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Interaction between transverse cracks and delamination during damage progress in CFRP cross-ply laminates
Composites
ELSEVIER
Science and Technology 54 (1995) 395-404 0 1995 Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0266-3538/95/$09.50
0266-3538(95)00084-4
INTERACTION DELAMINATION
BETWEEN TRANSVERSE CRACKS AND DURING DAMAGE PROGRESS IN CFRP CROSS-PLY LAMINATES
Shinji Ogihara”
& Nobuo
Takedab
“Department of Mechanical Engineering, bResearch Center for Advanced
Faculty of Science and Technology, Science University of Tokyo, 2641 Yamazaki, Noda, Chiba 278, Japan Science and Technology, The University of Tokyo, 4-6-l Komaba, Meguro-ku, Tokyo 153, Japan
(Received 21 July 1994; revised version received 19 December 1994; accepted 19 May 1995) Abstract In previous papers the microscopic failure process of (0/9OJO) (n =4, 8,12) cross-ply laminates was investigated. Progressive damage parameters, such as the transverse crack density and the delamination ratio, were measured. A simple modified shear-lag analysis including the thermal residual strains was conducted to predict the transverse crack density and the delamination length. The analysis did not consider the interaction between the transverse cracks and the delamination. In the present paper, a prediction is presented for the transverse crack density including the effect of delamination growth. The prediction shows better agreement with the experimental results, especially for laminates with thicker 90”plies in which extensive delamination occurs. Loading/unloading tests have also been performed to obtain the Young’s modulus reduction and the permanent strain as functions of the damage state. The shear-lag predictions of the YoungS modulus reduction and the permanent strain are compared with the experimental data. Better agreement is obtained when the interaction between transverse cracks and delamination is considered.
for light-weight structures, laminates are usually designed not to be loaded over a critical load which corresponds to the first-ply-failure (FPF) strain. The FPF strain is much smaller than the laminate ultimate strain in most cases, which means that the advantages of composites (high specific modulus and high specific strength) are not fully utilized. The design concept of FPF has been widely used, mainly because the growth behavior of microscopic damage is not fully understood, in order to establish a quantitative model. For more extensive use of CFRP for primary structures, damage-tolerant design has to be introduced. Damage-tolerant design will only be possible when a methodology for the prediction of the onset and growth of microscopic damage and the appropriate selection and execution of non-destructive inspection are available. The objective of the present study is to establish a methodology for predicting the growth behavior of microscopic damage in laminated composites as a basis for damage-tolerant design. Among laminated composites, cross-ply laminates have been studied extensively because they are basic laminate configurations. Previous studies on the failure process of cross-ply laminates have focused on the experimental measurement of the transverse crack density as a function of applied load or on the theoretical prediction of the onset of transverse cracking and its accumulation.‘-25 However, transverse cracks often induce interlaminar delaminations which relieve the local strain concentration at the transverse crack tips, but which cause other problems in the load-carrying plies. Only a few studies have been conducted on the initiation of delaminations from the tips of transverse cracks. O’Brien26 developed a simple analytical method to calculate the energy release rate associated with delamination growth from a transverse crack. In the analysis, the 90” plies were assumed to carry no
Keywords: cross-ply laminates, delamination, microscopic failure process, temperature effect, transverse crack, energy release rate, Young’s modulus reduction, permanent strain
1 INTRODUCTION
Fiber-reinforced plastics are usually used in the form of multidirectional laminates. Two unique types of damage in laminated composites are transverse cracks and delaminations. In current applications of CFRP 395
396
S. Ogihara,
load in the delamination regions. In the regions without delamination, the stresses in all plies were assumed to be the same as those in the undamaged laminate. The resulting energy release rate was independent of the delamination length because the interaction between the transverse cracks was not considered in the analysis. Dharani and Tangz7 conducted a consistent shear-lag analysis for both transverse cracking and delaminations from the tips of transverse cracks. They predicted the failure by using the numerical stress calculations and the point-stress failure criterion. As in the case of O’Brien’s analysis, however, analyses of Dharani and Tang were limited to delamination growth from a transverse crack in a laminate of infinite length. Wang et al.’ conducted a three-dimensional finite element stress analysis to calculate the energy release rate associated with delamination growth. Free-edge delamination, delamination near a freeedge/transverse-crack corner point, and delaminations at the cross cracks were studied. Salpekar and O’Brien28 investigated the effect of the free edge on the growth of delaminations initiating from a transverse crack by using a three-dimensional finite element analysis. The analysis indicated that high tensile interlaminar normal stresses were present at the intersection of the transverse crack with the free edge. Kim et a1.29 examined the stress redistribution due to the delamination from the transverse cracks based on the Stroh formulation for anisotropic elastic materials and on the method of eigenfunction expansion. The fracture mechanics parameters, the stress intensity factors and the energy release rates were calculated for a discussion of the stability of crack growth. Nairn and Hu3’ extended a variational analysis of transverse cracking in cross-ply laminates to account for the presence of delaminations initiating from the tips of transverse cracks in a 90” ply. They discussed competition between the transverse cracking and the delamination failure modes. In the above analyses, however, comparisons of predictions of delamination growth with experimental results were not conducted because of the lack of experimental data. Furthermore, no prediction has been made for transverse cracking and delamination growth where there is interaction between them, except for the discussion on the competition between transverse cracking and delamination.‘” Takeda and 0gihara31,“2 investigated the microscopic failure process of three different types of cross-ply laminates, (O/90,/0) (n = 4,8,12), at room temperature and at 80°C. Progressive damage parameters, such as the transverse crack density and the delamination ratio were measured. Experimental results similar to those of Crossman and Wang’ were obtained for the dependence of transverse cracking
N. Takeda and delamination onset strain on ply thickness. A simple modified shear-lag analysis, including residual conducted to obtain the thermal strains, was transverse crack density as a function of laminate strain, considering the constraint effect as well as the strength distribution of the transverse layer. The analysis was also extended to a system containing delamination to predict the delamination length. The transverse crack density and the delamination length were predicted separately. In the present paper, the prediction was extended for transverse crack density including the effect of delamination growth. The prediction showed better agreement with experimental results, especially for laminates with thicker 90” plies in which extensive delamination occurs. Loading/unloading tests were also performed to obtain the Young’s modulus reduction and the permanent strain as functions of the damage state. The shear-lag predictions of the Young’s modulus reduction and the permanent strain were compared with the experimental data. 2 EXPERIMENTAL 2.1 Materials Three different cross-ply stacking sequences, (O/90,/0), (O/90,/0), and (0/90,,/0), were used for both toughened TSOOH/3631 and conventional T300/3601 composite systems supplied by Toray Inc. The specimen size was 100 mm long and 3 mm wide. The narrow width was chosen so that in situ observation was possible under an optical microscope or a scanning acoustic microscope.” The thickness of each ply was approximately 0.132 mm. The fiber volume fraction was about 0.6. Both T300 and TSOOH are high-strength carbon fibers, but the latter has a higher modulus, strength, and failure strain. The resin system, 3601, is a TGDDM/DDS epoxy system, whereas 3631 is a modified epoxy system with improved toughness. GRP tabs were glued on the specimens, which resulted in a specimen gage length of 30 mm. All specimens were stored in a desiccator to make the tests under ‘dry’ conditions with the water content less than 1 wt%. 2.2 Tensile tests and observation of the failure process Tensile tests were performed at room temperature (20°C) and at 80°C. In situ microscopic observation of the failure process was conducted with both a scanning acoustic microscope (SAM, Olympus UH-3) and a scanning electron microscope (SEM, Hitachi S-2150) with ‘instrumented tensile loading devices’ on their stages.“” Furthermore, the replica technique was also used to support the above results. During tensile tests, the polished edge surface of a specimen was replicated on cellulose acetate film with methyl
