Damage Tolerance of Cracked Cylindrical Shells under

(c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

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Damage Tolerance of Cracked Cylindrical Shells underInternal Pressure

Dennis Hoogkamer, Anthony M. Waas* and Johann Arbocz^Composite Structures Laboratory,

Department of Aerospace Engineering,University of Michigan, Ann Arbor, MI48109-2140

February 9, 2001

AbstractIn the present work, a series of unstiffened, thin-

walled (radius to thickness ratio, R/t — 216), metalliccylindrical shells, containing centralized cracks at an-gles to the shell generators ranging from 0° to 90° atintervals of 15° and of two crack lengths, are internallypressurized in order to determine the effect of crack an-gle on the failure mechanism of the shells. This is doneunder two different sets of boundary conditions. In thefirst case the shell is free to move in the axial direction,in the second case this axial movement is constrained.

Curves of the maximum pressure leading to globalfailure of the shells as a function of crack angle for twodifferent crack lengths and for both cases of bound-aray consitions are presented. The present experimen-tal data can be used to find a governing predictive crackgrowth criterion as a function of crack angle. This crite-rion can be used in commercial finite element codes forpredicting the failure of more complex shell structureslike those used in transport fuselage shell structures.

The experimental data are explained via a fracturemechanics based analysis. For cracks which are inclinedwith respect to the applied stress, both the opening(tensile) mode I stress intensity factor and the sliding(shear) mode II stress intensity factor are significant.These depend on the angle of the crack with respectto the applied load. Fairly good agreement betweenanalysis and experiment is obtained if adjusted valuesof Kjc and KHC are used along with the assumptionthat crack growth is governed by a quadratic failurecriterion. The adjusted values account for shell curva-

* Graduate student and Professor of Aerospace Engineering re-spectively, University of Michigan. Assoc. Fellow AIAA, Copy-right ©2001 by Anthony M. Waas. Published by the AIAA, withpermission.

^ Chairman, Aerospace Structures & Computational Mechan-ics Group, Department of Aerospace Engineering, Delft Univer-sity of Technology, The Netherlands. Fellow AIAA.

ture through a bulging factor ft. Finite element simu-lations of selected shell configurations are carried outto obtain the crack flange deflection corresponding tothe burst pressures. An approximate analysis of 0° an-gle cracks based on technical beam theory and axisym-metric cylindrical shell analysis was presented earlier[1]. Results from this analysis are compared with thepresent finite element analysis for 0° angle cracks ina forthcoming paper [2]. Finally, images of the crackpropagation event acquired at high framing rates forselected shell configurations are included.

1 IntroductionDue to the aging of the commercial aircraft fleet of theworld's airlines, it is becoming more likely that aircraftdo not meet their original intended design requirementsdue to the onset of fatigue cracks and damage caused byeither corrosion or impact. The effects of this multiplesite damage can be devastating as was shown in Alohaflight 243 where a considerable part of the fuselage roofwas blown away due to multiple site fatigue damage[3]. The concept of damage tolerance is based on thenotion of assessing the departure in response of a struc-ture that contains a specified amount of damage (im-perfections) in comparison with the intended responseof the corresponding undamaged structure. Thus, in asense, damage tolerance provides an assessment of how'robust' a structural design or concept is. The 'amount'and 'type' of damage is specified a priori, based on theknowledge of the life-cycle that is specific to the partic-ular aircraft or spacecraft.

At The University of Michigan, Department ofAerospace Engineering, Composite Structures labora-tory, a program has been initiated with the aim of in-vestigating the static and dynamic failure of cylindricalshells containing cracks of different lengths and at dif-

1


ferent angles to the shell generators. In the present pa-per, experimental results are presented to characterizethe failure of a set of cylindrical metallic shells contain-ing a centralized crack at an angle with respect to thegenerators of the shell. Results from a finite elementanalysis for 0° angle cracks are also presented.

Previous work done by a handful of authors hasmainly been focussed towards the static loading (uni-and biaxial) of a cylindrical shell with a centralizedcrack at a zero angle with respect to the shell gener-ators. These previous studies can roughly be split upinto an experimental and a numerical group, howeverthe most significant difference between the different re-search initiatives is the use of a linear or a nonlinearapproach in analyzing the results, [4,5,6].

