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Mixed-mode fracture of notched composite laminates under uniaxial and multiaxial loading
EngineeringFractureMeckanics Vol. 31. No. 5, pp. 733-746, 1988 Printed in Great Britain.
0013-7944/88 s3.00+ .xl 0 1988 Pergamon Press pk.
MIXED-MODE FRACTURE OF NOTCHED COMPOSITE LAMINATES UNDER UNIAXIAL AND MULTIAXIAL LOADING SENG C. TAN AdTech System Research Inc. (On-site), US Air Force Materials Laboratory, Wright Patterson AFB, AFWAL/MLBM, Dayton, OH 45433-6533, U.S.A. Abstract-Two models are proposed to predict the mixed-mode fracture of notched composite laminates subjected to uniaxial and multiaxial loading. The basic elastic, strength properties and a characteristic length are utilized by these models. The characteristic length is determined differently for the two models. A first-ply-failure criterion or a fiber failure criterion is applied in conjunction with the models. It was found that failure of notched laminates for the fiber dominated mode, matrix dominated mode or a combination of the two modes under different loading conditions can all be predicted very well and the characteristic length is independent of loading conditions.
INTRODUCTION strength of orthotropic composite laminate with a circular opening or a straight crack under uniaxial tensile or compressive loading can generally be predicted well using some of the failure criteria documented in the literature. The mixed-mode failure of unnotched laminate on the first-ply-failure basis can also be handled quite well generally. On the ultimate strength basis, simplifying assumptions are needed to avoid the extreme complexity of the failure mechanisms under a single or combined load. In some cases, a mixed-mode failure can be predicted by using a simple model yet provides reasonably good result. For instance, a multidirectional laminate containing a slant crack under uniaxial normal loading has been modeled[l] by using an effective normal crack with a crack length equal to the normal projection of the slant crack. The point stress failure criterion[2] was then applied incorporating with their model. This approach, however, can only be applied in uniaxial normal loading case. The notched strength of composite laminate containing a circular hole or a center crack due to biaxial loading has been studied previously by some researches using the Tsai-Hill failure criterion[3-41 and the maximum stress, maximum strain and the tensor polynomial failure criterion[5]. It is critical to point out that although fiber failure is assumed the ultimate failure by special considerations of those criteria[3-51, different fiber failure criterion can result in significant different prediction. For instance, two distinct characteristic lengths are needed for tensile and compressive loadings if the Yamada-Sun’s criterion[6] is applied. However, only one characteristic length is required for those two loading conditions if the present fiber failure criterion is utilized. The approach in [5] is a special version of progressive failure model which involves a characteristic dimension and considers the redistribution of stresses only through the thickness of the laminate. The present research shows that the notched strength of composite laminates under multiaxial in-plane loading can be modeled in a compact-form solution yet the accuracy is assured by a number of comparisons. In addition, if a laminate fails in a fiber dominated mode due to one loading condition and matrix dominated mode due to some other loading conditions, the present model is able to predict the notched strength by using one characteristic length. This capability has not been shown in literature before by any other models.
THE NOTCHED
2. STRESS DISTRIBUTION The stress distribution of an infinite anisotropic laminate containing an elliptical opening under in-plane loading can be derived using Lekhnitskii’s complex potentials[7-81. The laminate is assumed homogeneous and symmetric with respect to its mid-plane (Fig. 1). The in-plane stresses (using index notation) given in principal coordinates of the opening are tFM 215-B
733
SENG C. TAN
734
Fig. 1. Coordinate system and the characteristic length, b,,, for the minimum strength model.
u1 =
(14
2 Re [d41(zl) + &&(z2)1
m = 2 Re [K(zd + 4X41
(lb)
a6
(ICI
=
-2
Re
where C#I;and C#J;denote the derivatives respectively, and
[h+;(zd
+
Pz~;(zz)I
of the functions
4r and & with respect to z1 and z2,
4*(z,) = p1- *o* -!_ Pl-
&(z2)
=
-
P2
PI
-
(24
51
*al
ccl-l-h
W-9
l r2
where cyr = - (~7,/2) cos tJ( a cos I&+ ib sin +I)- (~7~/2) sin $( a sin JI - ib cos 4) +(?,,/2)(asin2$-iibcos2+) p1 = -(&/2)
(34
/2) cos I++( a sin $ - ib cos I,!J) sin +( a cos I,$+ ib sin +) + (CT,, - ( T~,,/2)( a cos 2+ + ib sin t,G)
(3b)
and zi 5i =
+
Jz:
-
a-
a2 - p$b2 ipib ’
i= 1,2
(4)
where zi = n + /.Liy, i = 1,2 and where @I and I_L~are the principal roots of the characteristic equation:
aw4-
2a16p3
+(2a12+
a66)p2-2a26p+
a22
=
0
(5)
where a,, i, j = 1,2,6 are the laminate compliances with 1 and 2 in the major and minor axes of the opening, respectively. Substituting eq. (2a and b) into (1) yields
Ui=UT+Uq,
i=l,2,6
(6)
Mixed-mode fracture of notched composite laminates
735
where ai is the overall laminate stress, a? represents the stress components due to the opening and C$ designates the components due to the uniform stress field. The stress components, a:, of eq. (6) can be expressed in detail in the following: oT=Re
(74
aq=Re
(7b)
a: = Re
(7c)
where (1 - i&A)
h= PJP2-1-~~A2+P2-1-CL:A2
(84
gj=ihUy-pjO$-(l-i&A)Ug
@b)
where i = 1,2 and
The stress components
p=(l+(Y)cos0+&(A+~)sin8
(SC)
A = b/a
(84
a = bola.
