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Crack extension in unidirectional composite laminae
Pergamon
Engineering Fracture Mechanics Vol. 51, No. I, pp. 27-36, 1995 Copyright © 1995 Elsevier Science Ltd 0013-7944(94)00222-3 Printed in Great Britain. All rights reserved 0013-7944/95 $9.50 + 0.00
CRACK EXTENSION IN UNIDIRECTIONAL COMPOSITE LAMINAE H. EL KADIt and F. ELLYIN Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, T6G 2G8, Canada Abstract--The problem of predicting the crack extension behaviour of a centre-notched unidirectional graphite-epoxy coupon is investigated. Both the load required to cause initial crack growth and the direction of crack growth are of interest. Comparison between the theoretical predictions and the experimentalresults indicates that the present criterion, based on the critical strain energy, accurately predicts the direction of crack growth as well as the load causing crack growth initiation for the test cases considered.
INTRODUCTION A FUNDAMENTAL problem in predicting the failure of a laminated composite material is the determination of the load at which an existing crack (flaw) will begin to propagate as well as the direction of crack growth. This is a complex problem since, in general, a three-dimensional (3D) stress state exists in a notched laminate. A simpler and perhaps a more fundamental problem to deal with in the first instance, is to consider the constituent laminae. This paper is concerned with predicting both the direction of crack growth and the load causing that initial growth. The material under investigation will be assumed elastic, homogeneous and anisotropic. A number of theories has been proposed to predict the direction of crack growth in anisotropic materials, among them: the tensor polynomial criterion [1], the minimum strain energy density criterion [2] and the normal stress ratio criterion [3]. The present authors have also introduced a criterion based on the critical strain energy [4]. The results obtained using the latter criterion were shown to be in better agreement with the experimental data than those predicted by the tensor polynomial or the minimum strain energy density methods. On the other hand, the results obtained using the normal stress ratio method were comparable to those obtained using the critical strain energy criterion [4]. For certain cases, however, the normal stress ratio method fails to predict the proper direction of crack growth, e.g. pure shear loading. It will be shown here that the critical strain energy criterion does not suffer from this type of break-down. The determination of the load causing initial crack growth is of practical importance. For an off-axis unidirectional coupon this load will be synonymous to the failure load. It should be noted here, that the same load may not cause the lamina to fail if it is part of a multi-directional laminate. This is due to the constraining effect caused by the other layers. This paper is concerned with the unidirectional laminae, and in the following, the crack initiation load and the growth direction will be predicted and compared to the experimental results. CRACK T I P S T R E S S F I E L D S - - A N I S O T R O P I C E L A S T I C I T Y ANALYSIS The stress analysis of an elliptical hole in an anisotropic plate can be directly related to a crack by reducing the minor axis dimension to zero. This problem has been extensively investigated since the late 1930s. A summary of these investigations can be found in ref. [5]. Lekhnitskii's formulation of this problem, which is analogous to that of Muskhelishvili's counter-part in isotropic elasticity, is adapted in the following discussion. For the plane, anisotropic tPresent address: National Research Council Canada, Industrial Materials Institute, Boucherville, Quebec, J4B 6Y4, Canada. 27
28
H. EL KADI and F. ELLYIN
elastic material, shown in Fig. 1, the equations of stress equilibrium and strain compatibility can be represented in terms of Airy's stress function, q), in the familiar form
f:~22t "~'~4"X 4)
~
~x~.~) - - 2 A I 6 ( ~(~4(I) ) + ~ , , t T ) :°,
"'
where Ao are components of compliance tensor for plane stress (or plane strain). The solution of eq. (l) for an anisotropic plate containing a crack can be obtained in terms of two holomorphic functions, Ut (ZI) and U2(Z2), of two complex variables, ZI and Z:, in the following form
$=
2~[F, (Z,) + f 2(Z 2)],
(2)
where
dE, (Z,)
UI (Zl)
dZ~ dF2(Z2)
u2(z2)
-
-
-
dZ2
and the complex variables are defined as Z , = x + S~y
(3)
Z2= x + Szy.
Assuming (D = e -'+s-', the characteristic eq. (1) takes the form All S 4 - 2Ai6 $3 + (2AI2 + A66)S 2 - 2A26S + Az2 = 0.
(4)
The roots of the characteristic equation, S~ and $2, are complex, and are functions of the material properties and the orientation of the crack relative to the principal material direction. Considering the case where Si ~ $2, evaluation of the complex potential functions near the crack tip yields expressions for the stress and displacement distributions of the form
-:{, ~"'-~ ~ ° ~,i~ ~l I
s,s,
~J qs-~ ~JJ •
[ -~',]} 1
+
T ~N//a
, 91
r,
1
Cjr°~ ,T°o
Yl
I I I
I I
Fig. 1. Infinite centre cracked plate with far-field stresses.
'II SI
--_--
S2
(5)
Crack extension behaviour
29
and
u=a~x/~9~ v =a~x/~R
} }
[S,pz~b2-S2p,@~] + r ~ x / ~
{
S~_ s~ [S, q2qJ~- S2q,@~] + r ~ x / ~ R
{, {
} }
S~_s~[P2~b-~-p,~O~]
S~_ S~ [q2@z-q,@~] ,
(6)
where r, ~b are the polar coordinates with the origin at the crack tip (Fig. 1) and a ~, r~- are the far field applied stresses, and p, = A,, S~ + Ai2 - AI6SI
P2 = Atl S~ + Al2
-
AI2S~ + A22 -ql =
A2651
Si .412522 or- A 2 2
q2 =
AI6S2
-
A26S 2
$2
~bI = cos q~ + Si sin @2= cos ~b + $2 sin tk. In the case of a pair of equal roots S~ = 5'2, eq. (2) takes the form 4) = 2~tl[Zl F~(Zi)
+ F2(Z, )]
(7)
and the problem is reduced to the plane problem of the theory of elasticity of an isotropic body [5]. As in the isotropic case, the crack tip stresses exhibit a singularity of 1/x/~. However, the magnitude of the stresses is not simply a function of the stress intensity factors (K, = a ~-x//-~ and K, = z ~ x / ~ ) . The quantities S, and $2 also affect the magnitude of the stresses. This is an important difference between anisotropic and isotropic fracture. In anisotropic fracture, the magnitude of the crack tip stresses is a function of not only the applied load, specimen geometry and crack length, but also the material properties and the orientation of the crack relative to the principal material direction. CRITICAL STRAIN ENERGY CRITERION
The physical basis of this criterion is as follows [4]: when a material element is loaded by an external agency, part of the supplied energy is dissipated into heat and the remaining part is stored in the material. Damage is caused by the irrecoverable part of the stored energy, i.e. d Wdamag e = d Wsopo,i~d- (dQ + d Wr..... table)"
(8)
Each material has certain capacity to absorb damage, and failure results as a result of damage accumulation. Because of the difficulty in measuring the heat loss, dQ, it is generally assumed that damage is proportional to the supplied energy, i.e. d Wdamag c OC d Wsupplicd.
