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Probabilistic character of the microcrack growth in brittle layered materials
theoretical and applied fracture mechanics ELSEVIER
Theoretical and Applied Fracture Mechanics 27 (1997) 167-174
Probabilistic character of the microcrack growth in brittle layered materials Luoyu X u
*
Graduate Aeronautical Laboratories, California Institute of Technology, Mail Code 205-45, Pasadena, CA 91 I25, USA
Abstract Characteristics of microcrack initiation, multiplication and saturation in layered materials are discussed. A probabilisticanalytical method, the 'characteristic curve method (CCM)' is developed to correlate the initial defects and the microcrack evolution under static and cyclic loadings. The 'equivalent applied loading' and the 'equivalent crack density' concepts are introduced to describe different microcrack multiplication features in different layered materials. Microcrack multiplication processes in many layered materials with brittle matrices subjected to static and cyclic loadings can be easily predicted. © 1997 Elsevier Science B.V.
1. Introduction With the increasing applications of layered materials in engineering, for example, concrete pavements in highways, advanced fiber-reinforced composites in aerospace structures, the failure behaviors of layered materials have received attention in numerous scientific researches [1-5]. Unlike the traditional failure modes of isotropic homogeneous materials, the failure or damage mode in layered materials is often in the microcrack form. These microcracks are not self-similar; they are often progressive and discontinuous. Although microcracks do not lead to catastrophic failure, their presence often affects the structural integrity and durability. Repairing the cracking of highways, for example, could be very costly [6]. The microcrack multiplication behavior in layered materials is of practical importance. The general continuum mechanics theory is em-
* Tel.: + 1-626-3954758; fax: + 1-626-4492677; e-mail: [email protected].
ployed to study the microcrack growth problem. This leads to the complex stress fields of the layered material with a group of stochastic microcracks. Generally, these microcracks interact with one another [7], so the resulting stress solution is rather complex. While the strain energy release rate expression may be derived and used as a governing parameter of crack multiplication. Some results show that the critical strain energy release rate of the microcracking is not a material constant; it often varies from one cracking state to another [5,8,9]. Therefore, the prediction of this method is limited and the probabilistic method should be introduced [10].
2. Characteristic curve for microcraek multiplication Griffith pointed out that the material strength is the result of the propagation of an inherent material flaw [11]. Applying this concept in the microcrack multiplication problem of composite laminates, the
0167-8442/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S01 67-8442(97)0001 9-0
L. Xu / Theoretical and Apph'ed Fracture Mechanics 27 (1997) 167 174
168
Z
•
Microcracks I
.
//\,
-
J'
X
Cracked layer
~_ i !
,11
I.
I Constraining layer
Fig. l. Schematic diagram of a layered composite matcrJa] with microcracks.
'effective flaw hypothesis' was proposed [12]. The defects in the brittle matrices can be simulated by a distribution of the effective flaws. These flaws are assumed to be perpendicular to the loading direction and they may form microcracks under applied loadings as shown in Fig. 1. The size parameter of microcracks is characterized by the microcrack density D. It is defined by the number of microcracks existing over a unit length. Here, only the simplified plane problem is investigated such that the microcrack extends completely through in the direction normal in the plane. The length of the effective flaw 2 a in a weak layer is assumed to be an unknown distribution. These flaws act like actual cracks, which implies that principles of the classical linear elastic fracture mechanics (LEFM) may be used to describe these flaws [12]. According to the LEFM principle, a relationship prevail between the flaw length and the critical applied loading when these flaws form microcracks. For a layer with a group of flaws, the critical strain energy release rate or the stress intensity factor is chosen as the governing parameter for the formation of the microcrack: G c = Coc27ra
( 1)
where C is a correction parameter. G~ is the fracture toughness and crc is the critical stress. Using these results, the effective flaws and the microcrack density variation may be correlated qualitatively. The first microcrack is assumed to stem from the most severe initial flaw (defect). According to the flaw
distribution, the number of these severe flaws is small (1%). Those flaws having the average lengths are the majority. So, after the onset of the first microcrack, these flaws may form microcracks and the number of microcracks as well as the microcrack tnultiplication rate increase rapidly. However, flaws with the smaller length are few in the cracked layer. Therefore, after the rapid multiplication stage, the microcrack multiplication rate will decrease because few flaws will form microcracks. Hence, a final regular cracking state may be achieved. For example, Fig. 2 shows the rule of multiplication of transverse microcracks in highways [13]. In order to describe the microcrack multiplication rate quantitatively, it probabilistic-analytical method, the 'characteristic curve method ( C C M ) ' will be proposed. The lengths of effective flaws in a layer are assumed to obey an unknown distribution. The maximum length and minimum length are, respectively, aMAx (corresponding to the microcrack initiation) and aMl N (flaws have formed microcracks reaching the saturation state). A non-dimensional variable, the equivalent crack length aEQ , is first introduced:
aF.0 =
(1 --
¢JMIN
(2)
('/MAX -- U
For a layered material, the critical flaw length corresponding to a critical loading level can be con-
E
2.0
0
~DLY-60 MMY-60 , ,SLY-100 • DSY-100
E 0
~ 1.0 0 t~
(.9 1980.10
1981.10
1982.10
Time
Fig. 2. Transverse microcrack multiplication in a highway during two years.