We)
due to the uniform stress field are (9a) u~=&~os~JI+~~sin~*-~~~sin2*
(9b)
(Y:= (c?~- f?J sin I+% cos $ - fXYcos 21c,
(9c)
where a and b are the semi-major and semi-minor axes of the elliptical opening respectively; b. is a characteristic dimension which indicates a distance b. between the characteristic curve and the opening contour; I,Qdesignates the slant angle from the y-axis. c&.,a,, and fXYare the applied normal and shear stresses. The stress transformation between the laminate axes and the principal axes of the elliptical opening is given by eq. (9a-c). Equation (5), for anisotropic laminate, can be solved numerically. For orthotropic laminates, aI6 = a26 = 0, eq. (5) becomes
The solutions of eq. (10) are cL2=-
2&Z + a66 2a 11
(11)
Solving eq. (11) yields ,&,2,3,4=
&[COS
(0/2 + k?r) + i sin (8/2 + k?r)],
k=O,l
(12)
736
SENGC.TAN
where
0 = tan-’
(14) -@a12
+
&6)/(2&l)
’
Only the two principal roots of eq. (12) are useful for elasticity’s problem. If 8, eq. (14), lies in the second or third quadrant, it has to be subtracted from 7r before substituting into eq. (12). 3. FAILURE
MECHANISMS,
CRITERIA
AND MODELS
The failure of composite laminates can generally be classified into fiber mode and matrix mode. The ultimate fracture of a multidirectional laminate which contains longitudinal plies with fibers in the principal loading directions is dominated by fibers. From experimental results for composite laminates containing stress concentrations under tensile or compressive loading, one usually finds that the failure initiates with matrix cracking at a small region near the discontinuities. As the applied load increases, some degrees of fiber matrix splitting will appear at the opening edge for most laminates. The local damages release the intensed stresses and redistribute the stress concentration. Therefore, the stress concentration at the final stage of fracture is not as high as that for a virgin laminate (due to the degradation in laminate stiffness). Due to the extreme complexity of the fracture mechanisms, all solutions found in literatures are pursued with some simplifying assumptions. In this paper, a characteristic damage zone is utilized to compute the stress concentration of the laminate, which is considered as an “effective stress concentration” at the final stage of the failure process. The effective stress concentration (stress concentration at the point or points associated with the characteristic length) can be considered as a one-dimensional (point-wise) or two-dimensional (average over the characteristic length) parameters. The characteristic length was assumed as a constant and a two-parameter power function[8] to predict the strength reduction due to the hole size effect . Two criteria: FPF (First-Ply-Failure) and a FF (Fiber Failure) criteria[6] were applied to predict the mixed-mode fracture of multidirectional laminates under uniaxial and multiaxial loading. The Tsai-Wu quadratic failure criterion[9] with the stress interaction term, FT2, equal to -0.5 was chosen for the FPF analysis whereas the following criterion was applied for the fiber failure (FF) prediction.
4
-----A-cr,
XX’
(
i-i
>=
ef
(15)
where gl is the lamina normal stress in the fibers direction, X and X’ denote the longitudinal lamina tensile and compressive strength, respectively. If any ply within a laminate satisfies the condition: ef zz 1, the ultimate failure is considered occurred. Two models are described below for the notched strength prediction. 3.1. Point strength model (PSM) The characteristic length, b,, in this model is determined along the assumed fracture propagating plane, Fig. 2, and its magnitude is obtained by back substituting one notched strength data into the stress and strength analyses. In the case of an orthotropic laminate ember-controlled) containing a circular hole or a normal crack under uniaxial loading, the fracture plane is perpendicular to the loading direction and the stress state along the fracture propagating plane can be expressed in a compact form[ lo]. For cases other than that, eqs (1-14) will be applied. Once the laminate stresses are obtained, the lamina stresses can be calculated using the classical laminated plate theory. Then the FPF and the FF, eq. (15), criteria are utilized ply-by-ply to predict the notched strength reduction factor and the notched strength (absolute value) respectively.
Mixed-mode fracture of notched composite laminates
Fig. 2. The characteristic
3.2.
Minimum
strength model
131
length, b,, utilized for the point strength model.
(MSM)
When a laminate with stress concentrations is subjected to multiaxial loading, the direction of the fracture propagating plane depends on the stress ratios and the lay-ups of the laminate. Unlike the previous model, the fracture propagating direction will also be predicted. In this model, the strength distribution is analysed point-by-point along a characteristic curve which is at a distance, bo, away from the opening contour, Fig. 1. The characteristic curve can be written explicitly as 2 2 (16) (a:b,)2+&~= 1 where b. is determined by a similar method as the previous model. The laminate and lamina stresses are computed, for all points selected along this characteristic curve [using eqs (t-14) and laminated plate theory]. The minimum allowable stress that satisfies the FPF strength or the FF condition, eq. (15), is computed ply-by-ply. The point that first satisfies condition (15) indicates the direction of the fracture propagating plane. Under a combined loading, the following expressions are utilized to relate the notched strength reduction as a ratio of the uniaxial unnotched strength. When the FPF criterion is applied:
SRF=z=
x-component of FPF notched strength Uniaxial FPF unnotched strength (x-axis loading)
(17)
and when the fiber failure criterion, eq. (15), is utilized
SRF=%= aox
x-component of notched FF strength Uniaxial unnotched FF strength (x-axis loading)
(18)
where SRF denotes the strength reduction factor, uNX is the x-component of the ultimate notched strength under combined loading and CT ox is the ultimate unnotched strength due to a uniaxial loading in the x-direction. The characteristic length, bo, for a laminate under combined loading is assumed the same as that under uniaxial loading. Therefore, the characteristic length, bo, of a laminate can be determined using one notched strength data under uniaxial loading. 4. UNNOTCHED
ULTIMATE
STRENGTH
Two material systems, graphite/PEEK AS4/APC-2 and graphite/epoxy T300/SP-286 were studied. The unnotched ultimate strength of a laminate can be predicted using the laminated plate theory for the ply stresses and the fiber failure criterion, (eq. 15). The material properties are listed in Table l(a-b). The comparison, Table 2, shows that the predicted unnotched ultimate strength agrees reasonably well with the experimental data.
SENG C. TAN
738
Table l(a). Elastic properties of the graphite/PEEK Parameter
Longitudinal modulus, Et, Transverse modulus, Ez2 In-plane shear modulus, G,, Poission’s ratio, VIZ
and graphite/epoxy
lamina
AWAPC-2 * lo6 psi (CPa)
T300/SP-286 * lo6 psi @Pa)
19.435 (134.0) 1.291 ( 8.9) 0.739 ( 5.1)
21.6 (149.0) 1.53 ( 10.6) 0.93 ( 6.4)
0.28
0.31
Table l(b). Strength properties Parameter Longitudinal tensile strength, X Longitudinal compressive strength, X’
AWAPC-2 Ksi (MPa)
T3OO/SP-286 Ksi (MPa)
308.9 (2130) 159.5 (1100)
214 (1477) 164 (1132)
Table 2. Comparison of the predicted ultimate unnotched strength and the experimental data for the; (A) T300/SP-286 and; (B) AS4/APC-2 under uniaxial loading Material system
Lay-up
Prediction Ksi (MPa)
Experimental data Ksi (MPa)
(A)
10#451,
125 (863)
116 (800)
(A) (A)
[o/*45/90], [-30/15/-75/601,
84 (579) 51 (354)
73 (504) 60 (414)
(B)
[90/O/ f45] s
120 (824)
126 (870)
5. UNIAXIAL
LOADING
OF LAMINATES
WITH A SLANT CRACK
An earlier study[l l] has shown that an implanted crack (fabricated using a 0.005 in. diameter diamond wire) can be simulated very well by an elliptical opening with aspect ratio, a/b, equal to 50. The difference in stress profile for this elliptical opening and a theoretical crack is negligible except for a very small region near the crack tips, say (x - a)/a < 0.001. In addition to the PSM, a slant crack is also investigated by a normal crack which has a length equal to the normal projection of the slant crack. For this effective normal crack, a characteristic length, co, is defined along the normal crack plane. This plane coincides with the failure propagating direction at one end of the crack (Fig. 3). The strength reduction of the effective normal crack can be solved using eqs (l-15) or from [6] for the extreme case (b = 0):
where 0, is the applied stress and ap designates the laminate normal stress in the x-direction the point (0, co) and al/~,
= (Q,,S,,
+ Q12%)
cm*
4 + (Q12%1
+ Q22S12)
sin* +
at
4
2(Q1d%1 + Q&J
sin 4 cos 4
(20)
where Qij and gij, i, j = 1,2,6 are the lamina stiff nesses and laminate compliances, respectively. u1 has already been defined and related to the failure condition (eq. 15). The “effective normal crack” described above can be treated as a special case of the point strength model because it can only be applied under uniaxial loading while the point strength model can be applied under uniaxial and combined loading conditions. The predictions using these models and the experimental data are compared in Figs 4-5 for
Mixed-mode fracture of notched composite laminates
Fig. 3. Failed specimen of an AS4/APC-2
graphite/PEEK 45” crack.