Interaction between transverse cracks and delamination
acetate as a solvent. The most important advantage of this technique is that the damage state of a large area at each load level can be preserved after a test. Three specimens were used for the in situ observation of the failure process under each condition. The distributions of the transverse tensile strength of unidirectional laminates of each material system at room temperature and 80°C were also obtained to explain the failure properties of the 90” ply in cross-ply laminates. The details of the experiments are presented elsewhere.31,32 2.3 Loading/unloading Young’s
modulus
test-measurement of reduction and permanent strain
In addition to the above quantitative measurement of progressive damage by the observation of the failure process, the loading/unloading tests were also performed to obtain the Young’s modulus reduction and the permanent strain (the residual strain at the unloaded point) as functions of the damage state. The loading/unloading procedure cycles were repeated several times with continuous recording of the stress/strain curves. The specimen edge was replicated at each perfectly unloaded point. The Young’s modulus reduction and the permanent strain as functions of damage state were obtained from the combination of replica observation and measurement of stress/strain curves. 2.4 End notched flexure test (ENF)-measurement of mode II interlaminar
fracture toughness
ENF tests were performed at room temperature and 80°C to obtain the mode II interlaminar fracture toughness in order to understand the delamination growth behavior. The same material system, the toughened CFRP (T800H/3631), and conventional CFRP (T300/3601), were used in the experiment, but the laminate configuration was unidirectional (O,,). The detailed experimental procedure is given elsewhere.32
I
alarl:al:L c-wo.0
0.5
1.0
(a) (0/9Odo)
laminate
0.5
1.0
(c) (0/9O,fl)
Fig. 2. Transverse laminate
crack density strain for T300/3601
as a function of applied cross-ply laminates.31
3 RESULTS 3.1 Tensile tests and in situ observation failure process
of the
Figures 1 and 231 show experimental data for the transverse crack density variation as a function of applied laminate strain for the TSOOH/3631 and the T300/3601 laminates, respectively. The transverse crack density is defined as the number of cracks per unit specimen length. In the T800H/3631 system, the crack density decreases and the first cracking strain increases at 80°C. On the other hand, in the T300/3601 system, the crack density is relatively high at 80°C but the temperature dependence of the first cracking strain is small. In the T300/3601 (0/90,,/0) laminate, the first cracking strain and the ultimate laminate strain are comparable and progressive crack multiplication is not observed. In both systems, the first cracking strain is about the same as or larger than the 90” UD ultimate strain. This is due to the constraint effect.3 At a higher strain level, delaminations at the tips of the transverse cracks are observed in the T800H/3631 system. Delamination grows mainly as a consequence of fiber/matrix interfacial debonding. In this system,
20
0
8
0.0
0.5
1.0
1.5
(a) (O/904)
~xperimcntal Results:*R.T.0 WC
Fig. 1. Transverse
(4.0
40
@) (O/90$0)
Present Probabilistic Approach: without dctandnadoneffect including delaminationeffect -
@) “0.0 0.5 1.0 Laminate Strain (%) @) (O/90$0)
~xperimsntat ttcsu~ts l R.T. 0 BO’C Probabilistic Resent Approach;- R.T..... 8o’C
Applied Laminate Str,ain (a) (0/90&J)
397
R,T. . . 80’~ R.T.m-e8o’C
crack density as a function of applied strain for TSOOH/3631 cross-ply laminates.
0.0 0.5 1.0 1.5 0.0 Applied Laminate Strain (%) @) (O/908/0)
Experimental Results: l R.T. 0 80°C Theoretical Prediction: using (EF)“‘+‘-RT. wing (E~UT -R.T.
Fig. 3. Delamination
0.5
1.0
1.5
(c) (O/9012/0)
...... &?o’C ..---. 8O’C
ratio as a function of applied laminate strain for TSOOH/3631 cross-ply laminates.32
S. Ogihara, N. Takeda
398
delamination grows gradually and to a large extent. In the T300/3601 system, by contrast, delamination was not observed at room temperature, but very small delamination occurred at 80°C at the strain close to which was not measured the ultimate failure, quantitatively. ratio as a Figure 332 shows the delamination function of the applied laminate strain for the ratio was T800H/3631 system. The delamination defined as the ratio of the sum of the delamination length in a specimen to the total O/90 interlaminar length. The temperature dependence of delamination is similar to that of transverse cracking. That is, the delamination ratio decreases and the delamination onset strain increases at 80°C. Furthermore, in laminates with a thicker 90“ ply the delamination grows rapidly and extensively. For the (0/90,,/0) laminate, most specimens showed complete delamination at the ultimate failure. 3.2 Young’s modulus reduction and permanent strain Figures 4 and 5 show the Young’s modulus reduction as a function of transverse crack density obtained by the combination of loading/unloading tests and replica observations for the T800H/3631 laminates at room temperature and at 8O”C, respectively. The Young’s
h 0
5 10 Transverse Crack Density
15
@) (O/90$0)
~~~~~
0
5 10 Transverse crack deosity(/cm)
If
(4 (0~90,~O)
Fig. 4. Young’s modulus reduction as a function of transverse crack density for T800H/3631 cross-ply laminates at room temperature.
Tj 0.80 7@ 0.75 sj
no delamination alL=O.l ---_ with dslamination
0.70 0
5 Transverse Crack Density (/cm)
10
(a) (0/9OdO)
Transverse Crack Density (/cm) @) (OfwO)
z
0
5
10
Transverse Crack Density (/cm) (c) (O/90,2/0)
Fig. 5. Young’s transverse
modulus reduction crack density for T800H/3631 at 80°C.
as a function of cross-ply laminates
modulus reduction is larger in the laminates with thicker 90” plies at the same transverse crack density. It should be noted that the normalized Young’s modulus shown here is affected by the delamination. Figures 6 and 7 show the permanent strain as a function of the transverse crack density for T800H/3631 laminates at room temperature and at 8O”C, respectively. The permanent strain is larger in the laminates with thicker 90” plies at the same transverse crack density. The Young’s modulus reduction and the permanent strain for T300/3601 system were not measured because the strain for onset of transverse cracking is comparable to the laminate ultimate strain. 3.3 Mode II interlaminar fracture toughness The load/displacement curves obtained by the ENF tests showed slight nonlinearity.32 In the present study, crack initiation was defined as the onset of nonlinearity in the load/displacement curves.34 The critical energy release rate for crack initiation, G1,,, can be calculated from the load at the onset of nonlinearity, the initial crack length, and the initial compliance. The region from the load at the onset of nonlinearity up to the maximum load was assumed to be the subcritical crack growth region. In order to examine the fracture resistance, GIIR, in the subcritical
Interaction between transverse cracks and delamination
399
Table 1. Mode II interlaminar fracture toughness, and Gn_ (J/m’) Material
0
10 Transverse Crack Density (/cm) 5
T800H/3631 T300/3601
-o’2o l Experiment ap Rsdictbm -
RT = room
5 10 Transverse Crack Density (/cm)
15
(4 (0~00)
Fig. 6. Permanent strain as a function of transverse density for T800H/3631 cross-ply laminates at temperature.
I
’
0
.*
crack room
I
Fxpcrimcnt
Rcdicttom no delamination ---- with delamination olL=O.3
$ 0.05 B %
@ g 0.
G IImax
RT
80°C
RT
80°C
398 3.50
345 219
613 549
578 429
temperature.
no delamination
0
rp,
G nc
15
(a) (OHdo)
.=0.15 z
system
GrI,
crack growth region, the crack length was determined following a previous method.34 The critical energy release rates for crack initiation evaluated by the load at the onset of nonlinearity at each temperature, Gn,, are shown in Table 1. The values evaluated at the maximum load, GrImax,are also shown. The values are the average for three specimens. For the T800H/3631 system, Gnc at 80°C is slightly lower than at room temperature. For the T300/3601 system, Gnc at 80°C is much lower than at room temperature, as a consequence of the decrease in the fiber/matrix interfacial strength at higher temperature.31 Figure 8 shows the mode II interlaminar fracture toughness in the subcritical crack growth region, GnR, as a function of the crack extension, Aa. The crack extension, Aa, was not measured but calculated.34 The GnR value increases as Aa increases at both room temperature and 80°C which should be called R-curve behavior. In the present study, the R curves are fitted to a relationship of the form:
I 0
GIIR= Go + k%“%
(1)
5
Transverse Crack Density(/cm) (a) (OPOdO)
where Go and k are constants, determined by the least-squares method and their values are summarized in Table 2. In Fig. 8 the solid and dotted lines are the fitted curves for the data at room temperature and at 80°C respectively. The Gun curves will be used to predict delamination behavior in the following discussion.
Transverse Crack Density(/cm) @) (O~%w T8ooH/3631 700,
0 Transverse Crack D&sity(/cm) (4 (O~OIfl)
Fig. 7. Permanent strain as a function of transverse crack density for T800H/3631 cross-ply laminates at 80°C.