(9)
The above applies for materials which show hysteretic response under cyclic loading. Polymeric matrix reinforced composites display such a behaviour; however, in the following an elastic behaviour is assumed, which is a reasonable approximation for practical purposes. The critical strain energy method is based on the assumption that crack initiation takes place
30
H. EL KADI and F. ELLYIN 360
. (Degrees)
330 3OO 0 .~ 0 I.(
270
240
"~ 210 0 L4 180 0 150
~)
120
o
:~
9o 6O
2Po
30 0
(Degrees)
I
I
I
I
I
I
I
[
i
I
i
30
60
90
120
150
180
210
24-0
270
300
330
Experimental
360
Crack Growth Direction
Fig. 2. Direction of crack growth--experimental vs predicted.
w h e n t h e v a l u e o f t h e s t r a i n e n e r g y r e a c h e s a c r i t i c a l v a l u e , i.e.
(lO)
W = (axE,- + aye,. + rx;.7~,.)/2 = Wcrit.
T h e v a l u e o f W will d e p e n d
on the material properties
c r a c k g r o w t h will c o r r e s p o n d
to the radial direction having the maximum
and the fibre direction. The direction of value of the strain energy.
Table 1. Crack growth direction and crack initiation load--comparison between analytical predictions and experimental results Crack growth direction (degrees)
Test data Case (1): AS4/3501-6 0 = 0 <', a = 0 ° 2a = 5.08 mm 6 =0.11 mm Case (2): AS4/3501-6 0 = 105° , • = 0 '~ 2a = 5.08 mm 6 =0.11 mm Case (3): AS4/3501-6 0 = 105'7, a = 15 2a = 5.08 mm =0.11 mm Case (4): AS4/3501-6 0 = 135', • = 0' 2a = 5.08 mm 6 =0.11 mm Case (5): AS4/3501-6 0 = 90', ~ = 0 2a = 5.08 mm 6 =0.11 mm
Critical strain energy 0
Experimental results [9, 10] 0
Normal stress ratio [9, 10] 0
Crack initiation load (MPa) Minimum strain energy density [2] -90
Critical strain energy 36.3
Experimental results [9, 10] 38.2
Normal stress ratio [9, 10] 20.9
--76
-75
--73
60
8.15
8.64
8.48
-90
-90
--88
-40
8.76
7.35
7.56
-46
-45
-43
- 10
3.98
4.12
5.02
+88
+90
+87
15
Baseline test
2.81
Baseline test
31
Crack extension behaviour
% 4~
20.0
0.40 0
0
o
a:l
0.30
15.0
C~ C~
Experime.tal0.20
10.0
Experhneotal_
\
0.10
o~ 0 Z
5.0
I..,
0.00 0.0
90
180
Z?O
0
360
90
180
270
360
(Degrees)
~p (Degrees)
(b)
(a)
0.04
/~e
0.03 1.4 Exl~r%mt~ttal-0.02
<
I: -i..)
0.01
0.00
0
90
180
270
S60
(Degrees)
(c) Fig. 3. Variation of crack growth criteria as a function of 4) for 30' off-axis graphite-epoxy coupons (Case 2). (a) Normal stress ratio; (b) minimum strain energy and (c) critical strain energy (present model).
Thus the direction of crack extension ~b = 4),, is obtained from dW a2W = 0 and _-:--=-=< 0. d~b
(ll)
PREDICTION OF CRACK GROWTH DIRECTION For off-axis unidirectional tensile coupons and for unidirectional laminae subjected to mixed-mode loading, the results obtained using the critical strain energy criterion were reported in ref. [4] and the predictions were in very good agreement with the experimental results, see Fig. 2. In the present work, five cases (shown in Table 1) are considered. The properties of the material used (AS4/3501-6 graphite-epoxy) are shown in Table 2, extracted from ref. [6]. In Table I, the results obtained using the critical strain energy criterion are compared to the experimental results
32
H. EL KADI and F. ELLYIN
and are shown to be in good agreement. The results obtained using the minimum strain energy density [2] and those obtained using the normal stress ratio method [7-9] are also shown in Table I. For one of the cases presented in Table 1, the variation of the three crack growth criteria as a function of the angle, q~ is shown in Fig. 3. In this figure both the experimental results and the predicted values are shown. It should be stated that although both Sih's method (based on the minimum strain energy density [2]) and the present criterion use strain energy as a criterion for crack growth direction, each method utilizes a unique interpretation. This fact is shown by comparing Fig. 3(b, c). The case of an infinite unidirectional graphite-epoxy plate with a centre-crack along the fibres direction and subjected to pure shear loading in the coordinates system of the crack seems to be one of the limitations of the normal stress ratio theory [6]. For the positive shear case illustrated in Fig. 4 the direction of crack extension predicted by the normal stress ratio theory was - 1 2 ° when the theory was applied within the sharp crack analysis of Lekhnitskii. For this loading case, it appears that the normal stress ratio theory is incapable of correctly predicting the direction of crack extension obtained from the experiments. Beuth and Herakovich [6] mentioned that this result was valid for any notched anisotropic material regardless of its elastic and/or strength properties. They thus concluded that the discrepancy between the normal stress ratio theory's prediction and the experimental behaviour for this case was not due to potential inaccuracies, but instead due to a fundamental error in either the theory itself or in its method of application within the sharp crack analysis. A couple of explanations were offered for why the normal stress ratio method breaks down for the pure shear case. The first reason was a basic one where a theory based solely on local normal stress cannot yield a prediction of crack extension along the fibres. It was suggested that there was a need for the inclusion of stress components other than the normal stress defined with respect to the crack within a crack growth theory. This points out a basic deficiency of this theory. The second explanation for the discrepancy between the normal stress ratio theory and expected behaviour was that for some cases, the use of the stress solution by Lekhnitskii was incorrect since it models the actual notch as a line crack with an indefinitely sharp tip. They thus proceeded to apply the normal stress ratio theory within the elliptical flaw analysis of Savin [10]. The above assertion is inappropriate, since a crack obtained from a limit case of an elliptical hole is not "indefinitely sharp tip", see Goodier [11]. Furthermore, it seems that the nature of the problem has been changed from a crack problem to a notch problem which is not representative of the problem under consideration. For the same example (Fig. 4), the application of the critical strain energy criterion within the sharp crack analysis does not lead to an erroneous result. The crack growth direction obtained is parallel to the fibres and similar to the experimental findings (~b, = 0°).