L. Xu / Theoretical and Applied Fracture Mechanics 27 (1997) 167-174 veniently expressed in terms of the applied strain using Eq. (1): CM Cs ff
a
~gl.2
2 (CMCs/ 2)-(CMCsCS/
• GL/EP --Y = 0.435"X
(3)
where C M is a parameter of the layered material. It is a constant for different laminates of the same material. C s is a parameter of the laminate structure. It varies from one cracking state to another in the same laminate. For example, the critical strain energy release rate or stress intensity factor of a flaw which is included in this parameter is a function of the flaw length [8,9]. Substituting Eq. (3) into Eq. (2), there results:
aEQ
1.B
169
CI//ffCl) 2
S
-- (CMCs/E2 )
2 IffCS 2 __ Kcsse2 ~CI = e ~ s K c l s f f 2 E21 = EE2Q
0.4S 00
,
0.4
0.8
1.2 116 X=Inl/D
2
2.4
eq
Fig. 3. Parameter transformation relationship between the equivalent loading and the equivalent crack density of a [903/0] s composite laminate.
age microcrack density and the saturation crack density as D = N/L, Dcs = NT/L, Eq. (7) yields: (4)
ac
D=Dcsfa/(aEQ)daEQE where Kcs s = C scs / C s, Kcl s = CCJ/Cs . Note that CI and CS denote the microcrack initiation and the saturation states. '~EQ is called the 'equivalent applied loading':
(8)
Introducing another non-dimensional parameter, the equivalent crack density DEQ = D/Dcs, Eq. (8) becomes ::c
eo ~/e~ s _ K c s s • 2 ffEQ -- ~CS V KCISE2 -- eC2l
DEQ = (5)
"c, / e 2 s - e 2 (6)
Consider a long enough layer in which microcracks will appear; its length is L and its total number of effective flaws is N T. At an applied loading level e, the number of effective flaws which have formed microcracks is ao
N= NTfaE/(("/EQ) daeQ
aEQ ) daEQ = F ( ~ ) - Fa( aEQ )
-61 E
= 1 - F~( eEQ )
Based on recent theoretical and experimental resuits [8,9], the variation of the laminated structure parameter C s is small. Hence Kcs s = Kcl s = 1, Eq. (5) may be written as:
• EQ = --ecs V e'-'5--- e2-]"
f fe(
(7)
(9)
where F~(%Q) is the distribution function of the equivalent loading. Fig. 3 shows a parameter transformation of the equivalent loading and the equivalent crack density of a G L / E P [903/0] s composite laminate [14]. The distribution function F~(eEQ) can thus be described by the Rayleigh distribution function: F~( EEQ ) = 1 - e -(¢'~Q/~0)2
DE Q = e-(~E~/~')2
(10)
In the sequel, the fitting curve based on the Rayleigh distribution function will be referred to as the 'Rayleigh curve'. More generally, the Rayleigh distribution function should be replaced by the Weibull distribution function:
w h e r e f(aEQ) is the probabilistic density function of
F~(ffEQ) = 1 - e -(~EQ/¢~'~)CK
the equivalent crack length. Dividing Eq. (7) with respect to the layer length L and defining the aver-
DE Q = e - ( ~ Q / ~
)cK
( 1 1)
L. Xu / Theoretical and Applied Fracture Mechanics 27 (1997) 167 174
170
A relationship between the applied loading and the average microcrack density is established. The relationship between the equivalent crack density and the equivalent loading is called the 'characteristic curve for microcrack multiplication'. These new concepts are quite different from the actual applied loading and crack density. For example, when the actual applied loading tends to tile crack saturation loading, the equivalent loading tends to be zero. This means that, although the applied loading increases continuously, the microcrack density will not increase. Note that the obtained fitting parameters C~, e~'Q and eECQ w are laminate parameters. It should be verified by experimental results of different laminates with the same material to determine whether or not these parameters may be treated as the material constants. Fig. 4 shows the actual microcrack multiplication processes of two different G R / B M 1 composite laminates [15]. Fig. 5 shows their characteristic curves for microcrack growth. The results show that the characteristic curves for different laminates of the same constitute material will tend to a same curve. More examples will be reported in Refs. [16,17]. This may imply that the fitting parameters used in the characteristic curve can be treated as material constants which represent the total effect of the material defects. Some data-reduction techniques should be employed to establish the characteristic curve for microcrack growth [17].
1.0 G: [02/F/904] s 0.8 H: [0/F/904/F/0]s ": 0.6 E E o 0.4
-
0.2 •
o
0"00
\
,
1
~
2
3
4
eq
Fig. 5. The characteristic curves for microcrack multiplication of two T300/QY8911 GR/BMI interleaved composite laminates.
These results can be extended to the microcrack multiplication under the cyclic loading. The equivalent crack density is: DEQ
=e (Ne'~/u~w)~K uC~v = eEC~v
(12)
where the equivalent number of cycles NEQ is a function of the actual number of cycles similar to the equivalent static loading. If the maximum cyclic loading is less than the static microcrack initiation loading, the equivalent number of cycles is
NQ / N~s - N 2 NF~Q~ Nc~ V ~ 5 ~ ~
(13)
where N is the actual number of cycles, NQ and N(:s are numbers of cycles to the microcrack initiation and the microcrack saturation, respectively.
°
3.1. Prediction of the microcrack initiation
~og° ~ ° ° O
200
0.2
o Lam G • Lam H --Weibull curve -.Rayleigh curve
3. M i c r o c r a e k initiation and saturation states
rO
100
0.8 ~ . "~ c~g0.6 o~'k~ 0.4 - %0
= O
."
10r
300 400 a8 ( MPa )
500
600
Fig. 4. Different microcrack multiplication processes in two T300/QY8911 GR/BMI interleaved composite laminates under static tensile loading.
