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Effect of transverse cracks on the thermomechanical properties of cross-ply laminated composites
Composites Science and Technology 34 (1989) 145-162
Effect of Transverse Cracks on the Thermomechanicai Properties of Cross-Ply Laminated Composites S. G. Lim & C. S. H o n g * Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, PO Box 150, Cheongryang, Seoul, Korea (Received 6 April 1988; revised version received 3 June 1988; accepted 6 July 1988)
A BS TRA C T The development of transverse cracks can be detrimental to the stiffness and dimensional stability of composite laminates. In this investigation, a modified shear lag analysis, taking into account the concept of interlaminar shear layer, is employed to evaluate the effect of transverse cracks on the stiffness reduction and change in the coefficient of thermal expansion in cross-ply laminated composites. This analysis involves the construction of admissible displacement fields which satisfy equilibrium and boundary conditions on the laminate and transverse crack surfaces. The present results represent well the dependence of the degradation of thermomechanical properties on the laminate configuration.
1 INTRODUCTION The important design considerations for use of composite materials in lightweight structures are stiffness and dimensional stability. Transverse cracking, which is one of the most frequently encountered damage modes in composite laminates, can be detrimental to the stiffness and dimensional stability of composite laminates. Comprehensive understanding of the effect of transverse cracks is of fundamental importance for structural design and * To whom all correspondence should be addressed. 145
Composites Science and Technology 0266-3538/89/$03'50 © ! 989 Elsevier Science Publishers Ltd, England. Printed in Great Britain
146
S. G. Lira, C. S. Hong
integrity considerations of composite structures. Transverse cracks, for the purpose of this paper, are defined as matrix-dominated cracks which run through the thickness of a given layer and extend parallel to the fiber direction. Numerous models for predicting the initiation of a transverse crack in composite laminates have been developed and correlated with experimental data.1 - 10 The effect of transverse cracks on the stiffness loss of composite laminates has also been investigated experimentally. 11'12 Experimental results have shown that multiple transverse cracking by increasing the applied stress causes a progressive stiffness reduction in composite laminates that is more marked the greater the thickness of the cracked transverse layer. In recent years, various approaches have been developed to evaluate quantitatively the loss of stiffness of composite laminates due to transverse cracks.12-16 We cite briefly this previous work. Highsmith and Reifsnider modeled the stiffness loss of damaged laminates by reducing the transverse stiffness in the cracked layer using shear lag analysis. ~2 However, the mutual interaction of cracks due to multiple transverse cracking is not considered in their analysis. Dvorak, Laws and Hejazi employed the self-consistent scheme approximation to assess the stiffness reduction of a cracked lamina in a laminate. 13 Talreja evaluated the stiffness changes of transverse cracked laminates using continuous damage theory.l'~'15 Also, Hashin analyzed the stiffness reduction of damaged cross-ply laminates by a variational method on the basis of the principle of minimum complementary energy. 16 His work gave lower bounds on laminate stiffness reduction due to multiple transverse cracking and good agreement with experimental data 12 for an E-glassepoxy [ 0 / 9 0 3 ] s laminate. There has also been interest in the effect of transverse cracks on the dimensional stability of composite laminates. Recent numerical studies have shown that transverse cracks can result in a significant change in the coefficient of thermal expansion, CTE, of composite laminates. 17.18 The purpose of the present investigation is to evaluate consistently and more simply the stiffness reduction and change in the CTE due to transverse cracks in cross-ply laminates. A modified shear lag analysis, taking into account the concept of interlaminar shear layer, is formulated to evaluate the effect of transverse cracks on the thermomechanical property degradation of composite laminates. This analysis involves a construction of admissible displacement fields which satisfy equilibrium and boundary conditions on the laminate and crack surfaces. The mutual interaction of transverse cracks is also considered in this analysis as the transverse crack spacing becomes small. Numerical results for the thermomechanical property changes due to transverse cracks are compared with existing experimental data. The results represent well the dependence of the changes
Transverse cracks in cross-ply laminated composites
147
in thermomechanical properties on the laminate configuration. Also, the present analysis is simple, yet its predictions show reasonable agreement with numerical results and experimental data.
2 F O R M U L A T I O N OF T R A N S V E R S E C R A C K I N G
2.1 Shear lag analysis for multiple cracks Consider the cross-ply laminated composites subjected to mechanical and thermal loading as shown in Fig. 1, in which the two outside 0 ° layers are of equal thickness, b, the two interlaminar shear layers are of thickness do, and the central 90 ° layer is of thickness 2d. When extended in tension, transverse cracks which run through the thickness and extend parallel to the fiber direction occur in the central 90 ° layer. The stress distribution around the transverse crack can be approximated by using shear lag analysis. '~'5'12 The derivation of the present analysis follows a one-dimensional shear lag model used by Fukunaga e t al. 4 However, the boundary conditions in their analysis are not satisfied as the transverse crack spacing becomes small. In this paper, we consider that the boundary conditions for the initial transverse crack and another crack in the vicinity of the first are satisfied. Therefore, the multiple cracking problem can be handled for the evaluation of change of thermomechanical properties in laminated composites.
....
.-:__
_:=
iii :iiii iiiiiiii, iiiiii iliiiiii. iiiiiliiiiiiii ilililil ~"
0
O~
~ ~
,
,
I
=X
o,
I I'
I
~O,.o
A ~)T~ V I -×
~oo ,,=
Fig. 1.
2L
~L
=j
Transverse cracked cross-ply laminate and analytical model.
148
S. G. Lira, C. S. Hong
The displacement fields in the vicinity of a transverse crack are assumed to be
/'/I = (80 -[- 8xT0)x + UI(X) ~)1 =
--
~0 -~- ~yO ) + VI(X)
(~) ~ = (~o + afo)X + U2(x)
v2 =
(
A12
+
r)
yo. y + V2tx)
Displacements u and v denote the components of x- and y-directions, respectively, and are assumed to be uniform through the thickness in each layer. The subscripts 1 and 2 denote the 0 ° and 90 ° layers, respectively. The linear displacement results from uniaxial extension, eo and mid-plane r and eyO, ~ due to mechanical and thermal loading, thermal strains, e~o respectively. The A u denote the laminate extensional stiffnesses. Considering the damaged cross-ply laminates with a transverse crack spacing of 2L in the 90 ~ layer (see Fig. 1), the equilibrium equations around the transverse crack become
Q l l ~d2Ul - x 2 -t- ~ d 0 (g/2-g/1) = 0 d2bt2 Q22 dx 2
G (u 2 - t~l)=0 dd 0
(2) d2vl
Q66d~5-r2 +
d2v2
Q66 ~
b@°
(v2-v,)=0
G (v z _ vl ) = 0
ddo
where the Qij and G denote the reduced stiffnesses and the shear modulus of the interlaminar shear layer, respectively. The shear stress is assumed to be proportional to the difference in displacements of the 0 ° and 90 ° layers. The boundary conditions are o~~
+ ~'~" = 0
ba~uu = (b + d)O~ v~ = 0
dV~ _ 0 dx
(3)
Transverse cracks in cross-ply laminated composites
149
at x = 0 a n d x = 2L. The above b o u n d a r y conditions are satisfied for a n y crack spacing. The superscripts T a n d M d e n o t e the t h e r m a l a n d mechanical stress c o m p o n e n t s , respectively. After imposing the b o u n d a r y conditions, the displacement fields are o b t a i n e d as
1 do'~o) -0,,) ~bQ11 ( - / 3 , e = x + f l z e
t/1 : (g0 + 8xT0)x
1 a~
u~ ~ ~o + ~o~X + ~ - &
Vl = V2 =
e ~" + ~ e -
~)
(4)
+ eyo y
----gO
where
# 1 = (1 - - e - 2=L)/(e2aL - - e - 2aL)
(5)
f12 = (e2~L -- 1)/( e2~L -- e - 2aL) F r o m eqn (4) a n d classical l a m i n a t e t h e o r y (CLT), 23 the strain c o m p o n e n t s in each layer can be written as
e?)