•
~
_
~
,Too
Table 2. Mechanical properties of AS4/3501-6 graphite-epoxy [9] Property
Value
X t (GPa) lit (MPa) S (MPa) E: (GPa) E, (GPa) G (GPa) v
1.45 53.4 99.3 126 10 5.61 0.305
Fig. 4. Infinite centre cracked plate under pure shear far-field stress.
Crack extension behaviour
33
CRACK GROWTH INITIATION STRESS A major portion of the useful life of a composite structure/component involves subcritical damage accumulation which is finally manifested in various combinations of matrix cracking, fibre-matrix debonding, delamination, fibre breakage failure modes. A precise characterization of a composite material would therefore require a knowledge of the way the energy dissipates throughout the inhomogeneous structure as damage is being accumulated. The strain energy density is a parameter which can be related to this damage process. The critical strain energy criterion has been successfully used as a fatigue failure criterion for composite laminae [12]. The relationship between the cyclic strain energy and the number of reversals to failure (i cycle = 2 reversals) was presented in the form: AW = x (0)(2N/) ~l°~,
(! 2)
where AW =½(am,x~max- ammEmm) and ~c and a are material constants depending on the fibre orientation angle. For a monotonic loading 2N/= 1, reducing eq. (12) to AW = K ( 0 ) .
(13)
The variation of K with the fibre orientation angle, 0, was presented in ref. [12] as log ~c = log g0 + bO #,
(14)
where b and fl are material parameters, and x0 is the value of x at 0 = 0. The variation of K with the fibre orientation angle, 0, for a unidirectional glass-epoxy composite under cyclic loading is presented in ref. [13] for various values of the stress ratio (R = ami./am,x) as well as for the monotonic case. It is also shown that the relation between ~c and the fibre orientation angle, 0, does not show much variation between the cyclic and monotonic cases (Fig. 5). This implies that the monotonic value of ~c can be used to quantify damage without significant error, no matter what stress ratio is used. For the special case of 2Ny = 1, the monotonic value of K could then be used as a criterion for failure under static loads. 1.5
LEGEND x=R= 0 o=R=O.5 o = MONOTONIC
1.0
0.5
v
L~
0.0
O
-0.6
-I.0
-1.5
0
10
20
30
40
50
60
70
80
90
O (Degrees) Fig. 5. Variation of h with the fibre orientation angle, 0, for positive stress ratios. EFM51:I-C
34
H. EL KADI and F. ELLYIN
For the case of a unidirectional fibre-reinforced coupon with an existing crack (flaw), the same parameter could be used as an indication of crack growth initiation. The failure criterion (10) can then be written as W = ~:(0).
(15)
It should be noted here that the crack geometry changes significantly before failure in some of the tests that exhibited slow crack growth not collinear with the original notch. It is reasonable therefore to assume that the applied stress obtained from eq. (15) will cause initiation rather than coupon test failure as reported in ref. [14]. In the following, eq. (15) will be used to determine the stress required to initiate crack growth. Once more, the properties of the material under consideration (AS4/3501-6 graphite-epoxy) are shown in Table 2 extracted from ref. [6]. For a uniaxially loaded coupon in the y-direction, the strain energy, W, can be written as W=½ayey.
(16)
This can be written as
2 W-
ay
2Ev"
(17)
Using the strength and the stiffness of the material at three different values of the fibre orientation angle, the coefficients of eq. (14) are calculated as log ~c = 0.9212 - 0.361050 0.3529
(1 8)
(i) Baseline test. The normal stress ratio theory has to be applied at a distance ro from the crack tip. In ref. [9], Beuth and Herakovich proceeded to determine the value of r o experimentally from a so-called baseline test. Once this value was determined, it was used as a material constant. However they observed that for most tests, the applied stress causing crack initiation was significantly different from that causing failure, and a test was chosen such that these two values were the same. The baseline specimen had a notch oriented along the fibre direction and both fibres and notch were oriented at 90 ° to the loading direction (similar to Fig. 1 with 0 = 0 and z ~- = 0). They obtained an average value of the failure stress of 19.4MPa which led to a radius, ro = 1.064 mm. (ii) C r a z e zone. For metals, Kujawski and Ellyin [15] defined a small region which exists immediately ahead of the crack tip where micro-failure takes place. They termed that region, the fracture process zone. In this zone, the plastic deformation was found to be non-proportional and crack blunting occurred. Botsis et al. [16] showed that for polystyrene, during slow crack growth, an intensive zone of crazing surrounds and precedes the propagating crack. They called the zone ahead of the crack tip (where crazing accumulates prior to crack growth) the active zone. For composites, it would appear that the same phenomenon occurs. Ahead of the crack there is a zone where the deformation cannot be assumed elastic and therefore, the use of a criterion based on the elastic normal stress ratio or the elastic strain energy will not be appropriate. We will term this region craze zone of size 6. This will mean that for any elastic criterion to be successful it would have to be applied outside the craze zone. It would be desirable to relate the value of 6 to the microstructural parameters, however, at this stage, it may be acceptable to use a baseline test to determine its value. Using the critical strain energy criterion, with the same baseline test presented in ref. [9], a value of 6 = 0.11 mm was obtained. This value is almost l0 times smaller than that estimated for r,, in ref. [9]. P R E D I C T I O N B A S E D ON T H E CRITICAL STRAIN E N E R G Y An experimental investigation to study the crack growth of unidirectional graphite-epoxy coupons was conducted in ref. [9]. The predictions obtained using the critical strain energy criterion will be compared to the above experimental results.
