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A simplified analysis of transverse ply cracking in cross-ply laminates
Composites Science and Technology 31 (1988) 165-177
A Simplified Analysis of Transverse Ply Cracking in Cross-Ply Laminates Y. M. Han, H. T. Hahn Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA 16802 (USA)
and R. B. C r o m a n E. I. du Pont de Nemours & Co., Inc., Pioneering Research Laboratory, Experimental Station, Wilmington, DE 19898 (USA) (Received 16 July 1987; accepted 29 October 1987)
SUMMAR Y This paper presents a method o f analyzing transverse crack initiation and multiplication in symmetric cross-ply laminates. The method is based on the concept of a through-the-thickness inherent flaw and the energy balance principle. With a second-order polynomial assumed for the crack opening displacement, the perturbed stress.field due to the presence of ply cracks is determined from the equilibrium conditions. The energy released as a result of ply cracking is then calculated and used to predict the increase in crack density. Based on an experimental correlation of the analytical result, a resistance curve is proposed to be used as a measure of the resistance to crack multiplication. The resistance to crack multiplication is shown to increase with the increasing crack density.
1 INTRODUCTION
Since the early 1970s the transverse ply cracking in composite laminates and the corresponding mechanical property degradation have been a subject of extensive research, both experimental and theoretical.I-14 165 Composites Science and Technology 0266-3538/88/$03"50 © 1988 Elsevier Applied Science Publishers Ltd, England. Printed in Great Britain
166
Y. M. Hart, H. T. Hahn, R. B. Croman
As a result the applied strain at the cracking of the first ply, called the firstply cracking strain, is known to depend not only on the thickness of the cracked ply but also on the stiffness of the adjacent ply. 6'9 Furthermore, at a given strain, the crack spacing increases as the ply thickness increases and the crack spacing in the outer transverse ply is only half of that in the inner ply when both have the same thickness, t t.14 Several analytical models have been proposed to predict the first-ply cracking strain as well as the crack density. Some models are based on the use of strength, 1'3-5 while others rely on the linear fracture mechanics 8-13 as shown in Fig. 1. The latter models can account for the dependence of the first-ply cracking Strain on the ply thickness.
Anal,sis I [Strength
I
I
Deterministic
I
Fig. 1.
Fracture
Model [
I
Mechanics
Model
I
i Model
Vasil'ev [1] Hahn [2] Garrett [3]
of Ply Cracking ]
I
I
Through-the-thickness]lThrough'the'width [ -flaw Model -flaw Model
Manders [4] Fukunaga [5]
Parvizi[8] Nuismer [9] Hahn [10] Dvorak [13]
I
Wang [11] Flaggs [12] Dvorak [13]
Flow chart for the methods of analyzing ply cracks.
Fukunaga et al., 5 investigated the failure characteristics of cross-ply laminates based on a statistical strength analysis. A shear lag analysis was used to determine stress distribution. The transverse strength was assumed to obey a two-parameter Weibull distribution. It was assumed that a new crack occurred midway between any two adjacent cracks at 50% failure probability. Parvizi e t al., 8 and Nuismer and Tan 9 investigated the failure behavior of cross-ply laminates from the energy point of view. The former investigators used a one-dimensional shear lag analysis while the latter used an approximate elasticity solution for stress analysis. The work of crack closure was used as the energy released as a result of ply cracking. Several fracture-mechanics-based models have been proposed which assume the growth of an inherent flaw to be the mechanism for ply cracking.lO-13 The inherent flaw could be entirely through the width but partially through the thickness, 1~ or vice versa. 1° The through-the-width flaw model was further extended to account for the statistical nature of the
Analysis of transverse ply cracking in cross-ply laminates
167
ply cracking. 11 While a finite element method was originally used for stress analysis by Wang, 11 a two-dimensional shear lag analysis was also applied by Flaggs. 12 Hahn and Johannesson 1o showed that the energy release rate associated with the widthwise growth of a long through-the-thickness-flaw is equal to the work of crack closure per unit width of the fully developed flaw. The energy release rate is independent of the flaw length and hence the growth of the flaw is stable. Dvorak and Laws13 considered a flaw which is both partially through the thickness and partially through the width. The required stress analysis is three-dimensional by contrast with the aforementioned two models. Two directions of the crack growth and the corresponding energy release rates were calculated, i.e. parallel to the fiber axis and parallel to the thickness direction of the laminate. In the present paper a simple method of analyzing transverse crack initiation and multiplication in symmetric cross-ply laminates is presented. The method is based on the through-the-thickness-flaw concept proposed by Hahn and Johannesson. 1° The requisite stress analysis is simplified by introducing a displacement function which describes a parabolic crack opening displacement. A resistance curve is proposed for characterization of the resistance to the multiplication of transverse cracks.
2 ANALYSIS OF T R A N S V E R S E C R A C K I N G When a symmetric cross-ply laminate is under tensile loading, ply cracks appear parallel to the fibers in the transverse ply long before the ultimate laminate failure (Fig. 2). The origins of these ply cracks are believed to be inherent defects such as voids, microcracks, debonded fiber-matrix interfaces and broken fibers. As the load increases, these defects act as local stress raisers, and grow or coalesce to form a through-the-thickness-flaw. Since the matrix and interface are much weaker than the fiber, these flaws always propagate through the matrix and interface parallel to the fibers. The energy release rate associated with such flaw growth has been shown to be independent of the flaw length.I° Moreover, this energy release rate is equal to the work of crack closure per unit ply width of the fully developed transverse crack. To predict the initiation and multiplication of transverse cracks, consider an idealized case of a cross-ply laminate containing a series of equally spaced cracks in the 90 ° plies. A representative element of this laminate containing only one crack is shown in Fig. 3, where 2L is the crack spacing. As indicated in the figure, this problem can be solved in two steps: a laminate without a crack subjected to the far-field stresses and a laminate with a loaded crack.