1
0.0 0.5 1.0 1.5 2.0 2.5 3.0 Crack Extension, da (mm)
l3CW3601 7001
I
0.0 0.5 1.0 1.5 2.0 2.5 3.0 Crack Extension, dn (mm)
Fig. 8. Mode II interlaminar fracture toughness subcritical crack growth region, GnR, as a function extension, Au.
in the of crack
S. Ogihara,
400 Table 2. G,, and k in eqn (1)
Material system
G,, (J/m’)
T800H/3631 T300/3601
k (J/m3”)
RT
80°C
RT
80°C
394 339
357 206
5850 5470
4510 4650
N. Takeda
which the interlaminar shear layers are of thickness d “. The Young’s modulus reduction and the permanent strain due to transverse cracking can be derived from the obtained stress and strain distributions.“‘,32 The average axial strain of the cross-ply laminate with transverse cracks is given by:
RT = room temperature. 4 THEORETICAL 4.1 Modified
= E + 2dc2oP ptanh 0
BACKGROUND
shear-lag
r ( 2~ 1
bEi
analysis
Shear-lag analysis was first introduced for cross-ply laminates by Garrett and Bailey,l and a modified shear-lag analysis considering the interlaminar shear layer was conducted by Fukunaga et aZ.9 The boundary condition of the method was modified by Lim and Hong17,‘8 to satisfy the conditions for arbitrary crack spacings. In this analysis, the stress state in a cross-ply laminate is calculated by following their method. For simplicity, the method is reduced to one dimension.31,32 The following theoretical treatment is similar to that given previously,31,32 but is described here for completeness. Consider a cross-ply laminate subjected to mechanical and thermal loading, as shown in Fig. 9(a), in
where L is the half crack spacing, E, is the axial strain of the 0” ply, ET is the thermal residual stain in the 0 PlY> E,, is the undamaged laminate strain, b is the thickness of the 0” ply, d is the half thickness of the 90” ply, azo is the 90” ply stress in the undamaged laminate, p is the associated transverse crack density, E, is the Young’s modulus of the 0” ply, and
e=&J&+&J where E, is the Young’s modulus of the 90” ply, and G is the shear modulus of the interlaminar shear layer. By putting co = 0 in eqn (2), the permanent strain (the residual strain at the perfectly unloaded point), lp, can be expressed as: dE2& E = -tanh ’ bE,tL
(5L)
(4)
b do
where l 2T 1s ’ the thermal residual strain in the 90” ply. For a constant applied stress, a,,, the stress/strain relationship of the undamaged cross-ply laminate can be written as:
2d do
b
u. = EOeO
\ Transverse Cracks Interlaminar Shear Layer z n
L 245-u 245
E(< - Q,) = cr,,
b
0” El
do
2d*
90°E2 00, co
(5)
where E, is the Young’s modulus of the undamaged laminate. The stress/strain relationship of the laminate with transverse cracks is:
(a) t
(2)
d0
ao*to
(6)
where E is the Young’s modulus of the laminate with transverse cracks. From eqns (2)-(6), the normalized Young’s modulus reduction due to transverse cracking is:
b
0” _Delamination
Fig. 9. Analytical models of cross-ply laminates: (a) with only transverse cracks; (b) with both transverse cracks and delamination.
4.2 Extension of modified shear-lag analysis In this section, we consider the state where a delamination with a length 2a has developed at each
Interaction between transverse cracks and delamination
tip of a transverse crack (the crack spacing is 2L) as shown in Fig. 9(b). Assuming that there is no friction between the 0” and the 90” layers in the delaminated region, all stress components in the 90” ply and the interlaminar shear layer are zero in the region. The stress components in each layer were obtained previously.31~32 The Young’s modulus reduction and the permanent strain due to transverse cracking and delamination at the tips of transverse cracks can be derived by the same procedure as in Section 4.1. The axial strain of a laminate with delamination at the tips of the transverse cracks is: El&
-
ET=
+-----tanh 2dazop b& 5
+ 2&w ( bE, 1 FE2 2d&ap
Eg
-
$1
1 +
2ap)
By putting e0 = 0 in eqn (S), the permanent including the effect, can be expressed as: EP =Fp[a
T
~
+ $anh&l
- ap)}]
strain,
(9)
Substituting eqns (8) and (9) into eqn (7), the normalized Young’s modulus reduction due to transverse cracking and delamination at the tips of transverse cracks is obtained. The energy release rate associated with delamination growth, Gd, can also derived by using the obtained stress distribution31,32 as: G =
dE2(b&
dE,)(%
+
d
[
4 [exp&(L - a) + exp - 5(L - a)]’ I
4.3 Probabilistic
where L, and V, denote the element length and volume, respectively, CY,and p are the Weibull scale and shape parameters, respectively, and E, is the constraint strain defined in Ref. 31.
4.4 Prediction of delamination growth Assuming that the critical energy release rate for delamination growth is equal to the mode II interlaminar fracture toughness, GuR, obtained from ENF tests, the delamination length in the cross-ply laminates can be predicted as a function of applied laminate strain.3z In reality, the mode I component of energy release rate associated with the delamination growth is not zero and is very large near the free edge and when delamination is small. It is not easy, however, to determine the crack growth criterion under mixed-mode conditions, and a mode II failure criterion was therefore assumed for a simple delamination growth prediction. Letting Gd in eqn (10) be equal to GIIR in eqn (l), and rewriting Aa in eqn (1) as a, we have the following equation: dEz(Elb
+ E,d)(e,
+ E:)~
2E, b
x
lL
{exp{&-(1 -a)]
+tp(
prediction
of transverse
(lo)
crack
density
A novel model predicting the relationship between the applied laminate strain and the transverse crack density by making use of the statistical parameters of an experimentally-obtained 90” UD strength distribution has been presented previously.31 Following earlier studies,14-‘6 the 90” ply in the cross-ply laminate is divided into elements which are assumed to fail only once up to the ultimate failure. The strength distribution of the elements is assumed to obey the same two-parameter Weibull distribution as measured for the transverse strength of UD. Then the failure probability of the elements under a stress is known and the number of cracks can be calculated by multiplying by the number of elements:31 -V~(E2(‘o+-~‘~‘c))p]]
(11)
-a)lr] (12)
By solving this equation with respect to a, the total delamination length can be obtained at a given laminate strain and the corresponding transverse crack density.
4.5 Prediction of interaction cracks and delamination
between
transverse
In the discussion above, the interaction between transverse cracks and delamination has not been considered. In this section, a prediction is presented by considering this interaction. Before the onset of delamination (e. < e&i, where lo is the laminate strain and E&i is the delamination onset strain), only the transverse cracks are considered. Therefore, the transverse crack density, p, and the delamination ratio, D, are: p=2[
-exp(
-VC(E”‘o~:‘-rc)~}] D=O
p=+J-exp(
-&(l
=G,+k&
+ ‘:)’
2bE,
X l-
401
The delamination
(13) (14)
onset strain, E&i, can be obtained by
S. Ogihara,
402
N. Takeda Table 3. Values of G/d,,
solving the following equation with respect to eO: d&(E,
b + E,d)(q,
Material system
+ 6;)’
Temperature
9.07 x 7.64 x 1.04 x 8.71 x
RT
T800H/3631
2Elb
G/d, (Pa/m)
80°C
T300/3601
RT 80°C
lOI
10” lOI 1o13
RT = room temperature. By letting the transverse crack density after the onset of delamination be p’, the rate of p’, dp’/deO, can be assumed to be reduced by the proportion of the delamination ratio, D, i.e.