The prediction of static microcrack initiation (formation of the first microcrack) has been studied successfully and here we only cite some results. Two approaches were often employed. They are the fracture mechanics approach and the maximum stress approach. According to the linear elastic fracture mechanics principle, the critical strain energy release
L. Xu / Theoretical and Applied Fracture Mechanics 27 (1997) 167-174
171
rate is equal to the fracture toughness at the crack initiation:
1
U = ~ f v ~ i j s i j dV
dU G c - Wdaa--*
(14)
where U is the strain energy of the layered material with microcracks, which is often expressed by a function of the microcrack density or the flaw length [4,5,12]. In the maximum stress approach, the stress of the weak layer is assumed to reach its critical strength value when the first microcrack initiates [18]. For the microcrack initiation under the cyclic loading, the number of cycles of microcrack initiation is based on the experimental S - N curve [19,20]. It is a material property of the cracked layer and seems to be independent of the layered structure. O'MAX _ e - ( i o g No,/log NCc,y"c'
(15)
O'cl where NCI and log NcCl are material constants obtained from the S - N curve fitting. 3.2. Determination o f the saturation crack state
When the first microcrack appears in the cracked layer and the coordinate origin is located in the first microcrack, the total average normal stress of the cracked layer may be determined by many analytical models. For example, the typical shear-lag model gives [10,21]: ffcc = if°c(1 - e - X X )
0 C0
o
a t
(16)
where o'°c is the far-field normal stress of the cracked layer in the X direction and K is the so-called shear-lag parameter of the layered material [21]. According to Eq. (16), the approximate normal stress in the cracked layer is illustrated in Fig. 6. Fig. 6 is similar to the diagram reported in Ref. [22] in which an explanation was given for the determination of the saturation crack spacing (density) in composite laminates. The shortest distance were thought to be that from the original crack to the position at which the stress reached the original (undisturbed) level was the 'saturation crack spacing'. However, this explanation seems to be insufficient. Here, the definition is expanded and combined with the 'effective flaw hypothesis' [12].
tt) (tJ
E
/
F,aws
0 Z _
4"--
0
"~ (..)
[
Cracked layer
Fig. 6. Schematic diagram of the formation of the microcrack saturation state.
As shown in Fig. 5, after the first microcrack initiates, the normal stress at this position becomes zero. Due to the existence of the interlayer shear stress, the normal stress of the cracked layer will recover to the original level at a certain distance from the formed crack. Thus it leads to a so-called 'shear-lag zone' in the layered material. This zone has the significant effect on the microcrack multiplication. For example, in Fig. 5, two effective flaws exist inside and outside the shear-lag zone. Their lengths are assumed to be approximately the same. Obviously, the effective flaw outside the shear-lag zone is superior to forming a microcrack due to the higher stress. Only after the loading reaches a higher level, is the effective flaw inside the shear-lag zone able to form the microcrack. Therefore, the length of the shear-lag zone is regarded as the saturation crack spacing. This length is a probabilistic expectation value. Based on Eq. (16), the final saturation crack spacing of microcracks Lcs can be determined by: 1 - e -KL~~= 1 - D e D c s = g / l n De
De_<0.15) (17)
where De is a material constant with probabilistic meaning, which represents the total effect of the distributive defects which may form microcracks. It should be a constant for these laminates of the same material under the same process. The dependence of the saturation crack density on the shear-lag parame-
172
l,. Xu / Yheoretical and Applied Fracture Mechanics 27 (1997) 167 174
ter has been verified by many experimental results of composite laminates [ 14,15,21 ]. The static microcrack saturation loading should be measured by experiments and is assumed as a material constant approximately. Note that the maximum cyclic loading has the significant effect on the number of cycles of microcrack saturation. An empirical equation is employed [20]: (/'MAX
(h>g
,'v, ~
e
Ioe ~,'~ ~ \ /
1 0.8 0.6 E r'h 0.4 0.2
(18)
",
.....
O'cs
06
where N C S and log N~.s are material constants obtained from the S - N curve fitting.
__~_Jf=o prediction ,
50
100
o GL/EP [0/903]s
150 200 c ( MPa )
250
300
350
Fig. 8. Microcrack multiplication prediction for a GL/EP composite laminate under the static tension. 4. P r e d i c t i o n e x a m p l e s a n d discussion To predict the microcrack multiplication, one should predict the microcrack initiation and the saturation state and then utilize the characteristic curve for microcrack multiplication. Here, only some predictions of the microcrack multiplication in fibrous composite laminates are presented; the required characteristic curves are given in [17]. The microcrack multiplication predictions of G L / E P [14] and G L / B M I $ 2 / Q Y 8 9 1 1 [21] layered composite materials have been presented in Figs. 7 and 8. The current predicted crack growth curve is different from the theoretical curve based on the pure fracture mechanics model [23]. For the latter model, the crack density may tend to an infinite value, not a
2 i
~
saturation value reported by many experimental results [2,14,18,19]. Hence, using the critical energy release rate or static strength value as the material constant is not very reasonable in those predictions. Actually, those models are excellent in predicting the microcrack initiation [4,8,24]. To predict the microcrack growth, introducing the state of microcrack saturation may obtain good results [25]. Generally, for the microcracking in glass fiber composite or ceramic matrix composite laminates, crack saturation is significant. Figs. 9 and 10 show the microcrack multiplication predictions of two ceramic matrix composite laminates [17,26]. There appears to be a correlation between the microcrack saturation loading and the matrix toughness. The microcrack satura-
121
4
1.6 1.2 v
v
a
o
0.8 0.4
,.,/
gl O0
[ - - C C M prediction oGL/BMI $2/QY8911 [OJ90/0]
0.4
-~
0.8
1.2
1.6
3
2
.s
1
I / , // !'r
O0
50
--CCM prediction - SiC/CAS [0/90]+ Exp. range
, 2.4
(%)
+
2
8
Fig. 7. Microcrack multiplication prediction for a GL/BM] com posite laminate under the static tension.
100
150 200 (MPa)
250
--300 350
II
Fig. 9. Microcrack multiplication prediction for a ceramic matrix composite laminate under the static tension.
173
L. Xu / Theoretical and Applied Fracture Mechanics 27 (1997) 167-174 4
3
1.6
•
• GR/EP AS4/3501-6 [0/9021, --CCM prediction • ~ =33.7% max
"7.
