4516 0.4769 0.4162
13.97 2.97 26.17 8.44 5.79 8.10
4.164
4.120 4.031
•
0.4 "W 0.3 z 0.2
,
-
,
•
,
.
,
-
,
•
-
r
Sample No. 3 Thickness = 1/8" Overlay = 2"
,
•
,
.
=
0.6
.
,
.
.
,
,
I
,
,
z
Z
f
0.3 ,~ 0.2
0.0
0.0 I
i
i
0.05
0.10
0.15
i
0.00
i
i
.
0.20 0.25
=
.
i
.
i
i
0.00 0.05 0.10 0.15 0.20 0.25 Time (ms)
Time (ms) (a)
0.3
,
"G" 0.5
.~ 0.1
0.4
•
Sample No. 11 Thickness = 1/4" Overlay ffi2"
0a) "
Sample No. 14 Thickness= I/8" Overlay = 2.5"
0.6 ~ --"
0.5
,
.
,
-
i
-
,
Sample No. ?.4 Thickness = 1/4" Overlay = 2.5"
Z
0.2
0.3
.~0.1
~ 0.2
0.0
0.0 i
i
i
0.00
0.05
0.10
~
0.15
I
I
0.20
0.25
Time (ms)
0.00
0.05
0.10
0.15
Time (ms)
(c)
(d) Fig. 8. Impulse during impact.
0.20
0.2
Impact response of single-lap composite joints
1025
When measured by average values, a 7.60o70 deviation has been noticed for the 3.18 mm (1/8 in) thick samples with 50.8 mm (2 in) adhesive overlay, a 4.1007o deviation for the 6.35 mm (1/4 in) thick samples with 50.8 mm (2 in) overlay, a 2.57o7o deviation for the 3.18 mm (1/8 in) thick sample with 63.5 mm (2.5 in) overlay, and a 2.15070 deviation for the 6.35 mm (1/4 in) thick samples with 63.5 mm (2.5 in) overlay. There are many factors which may cause the analytical results to deviate from the experimental results. For example, the uneven friction between guiding rods and the bearings in the " H " shape beam which holds the impactor will cause the changes in acceleration from point to point, whereas the acceleration is assumed to be constant throughout the impact process in the calculation of the impact velocity. Damping is another cause for the deviation between the analytical and experimental results. In the analytical model, the damping effect has not been considered; while in the actual impact test, damping does exist due to the supporting conditions, adhesive layer, etc. Nethertheless, for the purpose of engineering design, the developed analytical model still provides a reasonably good prediction for the maximum impact force for such a complicated composite structure. From the impact force, a feedback calculation can be performed to predict the adhesive stresses, such as shear and peel. In order to accomplish these, the impact force is first used to calculate the P in the governing equations (eqns (30)-(32)) and the boundary conditions, then the six variables of the displacement field (u °U, u °L, WU, WE, wU, wE) in joint sections (3) and (4) can be calculated. Subsequently with eqns (34) and (37), the adhesive shear stress z and peel stress q can be calculated, respectively. CONCLUSIONS
The impact response of adhesive-bonded single-lap composite joints was investigated. A simplified spring-mass model has been applied to represent the impactor-target system. During the impact period of a single wave of the structure vibration, the transverse shock wave can travel at least 18 round trips for 6.35 mm (1/4 in) laminate or 36 round trips for 3.18 mm (l/8 in) laminate. The quasi-static equilibrium can then be assumed to be established during the impact process. An equivalent mass with equivalent stiffness has been used to represent the complicated single-lap composite joint in the model. An analytical model has been developed to determine the equivalent mass and the equivalent stiffness based on the laminated anisotropic plate theory in which the governing equations of the two bonded laminates have been derived, and the entire coupled system was solved in closed-form using Fourier series and appropriate boundary conditions. Combining the developed model with the quasi-static spring-mass model, a relationship between the impact duration and the impact force has been obtained. Low-velocity impact tests of single-lap joints with two different adherend thicknesses and two overlay lengths have been conducted to verify the proposed model. Analytical and experimental results of maximum impact forces, as well as impulses, have been provided. The relationship between the impact duration and the impact force obtained from the developed analytical model can be used to predict the maximum impact force during the low-velocity impact. The results show that the average deviations between the predicted values and test data range from 2.15o7o to 4.10070, depending upon the sample configurations. The maximum deviation for an individual case is around 17°70. The maximum impact force is affected by many factors such as the impact velocity and the adherend thickness. The impact velocity directly affects the maximum impact force. The higher the impact velocity, the greater the maximum impact force. The thickness of the adherends has a notable influence on the value of the maximum impact force because the thickness change will cause a significant change in the equivalent stiffness of the single-lap joint sample. On the other hand, the overlay length of the joint shows no obvious effect on the maximum impact force. Based on the current study, the spring-mass model can provide a reasonably good prediction for cases where impact failure occurs after a long enough time to allow wave reflections to establish quasi-static equilibrium. But it may not be adequate for situations that result from localized stress wave action near the point of impact.
1026
S.-S. Pang et al.