Y. M. Han, H. T. Hahn, R. B. Croman
168
P,c------
~/
(crack spacing)
l" " x
Z
90.Plies O-Plies
ho
t Transverse Fig. 2.
cracks
Coordinates for analysis of multiple transverse cracks.
-f
~
i T i'~
lllll lllll-Illlll Illlll"IIIIl lllll" IllltlIllll llllll " I111IIIIII. 11111 III1 ~*~, (a)
t~t
(b)
°°t f~t
:
(e) (f) Fig. 3. Stress analysis via superposition.
(c)
(d)
Analysis of transverse ply cracking in cross-ply laminates
169
The superposition of the two solutions will then provide the stress field in the laminate with a crack. The ply stresses before cracking due to the mechanical and thermal loading are determined by the classical laminated plate theory and are denoted by 0~, 0~o and 60T, 09T0, respectively. Here, subscripts 0 and 90 represent variables associated with the 0 ° and 90 ° plies, respectively. The applied load per unit specimen width, P, is simply given by P = 2(ff~h o + ~Moh9o)
(1)
since the thermal stresses are in self-equilibrium. The formation of a crack is equivalent to the application of a boundary stress, -090, on the crack surfaces, Fig. 3, where
¢~90 = ¢~Mo"[- ¢~9T0
(2)
To determine the resulting stress fields, ao and cr9o, in the laminate with a loaded crack, consider a quadrant of the representative element shown in Fig. 4. The perturbed stress, a o, in the 0 ° ply is assumed uniform through the
h~h~° ~ x I~
L
ill -I Fig. 4. A representative element for cracked laminate. thickness so that it is a function o f x only. If the shear stress at the interface is z, the equilibrium equation for the 0 ° ply becomes
do o T dx - ho
(3)
On the other hand, the equilibrium condition for the entire laminate at any cross-section requires that hoa o +
a9o dy = 0
(4)
In the x direction, the perturbed displacement, uo, is a function o f x only but u9o is assumed to be U9o(X,Y) = f l ( x ) Y 2 +f2(x)
The functions fa(X) and f2(x) are to be determined later.
(5)
170
Y. M. Hart, H. T. Hahn, R. B. Croman
The interfacial shear stress, z, follows from •U9o I
27= G T ~ - y
y=hgo = 2hgoGTfl
(6)
where GT is the transverse shear modulus. The 90 ° ply stress, 0-90, also follows from U9o as (~U90
a9o = ET--~-x = ET(y2f~ +.[2)
(7)
where a prime denotes differentiation with respect to x, and E T is the transverse Young's modulus of the lamina. The displacement continuity condition at the interface requires that b/0(X) : U90(x, Y)ly: ho,, =./1 h2o +./2
(8)
Therefore, the 0 ° ply stress, o o, becomes duo
ao = EL~--f = EL(h~of~ + f~)
(9)
where E L is the longitudinal Young's modulus of the lamina. Equations (4), (7) and (9) are now solved for J~ and./i to yield
.[~ = Floo
f~ = F20-o
(10)
where
3 hoE c + h9oEv F1 = 2 Ec Evh3o
F2 -
1 Ec
h2oF 1
(11)
Thus the governing equation for o o follows from eqns (3), (6) and (10) as -
~
ao
(12)
where ¢2 : 3GxhoEc + hooEy
(13)
hoEc Ev
2.1 First-ply cracking When there is only one crack in the 90 ° ply, the b o u n d a r y conditions to be satisfied are h,, ao = C 0 9 o Uo = 0 at x 0 (14) dao .[~ = 0 dx - 0 as x --* oc The solutions for the perturbed stress field, 6 o, and the displacement, Uoo, are
Analysis of transverse ply cracking in cross-ply laminates
171
then O"o = A f f 9 o
e-~
(15)
U90 = (A/flp)~90h90[Fl(h20 _ y2
e-4,~) +
]72(1 _
e-~X)]
where A = h9o/h o
.~ = x/h90
(16)
Now that the perturbed stress and displacement fields are known, one can calculate the energy release, A U, per unit laminate width associated with the formation of the crack. Specifically, AU is given by 1° AU = 2 f[9o ff90U90lx=Ody 2 -2 ,2 [hoEc + h90ET~ 1/2
(17)
=N//30"90n90~ hooEL-E-~T : The criterion for first-ply cracking is then AU = 4h9oTv
(18)
where Yv is the surface fracture energy density. Thus, the ply stress, (ff9o)vPc, required for the first cracking of the 90 ° ply and the corresponding ply strain, (gg0)vPC, are given by
(~90)FPC= (~F ~1/2 (12hoELETGT ~1/4 (19)
1
( 9o) Pc = (O9o) Pc Equation (19) shows the dependence of (tr90)FPC on the 90 ° ply thickness and the modulus of the adjacent plies. Note that (ff90)FPCincludes the residual ply stress as well.