•r
1.2
o
0.8
E E
r~ 1
prediction • SiC/CAS [03//903//03]T
=/
Oo-
0.2
'4
ols
0.4
o18
00
(%)
1
2
3
4
5
6
7
8
log N
a
Fig. 10. Microcrack multiplication prediction for a ceramic matrix composite laminate under the static tension.
tion strain of the more brittle ceramic matrix composites is about 0.6, compared with 2.2 for brittle G L / B M I polymeric matrix composites. However, the relationships between material constants obtained in the characteristic curve fitting and other standard material constants, such as the tensile strength, are not clear. For the microcrack multiplication in layered materials under cyclic loadings, two examples of the brittle polymeric matrix composites have been shown in Figs. 11 and 12, i.e. the graphite/epoxy materials T 3 0 0 / 9 1 4 [27] and A S 4 / 3 5 0 1 - 6 [19]. The material constants used in the characteristic curve are assumed to be the same values as those for static loading of the A S 4 / 3 5 0 1 - 6 material. But for the 5
Fig. 12. Microcrack multiplication prediction in a composite laminate under the cyclic loading.
T 3 0 0 / 9 1 4 material, those constants are based on the microcrack multiplication information of another baseline laminate. The equivalent concept has been widely used in chemistry because its advantage is revealing the same inner reality in different materials. For the layered material such as fibrous composite laminates used extensively in aerospace industries, because they are not the homogeneous materials and like structures, one may find a physical law obtained from one laminate may be not applicable to another laminate. By introducing the concepts of the equivalent crack density and the equivalent loading, we may obtain some laminate independent parameters. Then, we can describe the same physical nature of the microcrack multiplication in different laminates and materials.
[] GR/EP T300/914 [0290] --CCM
4
prediction
~
e
2
• °0
1
~, •
•
3
a
s
, [03/90101
5. Concluding remarks
e
[] []
[]
[][][] 0 [ ] [][] o
°o
~
~,
:~
4
5
s
-~
8
log N
Fig. I I. Microcrack multiplication prediction in a composite laminate under the cyclic loading.
The main characteristics of the microcrack multiplication in layered materials are progressive, discontinuous. Unlike the typical self-similar crack, microcracking has the initiation, multiplication and saturation stages. Microcrack multiplication is governed not only by the mechanics state (stress distribution), but also by the material state (initial defects). The proposed probabilistic-analytical method, the characteristic curve method, is effective in revealing the physical essence of microcracks, thus making micro-
174
L. Xu / Theoretical and Applied Fracture Mechanics 27 (1997) 167-174
cracking in different layered materials more predictable.
References [1] Z. Hashin, Analysis of cracked laminate: A variational ap proach, Mech. Mater. 4 (1985) 121 136. [2] C.T. Herakovich, M.W. Hyer, Damage-induced property change in composites subjected to cyclic thermal loading, Eng. Fracture Mech. 35 ( 5 / 6 ) (1986) 779-791. [3] D.H. Allen, C.E. Harris, S.E. Groves, A thermomechanical constitutive theory for elastic composites with distributed damage-part. 2. Application to matrix cracking in laminated composites, Int. J. Solids Structures 23 (9)(1987) 1319 1338. [4] G.J. Dvorak, N. Laws, Analysis of progressive matrix cracking in composite laminates. 2. First ply failure. J. Composite Mater. 21 (4) (1987) 309-329. [5] S. Ho, Z. Suo, Tunneling cracks in constrained layers, J. Appl. Mech. 60 (1993) 890 894. [6] E.J. Yoder, M.W. Witczak, Principles of Pavement Design, 2nd ed., John Wiley and Sons, New York, 1975, pp. 391-399. [7[ A.C. Wijeyewickrema, L.M. Keer, Matrix crack interaction in a fiber-reinforced brittle matrix composite, lnt. J. Solids Structures 29 (1991) 559-570. [8] S. Yalvac, L.D. Yats, D.G. Wetter, Transverse ply cracking in toughened and untoughened graphite/epoxy and graphite/polycyanate crossply laminates, J. Composite Mater. 25 (11) (1991) 1653-1667. [9] B. Wu, F. Erdogan, Fracture mechanics of orthotropic lami nated plates. 2. The plane strain crack problem for two bonded orthotropic layers, lnt. J. Solids Structures 30 (1993) 2379-2391. [10] N. Laws, G.J. Dvorak, Progressive transverse cracking in composite laminates, ]. Composite Mater. 22 (5) (1988) 900-916. [11] A.A. Griffith, The phenomena of rupture and flaw in solid, Philos. Trans. R. Soc. A 221 (1920) 163-172. [12] A.S.D. Wang, Fracture mechanics of sublaminate cracks ill composite materials, Composites Technol. Rev. 6 (2) (1984) 45 62. [13] L.Y. Xu et al.. The nonmechanical loading fracture and controlling in concrete pavements, Proc. of the Ist lnterna-
tional Conference on Fracture Mechanics of Concrete Structures, Elsevier, London, 1992, pp. 997-1002. [14] A.L. Highsmith, K.L. Reifsnider, Stiffness-reduction mechanisms in composite laminates, in: K.L. Reifsnider (Ed.), Damage ill Composite Materials, ASTM STP 775, Philadelphia, 1982, pp. 103-117. {15] L.Y. Xu, C.H. Kou, Effect of interracial interleaf to the interlaminar fracture and intralaminar fracture in a new bismaleimide matrix composite system, J. Reinforced Plastics Composites 13 (1994) 509-540. [16] L.Y. Xu, On the law of the transverse matrix cracking in brittle polymeric composite laminates under static loadings, .1. Composite Mater. 30 (3) (1996) 384-405. 117] L.Y. Xu, Some techniques to establish the characteristic curve for microcrack multiplication in laminated materials, Eng. Fracture Mech. 56 (6) (1997) 827 837. [18] A.W. Pryce, P.A. Smith, Behaviour of unidirectional and crossply ceramic matrix composites under quasi-static tensile loading, J. Mater. Sci. 27 (1992) 2695-2704. [19] I.M. Daniel, A. Charewicz, Fatigue damage mechanisms and residual properties of graphite/epoxy laminates, Eng. Frac ture Mech. 35 ( 5 / 6 ) (1986) 793-808. [2(/1 L.Y. Xu, Study on the characteristic stress-life curve of different materials and failure modes, submitted. [21] L.Y. Xu, Influence of stacking sequence on the transverse matrix cracking in continuous fiber crossply laminates, J. Composite Mater. 29 (10)(1995) 1337-1358. [22] K.L. Reifsnider, Damage and damage mechanics, in: K.L. Reifsnider (Ed.), Fatigue of Composite Materials, Elsevier Sci. Publisher, Amsterdam, 1990, pp. 11-12. [23] J.A. Nairn, S. Hu, The formation and effect of outer-ply microcracks in cross-ply laminates: A variational approach, Eng. Fracture Mech. 41 (2) (1992) 203-221. [24] L.R. Dharani, H. Tang, Micromechanics characterization of sublaminate damage, Int. J. Fracture 46 (1990) 123-140. [25] 1. Shahid, F.K. Chang, An accumulative damage model for tensile and shear failures of laminated composite plates, J. Composite Mater. 29 (7) (1995) 926-981. [26] P. Karandikar, T.W. Chou, Characterization and modeling of microcracking and elastic moduli changes in Nicalon/CAS composites, Composites Sci. Technol. 46 (1993) 253-263. [27] M.C. Lafarie-Frenot, C.H. Gardin, Formation and multiplica tion of 90 ° ply fatigue cracks in carbon/epoxy laminates, Composite Sci. Technol. 40 (1991) 307-324.