interaction between transverse tion are taken into account. 5.2 Delamination
!g=$-D) 0 The delamination
(16)
ratio, D, is:
dEz(EI b + E,d)(c,
+ 6;)’
2E,b 4
(exp{ &(I-
D)} + exp{ - $0
- D)]}*
= G,+ k&
1 (17)
The transverse crack density, p’, and the delamination ratio, D, after the delamination onset can be obtained by solving eqns (13) (16), and (17), as functions of the laminate strain, eO. By substituting the above results into eqns (7) and (9), the Young’s modulus reduction and the permanent strain can be obtained considering the damage growth. 5 DISCUSSION 5.1 Transverse
cracking behavior
Figures 1 and 2 show a comparison of the predictions of the transverse crack density and experimental results for the TSOOH/3631 and T300/3601 composite systems. The present probabilistic approach including delamination effect is shown for the T800H/3631, but is not shown for the T300/3601, because delamination did not occur in the latter system. The values of G/do are shown in Table 3. The value of G/L&, is a parameter that is difficult to determine experimentally, but the values used are similar to those which are often adopted.‘&r* As shown in the figures, for the T800H/3631 system the delamination effect is larger in the laminates with thicker 90” plies, which can be predicted by the present approach including the delamination effect. In summary, it can be concluded that the transverse crack behavior can be well predicted when the
cracks and delamina-
behavior
In the first place, the occurrence of delamination is discussed. The delamination onset strain, E+,, can be predicted from eqn (15). Comparing the delamination onset Strain, Ed& and the laminate ultimate strain, ls, the occurrence of delamination should be judged. That is, if lde1< eg delamination occurs before the laminate finally breaks, and if l&r > lu, delamination will not be observed in the failure process. The predicted delamination onset strain, (ede,)Pred, and laminate ultimate strain obtained by experiments are shown in Table 4. For the T800H/3631, (e&_i)P’edis smaller than lu only except for (O/90,/0) at 8o”C, which is consistent with the experimentally-observed delamination growth. For the T300/3601, (Edei)P’edis smaller than lu for (O/90,/0) at 80°C and for and therefore delamination may be (0/9OJO), observed in these laminates. But since the onset strain of transverse cracking for these laminates is comparable to the ultimate strain, no delamination was experimentally observed for all laminate configurations of the T300/3601 system. Fig. 3 shows a comparison of the predictions of delamination ratio with the experimental results. The predictions are made for both (.$jPSM, which was obtained by the ply separation method31*32 and (e;)CLT, which was calculated from the classical laminate theory with the catalog data for thermal expansion coefficients. The predictions in Fig. 3 show very good agreement with the experimental results when (eT)PSM 1s . used, which confirms the validity of the present prediction and the ply separation method. The effect of the interaction between transverse cracks and delamination was found to be negligible in the prediction of the delamination growth. 5.3 Prediction permanent
of Young’s strain
modulus
reduction
and
First, the prediction of the Young’s modulus reduction is compared with the experimental results. From eqns (2) (4), and (7), the normalized Young’s modulus can be predicted for cross-ply laminates only with
Interaction between transverse cracks and delamination Table 4. Predicted
delamination
onset
strain, (E,,~,)~~‘, and strain, lB
laminate
403 ultimate
RT
T800H/3631 Go=394 (N/m) Go = 357 (N/m) T300/3601 G,, = 339 (N/m) Go = 206 (N/m)
(RT) (SO’C) (RT) (80%)
(O/90,/0) (O/90,/0) (0/90,,/0) (O/90,/0) (O/90,/0) (0/90,,/0)
1.39 0.870 0.646 1.51 0.952 0.706
failure
80°C
1.46 1.42 1.36 0.890 0.860 0.915
1.55 0.995 0.757 1.36 0%30 0.666
1.52 1.49 1.25 0.995 0.970 1.01
RT = room temperature.
transverse cracks. For laminates with delamination from the tips of transverse cracks, eqns (8) and (9) should be used instead of eqns (2) and (4). In addition, use of the prediction in Section 4.6 enables a prediction of the Young’s modulus reduction considering damage growth to be made. Figures 4 and 5 show comparisons of the predictions with the experimental data at room temperature and at 80°C. The thermal residual strains are also necessary for the calculations. It is found that the influence of thermal residual strains is small in the present calculation, so the thermal residual strains obtained by the ply separation method31T32 are used for the predictions shown here. for the laminates with only The predictions transverse cracks seem to be accurate for (O/90,/0) at lower crack densities. But for (O/90,/0) at higher crack densities and for (O/90,/0), prediction for the laminates with only transverse cracks tends to be inadequate and a consideration of delamination proves to be necessary. For (O/9O,2/O) laminates, the modified shear-lag analysis for the laminate with only transverse cracks is no longer a useful tool for predicting the Young’s modulus reduction. For the laminates in which the delamination grows extenof the shear-lag analysis sively, the extension described in the present study has proved to be important for understanding the degradation of material properties in the failure process. Furthermore, the predictions considering damage growth showed excellent agreement with the experimental results. Next, the prediction of the permanent strain is compared with the experimental results. Figures 6 and 7 show comparisons of the predictions with the experimental data at room temperature and at So”C, respectively. The value of the thermal residual strain has a large influence on the predicted value of the permanent strain. The prediction shown here is obtained by using (e2T)PSM.Results were similar to the Young’s modulus reduction, i.e. analysis including delamination effects was necessary for the laminates
with thicker 90” plies, and the prediction considering damage growth showed good agreement with the experimental results. 6 CONCLUSIONS A damage progression model of cross-ply laminates has been developed on the basis of a modified shear-lag analysis and the probabilistic strength of 90” between transverse layers, considering interaction cracks and delaminations. The transverse crack density and the delamination ratio were obtained as functions of the laminate strain. The predictions showed excellent agreement with experimental results when the interaction was properly taken into account. Similarly, the modified shear-lag prediction of the Young’s modulus reduction and the permanent strain considering the interaction of transverse cracks and delamination were compared with the experimental results. It was found to be important to understand the interaction of transverse cracks and delamination in order to predict the Young’s modulus reduction and the permanent strain induced by the damage development.
ACKNOWLEDGEMENTS
The present study was supported financially by a Grant-in-Aid from the Ministry of Education, Japan. The authors thank sincerely Professor Akira Kobayashi, Science University of Tokyo, and Professor Tadashi Shioya, The University of Tokyo, for fruitful discussions throughout the study.
REFERENCES 1.
Garrett, K. W. & Bailey, J. E., Multiple transverse failure in 90” cross-ply laminate of a glass fiberreinforced polyester. J. Mater. Sci., 12 (1977) 157-68.
S. Ogihara, 2. Parvizi, A. & Bailey, J. E., On multiple transverse cracking in glass fiber epoxy cross-ply laminates. J. Mater. Sci., 13 (1978) 2131-6. 3. Bailey, J. E., Curtis, P. T. & Parvizi, A., On the transverse cracking and longitudinal splitting behavior of glass and carbon fiber reinforced epoxy cross-ply laminates and the effect of Poisson and thermally generated strain. Proc. R. Sot. Lend., A366 (1979) 599-623. 4. Flaggs, D. L. & Kural, M. H., Experimental determination of the in situ transverse lamina strength in graphite/epoxy laminates. J. Camp. Mater., 16 (1982) 103-16. 5. Crossman, F. W. & Wang, A. S. D., The dependence of transverse cracking and delamination on ply thickness in graphite/epoxy laminates. Damage in Composite Materials, ASTM STP 775, 1982, pp. 118-39. 6. Flaggs, D. L., Prediction of tensile matrix failure in composite laminates. J. Comp. Mater., 19 (1985) 29-51. 7. Wang, A. S. D., Chou, P. C. & Lei, S. C., A stochastic model for the growth of matrix cracks in composite laminates. J. Camp. Mater., 18 (1984) 339-56. 8. Wang, A. S. D., Kishore, N. N. & Li, C. A., Crack development in graphite-epoxy cross-ply laminates under uniaxial tension. Comp. Sci. Technol., 24 (1985) 1-31. 9. Fukunaga, H., Chou, T.-W., Peters, P. W. M. & Schulte, strength Probabilistic failure analyses of K., graphite/epoxy cross-ply laminates. J. Comp. Mater., 18 (1984) 339-56. 10. Laws, N. & Dvorak, G. J., Progressive transverse cracking in composite laminates. J. Camp. Mater., 22 (1988) 900-16. 11. Han, Y. M., Hahn, H. T. & Croman, R. B., A simplified analysis of transverse ply cracking in cross-ply laminates. Comp. Sci. Technol., 31 (1988) 165-77. 12. Lee, J. W. & Daniel, I. M., Progressive transverse cracking of crossply composite laminates. J. Comp. Mater., 24 (1990) 1225-43. 13. Nairn, J. A., The strain energy release rate of composite microcracking: a variational approach. J. Camp. Mater., 23 (1989) 1106-29. 14. Peters, P. W. M., The strength distribution of 90” plies in O/90/0 graphite epoxy laminates. J. Comp. Mater., 18 (1984) 545-56. 15. Peters, P. W. M., The influence of fiber, matrix and interface on transverse cracking in carbon fiber reinforced plastic cross-ply laminates. Composite Materials: Fatigue and Fracture, Vol. 2, ASTM STP 1012, 1989, pp. 103-17. 16. Peters, P. W. M. & Andersen, S. I., The influence of matrix fracture strain and interfacial strength on cross-ply cracking in CFRP in the temperature range of -100°C to +lOO”C. J. Comp. Mater., 23 (1989) 944-60. 17. Lim, S. G. & Hong, C. S., Prediction of transverse cracking and stiffness reduction in cross-ply laminated composites. J. Comp. Mater., 25 (1989) 695-713. 18. Lim, S. G. & Hong, C. S., Effect of transverse cracks on the thermomechanical properties of cross-ply laminated