Crack extension behaviour
{o}
35
(b)
o.:O ° ~t:15 o 0 : I05" 0 : to5* Fig. 6. Notch configurations used in experimental investigation: (a) configuration H (notch perpendicular to loading direction) and (b) configuration A (notch perpendicular to material fibre direction).
The 15 ° off-axis tensile coupons used had two notch configurations: one perpendicular to the loading direction (configuration H) and the other perpendicular to the material fibre direction (configuration A). Both configurations are shown in Fig. 6. Specimens with aspect ratios (length-to-width ratios) of 8, 4 and 1 were considered. Although apparent trends in the value of the stress causing initial crack growth with respect to the notch orientation, were obvious, no trends were seen with respect to the specimens' aspect ratios. Since the aspect ratio does not seem to have a noticeable effect on the experimental results, the average of the experimental data will be used for the purpose of comparison. The comparison between the experimental results and the predictions obtained using the critical strain energy criterion is given in Table 1. Also listed in Table 1 are the predictions obtained using the normal stress ratio criterion. Centre-notched 0 °, 45 ° and 90 ° coupons with notches oriented at 90 ° to the loading axis were tested under far-field uniaxial tension in ref. [6]. Again several tests were performed for every value of the fibre orientation angle (mainly due to the scatter observed in the experimental results). The comparisons between the average experimental data for each of the fibre orientation angles examined, and the predictions obtained using both theories are shown in Table 1. From this table, it is obvious that the results obtained using the critical strain energy criterion are consistent with experimental data and are in better agreement than those obtained using the normal stress ratio theory.
CONCLUSIONS A criterion based on the critical strain energy has been shown to accurately predict the crack extension behaviour in unidirectional composites within the elastic crack analysis. This criterion was successfully used to predict the crack growth direction of a centre-notched unidirectional coupon subjected to both tensile and shear loading. The use of the critical strain energy density criterion was also shown to be very consistent in predicting the value of the stress causing crack growth initiation.
36
H. EL KADI and F. ELLYIN
REFERENCES [1] S. W. Tsai and E. M. Wu, A general theory of strength for anisotropic materials. J. compos. Mater. 5, 58-80 (1971). [2] G.C. Sih, E. P. Chen, S. L. Huang and E. J. McQuillen, Material characterization on the fracture of filament-reinforced composites. J. compos. Mater. 9, 167 186 (1975). [3] M. B. Buczek and C. T. Herakovich, A normal stress criterion for crack extension direction in orthotropic composite materials. J. compos. Mater. 19, 544 553 (1985). [4] F. Ellyin and H. El Kadi, Predicting crack growth direction in unidirectional composite laminae. Engng Fracture Mech. 36, 27-37 (1990). [5] S. G. Lekhnitskii, Theory of Elasticity of an Anisotropic Elastic Body (Translated by P. Fern, Edited by J. J. Brandstatter). Holden-Day, San Francisco (1963). [6] J. L. Beuth, Jr and C. T. Herakovich, Analysis of crack extension in anisotropic materials based on local normal stress. Theor. appl. Fracture Mech. 11, 27-46 (1989). [7] M, A. Gregory and C. T. Herakovich, Predicting crack growth direction in unidirectional composites. J. eompos. Mater. 20, 67-85 (1986). [8] M. A. Gregory, J. L. Beuth, Jr, A. Barbe and C. T. Herakovich, Application of the normal stress ratio theory for predicting crack growth direction in unidirectional composites. Fracture Fibrous Compos. AMD, 74, 33-42 (1985). [9] J. L. Beuth, Jr and C. T. Herakovich, On fracture of fibrous composites. Composites '86: Recent Advances in Japan and the United States (Edited by K. Kawata, S. Umekawa and A. Kobayashi), Proc. Japan-U.S.A. CCM-HL pp. 267 277, Tokyo (1986). [10] G. Savin, Stress Concentration Around Holes (Translation editor W. Johnson). Pergamon Press, London (1961). [11] J. N. Goodier, Mathematical theory of equilibrium cracks, in Fracture: An Advanced Treatise (Edited by H. Liebowitz). Academic Press, New York (1968). [12] F. Ellyin and H. El Kadi, A fatigue failure criterion for fiber reinforced composite laminae. Compos. Structures 15, 61-74 (1990). [13] H. El Kadi and F. Ellyin, Effect of stress ratio on the fatigue of unidirectional fiberglass-epoxy composite laminae. Composites 25, 917-924 (1994). [14] J. L. Beuth, Jr, M. A. Gregory and C. T. Herakovich, Crack growth in unidirectional graphite-epoxy under biaxial loading. Exp. Mech. 26, 245 253 (1986). [15] D. Kujawski and F. Ellyin, A fatigue crack propagation model. Engng Fracture Mech. 20, 695-704 (1984). [16] J. Botsis, A. Chudnovsky and A. Moet, Fatigue crack layer propagation in polystyrene--I. Experimental observations. Int, J. Fracture 33, 263 276 (1987). (Received 24 December 1993)
Engineering Fracture Mechanics Vol. 51, No. I, pp. 27-36, 1995 Copyright © 1995 Elsevier Science Ltd 0013-7944(94)00222-3 Printed in Great Britain. All rights reserved 0013-7944/95 $9.50 + 0.00
CRACK EXTENSION IN UNIDIRECTIONAL COMPOSITE LAMINAE H. EL KADIt and F. ELLYIN Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, T6G 2G8, Canada Abstract--The problem of predicting the crack extension behaviour of a centre-notched unidirectional graphite-epoxy coupon is investigated. Both the load required to cause initial crack growth and the direction of crack growth are of interest. Comparison between the theoretical predictions and the experimentalresults indicates that the present criterion, based on the critical strain energy, accurately predicts the direction of crack growth as well as the load causing crack growth initiation for the test cases considered.