Theoretical and Applied Fracture Mechanics 27 (1997) 167-174
Probabilistic character of the microcrack growth in brittle layered materials Luoyu X u
*
Graduate Aeronautical Laboratories, California Institute of Technology, Mail Code 205-45, Pasadena, CA 91 I25, USA
Abstract Characteristics of microcrack initiation, multiplication and saturation in layered materials are discussed. A probabilisticanalytical method, the 'characteristic curve method (CCM)' is developed to correlate the initial defects and the microcrack evolution under static and cyclic loadings. The 'equivalent applied loading' and the 'equivalent crack density' concepts are introduced to describe different microcrack multiplication features in different layered materials. Microcrack multiplication processes in many layered materials with brittle matrices subjected to static and cyclic loadings can be easily predicted. © 1997 Elsevier Science B.V.
1. Introduction With the increasing applications of layered materials in engineering, for example, concrete pavements in highways, advanced fiber-reinforced composites in aerospace structures, the failure behaviors of layered materials have received attention in numerous scientific researches [1-5]. Unlike the traditional failure modes of isotropic homogeneous materials, the failure or damage mode in layered materials is often in the microcrack form. These microcracks are not self-similar; they are often progressive and discontinuous. Although microcracks do not lead to catastrophic failure, their presence often affects the structural integrity and durability. Repairing the cracking of highways, for example, could be very costly [6]. The microcrack multiplication behavior in layered materials is of practical importance. The general continuum mechanics theory is em-
* Tel.: + 1-626-3954758; fax: + 1-626-4492677; e-mail: [email protected].
ployed to study the microcrack growth problem. This leads to the complex stress fields of the layered material with a group of stochastic microcracks. Generally, these microcracks interact with one another [7], so the resulting stress solution is rather complex. While the strain energy release rate expression may be derived and used as a governing parameter of crack multiplication. Some results show that the critical strain energy release rate of the microcracking is not a material constant; it often varies from one cracking state to another [5,8,9]. Therefore, the prediction of this method is limited and the probabilistic method should be introduced [10].
2. Characteristic curve for microcraek multiplication Griffith pointed out that the material strength is the result of the propagation of an inherent material flaw [11]. Applying this concept in the microcrack multiplication problem of composite laminates, the
0167-8442/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S01 67-8442(97)0001 9-0
L. Xu / Theoretical and Apph'ed Fracture Mechanics 27 (1997) 167 174
168
Z
•
Microcracks I
.
//\,
-
J'
X
Cracked layer
~_ i !
,11
I.
I Constraining layer
Fig. l. Schematic diagram of a layered composite matcrJa] with microcracks.
'effective flaw hypothesis' was proposed [12]. The defects in the brittle matrices can be simulated by a distribution of the effective flaws. These flaws are assumed to be perpendicular to the loading direction and they may form microcracks under applied loadings as shown in Fig. 1. The size parameter of microcracks is characterized by the microcrack density D. It is defined by the number of microcracks existing over a unit length. Here, only the simplified plane problem is investigated such that the microcrack extends completely through in the direction normal in the plane. The length of the effective flaw 2 a in a weak layer is assumed to be an unknown distribution. These flaws act like actual cracks, which implies that principles of the classical linear elastic fracture mechanics (LEFM) may be used to describe these flaws [12]. According to the LEFM principle, a relationship prevail between the flaw length and the critical applied loading when these flaws form microcracks. For a layer with a group of flaws, the critical strain energy release rate or the stress intensity factor is chosen as the governing parameter for the formation of the microcrack: G c = Coc27ra
( 1)
where C is a correction parameter. G~ is the fracture toughness and crc is the critical stress. Using these results, the effective flaws and the microcrack density variation may be correlated qualitatively. The first microcrack is assumed to stem from the most severe initial flaw (defect). According to the flaw
distribution, the number of these severe flaws is small (1%). Those flaws having the average lengths are the majority. So, after the onset of the first microcrack, these flaws may form microcracks and the number of microcracks as well as the microcrack tnultiplication rate increase rapidly. However, flaws with the smaller length are few in the cracked layer. Therefore, after the rapid multiplication stage, the microcrack multiplication rate will decrease because few flaws will form microcracks. Hence, a final regular cracking state may be achieved. For example, Fig. 2 shows the rule of multiplication of transverse microcracks in highways [13]. In order to describe the microcrack multiplication rate quantitatively, it probabilistic-analytical method, the 'characteristic curve method ( C C M ) ' will be proposed. The lengths of effective flaws in a layer are assumed to obey an unknown distribution. The maximum length and minimum length are, respectively, aMAx (corresponding to the microcrack initiation) and aMl N (flaws have formed microcracks reaching the saturation state). A non-dimensional variable, the equivalent crack length aEQ , is first introduced:
aF.0 =
(1 --
¢JMIN
(2)
('/MAX -- U
For a layered material, the critical flaw length corresponding to a critical loading level can be con-
E
2.0
0
~DLY-60 MMY-60 , ,SLY-100 • DSY-100
E 0
~ 1.0 0 t~
(.9 1980.10
1981.10
1982.10
Time
Fig. 2. Transverse microcrack multiplication in a highway during two years.