N. Takeda composites. Comp. Sci. Technol., 34 (1989) 145-62. 19. Talreja, R., Yalvac, S., Yats, L. D. & Wetters, D. G., Transverse cracking and stiffness reduction in cross ply laminates of different matrix toughness. J. Comp. Mater., 26 (1992) 1644-63. 20. Varna, J. & Berglund, L., Multiple transverse cracking and stiffness reduction in cross-ply laminates. J. Comp. Tech. Rex, 13 (1991) 97-106. 21. Berglund, L. A., Varna, J. & Yuan, J., Transverse cracking and local delamination in [0,/90,], and [90,/04], carbon fiber/toughened epoxy laminates. J. Reinj Plast. Comp., 11 (1992) 643-60. 22. Varna, J. & Berglund, L. A., A model for prediction of the transverse cracking strain in cross-ply laminates. J. Reinf Plast. Comp., 11 (1992) 708-28. 23. Liu, S. & Nairn, J. A., The formation and propagation of matrix microcracks in cross-ply laminates during static loading. J. Reinf. Plast. Comp., 11 (1992)158-78. 24. Nairn, J. A., Hu, S. & Bark, J. S., A critical evaluation of theories for predicting microcracking in composite materials. J. Mater. Sci., 28 (1993) 5099-111. 25. Nairn, J. A. & Hu, S., Matrix microcracking. In Damage Mechanics of Composite Materials, ed. R. Talreja. Elsevier Science, 1994. 26. O’Brien, T. K., Analysis of local delaminations and their influence on composite laminate behavior. Delamination and Debonding of Materials, ASTM STP 876, ed. W. S. Johnson. American Society for Testing and Materials, Philadelphia, PA, 1985, pp. 282-97. 27. Dharani, L. R. & Tang, H., Micromechanics characterization of sublaminate damage. Znt. J. Fract., 46 (1990) 123-40. 28. Salpekar, S. A. & O’Brien, T. K., Combined effect of matrix cracking and stress-free edge on delamination. NASA TM 102591, 1990. 29. Kim, T. W., Kim, Y. J. & Im, S., Delamination cracks originating from transverse cracking in cross-ply laminates under various loadings. Proc. ZCCM/8, 28-H, 1991. 30. Nairn, J. A. & Hu, S., The initiation and growth of delaminations induced by matrix microcracks in laminated composites. Znt. J. Fract., 57 (1992) l-24. 31. Takeda, N. & Ogihara, S., In situ observation and probabilistic prediction of microscopic failure process in CFRP cross-ply laminates. Comp. Sci. Technol., 52 (1994) 183-95. 32. Takeda, N. & Ogihara, S., Initiation and growth of delamination from the tips of transverse cracks in CFRP cross-ply laminates. Comp. Sci. Technol., 52 (1994) 309-l 8. 33. Takeda, N., Miyasaka, C. & Nakata, K., In-situ scanning acoustic microscopy for damage progress characterization in thermoplastic composites. Nondestr. Test. Eval., 8-9 (1992) 813-24. 34. Chou, I., Kimpara, I., Kageyama, K. & Osawa, I., Effect of moisture absorption on mode I and mode II interlaminar fracture toughness of CFRP laminates. J. Jap. Sot. Comp. Mater., 19 (1993) 46-54 (in Japanese).
ELSEVIER
Science and Technology 54 (1995) 395-404 0 1995 Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0266-3538/95/$09.50
0266-3538(95)00084-4
INTERACTION DELAMINATION
BETWEEN TRANSVERSE CRACKS AND DURING DAMAGE PROGRESS IN CFRP CROSS-PLY LAMINATES
Shinji Ogihara”
& Nobuo
Takedab
“Department of Mechanical Engineering, bResearch Center for Advanced
Faculty of Science and Technology, Science University of Tokyo, 2641 Yamazaki, Noda, Chiba 278, Japan Science and Technology, The University of Tokyo, 4-6-l Komaba, Meguro-ku, Tokyo 153, Japan
(Received 21 July 1994; revised version received 19 December 1994; accepted 19 May 1995) Abstract In previous papers the microscopic failure process of (0/9OJO) (n =4, 8,12) cross-ply laminates was investigated. Progressive damage parameters, such as the transverse crack density and the delamination ratio, were measured. A simple modified shear-lag analysis including the thermal residual strains was conducted to predict the transverse crack density and the delamination length. The analysis did not consider the interaction between the transverse cracks and the delamination. In the present paper, a prediction is presented for the transverse crack density including the effect of delamination growth. The prediction shows better agreement with the experimental results, especially for laminates with thicker 90”plies in which extensive delamination occurs. Loading/unloading tests have also been performed to obtain the Young’s modulus reduction and the permanent strain as functions of the damage state. The shear-lag predictions of the YoungS modulus reduction and the permanent strain are compared with the experimental data. Better agreement is obtained when the interaction between transverse cracks and delamination is considered.
for light-weight structures, laminates are usually designed not to be loaded over a critical load which corresponds to the first-ply-failure (FPF) strain. The FPF strain is much smaller than the laminate ultimate strain in most cases, which means that the advantages of composites (high specific modulus and high specific strength) are not fully utilized. The design concept of FPF has been widely used, mainly because the growth behavior of microscopic damage is not fully understood, in order to establish a quantitative model. For more extensive use of CFRP for primary structures, damage-tolerant design has to be introduced. Damage-tolerant design will only be possible when a methodology for the prediction of the onset and growth of microscopic damage and the appropriate selection and execution of non-destructive inspection are available. The objective of the present study is to establish a methodology for predicting the growth behavior of microscopic damage in laminated composites as a basis for damage-tolerant design. Among laminated composites, cross-ply laminates have been studied extensively because they are basic laminate configurations. Previous studies on the failure process of cross-ply laminates have focused on the experimental measurement of the transverse crack density as a function of applied load or on the theoretical prediction of the onset of transverse cracking and its accumulation.‘-25 However, transverse cracks often induce interlaminar delaminations which relieve the local strain concentration at the transverse crack tips, but which cause other problems in the load-carrying plies. Only a few studies have been conducted on the initiation of delaminations from the tips of transverse cracks. O’Brien26 developed a simple analytical method to calculate the energy release rate associated with delamination growth from a transverse crack. In the analysis, the 90” plies were assumed to carry no
Keywords: cross-ply laminates, delamination, microscopic failure process, temperature effect, transverse crack, energy release rate, Young’s modulus reduction, permanent strain
1 INTRODUCTION
Fiber-reinforced plastics are usually used in the form of multidirectional laminates. Two unique types of damage in laminated composites are transverse cracks and delaminations. In current applications of CFRP 395
396
S. Ogihara,
load in the delamination regions. In the regions without delamination, the stresses in all plies were assumed to be the same as those in the undamaged laminate. The resulting energy release rate was independent of the delamination length because the interaction between the transverse cracks was not considered in the analysis. Dharani and Tangz7 conducted a consistent shear-lag analysis for both transverse cracking and delaminations from the tips of transverse cracks. They predicted the failure by using the numerical stress calculations and the point-stress failure criterion. As in the case of O’Brien’s analysis, however, analyses of Dharani and Tang were limited to delamination growth from a transverse crack in a laminate of infinite length. Wang et al.’ conducted a three-dimensional finite element stress analysis to calculate the energy release rate associated with delamination growth. Free-edge delamination, delamination near a freeedge/transverse-crack corner point, and delaminations at the cross cracks were studied. Salpekar and O’Brien28 investigated the effect of the free edge on the growth of delaminations initiating from a transverse crack by using a three-dimensional finite element analysis. The analysis indicated that high tensile interlaminar normal stresses were present at the intersection of the transverse crack with the free edge. Kim et a1.29 examined the stress redistribution due to the delamination from the transverse cracks based on the Stroh formulation for anisotropic elastic materials and on the method of eigenfunction expansion. The fracture mechanics parameters, the stress intensity factors and the energy release rates were calculated for a discussion of the stability of crack growth. Nairn and Hu3’ extended a variational analysis of transverse cracking in cross-ply laminates to account for the presence of delaminations initiating from the tips of transverse cracks in a 90” ply. They discussed competition between the transverse cracking and the delamination failure modes. In the above analyses, however, comparisons of predictions of delamination growth with experimental results were not conducted because of the lack of experimental data. Furthermore, no prediction has been made for transverse cracking and delamination growth where there is interaction between them, except for the discussion on the competition between transverse cracking and delamination.‘” Takeda and 0gihara31,“2 investigated the microscopic failure process of three different types of cross-ply laminates, (O/90,/0) (n = 4,8,12), at room temperature and at 80°C. Progressive damage parameters, such as the transverse crack density and the delamination ratio were measured. Experimental results similar to those of Crossman and Wang’ were obtained for the dependence of transverse cracking