(1)
da~o)
=~x0 + gQ~7,~ (/3,
e~ +
,,d2~
e(x2)= g(~2 o, -
~(#,
e "~ +
#2 e-'X)
#ze-=O (6)
g~1 )
e( I ) ~ oy 0
~2)
~(2) ~ ~yO
~(~y ~ ) = ~~) = 0 where
~1o~ = ~o + ~x~o)~ ~.(2) T g(x2d ~_ ~0 4- - °xO
"(~) - -
e(1)T A12 eo + ~yo
~(2) __ ~yO
~(2)T A 1 2 B0 -+- ~pO A2 2
t'YO
A2 2
(7)
S. G. Lira, C. S. Hong
150
~.(i)T xO .~t~ tt**..a e(i)T t,y 0 , i = 1, 2, denote the thermal strain components in the x- and y-
directions, respectively and the superscripts 1 and 2 denote the 0 ° and 90 ° layers, respectively. The stress components in each layer are (lj
~x
~
~1~ + ~ °'~2d(fll e'X + f12 e-'~)
ffxO
a ~ = %or _~2~1 -(fl~ e ~ + fie e ~)} G y~1~=
{1) dQ12 {2~ 6vo + ~ 6 x O ( f l l
~
~2)
=
-
(8)
e~x + f12 e -
Q~2 ~2~t~ e~X +
&*
~x)
-
where ~ and ~yO~ ~i~ i = 1~ 2 denote the summation of mechanical and thermal stress components of the undamaged cross-ply laminate in the x- and ydirections, respectively.
2.2 Transverse cracking criterion With the solution of a modified shear lag analysis, the next procedure is to formulate the transverse cracking criterion in cross-ply laminated composites. By making use of an energy concept for the development of transverse crack under mechanical and thermal loading, the transverse cracking criterion is given by d
(9)
~da ( W - U) >_ 2dG~c
where W is the work done by the applied stress, U is the stored elastic strain energy, G~c is the critical strain energy release rate in the fiber direction and a is the growth of transverse crack. This criterion is based upon the assumption of the growth of a transverse crack spanning the entire thickness of the 90 ° layer in the direction of parallel to the fiber direction. For unit crack growth, the transverse cracking criterion, eqn (9), can be rewritten as: 3,5
( A W - AU) _ 2dG~c
(10)
where A W is the work done by the applied stress and A U is the change of stored elastic strain energy during the formation of transverse crack. The development of the transverse crack is associated with a laminate extension, 6L, due to the applied stress, where 6 L is given by /~L . I ~ ( 2 )
6L = 2 t
-",,o t n e ~,,
jo bQ~l u.,
+ flz
e -~x)
dx
(11)
Transverse cracks in cross-ply laminated composites
151
Thus 6L - 2 d~o ) ~xbQli
{fl,(e~,L _ 1) - fl2(e -~
(12)
The work done by the applied stress during the development of a transverse crack is A W = 28x(b + d)3L
(13)
for 6L from eqn(12), and At2A~z)/A22 from C L T in eqn (13) gives
2ff~(b+d)=eo(AllA22-
Substituting
A W = 2e°A~xA22-A~2A~2 &rtx2°){fl~(e" t ' - 1)-flz(e - ' ~ - 1)} (~ A22 bQli
(14)
The change of stored elastic strain energy, AU, is given by
(15)
A U = AU 1 + AU 2 -+ AU 3
where A Ut is the change of stored elastic strain energy in the 0 ° layers, A U 2 the energy change in the 90 ° layer and AUa the energy change due to the shear stress built up in the interlaminar shear layer during the development of transverse crack. The change of stored elastic strain energy in the 0 ° layers, AUt, is as follows: AU~ = 2
;o
{%-"~l~x + Oy -mo.~ ~ l _ .~l~o~m~,~.. ~y _ °- m xO~xO ~y0~y0 ~ ~
(16)
Substituting eqns (6) and (8) into eqn (16) gives AU~
4d -
~xo~t ~)~t2)S ~ 0 t v ~wt°~-- 1 ) - fl2(e - ~ -
1)}
~11
+ ~2dZ
S~(2)t2~1 ~o, {~,~,~2=L - 1 ) + ~ , ~ - ~ fl~(e -z:~-
1)
}
(17)
Similarly, the strain energy change in the 90 ° layer, A U~ is
~-
4d
~ O = { ~ ° (}2 ){ ~2d e ~
2d
+~<
~ xo,~
B~
~L
-~-~(e
t 2 a w~-~ -
-eL
-~}
B~ _
1) + ~B~B~L - ~ ( e
z:e-1)
~
(1~)
)
The strain energy change in the interlaminar shear layer, AU3, is AU3 = 2 fo ; 2Z;~do dx G
(19)
S. G. Lirn, C. S. Hong
152 where
G = doo tu2 -
(20)
u,)
Substituting eqn (20) into eqn (19) gives 2G A U 3 = ~-o
f
' dQ22
t 2~el (e2~L- 1)
1 + b~ ~-Q-~ : atx~°~
f12
(2~
__2fl~fl2L_
2(e 2~L - 1)
}
t21) By substituting eqns (14), (17), (18) and (21) into eqn (10), the transverse cracking strain, e 0, with crack spacing c~f 2L can be determined.
3 L A M I N A T E S T I F F N E S S R E D U C T I O N D U E TO T R A N S V E R S E CRACKS Multiple transverse cracking in laminated composites has been known to cause a progressive reduction in laminate stiffness. ~1'1z The occurrence of multiple transverse cracking can be attributed to the existence of a shear lag zone in the vicinity of a transverse crack. The stiffness reduction of cross-ply laminates due to transverse cracks in the 90 ° layer is derived in this section. The average axial strain of the damaged cross-ply laminate with a transverse crack spacing of 2L is
gx=(du_~
=1~ f L ( d u ~ d x L Jo \ dx,/
\dx}averag e
(22)
Substituting eqn (4) into eqn (22) gives
g~=eo+e~%+~a~fl,(e ~bQ ~~ L
~-
1 ) - fl2(e - ' ~ -
1)}
(23)
k
g~ is the strain theoretically evaluated from the stress-free state during the fabrication of composite laminates. The strain measured by the strain sensor during mechanical testing at room temperature is (g~ - e~). For a constant applied stress, 6~, the stress-strain relation of the undamaged cross-ply laminate can be written as fix = ~oE~o
(24)
where E~o denotes the Young's modulus of the undamaged cross-ply laminate in the x-direction. Also, the stress-strain relationship of the damaged laminate with crack spacing of 2L is
= (gx -- ~o)Ex
(25)
153
Transverse cracks in cross-ply laminated composites
where E~, denotes the effective Young's modulus of the damaged cross-ply laminate. F r o m eqns (24) and (25), the normalized laminate stiffness reduction due to transverse cracks is defined by Ex/Exo = go/(gx - e~o) r
(26)
where t o is the transverse cracking strain of cross-ply laminates with crack spacing of 2L.
4 CHANGE OF THERMAL EXPANSION COEFFICIENT Transverse cracks can result in a significant change in the CTE of composite laminates. The change in the CTE due to transverse cracks in cross-ply laminate family is evaluated in this section.
4.1 Modified shear lag analysis Consider again the problem geometry shown in Fig. 1. For a uniform change in temperature without mechanical loading, the average thermal strain of the damaged cross-ply laminate with crack spacing of 2L can be obtained from eqn (23) such as -r r d ~;~ = gxo + ~
L ~'xO L {fll(eaL -- 1) -- flz(e -~L -- 1)} ,.v(2) T
(27)
The effective CTE of damaged cross-ply laminate may be defined as follows: ~x=~xoq
1 d AT~bQ11
a ~2)T1--yn teaL 1) ~o L tvI~ -fl2(e -~L-1)}
(28)
where (29)
~x = grx/A T
~xo = ~;x~o/AT ~,o is the effective CTE of an undamaged cross-ply laminate. The change in the CTE due to transverse cracks can be written as 1
d
%/~xO = 1 -~ exOT"o~bQ~l
a~2) T 1
~,o ~-{fl,( e~L- 1 ) - flz(e - ' L -
1)}
(30)
4.2 Finite element analysis The numerical results for changes in the CTE in cross-ply laminated composites are obtained by using a two-dimensional finite element analysis
154
S. G. Lim, C. S. Hang
Z}
Finite Element eglon
[
~ - ~
. Transverse _Crack. • L/ Spoclng(2L) (a)
Z U=O t on
dI
Failured _ Free Surface go m
°l
I
U= U* (Constrained Displacement)
d
=-X
W=O L
•
(b)
Fig. 2. Repeated transverse cracking and finite element modeled region. (a) Repeated transverse cracks in the 90 ° layer. (b) Boundary conditions of modeled region.
