THE ROLE OF THIN CYLINDER BULGING ON CRACK CURVATURE

I a n M. Fyfe ' a n d V i n i t S e t h i t IJniversity of Washington, Seattle, Washington

A b s t r a c t

Three different experimental configurations were ex- amined as a means of determining the mechanisms and parameters which control the direction of crack growth in thin pressurized cylinders. The systems were de- signed to cause failure under quasi-static pressure con- ditions, explosive loading and fatigue loading.

It was determined that lack of symmetry in a bulge introduces the conditions which will control the direc- tion of growth rather than the presence of the bulge itself. This unsvmmetric bulge can be caused by a num- ber of external and crack t ip loading conditions but in each case the potential for mode 111 deformation exists.

It was also determined tha t a liner in the cylinder strongly influences the degree of crack curving, but that the tendency of the crack to curve is reduced by increas- ing the cylinder radius.

1. I n t r o d u c t i o n

When a crack curves sufficiently t o allow the inter- nal pressure to cause a shell t o open in a small area we have a condition in which controlled decompression occurs by flapping '. In a cvlinder this phenomenon, which causes longitudinal cracks to turn in a circumfer- ential direction, may ensure rapid pressure relief with continued structural integrity. In aircraft this type of crack also makes fuselage pressurization difficult, and results in a drop in cabin pressure, thus an early warn- ing about possible structural failure is built into the system.

It has been known for sometime tha t which occurs when a pressurized cylinder has a longi- tudinal crack, plavs an important role in the initiation of crack growth. It has also been postulated ' tha t this bulging is one of the causes of crack curvature in tha t i t introduces an axial stress component ahead of the crack, and this, coupled with the hoop stress, gives rise to the conditions necessary for curving.

The importance of stress biaxiality on the behavior of cracks in plates was confirmed bv R a d o n b h o applied

'Prnfessor, Dept. Aeronautics and Astronautics, . Member AIAA

tcurrently Research Engineer, Conveyer Dynamics, Inc. . . Bellingham, WA.

various biaxiality ratios R A , where R A was defined as the ratio of remote stress transverse to the crack and normal to it. He observed that for low ratios (0 < RA < 1 ) the crack path was straight and normal to the maximum applied load. For a ratio RA > 1, the crack trajectory becomes increasingly curved into an antisymmetric S-shape, centered on the initial straight notch. However, Cotterell" and more recently Streit and Finnie6 in their studies of crack stability found that a lengthwise compressive stress was required to ensure directional stability. This lengthwise compressive stress is characterized in part by the nonsingular stress term (a,,,) in the classical crack-tip solutions.

The objective of this research was to investigate the factors influencing the directional stability of cracks in pressurized thin-walled cylinders, and if possible deter- mine which of the above stability theories might be ap- plicable t o the cylinder configuration. The approach taken was to conduct crack propagation tests under a variety of conditions, and use fractional factorial de- sign techniques to determine the interaction between the different parameters envolved. The details of which are described by Sethi7. T h e study was carried out with three different experimental configurations each designed to cover different aspects of thin cylinder fail- ure. Two were essentially burst type tests, one dynamic and the other quasi-static, while the third was a fatigue test in which the hoop stresses, axial stresses and the loading frequency could be varied independently.

2. S t a t i c B u r s t T e s t s

As a crack in a quasi-statically pressurized thin cylin- der is invariably unstable i t might be expected that the axial stress due to bulging (ah,,l,,) would approach the magnitude of the hoop stresses. To investigate this hy- pothesis a series of tests were conducted to determine the role of thin cylinder bulging on the stress.

The apparatus used in the quasi-static thin cylinders tests is shown in Figure 1, and was a simple oil pressure system designed so tha t the a x i d loads could be applied independently of the internal pressure The specimens were 6061-T4 aluminum cylinders, 4-6 ins. long with a

wall thickness of 0.028 ins. and a diameter of 1-1.5 ins. T o prevent the oil from leaking out of the machined slot a n inner vinyl sleeve was used. The pressure was

Copyright @ 1991 American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

1341

Figure 1: Configuration for Static Burst Tests

inc r~ased until t,he crack in the cylinder opened up and extended a specific distance.