2.2 Crack multiplication When there are several cracks with a uniform spacing 2L, the boundary conditions at infinity are replaced by fl=0
dcr o
dx - 0 a t x = L
(20)
The solutions are given by 6 o = A09o(e -~-~ + e-2~Ce ~s) -~~ + e 2,c eO.~)(Fly2 + F2 ) + (1 - e 2~C)(Flh2 o +/72) ]
/290~---(A/~))69oh9o[(--e
(21)
Y. M. Han, H. T. Hahn, R. B. Croman
172
where
1 h9o A = 1 + e -24'L h o
x x = hgo
E=
L
(22)
h9o
The energy release, AU per crack, is given by AU= 2
~0hg°~9oU9olx=odY 2
--
(23)
-2 " 2 I/hoEL + h9o Ev'~I/2
- - G90~90 . . . ,,~ t .hoELErGr )
tanh(q~E)
When there are Ncracks per unit length, the total energy release is N. AU. To account for crack multiplication the authors postulate that at a given value of 690, the crack density changes from N to N + d N when the additional energy released is equal to the energy absorbed. That is, d(N. AU) = 4h9o7 F d N
(24)
Noting that N = 1/(2L) and using eqn (23), one can obtain the ply stress, (690)N, required to produce N cracks as
l/.;F,,/21/ 12hoELETGT'~,,4[ n. Ck 4) (fY90)N=t~90) th0~-~-L~- ) tta n ~ - ~ s e c h
2
~)-~
(25)
where IV is a normalized crack density defined by :7 = hgo. N
(26)
Note that N represents the number of cracks over a length equal to the thickness of the 90 ° ply. Equation (25) reduces to eqn (19) when the crack density goes to zero, i.e. for the case of the crack initiation. The additional average axial strain due to cracking can also be obtained from
Ag= u°(L) - 1 590 h9o ~ hoELET ]1/2 L x~ Ec ~oE[_G-r(ho-E77h9oE-r) tanh(q~/~)
(27)
The total average axial strain is thus 0"9O
= E 7 + Ae
t28)
The applied load, P, and applied stress, %, corresponding to a crack density,
Analysis of transverse ply cracking in cross-ply laminates
173
.~', are obtained from eqns (1), (2) and (25) as P = 2 (h°Er + hg°Er) [( 9o)N - a o] ET P ~ra - 2(ho + h 9 o )
(29)
3 DISCUSSIONS
To check the validity of using a second-order polynomial for the crack opening displacement, a finite element code 'ANSYS' was used to determine the displacement field in a [0/90] s laminate. I s Because of symmetry, only one quadrant of the section was modeled by using 2-D plane strain elements Y
J" ~
~, Fig. 5.
h0
,777777774 A finite element model for the representative element.
(Fig. 5). The material constants for the T300/934 graphite/epoxy composite used in the analysis are in Table 1. Figure 6 shows the crack opening displacements of [0/90]~ laminate under 1 MPa tensile loading. The displacement from eqn (15) agrees fairly well with TABLE 1
Mechanical Properties of T300/93412 Longitudinal modulus, Ell Transverse modulus, E22 In-plane shear modulus, G12 Transverse shear modulus, Gz3 In-plane Poisson's ratio, vl2 Out-plane Poisson's ratio, v23 Longitudinal tensile strength, X Transverse tensile strength, Y Thermal expansion coefficient, ~l Thermal expansion coefficient, ~2 Stress free temperature, AT Critical energy release rate, Glc (Re£ 11) Lamina thickness, t
138 GPa 11.7 GPa 4.56 GPa 4,18 GPa 0.29 0.4 1 724 MPa (Ref. 11) 44.8 MPa (Ref. 11) 0.09 pm m - 1 °C- l 28"8 p m m - 1 ° C - l - 129.4oc 158 J m - 2 (0/0 interface) 228 J m - 2 (90/90 interface) 0-132 mm
174
Y. M. Han, H. T. Hahn, R. B. Croman
2.5
~" 2.0 & 1.5
1.o }5 o.~
•-m- Eq.(15) Infinite plate -0- FEM
O.C
- 1 . 0 - 0 . 8 - 0 . 6 - 0 , 4 - 0 . 2 - 0 . 0 0.2 0.4 0.6 0.8 1.0 Fig. 6.
Y~90 Crack opening displacement in T300/934 [0/90]~ laminate under I MPa tensile load.
the finite element result. Since the energy release rate due to cracking is directly proportional to the area beneath the crack opening displacement, the calculated energy release from eqn (17) is seen to be only 8 % smaller than that from the finite element analysis. For comparison purposes, the displacement for an unconstrained crack in an infinite plate is also shown in Fig. 6. The constraining effect of the adjacent plies on the crack opening displacement is obvious. The assumed displacement field has a shear stress discontinuity across the interface, a shortcoming of the proposed approximation. However, it should be noted that most shear lag analyses have the same shear stress discontinuity. Furthermore, the crack opening displacement resulting from the assumed displacement field is in good agreement with that from a finite element stress analysis. Therefore, the assumed displacement field is expected to be a good approximation. The fracture criteria in eqns (18) and (24) apply to flaws propagating parallel to the fibers. It is noted that Wang's model ' ' uses an energy release rate for normal crack propagation. In Fig. 7, the ply strains, (gg0)vPC, from eqn (19) are compared with the experimental results 6 for [02/90n] s T300/934 laminate family. The analytical results based on the 1-D shear lag model of Parvizi) the 2-D elasticity model of Nuismer, 9 and the 2-D shear lag model of Flaggs ,2 are also shown in Fig. 7. The predictions of the present model fall between those of the two models from Refs 8 and 12. When there are fewer than 6 transverse plies, the predictions agree well with the experimental results. But the deviation from the experimental results increases as the transverse ply thickness increases. This observation indicates that the mode of transverse ply cracking changes as the thickness of the transverse ply changes. When the 90 ° ply is thin, the assumption of the inherent flaw being a through-the-thickness type is reasonable. However, as the 90 '~ply becomes thicker, the same assumption is
Analysis of transverse ply cracking in cross-ply laminates
175
1.2
\
1.0 {J
0.8
iw~, 0.6 ~
v
"---. ~ ""
~
:e- Parvizi t8] ~ Nuismer [9]
!
4-
0.4
I
~ ~ , * -
0,2
I
]
1
2
3
[
Faag~s t121 Eq. (19)
5 6 7 8 Number of transverse plies (ra)
4
Predictions of first ply cracking strains (ggo)vPcin T300/934 [0z/90,], laminates.