based upon the constrained nodal displacement formulation. The damaged cross-ply laminate in this analysis is modeled as an assembly of homogeneous orthotropic layers separated by thin interlaminar shear layers as shown in Fig. 2. The finite element modeled region and appropriate boundary and constraint conditions are depicted in Fig. 2(a) and (b). The effective CTE of a damaged cross-ply laminate with crack spacing of 2L, ~x, is given in terms of the unknown constrained displacement, U*, as o~x =
U*/LAT
(31)
where the displacement, U*, is constrained to be uniform. This problem is treated as a plane strain problem. Eight-node isoparametric, quadrilateral elements are used to model the region of interest. The finite element mesh has 180 elements, 597 nodes and 1148 unknown degrees of freedom. A quasi three-dimensional finite element code developed to investigate the delamination growth in angle-ply laminated composites by Kim and Hang 2~ is modified to handle this problem. 5 RESULTS AND DISCUSSION Material properties of typical graphite-epoxy and glass-epoxy composite systems used in the present analysis are given in Table 1. The values of elastic
155
Transverse cracks in cross-ply laminated composites TABLE 1 Material Properties o f Used Composite Systems
AS4-35021~ E 1 (GPa) E z (GPa) vi 2
G12 (GPa) G1c (J/m 2) ~q (pU°C) ~ (p~,/°C) AT (~C) h (mm) G/do (Pa/m)
144-78 9"58 0"31 4"79 130 -0-30 28-10 - 147 0"127 1.04 x l0 ~4
E-Glass-Epoxy 12,24 41"7 13"0 0"30 3"4 240 3"8 16.7 - 125 0"203 3'37 x 10 la
T300-520817,21 132-0 10"8 0"24 5"7 --0-11 27.2 -0"140 9-14 x l0 ~a
constants and thermal expansion coefficients are well documented in many publications, but experimental data for G/do for the interlaminar shear layer are not generally available owing to experimental difficulties. Values similar to those of Refs 4, 12 and 19-21 for the shear layer parameter, G/do, are used in the present analysis. Also, the critical strain energy release rate of a unidirectional lamina is not well esta.blished. In this analysis, the values of G~c of graphite-epoxy and glass-epoxy systems are taken as 130 and 240 J/m 2 from experimental data, respectively. L22 The effect of transverse cracks on the stiffness reduction and dimensional stability of the [0n/90m] s cross-ply laminate family is evaluated by a modified shear-lag analysis. The present quantitative predictions of the thermomechanical property changes are compared with numerical results and existing experimental data. Figure 3 shows the comparison of the present predictions of stiffness reduction due to transverse cracks with experimental data for the AS4-3502 graphite-epoxy cross-ply laminate family. The stiffness of the damaged laminates is normalized by the stiffness of the undamaged laminates and plotted with respect to the transverse crack density, i.e. the number of cracks per unit length. The stiffness reduction becomes more marked with increasing 90 ° layer thickness. This is explained by the fact that by increasing the 90 ° layer thickness, more load is carried by the 90 ° layer and a larger COD (i.e. crack opening displacement) of a transverse crack results for a given applied load. The agreement between the present analysis and experimental data is reasonable except for the [02/902] s laminate. Figure 4 shows the stiffness reduction of a [0/903]s E-glass-epoxy cross-ply laminate. The discrepancy between the present predictions and the experimental data is greater than for the graphite-~epoxy cross-ply laminate family of Fig. 3. However, the general character of the predicted curve
S. G. Lim, C. S. Hong
156
AS4/3502 Grephite/Epoxy
o x w
''1''
1.00 w
0.98 0.96 ._
[o/9o] s
0.94
~9o~1s 0.92 0.90
•
[0/903.] S
~
[0/902] s
o z
0.0
I
I
I
I
I
0.5
1.0
1.5
2.0
2.5
3.0
Transverse crack density, crocks/mm Fig. 3.
Stiffness reduction due to transverse cracks in the graphite-epoxy laminate family.
[0,/90.,)s
AS4-3502
suggests similarity with experimental data. Some difference between the analytical model and experiment is expected in any case, since the damage mode in the present analysis is restricted to uniformly distributed straight transverse cracks. In practice, because of the random nature of the transverse cracking behavior, the uniform distribution of straight transverse cracks does not occur exactly in thick cracking layers. It was observed experimentally that there are more curved transverse cracks than straight transverse cracks with increasing thickness of cracking layer in cross-ply laminated composites. ~ The thickness of the transverse cracking layer of E - g l a s s - e p o x y [ 0 / 9 0 3 ] s laminates is thicker than that of the AS4-3502 graphite-epoxy cross-ply laminate family in Fig. 3.
E-Glass/Epoxy E0/90.3]s ¢n ,~
0.8
• •
LG ~3 0.6 -~ ®
0.4
._N -O ~ 0.2
o Z
~
•
Analysis
: Ply Discounting Method :
Experimental
12
1
I
I
I
0.2
0.4
0.6
0.8
Transverse
Fig. 4.
•
: Pre~ent . . . . . •
0.0 0.0
••
crack
1.0
density, c r a c k s / r a m
Stiffness reduction due to transverse cracks in a [0/903] s E-glass-epoxy laminate.
Transverse cracks in cross-ply laminated composites
157
The numerical results for %, for the damaged [-0n/90m] s laminate family are obtained by determination of the constrained displacement, U*, per unit temperature change with respect to the transverse crack density. The CTE of undamaged laminate, %,0, is determined by requiring that U = 0 along the entire edge X = 0 (see Fig. 2). The change in the CTE due to transverse cracks is obtained using material properties of T300-5208 graphite-epoxy composites. As expected, the largest change in the CTE occurs in the laminate configuration with the largest percentage of 90 ° plies as shown in Fig. 5. For laminates with identical 0/90 ratios such as [0/90]s, [02/902] s and [-03/903]s, the laminate with the higher repeating number of 90 ° plies shows
phf./Epoxy[0./~0~]~
1.0
0.8
Is
xo 0.6
[o,/~o~]~ 0.2
[o/903]s 0.0 0.0
I
I
l
i
I
I
0.5
1.0
1.5
2.0
2.5
3.0
crock
Tronsverse
density,
3.5
crocks/ram
(a)
1.0
o.,~
~~/Epo×~,
.~
x
~
[oo/9o,~1,
~.~90~
0.4
o.~ 0.0
o.o
]s
[0/9% ]~ ~ o.~
~ ~.o
Tr~n~veKee
~ ~.~ crock
~ ~o den~ffy,
~ ~
~ ~.o
~.~
crock~/~
(b) F~g. ~. Prediction of the effect of transverse cracks on e~ of the [0n/90m]~ T3~-5208 graphite~poxy laminate family (a) using finite element method; {b) using modi~ed shear lag analysis.
S. G. Lira, C. S, Hong
158 1.0
e / E p o x y ['On/9Ore.Is 0.8
xo 0 . 8 ,~, x ~ 0.4
~]s
[03/90.~Is
0.2 0.0 0.0
i
I
i
I
i
i
0.5 1.0 1.5 2.0 2.5 3,0 Transverse crack density, c r a c k s / m m
3.5
(a)
1.0 O.B o
0.6
~
Epoxy [On/BOrn.1 s
~.
g o.,
"~~~/9o2]~
[%/9%;
o.~ 0.0
0.0
'
~
'
'
'
~
0.5
~.0
1.5
~0
2"5
3.0
3.5
Transverse crock 6entRy, c r ~ c k ~ / m m (bl Fig. 6. gffect o f transverse cracks on ~ o f the [0~/90~]~ T300-5208 g r a p h i t e ~ p o x y laminate family (a) using finite e l e m e n t m e t h o d ; (b) using modified s h e a r lag analysis.
