Stress failure criteria for laminated composites

Damage and Interfacial Debonding in Composites G.Z. Voyiadjis and D.H. Allen 1996 Elsevier Science B.V.

133

Stress failure criteria for laminated composites Hsien-Yang Yeh a and A.K. Feng Mechanical Engineering Department, California State University, Long Beach Long Beach, CA 90840

1. ABSTRACT The generalized Yeh-Stratton criterion is applied to calculate the fracture stress of multidirectional fibrous composite material with existing cracks or circular holes under uniaxial tensile load. The failure stresses were predicted and compared with the prediction by the theory of Nuismer and Whitney and the experimental data. Based on this study, it is recommended that for the cracks and circular holes with larger crack size and radius ( > 0.6 inch), the Yeh-Stratton failure criterion could be used as a proper design guide.

2. INTRODUCTION Several failure theories have been developed for composite materials. One of the theories is the Yeh-Stratton Criterion (simply called the Y-S criterion) [1,2] which has been newly developed and proved reliable in the prediction of failure in both isotropic and anisotropic materials. The Y-S criterion, like many other failure theories of anisotropic materials, were developed from those of isotropic materials and generalizexl to composite materials. One unique feature of the Y-S criterion is in its ability to change its format depending upon what types of stresses are applied and what types of material strengths are considered. Thus, it is shown that the entire closed failure surface by the Y-S criterion is composed of piecewise surfaces. Instead of defining the failure surface by a single function, the Y-S criterion can form the similar functions for each quadrant of the stress space and all constants would be evaluated by experimental values from that quadrant. Even though each surface may be defined to be hyperboloid, the final failure surfaces must be closed. This study is a further development of the Y-S criterion to calibrate its analytical results by experimental data and apply to the prediction of the uniaxial failure of a multi-directional and multi-ply laminated composite structures when involved with stress concentrations. In this paper, the experimental data from the report by R.J. Nuismer and J.M. Whitney [3] will be used. Since the theory of normal stress distribution in predicting the failure and showed a high accuracy in comparison with the testing results, it would be very interesting to see the comparison of the Y-S criterion to that of Nuismer and Whitney and evaluate the difference in

a Author to whom all correspondence should be addressed.

134 those theories. On the other hand, if the Y-S criterion could provide to the same level of accuracy or even better in prediction of fracture failure when involved with stress concentration, it would be a further prove of the application of Y-S criterion in the field of composite materials.

3. ANALYSIS The main focus of the fracture mechanics is to calculate the stresses at the immediate vicinity of the crack tip because the region near the tip of the crack is most likely to fail or fracture first due to the nature of stress singularity. The crack problem in the anisotropic elastic body, as shown in Figure 1, has already been extensively studied by many scholars such as Lekhnitskii [4], Sih et al. [5,6], Tohgo et al. [7], Waddoups et al. [8], Whitney, and Nuismer [9] and Konish and Cruse [10] etc. The constitutive relations of any lamina, referring to the material principal coordinate 1-2, are given by

(1)

o[ The component of the compliance matrix are as follows:

Sl l= l/E ll , $22=1lE22, S12=-V12[Ell, $66=1/G12

(2)

where En and E22 are Young's moduli in the principal direction; G12 is the shear modules; and Vn is Poisson's ratio. The constitutive relation, referring to the physical coordinate x-y, is obtained from Equation (1) by coordinate Transformation.

ey = [b12 b22 b26 [b16 b26 b~

%

(3)

where bll =S11c~

+S66)sin2acos2a+S22sin4a

b n=S 12(sin4a+C0S40~)+(S 11+822-S66)Sin2acOs20~ b22=Sllsin4a +(2S12+S66)sin2acos2a+S22c~ b16=(2S11-2Sn-S66)sinacos3a-(2S22-2Sn-S66)sin3acosa

(4)

135

b26=(2Sll-2S12-S66)sin3acosa-(2S22-2S12-S66)sinacos3a b66 =2 ( 2S l l 4-2S22-4S12 -$ 66)sin2ac~

4-S66(sin4a4-c~

The stresses around the crack tip are expressed by the stress intensity factor kt and k2 r x=Kl/(2zrr)-gRe [(x I -x2) * (x2F2-x2F2-x lF1) ] +K 2(2 zrr) -~ Re [ l l (x I-x2)* (x22F2-x~F O ] Cry=Kll(2zrr)-gRe [ l/(x 1-x2)*(xlF2-xlF1)]+K2(27rr) -~

(5)

Re [l/(x 1-x2), (F2-F1)]