Fig. 7.
not likely to hold true. Since the analytical models s'9 are based on the presence of a through-the-thickness flaw, their predictions are expected to become worse as the 90 ° ply thickness increases. The transition occurs when n is about 3. For thick 90 ° plies, one may use an inherent flaw which is neither through-the-thickness nor through-the-width, as was studied by D v o r a k and Laws.13 In Fig. 8, the predicted stress, u,, required to produce a crack density, N (eqn (29)), is c o m p a r e d with the experimental data 11 for uniaxially loaded [0/90,/0]x T300/934 laminates. By contrast with the experimental results, the predicted stress, a a, remains fairly independent of the crack density. Equation (23) indicates that the energy release per crack does not increase m u c h until the crack spacing becomes small enough to allow interaction between neighboring cracks. Consequently, the model predicts the formation of m a n y cracks almost simultaneously without m u c h increase in 900
:
~', 600
..
[]
,n[n-2] I:l
[] ~
.1.-31 Eq. (29, mmL.-Q
=~ 300 •
< 0
0
•
•
•
•
I
I
m
3
6
9
•
•
i
•
[
•
i
°p-q °lo-d
12 15 18 Crack density (era"1)
Fig. 8. Applied stress o-, vs crack density in T300/934 [O/90n/O]Tlaminates.
176
Y. M. Han, H. T. Hahn, R. B. Croman
300 •
u nn
•
200 [] • m •me [] 100
rnm
0
i,
g
[]
[][]
[]
[]
[] n=2
I
,
3
I
6
n
,
l
,
I
,,
,I
•
I1=3
•
11=4
i
12 15 18 Crack density (cm"1) Resistance curves for crack multiplication obtained for T300/934 [0/90./0]T laminates. 0
Fig. 9,
[]
9
load. However, the experimental data indicate more gradual crack multiplication. One possible reason is that the material is not really homogeneous in fracture energy, and hence 7v will be an increasing function of N. Thus, eqn (29) can be used to determine }'v from the measured P-N relations. The results are shown in Fig. 9. It is interesting that 7v is almost a linear function of N and the slope increases as the transverse ply thickness increases. The relation between 7r and N has a similar meaning as the relation between fracture energy density and crack length during crack growth in homogeneous materials. Therefore, the 7v-N relations shown in Fig. 9 are called the resistance curves for crack multiplication. Whereas the resistance curve for homogeneous materials characterizes the increasing resistance to crack growth due to crack tip damage, the new resistance curve for crack multiplication in composite laminates represents the inhomogeneity of fracture energy. 4 CONCLUSIONS A simple method has been proposed to calculate the energy released as a result of ply cracking. The method is based on the use of a second order polynomial to represent the crack opening displacement. The calculated energy release per ply crack is then used in conjunction with the energy balance principle to predict the multiplication of ply cracks. A resistance curve for crack multiplication is proposed as a means of characterizing the resistance to ply cracking in composite laminates.
Analysis of transverse ply cracking in cross-ply laminates
177
ACKNOWLEDGEMENTS This paper is based on work supported by a grant from E. I. du Pont de Nemours Co. REFERENCES 1. V. V. Vasil'ev, A. A. Dudchenko and A. N. Elpat'evskii, Analysis of the tensile deformation of glass-reinforced plastics, Mekhanika Polimerov, 1 (1970), pp. 144-7. 2. H. T. Hahn and S. W. Tsai, On the behavior of composite laminates after initial failure, J. Composite Materials, 8 (1974), pp. 288-305. 3. K. W. Garrett and J. E. Bailey, Multiple transverse fracture in 90° cross-ply laminates of a glass fiber-reinforced polyester, J. Material Science, 12 (1977), pp. 157-68. 4. P. W. Manders, T. W. Chou, F. R. Jones and J. W. Rock, Statistical analysis of multiple fracture in 0/90/0 glass fiber/epoxy resin laminates, J. Materials Science, 18 (1983), pp. 2876-89. 5. H. Fukunaga, T. W. Chou, P. W. M. Peters and K. Schulte, Probabilistic failure strength analysis of graphite/epoxy cross-ply laminates, J. Composite Materials, 18 (1984), pp. 339-56. 6. D. L. Flaggs and M. H. Kural, Experimental determination of the in situ transverse lamina strength in graphite/epoxy laminates. J. Composite Materials, 16 (1982), pp. 103 16. 7. A. L. Highsmith and K. L. Reifsnider, Stiffness-Reduction Mechanisms in Composite Laminates, ASTM STP 775 (1982), pp. 103-17. 8. A. Parvizi, K. W. Garrett and J. E. Bailey, Constrained cracking in glass fiberreinforced epoxy cross-ply laminates, J. Materials Science, 13 (1978), pp. 195 201. 9. R. J. Nuismer and S. C. Tan, The role of matrix cracking in the continuum constitutive behavior of a damaged composite ply, Proc. of the IUTAM Symposium on Mechanics of Composite Material Virginia Polytechnic Institute and State University, Blacksburg, VA, 1982. 10. H. T. Hahn and T. Johannesson, Fracture of unidirectional composites: theory and applications, in Mechanics of Composite Materials, AMD-Vol. 58, !983, ASME, New York, pp. 135-42. 11. A. S. D. Wang, Fracture Analysis of Matrix Cracking in Laminated Composites, NADC-85118-60, 1985. 12. D. L. Flaggs, Prediction of tensile matrix failure in composite laminates, J. Composite Materials, 19 (1985), pp. 29-50. 13. G.J. Dvorak and N. Laws, Mechanics of first ply failure in composite laminates, in Fracture of Fibrous Composites, AMD-Vol. 74, 1985, ASME, New York, pp. 59-69. 14. Y. M. Han and H. T. Hahn, Ply Cracking in Thermoplastic Composites, Interim Report prepared for E. I. du Pont de Nemours & Co., Dept of ESM, Pennsylvania State, University, University Park, PA, 1987. 15. Swanson Analysis Systems,Inc., ANSYS Engineering Analysis System User's Manual Vol. 1-2, March 1983.