[90/O/*45& laminate containing
739
a slant
Mixed-mode fracture of notched composite laminates
C90/0/:45ls.
0.
.
EXPERIMENTAL
a/b=50
DATA
0.40
CRACK
z
Fig. 4. Predicted
HALF
741
LENGTH,
a
I 3. 50
(INCHES)
(FF criterion) and experimental notched strength for the Gr/PEEK [90/O/*451, laminate with a slanted 45” crack.
AS4/APC-2
the AS4/APC-2 graphite/PEEK [90/O/*45], laminate with a slant 45” crack and the T300/SP286 graphite/epoxy [-19/61/26/-641, laminate with a slant 19” crack, respectively. The [-19/71/26/-641, laminate is in fact a lr/4 laminate with respect to the principal axes of the opening. From these comparisons, it is of interest to see that the characteristic lengths, b. = 0.05 in. (MSM), for these two laminates (different material systems and different slant angles) are the same as the one for the graphite/epoxy AS4/3502 [O/90/*451, laminate with a normal crack[6]. The best fit characteristic lengths using the PSM are: bl = 0.025 in. for the [-19/71/26/-641, and b, = AS4/APC-2 [90/O/*45],, bl = 0.022 in. for the T300/SP-286 0.025 in. for the AS4/3502 [O/90/*45], and [O/90/0], laminates[6]. This reveals that constant characteristic lengths, b. = 0.05 in. (MSM) and bl = 0.025 in. (PSM), are applicable to many material systems of quasi-isotropic laminates, containing a crack, with reasonably good result. If the characteristic length, bl, is taken along the crack plane, the notched strength can be predicted quite accurately. The predicted locations of initiation of failure by the MSM lie in the middle of the crack plane and the axis normal to the applied load. Whereas the experimental result is along the crack plane and on the axis perpendicular to the applied load, Fig. 3. C-19/71/26/-641sra/b=50 0"
.
EXPERIMENTAL
!jzjj 0.00
0.10
CRACK
Fig. 5. Predicted
0.20
HALF
DATA
1
0.30
LENGTH.
0.40
a
0.50
(INCHES)
(FF criterion) and measured tensile strength for the Gr/EP [-19/71/26/-641. laminate with a slanted 19” crack.
T3OO/SP-286
SENG C. TAN
742
SP-2861300,
(O/-+45/90)
5,
a/b=1
[b,=0.04b”l
EXPERIMENTAL
~d+.~~.~~.-~~...~~....~ 0.30 0.10 0.20 HOLE
RADIUS.
0.30
a
0.40
DATA
.‘.I
0.50
(INCHES)
Fig. 6. Predicted ultimate strength reduction (MSM-FPF) and observed data for the Gr/Ep T3OO/SP286 [O/~k45/90], laminate with a circular opening.
6, FRACTURE
UNDER MULTIAXIAL
LOADING
The predicted notched strengths for some graphite/epoxy .r3OO/SP-286 laminates subjected to combined in-plane loadings were predicted and compared to experimental data[12-131 reported by Daniel. The MSM is applied in conjunction with both the FPF criterion and the fiber failure criterion. 6.1. Result by the FPF cr~ter~~~ In the present analysis, each two successive points along the characteristic curve is 2-3” apart for the circular hole case and 1” apart from the crack case. Figures 6 and ‘7 illustrate the results for the [0/~1~45/90], and [OJrt45], laminates containing a circular opening respectively. The former laminate was subjected to a uniaxial and an equal biaxial loading while the later was loaded uniaxialiy and biaxially with a stress ratio 6 + : cFy= 2 : 1. Due to the symmetry of the laminate and the loading condition, only one-quarter of the laminate needs to be analysed. The experimental data agrees reasonably well with the predictions using characteristic lengths, bo = 0.045 in. and 0.065 in., respectively, determined by uniaxial data. It is very important to SP-286T300.
(02/i45)
EXPERIMENTAL IO. s. c
0
ap*=i:o
A
i&:+Z:I
s
F
a/b=1
DATA
$00 ~Do.oo
0.10
0.20
HOLE
Fig. 7. Ultimate strength
RADIUS,
0.30
a
0.40
0.50
(INCHES1
prediction (MSM-FPF) and observed data for the Gr/Ep T3OO/SP-286 [0,/*45], laminate with a circular hole.
Mixed-mode fracture of notched composite laminates
GR/EP,
743
(-30/15/-75/60)s.a/b=50
;n”o:oo
0:40 v
a
0150
(INCHES)
Fig. 8. Predicted ultimate strength (MSM-FPF) and measured data for the Gr/Ep [-30/15/-75/60]s laminate with a slanted crack.
point out that the [O&45], laminate fails at different modes (fiber dominated and matrix dominated) under the prescript uniaxial and biaxial loading conditions. The comparison of a [-30/15/-75/601, laminate (quasi-isotropic laminate with respect to the principal axes of the opening) containing an inclined crack (30” from the y-axis) is shown in Fig. 8. The characteristic length, b0 = 0.11 in., is shown the same for different biaxial loading conditions. 6.2. Result by the fiber failure criterion The application of fiber failure criterion directly determines the ultimate strength and strength ratio. The notched strength of the [O/*45/90], laminate, Fig. 9, under a uniaxial and biaxial loading can be predicted using a same bO.The predicted initiation locations of failure under a uniaxial load is between 45” (small hole) and 0” (large hole) from the axis perpendicular to the loading direction. The observed result is 22.5”. Under equal biaxial loading, the predicted failure initiates along the laminate axes and the &45” directions, whereas the experimental results are at a 22.5“ from the predicted locations. In Fig. 10, the observed failure locations of the [O&45], laminate under uniaxial loading (fiber dominated) initiate at 18” directions from the CO/+45/9Ols
T300/SP-286,
EXPERIMENTAL
HOLE
RADIUS.
a
DATA
(INCH)
Fig. 9. Predicted ultimate strength [MSM-FF, eq. (15)] and observed data for the Gr/Ep T300/SP-286 [O/*45/90], laminate with a circular opening.