In a n early test the internal pressure was gradually increased, and the hoop and axial strains close to the crack tip were measured a t discrete pressure intervals. The axial st.resses a n d hoop stresses obtained from these measurements are plotted as a function of the internal pressure in Figure 2. There was no applied axial load so the stress ratio RA was zero. Due to bulging the axial stress increased as the pressure increased, and rzs is t o be expected with the magnitude of the axial stress eventually exceeding that of the hoop stress, the path of the crack curved.

Since, the total axial stress can usuallv exceed t,he hoop stress, crack curvatlure a t failure is t o be expected. The axial tensile stress levels which create an unstable condition can be obtained either from bulging or ap- plied tensile loads. An exception to this situation oc- curs in the fatigue tests where the pressures are quite low, and little bulging occurs. If the requirement tha t RA > 1 is used as a stability criterion for cylinders, then fatigue tests should be stable until quite high ax- ial tensile loads are applied. This was not the case, in tha t the crack propagation was unstable regardless of the applied axial load. These results tend t o show that using the R A ratio as a criterion for crack curving4 is too restrictive, in tha t instability is seen to occur before RA > 1, and so other conditions can initiate crack curv- ing. These results then confirm the findings of Streit and FinnieG, that axial compression is required to ob- tain stable crack growth, a condition which rarely oc- curs in the cylinder configuration.

Ax~al Stress , d

8000' \ I

Pressure (psi)

Fignre 2: Axial and Hoop Stresses as a Function of Internal Pressure ( R A = 0, a / J ( ~ t ) = 2.58)

3. F a c t o r i a l D e s i g n T e s t i n g

As one of the main objectives of this research was to determine the factors responsible for crack curving in thin cylinders i t is particularly desirable to estimate their relative significance. Factorial design techniques were used to statistically analyse the data. For the burst test six parameters or factors were initially ex- pected to have an influence on the direction of crack propagation. These factors were the thickness of the vinyl liner used to prevent leakage through the slot, the presence of axial compression, the initial slot length 2a, the radius R of the aluminum cylinder, the thickness t of t,he cylinder wall and the crack t ip condition.

The factors were studied at two different levels, a high (+) and a low (-) setting. The levels can be quan- titative (continuous) or qualitative (discrete). In such a design it is assumed that the effects are linear over the range of the factor levels chosen. In order to reduce the number of experiments required due to time and mate- rial constraints, the number of factors was reduced from k=6 to k=4 by combining the initial slot length 2a, alu- minum cylinder thickness t and radius R to form one factor a l d ~ t ) which for convenience has been named the geometric intensity . T h e justification behind this was that the stress intensity factor for cracks in pres- surized cylinders is a. function of the ratio a l d ~ t ) . A computer program called SQC' was utilized to analyze the data.

The deviation of the crack tip over a specific axial dis- tance from the t ip of the initial slot was used and the responses of the top and bottom crack t ip were recorded

as two s r p a r a t , ~ set.s of ditt,a (i.e. corresponding to two different runs of the same set of experiments). By doing this we =assume that the two cracks propagate indepen- dently of each other. T h e high (+) and low (-) settings are given in Table 1.

Table 1 Factor values for a 2k design

Crack T iv Condition

The factors don't have to be numerically specified, but are done so here for identication purposes. The axial compression is specified by the holding down torque applied t o the apparatus, and the settings for the crack t,ip condition were qualitative rather than quantit,ative. The (-) setting was cbt,ained by machining the initial slot with a 0.015 ins. thin radial saw blade. T h e (+) setting was obtained by using a very fine ( 0.013 ins. thick) saw blade to square the ends.

