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Stiffness reduction and energy release rate of cross-ply laminates during fatigue tests
Composite Srrucrures 30 (1995) 123-130 0 1995 Elsevier Science Limited Printed in Great Britain. All rights reserved 0263-8223/95/$9.50 0263-8223(94)00061-l
ELSEVIER
Stiffness reduction and energy release rate of cross-ply laminates during fatigue tests A. El Mahi,” J.-M. BertheloF &J. Brillaudb * Laboratoire de Mkanique
Productique et Mat&riaux, Universite’ du Maine, avenue Olivier Messiaen, B.P. 535, 72017 Le Mans Ckdex, France “Laboratoire de Mkcanique et Physique des Ma&am, ENSMA, site du Futuroscope, 86360 Chasseneuil du Poitou, France
The purpose of the paper is to investigate the experimental results obtained during cyclic tensile tests in the case of two equivalent stacking sequences: (O,/ 90), and (0,/90/O,),, of carbon fibre laminates. During the fatigue tests, matrix cracking in 90” plies has been characterized by the crack distributions in the width of the test specimen, giving the S-N distributions. Then, the longitudinal stiffness reduction has been evaluated from these crack distributions and by a finite element analysis. The results obtained show a linear relationship, in the domain studied, between the stiffness reduction and the crack area, leading to a constant energy release rate during the fatigue tests.
degradation. The shear lag analysis was used to predict the stiffness reduction according to the transverse crack density.2-5 A variational approach was used by Hashin6 to study the elastic property degradation and the stress distribution in a cracked cross-ply laminate. Lee and Aboudi’ investigated an analytical solution to analyse the effects of transverse matrix cracking, where the displacement field was estimated by Legendre polynomials. Sun and Jens studied the influence of the stacking sequences by finite element analysis, in the case of [0,/90,], and [90./O,], laminates. Brillaud and El Mahi proposed a numerical simulation of the influence of stacking sequence on transverse ply cracking of cross-ply laminates. They studied two particular laminates [(O,/ 90&l, and ](OmlN190plN)S]~hi,which have the same total ply number but differ by the layer thicknesses. They established a relationship between their respective stiffness and energy release rate, involving the aspect ratio of the representative elementary cell of the cracked laminates. Extensive experimental studies have been achieved in the case of cross-ply carbon/epoxy (T 300/914) laminates, subjected to fatigue tests.“’ Two stacking sequences have been studied: [O,/ 90/O,], and [0,/90], (Fig. l), which differ by the position of 90” plies. All the plies are similar (thickness of O-125 mm) and the difference
1 INTRODUCTION In the case of cross-ply laminates subjected to uniaxial quasi-static or cyclic loading, the early stage of damage is dominated by matrix cracking in the 90” plies. These matrix cracks develop in the fibre direction and extend across the laminates from the free edges of test specimens. The number of cracks increases with the increase of the load or the number of cycles until a saturation crack density, depending on the thickness of the transverse plies.’ Since the matrix cracks extend in the transverse direction only, it is possible to use a two-dimensional analysis to investigate the transverse cracking: Many authors studied the experimental initiation and propagation of transverse cracks in cross-ply laminates subjected to tensile tests. The results obtained show that, in the case of quasistatic tests, most cracks cross instantaneously the test specimen width from edge to edge. In this case, the transverse cracking is characterized only by the crack density. In return, in the case of cyclic tensile tests, the crack lengths and the cracking density increase progressively with increasing cycle number. So in this case, the transverse cracking must be characterized by the distribution of the crack lengths. Analytical and numerical procedures have been developed for predicting the laminate 123
124
A. Et Mahi, J.-M. Berthetot, J. Brittaud
between the two stacking sequences lies in the number and the thickness of the 90” plies: two plies of a single thickness for the first sequence, noted S, and one ply of a double thickness for the second sequence, noted D. The purpose of this paper is to evaluate, from the experimental results observed, the stiffness reduction of these laminates and the energy release rate as a function of the cycle number.
Throughout the fatigue tests, the crack distributions have been evaluated by X-ray observations. Figure 3 gives a typical distribution for the cycle number Nj. On the other hand, the different analyses2-9 of the stiffness reduction suppose the cracks cross the whole width of the test specimen and are uniformly distributed along the xdirection. So a simplified procedure can be suggested to evaluate the stiffness reduction. The test specimen is divided in n bands along the width w (Fig. 4). For the cycle number Ni, the crack distribution can be evaluated by the number n, of cracks crossing the band i (Fig. 4). Then, the stiffness reduction in band i can be evaluated using one of the analyses2-9 and supposing the cracks are uniformly distributed along the length L of the test specimen, with the uniform crack density di( N,) given by:
2 MODELLING THE STIFFNESS REDUCTION During the fatigue tests, matrix cracks in 90” plies initiate at the free edges of test specimens and grow slowly across the specimen edges (Fig. 2). For a given cycle number Nj, the matrix cracking pattern is characterized by the edge of initiation (edge 1 or 2), the position xi and the length a, of each crack. So the exact analyses of the stress distribution and the stiffness reduction are very complex, and need the resolution of a three-dimensional problem.
laminate
D
[ 0,190
Is
Fig. 1.
&(N;)=
nANi) L
This approach neglects also the crack tip effect on the stress distribution. The longitudinal Young’s modulus Exi of the band i can be deduced
laminate
5
[ 0,/90/O,
Is
The two stacking sequences investigated.
i________________________________ )___________________ ____________
Fig. 2.