greater change in the CTE as shown in Fig. 6. These changes in the CTE due to transverse cracks are similar to the stiffness reduction as shown in Fig. 3. The constraining effect of 0 ° layers on the change in the CTE is represented in Fig. 7. The change in the CTE is relatively insensitive to the constraint of 0 ° layers for the laminates with constant 90 ° layer thickness. However, the change in the CTE due to transverse cracks is profoundly influenced by the thickness of the transverse cracking layer. In Figs 5-7, a comparison is made between changes in the CTE due to transverse cracks evaluated by the modified shear lag analysis and finite element analysis. The finite element results show larger and sharper changes in the CTE than the predictions of modified shear lag analysis. These
Transverse cracks in cross-ply laminated composites
159
1.0
p,,t,/Epoxy 0.8
o 0.6 g,~ ~ 0.4
0.2 [o/90~] s 0,0
0.0
I
I
0.5
1.0
Transverse
I
I
I
I
1.5 2.0 2.5 3.0 crack density, crocks/ram
3.5
(a) 1.0
0.8 ~ 5 2 0 8
x ,~ x
o 0.6
Graphite/Epoxy i'On/gOnOs
~ ~ 0 3 / 9 0 3] s
~ 0,4
[o/~%] s
0.2 0.0
I
0.0
~
I
~
I
~
0.5 1.0 1.5 2.0 2.5 &O Transverse crock density, cracks/ram
3.5
(b) Fig. 7. Effect of transverse cracks on ~x of the [0./903] ~ T300-5208 graphite-epoxy laminate family (a) using finite element analysis: (b) using modified shear lag analysis.
differences result from the shear deformation restricted to the interlaminar shear layer in the modified shear lag model. However, the general character of the predicted curves shows similarity with the finite element results. CLT with reduced properties of the cracked layer has been suggested as a method to predict the effect of damage on laminate properties, z4 Changes in the CTE predicted for a physically reasonable crack density of 3 cracks/mm by various approaches are compared in Table 2. Present results using modified shear lag analysis agree well with those of numerical methods and indicate well the dependence of change in the CTE on the laminate configuration. However, CLT results with a 100% reduction in the transverse stiffness of cracked layer overpredict the effect of transverse
S. G. Lira, C. S. Hong
160
TABLE 2 The Change of the Coefficient of Thermal Expansion at the Transverse Crack Density of 3 Cracks/mm in T300-5208 Cross-ply Laminate Family Laminate .[hmily
[03/90]~ [02/902] ~ [0/903] ~ [0,/90]~ [03/903]
s
~x/~xO (%) Generalized plane" strain finite element analysis ~~
Plane strain ~'~ finite element analysis
Present analysis ~'~ (shear lag model)
Laminate ~ analysis
53"9 30"0 18"9 --
56"0 30"9 19"4 44" 1 26"5
54"9 34"6 24.0 45'5 29'6
32"2 16"2 7"3 16"2 16"2
--
a Material properties: E 1 = 132 GPh, E2 = E 3 = 10"8 GPa, G~2 = G13 = 5"7 GPa, Gz3=3"4GPa, v12=v13=0"24, v23=0"49, ~ = 0 " l l / ~ e , , / ° C , ~ 2 2 = % 3 = 2 7 . 2 # g , / ~ C , h = 0"140mm, G1c = 130J/m z. ~ lnterlaminar shear layer: E = 3 . 4 5 G P a , G = 1.28GPa, v = 0 . 3 5 , 7=57.6/~,/°C, G/do = 9 . 1 4 x 10~3 Pa/m. c With 100% reduction in Ez of the 90 ~ layer (ply discounting method).
cracks and cannot describe the dependence of change in the CTE on the thickness of transverse cracking layer.
6 CONCLUSIONS
A modified shear lag analysis has been used to evaluate the effect of transverse cracks on the stiffness reduction and change in laminate thermal expansion coefficient in a family of cross-ply laminates. Predictions of the thermomechanical property changes due to transverse cracks are compared with numerical results and existing experimental data. On the basis of the present results the following conclusions can be drawn: (1) (2)
(3)
(4)
Transverse cracks reduce the effective stiffness and laminate thermal expansion coefficient in composite laminates. The stiffness reduction and change in laminate thermal expansion coefficient due to transverse cracks are profoundly influenced by the laminate configuration. The present modified shear lag model represents well the dependence of the degradation of thermomechanical properties on the laminate configuration. The present analysis is simple, yet its results show reasonable agreement with numerical results and experimental data.
Transverse cracks in cross-ply laminated composites
161
ACKNOWLEDGEMENT The authors wish to thank the K o r e a Science and Engineering F o u n d a t i o n (KOSEF) for the support of this study.
REFERENCES 1. Parvizi, A., Garrett, K. W. & Bailey, J. E., Constrained cracking in glass fibrereinforced epoxy cross-ply laminates. J. Mater. Sci., 13 (1978) 195-201. 2. Bader, M. G., Bailey, J. E., Curtis, P. T. & Parvizi, A., The mechanisms of initiation and development of damage in multi-axial fibre-reinforced plastics laminates. Proceedings of the Third International Conference, Cambridge, UK, Mechanical Behaviours of Materials, 3 (1979) 227-39. 3. Bailey, J. E., Curtis, P. T. & Parvizi, A., On the transverse cracking and longitudinal splitting behavior of glass and carbon fibre reinforced epoxy crossply laminates and the effect of Poisson and thermally generated strain. Proc. R. Soc. Lond., A, 366 (1979) 599-623. 4. Fukunaga, H., Chou, T. W., Peters, P. W. M. & Schulte, K., Probabilistic failure strength analyses of graphite/epoxy cross-ply laminates, J. Comp. Mater., 18 (1984) 339-56. 5. Flaggs, D. L., Prediction of tensile matrix failure in composite laminates. J. Comp. Mater., 19 (1985) 29-50. 6. Korczynskyj, Y. & Morley, J. G., Constrained cracking in cross-ply laminates, J. Mater. Sci., 16 (1981) 1785-95. 7. Wang, A. S. D., Fracture mechanics of sublaminate cracks in composite materials. Composite Technol. Rev., 6 (1984) 45-62. 8. Dvorak, G. J. & Laws, N. Analysis of first ply failure in composite laminates. Eng. Fract. Mech., 25 (1986) 763-70. 9. Dvorak, G. J. & Laws, N., Analysis of progressive matrix cracking in composite laminates II. First ply failure. J. Comp. Mater., 21 (1987) 309-29. 10. Hart, Y. M., Hahn, H. T. & Croman, R. B., A simplified analysis of transverse ply cracking in cross-ply laminates. Composites Sci. Technol., 31 (1988) 165-77. 11. Groves, S. E., Harris, C. E., Highsmith, A. L., Allen, D. H. & Norvell, R. G., An experimental and analytical treatment of matrix cracking in cross-ply laminates, Exp. Mech., 27 (1987) 73-9. 12. Highsmith, A. L. & Reifsnider, K. L., Stiffness reduction mechanisms in composite laminates. Damage in Composite Materials, A S T M STP, 775 (1982) 103-17. 13. Dvorak, G. J., Laws, N. & Hejazi, M., Analysis of progressive matrix cracking in composite laminates I. Thermoelastic properties of a ply with cracks. J. Comp. Mater., 19 (1985) 216-34. 14. Talreja, R., Transverse cracking and stiffness reduction in composite laminates. J. Comp. Mater., 19 (1985) 355-75. 15. Talreja, R., Stiffness properties of composite laminates with matrix cracking and interior delamination, Eng. Fract. Mech., 25 (1986) 751-62. 16. Hashin, Z., Analysis of stiffness reduction of cracked cross-ply laminates. Eng. Fract. Mech., 25 (1986) 771-8.
162
S. G. Lim, C. S. Hong
17. Bowles, D., Effect of microcracks on the thermal expansion of composite laminates. J. Comp. Mater., 17 (1984) 173-87. 18. Herakovich, C. T. & Hyer, M. W., Damage induced property changes in composites subjected to cyclic thermal loading. Eng. Fract. Mech., 25 (1986) 779-91. 19. Peters, P. W. M., The strength distribution of 90 ° plies in 0/90/0 graphite-epoxy laminates. J. Comp. Mater., 18 (1984) 545-56. 20. Peters, P. W. M. & Chou, T. W., On cross-ply cracking in glass and carbon fibrereinforced epoxy laminates. Composites, 18 (1987) 4(~6. 21. Kim, K. S. & Hong, C. S., Delamination growth in angle-ply laminated composites. J. Comp. Mater., 20 (1986) 423-38. 22. Wilkins, D. J., A comparison of the delamination and environmental resistance of a graphite-epoxy and graphite-bismaleimide. NAV-GD-0037, Naval Air Command, Washington, DC, 1981. 23. Jones, R. M., Mechanics of Composite Materials. McGraw-Hill, New York, 1975. 24. Tsai, S. W., Composites Design--1986, Think Composites. Dayton, Ohio, USA, 1986.