A Simplified Analysis of Transverse Ply Cracking in Cross-Ply Laminates Y. M. Han, H. T. Hahn Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA 16802 (USA)
and R. B. C r o m a n E. I. du Pont de Nemours & Co., Inc., Pioneering Research Laboratory, Experimental Station, Wilmington, DE 19898 (USA) (Received 16 July 1987; accepted 29 October 1987)
SUMMAR Y This paper presents a method o f analyzing transverse crack initiation and multiplication in symmetric cross-ply laminates. The method is based on the concept of a through-the-thickness inherent flaw and the energy balance principle. With a second-order polynomial assumed for the crack opening displacement, the perturbed stress.field due to the presence of ply cracks is determined from the equilibrium conditions. The energy released as a result of ply cracking is then calculated and used to predict the increase in crack density. Based on an experimental correlation of the analytical result, a resistance curve is proposed to be used as a measure of the resistance to crack multiplication. The resistance to crack multiplication is shown to increase with the increasing crack density.
1 INTRODUCTION
Since the early 1970s the transverse ply cracking in composite laminates and the corresponding mechanical property degradation have been a subject of extensive research, both experimental and theoretical.I-14 165 Composites Science and Technology 0266-3538/88/$03"50 © 1988 Elsevier Applied Science Publishers Ltd, England. Printed in Great Britain
166
Y. M. Hart, H. T. Hahn, R. B. Croman
As a result the applied strain at the cracking of the first ply, called the firstply cracking strain, is known to depend not only on the thickness of the cracked ply but also on the stiffness of the adjacent ply. 6'9 Furthermore, at a given strain, the crack spacing increases as the ply thickness increases and the crack spacing in the outer transverse ply is only half of that in the inner ply when both have the same thickness, t t.14 Several analytical models have been proposed to predict the first-ply cracking strain as well as the crack density. Some models are based on the use of strength, 1'3-5 while others rely on the linear fracture mechanics 8-13 as shown in Fig. 1. The latter models can account for the dependence of the first-ply cracking Strain on the ply thickness.
Anal,sis I [Strength
I
I
Deterministic
I
Fig. 1.
Fracture
Model [
I
Mechanics
Model
I
i Model
Vasil'ev [1] Hahn [2] Garrett [3]
of Ply Cracking ]
I
I
Through-the-thickness]lThrough'the'width [ -flaw Model -flaw Model
Manders [4] Fukunaga [5]
Parvizi[8] Nuismer [9] Hahn [10] Dvorak [13]
I
Wang [11] Flaggs [12] Dvorak [13]
Flow chart for the methods of analyzing ply cracks.
Fukunaga et al., 5 investigated the failure characteristics of cross-ply laminates based on a statistical strength analysis. A shear lag analysis was used to determine stress distribution. The transverse strength was assumed to obey a two-parameter Weibull distribution. It was assumed that a new crack occurred midway between any two adjacent cracks at 50% failure probability. Parvizi e t al., 8 and Nuismer and Tan 9 investigated the failure behavior of cross-ply laminates from the energy point of view. The former investigators used a one-dimensional shear lag analysis while the latter used an approximate elasticity solution for stress analysis. The work of crack closure was used as the energy released as a result of ply cracking. Several fracture-mechanics-based models have been proposed which assume the growth of an inherent flaw to be the mechanism for ply cracking.lO-13 The inherent flaw could be entirely through the width but partially through the thickness, 1~ or vice versa. 1° The through-the-width flaw model was further extended to account for the statistical nature of the
Analysis of transverse ply cracking in cross-ply laminates
167
ply cracking. 11 While a finite element method was originally used for stress analysis by Wang, 11 a two-dimensional shear lag analysis was also applied by Flaggs. 12 Hahn and Johannesson 1o showed that the energy release rate associated with the widthwise growth of a long through-the-thickness-flaw is equal to the work of crack closure per unit width of the fully developed flaw. The energy release rate is independent of the flaw length and hence the growth of the flaw is stable. Dvorak and Laws13 considered a flaw which is both partially through the thickness and partially through the width. The required stress analysis is three-dimensional by contrast with the aforementioned two models. Two directions of the crack growth and the corresponding energy release rates were calculated, i.e. parallel to the fiber axis and parallel to the thickness direction of the laminate. In the present paper a simple method of analyzing transverse crack initiation and multiplication in symmetric cross-ply laminates is presented. The method is based on the through-the-thickness-flaw concept proposed by Hahn and Johannesson. 1° The requisite stress analysis is simplified by introducing a displacement function which describes a parabolic crack opening displacement. A resistance curve is proposed for characterization of the resistance to the multiplication of transverse cracks.
2 ANALYSIS OF T R A N S V E R S E C R A C K I N G When a symmetric cross-ply laminate is under tensile loading, ply cracks appear parallel to the fibers in the transverse ply long before the ultimate laminate failure (Fig. 2). The origins of these ply cracks are believed to be inherent defects such as voids, microcracks, debonded fiber-matrix interfaces and broken fibers. As the load increases, these defects act as local stress raisers, and grow or coalesce to form a through-the-thickness-flaw. Since the matrix and interface are much weaker than the fiber, these flaws always propagate through the matrix and interface parallel to the fibers. The energy release rate associated with such flaw growth has been shown to be independent of the flaw length.I° Moreover, this energy release rate is equal to the work of crack closure per unit ply width of the fully developed transverse crack. To predict the initiation and multiplication of transverse cracks, consider an idealized case of a cross-ply laminate containing a series of equally spaced cracks in the 90 ° plies. A representative element of this laminate containing only one crack is shown in Fig. 3, where 2L is the crack spacing. As indicated in the figure, this problem can be solved in two steps: a laminate without a crack subjected to the far-field stresses and a laminate with a loaded crack.