INTRODUCTION A FUNDAMENTAL problem in predicting the failure of a laminated composite material is the determination of the load at which an existing crack (flaw) will begin to propagate as well as the direction of crack growth. This is a complex problem since, in general, a three-dimensional (3D) stress state exists in a notched laminate. A simpler and perhaps a more fundamental problem to deal with in the first instance, is to consider the constituent laminae. This paper is concerned with predicting both the direction of crack growth and the load causing that initial growth. The material under investigation will be assumed elastic, homogeneous and anisotropic. A number of theories has been proposed to predict the direction of crack growth in anisotropic materials, among them: the tensor polynomial criterion [1], the minimum strain energy density criterion [2] and the normal stress ratio criterion [3]. The present authors have also introduced a criterion based on the critical strain energy [4]. The results obtained using the latter criterion were shown to be in better agreement with the experimental data than those predicted by the tensor polynomial or the minimum strain energy density methods. On the other hand, the results obtained using the normal stress ratio method were comparable to those obtained using the critical strain energy criterion [4]. For certain cases, however, the normal stress ratio method fails to predict the proper direction of crack growth, e.g. pure shear loading. It will be shown here that the critical strain energy criterion does not suffer from this type of break-down. The determination of the load causing initial crack growth is of practical importance. For an off-axis unidirectional coupon this load will be synonymous to the failure load. It should be noted here, that the same load may not cause the lamina to fail if it is part of a multi-directional laminate. This is due to the constraining effect caused by the other layers. This paper is concerned with the unidirectional laminae, and in the following, the crack initiation load and the growth direction will be predicted and compared to the experimental results. CRACK T I P S T R E S S F I E L D S - - A N I S O T R O P I C E L A S T I C I T Y ANALYSIS The stress analysis of an elliptical hole in an anisotropic plate can be directly related to a crack by reducing the minor axis dimension to zero. This problem has been extensively investigated since the late 1930s. A summary of these investigations can be found in ref. [5]. Lekhnitskii's formulation of this problem, which is analogous to that of Muskhelishvili's counter-part in isotropic elasticity, is adapted in the following discussion. For the plane, anisotropic tPresent address: National Research Council Canada, Industrial Materials Institute, Boucherville, Quebec, J4B 6Y4, Canada. 27
28
H. EL KADI and F. ELLYIN
elastic material, shown in Fig. 1, the equations of stress equilibrium and strain compatibility can be represented in terms of Airy's stress function, q), in the familiar form
f:~22t "~'~4"X 4)
~
~x~.~) - - 2 A I 6 ( ~(~4(I) ) + ~ , , t T ) :°,
"'
where Ao are components of compliance tensor for plane stress (or plane strain). The solution of eq. (l) for an anisotropic plate containing a crack can be obtained in terms of two holomorphic functions, Ut (ZI) and U2(Z2), of two complex variables, ZI and Z:, in the following form
$=
2~[F, (Z,) + f 2(Z 2)],
(2)
where
dE, (Z,)
UI (Zl)
dZ~ dF2(Z2)
u2(z2)
-
-
-
dZ2
and the complex variables are defined as Z , = x + S~y
(3)
Z2= x + Szy.
Assuming (D = e -'+s-', the characteristic eq. (1) takes the form All S 4 - 2Ai6 $3 + (2AI2 + A66)S 2 - 2A26S + Az2 = 0.
(4)
The roots of the characteristic equation, S~ and $2, are complex, and are functions of the material properties and the orientation of the crack relative to the principal material direction. Considering the case where Si ~ $2, evaluation of the complex potential functions near the crack tip yields expressions for the stress and displacement distributions of the form
-:{, ~"'-~ ~ ° ~,i~ ~l I
s,s,
~J qs-~ ~JJ •
[ -~',]} 1
+
T ~N//a
, 91
r,
1
Cjr°~ ,T°o
Yl
I I I
I I
Fig. 1. Infinite centre cracked plate with far-field stresses.
'II SI
--_--
S2
(5)
Crack extension behaviour
29
and
u=a~x/~9~ v =a~x/~R
} }
[S,pz~b2-S2p,@~] + r ~ x / ~
{
S~_ s~ [S, q2qJ~- S2q,@~] + r ~ x / ~ R
{, {
} }
S~_s~[P2~b-~-p,~O~]
S~_ S~ [q2@z-q,@~] ,
(6)
where r, ~b are the polar coordinates with the origin at the crack tip (Fig. 1) and a ~, r~- are the far field applied stresses, and p, = A,, S~ + Ai2 - AI6SI
P2 = Atl S~ + Al2
-
AI2S~ + A22 -ql =
A2651
Si .412522 or- A 2 2
q2 =
AI6S2
-
A26S 2
$2
~bI = cos q~ + Si sin @2= cos ~b + $2 sin tk. In the case of a pair of equal roots S~ = 5'2, eq. (2) takes the form 4) = 2~tl[Zl F~(Zi)
+ F2(Z, )]
(7)
and the problem is reduced to the plane problem of the theory of elasticity of an isotropic body [5]. As in the isotropic case, the crack tip stresses exhibit a singularity of 1/x/~. However, the magnitude of the stresses is not simply a function of the stress intensity factors (K, = a ~-x//-~ and K, = z ~ x / ~ ) . The quantities S, and $2 also affect the magnitude of the stresses. This is an important difference between anisotropic and isotropic fracture. In anisotropic fracture, the magnitude of the crack tip stresses is a function of not only the applied load, specimen geometry and crack length, but also the material properties and the orientation of the crack relative to the principal material direction. CRITICAL STRAIN ENERGY CRITERION
The physical basis of this criterion is as follows [4]: when a material element is loaded by an external agency, part of the supplied energy is dissipated into heat and the remaining part is stored in the material. Damage is caused by the irrecoverable part of the stored energy, i.e. d Wdamag e = d Wsopo,i~d- (dQ + d Wr..... table)"
(8)
Each material has certain capacity to absorb damage, and failure results as a result of damage accumulation. Because of the difficulty in measuring the heat loss, dQ, it is generally assumed that damage is proportional to the supplied energy, i.e. d Wdamag c OC d Wsupplicd.