(1)
Transverse matrix cracking.
125
Fatigue tests of cross-ply laminates
from a shear lag analysis or a finite element analysis, as a function of di: E,;=E,Wi)
(2
Equality of the displacements leads to:
of the YIbands
iEX;;S; I=1
E+=
(3)
where E, is the longitudinal Young’s modulus of the cracked laminate, u the imposed displace-
T
$ (NJ= & t Y;Ex[d,(Nj)I x
XI
I
where E,[dj( Nj)] is the Young’s modulus of a cracked band i with the crack density equal to di(Nj )* In the case of bands of the same width (y,= W/ n), relation 4 becomes:
(5)
1
Fig. 3.
ment, S the area of the cross-section of specimen and Si the area of the cross-section of band i. Introducing the width yi of band i and the longitudinal Young’s modulus EO, of the undamaged laminate, relation 3 yields:
Figures 5 and 6 show examples of crack distributions in the two laminates studied, evaluated
Typical transverse matrix crack distribution.
W
position
Fig. 4.
Evaluation
of the bands
of the crack distribution.
5 (x103) 4
l
laminates D
D
laminates D 0.6~3,
0.813,
7 6 h .e
3
10 cycles
r 8
2
:: E "
l 1
l
l
l
l
l
l
l
I 0
0
0’ 4
Fig. 5.
0
0 I
w=
I
OO
15
15 mm
Crack distributions obtained after 10 cycles.
0
0
0
_
+
for the D laminates,
4
Fig. 6.
Crack
_
15
w=lSmm
W
distributions obtained for the D and laminates, after 1O4cycles.
S
126
A. El Mahi, J.-M. Berthelot, J. Brillaud
Fig. 7.
Crack distribution
evaluated as a two crack band distribution.
in the case where the specimen width (15 mm) is divided in 8 bands, for two cycle numbers ( 10 and 104) and for a maximum fatigue tensile stress of 0*6a,, and O*So,,, CJ~,being the static failure stress of laminates ( CT[,= 1, 440 MPa). The bands being equal, the distributions of the cracked area is simply deduced from these crack distributions by multiplying the crack number scale by the area of an elementary crack. So in the case of equal bands, the crack distributions represent the ‘S-N distributions’ of the fatigue tests. On the other hand, it can be observed (Fig. 6) that, in the case of S laminates and for O*Sa,,, the length of the cracks are lower than half the specimen width. In this case and according to the of Lafaire-Frenot and approach HenaffGardin ‘(),’l the crack distributions can be evaluated (Fig. 7) as two cracked bands from each edge of the test specimen, of width equal to the average length j of the actual cracks and with cracks uniformly distributed in the two bands with an average density d.
Thus, the longitudinal Young’s modulus can be calculated, supposing the equality of the displacement in the three bands, by:
where E,[d(Nj)] is the cracked band with the a(Nj). At last, it can be also the case of D laminates
Young’s modulus of a crack denstiy equal to observed (Fig. 6) that, in the crack density is the
same in all the bands, for a large enough number of cycles ( lo4 cycles for a maximum fatigue stress of O.Sa,,). So in this case, the stiffness reduction can be estimated by considering that the cracks created cross all the test specimen by:
where E,[d( N,)] is the Young’s modulus of a cracked band with the crack density equal to d(N,), the crack density in all the bands for the cycle number N,. Thus in the most general case, the stiffness reduction is evaluated by relations 4 or 5, with two extreme crack distributions: a two-cracked band distribution and a single-band distribution across all the specimen width, where the stiffness reduction can be approximated respectively by eqns (8) and (9). 2.1 Stiffness reduction as a function of crack density Equations (4), (5), (8) and (9) point out that the basic result to be estimated is the stiffness reduction E,( d)/Et as a function of the transverse crack density. Figure 8 shows the comparison between the predicted and the experimental results obtained for the longitudinal Young’s modulus reduction as a function of the crack density, in the case of glass fibre epoxy cross-ply laminates. The experimental results were reported by Highsmith and Reifsnider.* The predicted results are derived from the shear analyses developed by Highsmith andd Reifsnider* and by Lee and Daniel,’ and from the finite element analysis implemented in Ref. 9. The results reported in Fig. 8 show that values obtained by finite element analysis are in good agreement with the experimental values. The predictions obtained by the shear lag analyses are close to the experimental results, although closer in the case of analysis developed by Lee and Daniel. So the shear lag analysis can
127
Fatigue tests of cross-ply laminates
6, +! G 3 9 B E _m F >a -z a a ‘zl s _I
1.25
1 .oo
z
1.002
r s
1.000
.z +z
0.998
l
laminates
S
I
Fz
0.75
-;
0.996
5 0.50 l
.
l
--- --
0.25
0
0
experimental shear
lag analysis
shear lag analysis finite element analysis
[ 9 ]
1
I
200
400
600
density
0.994 0.992
F
I crack
Fig. 8.