L. Xu / Theoretical and Applied Fracture Mechanics 27 (1997) 167-174 veniently expressed in terms of the applied strain using Eq. (1): CM Cs ff
a
~gl.2
2 (CMCs/ 2)-(CMCsCS/
• GL/EP --Y = 0.435"X
(3)
where C M is a parameter of the layered material. It is a constant for different laminates of the same material. C s is a parameter of the laminate structure. It varies from one cracking state to another in the same laminate. For example, the critical strain energy release rate or stress intensity factor of a flaw which is included in this parameter is a function of the flaw length [8,9]. Substituting Eq. (3) into Eq. (2), there results:
aEQ
1.B
169
CI//ffCl) 2
S
-- (CMCs/E2 )
2 IffCS 2 __ Kcsse2 ~CI = e ~ s K c l s f f 2 E21 = EE2Q
0.4S 00
,
0.4
0.8
1.2 116 X=Inl/D
2
2.4
eq
Fig. 3. Parameter transformation relationship between the equivalent loading and the equivalent crack density of a [903/0] s composite laminate.
age microcrack density and the saturation crack density as D = N/L, Dcs = NT/L, Eq. (7) yields: (4)
ac
D=Dcsfa/(aEQ)daEQE where Kcs s = C scs / C s, Kcl s = CCJ/Cs . Note that CI and CS denote the microcrack initiation and the saturation states. '~EQ is called the 'equivalent applied loading':
(8)
Introducing another non-dimensional parameter, the equivalent crack density DEQ = D/Dcs, Eq. (8) becomes ::c
eo ~/e~ s _ K c s s • 2 ffEQ -- ~CS V KCISE2 -- eC2l
DEQ = (5)
"c, / e 2 s - e 2 (6)
Consider a long enough layer in which microcracks will appear; its length is L and its total number of effective flaws is N T. At an applied loading level e, the number of effective flaws which have formed microcracks is ao
N= NTfaE/(("/EQ) daeQ
aEQ ) daEQ = F ( ~ ) - Fa( aEQ )
-61 E
= 1 - F~( eEQ )
Based on recent theoretical and experimental resuits [8,9], the variation of the laminated structure parameter C s is small. Hence Kcs s = Kcl s = 1, Eq. (5) may be written as:
• EQ = --ecs V e'-'5--- e2-]"
f fe(
(7)
(9)
where F~(%Q) is the distribution function of the equivalent loading. Fig. 3 shows a parameter transformation of the equivalent loading and the equivalent crack density of a G L / E P [903/0] s composite laminate [14]. The distribution function F~(eEQ) can thus be described by the Rayleigh distribution function: F~( EEQ ) = 1 - e -(¢'~Q/~0)2
DE Q = e-(~E~/~')2
(10)
In the sequel, the fitting curve based on the Rayleigh distribution function will be referred to as the 'Rayleigh curve'. More generally, the Rayleigh distribution function should be replaced by the Weibull distribution function:
w h e r e f(aEQ) is the probabilistic density function of
F~(ffEQ) = 1 - e -(~EQ/¢~'~)CK
the equivalent crack length. Dividing Eq. (7) with respect to the layer length L and defining the aver-
DE Q = e - ( ~ Q / ~
)cK
( 1 1)
L. Xu / Theoretical and Applied Fracture Mechanics 27 (1997) 167 174
170
A relationship between the applied loading and the average microcrack density is established. The relationship between the equivalent crack density and the equivalent loading is called the 'characteristic curve for microcrack multiplication'. These new concepts are quite different from the actual applied loading and crack density. For example, when the actual applied loading tends to tile crack saturation loading, the equivalent loading tends to be zero. This means that, although the applied loading increases continuously, the microcrack density will not increase. Note that the obtained fitting parameters C~, e~'Q and eECQ w are laminate parameters. It should be verified by experimental results of different laminates with the same material to determine whether or not these parameters may be treated as the material constants. Fig. 4 shows the actual microcrack multiplication processes of two different G R / B M 1 composite laminates [15]. Fig. 5 shows their characteristic curves for microcrack growth. The results show that the characteristic curves for different laminates of the same constitute material will tend to a same curve. More examples will be reported in Refs. [16,17]. This may imply that the fitting parameters used in the characteristic curve can be treated as material constants which represent the total effect of the material defects. Some data-reduction techniques should be employed to establish the characteristic curve for microcrack growth [17].
1.0 G: [02/F/904] s 0.8 H: [0/F/904/F/0]s ": 0.6 E E o 0.4
-
0.2 •
o
0"00
\
,
1
~
2
3
4
eq
Fig. 5. The characteristic curves for microcrack multiplication of two T300/QY8911 GR/BMI interleaved composite laminates.
These results can be extended to the microcrack multiplication under the cyclic loading. The equivalent crack density is: DEQ
=e (Ne'~/u~w)~K uC~v = eEC~v
(12)
where the equivalent number of cycles NEQ is a function of the actual number of cycles similar to the equivalent static loading. If the maximum cyclic loading is less than the static microcrack initiation loading, the equivalent number of cycles is
NQ / N~s - N 2 NF~Q~ Nc~ V ~ 5 ~ ~
(13)
where N is the actual number of cycles, NQ and N(:s are numbers of cycles to the microcrack initiation and the microcrack saturation, respectively.
°
3.1. Prediction of the microcrack initiation
~og° ~ ° ° O
200
0.2
o Lam G • Lam H --Weibull curve -.Rayleigh curve
3. M i c r o c r a e k initiation and saturation states
rO
100
0.8 ~ . "~ c~g0.6 o~'k~ 0.4 - %0
= O
."