The design matrix which controls the order and com- bination of factors to be tested was generated by SQC. Each combination of factor settings, i.e., each row of the design matrix was run only once without replica- tion, but by assuming that the path of the upper crack was independent of the lower it was possible t o record the responses of the top and bottom crack t,ip as two spperate sets of d a t a giving a replication of two for each test. Factorial Analysis

The SQC program provides a means for performing the statistical operations associated with the 2k full and fractional design and resolution. The program gives as output , graphs indicating the influence of the main fac- tors and the interactions between them. It also provides numerical estimates of their relative influence on crack curving through ANOVA (Analysis of Variance) tables, and effect probabilitv plots. The ANOVA for a full factorial 2k and a replicate of 2 responses are given in Table 2. The influence of each factor is indicated in the last two columns, where a large value of F, or a small value of P. indicates the factors with the greatest influ- ence on the crack path. In this first series of tests the factor tha t had the largest influence on crack curving was the thickness of the vinyl liner. T h e thinner and hence less stiff the liner, the more the likelihood of crack curving. The probable reason the thickness of the liner plays such an important role is due to it's influence on the moments acting on the sides of the crack. A small lack of symmetrv in moments is considerably amplified by the use of a thin liner.

T h e next most significant fartor, though nowhere quite as significant a s the vinyl liner thickness, was axial compression. The presence of axial compression resulted in a crack with increased curvature. At first glance this result appears to be contrary to the observa- tions of Cotterell as well as those of Streit and Finnie, where compression in flat specimens was found to de- crease instability and crack curvature. The reason for the results obtained in this study could be linked to the fact tha t in the experimental arrangement used it was not possible to ensure tha t the axial compression being applied was uniform along the circumference of the cylinder as the crack opened. I t was suspected that if the axial compression was not applied symmetrically the bulging also would not be symmetric, and a direc- tional instability would exist.

The geometric factor which combined the crack length, cylinder radius and thickness was, like the axial compression, also a significant factor. It seemed that a - larger value of a / d ( R t ) is more likely to cause a crack to curve, which by its definition indicates that crack curving in a large radius cylinder is likely to be less. A fact borne out by fuselage crack growth.

The crack tip condition was the least significant fac- tor, which is not too surprising in tha t , although the initial condition is obviously important in determining the pressure required to cause growth, the actual path is controlled by the loading conditions on the moving or established crack. I t was felt, however, tha t one aspect of the crack tip could be important , that is the plane stress shear lip, whose angle (h45") influences the stress distribution a t the crack tip.

4. Non-Uniform Axial S t r e s s T e s t s

Unsymmetrical axial compression was applied by spacers t,ransmitting load to one side of the speci- men. The crack curving response was measured as in the factorial design experiments, but with the an- gle 6 , a t which the crack initially started off, also be- ing measured. This angle was obtained from 6, = ta11'(Dn.~/0.5) where was the crack t,ip deviation a t a distance of 0.5 cm. from the tip of the machined slot. A negative value of the angle 6, indicates that the crack turned away from the side in compression. A total of five cylinders were tested in this manner and the observed results are given in Table 3.

It was observed in four out of the five cylinders tha t the crack curved initially towards the side which was under axial compression. In cylinder #4 it is quite possible tha t the initial machined slot may have been slightly misaligned or the axial compression was misap- plied. Comparing these results with those of the fac- torial design experiments, which had almost symmetri-

Table 2. Statistical Analysis (ANOVA) by SQC

Source I Sum Square [ DF I Mean Square I F s t a t I P value

Table 3. Crack Deviations in Unsymmetric Axial Compression Tests

Liner Thickness Axid Compression Geometric Intensity Crack T ip Condition Error

cal axial loading, indicates tha t the lack of svmmetry in t,he axial compression, and hence the iinsymmetric bulge, tends to increase crack curving, and that the crack turns towards the side which is under axial com- pression.