%=K1(2r

(F 1-F2)]4-K2(2zrr) -~

Re [ I](xI-X2),(xIFI-x2F2) ]

where Re indicates the real part of the complex number. F1 and F2 are given by F1 =(cos04-xlsin0)-~

(6)

F2=(cosO+x2sinO) -v" where xl and x2 are roots of the following characteristicequation: b11x 4-2b10Jr;34-(2b12+b66)x2-262ax+b22=0

(7)

and it is the function of material properties. The roots of the characteristicequation consist of two complex conjugate pairs, m

XI~ XI~ X2~ X 2

The stressintensityfactors for mode I and mode 2 are given as following,

K~=o*(a)~sin2/~

(S)

K2=o|

(9)

136 where a is the crack length and 3 is the crack angle. The stresses al, a2 and rn shown in equation (5) are the stresses along the material axes. Substitute the equation (8) and (9) into (5) it can be shown that,

o l=o | ( 2rla )-~(x2-x l)- 1(sin2flzrr2~ l +l / 2 s i n ~ 3)

(10)

02=0 00(2flu)-~(x 2-x 1)- 1(sin2~'~4 + l/2sin2fl~ 2)

(11)

,rn=o| (2rla )-~ (x2-x 1)- l ( sin2flx rr2T/2+l l2sin2fl r/2)

(12)

where 7/1=x2cos(~ 2/2) tF 2J4-x 1COS(t~ 1/2)tF 1~

(14)

r/2=sin(ff2/2)~F~-sin(ffl[2) ~F1-~ 7/3=x~sin(4,~/2)~F~-x2sin(4)~J2)*F2-~ 7/4=x2cos(~bl[2) tFl~-xlc~

tF i=(cos20+xisin20)~ d?i=tan-l(xitanO)

-~

i= 1,2 i=1,2

(13)

(15) (16) (17) (18)

substitute the equation (10), (11), and (12) into general Y-S criterion, the fracture under the uniaxial tensile load is initiated if the following is met.

011X+B1201ty2+o'2/Y+(7"12/S)2-1

(19)

The above stress field will be substituted into the fracture criterion to compute the fracture stresses. To evaluate the analytical results, it is necessary to compare with the existing test data and the developed model based on the normal stress distribution from the report presented by Nuismer and Whitney [3] The material systems investigated are: 9 Scotchply 1002 9 Thomel 300/Narmco 5208 Two laminates of each material were used: 9 (0/+45/90)2, 9 (0/90)4,

137

The mechanical properties of above materials are shown in table [1] and all laminates are made of 16 plies. The effect of notch size were examined by considering sharp-tipped center cracks of sizes 0.1, 0.3, 0.6 and 1.0 inc (crack length). The particular sizes were chosen because the previous testing data from reference [8], [9], [ 10] have indicated that in this range of notch sizes there is a transition behavior from unnotched to large-notch size laminates.

4. DISCUSSION

In the laminate of (0/90)4,, the failure is initiated in the layers of 0 degree. Failure initiation takes the form of distributed micro-cracks and will extend across the thickness of the layers. When the 0 degree plies failed, it doesn't cause the failure of the laminates but the 90 degree will take all the stresses after the FPF (First Ply Failure). Same assumption would apply to the case of (0/-45/45/90)2, laminates. In the study of Nuismer and Whitney [3], the actual fracture was adjusted by the isotropic finited width correction factor which K r / K T =[2 +( 1-2a/W) 3113( 1- 2a/W) KI/K l =[( W[ 2a )tan (2a / W) ]~

for holes for cracks

where a is the half crack length and W is the width of specimen. It is understood that the correction factor of geometry should be influenced by the crack angle near the crack tip, because under the same stress, the relative length between crack and specimen width would cause different opening in the crack area, therefore, change the micro crack angle around the core region (damage zone). The correction angle 0c^ should have the form: 0ca =k,sin-~ {(2+( 1-2a/B03)/3 ( 1-2a/B0} where k is the factor should calibrate with the experiment results. Using the k=0.05, 0.08, 0.09, 0.10, 0.11, 0.12, 0.15, 0.20 to verify the prediction by Y-S criterion. Table 2 listed the different value of characteristic "k" compared with test data and it was found that k=0.10 would be the best fit for scotchply material system and k=0.09 for the graphite 5208/T300 material system. Compare the developed data with both center crack and circular hole, it was indicated that even through in case of circular hole, due to the nature of weakest ply and preferable direction exist in the composite material, the fracture would occur in the same tendency as the center crack that was set up in this paper. Figure 2 to 5 shows the results from the Y-S criterion and the comparison to the results of testing data from specimen with center crack and those associated data predicted by Nuismer & Whitney. For the prediction of 5208/T300 graphite composite that the results from the Y-S criterion were shown pretty good agreement to the testing data. In the case of center crack of (0/90)