10r
300 400 a8 ( MPa )
500
600
Fig. 4. Different microcrack multiplication processes in two T300/QY8911 GR/BMI interleaved composite laminates under static tensile loading.
The prediction of static microcrack initiation (formation of the first microcrack) has been studied successfully and here we only cite some results. Two approaches were often employed. They are the fracture mechanics approach and the maximum stress approach. According to the linear elastic fracture mechanics principle, the critical strain energy release
L. Xu / Theoretical and Applied Fracture Mechanics 27 (1997) 167-174
171
rate is equal to the fracture toughness at the crack initiation:
1
U = ~ f v ~ i j s i j dV
dU G c - Wdaa--*
(14)
where U is the strain energy of the layered material with microcracks, which is often expressed by a function of the microcrack density or the flaw length [4,5,12]. In the maximum stress approach, the stress of the weak layer is assumed to reach its critical strength value when the first microcrack initiates [18]. For the microcrack initiation under the cyclic loading, the number of cycles of microcrack initiation is based on the experimental S - N curve [19,20]. It is a material property of the cracked layer and seems to be independent of the layered structure. O'MAX _ e - ( i o g No,/log NCc,y"c'
(15)
O'cl where NCI and log NcCl are material constants obtained from the S - N curve fitting. 3.2. Determination o f the saturation crack state
When the first microcrack appears in the cracked layer and the coordinate origin is located in the first microcrack, the total average normal stress of the cracked layer may be determined by many analytical models. For example, the typical shear-lag model gives [10,21]: ffcc = if°c(1 - e - X X )
0 C0
o
a t
(16)
where o'°c is the far-field normal stress of the cracked layer in the X direction and K is the so-called shear-lag parameter of the layered material [21]. According to Eq. (16), the approximate normal stress in the cracked layer is illustrated in Fig. 6. Fig. 6 is similar to the diagram reported in Ref. [22] in which an explanation was given for the determination of the saturation crack spacing (density) in composite laminates. The shortest distance were thought to be that from the original crack to the position at which the stress reached the original (undisturbed) level was the 'saturation crack spacing'. However, this explanation seems to be insufficient. Here, the definition is expanded and combined with the 'effective flaw hypothesis' [12].
tt) (tJ
E
/
F,aws
0 Z _
4"--
0
"~ (..)
[
Cracked layer
Fig. 6. Schematic diagram of the formation of the microcrack saturation state.
As shown in Fig. 5, after the first microcrack initiates, the normal stress at this position becomes zero. Due to the existence of the interlayer shear stress, the normal stress of the cracked layer will recover to the original level at a certain distance from the formed crack. Thus it leads to a so-called 'shear-lag zone' in the layered material. This zone has the significant effect on the microcrack multiplication. For example, in Fig. 5, two effective flaws exist inside and outside the shear-lag zone. Their lengths are assumed to be approximately the same. Obviously, the effective flaw outside the shear-lag zone is superior to forming a microcrack due to the higher stress. Only after the loading reaches a higher level, is the effective flaw inside the shear-lag zone able to form the microcrack. Therefore, the length of the shear-lag zone is regarded as the saturation crack spacing. This length is a probabilistic expectation value. Based on Eq. (16), the final saturation crack spacing of microcracks Lcs can be determined by: 1 - e -KL~~= 1 - D e D c s = g / l n De
De_<0.15) (17)
where De is a material constant with probabilistic meaning, which represents the total effect of the distributive defects which may form microcracks. It should be a constant for these laminates of the same material under the same process. The dependence of the saturation crack density on the shear-lag parame-
172
l,. Xu / Yheoretical and Applied Fracture Mechanics 27 (1997) 167 174
ter has been verified by many experimental results of composite laminates [ 14,15,21 ]. The static microcrack saturation loading should be measured by experiments and is assumed as a material constant approximately. Note that the maximum cyclic loading has the significant effect on the number of cycles of microcrack saturation. An empirical equation is employed [20]: (/'MAX
(h>g
,'v, ~
e
Ioe ~,'~ ~ \ /
1 0.8 0.6 E r'h 0.4 0.2
(18)
",
.....
O'cs
06
where N C S and log N~.s are material constants obtained from the S - N curve fitting.
__~_Jf=o prediction ,
50
100
o GL/EP [0/903]s
150 200 c ( MPa )
250
300
350
Fig. 8. Microcrack multiplication prediction for a GL/EP composite laminate under the static tension. 4. P r e d i c t i o n e x a m p l e s a n d discussion To predict the microcrack multiplication, one should predict the microcrack initiation and the saturation state and then utilize the characteristic curve for microcrack multiplication. Here, only some predictions of the microcrack multiplication in fibrous composite laminates are presented; the required characteristic curves are given in [17]. The microcrack multiplication predictions of G L / E P [14] and G L / B M I $ 2 / Q Y 8 9 1 1 [21] layered composite materials have been presented in Figs. 7 and 8. The current predicted crack growth curve is different from the theoretical curve based on the pure fracture mechanics model [23]. For the latter model, the crack density may tend to an infinite value, not a
2 i
~
saturation value reported by many experimental results [2,14,18,19]. Hence, using the critical energy release rate or static strength value as the material constant is not very reasonable in those predictions. Actually, those models are excellent in predicting the microcrack initiation [4,8,24]. To predict the microcrack growth, introducing the state of microcrack saturation may obtain good results [25]. Generally, for the microcracking in glass fiber composite or ceramic matrix composite laminates, crack saturation is significant. Figs. 9 and 10 show the microcrack multiplication predictions of two ceramic matrix composite laminates [17,26]. There appears to be a correlation between the microcrack saturation loading and the matrix toughness. The microcrack satura-
121
4
1.6 1.2 v
v
a
o
0.8 0.4
,.,/
gl O0
[ - - C C M prediction oGL/BMI $2/QY8911 [OJ90/0]
0.4
-~
0.8
1.2
1.6
3
2
.s
1
I / , // !'r
O0
50
--CCM prediction - SiC/CAS [0/90]+ Exp. range
, 2.4
(%)
+
2
8
Fig. 7. Microcrack multiplication prediction for a GL/BM] com posite laminate under the static tension.