Y. M. Han, H. T. Hahn, R. B. Croman
168
P,c------
~/
(crack spacing)
l" " x
Z
90.Plies O-Plies
ho
t Transverse Fig. 2.
cracks
Coordinates for analysis of multiple transverse cracks.
-f
~
i T i'~
lllll lllll-Illlll Illlll"IIIIl lllll" IllltlIllll llllll " I111IIIIII. 11111 III1 ~*~, (a)
t~t
(b)
°°t f~t
:
(e) (f) Fig. 3. Stress analysis via superposition.
(c)
(d)
Analysis of transverse ply cracking in cross-ply laminates
169
The superposition of the two solutions will then provide the stress field in the laminate with a crack. The ply stresses before cracking due to the mechanical and thermal loading are determined by the classical laminated plate theory and are denoted by 0~, 0~o and 60T, 09T0, respectively. Here, subscripts 0 and 90 represent variables associated with the 0 ° and 90 ° plies, respectively. The applied load per unit specimen width, P, is simply given by P = 2(ff~h o + ~Moh9o)
(1)
since the thermal stresses are in self-equilibrium. The formation of a crack is equivalent to the application of a boundary stress, -090, on the crack surfaces, Fig. 3, where
¢~90 = ¢~Mo"[- ¢~9T0
(2)
To determine the resulting stress fields, ao and cr9o, in the laminate with a loaded crack, consider a quadrant of the representative element shown in Fig. 4. The perturbed stress, a o, in the 0 ° ply is assumed uniform through the
h~h~° ~ x I~
L
ill -I Fig. 4. A representative element for cracked laminate. thickness so that it is a function o f x only. If the shear stress at the interface is z, the equilibrium equation for the 0 ° ply becomes
do o T dx - ho
(3)
On the other hand, the equilibrium condition for the entire laminate at any cross-section requires that hoa o +
a9o dy = 0
(4)
In the x direction, the perturbed displacement, uo, is a function o f x only but u9o is assumed to be U9o(X,Y) = f l ( x ) Y 2 +f2(x)
The functions fa(X) and f2(x) are to be determined later.
(5)
170
Y. M. Hart, H. T. Hahn, R. B. Croman
The interfacial shear stress, z, follows from •U9o I
27= G T ~ - y
y=hgo = 2hgoGTfl
(6)
where GT is the transverse shear modulus. The 90 ° ply stress, 0-90, also follows from U9o as (~U90
a9o = ET--~-x = ET(y2f~ +.[2)
(7)
where a prime denotes differentiation with respect to x, and E T is the transverse Young's modulus of the lamina. The displacement continuity condition at the interface requires that b/0(X) : U90(x, Y)ly: ho,, =./1 h2o +./2
(8)
Therefore, the 0 ° ply stress, o o, becomes duo
ao = EL~--f = EL(h~of~ + f~)
(9)
where E L is the longitudinal Young's modulus of the lamina. Equations (4), (7) and (9) are now solved for J~ and./i to yield
.[~ = Floo
f~ = F20-o
(10)
where
3 hoE c + h9oEv F1 = 2 Ec Evh3o
F2 -
1 Ec
h2oF 1
(11)
Thus the governing equation for o o follows from eqns (3), (6) and (10) as -
~
ao
(12)
where ¢2 : 3GxhoEc + hooEy
(13)
hoEc Ev
2.1 First-ply cracking When there is only one crack in the 90 ° ply, the b o u n d a r y conditions to be satisfied are h,, ao = C 0 9 o Uo = 0 at x 0 (14) dao .[~ = 0 dx - 0 as x --* oc The solutions for the perturbed stress field, 6 o, and the displacement, Uoo, are
Analysis of transverse ply cracking in cross-ply laminates
171
then O"o = A f f 9 o
e-~
(15)
U90 = (A/flp)~90h90[Fl(h20 _ y2
e-4,~) +
]72(1 _
e-~X)]
where A = h9o/h o
.~ = x/h90
(16)
Now that the perturbed stress and displacement fields are known, one can calculate the energy release, A U, per unit laminate width associated with the formation of the crack. Specifically, AU is given by 1° AU = 2 f[9o ff90U90lx=Ody 2 -2 ,2 [hoEc + h90ET~ 1/2
(17)
=N//30"90n90~ hooEL-E-~T : The criterion for first-ply cracking is then AU = 4h9oTv
(18)
where Yv is the surface fracture energy density. Thus, the ply stress, (ff9o)vPc, required for the first cracking of the 90 ° ply and the corresponding ply strain, (gg0)vPC, are given by
(~90)FPC= (~F ~1/2 (12hoELETGT ~1/4 (19)
1
( 9o) Pc = (O9o) Pc Equation (19) shows the dependence of (tr90)FPC on the 90 ° ply thickness and the modulus of the adjacent plies. Note that (ff90)FPCincludes the residual ply stress as well.