Effect of Transverse Cracks on the Thermomechanicai Properties of Cross-Ply Laminated Composites S. G. Lim & C. S. H o n g * Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, PO Box 150, Cheongryang, Seoul, Korea (Received 6 April 1988; revised version received 3 June 1988; accepted 6 July 1988)
A BS TRA C T The development of transverse cracks can be detrimental to the stiffness and dimensional stability of composite laminates. In this investigation, a modified shear lag analysis, taking into account the concept of interlaminar shear layer, is employed to evaluate the effect of transverse cracks on the stiffness reduction and change in the coefficient of thermal expansion in cross-ply laminated composites. This analysis involves the construction of admissible displacement fields which satisfy equilibrium and boundary conditions on the laminate and transverse crack surfaces. The present results represent well the dependence of the degradation of thermomechanical properties on the laminate configuration.
1 INTRODUCTION The important design considerations for use of composite materials in lightweight structures are stiffness and dimensional stability. Transverse cracking, which is one of the most frequently encountered damage modes in composite laminates, can be detrimental to the stiffness and dimensional stability of composite laminates. Comprehensive understanding of the effect of transverse cracks is of fundamental importance for structural design and * To whom all correspondence should be addressed. 145
Composites Science and Technology 0266-3538/89/$03'50 © ! 989 Elsevier Science Publishers Ltd, England. Printed in Great Britain
146
S. G. Lira, C. S. Hong
integrity considerations of composite structures. Transverse cracks, for the purpose of this paper, are defined as matrix-dominated cracks which run through the thickness of a given layer and extend parallel to the fiber direction. Numerous models for predicting the initiation of a transverse crack in composite laminates have been developed and correlated with experimental data.1 - 10 The effect of transverse cracks on the stiffness loss of composite laminates has also been investigated experimentally. 11'12 Experimental results have shown that multiple transverse cracking by increasing the applied stress causes a progressive stiffness reduction in composite laminates that is more marked the greater the thickness of the cracked transverse layer. In recent years, various approaches have been developed to evaluate quantitatively the loss of stiffness of composite laminates due to transverse cracks.12-16 We cite briefly this previous work. Highsmith and Reifsnider modeled the stiffness loss of damaged laminates by reducing the transverse stiffness in the cracked layer using shear lag analysis. ~2 However, the mutual interaction of cracks due to multiple transverse cracking is not considered in their analysis. Dvorak, Laws and Hejazi employed the self-consistent scheme approximation to assess the stiffness reduction of a cracked lamina in a laminate. 13 Talreja evaluated the stiffness changes of transverse cracked laminates using continuous damage theory.l'~'15 Also, Hashin analyzed the stiffness reduction of damaged cross-ply laminates by a variational method on the basis of the principle of minimum complementary energy. 16 His work gave lower bounds on laminate stiffness reduction due to multiple transverse cracking and good agreement with experimental data 12 for an E-glassepoxy [ 0 / 9 0 3 ] s laminate. There has also been interest in the effect of transverse cracks on the dimensional stability of composite laminates. Recent numerical studies have shown that transverse cracks can result in a significant change in the coefficient of thermal expansion, CTE, of composite laminates. 17.18 The purpose of the present investigation is to evaluate consistently and more simply the stiffness reduction and change in the CTE due to transverse cracks in cross-ply laminates. A modified shear lag analysis, taking into account the concept of interlaminar shear layer, is formulated to evaluate the effect of transverse cracks on the thermomechanical property degradation of composite laminates. This analysis involves a construction of admissible displacement fields which satisfy equilibrium and boundary conditions on the laminate and crack surfaces. The mutual interaction of transverse cracks is also considered in this analysis as the transverse crack spacing becomes small. Numerical results for the thermomechanical property changes due to transverse cracks are compared with existing experimental data. The results represent well the dependence of the changes
Transverse cracks in cross-ply laminated composites
147
in thermomechanical properties on the laminate configuration. Also, the present analysis is simple, yet its predictions show reasonable agreement with numerical results and experimental data.
2 F O R M U L A T I O N OF T R A N S V E R S E C R A C K I N G
2.1 Shear lag analysis for multiple cracks Consider the cross-ply laminated composites subjected to mechanical and thermal loading as shown in Fig. 1, in which the two outside 0 ° layers are of equal thickness, b, the two interlaminar shear layers are of thickness do, and the central 90 ° layer is of thickness 2d. When extended in tension, transverse cracks which run through the thickness and extend parallel to the fiber direction occur in the central 90 ° layer. The stress distribution around the transverse crack can be approximated by using shear lag analysis. '~'5'12 The derivation of the present analysis follows a one-dimensional shear lag model used by Fukunaga e t al. 4 However, the boundary conditions in their analysis are not satisfied as the transverse crack spacing becomes small. In this paper, we consider that the boundary conditions for the initial transverse crack and another crack in the vicinity of the first are satisfied. Therefore, the multiple cracking problem can be handled for the evaluation of change of thermomechanical properties in laminated composites.
....
.-:__
_:=
iii :iiii iiiiiiii, iiiiii iliiiiii. iiiiiliiiiiiii ilililil ~"
0
O~
~ ~
,
,
I
=X
o,
I I'
I
~O,.o
A ~)T~ V I -×
~oo ,,=
Fig. 1.
2L
~L
=j
Transverse cracked cross-ply laminate and analytical model.
148
S. G. Lira, C. S. Hong
The displacement fields in the vicinity of a transverse crack are assumed to be
/'/I = (80 -[- 8xT0)x + UI(X) ~)1 =
--
~0 -~- ~yO ) + VI(X)
(~) ~ = (~o + afo)X + U2(x)
v2 =
(
A12
+
r)
yo. y + V2tx)
Displacements u and v denote the components of x- and y-directions, respectively, and are assumed to be uniform through the thickness in each layer. The subscripts 1 and 2 denote the 0 ° and 90 ° layers, respectively. The linear displacement results from uniaxial extension, eo and mid-plane r and eyO, ~ due to mechanical and thermal loading, thermal strains, e~o respectively. The A u denote the laminate extensional stiffnesses. Considering the damaged cross-ply laminates with a transverse crack spacing of 2L in the 90 ~ layer (see Fig. 1), the equilibrium equations around the transverse crack become
Q l l ~d2Ul - x 2 -t- ~ d 0 (g/2-g/1) = 0 d2bt2 Q22 dx 2
G (u 2 - t~l)=0 dd 0
(2) d2vl
Q66d~5-r2 +
d2v2
Q66 ~
b@°
(v2-v,)=0
G (v z _ vl ) = 0
ddo
where the Qij and G denote the reduced stiffnesses and the shear modulus of the interlaminar shear layer, respectively. The shear stress is assumed to be proportional to the difference in displacements of the 0 ° and 90 ° layers. The boundary conditions are o~~
+ ~'~" = 0
ba~uu = (b + d)O~ v~ = 0
dV~ _ 0 dx
(3)
Transverse cracks in cross-ply laminated composites
149
at x = 0 a n d x = 2L. The above b o u n d a r y conditions are satisfied for a n y crack spacing. The superscripts T a n d M d e n o t e the t h e r m a l a n d mechanical stress c o m p o n e n t s , respectively. After imposing the b o u n d a r y conditions, the displacement fields are o b t a i n e d as
1 do'~o) -0,,) ~bQ11 ( - / 3 , e = x + f l z e
t/1 : (g0 + 8xT0)x
1 a~
u~ ~ ~o + ~o~X + ~ - &
Vl = V2 =
e ~" + ~ e -
~)
(4)
+ eyo y
----gO
where
# 1 = (1 - - e - 2=L)/(e2aL - - e - 2aL)
(5)
f12 = (e2~L -- 1)/( e2~L -- e - 2aL) F r o m eqn (4) a n d classical l a m i n a t e t h e o r y (CLT), 23 the strain c o m p o n e n t s in each layer can be written as
e?)