138 laminate, the predicated failure stress would be 15.57% higher for a=0.05" and -22.41% lower for a=0.5". When compared the same result with Nuismer & Whitney theory, from 9.79% to -12.36% of error were found. According to the explanation in the paper [3] that the specimen could be made by materials with different quality or the different machining accuracy when cutting the cracks. No good explanation could be made at this point, according to the observation of Nuismer & Whitney that prior to final failure, no damage of any sort visible to the unaided eyes was seen in any of the 5208/T300 specimen. The process of accumulated internal damage until a total final crack might be owing to the extra stress the specimen could take before sudden failure. Figure 6 to 9 shows the comparison between the results from the Y-S criterion, the testing data of specimens for circular hole and the associated data from the Nuismer & Whitney. For specimen of scotch plies, good agreement between the Y-S criterion and testing results were obtained again. The Y-S criterion predicted higher fracture stress when the circular was small (0.05"r), getting very close for the circular hole with radius 0.15", and turned to be lower and conservative when the initial radius of circular increasing to 0.3" and 0.5". By observing the prediction of the Nuismer & Whitney theory for scotch plies, the fracture stress would be average higher. The prediction from both theory were pretty similar in the data reduction trend but the one from the Y-S criterion provides better prediction toward the safe side. For the specimen of 5208/T300, in the group of (0/90)4, the prediction from the Y-S criterion and Nuismer & Whitney theory are getting similar but both of them were lower than the testing results. In the group of (0/-45/45/90)2,, the Nuismer & Whitney theory showed higher value compared with the testing data and the Y-S criterion showed very close to the testing data [11]. It was assumed that the center crack and circular hole should have the similar fracture characteristics for composite materials. For scotch ply in general, the circular hole specimen had higher fracture stress compared with the counterpart of center crack specimen, but the difference is relatively small. For the graphite 5208/T300 in specimen (0/90)4,, the comparison between center crack and circular hole indicated that the difference between two groups of specimen is also small.

5. CONCLUSION

0

0

"

In the scotch ply material system, the Y-S criterion had showed very good agreement with the test data. (figures 2,3,6,7) During the process of calculation, it was shown that the modification or change of crack angle or fracture angle could directly influence the value of failure stress even though the theoretical prediction showed pretty good agreement with this selected testing data, the crack angle correction should require further investigation by more experimental verifications. In this paper, it was based on uniaxial load for each ply and assumed the total failure stress would be the accumulation of failure stress from each ply due to its individual fracture properties, therefore the interaction between plies was ignored. In the comparison, the point stress failure criterion was used to represent the prediction

139 of Nuismer & Whitney failure theory because this criterion would show better matching with more testing data. Based on this study, it is recommended that for the crack or circular holes with larger crack size and radius (> 0.6 inch), the Y-S failure criterion will probably be used as a proper design guide.

.

6. REFERENCES

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@

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10. 11.

Hsien-Yang Yeh and Chang H. Kim, The Mixed Mode Fracture Analysis of Unidirectional Composites, Journal of Reinforced Plastic and Composites, Vol. 13, June 1994, PP 498--508. Hsien-Yang Yeh and Chang H. Kim, The Yeh-Stratton Criterion For Composite Materials, Journal of Composite Materials, Vol. 28, No. 10, 1994, PP 926--1239. R.J. Nuismer and J.M. Whitney, Uniaxial Failure of Composite Laminates Containing Stress Concentrations, Fracture Mechanics Of Composites ASTM STP 593, American Society for Testing and Material, 1975 PP 117--142. Lekhnitskii, S.G., Theory of Elasticity of an Anisotropic Elastic Body, Trans. P. Fern San Francisco, CA: Holden Day, 1963. Sih, G.C., P.C. Paris and G.R. Irwin, On Crack in Rectilinearly Anisotropic Bodies, Int. J. Fract. Mech., Vol. 1:1965 PP 189--203. Sih, G.C. and H. Liebowitz, Mathematical Theories of Brittle Fracture, Chapter in Fracture, Vol. 2, H. Liebowitz, ed., New York: Academic Press, 1968 PP 67--189. K. Tohgo, Albert S.D. Wang and Tsu-Wei Chou, A Criterion for Splitting Crack initiation in Unidirectional Fiber-Reinforced Composites, University of Delaware Newark, DE, 1993. Waddoups, M.E., Eisenmann, J.R., and Kaminski, B.E., Journal of Composite Materials, Vol. 5, 1971, PP 446--454. Whitney, J.M. and Nuismer, R.J., Journal of Composite Material, Vol. 8, 1974. PP 253--265. Konish, H.J., Jr. and Cruse, T.A. in Composite Reliability, ASTM STP 580, American Society For Testing and Materials, 1975. Feng, A.K. "Stress Fracture Criterion For Laminated Composite" Master's thesis, California State University, Long Beach, May 1995.