2.2 Crack multiplication When there are several cracks with a uniform spacing 2L, the boundary conditions at infinity are replaced by fl=0
dcr o
dx - 0 a t x = L
(20)
The solutions are given by 6 o = A09o(e -~-~ + e-2~Ce ~s) -~~ + e 2,c eO.~)(Fly2 + F2 ) + (1 - e 2~C)(Flh2 o +/72) ]
/290~---(A/~))69oh9o[(--e
(21)
Y. M. Han, H. T. Hahn, R. B. Croman
172
where
1 h9o A = 1 + e -24'L h o
x x = hgo
E=
L
(22)
h9o
The energy release, AU per crack, is given by AU= 2
~0hg°~9oU9olx=odY 2
--
(23)
-2 " 2 I/hoEL + h9o Ev'~I/2
- - G90~90 . . . ,,~ t .hoELErGr )
tanh(q~E)
When there are Ncracks per unit length, the total energy release is N. AU. To account for crack multiplication the authors postulate that at a given value of 690, the crack density changes from N to N + d N when the additional energy released is equal to the energy absorbed. That is, d(N. AU) = 4h9o7 F d N
(24)
Noting that N = 1/(2L) and using eqn (23), one can obtain the ply stress, (690)N, required to produce N cracks as
l/.;F,,/21/ 12hoELETGT'~,,4[ n. Ck 4) (fY90)N=t~90) th0~-~-L~- ) tta n ~ - ~ s e c h
2
~)-~
(25)
where IV is a normalized crack density defined by :7 = hgo. N
(26)
Note that N represents the number of cracks over a length equal to the thickness of the 90 ° ply. Equation (25) reduces to eqn (19) when the crack density goes to zero, i.e. for the case of the crack initiation. The additional average axial strain due to cracking can also be obtained from
Ag= u°(L) - 1 590 h9o ~ hoELET ]1/2 L x~ Ec ~oE[_G-r(ho-E77h9oE-r) tanh(q~/~)
(27)
The total average axial strain is thus 0"9O
= E 7 + Ae
t28)
The applied load, P, and applied stress, %, corresponding to a crack density,
Analysis of transverse ply cracking in cross-ply laminates
173
.~', are obtained from eqns (1), (2) and (25) as P = 2 (h°Er + hg°Er) [( 9o)N - a o] ET P ~ra - 2(ho + h 9 o )
(29)
3 DISCUSSIONS
To check the validity of using a second-order polynomial for the crack opening displacement, a finite element code 'ANSYS' was used to determine the displacement field in a [0/90] s laminate. I s Because of symmetry, only one quadrant of the section was modeled by using 2-D plane strain elements Y
J" ~
~, Fig. 5.
h0
,777777774 A finite element model for the representative element.
(Fig. 5). The material constants for the T300/934 graphite/epoxy composite used in the analysis are in Table 1. Figure 6 shows the crack opening displacements of [0/90]~ laminate under 1 MPa tensile loading. The displacement from eqn (15) agrees fairly well with TABLE 1
Mechanical Properties of T300/93412 Longitudinal modulus, Ell Transverse modulus, E22 In-plane shear modulus, G12 Transverse shear modulus, Gz3 In-plane Poisson's ratio, vl2 Out-plane Poisson's ratio, v23 Longitudinal tensile strength, X Transverse tensile strength, Y Thermal expansion coefficient, ~l Thermal expansion coefficient, ~2 Stress free temperature, AT Critical energy release rate, Glc (Re£ 11) Lamina thickness, t
138 GPa 11.7 GPa 4.56 GPa 4,18 GPa 0.29 0.4 1 724 MPa (Ref. 11) 44.8 MPa (Ref. 11) 0.09 pm m - 1 °C- l 28"8 p m m - 1 ° C - l - 129.4oc 158 J m - 2 (0/0 interface) 228 J m - 2 (90/90 interface) 0-132 mm
174
Y. M. Han, H. T. Hahn, R. B. Croman
2.5
~" 2.0 & 1.5
1.o }5 o.~
•-m- Eq.(15) Infinite plate -0- FEM
O.C
- 1 . 0 - 0 . 8 - 0 . 6 - 0 , 4 - 0 . 2 - 0 . 0 0.2 0.4 0.6 0.8 1.0 Fig. 6.
Y~90 Crack opening displacement in T300/934 [0/90]~ laminate under I MPa tensile load.
the finite element result. Since the energy release rate due to cracking is directly proportional to the area beneath the crack opening displacement, the calculated energy release from eqn (17) is seen to be only 8 % smaller than that from the finite element analysis. For comparison purposes, the displacement for an unconstrained crack in an infinite plate is also shown in Fig. 6. The constraining effect of the adjacent plies on the crack opening displacement is obvious. The assumed displacement field has a shear stress discontinuity across the interface, a shortcoming of the proposed approximation. However, it should be noted that most shear lag analyses have the same shear stress discontinuity. Furthermore, the crack opening displacement resulting from the assumed displacement field is in good agreement with that from a finite element stress analysis. Therefore, the assumed displacement field is expected to be a good approximation. The fracture criteria in eqns (18) and (24) apply to flaws propagating parallel to the fibers. It is noted that Wang's model ' ' uses an energy release rate for normal crack propagation. In Fig. 7, the ply strains, (gg0)vPC, from eqn (19) are compared with the experimental results 6 for [02/90n] s T300/934 laminate family. The analytical results based on the 1-D shear lag model of Parvizi) the 2-D elasticity model of Nuismer, 9 and the 2-D shear lag model of Flaggs ,2 are also shown in Fig. 7. The predictions of the present model fall between those of the two models from Refs 8 and 12. When there are fewer than 6 transverse plies, the predictions agree well with the experimental results. But the deviation from the experimental results increases as the transverse ply thickness increases. This observation indicates that the mode of transverse ply cracking changes as the thickness of the transverse ply changes. When the 90 ° ply is thin, the assumption of the inherent flaw being a through-the-thickness type is reasonable. However, as the 90 '~ply becomes thicker, the same assumption is
Analysis of transverse ply cracking in cross-ply laminates
175
1.2
\
1.0 {J
0.8
iw~, 0.6 ~
v
"---. ~ ""
~
:e- Parvizi t8] ~ Nuismer [9]
!
4-
0.4
I
~ ~ , * -
0,2
I
]
1
2
3
[
Faag~s t121 Eq. (19)
5 6 7 8 Number of transverse plies (ra)
4
Predictions of first ply cracking strains (ggo)vPcin T300/934 [0z/90,], laminates.