(9)
The above applies for materials which show hysteretic response under cyclic loading. Polymeric matrix reinforced composites display such a behaviour; however, in the following an elastic behaviour is assumed, which is a reasonable approximation for practical purposes. The critical strain energy method is based on the assumption that crack initiation takes place
30
H. EL KADI and F. ELLYIN 360
. (Degrees)
330 3OO 0 .~ 0 I.(
270
240
"~ 210 0 L4 180 0 150
~)
120
o
:~
9o 6O
2Po
30 0
(Degrees)
I
I
I
I
I
I
I
[
i
I
i
30
60
90
120
150
180
210
24-0
270
300
330
Experimental
360
Crack Growth Direction
Fig. 2. Direction of crack growth--experimental vs predicted.
w h e n t h e v a l u e o f t h e s t r a i n e n e r g y r e a c h e s a c r i t i c a l v a l u e , i.e.
(lO)
W = (axE,- + aye,. + rx;.7~,.)/2 = Wcrit.
T h e v a l u e o f W will d e p e n d
on the material properties
c r a c k g r o w t h will c o r r e s p o n d
to the radial direction having the maximum
and the fibre direction. The direction of value of the strain energy.
Table 1. Crack growth direction and crack initiation load--comparison between analytical predictions and experimental results Crack growth direction (degrees)
Test data Case (1): AS4/3501-6 0 = 0 <', a = 0 ° 2a = 5.08 mm 6 =0.11 mm Case (2): AS4/3501-6 0 = 105° , • = 0 '~ 2a = 5.08 mm 6 =0.11 mm Case (3): AS4/3501-6 0 = 105'7, a = 15 2a = 5.08 mm =0.11 mm Case (4): AS4/3501-6 0 = 135', • = 0' 2a = 5.08 mm 6 =0.11 mm Case (5): AS4/3501-6 0 = 90', ~ = 0 2a = 5.08 mm 6 =0.11 mm
Critical strain energy 0
Experimental results [9, 10] 0
Normal stress ratio [9, 10] 0
Crack initiation load (MPa) Minimum strain energy density [2] -90
Critical strain energy 36.3
Experimental results [9, 10] 38.2
Normal stress ratio [9, 10] 20.9
--76
-75
--73
60
8.15
8.64
8.48
-90
-90
--88
-40
8.76
7.35
7.56
-46
-45
-43
- 10
3.98
4.12
5.02
+88
+90
+87
15
Baseline test
2.81
Baseline test
31
Crack extension behaviour
% 4~
20.0
0.40 0
0
o
a:l
0.30
15.0
C~ C~
Experime.tal0.20
10.0
Experhneotal_
\
0.10
o~ 0 Z
5.0
I..,
0.00 0.0
90
180
Z?O
0
360
90
180
270
360
(Degrees)
~p (Degrees)
(b)
(a)
0.04
/~e
0.03 1.4 Exl~r%mt~ttal-0.02
<
I: -i..)
0.01
0.00
0
90
180
270
S60
(Degrees)
(c) Fig. 3. Variation of crack growth criteria as a function of 4) for 30' off-axis graphite-epoxy coupons (Case 2). (a) Normal stress ratio; (b) minimum strain energy and (c) critical strain energy (present model).
Thus the direction of crack extension ~b = 4),, is obtained from dW a2W = 0 and _-:--=-=< 0. d~b
(ll)
PREDICTION OF CRACK GROWTH DIRECTION For off-axis unidirectional tensile coupons and for unidirectional laminae subjected to mixed-mode loading, the results obtained using the critical strain energy criterion were reported in ref. [4] and the predictions were in very good agreement with the experimental results, see Fig. 2. In the present work, five cases (shown in Table 1) are considered. The properties of the material used (AS4/3501-6 graphite-epoxy) are shown in Table 2, extracted from ref. [6]. In Table I, the results obtained using the critical strain energy criterion are compared to the experimental results
32
H. EL KADI and F. ELLYIN
and are shown to be in good agreement. The results obtained using the minimum strain energy density [2] and those obtained using the normal stress ratio method [7-9] are also shown in Table I. For one of the cases presented in Table 1, the variation of the three crack growth criteria as a function of the angle, q~ is shown in Fig. 3. In this figure both the experimental results and the predicted values are shown. It should be stated that although both Sih's method (based on the minimum strain energy density [2]) and the present criterion use strain energy as a criterion for crack growth direction, each method utilizes a unique interpretation. This fact is shown by comparing Fig. 3(b, c). The case of an infinite unidirectional graphite-epoxy plate with a centre-crack along the fibres direction and subjected to pure shear loading in the coordinates system of the crack seems to be one of the limitations of the normal stress ratio theory [6]. For the positive shear case illustrated in Fig. 4 the direction of crack extension predicted by the normal stress ratio theory was - 1 2 ° when the theory was applied within the sharp crack analysis of Lekhnitskii. For this loading case, it appears that the normal stress ratio theory is incapable of correctly predicting the direction of crack extension obtained from the experiments. Beuth and Herakovich [6] mentioned that this result was valid for any notched anisotropic material regardless of its elastic and/or strength properties. They thus concluded that the discrepancy between the normal stress ratio theory's prediction and the experimental behaviour for this case was not due to potential inaccuracies, but instead due to a fundamental error in either the theory itself or in its method of application within the sharp crack analysis. A couple of explanations were offered for why the normal stress ratio method breaks down for the pure shear case. The first reason was a basic one where a theory based solely on local normal stress cannot yield a prediction of crack extension along the fibres. It was suggested that there was a need for the inclusion of stress components other than the normal stress defined with respect to the crack within a crack growth theory. This points out a basic deficiency of this theory. The second explanation for the discrepancy between the normal stress ratio theory and expected behaviour was that for some cases, the use of the stress solution by Lekhnitskii was incorrect since it models the actual notch as a line crack with an indefinitely sharp tip. They thus proceeded to apply the normal stress ratio theory within the elliptical flaw analysis of Savin [10]. The above assertion is inappropriate, since a crack obtained from a limit case of an elliptical hole is not "indefinitely sharp tip", see Goodier [11]. Furthermore, it seems that the nature of the problem has been changed from a crack problem to a notch problem which is not representative of the problem under consideration. For the same example (Fig. 4), the application of the critical strain energy criterion within the sharp crack analysis does not lead to an erroneous result. The crack growth direction obtained is parallel to the fibres and similar to the experimental findings (~b, = 0°).