5 C a .s
( 2] [2 ] [5 ]
results
800
(cracks/m)
Stiffness reduction of [O/90,], laminates as a function of transverse crack density.
be used as a first practical estimation of the stiffness reduction. Figure 9 shows the stiffness reduction as a function of the crack density evaluated by the finite element analysis in the case of II laminates [0,/90]s and S laminates [0,/90/O,],. The results show a more pronounced reduction (about twice) in the case of D laminates. This fact is the consequence of the simihtude property of S and D laminates: the stiffness reduction of D laminates for a crack density is equal to the stiffness reduction of S laminates for a double crack density. At last, it must be noted that, for a practical use (following sections), the results obtained by finite element analysis and reported in Fig. 9, can be fitted by polynomials. 2.2 Stiffness reduction as a function of cycle number Stiffness reduction as a function of cycle number is deduced from the stiffness reduction as a function of the crack density (Fig. 9) and from the crack distributions as a function of the cycle number (Figs 5 and 6). Figure 10 shows the longitudinal Young’s modulus reduction, calculated by relation 5 in the case of D and S laminates, subjected to a maximum fatigue tensile stress of 0.60, and 0.80,. The longitudinal Young’s modulus E,( di) as a function of the crack density in the band i has been derived from the results reported in Fig. 9. For comparison, the stiffness reduction was also evaluated by eqn (8) (two-cracked band distributions) in the case of S laminates in Figs 10(c) and 10(d) and by eqn (9) (single-band distribution) in the case of D lami-
crack
density
(cracks/m)
Fig. 9. Stiffness reduction of laminates 10,/90]J (D Iaminates) and [0,/90/0,],Y (S laminates) as a function of crack density.
nates and for a maximum fatigue stress of Oga, (Fig. 10(b)). The different evaluations are very close. According to the crack distributions, the stiffness reduction occurs with the initiation of transverse matrix cracking, since the first cycles for the D laminates and for lo3 to lo4 cycles in the case of S laminates. Then, the longitudinal Young’s modulus of laminates decreases with the crack propagation and tends to be constant with the effect of crack saturation, when the cycle number increases. Moreover, the stiffness reduction is very low: reduction of about 0.5% in the case of D laminates and 0.15% in the case of S laminates. Such low variations result from the low thickness of 90” plies compared to the thickness of 0” plies, and are obviously difficult to measure for the laminates studied. 2.3 Stiffness reduction as a function of the crack area The longitudinal Young’s modulus reduction as a function of the crack area per unit length of the test specimen is deduced from the curves of Fig. 10, by multiplying, for the cycle number Nj, the crack density in each band i by the thickness of the 90” plies and the width yj of band i. The results obtained, independent of the maximum fatigue stress, are reported in Fig. 11 for the S and D laminates. These results show a linear relationship (in the domain studied) between the stiffness reduction and the crack area S,:
128
A. El Mahi, J.-M. Berthelot, J. B&laud
1.000 0.999 0.998 0.997
~~~~~
0.996 0.995
I 1
h 10
,
102
Cycle number
103
104
105
1
( 106
10'
Cycle number
‘.ooo’-
a, “OOOIY % 1.0000%
“t. 2
0.9995-
z P
E : 0.9990>
0.998d
’ 10
1
I
102
1
103
L
105
104
I
1
106
107
0.99851 1
I
10
L
102
Cycle number
o^,
r
1.000
l
9, 0
I
I
I
I
I
I
1
2
3
4
5
6
7
10'
s=f(N)
1
can be obtained by a simple law of the crack area growth as a function of cycle number. Equations (10) and (12) sh ow that the function f{ N) can be deduced from the rate dS,./dN of crack growth given by:
_* F 2 0.9990 >:Y! ux 1.0000 K
7
(12)
I
0.9995
0
(11)
(m’/m)
,
2
dS. L=-!df 1
2
3
4
Crack area S,
Fig. 11.
106
(x10-3)
a*
0.9985
105
Equation (10) relates the Young’s modulus reduction as a function of crack area being established, and it is possible to consider whether the Young’s modulus reduction as a function of cycle number N (Figs 10):
0.996
Crack area S,
2z
104
a = O-667 m- ’ for the D laminates.
0.997
1.0001
103
I
a = 0.357 m- ’ for the S laminates,
0.998
0.995
I
with:
0.80,,
52 0.999
F :
I
Cycle number
2
$ f _w
I
Stiffness reduction of [0,/90],5 and [0,/90/O,] ( laminates as a function of cycle number.
Fig. 10.
1.001
0
\
9
I
5
6
7
(x10-3)
(mz/m)
Stiffness reduction of [0,/90],s and [03/90/0,],s laminates as a function of the crack area.
dN
a dN
(13)
The rate of crack area growth versus the cycle number is reported in Fig. 12. This figure shows that the crack propagation rate is decreasing
129
Fatigue tests of cross-ply laminates 1 o-2
3
u
lo-' -
B E
+
Laminates .S
l
0.80, 0.80,
expressed as:
o 0.60, q
0.6~7,
I
(18)
l l l
0
N‘ 5 2 ?
Laminates D
l *
l*
00
10-G -
0
In the case of laminates studied: US,+ 1, and the energy release rate is simply approximated by:
%b
O Oo
000
00
0
a O.
0
G- aW’,‘,
‘HO a
1O-8 -
:
O
Z” 1O-10
I
I
I
,
,
1
10
102
103
104
10s
a$ 106
10'
Cycle number
Fig. 12.
Rate of crack area growth versus cycle number.
(19)
Thus the energy release rate is found to be independent of the crack area created in the domain studied. Finally, the energy release rate can be expressed as: ,
throughout the fatigue tests and that the results are approximately distributed on a straight line. However any general law does not seem to be deduced from these results.