(1)
da~o)
=~x0 + gQ~7,~ (/3,
e~ +
,,d2~
e(x2)= g(~2 o, -
~(#,
e "~ +
#2 e-'X)
#ze-=O (6)
g~1 )
e( I ) ~ oy 0
~2)
~(2) ~ ~yO
~(~y ~ ) = ~~) = 0 where
~1o~ = ~o + ~x~o)~ ~.(2) T g(x2d ~_ ~0 4- - °xO
"(~) - -
e(1)T A12 eo + ~yo
~(2) __ ~yO
~(2)T A 1 2 B0 -+- ~pO A2 2
t'YO
A2 2
(7)
S. G. Lira, C. S. Hong
150
~.(i)T xO .~t~ tt**..a e(i)T t,y 0 , i = 1, 2, denote the thermal strain components in the x- and y-
directions, respectively and the superscripts 1 and 2 denote the 0 ° and 90 ° layers, respectively. The stress components in each layer are (lj
~x
~
~1~ + ~ °'~2d(fll e'X + f12 e-'~)
ffxO
a ~ = %or _~2~1 -(fl~ e ~ + fie e ~)} G y~1~=
{1) dQ12 {2~ 6vo + ~ 6 x O ( f l l
~
~2)
=
-
(8)
e~x + f12 e -
Q~2 ~2~t~ e~X +
&*
~x)
-
where ~ and ~yO~ ~i~ i = 1~ 2 denote the summation of mechanical and thermal stress components of the undamaged cross-ply laminate in the x- and ydirections, respectively.
2.2 Transverse cracking criterion With the solution of a modified shear lag analysis, the next procedure is to formulate the transverse cracking criterion in cross-ply laminated composites. By making use of an energy concept for the development of transverse crack under mechanical and thermal loading, the transverse cracking criterion is given by d
(9)
~da ( W - U) >_ 2dG~c
where W is the work done by the applied stress, U is the stored elastic strain energy, G~c is the critical strain energy release rate in the fiber direction and a is the growth of transverse crack. This criterion is based upon the assumption of the growth of a transverse crack spanning the entire thickness of the 90 ° layer in the direction of parallel to the fiber direction. For unit crack growth, the transverse cracking criterion, eqn (9), can be rewritten as: 3,5
( A W - AU) _ 2dG~c
(10)
where A W is the work done by the applied stress and A U is the change of stored elastic strain energy during the formation of transverse crack. The development of the transverse crack is associated with a laminate extension, 6L, due to the applied stress, where 6 L is given by /~L . I ~ ( 2 )
6L = 2 t
-",,o t n e ~,,
jo bQ~l u.,
+ flz
e -~x)
dx
(11)
Transverse cracks in cross-ply laminated composites
151
Thus 6L - 2 d~o ) ~xbQli
{fl,(e~,L _ 1) - fl2(e -~
(12)
The work done by the applied stress during the development of a transverse crack is A W = 28x(b + d)3L
(13)
for 6L from eqn(12), and At2A~z)/A22 from C L T in eqn (13) gives
2ff~(b+d)=eo(AllA22-
Substituting
A W = 2e°A~xA22-A~2A~2 &rtx2°){fl~(e" t ' - 1)-flz(e - ' ~ - 1)} (~ A22 bQli
(14)
The change of stored elastic strain energy, AU, is given by
(15)
A U = AU 1 + AU 2 -+ AU 3
where A Ut is the change of stored elastic strain energy in the 0 ° layers, A U 2 the energy change in the 90 ° layer and AUa the energy change due to the shear stress built up in the interlaminar shear layer during the development of transverse crack. The change of stored elastic strain energy in the 0 ° layers, AUt, is as follows: AU~ = 2
;o
{%-"~l~x + Oy -mo.~ ~ l _ .~l~o~m~,~.. ~y _ °- m xO~xO ~y0~y0 ~ ~
(16)
Substituting eqns (6) and (8) into eqn (16) gives AU~
4d -
~xo~t ~)~t2)S ~ 0 t v ~wt°~-- 1 ) - fl2(e - ~ -
1)}
~11
+ ~2dZ
S~(2)t2~1 ~o, {~,~,~2=L - 1 ) + ~ , ~ - ~ fl~(e -z:~-
1)
}
(17)
Similarly, the strain energy change in the 90 ° layer, A U~ is
~-
4d
~ O = { ~ ° (}2 ){ ~2d e ~
2d
+~<
~ xo,~
B~
~L
-~-~(e
t 2 a w~-~ -
-eL
-~}
B~ _
1) + ~B~B~L - ~ ( e
z:e-1)
~
(1~)
)
The strain energy change in the interlaminar shear layer, AU3, is AU3 = 2 fo ; 2Z;~do dx G
(19)
S. G. Lirn, C. S. Hong
152 where
G = doo tu2 -
(20)
u,)
Substituting eqn (20) into eqn (19) gives 2G A U 3 = ~-o
f
' dQ22
t 2~el (e2~L- 1)
1 + b~ ~-Q-~ : atx~°~
f12
(2~
__2fl~fl2L_
2(e 2~L - 1)
}
t21) By substituting eqns (14), (17), (18) and (21) into eqn (10), the transverse cracking strain, e 0, with crack spacing c~f 2L can be determined.
3 L A M I N A T E S T I F F N E S S R E D U C T I O N D U E TO T R A N S V E R S E CRACKS Multiple transverse cracking in laminated composites has been known to cause a progressive reduction in laminate stiffness. ~1'1z The occurrence of multiple transverse cracking can be attributed to the existence of a shear lag zone in the vicinity of a transverse crack. The stiffness reduction of cross-ply laminates due to transverse cracks in the 90 ° layer is derived in this section. The average axial strain of the damaged cross-ply laminate with a transverse crack spacing of 2L is
gx=(du_~
=1~ f L ( d u ~ d x L Jo \ dx,/
\dx}averag e
(22)
Substituting eqn (4) into eqn (22) gives
g~=eo+e~%+~a~fl,(e ~bQ ~~ L
~-
1 ) - fl2(e - ' ~ -
1)}
(23)
k
g~ is the strain theoretically evaluated from the stress-free state during the fabrication of composite laminates. The strain measured by the strain sensor during mechanical testing at room temperature is (g~ - e~). For a constant applied stress, 6~, the stress-strain relation of the undamaged cross-ply laminate can be written as fix = ~oE~o
(24)
where E~o denotes the Young's modulus of the undamaged cross-ply laminate in the x-direction. Also, the stress-strain relationship of the damaged laminate with crack spacing of 2L is
= (gx -- ~o)Ex
(25)
153
Transverse cracks in cross-ply laminated composites
where E~, denotes the effective Young's modulus of the damaged cross-ply laminate. F r o m eqns (24) and (25), the normalized laminate stiffness reduction due to transverse cracks is defined by Ex/Exo = go/(gx - e~o) r
(26)
where t o is the transverse cracking strain of cross-ply laminates with crack spacing of 2L.
4 CHANGE OF THERMAL EXPANSION COEFFICIENT Transverse cracks can result in a significant change in the CTE of composite laminates. The change in the CTE due to transverse cracks in cross-ply laminate family is evaluated in this section.
4.1 Modified shear lag analysis Consider again the problem geometry shown in Fig. 1. For a uniform change in temperature without mechanical loading, the average thermal strain of the damaged cross-ply laminate with crack spacing of 2L can be obtained from eqn (23) such as -r r d ~;~ = gxo + ~
L ~'xO L {fll(eaL -- 1) -- flz(e -~L -- 1)} ,.v(2) T
(27)
The effective CTE of damaged cross-ply laminate may be defined as follows: ~x=~xoq
1 d AT~bQ11
a ~2)T1--yn teaL 1) ~o L tvI~ -fl2(e -~L-1)}
(28)
where (29)
~x = grx/A T
~xo = ~;x~o/AT ~,o is the effective CTE of an undamaged cross-ply laminate. The change in the CTE due to transverse cracks can be written as 1
d
%/~xO = 1 -~ exOT"o~bQ~l
a~2) T 1
~,o ~-{fl,( e~L- 1 ) - flz(e - ' L -
1)}
(30)
4.2 Finite element analysis The numerical results for changes in the CTE in cross-ply laminated composites are obtained by using a two-dimensional finite element analysis
154
S. G. Lim, C. S. Hang
Z}
Finite Element eglon
[
~ - ~
. Transverse _Crack. • L/ Spoclng(2L) (a)
Z U=O t on
dI
Failured _ Free Surface go m
°l
I
U= U* (Constrained Displacement)
d
=-X
W=O L
•
(b)
Fig. 2. Repeated transverse cracking and finite element modeled region. (a) Repeated transverse cracks in the 90 ° layer. (b) Boundary conditions of modeled region.