•
~
_
~
,Too
Table 2. Mechanical properties of AS4/3501-6 graphite-epoxy [9] Property
Value
X t (GPa) lit (MPa) S (MPa) E: (GPa) E, (GPa) G (GPa) v
1.45 53.4 99.3 126 10 5.61 0.305
Fig. 4. Infinite centre cracked plate under pure shear far-field stress.
Crack extension behaviour
33
CRACK GROWTH INITIATION STRESS A major portion of the useful life of a composite structure/component involves subcritical damage accumulation which is finally manifested in various combinations of matrix cracking, fibre-matrix debonding, delamination, fibre breakage failure modes. A precise characterization of a composite material would therefore require a knowledge of the way the energy dissipates throughout the inhomogeneous structure as damage is being accumulated. The strain energy density is a parameter which can be related to this damage process. The critical strain energy criterion has been successfully used as a fatigue failure criterion for composite laminae [12]. The relationship between the cyclic strain energy and the number of reversals to failure (i cycle = 2 reversals) was presented in the form: AW = x (0)(2N/) ~l°~,
(! 2)
where AW =½(am,x~max- ammEmm) and ~c and a are material constants depending on the fibre orientation angle. For a monotonic loading 2N/= 1, reducing eq. (12) to AW = K ( 0 ) .
(13)
The variation of K with the fibre orientation angle, 0, was presented in ref. [12] as log ~c = log g0 + bO #,
(14)
where b and fl are material parameters, and x0 is the value of x at 0 = 0. The variation of K with the fibre orientation angle, 0, for a unidirectional glass-epoxy composite under cyclic loading is presented in ref. [13] for various values of the stress ratio (R = ami./am,x) as well as for the monotonic case. It is also shown that the relation between ~c and the fibre orientation angle, 0, does not show much variation between the cyclic and monotonic cases (Fig. 5). This implies that the monotonic value of ~c can be used to quantify damage without significant error, no matter what stress ratio is used. For the special case of 2Ny = 1, the monotonic value of K could then be used as a criterion for failure under static loads. 1.5
LEGEND x=R= 0 o=R=O.5 o = MONOTONIC
1.0
0.5
v
L~
0.0
O
-0.6
-I.0
-1.5
0
10
20
30
40
50
60
70
80
90
O (Degrees) Fig. 5. Variation of h with the fibre orientation angle, 0, for positive stress ratios. EFM51:I-C
34
H. EL KADI and F. ELLYIN
For the case of a unidirectional fibre-reinforced coupon with an existing crack (flaw), the same parameter could be used as an indication of crack growth initiation. The failure criterion (10) can then be written as W = ~:(0).
(15)
It should be noted here that the crack geometry changes significantly before failure in some of the tests that exhibited slow crack growth not collinear with the original notch. It is reasonable therefore to assume that the applied stress obtained from eq. (15) will cause initiation rather than coupon test failure as reported in ref. [14]. In the following, eq. (15) will be used to determine the stress required to initiate crack growth. Once more, the properties of the material under consideration (AS4/3501-6 graphite-epoxy) are shown in Table 2 extracted from ref. [6]. For a uniaxially loaded coupon in the y-direction, the strain energy, W, can be written as W=½ayey.
(16)
This can be written as
2 W-
ay
2Ev"
(17)
Using the strength and the stiffness of the material at three different values of the fibre orientation angle, the coefficients of eq. (14) are calculated as log ~c = 0.9212 - 0.361050 0.3529
(1 8)
(i) Baseline test. The normal stress ratio theory has to be applied at a distance ro from the crack tip. In ref. [9], Beuth and Herakovich proceeded to determine the value of r o experimentally from a so-called baseline test. Once this value was determined, it was used as a material constant. However they observed that for most tests, the applied stress causing crack initiation was significantly different from that causing failure, and a test was chosen such that these two values were the same. The baseline specimen had a notch oriented along the fibre direction and both fibres and notch were oriented at 90 ° to the loading direction (similar to Fig. 1 with 0 = 0 and z ~- = 0). They obtained an average value of the failure stress of 19.4MPa which led to a radius, ro = 1.064 mm. (ii) C r a z e zone. For metals, Kujawski and Ellyin [15] defined a small region which exists immediately ahead of the crack tip where micro-failure takes place. They termed that region, the fracture process zone. In this zone, the plastic deformation was found to be non-proportional and crack blunting occurred. Botsis et al. [16] showed that for polystyrene, during slow crack growth, an intensive zone of crazing surrounds and precedes the propagating crack. They called the zone ahead of the crack tip (where crazing accumulates prior to crack growth) the active zone. For composites, it would appear that the same phenomenon occurs. Ahead of the crack there is a zone where the deformation cannot be assumed elastic and therefore, the use of a criterion based on the elastic normal stress ratio or the elastic strain energy will not be appropriate. We will term this region craze zone of size 6. This will mean that for any elastic criterion to be successful it would have to be applied outside the craze zone. It would be desirable to relate the value of 6 to the microstructural parameters, however, at this stage, it may be acceptable to use a baseline test to determine its value. Using the critical strain energy criterion, with the same baseline test presented in ref. [9], a value of 6 = 0.11 mm was obtained. This value is almost l0 times smaller than that estimated for r,, in ref. [9]. P R E D I C T I O N B A S E D ON T H E CRITICAL STRAIN E N E R G Y An experimental investigation to study the crack growth of unidirectional graphite-epoxy coupons was conducted in ref. [9]. The predictions obtained using the critical strain energy criterion will be compared to the above experimental results.