The maximum energy release rate G,,, associated with the maximum fatigue stress a,,, = Rg,, is then given by: G,,,=R’G,
3 ENERGYRELEASERATE
with
GU=t$
V,, x
As the fatigue tests are performed under stress control, the energy release rate G is related to the stored elastic energy W,, by the expression:
(14) where S, is the surface area created and cr the prescribed stress. The elastic energy is given by:
(21)
In this expression V, is the volume of a test specimen of one meter length (a is expressed in m/m2): V, = 30 10e6 m3/m. E!j is the Young’s modulus of the undamaged laminates: E’j = 125% GPa, deduced from tensile tests. These values associated with values of a given by eqn ( 11) yields to:
G,, = 88 (J/m3)/m for the S laminates, G,, = 165 (J/m”)/m for the D laminates. These values give:
introducing the longitudinal Young’s modulus E, and the volume V, of the test specimen. The substitution of relation 15 in relation 14 yields: G=
wo
a@:/%) e’ as,
(16)
where W$ is the elastic energy stored in the undamaged test specimen:
for S laminates G,,, = 56 (J/m3)/m for a,,, = 0% (T,,, G,,, = 32 (J/m”)/m for a,,, = O-6(7,,; for D laminates G,,,,, = 106 (J/m3)/m for o,,, = 0.8 or,, G,,, = 59 (J/m”)/m for a,,, = O-6(T,,.
(17) Differentiating with respect to the crack area, S,, the function E!j/E, given by eqn (lo), the energy relase rate as a function of the crack area is
These values allow an explanation of the delay in crack initiation and propagation, observed in the case of S laminates and in the case of D laminates for a loading amplitude equal to O-60, (Fig.
130
A. El Mahi, J.-M. Berthelot, J. Brillaud
10). Indeed, only the D laminates for a load level of O*Sa,, show crack initiation and propagation since the first cycles. This process can be explained by the value of the maximum energy release rate G,, equal to 106 (J/m”)/m and practically the same as the critical value G,, equal to 110 (J/m”)/m, obtained at the crack initiation in the case of quasi-static tests.i3 The delay in crack initiation is observed in the three other cases (D laminates at a load level of 0.6~7,~and S laminates at the two loading amplitudes of 0.6 ou and 0.8 oU), for which G,,,,, is lower than G,.. This delay is all the more important as the value of G,,,,, is small.
REFERENCES 1. Garett, K. W. & Bailey, J. E., Multiple transverse fracture in 90” cross-ply laminates of a glass fibre-reinforced polyester. Journal of Materials Science, 12 (1977) 157-68.
2. Highsmith, A. L. & Reifsnider, K. L., Stiffness reduction mechanisms in composite laminates. Damage in Composite Materials. ASTM STP 775, ed. K.-L. Reifsnider. American Society for Testing and Materials (1982), 103-17. 3. Laws, N. & Dvorak, G. J., Progressive transverse cracking in composite laminates. Journal of Composite Materials, 22 (1988) 900-16.
4. Lim, S. G. & Hong, C. S., Prediction of transverse cracking and stiffness reduction in cross-ply laminated composites. Journal of Composite Materials, 23 (1989) 695-713.
CONCLUSION In this paper, stiffness reduction and energy release have been investigated in the case of crossply laminates subjected to fatigue tensile tests. Two stacking sequences [0,/90], and [0,/90/O,], have been considered. First, during fatigue tests, the damage progression has been characterized by the crack distributions in the width of the test specimen, giving the S-N distributions. According to the S-N distributions observed during the fatigue tests, the stiffness reduction occurs with the initiation of transverse matrix cracking, over the first cycles for the [0,/90], laminates and for lo3 to lo4 cycles in the case of [03/90/041s laminates. Stiffness reduction as a function of the crack area has been evaluated by finite element analysis, from the S-N distributions. The results obtained show a linear relationship, in the domain studied, between the stiffness reduction and the crack area, which is depending on the stacking sequence. Thus, the energy release rate, associated to the crack propagation, is found to be independent of the crack area created, during fatigue tests.
5. Lee, J. W. & Daniel, I. M., Progressive transverse cracking of cross-ply composites laminates. Journal of Composite Materials, 24 (1990) 1225-43.
6. Haskin, Z., Analysis of cracked laminates: a variational approach. Mechanics of Materials, 4 (1985) 121-36. 7. Lee, S. W. & Aboudi, J., Analysis of composite laminates with matrix cracks. Report CCMS-88-03. (1988) College of Engineering Virginia Polytechnic Institute and State University, Blacksburg, Virginia. 8. Sun, C. T. & Jen, K. C., On the effect of matrix cracks on laminate strength. Journal of Reinforced Plastics and Composites. 6 (1987) 208-23.
9. Brillaud, J. & El Mahi, A., Numerical simulation of the influence of the stacking sequence on transverse ply cracking in composites laminates. Composite Structures, 17 (1991) 23-35.
10. Henaff-Gardin, C., Lafarie-Frenot, M. C., Brillaud, J. & El Mahi, A., Influence of the stacking sequence on fatigue transverse ply cracking in cross-ply laminates. ASTM STP 1128, ed. J. E. Masters, (1992), Philadelphia, USA, 236-55. 11 Lafarie-Frenot, M. C. & Henaff-Gardin, C., Analyse locale et modtlisation de la fissuration transverse de stratifies croids carbone/epoxyde. Journtes Nationales des Composites, ( 1990), JNC7, Lyon, 63 l-42. 12 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. Experimental Mechanics, 27 (1987) 73-9. 13 El Mahi, A., Brillaud, J. & Berthelot, J.-M., Evaluation numerique du module longitudinal et du taux de restitution d’energie de stratifies endommages au tours des essais de fatigue. Journees Nationales des Composites, (1992), JNC8, Paris, 461-72.