N. Takeda and delamination onset strain on ply thickness. A simple modified shear-lag analysis, including residual conducted to obtain the thermal strains, was transverse crack density as a function of laminate strain, considering the constraint effect as well as the strength distribution of the transverse layer. The analysis was also extended to a system containing delamination to predict the delamination length. The transverse crack density and the delamination length were predicted separately. In the present paper, the prediction was extended for transverse crack density including the effect of delamination growth. The prediction showed better agreement with experimental results, especially for laminates with thicker 90” plies in which extensive delamination occurs. Loading/unloading tests were also performed to obtain the Young’s modulus reduction and the permanent strain as functions of the damage state. The shear-lag predictions of the Young’s modulus reduction and the permanent strain were compared with the experimental data. 2 EXPERIMENTAL 2.1 Materials Three different cross-ply stacking sequences, (O/90,/0), (O/90,/0), and (0/90,,/0), were used for both toughened TSOOH/3631 and conventional T300/3601 composite systems supplied by Toray Inc. The specimen size was 100 mm long and 3 mm wide. The narrow width was chosen so that in situ observation was possible under an optical microscope or a scanning acoustic microscope.” The thickness of each ply was approximately 0.132 mm. The fiber volume fraction was about 0.6. Both T300 and TSOOH are high-strength carbon fibers, but the latter has a higher modulus, strength, and failure strain. The resin system, 3601, is a TGDDM/DDS epoxy system, whereas 3631 is a modified epoxy system with improved toughness. GRP tabs were glued on the specimens, which resulted in a specimen gage length of 30 mm. All specimens were stored in a desiccator to make the tests under ‘dry’ conditions with the water content less than 1 wt%. 2.2 Tensile tests and observation of the failure process Tensile tests were performed at room temperature (20°C) and at 80°C. In situ microscopic observation of the failure process was conducted with both a scanning acoustic microscope (SAM, Olympus UH-3) and a scanning electron microscope (SEM, Hitachi S-2150) with ‘instrumented tensile loading devices’ on their stages.“” Furthermore, the replica technique was also used to support the above results. During tensile tests, the polished edge surface of a specimen was replicated on cellulose acetate film with methyl
Interaction between transverse cracks and delamination
acetate as a solvent. The most important advantage of this technique is that the damage state of a large area at each load level can be preserved after a test. Three specimens were used for the in situ observation of the failure process under each condition. The distributions of the transverse tensile strength of unidirectional laminates of each material system at room temperature and 80°C were also obtained to explain the failure properties of the 90” ply in cross-ply laminates. The details of the experiments are presented elsewhere.31,32 2.3 Loading/unloading Young’s
modulus
test-measurement of reduction and permanent strain
In addition to the above quantitative measurement of progressive damage by the observation of the failure process, the loading/unloading tests were also performed to obtain the Young’s modulus reduction and the permanent strain (the residual strain at the unloaded point) as functions of the damage state. The loading/unloading procedure cycles were repeated several times with continuous recording of the stress/strain curves. The specimen edge was replicated at each perfectly unloaded point. The Young’s modulus reduction and the permanent strain as functions of damage state were obtained from the combination of replica observation and measurement of stress/strain curves. 2.4 End notched flexure test (ENF)-measurement of mode II interlaminar
fracture toughness
ENF tests were performed at room temperature and 80°C to obtain the mode II interlaminar fracture toughness in order to understand the delamination growth behavior. The same material system, the toughened CFRP (T800H/3631), and conventional CFRP (T300/3601), were used in the experiment, but the laminate configuration was unidirectional (O,,). The detailed experimental procedure is given elsewhere.32
I
alarl:al:L c-wo.0
0.5
1.0
(a) (0/9Odo)
laminate
0.5
1.0
(c) (0/9O,fl)
Fig. 2. Transverse laminate
crack density strain for T300/3601
as a function of applied cross-ply laminates.31
3 RESULTS 3.1 Tensile tests and in situ observation failure process
of the
Figures 1 and 231 show experimental data for the transverse crack density variation as a function of applied laminate strain for the TSOOH/3631 and the T300/3601 laminates, respectively. The transverse crack density is defined as the number of cracks per unit specimen length. In the T800H/3631 system, the crack density decreases and the first cracking strain increases at 80°C. On the other hand, in the T300/3601 system, the crack density is relatively high at 80°C but the temperature dependence of the first cracking strain is small. In the T300/3601 (0/90,,/0) laminate, the first cracking strain and the ultimate laminate strain are comparable and progressive crack multiplication is not observed. In both systems, the first cracking strain is about the same as or larger than the 90” UD ultimate strain. This is due to the constraint effect.3 At a higher strain level, delaminations at the tips of the transverse cracks are observed in the T800H/3631 system. Delamination grows mainly as a consequence of fiber/matrix interfacial debonding. In this system,
20
0
8
0.0
0.5
1.0
1.5
(a) (O/904)
~xperimcntal Results:*R.T.0 WC
Fig. 1. Transverse
(4.0
40
@) (O/90$0)
Present Probabilistic Approach: without dctandnadoneffect including delaminationeffect -
@) “0.0 0.5 1.0 Laminate Strain (%) @) (O/90$0)
~xperimsntat ttcsu~ts l R.T. 0 BO’C Probabilistic Resent Approach;- R.T..... 8o’C
Applied Laminate Str,ain (a) (0/90&J)
397
R,T. . . 80’~ R.T.m-e8o’C
crack density as a function of applied strain for TSOOH/3631 cross-ply laminates.
0.0 0.5 1.0 1.5 0.0 Applied Laminate Strain (%) @) (O/908/0)
Experimental Results: l R.T. 0 80°C Theoretical Prediction: using (EF)“‘+‘-RT. wing (E~UT -R.T.
Fig. 3. Delamination
0.5
1.0
1.5
(c) (O/9012/0)
...... &?o’C ..---. 8O’C
ratio as a function of applied laminate strain for TSOOH/3631 cross-ply laminates.32
S. Ogihara, N. Takeda
398
delamination grows gradually and to a large extent. In the T300/3601 system, by contrast, delamination was not observed at room temperature, but very small delamination occurred at 80°C at the strain close to which was not measured the ultimate failure, quantitatively. ratio as a Figure 332 shows the delamination function of the applied laminate strain for the ratio was T800H/3631 system. The delamination defined as the ratio of the sum of the delamination length in a specimen to the total O/90 interlaminar length. The temperature dependence of delamination is similar to that of transverse cracking. That is, the delamination ratio decreases and the delamination onset strain increases at 80°C. Furthermore, in laminates with a thicker 90“ ply the delamination grows rapidly and extensively. For the (0/90,,/0) laminate, most specimens showed complete delamination at the ultimate failure. 3.2 Young’s modulus reduction and permanent strain Figures 4 and 5 show the Young’s modulus reduction as a function of transverse crack density obtained by the combination of loading/unloading tests and replica observations for the T800H/3631 laminates at room temperature and at 8O”C, respectively. The Young’s
h 0
5 10 Transverse Crack Density
15
@) (O/90$0)
~~~~~
0
5 10 Transverse crack deosity(/cm)
If
(4 (0~90,~O)
Fig. 4. Young’s modulus reduction as a function of transverse crack density for T800H/3631 cross-ply laminates at room temperature.