744
SENG C. TAN
T3OO/SP-286,
HOLE
Fig. 10. Predicted
(02/+45)s
RADIUS.
a
(INCH)
ultimate strength (MSM-FF) and observed data for the Gr/Ep T300/SP-286 [0+451, laminate with a circular hole.
axis perpendicular to the loading axis. This value is about the average of the predictions: 44” for small holes and 0” for large holes. The fracture mechanism resulting from the biaxial loading, 6, : 6,, = 2: 1, is a combination of matrix cracking in the 0” ply and the *45” plies and some amounts of interlaminar shear. This loading case is not within the “fiber dominated” domain and, therefore, the notched strength is not possible to be predicted by using a b,, determined from a uniaxial test along the fiber dominated axis (no matter what kind of fiber failure criterion). The result of the [-30/15/-75/60], laminate containing a slant crack under biaxial loading is shown in Fig. 11. Excellent correlation of prediction with experimental data is obtained. A [-30/15/-75/60], laminate with an inclined crack (30” from the y-axis) under a far field biaxial normal loading (0%: ay = 2 : 1) is equivalent to a quasi-isotropic laminate with a horizontal crack under multiaxial loading, C?~: Oy : TX,,= 2: 1.4:0.35 (Fig. 12). The notched strength predicted by using a characteristic length b. = 0.11 in. agrees reasonably well with the observed data. Although a slightly better fit yields if b. = 0.08 in. is applied for the multiaxial loading with non-zero shear, the difference between the prediction with b0 = 0.11 in. and the data is not significant. The predicted locations of the initiation of failure under uniaxial loading are at 34-37” from the crack direction, which correlates well with experimental value, 40”. The
F.l....,....,....,....,....,
~“0.00
0.10 CRACK
0.20
0.30
HALF
LENGTH,
0.40
a
0.50
(INCH)
Fig Il. Comparison of the predicted strength reduction (MSM-FF) and experimental Gr/Ep T30O/SP-286 [-30/15/-75/60], laminate with a slanted crack.
data for the
Mixed-mode
fracture
of notched
T3OO/SP-286,
EXPENIPIERTAL 0
+-d
0-z 0.00
12. Predicted
laminates
745
C0/+45/901sva/b=50
DATA
a;:iffy:?~'l:o:o
0.10
CRACK Fig.
composite
0.20
HALF
0.30
LENGTH.
0.40
a
0.50
(INCH1
ultimate strength (MSM-FF) and observed data [O/+45/90], laminate with a crack under in-plane
for the Gr/Ep loading.
T3OO/SP-286
predicted crack propagation directions under the given combined stresses are at an angle between 67” (small cracks) and 38“ (large cracks) from the crack plane, whereas the experimental result is at 45” directions from the crack plane. 7. DISCUSSIONS
AND CONCLUSIONS
Mixed-mode fracture of unnotched and notched composite laminates can occur due to uniaxial or multiaxial loading. In the case of a multidirectional laminate (such as [90/O/*45],) with a slant crack and subjected to uniaxial loading, the failure is mainly controlled by the 0” fibers along the loading direction. The prediction using the point strength model (with the fiber failure criterion) is very accurate if the characteristic length, b,, is taken along the crack plane. The characteristic length, bl, was found independent of material systems for a 7r/4 and a rotated m/4 laminates with a crack and uniaxial loading condition. The predictions using the effective normal crack approach under uniaxial load also agree well with the observed data. This is mainly because: (1) the volumes of the continuous 0” fibers (which control the failure) for an inclined crack and its effective normal crack are the same; (2) the characteristic length is determined along one of the fracture propagating planes (Fig. 3). Under combined loading, the direction of the fracture propagating plane depends on the ratio of the applied loads and the laminate configuration. In this case, the concept of “effective normal crack” cannot be applied. If the fracture plane of a laminate under combined loading is different from that under uniaxial loading, the application of the point stress or average stress failure criteria (on the laminate basis[2, lo]) for these two loading cases are based on different laminate configurations with respect to the crack propagating plane. Thus, the correlation between the stress concentration and the notched strength under uniaxial loading does not link to that under combined loading. This incapability is overcome if the point strength model is applied in conjunction with the fiber failure criterion (eq. 15). If a laminate fails in a fiber dominated mode under one loading condition and matrix dominated mode under some other loading conditions, then a model with fiber failure criteria is not able to predict the strength under general loading conditions. However, the present model with a FPF criterion can predict the notched strength for that kind of bi-failure mode (Fig. 7 is one of the examples). The matrix dominated mixed-mode fracture has also been predicted successfully[l4] using this model. The only drawback about the FPF criterion is that some of the strength curves exhibit somewhat kinking behavior under compressive loading. If a multidirectional laminate containing fibers in the o”, 90” and *45” directions, the failure is dominated by fibers under general in-plane loading conditions, then the minimum strength model with the fiber failure criterion, eq. (15), gives better result than the FPF criterion. The MSM also has the
746
SENG C. TAN
features: (1) simple (comparing to some other methods) and accurate; (2) from the present study and some earlier results[6], the characteristic length, bo, is independent of loading conditions. REFERENCES [1] D. H. Morris, H. T. Hahn, Mixed-mode fracture of graphite/epoxy composites: fracture strength. J. Compos. Mater. II, 124-138 (1977). [2] J. M. Whitney and R. J. Nuismer, Stress fracture criteria for laminated composites containing stress concentrations. J. Compos. Mater. 8, 253-265 (July 1974). [3] S. P. Garbo and J. M. Ogonowskii, Strength predictions of composite laminates with unloaded fastener holes. AIAA J. 18,585-9 (May 1980). [4] S. P. Garbo, Compression strength of laminates with unloaded fastener holes. AIAA/ASME/ASCE/AHS 21~1 SDM Conference, 291-4 (1980). [5] K. H. Lo, E. M. Wu and D. Y. Konishi, Failure strength of notched composite laminates, J. Compos. Mater. 17. 384-98 (September 1983). [6] S. C. Tan, Effective stress fracture models for unnotched and notched multidirectional laminates, to appear in J. Compos. Mater. (April 1988). [7] S. G. Lekhnitskii, Anisotropic Plates, Translated from the 2nd Russian edition by S. W. Tsai and T. Cheron. Gordon and Breach (1968). [8] S. C. Tan, Notched strength prediction and design of laminated composites under in-plane loadings. J. Camps. Mater. 21, 750-780 (1987). [9] S. W. Tsai and E. M. Wu, A general theory of strength for anisotropic materials, J. Compos. Mater. 58-80 (1971). [lo] S. C. Tan, Laminated composites containing an elliptical opening. I. Approximate stress analysis and fracture models. /. Compos. Mater. 21, 925-948 (October 1987). [11] S. C. Tan, Fracture strength of composite laminates with an elliptical opening, Compos. Sci. Technol. 29, 133-152 (1987). [12] I. M. Daniel, Biaxial testing of graphite/epoxy composites containing stress concentrations-Part I, Air Force Report, AFML-TR-76-244 (December 1976). [13] I. M. Daniel, Biaxial testing of graphite/epoxy composites containing stress concentrations-Part II, Air Force Report, AFML-TR-76-244 (June 1976). [14] S. C. Tan, Mixed-mode fracture of notched uniaxial and off-axis laminates. Submitted to J. Compos. Maw. (Received 30 October 1987)
0013-7944/88 s3.00+ .xl 0 1988 Pergamon Press pk.