ELSEVIER
Stiffness reduction and energy release rate of cross-ply laminates during fatigue tests A. El Mahi,” J.-M. BertheloF &J. Brillaudb * Laboratoire de Mkanique
Productique et Mat&riaux, Universite’ du Maine, avenue Olivier Messiaen, B.P. 535, 72017 Le Mans Ckdex, France “Laboratoire de Mkcanique et Physique des Ma&am, ENSMA, site du Futuroscope, 86360 Chasseneuil du Poitou, France
The purpose of the paper is to investigate the experimental results obtained during cyclic tensile tests in the case of two equivalent stacking sequences: (O,/ 90), and (0,/90/O,),, of carbon fibre laminates. During the fatigue tests, matrix cracking in 90” plies has been characterized by the crack distributions in the width of the test specimen, giving the S-N distributions. Then, the longitudinal stiffness reduction has been evaluated from these crack distributions and by a finite element analysis. The results obtained show a linear relationship, in the domain studied, between the stiffness reduction and the crack area, leading to a constant energy release rate during the fatigue tests.
degradation. The shear lag analysis was used to predict the stiffness reduction according to the transverse crack density.2-5 A variational approach was used by Hashin6 to study the elastic property degradation and the stress distribution in a cracked cross-ply laminate. Lee and Aboudi’ investigated an analytical solution to analyse the effects of transverse matrix cracking, where the displacement field was estimated by Legendre polynomials. Sun and Jens studied the influence of the stacking sequences by finite element analysis, in the case of [0,/90,], and [90./O,], laminates. Brillaud and El Mahi proposed a numerical simulation of the influence of stacking sequence on transverse ply cracking of cross-ply laminates. They studied two particular laminates [(O,/ 90&l, and ](OmlN190plN)S]~hi,which have the same total ply number but differ by the layer thicknesses. They established a relationship between their respective stiffness and energy release rate, involving the aspect ratio of the representative elementary cell of the cracked laminates. Extensive experimental studies have been achieved in the case of cross-ply carbon/epoxy (T 300/914) laminates, subjected to fatigue tests.“’ Two stacking sequences have been studied: [O,/ 90/O,], and [0,/90], (Fig. l), which differ by the position of 90” plies. All the plies are similar (thickness of O-125 mm) and the difference
1 INTRODUCTION In the case of cross-ply laminates subjected to uniaxial quasi-static or cyclic loading, the early stage of damage is dominated by matrix cracking in the 90” plies. These matrix cracks develop in the fibre direction and extend across the laminates from the free edges of test specimens. The number of cracks increases with the increase of the load or the number of cycles until a saturation crack density, depending on the thickness of the transverse plies.’ Since the matrix cracks extend in the transverse direction only, it is possible to use a two-dimensional analysis to investigate the transverse cracking: Many authors studied the experimental initiation and propagation of transverse cracks in cross-ply laminates subjected to tensile tests. The results obtained show that, in the case of quasistatic tests, most cracks cross instantaneously the test specimen width from edge to edge. In this case, the transverse cracking is characterized only by the crack density. In return, in the case of cyclic tensile tests, the crack lengths and the cracking density increase progressively with increasing cycle number. So in this case, the transverse cracking must be characterized by the distribution of the crack lengths. Analytical and numerical procedures have been developed for predicting the laminate 123
124
A. Et Mahi, J.-M. Berthetot, J. Brittaud
between the two stacking sequences lies in the number and the thickness of the 90” plies: two plies of a single thickness for the first sequence, noted S, and one ply of a double thickness for the second sequence, noted D. The purpose of this paper is to evaluate, from the experimental results observed, the stiffness reduction of these laminates and the energy release rate as a function of the cycle number.
Throughout the fatigue tests, the crack distributions have been evaluated by X-ray observations. Figure 3 gives a typical distribution for the cycle number Nj. On the other hand, the different analyses2-9 of the stiffness reduction suppose the cracks cross the whole width of the test specimen and are uniformly distributed along the xdirection. So a simplified procedure can be suggested to evaluate the stiffness reduction. The test specimen is divided in n bands along the width w (Fig. 4). For the cycle number Ni, the crack distribution can be evaluated by the number n, of cracks crossing the band i (Fig. 4). Then, the stiffness reduction in band i can be evaluated using one of the analyses2-9 and supposing the cracks are uniformly distributed along the length L of the test specimen, with the uniform crack density di( N,) given by:
2 MODELLING THE STIFFNESS REDUCTION During the fatigue tests, matrix cracks in 90” plies initiate at the free edges of test specimens and grow slowly across the specimen edges (Fig. 2). For a given cycle number Nj, the matrix cracking pattern is characterized by the edge of initiation (edge 1 or 2), the position xi and the length a, of each crack. So the exact analyses of the stress distribution and the stiffness reduction are very complex, and need the resolution of a three-dimensional problem.
laminate
D
[ 0,190
Is
Fig. 1.
&(N;)=
nANi) L
This approach neglects also the crack tip effect on the stress distribution. The longitudinal Young’s modulus Exi of the band i can be deduced
laminate
5
[ 0,/90/O,
Is
The two stacking sequences investigated.
i________________________________ )___________________ ____________
Fig. 2.
(1)
Transverse matrix cracking.