Tj 0.80 7@ 0.75 sj
no delamination alL=O.l ---_ with dslamination
0.70 0
5 Transverse Crack Density (/cm)
10
(a) (0/9OdO)
Transverse Crack Density (/cm) @) (OfwO)
z
0
5
10
Transverse Crack Density (/cm) (c) (O/90,2/0)
Fig. 5. Young’s transverse
modulus reduction crack density for T800H/3631 at 80°C.
as a function of cross-ply laminates
modulus reduction is larger in the laminates with thicker 90” plies at the same transverse crack density. It should be noted that the normalized Young’s modulus shown here is affected by the delamination. Figures 6 and 7 show the permanent strain as a function of the transverse crack density for T800H/3631 laminates at room temperature and at 8O”C, respectively. The permanent strain is larger in the laminates with thicker 90” plies at the same transverse crack density. The Young’s modulus reduction and the permanent strain for T300/3601 system were not measured because the strain for onset of transverse cracking is comparable to the laminate ultimate strain. 3.3 Mode II interlaminar fracture toughness The load/displacement curves obtained by the ENF tests showed slight nonlinearity.32 In the present study, crack initiation was defined as the onset of nonlinearity in the load/displacement curves.34 The critical energy release rate for crack initiation, G1,,, can be calculated from the load at the onset of nonlinearity, the initial crack length, and the initial compliance. The region from the load at the onset of nonlinearity up to the maximum load was assumed to be the subcritical crack growth region. In order to examine the fracture resistance, GIIR, in the subcritical
Interaction between transverse cracks and delamination
399
Table 1. Mode II interlaminar fracture toughness, and Gn_ (J/m’) Material
0
10 Transverse Crack Density (/cm) 5
T800H/3631 T300/3601
-o’2o l Experiment ap Rsdictbm -
RT = room
5 10 Transverse Crack Density (/cm)
15
(4 (0~00)
Fig. 6. Permanent strain as a function of transverse density for T800H/3631 cross-ply laminates at temperature.
I
’
0
.*
crack room
I
Fxpcrimcnt
Rcdicttom no delamination ---- with delamination olL=O.3
$ 0.05 B %
@ g 0.
G IImax
RT
80°C
RT
80°C
398 3.50
345 219
613 549
578 429
temperature.
no delamination
0
rp,
G nc
15
(a) (OHdo)
.=0.15 z
system
GrI,
crack growth region, the crack length was determined following a previous method.34 The critical energy release rates for crack initiation evaluated by the load at the onset of nonlinearity at each temperature, Gn,, are shown in Table 1. The values evaluated at the maximum load, GrImax,are also shown. The values are the average for three specimens. For the T800H/3631 system, Gnc at 80°C is slightly lower than at room temperature. For the T300/3601 system, Gnc at 80°C is much lower than at room temperature, as a consequence of the decrease in the fiber/matrix interfacial strength at higher temperature.31 Figure 8 shows the mode II interlaminar fracture toughness in the subcritical crack growth region, GnR, as a function of the crack extension, Aa. The crack extension, Aa, was not measured but calculated.34 The GnR value increases as Aa increases at both room temperature and 80°C which should be called R-curve behavior. In the present study, the R curves are fitted to a relationship of the form:
I 0
GIIR= Go + k%“%
(1)
5
Transverse Crack Density(/cm) (a) (OPOdO)
where Go and k are constants, determined by the least-squares method and their values are summarized in Table 2. In Fig. 8 the solid and dotted lines are the fitted curves for the data at room temperature and at 80°C respectively. The Gun curves will be used to predict delamination behavior in the following discussion.
Transverse Crack Density(/cm) @) (O~%w T8ooH/3631 700,
0 Transverse Crack D&sity(/cm) (4 (O~OIfl)
Fig. 7. Permanent strain as a function of transverse crack density for T800H/3631 cross-ply laminates at 80°C.
1
0.0 0.5 1.0 1.5 2.0 2.5 3.0 Crack Extension, da (mm)
l3CW3601 7001
I
0.0 0.5 1.0 1.5 2.0 2.5 3.0 Crack Extension, dn (mm)
Fig. 8. Mode II interlaminar fracture toughness subcritical crack growth region, GnR, as a function extension, Au.
in the of crack
S. Ogihara,
400 Table 2. G,, and k in eqn (1)
Material system
G,, (J/m’)
T800H/3631 T300/3601
k (J/m3”)
RT
80°C
RT
80°C
394 339
357 206
5850 5470
4510 4650
N. Takeda
which the interlaminar shear layers are of thickness d “. The Young’s modulus reduction and the permanent strain due to transverse cracking can be derived from the obtained stress and strain distributions.“‘,32 The average axial strain of the cross-ply laminate with transverse cracks is given by:
RT = room temperature. 4 THEORETICAL 4.1 Modified
= E + 2dc2oP ptanh 0
BACKGROUND
shear-lag
r ( 2~ 1
bEi
analysis
Shear-lag analysis was first introduced for cross-ply laminates by Garrett and Bailey,l and a modified shear-lag analysis considering the interlaminar shear layer was conducted by Fukunaga et aZ.9 The boundary condition of the method was modified by Lim and Hong17,‘8 to satisfy the conditions for arbitrary crack spacings. In this analysis, the stress state in a cross-ply laminate is calculated by following their method. For simplicity, the method is reduced to one dimension.31,32 The following theoretical treatment is similar to that given previously,31,32 but is described here for completeness. Consider a cross-ply laminate subjected to mechanical and thermal loading, as shown in Fig. 9(a), in
where L is the half crack spacing, E, is the axial strain of the 0” ply, ET is the thermal residual stain in the 0 PlY> E,, is the undamaged laminate strain, b is the thickness of the 0” ply, d is the half thickness of the 90” ply, azo is the 90” ply stress in the undamaged laminate, p is the associated transverse crack density, E, is the Young’s modulus of the 0” ply, and
e=&J&+&J where E, is the Young’s modulus of the 90” ply, and G is the shear modulus of the interlaminar shear layer. By putting co = 0 in eqn (2), the permanent strain (the residual strain at the perfectly unloaded point), lp, can be expressed as: dE2& E = -tanh ’ bE,tL
(5L)
(4)
b do
where l 2T 1s ’ the thermal residual strain in the 90” ply. For a constant applied stress, a,,, the stress/strain relationship of the undamaged cross-ply laminate can be written as:
2d do
b
u. = EOeO
\ Transverse Cracks Interlaminar Shear Layer z n
L 245-u 245
E(< - Q,) = cr,,
b
0” El
do
2d*
90°E2 00, co
(5)
where E, is the Young’s modulus of the undamaged laminate. The stress/strain relationship of the laminate with transverse cracks is:
(a) t
(2)
d0
ao*to
(6)
where E is the Young’s modulus of the laminate with transverse cracks. From eqns (2)-(6), the normalized Young’s modulus reduction due to transverse cracking is:
b
0” _Delamination
Fig. 9. Analytical models of cross-ply laminates: (a) with only transverse cracks; (b) with both transverse cracks and delamination.
4.2 Extension of modified shear-lag analysis In this section, we consider the state where a delamination with a length 2a has developed at each
Interaction between transverse cracks and delamination
tip of a transverse crack (the crack spacing is 2L) as shown in Fig. 9(b). Assuming that there is no friction between the 0” and the 90” layers in the delaminated region, all stress components in the 90” ply and the interlaminar shear layer are zero in the region. The stress components in each layer were obtained previously.31~32 The Young’s modulus reduction and the permanent strain due to transverse cracking and delamination at the tips of transverse cracks can be derived by the same procedure as in Section 4.1. The axial strain of a laminate with delamination at the tips of the transverse cracks is: El&
-
ET=
+-----tanh 2dazop b& 5
+ 2&w ( bE, 1 FE2 2d&ap
Eg
-
$1
1 +
2ap)
By putting e0 = 0 in eqn (S), the permanent including the effect, can be expressed as: EP =Fp[a
T
~
+ $anh&l
- ap)}]
strain,
(9)
Substituting eqns (8) and (9) into eqn (7), the normalized Young’s modulus reduction due to transverse cracking and delamination at the tips of transverse cracks is obtained. The energy release rate associated with delamination growth, Gd, can also derived by using the obtained stress distribution31,32 as: G =
dE2(b&
dE,)(%
+
d
[
4 [exp&(L - a) + exp - 5(L - a)]’ I
4.3 Probabilistic
where L, and V, denote the element length and volume, respectively, CY,and p are the Weibull scale and shape parameters, respectively, and E, is the constraint strain defined in Ref. 31.