100
150 200 (MPa)
250
--300 350
II
Fig. 9. Microcrack multiplication prediction for a ceramic matrix composite laminate under the static tension.
173
L. Xu / Theoretical and Applied Fracture Mechanics 27 (1997) 167-174 4
3
1.6
•
• GR/EP AS4/3501-6 [0/9021, --CCM prediction • ~ =33.7% max
"7.
•r
1.2
o
0.8
E E
r~ 1
prediction • SiC/CAS [03//903//03]T
=/
Oo-
0.2
'4
ols
0.4
o18
00
(%)
1
2
3
4
5
6
7
8
log N
a
Fig. 10. Microcrack multiplication prediction for a ceramic matrix composite laminate under the static tension.
tion strain of the more brittle ceramic matrix composites is about 0.6, compared with 2.2 for brittle G L / B M I polymeric matrix composites. However, the relationships between material constants obtained in the characteristic curve fitting and other standard material constants, such as the tensile strength, are not clear. For the microcrack multiplication in layered materials under cyclic loadings, two examples of the brittle polymeric matrix composites have been shown in Figs. 11 and 12, i.e. the graphite/epoxy materials T 3 0 0 / 9 1 4 [27] and A S 4 / 3 5 0 1 - 6 [19]. The material constants used in the characteristic curve are assumed to be the same values as those for static loading of the A S 4 / 3 5 0 1 - 6 material. But for the 5
Fig. 12. Microcrack multiplication prediction in a composite laminate under the cyclic loading.
T 3 0 0 / 9 1 4 material, those constants are based on the microcrack multiplication information of another baseline laminate. The equivalent concept has been widely used in chemistry because its advantage is revealing the same inner reality in different materials. For the layered material such as fibrous composite laminates used extensively in aerospace industries, because they are not the homogeneous materials and like structures, one may find a physical law obtained from one laminate may be not applicable to another laminate. By introducing the concepts of the equivalent crack density and the equivalent loading, we may obtain some laminate independent parameters. Then, we can describe the same physical nature of the microcrack multiplication in different laminates and materials.
[] GR/EP T300/914 [0290] --CCM
4
prediction
~
e
2
• °0
1
~, •
•
3
a
s
, [03/90101
5. Concluding remarks
e
[] []
[]
[][][] 0 [ ] [][] o
°o
~
~,
:~
4
5
s
-~
8
log N
Fig. I I. Microcrack multiplication prediction in a composite laminate under the cyclic loading.
The main characteristics of the microcrack multiplication in layered materials are progressive, discontinuous. Unlike the typical self-similar crack, microcracking has the initiation, multiplication and saturation stages. Microcrack multiplication is governed not only by the mechanics state (stress distribution), but also by the material state (initial defects). The proposed probabilistic-analytical method, the characteristic curve method, is effective in revealing the physical essence of microcracks, thus making micro-
174
L. Xu / Theoretical and Applied Fracture Mechanics 27 (1997) 167-174
cracking in different layered materials more predictable.
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tional Conference on Fracture Mechanics of Concrete Structures, Elsevier, London, 1992, pp. 997-1002. [14] A.L. Highsmith, K.L. Reifsnider, Stiffness-reduction mechanisms in composite laminates, in: K.L. Reifsnider (Ed.), Damage ill Composite Materials, ASTM STP 775, Philadelphia, 1982, pp. 103-117. {15] L.Y. Xu, C.H. Kou, Effect of interracial interleaf to the interlaminar fracture and intralaminar fracture in a new bismaleimide matrix composite system, J. Reinforced Plastics Composites 13 (1994) 509-540. [16] L.Y. Xu, On the law of the transverse matrix cracking in brittle polymeric composite laminates under static loadings, .1. Composite Mater. 30 (3) (1996) 384-405. 117] L.Y. Xu, Some techniques to establish the characteristic curve for microcrack multiplication in laminated materials, Eng. Fracture Mech. 56 (6) (1997) 827 837. [18] A.W. Pryce, P.A. Smith, Behaviour of unidirectional and crossply ceramic matrix composites under quasi-static tensile loading, J. Mater. Sci. 27 (1992) 2695-2704. [19] I.M. Daniel, A. Charewicz, Fatigue damage mechanisms and residual properties of graphite/epoxy laminates, Eng. Frac ture Mech. 35 ( 5 / 6 ) (1986) 793-808. [2(/1 L.Y. Xu, Study on the characteristic stress-life curve of different materials and failure modes, submitted. [21] L.Y. Xu, Influence of stacking sequence on the transverse matrix cracking in continuous fiber crossply laminates, J. Composite Mater. 29 (10)(1995) 1337-1358. [22] K.L. Reifsnider, Damage and damage mechanics, in: K.L. Reifsnider (Ed.), Fatigue of Composite Materials, Elsevier Sci. Publisher, Amsterdam, 1990, pp. 11-12. [23] J.A. Nairn, S. Hu, The formation and effect of outer-ply microcracks in cross-ply laminates: A variational approach, Eng. Fracture Mech. 41 (2) (1992) 203-221. [24] L.R. Dharani, H. Tang, Micromechanics characterization of sublaminate damage, Int. J. Fracture 46 (1990) 123-140. [25] 1. Shahid, F.K. Chang, An accumulative damage model for tensile and shear failures of laminated composite plates, J. Composite Mater. 29 (7) (1995) 926-981. [26] P. Karandikar, T.W. Chou, Characterization and modeling of microcracking and elastic moduli changes in Nicalon/CAS composites, Composites Sci. Technol. 46 (1993) 253-263. [27] M.C. Lafarie-Frenot, C.H. Gardin, Formation and multiplica tion of 90 ° ply fatigue cracks in carbon/epoxy laminates, Composite Sci. Technol. 40 (1991) 307-324.