125
Fatigue tests of cross-ply laminates
from a shear lag analysis or a finite element analysis, as a function of di: E,;=E,Wi)
(2
Equality of the displacements leads to:
of the YIbands
iEX;;S; I=1
E+=
(3)
where E, is the longitudinal Young’s modulus of the cracked laminate, u the imposed displace-
T
$ (NJ= & t Y;Ex[d,(Nj)I x
XI
I
where E,[dj( Nj)] is the Young’s modulus of a cracked band i with the crack density equal to di(Nj )* In the case of bands of the same width (y,= W/ n), relation 4 becomes:
(5)
1
Fig. 3.
ment, S the area of the cross-section of specimen and Si the area of the cross-section of band i. Introducing the width yi of band i and the longitudinal Young’s modulus EO, of the undamaged laminate, relation 3 yields:
Figures 5 and 6 show examples of crack distributions in the two laminates studied, evaluated
Typical transverse matrix crack distribution.
W
position
Fig. 4.
Evaluation
of the bands
of the crack distribution.
5 (x103) 4
l
laminates D
D
laminates D 0.6~3,
0.813,
7 6 h .e
3
10 cycles
r 8
2
:: E "
l 1
l
l
l
l
l
l
l
I 0
0
0’ 4
Fig. 5.
0
0 I
w=
I
OO
15
15 mm
Crack distributions obtained after 10 cycles.
0
0
0
_
+
for the D laminates,
4
Fig. 6.
Crack
_
15
w=lSmm
W
distributions obtained for the D and laminates, after 1O4cycles.
S
126
A. El Mahi, J.-M. Berthelot, J. Brillaud
Fig. 7.
Crack distribution
evaluated as a two crack band distribution.
in the case where the specimen width (15 mm) is divided in 8 bands, for two cycle numbers ( 10 and 104) and for a maximum fatigue tensile stress of 0*6a,, and O*So,,, CJ~,being the static failure stress of laminates ( CT[,= 1, 440 MPa). The bands being equal, the distributions of the cracked area is simply deduced from these crack distributions by multiplying the crack number scale by the area of an elementary crack. So in the case of equal bands, the crack distributions represent the ‘S-N distributions’ of the fatigue tests. On the other hand, it can be observed (Fig. 6) that, in the case of S laminates and for O*Sa,,, the length of the cracks are lower than half the specimen width. In this case and according to the of Lafaire-Frenot and approach HenaffGardin ‘(),’l the crack distributions can be evaluated (Fig. 7) as two cracked bands from each edge of the test specimen, of width equal to the average length j of the actual cracks and with cracks uniformly distributed in the two bands with an average density d.
Thus, the longitudinal Young’s modulus can be calculated, supposing the equality of the displacement in the three bands, by:
where E,[d(Nj)] is the cracked band with the a(Nj). At last, it can be also the case of D laminates
Young’s modulus of a crack denstiy equal to observed (Fig. 6) that, in the crack density is the
same in all the bands, for a large enough number of cycles ( lo4 cycles for a maximum fatigue stress of O.Sa,,). So in this case, the stiffness reduction can be estimated by considering that the cracks created cross all the test specimen by:
where E,[d( N,)] is the Young’s modulus of a cracked band with the crack density equal to d(N,), the crack density in all the bands for the cycle number N,. Thus in the most general case, the stiffness reduction is evaluated by relations 4 or 5, with two extreme crack distributions: a two-cracked band distribution and a single-band distribution across all the specimen width, where the stiffness reduction can be approximated respectively by eqns (8) and (9). 2.1 Stiffness reduction as a function of crack density Equations (4), (5), (8) and (9) point out that the basic result to be estimated is the stiffness reduction E,( d)/Et as a function of the transverse crack density. Figure 8 shows the comparison between the predicted and the experimental results obtained for the longitudinal Young’s modulus reduction as a function of the crack density, in the case of glass fibre epoxy cross-ply laminates. The experimental results were reported by Highsmith and Reifsnider.* The predicted results are derived from the shear analyses developed by Highsmith andd Reifsnider* and by Lee and Daniel,’ and from the finite element analysis implemented in Ref. 9. The results reported in Fig. 8 show that values obtained by finite element analysis are in good agreement with the experimental values. The predictions obtained by the shear lag analyses are close to the experimental results, although closer in the case of analysis developed by Lee and Daniel. So the shear lag analysis can
127
Fatigue tests of cross-ply laminates
6, +! G 3 9 B E _m F >a -z a a ‘zl s _I
1.25
1 .oo
z
1.002
r s
1.000
.z +z
0.998
l
laminates
S
I
Fz
0.75
-;
0.996
5 0.50 l
.
l
--- --
0.25
0
0
experimental shear
lag analysis
shear lag analysis finite element analysis
[ 9 ]
1
I
200
400
600
density
0.994 0.992
F
I crack
Fig. 8.