140

Table 1 Material Properties Material

Material Constant

THORNEL300/NARMCO5208

Eu =21.4"103KSI Et2 = 1.6*10ZKSI Gn =0.77"103KSI V12=0.29

SCOTCHPLY

xl =0.708 x2=5.180

for 0 degree

xx=0.913 x2= 1.420

for 90 degree

xl =-0.928 x2=0.333

for 45 degree

Eu =5.6*I&KSI Es2= 1.2*103KSI Gt2=0.6*103KSI V12=0.26 xl =0.760 x2=2.700

for 0 degree

xl =0.348 x2= 1.330

for 90 degree

x I =-0.785 x2=0.281

for 45 degree

141

Table 2 Parameter "K" for the Material Sotchplies and Material 5208/T300

K Value

Average Deviation in Scotchply

Average Deviation in 5208/T300

0.05

96.42%

70.20%

0.08

25.03 %

8.51%

0.09

12.69%

0.10

[

2.79%]

-2.89% I -11.99%

0.11

-5.31%

-19.42%

0.12

-12.06%

-25.59%

0.15

-26.91%

-39.08%

0.20

-41.55 %

-52.28 %

Remarks: I Shows the selected K Value for corresponding material system

l

_

y 2

1

r

Figure 1. Geometry of the crack in uniaxial tension.

142

1O0

60-

70

-

60

-

50

-' I

4.0

30

20-

,~,

!

O. Y--S

0

!

0.30

CRITERION

0.60

CRACK LENGTH TEST DATA

+

o

1.00 W H I T N E Y &: N E I S M E R

Fig. 2. Test data, predictions comparison for scotchply with orientation (0190)4s and center crack

~"

70

-

60u') uJ r

;2

50

-

40 t 30

20

!

0 Y--S

CRITERION

!

0.30 -I-

CRACK LENGTH TEST DATA

0.60 o

Fig. 3. Test data, predictions comparison for scotchply with orientation

1.00 WHITNEY k

NEISMER

(01451-45190)2s and center crack

143

IO0

t

9O

70

6O

5O

40

ZO

10

"t

i

0.10 Y - - S CRITERION

i

0.30

0.60

CRACK LENGTH TEST DATA

4-

O

1 .DO

WHITNEY &

NEISMER

Fig. 4. Test data, predictions comparison for 5208/T300 with orientation (0/90)4s and center crack

1 O0

90

80

70

60

50

40

30

20

10 O. 10 Y - - S CRITERION

0.30 §

CRACK LENGTH TEST DATA

0.60 o

Fig. 5. Test data, predictions ~:0mparison fo~ 5208/T300 with orientation

O0

WHITNEY & NEISMER

(0/451-45/90)2s and center crack

144

90

-

80

-

70

-

60

-'

,50

'

t/') v v

~

30

2C ~

! |

"

0

0.,.30

Y--S C R I T E R I C N

CRACK T E S T DATA

+

0.60

1.00

LENGTH o

W H I T N E Y &: N E I S M E R

Fig. 6. Test data, predictions comparison for scotchply with orientation (0190)4s and circular hole

1 O0

u')

90

-

80

-

70

-

v1

E

r i.,J

60-

50-

40

"t

30

2o10

# O. 1 0

Y--S

CRITERION

0.30

+

CRACK LENGTH T E S T DATA

l 0.60

o

1.00

W H I T N E Y &: N E I S M E R

Fig. 7. Test data, predictions comparison for scotchply with orientation (01451-45190)2s and circular hole

145

-1

u')

7O

,.,

60

~,J

50

4O

10

O. O

0.30

Y--S CRITERION

0.60

CRACK LENGTH TEST DATA

+

O

1.00 WHITNEY &: NEISMER

Fig. 8. Test data, predictions comparison for 5208/T300 with orientation (0/90)4s and circular hole

80 70 60

-I

50 40 30

10

O.

Y--S CRITERION

4.

i

i

0.30

0.60

CRACK LENGTH TEST DATA

o

1.00

WHITNEY & NEISMER

Fig. 9. Test data, predictions comparison for 5208/T300 with orientation (0/45/-45/90)2s and circular hole