1.330 1 1 1 1.330 1 41.964 1 0.001

Reflection Photoelastic Tests

Having est,al,lished that the crack t,urns towards the side which is in axial compression i t was decided to test a few cylinders with photoelastic coatings t,o deter- mine the stress intensity factors, and a,,. The chang- ing fringe pattern was recorded by video as the pressure increased, and a typical fringe pattern showing the dis- torsion a t the crack tip due to mode I1 deformations is presented in Figure 3.

Three cylinders were tested and in two out of the three, the crack curved towards the side in which ax- ial compression was acting. The results (*), are also presented in Table 3.

In each of the three configurations lack of svmmetry resulted in deformations which produced mode I1 crack t ip conditions.

0.123 0.108 0.007 0.856

5. Dvnamic B u r s t T e s t s

1 1 1

2 7

0.123 1.108 0.007 0.032

The difference between the crack path in a quasi- static experiment and a dynamic one is often quite ~i~~~~ 3: ~~i~~~ pattern from an unsymmetric corn- dramatic" and to investigate how t.his is related to pression T~~~ bulging the following series of dynamic burst tests were

1344

3.878 3.408 0.223

0.059 0.076

n/s

COPPER WIRE I4 V

WGLE SECnON STRHGER

-1

SLOT S P K N G

'-2 GtWLND

ELECTRODE

Figure 4: Dynamic Test Specimen Configuration

carried out using a t echn iq~~e l " , that had been used earlier to study thin cylinder failure. The specimen configuration for this type of test is shown in Figure 4. In this case the tests were mainly qnalitative in nature, and were used to ascertain how sensitive crack stability is to various conditions of stress ratio and bulging. An exploding wire system was used which consisted of a set of capacitors which can be discharged through a cop- per wire. These capacitors can store up to 6000 joules of energy when fully charged to 20 kilovolts. When the stored energy is discharged through the wire, rapid heating and vaporization of the wire takes place. The rapid expansion of the heated vapor generates an ax- isvmmet ric blast wave which propagates through the surrounding air t o impinge uniformly on the inner wall of the cylinder. T h e impingement of this blast wave causes a radial expansion uniformily along the length of the cvlinder, which results in rapid crack growth.

The specimens used were chosen to be identical to those used in the quasi-static burst tests, but air was used instead of oil as the pressurizing medium with the absence of a vinvl liner. Also in the dynamic tests the pressure pulse lasts for a very short period of time and leads to fast crack propagation (crack speeds measured from high speed photographs varied between 450 and 915 feet/sec, but were well below the speed of sound in the material). In dynamic tests even after the crack begins to propagate, the hoop loading does not dimin- ish as inertia effects cause the cylinder to continue to expand.

T h e loading in this kind of configuration is com- pletely symmetric and the crack invariably runs straight until the cylinder opens up, almost into a plate. High

Figure 5: Explosive Failure of a Thin Cylinder

speed photographs taken of tlhe cvlinder profile as the crack propagated, like the sequence in Figure 5, re- vealed a significant amount of bulging as in quasi-static tests. This indicated tha t the presence of bulging alone was not sufficient to make the crack turn. In this case the bulge shape is such that one would expect the bulge induced stress ahead of the crack to be quite small, and so with RA < 1 the stress ratio concept of path stabil- i ty could apply. However, with only a large hoop stress, Poisson's rat,io results in a compressive axial stress, and so the Streit and Finnie criterion for stability also is mr t .

To divert the crack from it's straight path, an angle section longitudinal stiffening member was held in place on one side of the initial slot (Figure 4). Bulging was substantially less due to the constraining effect of the 'stringer'. This unsymmetric axial loading resulted in the crack consistently turning away from the 'stringer'. Tests were performed with different stringer to slot dis- tances. The extent of the straight crack propagation increased as the 'stringer' was moved away from the initial slot. The actual distance the crack grows before the 'stringer1 influence is felt is of course a function of the crack propagation speed and the speed of sound in the cylinder. In most of these tests the cylinders exhibit a typical c~ase of flapping.