A methodology of predicting adhesive stresses during impact was proposed. The maximum adhesive stresses can be utilized to predict the joint strength if the adhesive failure criterion is available. DEDICATION
This paper is specially dedicated to Professor Werner Goldsmith's 70th birthday on May 23, 1994. Professor Goldsmith is the Professor Emeritus of the Mechanical Engineering Department at the University of California, Berkeley, and served as the PhD advisor of Su-Seng Pang, the first author of this paper, during the period of 1983-86. REFERENCES Adams, R. D. and Wake, W. C. (1984). Structural Adhesive Joints in Engineering. Elsevier Applied Science Publishers, London and New York. Bert, C. W. (1973). Simplified analysis of static shear factors for beams of nonhomogeneous cross section. J. Comp. Mater. 7, 525-529. Chow, J. S. (1971). On the propagation of flexural waves in an orthotropic plate and its response to an impulsive load. J. Comp. Mater. 5, 306-319. Dharmarajan, S. and McCutchen, H. Jr (1973). Shear coefficients for orthotropic beams. J. Comp. Mater. 7, 530-535. Goland, M. and Reissner, E. (1944). The stresses in cemented joints. J. Appl. Mech. 11, AI7-A27. Goldsmith, W. (1960). Impact: The Theory and Physical Behavior of Colliding Solids. Edward Arnold, London. Griffin, S. A., Pang, S. S. and Yang, C. (1991). Strength model of adhesive bonded composite pipe joints under tension. Polym. Engng Sci. 31(7), 533-538. Hart-Smith, L. J. (1973a). Adhesive-bonded single-lap joints. Douglas Aircraft Co., NASA Langley Report CR-112236. Hart-Smith, L. J. (1973b). Adhesive-bonded double-lap joints. Douglas Aircraft Co., NASA Langley Report CR-112235. Hart-Smith, L. J. (1973c). Adhesive-bonded scarf and stepped-lap joints. Douglas Aircraft Co., NASA Langley Report CR- 112237. Hart-Smith, L. J. (1974). Analysis and design of advanced composite bonded joints. Douglas Aircraft Co., NASA Langley Report CR-2218. Kutscha, D. (1964). Mechanics of adhesive-bonded lap-type joints: survey and review. Technical Report AFML-TDR-64-298. Kutscha, D. and Hofer, K. E. Jr (1969). Feasibility of joining advanced composite flight vehicles. Technical Report AFML-TR-68-391. Matthews, F. L., Kiity, P. F. and Goodwin, E. W. (1982). A review of the strength of joints in fibre-reinforced plastics: Part 2. Adhesively bonded joints. Composites 13, 29-37. Mindlin, R. D. (1951). Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J. AppL Mech. 18, 336-343. Pang, S. S. and Kailasam, A. A. (1991). A study of impact response of composite pipe. J. Energy Resource TechnoL 113, 182-188. Pang, S. S., Zhao, Y., Yang, C. and Griffin, S. A. (1991). Impact response of composite laminates with a hemispherical indenter. Polym. Engng Sci. 31(20), 1461-1466. Reddy, J. N. (1984). A refined nonlinear theory of plates with transverse shear deformation. Int. J. Solids Struct. 20(9/10), 881-896. Reissner, E. (1945). The effect of transverse shear deformation on the bending of elastic plates. J. AppL Mech. 12, A69-A77. Shivakumar, K. N., Elber, W. and lllg, W. (1983). Prediction of impact force and duration during low velocity impact on circular composite laminates. NASA-TM-85703. Sun, C. T. and Yang, S. H. (1980). Contact law and impact response of laminated composites. NASACR-159884. Tsai, S. W. and Hahn, H. T. (1980). Introduction to Composite Materials. Technomic Publishing Company, Inc., Lancaster, PA. Vinson, J. R. (1989). Adhesive bonding of polymer composites. Polym. Engng Sci. 29(19), 1325-1331. Whitney, J. M. (1972). Stress analysis of thick laminated composite and sandwich plates. J. Comp. Mater. 6, July, 426-440. Whitney, J. M. (1973). Shear correction factors for orthotropic laminates under static load. J. Appl. Mech. 40, 302-304. Whitney, J. M. (1987). Structural Analysis of Laminated Anisotropic Plates. Technomic Publishing Company, Inc., Lancaster, PA. Whitney, J. M. and Pagano, N. J. (1970). Shear deformation in heterogeneous anisotropic plates. J. Appl. Mech. 37, 1031-1036. Yang, C. and Pang, S. S. (1993). Stress-strain analysis of single-lap composite joints under tension. Proc. 1993 A S M E Energy-Sources Technology Conference & Exhibition, Composite Material Technology (Edited by D. Hui, T. J. Kozik and O. O. Ochoa). PD-Vol. 53, pp. 85-94.
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Yang, C. and Pang, S. S. (1993). Stress-strain analysis of adhesive-bonded single-lap composite joints under cylindrical bending. Comp. Engng 3(1 l), 1051-1063. Yang, C., Pang, S. S. and Griffin, S. A. (1992). Failure analysis of adhesive-bonded double-lap joints under cantilevered bending. Polym. Engng Sci. 32(9), 632-640.
COE 5-8-E