Crack extension behaviour
{o}
35
(b)
o.:O ° ~t:15 o 0 : I05" 0 : to5* Fig. 6. Notch configurations used in experimental investigation: (a) configuration H (notch perpendicular to loading direction) and (b) configuration A (notch perpendicular to material fibre direction).
The 15 ° off-axis tensile coupons used had two notch configurations: one perpendicular to the loading direction (configuration H) and the other perpendicular to the material fibre direction (configuration A). Both configurations are shown in Fig. 6. Specimens with aspect ratios (length-to-width ratios) of 8, 4 and 1 were considered. Although apparent trends in the value of the stress causing initial crack growth with respect to the notch orientation, were obvious, no trends were seen with respect to the specimens' aspect ratios. Since the aspect ratio does not seem to have a noticeable effect on the experimental results, the average of the experimental data will be used for the purpose of comparison. The comparison between the experimental results and the predictions obtained using the critical strain energy criterion is given in Table 1. Also listed in Table 1 are the predictions obtained using the normal stress ratio criterion. Centre-notched 0 °, 45 ° and 90 ° coupons with notches oriented at 90 ° to the loading axis were tested under far-field uniaxial tension in ref. [6]. Again several tests were performed for every value of the fibre orientation angle (mainly due to the scatter observed in the experimental results). The comparisons between the average experimental data for each of the fibre orientation angles examined, and the predictions obtained using both theories are shown in Table 1. From this table, it is obvious that the results obtained using the critical strain energy criterion are consistent with experimental data and are in better agreement than those obtained using the normal stress ratio theory.
CONCLUSIONS A criterion based on the critical strain energy has been shown to accurately predict the crack extension behaviour in unidirectional composites within the elastic crack analysis. This criterion was successfully used to predict the crack growth direction of a centre-notched unidirectional coupon subjected to both tensile and shear loading. The use of the critical strain energy density criterion was also shown to be very consistent in predicting the value of the stress causing crack growth initiation.
36
H. EL KADI and F. ELLYIN
REFERENCES [1] S. W. Tsai and E. M. Wu, A general theory of strength for anisotropic materials. J. compos. Mater. 5, 58-80 (1971). [2] G.C. Sih, E. P. Chen, S. L. Huang and E. J. McQuillen, Material characterization on the fracture of filament-reinforced composites. J. compos. Mater. 9, 167 186 (1975). [3] M. B. Buczek and C. T. Herakovich, A normal stress criterion for crack extension direction in orthotropic composite materials. J. compos. Mater. 19, 544 553 (1985). [4] F. Ellyin and H. El Kadi, Predicting crack growth direction in unidirectional composite laminae. Engng Fracture Mech. 36, 27-37 (1990). [5] S. G. Lekhnitskii, Theory of Elasticity of an Anisotropic Elastic Body (Translated by P. Fern, Edited by J. J. Brandstatter). Holden-Day, San Francisco (1963). [6] J. L. Beuth, Jr and C. T. Herakovich, Analysis of crack extension in anisotropic materials based on local normal stress. Theor. appl. Fracture Mech. 11, 27-46 (1989). [7] M, A. Gregory and C. T. Herakovich, Predicting crack growth direction in unidirectional composites. J. eompos. Mater. 20, 67-85 (1986). [8] M. A. Gregory, J. L. Beuth, Jr, A. Barbe and C. T. Herakovich, Application of the normal stress ratio theory for predicting crack growth direction in unidirectional composites. Fracture Fibrous Compos. AMD, 74, 33-42 (1985). [9] J. L. Beuth, Jr and C. T. Herakovich, On fracture of fibrous composites. Composites '86: Recent Advances in Japan and the United States (Edited by K. Kawata, S. Umekawa and A. Kobayashi), Proc. Japan-U.S.A. CCM-HL pp. 267 277, Tokyo (1986). [10] G. Savin, Stress Concentration Around Holes (Translation editor W. Johnson). Pergamon Press, London (1961). [11] J. N. Goodier, Mathematical theory of equilibrium cracks, in Fracture: An Advanced Treatise (Edited by H. Liebowitz). Academic Press, New York (1968). [12] F. Ellyin and H. El Kadi, A fatigue failure criterion for fiber reinforced composite laminae. Compos. Structures 15, 61-74 (1990). [13] H. El Kadi and F. Ellyin, Effect of stress ratio on the fatigue of unidirectional fiberglass-epoxy composite laminae. Composites 25, 917-924 (1994). [14] J. L. Beuth, Jr, M. A. Gregory and C. T. Herakovich, Crack growth in unidirectional graphite-epoxy under biaxial loading. Exp. Mech. 26, 245 253 (1986). [15] D. Kujawski and F. Ellyin, A fatigue crack propagation model. Engng Fracture Mech. 20, 695-704 (1984). [16] J. Botsis, A. Chudnovsky and A. Moet, Fatigue crack layer propagation in polystyrene--I. Experimental observations. Int, J. Fracture 33, 263 276 (1987). (Received 24 December 1993)