These experiments show tha t crack curving is possi- ble by changing the bulge shape, so tha t the mode I1 component of the deformations is greatly enhanced by the reduction in symmetry. This deformation also sug- gests that a mode 111 type deformation could also play a role.

A series of experiments were conducted with the ini-

C 3 3 3 C 9 C

Z'ACX A'IG-E

Figure 6: Fracture Angle versils Intial Slot Angle

tin1 slot placed at, various angles to the cvlinder axis. These tests resulted in the crack starting in a direc- tion 0 , . A plot of the results from these experiments is shown in Figure 6 where thev are compared with the Rat plate values reported in ".

In this case when the crack angle varied from 90" to about 45" degrees the cvlinder response followed that of the flat plate, but as the angle tended to parallel that o l the hoop trnsion the fracture angle decreased. until extension could onlv be achieved bv applyinq more pressure. .It 45" the bi~lging had reached the point where the lack of svrnmetrv was greatest. From that point on as the svmmetrv increased the crack angle decreased until a t 0" the bulge was again svmmetric in rplation t o the crack but a t 90" to i t , that is still in the axial direction. Also a t some point close to 20" the bulge was too small to drive the crack.

6. F a t i g u e Induced Fa i lu re

Verv few elcperiment.al studies have been reported on cylinder fatigue, and those that d o exist are mainlv concerned with crack ra te and initiation of that growth3.

The appara tus designed for this part of the studv consisted of a M'TS svstem in which the cyclic hvdrauiic pressure to the specimen is controlled bv an electronic servo svstem and the axial load was a piston activated bv a t ap from the main hvdraulic supply. The sprci- mens for this series varied from 4.5-9.0 ins. in lenqth and the diameter was varied from 1.0-1.5 ins.

Followinq the experimental design concept used for

the quasi-static burst tests a factorial design experi- ment was conducted to investigate the main factors controlling the crack curvature in a fatigue situation.

Five factors were considered for this set of experi- ments, these being pressure, frequency, applied stress ratio, crack and cylinder lengths

Analyzing these results using SQC software it was found that none of the factors significantly influenced the crack curvature. This r e~ i i l t was somewhat disap- pointing, but it did indicate tha t the range of the factors was not sufficient to overcome the statistical variation in the test results.

However, t he results did show that the direction of crack curvature was again controlled by a lack of sym- metry in the bulge. In this case internal pressure acting an the shear-lip, which is expected in plane stress thin cvlinder failure, appeared to be the controlling mecha- nism.

7. C r a c k C u r v a t u r e P r e d i c t i o n

As is t o be expected a number of failure theories have been developed to predict crack propagation under mixed mode loading. Details relating to these are be- yond the scope of this paper, but two of the most relevent are brieflv outlined.

Maximum Circumferential Stress Theorv

According t o this theorv, bv Erdogan and Sih", a crack propagates in a radial direction perpendicular to the maximum circumferential stress un. The angle 8, a t which the crack propagates with respect to the initial crack plane is obtained from:

Kr sin 0, + Krr ( 3 cos 8, - 1) = 0

which gives

When the crack extends in this direction.

1 6,: - cos - [ K r ( l + cos 0,:) - 3Kr1 sine,] = Kr, 2 2

where Kr , is the critical stress intensitv factor a t which fracture occurs in mode I. This theorv does not account for the influence of material properties around the crack tip on the stress distribution, or the influence of the nonsingular stress term

\laximum Strain Enernv Release Rate

Thig throrv proporred by ~ t . r i f n r ~ ~ ' is Imsrd on the generalized J integral concept. T h e strain energy re- lease rate for a virtual displacement in terms of a gen- eralized J-integral quantity is given by:

K.+ 1 J H -- - [(K12 + ~ 3 ~ ) cos 0 - 2 K r K r r sin 6)

8 P

where K = ( 3 - v ) / ( l + v) for the plane stress case, and p and v are the shear modulus and Poisson's ratio respectively. According to this theory, which is considered to be ap- plicable in the case of ductile failure the direction of crack p r ~ p a g a t ~ i o n 6,. occurs when Jo is a maximum.