MIXED-MODE FRACTURE OF NOTCHED COMPOSITE LAMINATES UNDER UNIAXIAL AND MULTIAXIAL LOADING SENG C. TAN AdTech System Research Inc. (On-site), US Air Force Materials Laboratory, Wright Patterson AFB, AFWAL/MLBM, Dayton, OH 45433-6533, U.S.A. Abstract-Two models are proposed to predict the mixed-mode fracture of notched composite laminates subjected to uniaxial and multiaxial loading. The basic elastic, strength properties and a characteristic length are utilized by these models. The characteristic length is determined differently for the two models. A first-ply-failure criterion or a fiber failure criterion is applied in conjunction with the models. It was found that failure of notched laminates for the fiber dominated mode, matrix dominated mode or a combination of the two modes under different loading conditions can all be predicted very well and the characteristic length is independent of loading conditions.
INTRODUCTION strength of orthotropic composite laminate with a circular opening or a straight crack under uniaxial tensile or compressive loading can generally be predicted well using some of the failure criteria documented in the literature. The mixed-mode failure of unnotched laminate on the first-ply-failure basis can also be handled quite well generally. On the ultimate strength basis, simplifying assumptions are needed to avoid the extreme complexity of the failure mechanisms under a single or combined load. In some cases, a mixed-mode failure can be predicted by using a simple model yet provides reasonably good result. For instance, a multidirectional laminate containing a slant crack under uniaxial normal loading has been modeled[l] by using an effective normal crack with a crack length equal to the normal projection of the slant crack. The point stress failure criterion[2] was then applied incorporating with their model. This approach, however, can only be applied in uniaxial normal loading case. The notched strength of composite laminate containing a circular hole or a center crack due to biaxial loading has been studied previously by some researches using the Tsai-Hill failure criterion[3-41 and the maximum stress, maximum strain and the tensor polynomial failure criterion[5]. It is critical to point out that although fiber failure is assumed the ultimate failure by special considerations of those criteria[3-51, different fiber failure criterion can result in significant different prediction. For instance, two distinct characteristic lengths are needed for tensile and compressive loadings if the Yamada-Sun’s criterion[6] is applied. However, only one characteristic length is required for those two loading conditions if the present fiber failure criterion is utilized. The approach in [5] is a special version of progressive failure model which involves a characteristic dimension and considers the redistribution of stresses only through the thickness of the laminate. The present research shows that the notched strength of composite laminates under multiaxial in-plane loading can be modeled in a compact-form solution yet the accuracy is assured by a number of comparisons. In addition, if a laminate fails in a fiber dominated mode due to one loading condition and matrix dominated mode due to some other loading conditions, the present model is able to predict the notched strength by using one characteristic length. This capability has not been shown in literature before by any other models.
THE NOTCHED
2. STRESS DISTRIBUTION The stress distribution of an infinite anisotropic laminate containing an elliptical opening under in-plane loading can be derived using Lekhnitskii’s complex potentials[7-81. The laminate is assumed homogeneous and symmetric with respect to its mid-plane (Fig. 1). The in-plane stresses (using index notation) given in principal coordinates of the opening are tFM 215-B
733
SENG C. TAN
734
Fig. 1. Coordinate system and the characteristic length, b,,, for the minimum strength model.
u1 =
(14
2 Re [d41(zl) + &&(z2)1
m = 2 Re [K(zd + 4X41
(lb)
a6
(ICI
=
-2
Re
where C#I;and C#J;denote the derivatives respectively, and
[h+;(zd
+
Pz~;(zz)I
of the functions
4r and & with respect to z1 and z2,
4*(z,) = p1- *o* -!_ Pl-
&(z2)
=
-
P2
PI
-
(24
51
*al
ccl-l-h
W-9
l r2
where cyr = - (~7,/2) cos tJ( a cos I&+ ib sin +I)- (~7~/2) sin $( a sin JI - ib cos 4) +(?,,/2)(asin2$-iibcos2+) p1 = -(&/2)
(34
/2) cos I++( a sin $ - ib cos I,!J) sin +( a cos I,$+ ib sin +) + (CT,, - ( T~,,/2)( a cos 2+ + ib sin t,G)
(3b)
and zi 5i =
+
Jz:
-
a-
a2 - p$b2 ipib ’
i= 1,2
(4)
where zi = n + /.Liy, i = 1,2 and where @I and I_L~are the principal roots of the characteristic equation:
aw4-
2a16p3
+(2a12+
a66)p2-2a26p+
a22
=
0
(5)
where a,, i, j = 1,2,6 are the laminate compliances with 1 and 2 in the major and minor axes of the opening, respectively. Substituting eq. (2a and b) into (1) yields
Ui=UT+Uq,
i=l,2,6
(6)
Mixed-mode fracture of notched composite laminates
735
where ai is the overall laminate stress, a? represents the stress components due to the opening and C$ designates the components due to the uniform stress field. The stress components, a:, of eq. (6) can be expressed in detail in the following: oT=Re
(74
aq=Re
(7b)
a: = Re
(7c)
where (1 - i&A)
h= PJP2-1-~~A2+P2-1-CL:A2
(84
gj=ihUy-pjO$-(l-i&A)Ug
@b)
where i = 1,2 and
The stress components
p=(l+(Y)cos0+&(A+~)sin8
(SC)
A = b/a
(84
a = bola.
We)
due to the uniform stress field are (9a) u~=&~os~JI+~~sin~*-~~~sin2*
(9b)
(Y:= (c?~- f?J sin I+% cos $ - fXYcos 21c,
(9c)
where a and b are the semi-major and semi-minor axes of the elliptical opening respectively; b. is a characteristic dimension which indicates a distance b. between the characteristic curve and the opening contour; I,Qdesignates the slant angle from the y-axis. c&.,a,, and fXYare the applied normal and shear stresses. The stress transformation between the laminate axes and the principal axes of the elliptical opening is given by eq. (9a-c). Equation (5), for anisotropic laminate, can be solved numerically. For orthotropic laminates, aI6 = a26 = 0, eq. (5) becomes
The solutions of eq. (10) are cL2=-
2&Z + a66 2a 11
(11)
Solving eq. (11) yields ,&,2,3,4=
&[COS
(0/2 + k?r) + i sin (8/2 + k?r)],
k=O,l
(12)
736
SENGC.TAN
where
0 = tan-’
(14) -@a12
+
&6)/(2&l)
’
Only the two principal roots of eq. (12) are useful for elasticity’s problem. If 8, eq. (14), lies in the second or third quadrant, it has to be subtracted from 7r before substituting into eq. (12). 3. FAILURE
MECHANISMS,
CRITERIA
AND MODELS
The failure of composite laminates can generally be classified into fiber mode and matrix mode. The ultimate fracture of a multidirectional laminate which contains longitudinal plies with fibers in the principal loading directions is dominated by fibers. From experimental results for composite laminates containing stress concentrations under tensile or compressive loading, one usually finds that the failure initiates with matrix cracking at a small region near the discontinuities. As the applied load increases, some degrees of fiber matrix splitting will appear at the opening edge for most laminates. The local damages release the intensed stresses and redistribute the stress concentration. Therefore, the stress concentration at the final stage of fracture is not as high as that for a virgin laminate (due to the degradation in laminate stiffness). Due to the extreme complexity of the fracture mechanisms, all solutions found in literatures are pursued with some simplifying assumptions. In this paper, a characteristic damage zone is utilized to compute the stress concentration of the laminate, which is considered as an “effective stress concentration” at the final stage of the failure process. The effective stress concentration (stress concentration at the point or points associated with the characteristic length) can be considered as a one-dimensional (point-wise) or two-dimensional (average over the characteristic length) parameters. The characteristic length was assumed as a constant and a two-parameter power function[8] to predict the strength reduction due to the hole size effect . Two criteria: FPF (First-Ply-Failure) and a FF (Fiber Failure) criteria[6] were applied to predict the mixed-mode fracture of multidirectional laminates under uniaxial and multiaxial loading. The Tsai-Wu quadratic failure criterion[9] with the stress interaction term, FT2, equal to -0.5 was chosen for the FPF analysis whereas the following criterion was applied for the fiber failure (FF) prediction.