4.4 Prediction of delamination growth Assuming that the critical energy release rate for delamination growth is equal to the mode II interlaminar fracture toughness, GuR, obtained from ENF tests, the delamination length in the cross-ply laminates can be predicted as a function of applied laminate strain.3z In reality, the mode I component of energy release rate associated with the delamination growth is not zero and is very large near the free edge and when delamination is small. It is not easy, however, to determine the crack growth criterion under mixed-mode conditions, and a mode II failure criterion was therefore assumed for a simple delamination growth prediction. Letting Gd in eqn (10) be equal to GIIR in eqn (l), and rewriting Aa in eqn (1) as a, we have the following equation: dEz(Elb
+ E,d)(e,
+ E:)~
2E, b
x
lL
{exp{&-(1 -a)]
+tp(
prediction
of transverse
(lo)
crack
density
A novel model predicting the relationship between the applied laminate strain and the transverse crack density by making use of the statistical parameters of an experimentally-obtained 90” UD strength distribution has been presented previously.31 Following earlier studies,14-‘6 the 90” ply in the cross-ply laminate is divided into elements which are assumed to fail only once up to the ultimate failure. The strength distribution of the elements is assumed to obey the same two-parameter Weibull distribution as measured for the transverse strength of UD. Then the failure probability of the elements under a stress is known and the number of cracks can be calculated by multiplying by the number of elements:31 -V~(E2(‘o+-~‘~‘c))p]]
(11)
-a)lr] (12)
By solving this equation with respect to a, the total delamination length can be obtained at a given laminate strain and the corresponding transverse crack density.
4.5 Prediction of interaction cracks and delamination
between
transverse
In the discussion above, the interaction between transverse cracks and delamination has not been considered. In this section, a prediction is presented by considering this interaction. Before the onset of delamination (e. < e&i, where lo is the laminate strain and E&i is the delamination onset strain), only the transverse cracks are considered. Therefore, the transverse crack density, p, and the delamination ratio, D, are: p=2[
-exp(
-VC(E”‘o~:‘-rc)~}] D=O
p=+J-exp(
-&(l
=G,+k&
+ ‘:)’
2bE,
X l-
401
The delamination
(13) (14)
onset strain, E&i, can be obtained by
S. Ogihara,
402
N. Takeda Table 3. Values of G/d,,
solving the following equation with respect to eO: d&(E,
b + E,d)(q,
Material system
+ 6;)’
Temperature
9.07 x 7.64 x 1.04 x 8.71 x
RT
T800H/3631
2Elb
G/d, (Pa/m)
80°C
T300/3601
RT 80°C
lOI
10” lOI 1o13
RT = room temperature. By letting the transverse crack density after the onset of delamination be p’, the rate of p’, dp’/deO, can be assumed to be reduced by the proportion of the delamination ratio, D, i.e.
interaction between transverse tion are taken into account. 5.2 Delamination
!g=$-D) 0 The delamination
(16)
ratio, D, is:
dEz(EI b + E,d)(c,
+ 6;)’
2E,b 4
(exp{ &(I-
D)} + exp{ - $0
- D)]}*
= G,+ k&
1 (17)
The transverse crack density, p’, and the delamination ratio, D, after the delamination onset can be obtained by solving eqns (13) (16), and (17), as functions of the laminate strain, eO. By substituting the above results into eqns (7) and (9), the Young’s modulus reduction and the permanent strain can be obtained considering the damage growth. 5 DISCUSSION 5.1 Transverse
cracking behavior
Figures 1 and 2 show a comparison of the predictions of the transverse crack density and experimental results for the TSOOH/3631 and T300/3601 composite systems. The present probabilistic approach including delamination effect is shown for the T800H/3631, but is not shown for the T300/3601, because delamination did not occur in the latter system. The values of G/do are shown in Table 3. The value of G/L&, is a parameter that is difficult to determine experimentally, but the values used are similar to those which are often adopted.‘&r* As shown in the figures, for the T800H/3631 system the delamination effect is larger in the laminates with thicker 90” plies, which can be predicted by the present approach including the delamination effect. In summary, it can be concluded that the transverse crack behavior can be well predicted when the
cracks and delamina-
behavior
In the first place, the occurrence of delamination is discussed. The delamination onset strain, E+,, can be predicted from eqn (15). Comparing the delamination onset Strain, Ed& and the laminate ultimate strain, ls, the occurrence of delamination should be judged. That is, if lde1< eg delamination occurs before the laminate finally breaks, and if l&r > lu, delamination will not be observed in the failure process. The predicted delamination onset strain, (ede,)Pred, and laminate ultimate strain obtained by experiments are shown in Table 4. For the T800H/3631, (e&_i)P’edis smaller than lu only except for (O/90,/0) at 8o”C, which is consistent with the experimentally-observed delamination growth. For the T300/3601, (Edei)P’edis smaller than lu for (O/90,/0) at 80°C and for and therefore delamination may be (0/9OJO), observed in these laminates. But since the onset strain of transverse cracking for these laminates is comparable to the ultimate strain, no delamination was experimentally observed for all laminate configurations of the T300/3601 system. Fig. 3 shows a comparison of the predictions of delamination ratio with the experimental results. The predictions are made for both (.$jPSM, which was obtained by the ply separation method31*32 and (e;)CLT, which was calculated from the classical laminate theory with the catalog data for thermal expansion coefficients. The predictions in Fig. 3 show very good agreement with the experimental results when (eT)PSM 1s . used, which confirms the validity of the present prediction and the ply separation method. The effect of the interaction between transverse cracks and delamination was found to be negligible in the prediction of the delamination growth. 5.3 Prediction permanent
of Young’s strain
modulus
reduction
and
First, the prediction of the Young’s modulus reduction is compared with the experimental results. From eqns (2) (4), and (7), the normalized Young’s modulus can be predicted for cross-ply laminates only with
Interaction between transverse cracks and delamination Table 4. Predicted
delamination
onset
strain, (E,,~,)~~‘, and strain, lB
laminate
403 ultimate
RT
T800H/3631 Go=394 (N/m) Go = 357 (N/m) T300/3601 G,, = 339 (N/m) Go = 206 (N/m)
(RT) (SO’C) (RT) (80%)
(O/90,/0) (O/90,/0) (0/90,,/0) (O/90,/0) (O/90,/0) (0/90,,/0)
1.39 0.870 0.646 1.51 0.952 0.706
failure
80°C
1.46 1.42 1.36 0.890 0.860 0.915
1.55 0.995 0.757 1.36 0%30 0.666
1.52 1.49 1.25 0.995 0.970 1.01
RT = room temperature.
transverse cracks. For laminates with delamination from the tips of transverse cracks, eqns (8) and (9) should be used instead of eqns (2) and (4). In addition, use of the prediction in Section 4.6 enables a prediction of the Young’s modulus reduction considering damage growth to be made. Figures 4 and 5 show comparisons of the predictions with the experimental data at room temperature and at 80°C. The thermal residual strains are also necessary for the calculations. It is found that the influence of thermal residual strains is small in the present calculation, so the thermal residual strains obtained by the ply separation method31T32 are used for the predictions shown here. for the laminates with only The predictions transverse cracks seem to be accurate for (O/90,/0) at lower crack densities. But for (O/90,/0) at higher crack densities and for (O/90,/0), prediction for the laminates with only transverse cracks tends to be inadequate and a consideration of delamination proves to be necessary. For (O/9O,2/O) laminates, the modified shear-lag analysis for the laminate with only transverse cracks is no longer a useful tool for predicting the Young’s modulus reduction. For the laminates in which the delamination grows extenof the shear-lag analysis sively, the extension described in the present study has proved to be important for understanding the degradation of material properties in the failure process. Furthermore, the predictions considering damage growth showed excellent agreement with the experimental results. Next, the prediction of the permanent strain is compared with the experimental results. Figures 6 and 7 show comparisons of the predictions with the experimental data at room temperature and at So”C, respectively. The value of the thermal residual strain has a large influence on the predicted value of the permanent strain. The prediction shown here is obtained by using (e2T)PSM.Results were similar to the Young’s modulus reduction, i.e. analysis including delamination effects was necessary for the laminates
with thicker 90” plies, and the prediction considering damage growth showed good agreement with the experimental results. 6 CONCLUSIONS A damage progression model of cross-ply laminates has been developed on the basis of a modified shear-lag analysis and the probabilistic strength of 90” between transverse layers, considering interaction cracks and delaminations. The transverse crack density and the delamination ratio were obtained as functions of the laminate strain. The predictions showed excellent agreement with experimental results when the interaction was properly taken into account. Similarly, the modified shear-lag prediction of the Young’s modulus reduction and the permanent strain considering the interaction of transverse cracks and delamination were compared with the experimental results. It was found to be important to understand the interaction of transverse cracks and delamination in order to predict the Young’s modulus reduction and the permanent strain induced by the damage development.
ACKNOWLEDGEMENTS
The present study was supported financially by a Grant-in-Aid from the Ministry of Education, Japan. The authors thank sincerely Professor Akira Kobayashi, Science University of Tokyo, and Professor Tadashi Shioya, The University of Tokyo, for fruitful discussions throughout the study.
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