The stress intensity factors K r and K I I , required to apply the above theories, were determined from the ex- perimrntal photoelastic results described earlier using a technique developed by Sanford and ally'^.

K r and Kr r Determination

A ' non-linear equation was developed by San- ford and Dally relating the stress field in terms of K r , KIr and u,,,, t o the coordinates, r and 0, defin- ing the location of a point on an isochronlatic fringe of order N . This was done by considering the mixed- mode stresses in the local neighborhood of the crack tip ( r /2a(<l) together with the standard equations for the maximum in-plane shear stress r,, related to the Cartesian components of stress by

The governing equation for isochromatic fringe patterns obtained through reflection photoelasticitv is

where E and v are the elastic modr~lus and Poisson's ratio respectively of the test specimen ( i.e., of the aluminum cylinder), while f is the fringe value of the coating. The fringe value f can be computcd from f =A/(2tK) , where A = 2 2 . 7 ~ 1 0 - 6 inches for white light, t is the photoelastic coating thickness and K is the st,rain-opt,ical coefficient. of the photoelastic plastic (di- mensionless quanti tv). For the coating used t = 0.78 mm, K = 0.02 and thus f = 0.0183. When the crack tip stress equations are coml~ined with T,, and the fringe equation a non-linear relation which defines the isochro- matic fringe pattern in the local field near the crack tip is obtained

T h e det,ails of t,he above equation are given in7! 13 . An over-deterministic approach, where n(> 3) arbitrarily

located points arc ~ r l r r t r d from the fringe field, was ~rsed to solve the non-linear equation. The method of least squares was employed to fit the K-N relation to the field data .

It was found that reinforrrmrnt rffects due to the coating can be neglected, but a photoelastic correction factor for bending stress is required. The correction factor in this case was 0.5.

Crack P r o ~ a g a t i o n Annie Predictions

The program described above was used to compute K r , KIr and a,,, for three cylinders which were tested with photoelastic coatings. Twenty d a t a points from the fringe patterns were used in each case.

As the fringe pattern reflected the true stresses a t the crack t ip the usual cylinder curvature n~agnifica- tion factors are not required. The K I , KII and u,,, values determined by t,he above method are given in Ta- ble 4, together with the predicted angle of initial crack propagation. I t can be seen that both the maximum stress theory and the maximum strain energy release rate gave similar results. T h e values obtained for the K's appeared to be high, and the corresponding ratio of K I r / K r low. I t was recognized that plasticity and the blunt crack condition represented by the slot could in- troduce considerable error. However, it is also possible tha t with a n unsymmetric bulge mode I11 deformations should be considered.

Though the predicted and experimental values of 6,: differ, the predictions regarding direction were accu- rate in each of the three tests. The results of test 2 are of particular interest. In this test the crack turned away from the side in compression, probably due to the load being applied improperly. However, the theoretical prediction from the fringe pattern also indicated that direction. Although, the Mode I1 component is respon- sible for crack turning, in all three configurations it was the differential bulging tha t introduced the conditions which controlled i ts magnitude. The deformations a t the crack t ip might therefore he sufficiently complex that a more sophisticated model is required to describe some of the practical situations tha t could arise.

8. CONCLUSIONS

Crack propagation in a quasi-stat8icallv pressurized thin cylinder is inherently unstable, and the results of this study both confirm this fact, and help determine the factors which contribute to the instability and sub- sequent crack path curvature.

One result tha t was consistent with ail three exper- iments was the importance of the bulge configuration. It was determined that bulging in itself does not cause

'Table 4 . Computed \:slues of K I , K I I , B,,, and Predicted and Experimental 6,:

the crack to curve but introduces the conditions where the crack growth can be unstable.

In the static cases the bulging produced axial tension, and hence these configurations were always unstable, even in the fatigue tests with R A < 1 with low pressure levels such that Ra < 1, (Fig.2) so all the results are in accord with the crit,erion of Streit and FinnieG, rather than that of Radon4.