4
-----A-cr,
XX’
(
i-i
>=
ef
(15)
where gl is the lamina normal stress in the fibers direction, X and X’ denote the longitudinal lamina tensile and compressive strength, respectively. If any ply within a laminate satisfies the condition: ef zz 1, the ultimate failure is considered occurred. Two models are described below for the notched strength prediction. 3.1. Point strength model (PSM) The characteristic length, b,, in this model is determined along the assumed fracture propagating plane, Fig. 2, and its magnitude is obtained by back substituting one notched strength data into the stress and strength analyses. In the case of an orthotropic laminate ember-controlled) containing a circular hole or a normal crack under uniaxial loading, the fracture plane is perpendicular to the loading direction and the stress state along the fracture propagating plane can be expressed in a compact form[ lo]. For cases other than that, eqs (1-14) will be applied. Once the laminate stresses are obtained, the lamina stresses can be calculated using the classical laminated plate theory. Then the FPF and the FF, eq. (15), criteria are utilized ply-by-ply to predict the notched strength reduction factor and the notched strength (absolute value) respectively.
Mixed-mode fracture of notched composite laminates
Fig. 2. The characteristic
3.2.
Minimum
strength model
131
length, b,, utilized for the point strength model.
(MSM)
When a laminate with stress concentrations is subjected to multiaxial loading, the direction of the fracture propagating plane depends on the stress ratios and the lay-ups of the laminate. Unlike the previous model, the fracture propagating direction will also be predicted. In this model, the strength distribution is analysed point-by-point along a characteristic curve which is at a distance, bo, away from the opening contour, Fig. 1. The characteristic curve can be written explicitly as 2 2 (16) (a:b,)2+&~= 1 where b. is determined by a similar method as the previous model. The laminate and lamina stresses are computed, for all points selected along this characteristic curve [using eqs (t-14) and laminated plate theory]. The minimum allowable stress that satisfies the FPF strength or the FF condition, eq. (15), is computed ply-by-ply. The point that first satisfies condition (15) indicates the direction of the fracture propagating plane. Under a combined loading, the following expressions are utilized to relate the notched strength reduction as a ratio of the uniaxial unnotched strength. When the FPF criterion is applied:
SRF=z=
x-component of FPF notched strength Uniaxial FPF unnotched strength (x-axis loading)
(17)
and when the fiber failure criterion, eq. (15), is utilized
SRF=%= aox
x-component of notched FF strength Uniaxial unnotched FF strength (x-axis loading)
(18)
where SRF denotes the strength reduction factor, uNX is the x-component of the ultimate notched strength under combined loading and CT ox is the ultimate unnotched strength due to a uniaxial loading in the x-direction. The characteristic length, bo, for a laminate under combined loading is assumed the same as that under uniaxial loading. Therefore, the characteristic length, bo, of a laminate can be determined using one notched strength data under uniaxial loading. 4. UNNOTCHED
ULTIMATE
STRENGTH
Two material systems, graphite/PEEK AS4/APC-2 and graphite/epoxy T300/SP-286 were studied. The unnotched ultimate strength of a laminate can be predicted using the laminated plate theory for the ply stresses and the fiber failure criterion, (eq. 15). The material properties are listed in Table l(a-b). The comparison, Table 2, shows that the predicted unnotched ultimate strength agrees reasonably well with the experimental data.
SENG C. TAN
738
Table l(a). Elastic properties of the graphite/PEEK Parameter
Longitudinal modulus, Et, Transverse modulus, Ez2 In-plane shear modulus, G,, Poission’s ratio, VIZ
and graphite/epoxy
lamina
AWAPC-2 * lo6 psi (CPa)
T300/SP-286 * lo6 psi @Pa)
19.435 (134.0) 1.291 ( 8.9) 0.739 ( 5.1)
21.6 (149.0) 1.53 ( 10.6) 0.93 ( 6.4)
0.28
0.31
Table l(b). Strength properties Parameter Longitudinal tensile strength, X Longitudinal compressive strength, X’
AWAPC-2 Ksi (MPa)
T3OO/SP-286 Ksi (MPa)
308.9 (2130) 159.5 (1100)
214 (1477) 164 (1132)
Table 2. Comparison of the predicted ultimate unnotched strength and the experimental data for the; (A) T300/SP-286 and; (B) AS4/APC-2 under uniaxial loading Material system
Lay-up
Prediction Ksi (MPa)
Experimental data Ksi (MPa)
(A)
10#451,
125 (863)
116 (800)
(A) (A)
[o/*45/90], [-30/15/-75/601,
84 (579) 51 (354)
73 (504) 60 (414)
(B)
[90/O/ f45] s
120 (824)
126 (870)
5. UNIAXIAL
LOADING
OF LAMINATES
WITH A SLANT CRACK
An earlier study[l l] has shown that an implanted crack (fabricated using a 0.005 in. diameter diamond wire) can be simulated very well by an elliptical opening with aspect ratio, a/b, equal to 50. The difference in stress profile for this elliptical opening and a theoretical crack is negligible except for a very small region near the crack tips, say (x - a)/a < 0.001. In addition to the PSM, a slant crack is also investigated by a normal crack which has a length equal to the normal projection of the slant crack. For this effective normal crack, a characteristic length, co, is defined along the normal crack plane. This plane coincides with the failure propagating direction at one end of the crack (Fig. 3). The strength reduction of the effective normal crack can be solved using eqs (l-15) or from [6] for the extreme case (b = 0):
where 0, is the applied stress and ap designates the laminate normal stress in the x-direction the point (0, co) and al/~,
= (Q,,S,,
+ Q12%)
cm*
4 + (Q12%1
+ Q22S12)
sin* +
at
4
2(Q1d%1 + Q&J
sin 4 cos 4
(20)
where Qij and gij, i, j = 1,2,6 are the lamina stiff nesses and laminate compliances, respectively. u1 has already been defined and related to the failure condition (eq. 15). The “effective normal crack” described above can be treated as a special case of the point strength model because it can only be applied under uniaxial loading while the point strength model can be applied under uniaxial and combined loading conditions. The predictions using these models and the experimental data are compared in Figs 4-5 for
Mixed-mode fracture of notched composite laminates
Fig. 3. Failed specimen of an AS4/APC-2
graphite/PEEK 45” crack.