based upon the constrained nodal displacement formulation. The damaged cross-ply laminate in this analysis is modeled as an assembly of homogeneous orthotropic layers separated by thin interlaminar shear layers as shown in Fig. 2. The finite element modeled region and appropriate boundary and constraint conditions are depicted in Fig. 2(a) and (b). The effective CTE of a damaged cross-ply laminate with crack spacing of 2L, ~x, is given in terms of the unknown constrained displacement, U*, as o~x =
U*/LAT
(31)
where the displacement, U*, is constrained to be uniform. This problem is treated as a plane strain problem. Eight-node isoparametric, quadrilateral elements are used to model the region of interest. The finite element mesh has 180 elements, 597 nodes and 1148 unknown degrees of freedom. A quasi three-dimensional finite element code developed to investigate the delamination growth in angle-ply laminated composites by Kim and Hang 2~ is modified to handle this problem. 5 RESULTS AND DISCUSSION Material properties of typical graphite-epoxy and glass-epoxy composite systems used in the present analysis are given in Table 1. The values of elastic
155
Transverse cracks in cross-ply laminated composites TABLE 1 Material Properties o f Used Composite Systems
AS4-35021~ E 1 (GPa) E z (GPa) vi 2
G12 (GPa) G1c (J/m 2) ~q (pU°C) ~ (p~,/°C) AT (~C) h (mm) G/do (Pa/m)
144-78 9"58 0"31 4"79 130 -0-30 28-10 - 147 0"127 1.04 x l0 ~4
E-Glass-Epoxy 12,24 41"7 13"0 0"30 3"4 240 3"8 16.7 - 125 0"203 3'37 x 10 la
T300-520817,21 132-0 10"8 0"24 5"7 --0-11 27.2 -0"140 9-14 x l0 ~a
constants and thermal expansion coefficients are well documented in many publications, but experimental data for G/do for the interlaminar shear layer are not generally available owing to experimental difficulties. Values similar to those of Refs 4, 12 and 19-21 for the shear layer parameter, G/do, are used in the present analysis. Also, the critical strain energy release rate of a unidirectional lamina is not well esta.blished. In this analysis, the values of G~c of graphite-epoxy and glass-epoxy systems are taken as 130 and 240 J/m 2 from experimental data, respectively. L22 The effect of transverse cracks on the stiffness reduction and dimensional stability of the [0n/90m] s cross-ply laminate family is evaluated by a modified shear-lag analysis. The present quantitative predictions of the thermomechanical property changes are compared with numerical results and existing experimental data. Figure 3 shows the comparison of the present predictions of stiffness reduction due to transverse cracks with experimental data for the AS4-3502 graphite-epoxy cross-ply laminate family. The stiffness of the damaged laminates is normalized by the stiffness of the undamaged laminates and plotted with respect to the transverse crack density, i.e. the number of cracks per unit length. The stiffness reduction becomes more marked with increasing 90 ° layer thickness. This is explained by the fact that by increasing the 90 ° layer thickness, more load is carried by the 90 ° layer and a larger COD (i.e. crack opening displacement) of a transverse crack results for a given applied load. The agreement between the present analysis and experimental data is reasonable except for the [02/902] s laminate. Figure 4 shows the stiffness reduction of a [0/903]s E-glass-epoxy cross-ply laminate. The discrepancy between the present predictions and the experimental data is greater than for the graphite-~epoxy cross-ply laminate family of Fig. 3. However, the general character of the predicted curve
S. G. Lim, C. S. Hong
156
AS4/3502 Grephite/Epoxy
o x w
''1''
1.00 w
0.98 0.96 ._
[o/9o] s
0.94
~9o~1s 0.92 0.90
•
[0/903.] S
~
[0/902] s
o z
0.0
I
I
I
I
I
0.5
1.0
1.5
2.0
2.5
3.0
Transverse crack density, crocks/mm Fig. 3.
Stiffness reduction due to transverse cracks in the graphite-epoxy laminate family.
[0,/90.,)s
AS4-3502
suggests similarity with experimental data. Some difference between the analytical model and experiment is expected in any case, since the damage mode in the present analysis is restricted to uniformly distributed straight transverse cracks. In practice, because of the random nature of the transverse cracking behavior, the uniform distribution of straight transverse cracks does not occur exactly in thick cracking layers. It was observed experimentally that there are more curved transverse cracks than straight transverse cracks with increasing thickness of cracking layer in cross-ply laminated composites. ~ The thickness of the transverse cracking layer of E - g l a s s - e p o x y [ 0 / 9 0 3 ] s laminates is thicker than that of the AS4-3502 graphite-epoxy cross-ply laminate family in Fig. 3.
E-Glass/Epoxy E0/90.3]s ¢n ,~
0.8
• •
LG ~3 0.6 -~ ®
0.4
._N -O ~ 0.2
o Z
~
•
Analysis
: Ply Discounting Method :
Experimental
12
1
I
I
I
0.2
0.4
0.6
0.8
Transverse
Fig. 4.
•
: Pre~ent . . . . . •
0.0 0.0
••
crack
1.0
density, c r a c k s / r a m
Stiffness reduction due to transverse cracks in a [0/903] s E-glass-epoxy laminate.
Transverse cracks in cross-ply laminated composites
157
The numerical results for %, for the damaged [-0n/90m] s laminate family are obtained by determination of the constrained displacement, U*, per unit temperature change with respect to the transverse crack density. The CTE of undamaged laminate, %,0, is determined by requiring that U = 0 along the entire edge X = 0 (see Fig. 2). The change in the CTE due to transverse cracks is obtained using material properties of T300-5208 graphite-epoxy composites. As expected, the largest change in the CTE occurs in the laminate configuration with the largest percentage of 90 ° plies as shown in Fig. 5. For laminates with identical 0/90 ratios such as [0/90]s, [02/902] s and [-03/903]s, the laminate with the higher repeating number of 90 ° plies shows
phf./Epoxy[0./~0~]~
1.0
0.8
Is
xo 0.6
[o,/~o~]~ 0.2
[o/903]s 0.0 0.0
I
I
l
i
I
I
0.5
1.0
1.5
2.0
2.5
3.0
crock
Tronsverse
density,
3.5
crocks/ram
(a)
1.0
o.,~
~~/Epo×~,
.~
x
~
[oo/9o,~1,
~.~90~
0.4
o.~ 0.0
o.o
]s
[0/9% ]~ ~ o.~
~ ~.o
Tr~n~veKee
~ ~.~ crock
~ ~o den~ffy,
~ ~
~ ~.o
~.~
crock~/~
(b) F~g. ~. Prediction of the effect of transverse cracks on e~ of the [0n/90m]~ T3~-5208 graphite~poxy laminate family (a) using finite element method; {b) using modi~ed shear lag analysis.
S. G. Lira, C. S, Hong
158 1.0
e / E p o x y ['On/9Ore.Is 0.8
xo 0 . 8 ,~, x ~ 0.4
~]s
[03/90.~Is
0.2 0.0 0.0
i
I
i
I
i
i
0.5 1.0 1.5 2.0 2.5 3,0 Transverse crack density, c r a c k s / m m
3.5
(a)
1.0 O.B o
0.6
~
Epoxy [On/BOrn.1 s
~.
g o.,
"~~~/9o2]~
[%/9%;
o.~ 0.0
0.0
'
~
'
'
'
~
0.5
~.0
1.5
~0
2"5
3.0
3.5
Transverse crock 6entRy, c r ~ c k ~ / m m (bl Fig. 6. gffect o f transverse cracks on ~ o f the [0~/90~]~ T300-5208 g r a p h i t e ~ p o x y laminate family (a) using finite e l e m e n t m e t h o d ; (b) using modified s h e a r lag analysis.