5 C a .s
( 2] [2 ] [5 ]
results
800
(cracks/m)
Stiffness reduction of [O/90,], laminates as a function of transverse crack density.
be used as a first practical estimation of the stiffness reduction. Figure 9 shows the stiffness reduction as a function of the crack density evaluated by the finite element analysis in the case of II laminates [0,/90]s and S laminates [0,/90/O,],. The results show a more pronounced reduction (about twice) in the case of D laminates. This fact is the consequence of the simihtude property of S and D laminates: the stiffness reduction of D laminates for a crack density is equal to the stiffness reduction of S laminates for a double crack density. At last, it must be noted that, for a practical use (following sections), the results obtained by finite element analysis and reported in Fig. 9, can be fitted by polynomials. 2.2 Stiffness reduction as a function of cycle number Stiffness reduction as a function of cycle number is deduced from the stiffness reduction as a function of the crack density (Fig. 9) and from the crack distributions as a function of the cycle number (Figs 5 and 6). Figure 10 shows the longitudinal Young’s modulus reduction, calculated by relation 5 in the case of D and S laminates, subjected to a maximum fatigue tensile stress of 0.60, and 0.80,. The longitudinal Young’s modulus E,( di) as a function of the crack density in the band i has been derived from the results reported in Fig. 9. For comparison, the stiffness reduction was also evaluated by eqn (8) (two-cracked band distributions) in the case of S laminates in Figs 10(c) and 10(d) and by eqn (9) (single-band distribution) in the case of D lami-
crack
density
(cracks/m)
Fig. 9. Stiffness reduction of laminates 10,/90]J (D Iaminates) and [0,/90/0,],Y (S laminates) as a function of crack density.
nates and for a maximum fatigue stress of Oga, (Fig. 10(b)). The different evaluations are very close. According to the crack distributions, the stiffness reduction occurs with the initiation of transverse matrix cracking, since the first cycles for the D laminates and for lo3 to lo4 cycles in the case of S laminates. Then, the longitudinal Young’s modulus of laminates decreases with the crack propagation and tends to be constant with the effect of crack saturation, when the cycle number increases. Moreover, the stiffness reduction is very low: reduction of about 0.5% in the case of D laminates and 0.15% in the case of S laminates. Such low variations result from the low thickness of 90” plies compared to the thickness of 0” plies, and are obviously difficult to measure for the laminates studied. 2.3 Stiffness reduction as a function of the crack area The longitudinal Young’s modulus reduction as a function of the crack area per unit length of the test specimen is deduced from the curves of Fig. 10, by multiplying, for the cycle number Nj, the crack density in each band i by the thickness of the 90” plies and the width yj of band i. The results obtained, independent of the maximum fatigue stress, are reported in Fig. 11 for the S and D laminates. These results show a linear relationship (in the domain studied) between the stiffness reduction and the crack area S,:
128
A. El Mahi, J.-M. Berthelot, J. B&laud
1.000 0.999 0.998 0.997
~~~~~
0.996 0.995
I 1
h 10
,
102
Cycle number
103
104
105
1
( 106
10'
Cycle number
‘.ooo’-
a, “OOOIY % 1.0000%
“t. 2
0.9995-
z P
E : 0.9990>
0.998d
’ 10
1
I
102
1
103
L
105
104
I
1
106
107
0.99851 1
I
10
L
102
Cycle number
o^,
r
1.000
l
9, 0
I
I
I
I
I
I
1
2
3
4
5
6
7
10'
s=f(N)
1
can be obtained by a simple law of the crack area growth as a function of cycle number. Equations (10) and (12) sh ow that the function f{ N) can be deduced from the rate dS,./dN of crack growth given by:
_* F 2 0.9990 >:Y! ux 1.0000 K
7
(12)
I
0.9995
0
(11)
(m’/m)
,
2
dS. L=-!df 1
2
3
4
Crack area S,
Fig. 11.
106
(x10-3)
a*
0.9985
105
Equation (10) relates the Young’s modulus reduction as a function of crack area being established, and it is possible to consider whether the Young’s modulus reduction as a function of cycle number N (Figs 10):
0.996
Crack area S,
2z
104
a = O-667 m- ’ for the D laminates.
0.997
1.0001
103
I
a = 0.357 m- ’ for the S laminates,
0.998
0.995
I
with:
0.80,,
52 0.999
F :
I
Cycle number
2
$ f _w
I
Stiffness reduction of [0,/90],5 and [0,/90/O,] ( laminates as a function of cycle number.
Fig. 10.
1.001
0
\
9
I
5
6
7
(x10-3)
(mz/m)
Stiffness reduction of [0,/90],s and [03/90/0,],s laminates as a function of the crack area.
dN
a dN
(13)
The rate of crack area growth versus the cycle number is reported in Fig. 12. This figure shows that the crack propagation rate is decreasing
129
Fatigue tests of cross-ply laminates 1 o-2
3
u
lo-' -
B E
+
Laminates .S
l
0.80, 0.80,
expressed as:
o 0.60, q
0.6~7,
I
(18)
l l l
0
N‘ 5 2 ?
Laminates D
l *
l*
00
10-G -
0
In the case of laminates studied: US,+ 1, and the energy release rate is simply approximated by:
%b
O Oo
000
00
0
a O.
0
G- aW’,‘,
‘HO a
1O-8 -
:
O
Z” 1O-10
I
I
I
,
,
1
10
102
103
104
10s
a$ 106
10'
Cycle number
Fig. 12.
Rate of crack area growth versus cycle number.
(19)
Thus the energy release rate is found to be independent of the crack area created in the domain studied. Finally, the energy release rate can be expressed as: ,
throughout the fatigue tests and that the results are approximately distributed on a straight line. However any general law does not seem to be deduced from these results.