[90/O/*45& laminate containing
739
a slant
Mixed-mode fracture of notched composite laminates
C90/0/:45ls.
0.
.
EXPERIMENTAL
a/b=50
DATA
0.40
CRACK
z
Fig. 4. Predicted
HALF
741
LENGTH,
a
I 3. 50
(INCHES)
(FF criterion) and experimental notched strength for the Gr/PEEK [90/O/*451, laminate with a slanted 45” crack.
AS4/APC-2
the AS4/APC-2 graphite/PEEK [90/O/*45], laminate with a slant 45” crack and the T300/SP286 graphite/epoxy [-19/61/26/-641, laminate with a slant 19” crack, respectively. The [-19/71/26/-641, laminate is in fact a lr/4 laminate with respect to the principal axes of the opening. From these comparisons, it is of interest to see that the characteristic lengths, b. = 0.05 in. (MSM), for these two laminates (different material systems and different slant angles) are the same as the one for the graphite/epoxy AS4/3502 [O/90/*451, laminate with a normal crack[6]. The best fit characteristic lengths using the PSM are: bl = 0.025 in. for the [-19/71/26/-641, and b, = AS4/APC-2 [90/O/*45],, bl = 0.022 in. for the T300/SP-286 0.025 in. for the AS4/3502 [O/90/*45], and [O/90/0], laminates[6]. This reveals that constant characteristic lengths, b. = 0.05 in. (MSM) and bl = 0.025 in. (PSM), are applicable to many material systems of quasi-isotropic laminates, containing a crack, with reasonably good result. If the characteristic length, bl, is taken along the crack plane, the notched strength can be predicted quite accurately. The predicted locations of initiation of failure by the MSM lie in the middle of the crack plane and the axis normal to the applied load. Whereas the experimental result is along the crack plane and on the axis perpendicular to the applied load, Fig. 3. C-19/71/26/-641sra/b=50 0"
.
EXPERIMENTAL
!jzjj 0.00
0.10
CRACK
Fig. 5. Predicted
0.20
HALF
DATA
1
0.30
LENGTH.
0.40
a
0.50
(INCHES)
(FF criterion) and measured tensile strength for the Gr/EP [-19/71/26/-641. laminate with a slanted 19” crack.
T3OO/SP-286
SENG C. TAN
742
SP-2861300,
(O/-+45/90)
5,
a/b=1
[b,=0.04b”l
EXPERIMENTAL
~d+.~~.~~.-~~...~~....~ 0.30 0.10 0.20 HOLE
RADIUS.
0.30
a
0.40
DATA
.‘.I
0.50
(INCHES)
Fig. 6. Predicted ultimate strength reduction (MSM-FPF) and observed data for the Gr/Ep T3OO/SP286 [O/~k45/90], laminate with a circular opening.
6, FRACTURE
UNDER MULTIAXIAL
LOADING
The predicted notched strengths for some graphite/epoxy .r3OO/SP-286 laminates subjected to combined in-plane loadings were predicted and compared to experimental data[12-131 reported by Daniel. The MSM is applied in conjunction with both the FPF criterion and the fiber failure criterion. 6.1. Result by the FPF cr~ter~~~ In the present analysis, each two successive points along the characteristic curve is 2-3” apart for the circular hole case and 1” apart from the crack case. Figures 6 and ‘7 illustrate the results for the [0/~1~45/90], and [OJrt45], laminates containing a circular opening respectively. The former laminate was subjected to a uniaxial and an equal biaxial loading while the later was loaded uniaxialiy and biaxially with a stress ratio 6 + : cFy= 2 : 1. Due to the symmetry of the laminate and the loading condition, only one-quarter of the laminate needs to be analysed. The experimental data agrees reasonably well with the predictions using characteristic lengths, bo = 0.045 in. and 0.065 in., respectively, determined by uniaxial data. It is very important to SP-286T300.
(02/i45)
EXPERIMENTAL IO. s. c
0
ap*=i:o
A
i&:+Z:I
s
F
a/b=1
DATA
$00 ~Do.oo
0.10
0.20
HOLE
Fig. 7. Ultimate strength
RADIUS,
0.30
a
0.40
0.50
(INCHES1
prediction (MSM-FPF) and observed data for the Gr/Ep T3OO/SP-286 [0,/*45], laminate with a circular hole.
Mixed-mode fracture of notched composite laminates
GR/EP,
743
(-30/15/-75/60)s.a/b=50
;n”o:oo
0:40 v
a
0150
(INCHES)
Fig. 8. Predicted ultimate strength (MSM-FPF) and measured data for the Gr/Ep [-30/15/-75/60]s laminate with a slanted crack.
point out that the [O&45], laminate fails at different modes (fiber dominated and matrix dominated) under the prescript uniaxial and biaxial loading conditions. The comparison of a [-30/15/-75/601, laminate (quasi-isotropic laminate with respect to the principal axes of the opening) containing an inclined crack (30” from the y-axis) is shown in Fig. 8. The characteristic length, b0 = 0.11 in., is shown the same for different biaxial loading conditions. 6.2. Result by the fiber failure criterion The application of fiber failure criterion directly determines the ultimate strength and strength ratio. The notched strength of the [O/*45/90], laminate, Fig. 9, under a uniaxial and biaxial loading can be predicted using a same bO.The predicted initiation locations of failure under a uniaxial load is between 45” (small hole) and 0” (large hole) from the axis perpendicular to the loading direction. The observed result is 22.5”. Under equal biaxial loading, the predicted failure initiates along the laminate axes and the &45” directions, whereas the experimental results are at a 22.5“ from the predicted locations. In Fig. 10, the observed failure locations of the [O&45], laminate under uniaxial loading (fiber dominated) initiate at 18” directions from the CO/+45/9Ols
T300/SP-286,
EXPERIMENTAL
HOLE
RADIUS.
a
DATA
(INCH)
Fig. 9. Predicted ultimate strength [MSM-FF, eq. (15)] and observed data for the Gr/Ep T300/SP-286 [O/*45/90], laminate with a circular opening.