When.the cvlinders were constrained with 'stringers', t,hey limited the extent of bulging on the side with the dr inqer , leading to differential bulging, and hence the crack path was no longer st,raight. These results also indicate tha t unsvmmetric bulging introduces a mixed mode loading condition where the crack propagation, in addition t o being unstable, is also required to curve. The direction taken by the propagating crack is re- flected in the sign of the stress intensity ratio K I I j K I . In t,he dynamic case this ratio was clearly increased by the unsvmmetric bulging, but in the other two configu- rations unsvmmetric bulging was also cawed, by either the unsvmmetric axial loads, or the shear lip in combi- nation with the internal pressure.

The results show that in the burst tests a liner has a pronounced affect on crack curvature. and that a thin aud more flexible liner will increase the degree of curva- ture. It was also confirmed that as t,he cylinder radius increases the crack has less tendency to curve.

Internallv pressurized cylinders are inherentlv unst,a- ble, with the crack path controlled by conditions that could introduce mode I1 and mode 111 deformations.

Aknowledgements

This work was supported in part by the Boeing Corn- panv. The authors would like to thank Dr. hl. Miller and the staff of the Structural Damage Technology Group of Boeing for their helpful discussions.

Exprtl.

0,

3 1

R e f e r e n c e s

"Test

1

1. Swift, T . , "Damage Tolerance in Pressurized Fuse- lages". 11 th Plantema Memorial Lecture presented to 14th Symposium of the International Com- mittee on Aeronautical Fatigue (ICAF), Ottawa, Canada. June 10-12, 1987.

C o m p i ~ t e d V a l i ~ e s o f K ~ ~ K ~ ~ , ~ , from Photoelastic Fringe ( k s i 6

K I I Krr I firm 171.7 1 -24.1 1 -81.6

Predicted Angles of Initial Crack Propagation 8, (degrees)

Max. Si r . Stress I Min. Energy Density

2. Folias, E.S. "An Axial Crack in a Pressurized Cylindrical Shell", Int. Jl. Fracture Mechanics, 1, 104-113, 1965.

3. Erdogan, F. and Ratwani, M., "Fracture of Cylin- drical and Spherical Shells Containing a Crack", Nuclear Engr. and Design, 20, 265-286, 1972.

4. Radon, J . C., Leevers, P. S. and Culver, L. E., "Fracture Toughness of PMMA Under Biaxial Stress", Fracture 1972, Vol. 3, University of Wa- terloo Press, 1113, 1977.

5 . Cotterell, B., "Not,es on the Path and Stability of Cracks", Int. Jl. of Fracture Mech., 2, 3, 526-533, 1966.

6. Streit, R. and Finnie, I. "An Experimental Inves- t,igation of Crack-Path Directional Stability", Ex- perimental Mechanics, 20, 1, 17, 1980.

7. Set.hi, V. "An Jnveat,igat,ion of Initial Crack Direc- tion in Thin-Walled Cylinders," M.S. Thesis, Dept. Aero. & Astro., University of Washington, 1990.

8. Boeing SQC Instruct,ion Manual, Design of Exper- iment Program, 1989.

9. Erdogan F. and Sih, G.C, "On the Crack Exten- sion in Plates Under Plane Loading and Transverse Shear", J1. of Basic Engr., Trans. A.S.M.E. 519- 527, Dec. 1963.

10. Strifors, H., "A Generalized Force Measure of Con- ditions a t Crack Tips", Int . Jl. of Solids and Struc- tures", Vol. 10, 1389-1404, 1974.

11. Fyfe, I.M. and Rajendran, A.M., "Dynamic and Pre-Strain and Inertia Effects on the Fracture of Metals", J . Mech. Phys. Solids, 28, p17, 1980.

12. Peters, R. W. and Kuhn, P., "Bursting Strength of Unstiffened Pressure Cylinders with Slits", NACA Tech. Note 3993. 1957

15.4 20.3