Fig. 7.
not likely to hold true. Since the analytical models s'9 are based on the presence of a through-the-thickness flaw, their predictions are expected to become worse as the 90 ° ply thickness increases. The transition occurs when n is about 3. For thick 90 ° plies, one may use an inherent flaw which is neither through-the-thickness nor through-the-width, as was studied by D v o r a k and Laws.13 In Fig. 8, the predicted stress, u,, required to produce a crack density, N (eqn (29)), is c o m p a r e d with the experimental data 11 for uniaxially loaded [0/90,/0]x T300/934 laminates. By contrast with the experimental results, the predicted stress, a a, remains fairly independent of the crack density. Equation (23) indicates that the energy release per crack does not increase m u c h until the crack spacing becomes small enough to allow interaction between neighboring cracks. Consequently, the model predicts the formation of m a n y cracks almost simultaneously without m u c h increase in 900
:
~', 600
..
[]
,n[n-2] I:l
[] ~
.1.-31 Eq. (29, mmL.-Q
=~ 300 •
< 0
0
•
•
•
•
I
I
m
3
6
9
•
•
i
•
[
•
i
°p-q °lo-d
12 15 18 Crack density (era"1)
Fig. 8. Applied stress o-, vs crack density in T300/934 [O/90n/O]Tlaminates.
176
Y. M. Han, H. T. Hahn, R. B. Croman
300 •
u nn
•
200 [] • m •me [] 100
rnm
0
i,
g
[]
[][]
[]
[]
[] n=2
I
,
3
I
6
n
,
l
,
I
,,
,I
•
I1=3
•
11=4
i
12 15 18 Crack density (cm"1) Resistance curves for crack multiplication obtained for T300/934 [0/90./0]T laminates. 0
Fig. 9,
[]
9
load. However, the experimental data indicate more gradual crack multiplication. One possible reason is that the material is not really homogeneous in fracture energy, and hence 7v will be an increasing function of N. Thus, eqn (29) can be used to determine }'v from the measured P-N relations. The results are shown in Fig. 9. It is interesting that 7v is almost a linear function of N and the slope increases as the transverse ply thickness increases. The relation between 7r and N has a similar meaning as the relation between fracture energy density and crack length during crack growth in homogeneous materials. Therefore, the 7v-N relations shown in Fig. 9 are called the resistance curves for crack multiplication. Whereas the resistance curve for homogeneous materials characterizes the increasing resistance to crack growth due to crack tip damage, the new resistance curve for crack multiplication in composite laminates represents the inhomogeneity of fracture energy. 4 CONCLUSIONS A simple method has been proposed to calculate the energy released as a result of ply cracking. The method is based on the use of a second order polynomial to represent the crack opening displacement. The calculated energy release per ply crack is then used in conjunction with the energy balance principle to predict the multiplication of ply cracks. A resistance curve for crack multiplication is proposed as a means of characterizing the resistance to ply cracking in composite laminates.
Analysis of transverse ply cracking in cross-ply laminates
177
ACKNOWLEDGEMENTS This paper is based on work supported by a grant from E. I. du Pont de Nemours Co. REFERENCES 1. V. V. Vasil'ev, A. A. Dudchenko and A. N. Elpat'evskii, Analysis of the tensile deformation of glass-reinforced plastics, Mekhanika Polimerov, 1 (1970), pp. 144-7. 2. H. T. Hahn and S. W. Tsai, On the behavior of composite laminates after initial failure, J. Composite Materials, 8 (1974), pp. 288-305. 3. K. W. Garrett and J. E. Bailey, Multiple transverse fracture in 90° cross-ply laminates of a glass fiber-reinforced polyester, J. Material Science, 12 (1977), pp. 157-68. 4. P. W. Manders, T. W. Chou, F. R. Jones and J. W. Rock, Statistical analysis of multiple fracture in 0/90/0 glass fiber/epoxy resin laminates, J. Materials Science, 18 (1983), pp. 2876-89. 5. H. Fukunaga, T. W. Chou, P. W. M. Peters and K. Schulte, Probabilistic failure strength analysis of graphite/epoxy cross-ply laminates, J. Composite Materials, 18 (1984), pp. 339-56. 6. D. L. Flaggs and M. H. Kural, Experimental determination of the in situ transverse lamina strength in graphite/epoxy laminates. J. Composite Materials, 16 (1982), pp. 103 16. 7. A. L. Highsmith and K. L. Reifsnider, Stiffness-Reduction Mechanisms in Composite Laminates, ASTM STP 775 (1982), pp. 103-17. 8. A. Parvizi, K. W. Garrett and J. E. Bailey, Constrained cracking in glass fiberreinforced epoxy cross-ply laminates, J. Materials Science, 13 (1978), pp. 195 201. 9. R. J. Nuismer and S. C. Tan, The role of matrix cracking in the continuum constitutive behavior of a damaged composite ply, Proc. of the IUTAM Symposium on Mechanics of Composite Material Virginia Polytechnic Institute and State University, Blacksburg, VA, 1982. 10. H. T. Hahn and T. Johannesson, Fracture of unidirectional composites: theory and applications, in Mechanics of Composite Materials, AMD-Vol. 58, !983, ASME, New York, pp. 135-42. 11. A. S. D. Wang, Fracture Analysis of Matrix Cracking in Laminated Composites, NADC-85118-60, 1985. 12. D. L. Flaggs, Prediction of tensile matrix failure in composite laminates, J. Composite Materials, 19 (1985), pp. 29-50. 13. G.J. Dvorak and N. Laws, Mechanics of first ply failure in composite laminates, in Fracture of Fibrous Composites, AMD-Vol. 74, 1985, ASME, New York, pp. 59-69. 14. Y. M. Han and H. T. Hahn, Ply Cracking in Thermoplastic Composites, Interim Report prepared for E. I. du Pont de Nemours & Co., Dept of ESM, Pennsylvania State, University, University Park, PA, 1987. 15. Swanson Analysis Systems,Inc., ANSYS Engineering Analysis System User's Manual Vol. 1-2, March 1983.