greater change in the CTE as shown in Fig. 6. These changes in the CTE due to transverse cracks are similar to the stiffness reduction as shown in Fig. 3. The constraining effect of 0 ° layers on the change in the CTE is represented in Fig. 7. The change in the CTE is relatively insensitive to the constraint of 0 ° layers for the laminates with constant 90 ° layer thickness. However, the change in the CTE due to transverse cracks is profoundly influenced by the thickness of the transverse cracking layer. In Figs 5-7, a comparison is made between changes in the CTE due to transverse cracks evaluated by the modified shear lag analysis and finite element analysis. The finite element results show larger and sharper changes in the CTE than the predictions of modified shear lag analysis. These
Transverse cracks in cross-ply laminated composites
159
1.0
p,,t,/Epoxy 0.8
o 0.6 g,~ ~ 0.4
0.2 [o/90~] s 0,0
0.0
I
I
0.5
1.0
Transverse
I
I
I
I
1.5 2.0 2.5 3.0 crack density, crocks/ram
3.5
(a) 1.0
0.8 ~ 5 2 0 8
x ,~ x
o 0.6
Graphite/Epoxy i'On/gOnOs
~ ~ 0 3 / 9 0 3] s
~ 0,4
[o/~%] s
0.2 0.0
I
0.0
~
I
~
I
~
0.5 1.0 1.5 2.0 2.5 &O Transverse crock density, cracks/ram
3.5
(b) Fig. 7. Effect of transverse cracks on ~x of the [0./903] ~ T300-5208 graphite-epoxy laminate family (a) using finite element analysis: (b) using modified shear lag analysis.
differences result from the shear deformation restricted to the interlaminar shear layer in the modified shear lag model. However, the general character of the predicted curves shows similarity with the finite element results. CLT with reduced properties of the cracked layer has been suggested as a method to predict the effect of damage on laminate properties, z4 Changes in the CTE predicted for a physically reasonable crack density of 3 cracks/mm by various approaches are compared in Table 2. Present results using modified shear lag analysis agree well with those of numerical methods and indicate well the dependence of change in the CTE on the laminate configuration. However, CLT results with a 100% reduction in the transverse stiffness of cracked layer overpredict the effect of transverse
S. G. Lira, C. S. Hong
160
TABLE 2 The Change of the Coefficient of Thermal Expansion at the Transverse Crack Density of 3 Cracks/mm in T300-5208 Cross-ply Laminate Family Laminate .[hmily
[03/90]~ [02/902] ~ [0/903] ~ [0,/90]~ [03/903]
s
~x/~xO (%) Generalized plane" strain finite element analysis ~~
Plane strain ~'~ finite element analysis
Present analysis ~'~ (shear lag model)
Laminate ~ analysis
53"9 30"0 18"9 --
56"0 30"9 19"4 44" 1 26"5
54"9 34"6 24.0 45'5 29'6
32"2 16"2 7"3 16"2 16"2
--
a Material properties: E 1 = 132 GPh, E2 = E 3 = 10"8 GPa, G~2 = G13 = 5"7 GPa, Gz3=3"4GPa, v12=v13=0"24, v23=0"49, ~ = 0 " l l / ~ e , , / ° C , ~ 2 2 = % 3 = 2 7 . 2 # g , / ~ C , h = 0"140mm, G1c = 130J/m z. ~ lnterlaminar shear layer: E = 3 . 4 5 G P a , G = 1.28GPa, v = 0 . 3 5 , 7=57.6/~,/°C, G/do = 9 . 1 4 x 10~3 Pa/m. c With 100% reduction in Ez of the 90 ~ layer (ply discounting method).
cracks and cannot describe the dependence of change in the CTE on the thickness of transverse cracking layer.
6 CONCLUSIONS
A modified shear lag analysis has been used to evaluate the effect of transverse cracks on the stiffness reduction and change in laminate thermal expansion coefficient in a family of cross-ply laminates. Predictions of the thermomechanical property changes due to transverse cracks are compared with numerical results and existing experimental data. On the basis of the present results the following conclusions can be drawn: (1) (2)
(3)
(4)
Transverse cracks reduce the effective stiffness and laminate thermal expansion coefficient in composite laminates. The stiffness reduction and change in laminate thermal expansion coefficient due to transverse cracks are profoundly influenced by the laminate configuration. The present modified shear lag model represents well the dependence of the degradation of thermomechanical properties on the laminate configuration. The present analysis is simple, yet its results show reasonable agreement with numerical results and experimental data.
Transverse cracks in cross-ply laminated composites
161
ACKNOWLEDGEMENT The authors wish to thank the K o r e a Science and Engineering F o u n d a t i o n (KOSEF) for the support of this study.
REFERENCES 1. Parvizi, A., Garrett, K. W. & Bailey, J. E., Constrained cracking in glass fibrereinforced epoxy cross-ply laminates. J. Mater. Sci., 13 (1978) 195-201. 2. Bader, M. G., Bailey, J. E., Curtis, P. T. & Parvizi, A., The mechanisms of initiation and development of damage in multi-axial fibre-reinforced plastics laminates. Proceedings of the Third International Conference, Cambridge, UK, Mechanical Behaviours of Materials, 3 (1979) 227-39. 3. Bailey, J. E., Curtis, P. T. & Parvizi, A., On the transverse cracking and longitudinal splitting behavior of glass and carbon fibre reinforced epoxy crossply laminates and the effect of Poisson and thermally generated strain. Proc. R. Soc. Lond., A, 366 (1979) 599-623. 4. Fukunaga, H., Chou, T. W., Peters, P. W. M. & Schulte, K., Probabilistic failure strength analyses of graphite/epoxy cross-ply laminates, J. Comp. Mater., 18 (1984) 339-56. 5. Flaggs, D. L., Prediction of tensile matrix failure in composite laminates. J. Comp. Mater., 19 (1985) 29-50. 6. Korczynskyj, Y. & Morley, J. G., Constrained cracking in cross-ply laminates, J. Mater. Sci., 16 (1981) 1785-95. 7. Wang, A. S. D., Fracture mechanics of sublaminate cracks in composite materials. Composite Technol. Rev., 6 (1984) 45-62. 8. Dvorak, G. J. & Laws, N. Analysis of first ply failure in composite laminates. Eng. Fract. Mech., 25 (1986) 763-70. 9. Dvorak, G. J. & Laws, N., Analysis of progressive matrix cracking in composite laminates II. First ply failure. J. Comp. Mater., 21 (1987) 309-29. 10. Hart, Y. M., Hahn, H. T. & Croman, R. B., A simplified analysis of transverse ply cracking in cross-ply laminates. Composites Sci. Technol., 31 (1988) 165-77. 11. Groves, S. E., Harris, C. E., Highsmith, A. L., Allen, D. H. & Norvell, R. G., An experimental and analytical treatment of matrix cracking in cross-ply laminates, Exp. Mech., 27 (1987) 73-9. 12. Highsmith, A. L. & Reifsnider, K. L., Stiffness reduction mechanisms in composite laminates. Damage in Composite Materials, A S T M STP, 775 (1982) 103-17. 13. Dvorak, G. J., Laws, N. & Hejazi, M., Analysis of progressive matrix cracking in composite laminates I. Thermoelastic properties of a ply with cracks. J. Comp. Mater., 19 (1985) 216-34. 14. Talreja, R., Transverse cracking and stiffness reduction in composite laminates. J. Comp. Mater., 19 (1985) 355-75. 15. Talreja, R., Stiffness properties of composite laminates with matrix cracking and interior delamination, Eng. Fract. Mech., 25 (1986) 751-62. 16. Hashin, Z., Analysis of stiffness reduction of cracked cross-ply laminates. Eng. Fract. Mech., 25 (1986) 771-8.
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S. G. Lim, C. S. Hong
17. Bowles, D., Effect of microcracks on the thermal expansion of composite laminates. J. Comp. Mater., 17 (1984) 173-87. 18. Herakovich, C. T. & Hyer, M. W., Damage induced property changes in composites subjected to cyclic thermal loading. Eng. Fract. Mech., 25 (1986) 779-91. 19. Peters, P. W. M., The strength distribution of 90 ° plies in 0/90/0 graphite-epoxy laminates. J. Comp. Mater., 18 (1984) 545-56. 20. Peters, P. W. M. & Chou, T. W., On cross-ply cracking in glass and carbon fibrereinforced epoxy laminates. Composites, 18 (1987) 4(~6. 21. Kim, K. S. & Hong, C. S., Delamination growth in angle-ply laminated composites. J. Comp. Mater., 20 (1986) 423-38. 22. Wilkins, D. J., A comparison of the delamination and environmental resistance of a graphite-epoxy and graphite-bismaleimide. NAV-GD-0037, Naval Air Command, Washington, DC, 1981. 23. Jones, R. M., Mechanics of Composite Materials. McGraw-Hill, New York, 1975. 24. Tsai, S. W., Composites Design--1986, Think Composites. Dayton, Ohio, USA, 1986.