744
SENG C. TAN
T3OO/SP-286,
HOLE
Fig. 10. Predicted
(02/+45)s
RADIUS.
a
(INCH)
ultimate strength (MSM-FF) and observed data for the Gr/Ep T300/SP-286 [0+451, laminate with a circular hole.
axis perpendicular to the loading axis. This value is about the average of the predictions: 44” for small holes and 0” for large holes. The fracture mechanism resulting from the biaxial loading, 6, : 6,, = 2: 1, is a combination of matrix cracking in the 0” ply and the *45” plies and some amounts of interlaminar shear. This loading case is not within the “fiber dominated” domain and, therefore, the notched strength is not possible to be predicted by using a b,, determined from a uniaxial test along the fiber dominated axis (no matter what kind of fiber failure criterion). The result of the [-30/15/-75/60], laminate containing a slant crack under biaxial loading is shown in Fig. 11. Excellent correlation of prediction with experimental data is obtained. A [-30/15/-75/60], laminate with an inclined crack (30” from the y-axis) under a far field biaxial normal loading (0%: ay = 2 : 1) is equivalent to a quasi-isotropic laminate with a horizontal crack under multiaxial loading, C?~: Oy : TX,,= 2: 1.4:0.35 (Fig. 12). The notched strength predicted by using a characteristic length b. = 0.11 in. agrees reasonably well with the observed data. Although a slightly better fit yields if b. = 0.08 in. is applied for the multiaxial loading with non-zero shear, the difference between the prediction with b0 = 0.11 in. and the data is not significant. The predicted locations of the initiation of failure under uniaxial loading are at 34-37” from the crack direction, which correlates well with experimental value, 40”. The
F.l....,....,....,....,....,
~“0.00
0.10 CRACK
0.20
0.30
HALF
LENGTH,
0.40
a
0.50
(INCH)
Fig Il. Comparison of the predicted strength reduction (MSM-FF) and experimental Gr/Ep T30O/SP-286 [-30/15/-75/60], laminate with a slanted crack.
data for the
Mixed-mode
fracture
of notched
T3OO/SP-286,
EXPENIPIERTAL 0
+-d
0-z 0.00
12. Predicted
laminates
745
C0/+45/901sva/b=50
DATA
a;:iffy:?~'l:o:o
0.10
CRACK Fig.
composite
0.20
HALF
0.30
LENGTH.
0.40
a
0.50
(INCH1
ultimate strength (MSM-FF) and observed data [O/+45/90], laminate with a crack under in-plane
for the Gr/Ep loading.
T3OO/SP-286
predicted crack propagation directions under the given combined stresses are at an angle between 67” (small cracks) and 38“ (large cracks) from the crack plane, whereas the experimental result is at 45” directions from the crack plane. 7. DISCUSSIONS
AND CONCLUSIONS
Mixed-mode fracture of unnotched and notched composite laminates can occur due to uniaxial or multiaxial loading. In the case of a multidirectional laminate (such as [90/O/*45],) with a slant crack and subjected to uniaxial loading, the failure is mainly controlled by the 0” fibers along the loading direction. The prediction using the point strength model (with the fiber failure criterion) is very accurate if the characteristic length, b,, is taken along the crack plane. The characteristic length, bl, was found independent of material systems for a 7r/4 and a rotated m/4 laminates with a crack and uniaxial loading condition. The predictions using the effective normal crack approach under uniaxial load also agree well with the observed data. This is mainly because: (1) the volumes of the continuous 0” fibers (which control the failure) for an inclined crack and its effective normal crack are the same; (2) the characteristic length is determined along one of the fracture propagating planes (Fig. 3). Under combined loading, the direction of the fracture propagating plane depends on the ratio of the applied loads and the laminate configuration. In this case, the concept of “effective normal crack” cannot be applied. If the fracture plane of a laminate under combined loading is different from that under uniaxial loading, the application of the point stress or average stress failure criteria (on the laminate basis[2, lo]) for these two loading cases are based on different laminate configurations with respect to the crack propagating plane. Thus, the correlation between the stress concentration and the notched strength under uniaxial loading does not link to that under combined loading. This incapability is overcome if the point strength model is applied in conjunction with the fiber failure criterion (eq. 15). If a laminate fails in a fiber dominated mode under one loading condition and matrix dominated mode under some other loading conditions, then a model with fiber failure criteria is not able to predict the strength under general loading conditions. However, the present model with a FPF criterion can predict the notched strength for that kind of bi-failure mode (Fig. 7 is one of the examples). The matrix dominated mixed-mode fracture has also been predicted successfully[l4] using this model. The only drawback about the FPF criterion is that some of the strength curves exhibit somewhat kinking behavior under compressive loading. If a multidirectional laminate containing fibers in the o”, 90” and *45” directions, the failure is dominated by fibers under general in-plane loading conditions, then the minimum strength model with the fiber failure criterion, eq. (15), gives better result than the FPF criterion. The MSM also has the
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SENG C. TAN
features: (1) simple (comparing to some other methods) and accurate; (2) from the present study and some earlier results[6], the characteristic length, bo, is independent of loading conditions. REFERENCES [1] D. H. Morris, H. T. Hahn, Mixed-mode fracture of graphite/epoxy composites: fracture strength. J. Compos. Mater. II, 124-138 (1977). [2] J. M. Whitney and R. J. Nuismer, Stress fracture criteria for laminated composites containing stress concentrations. J. Compos. Mater. 8, 253-265 (July 1974). [3] S. P. Garbo and J. M. Ogonowskii, Strength predictions of composite laminates with unloaded fastener holes. AIAA J. 18,585-9 (May 1980). [4] S. P. Garbo, Compression strength of laminates with unloaded fastener holes. AIAA/ASME/ASCE/AHS 21~1 SDM Conference, 291-4 (1980). [5] K. H. Lo, E. M. Wu and D. Y. Konishi, Failure strength of notched composite laminates, J. Compos. Mater. 17. 384-98 (September 1983). [6] S. C. Tan, Effective stress fracture models for unnotched and notched multidirectional laminates, to appear in J. Compos. Mater. (April 1988). [7] S. G. Lekhnitskii, Anisotropic Plates, Translated from the 2nd Russian edition by S. W. Tsai and T. Cheron. Gordon and Breach (1968). [8] S. C. Tan, Notched strength prediction and design of laminated composites under in-plane loadings. J. Camps. Mater. 21, 750-780 (1987). [9] S. W. Tsai and E. M. Wu, A general theory of strength for anisotropic materials, J. Compos. Mater. 58-80 (1971). [lo] S. C. Tan, Laminated composites containing an elliptical opening. I. Approximate stress analysis and fracture models. /. Compos. Mater. 21, 925-948 (October 1987). [11] S. C. Tan, Fracture strength of composite laminates with an elliptical opening, Compos. Sci. Technol. 29, 133-152 (1987). [12] I. M. Daniel, Biaxial testing of graphite/epoxy composites containing stress concentrations-Part I, Air Force Report, AFML-TR-76-244 (December 1976). [13] I. M. Daniel, Biaxial testing of graphite/epoxy composites containing stress concentrations-Part II, Air Force Report, AFML-TR-76-244 (June 1976). [14] S. C. Tan, Mixed-mode fracture of notched uniaxial and off-axis laminates. Submitted to J. Compos. Maw. (Received 30 October 1987)