The maximum energy release rate G,,, associated with the maximum fatigue stress a,,, = Rg,, is then given by: G,,,=R’G,
3 ENERGYRELEASERATE
with
GU=t$
V,, x
As the fatigue tests are performed under stress control, the energy release rate G is related to the stored elastic energy W,, by the expression:
(14) where S, is the surface area created and cr the prescribed stress. The elastic energy is given by:
(21)
In this expression V, is the volume of a test specimen of one meter length (a is expressed in m/m2): V, = 30 10e6 m3/m. E!j is the Young’s modulus of the undamaged laminates: E’j = 125% GPa, deduced from tensile tests. These values associated with values of a given by eqn ( 11) yields to:
G,, = 88 (J/m3)/m for the S laminates, G,, = 165 (J/m”)/m for the D laminates. These values give:
introducing the longitudinal Young’s modulus E, and the volume V, of the test specimen. The substitution of relation 15 in relation 14 yields: G=
wo
a@:/%) e’ as,
(16)
where W$ is the elastic energy stored in the undamaged test specimen:
for S laminates G,,, = 56 (J/m3)/m for a,,, = 0% (T,,, G,,, = 32 (J/m”)/m for a,,, = O-6(7,,; for D laminates G,,,,, = 106 (J/m3)/m for o,,, = 0.8 or,, G,,, = 59 (J/m”)/m for a,,, = O-6(T,,.
(17) Differentiating with respect to the crack area, S,, the function E!j/E, given by eqn (lo), the energy relase rate as a function of the crack area is
These values allow an explanation of the delay in crack initiation and propagation, observed in the case of S laminates and in the case of D laminates for a loading amplitude equal to O-60, (Fig.
130
A. El Mahi, J.-M. Berthelot, J. Brillaud
10). Indeed, only the D laminates for a load level of O*Sa,, show crack initiation and propagation since the first cycles. This process can be explained by the value of the maximum energy release rate G,, equal to 106 (J/m”)/m and practically the same as the critical value G,, equal to 110 (J/m”)/m, obtained at the crack initiation in the case of quasi-static tests.i3 The delay in crack initiation is observed in the three other cases (D laminates at a load level of 0.6~7,~and S laminates at the two loading amplitudes of 0.6 ou and 0.8 oU), for which G,,,,, is lower than G,.. This delay is all the more important as the value of G,,,,, is small.
REFERENCES 1. Garett, K. W. & Bailey, J. E., Multiple transverse fracture in 90” cross-ply laminates of a glass fibre-reinforced polyester. Journal of Materials Science, 12 (1977) 157-68.
2. Highsmith, A. L. & Reifsnider, K. L., Stiffness reduction mechanisms in composite laminates. Damage in Composite Materials. ASTM STP 775, ed. K.-L. Reifsnider. American Society for Testing and Materials (1982), 103-17. 3. Laws, N. & Dvorak, G. J., Progressive transverse cracking in composite laminates. Journal of Composite Materials, 22 (1988) 900-16.
4. Lim, S. G. & Hong, C. S., Prediction of transverse cracking and stiffness reduction in cross-ply laminated composites. Journal of Composite Materials, 23 (1989) 695-713.
CONCLUSION In this paper, stiffness reduction and energy release have been investigated in the case of crossply laminates subjected to fatigue tensile tests. Two stacking sequences [0,/90], and [0,/90/O,], have been considered. First, during fatigue tests, the damage progression has been characterized by the crack distributions in the width of the test specimen, giving the S-N distributions. According to the S-N distributions observed during the fatigue tests, the stiffness reduction occurs with the initiation of transverse matrix cracking, over the first cycles for the [0,/90], laminates and for lo3 to lo4 cycles in the case of [03/90/041s laminates. Stiffness reduction as a function of the crack area has been evaluated by finite element analysis, from the S-N distributions. The results obtained show a linear relationship, in the domain studied, between the stiffness reduction and the crack area, which is depending on the stacking sequence. Thus, the energy release rate, associated to the crack propagation, is found to be independent of the crack area created, during fatigue tests.
5. Lee, J. W. & Daniel, I. M., Progressive transverse cracking of cross-ply composites laminates. Journal of Composite Materials, 24 (1990) 1225-43.
6. Haskin, Z., Analysis of cracked laminates: a variational approach. Mechanics of Materials, 4 (1985) 121-36. 7. Lee, S. W. & Aboudi, J., Analysis of composite laminates with matrix cracks. Report CCMS-88-03. (1988) College of Engineering Virginia Polytechnic Institute and State University, Blacksburg, Virginia. 8. Sun, C. T. & Jen, K. C., On the effect of matrix cracks on laminate strength. Journal of Reinforced Plastics and Composites. 6 (1987) 208-23.
9. Brillaud, J. & El Mahi, A., Numerical simulation of the influence of the stacking sequence on transverse ply cracking in composites laminates. Composite Structures, 17 (1991) 23-35.
10. Henaff-Gardin, C., Lafarie-Frenot, M. C., Brillaud, J. & El Mahi, A., Influence of the stacking sequence on fatigue transverse ply cracking in cross-ply laminates. ASTM STP 1128, ed. J. E. Masters, (1992), Philadelphia, USA, 236-55. 11 Lafarie-Frenot, M. C. & Henaff-Gardin, C., Analyse locale et modtlisation de la fissuration transverse de stratifies croids carbone/epoxyde. Journtes Nationales des Composites, ( 1990), JNC7, Lyon, 63 l-42. 12 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. Experimental Mechanics, 27 (1987) 73-9. 13 El Mahi, A., Brillaud, J. & Berthelot, J.-M., Evaluation numerique du module longitudinal et du taux de restitution d’energie de stratifies endommages au tours des essais de fatigue. Journees Nationales des Composites, (1992), JNC8, Paris, 461-72.