Linear fracture mechanics suggest that the likelihoodof fracture depends on the crack tip stress intensity fac-tor reaching a critical value. It is assumed initially thatchanges in the shell geometry during the event of load-ing do not cause any changes in the crack tip stressintensity factor. After solving the problem using theseassumptions, a so-called bulging factor is implementedto account for nonlinear effects which occur due to localshell bending deformation. The bulging factor providesa characterization of the departure in the stress inten-sity factor due to shell wall curvature. Thus,

Vshell

"- kp 'where kp is the stress intensity factor for the corre-

sponding flat plate crack problem.Analytical methods that have been reported previ-

ously are generally only valid for small (< 5) shell cur-vature parameters Af50, where

In the above expression, 2a is the crack length, R, theshell radius, t is the shell wall thickness and v the pois-son's ratio of the isotropic shell wall material. Thesemethods tend to overestimate the physical bulging un-less the investigated cracks are small. The cause of theover estimation is the linearization of shell equilibriumequations. The linearization does not account for thetensile stresses which develop parallel to the crack lineand increase the resistance to bulging and crack open-ing [7].

All nonlinear work that has been done in the field ofcylindrical shells with centralized cracks are focussedtowards specific materials and are valid under specialconditions. Previous work by Starnes and Rose [7] sum-marizes the resulting semi-empirical relations of theseresearch initiatives and refutes or verifies them using

a nonlinear Finite Element code-STAGS (StructuralAnalysis of General Shells).

Starnes and Rose [7] have shown that for a shell witha small \iso < 5, the bulging radial displacement wc,measured at the crack flange center is a linear func-tion of the far-field hoop stress. They also found thatthe bulging factor was independent of the hoop stress(pressure) for shells with a small \iso and the nonlinearstress intensity factor of the shell is a linear function ofthe far-field hoop stress. For shells with a large shellcurvature parameter(> 10), the results obtained weresignificantly different. The radial bulging displacementnow is a nonlinear function of the circumferential stress.Further, as the load is increased, axial stresses along thecrack edge become more dominant resulting in a mem-brane tensile axial stress, which increases the resistanceto bulging even more.

The results presented by Starnes, Rose and Young[8] includes finite element results of the effect of com-bined loading on the nonlinear bulging effect in shells ofdifferent configurations, with cracks in both axial andcircumferential directions.

The present paper is organized as follows. In section2, details of the experimental program carried out toobtain a failure envelope for cracked cylindrical shellsare presented. Included in this section is also a descrip-tion of the work that was done to obtain insight intothe dynamics of the failure process. Crack path direc-tion and crack velocity have been measured for selectedshell configurations.

In section 3, details of the finite element simulationthat was carried out to understand the cracked shellresponse are presented. Results are presented for crackorientations at 0 = 0°, where 9 is measured from theshell generators.

For this case, the predictions of an analytical modelfor crack face bulging developed earlier [1], using tech-nical beam theory and an assumption of axisymmetryare compared against the results presented by Starnesand Rose [7] and the present finite element work in [2].A Fracture Mechanics based failure criterion is intro-duced in section 4. Finally, some concluding remarksare presented in section 5.

2 Experimental WorkFor all of the experimental work reported in this pa-per, seamless shells, previously used as containers forCoca-Cola beverages were used. Due to the high man-ufacturing tolerances of the containers, the shells areeasily machined into a cylindrical shell by removing thetwo ends of the containers. As a result of the produc-tion process that is used to produce beverage contain-


ers, a certain thickness distribution over the length ofthe shell is apparent. The thickness distribution is dis-played in Figure 2.1. The 'nominal' wall thickness is0.006 inches and the 'nominal' internal radius is 1.3inches. This thickness distribution does not seem tobe of high significance due to the fact that most of thethickness variation falls inside the clamps holding theshell. In addition, the presence of a relatively largecrack overwhelms the presence of other unintended im-perfections such as a non-uniform thickness distribu-tion. As shown in the results section of this paper,consistent burst pressures for several of the shells arerepeatedly obtained.

The shells are placed between a static base and amobile top plate, as shown in Figure 2.2. Two typesof tests were conducted. In the first, the top platewas free to move when the shells were pressurized. Inthe second, the top plate was held from moving out-wards during the pressurization. The shell is clampedto the base and top plate by accurately manufactured,adjustable clamps leaving a free length of the shells inbetween the clamps of 3.5 inches. Nitrogen gas is usedto pressurize the shell and kept from flowing out bytwo o-rings, placed between the shells and the clampsat both ends as shown schematically in Figure 2.2. Theadjustable rings are kept from damaging the outside ofthe shell by wrapping a piece of Teflon tape at the out-side end of the shell. Note that in some of the tests,where the top plug is free to move, results obtained cor-respond to a situation of pure internal pressure. Thissituation is obtained by allowing the top clamp to ax-ially move outward during the pressurization process.However it was found that the entire shell (and thetop clamp) "slid out" from the bottom clamp duringsome of the tests, thereby relieving the axial stress inthe shell. This observation was captured by the highspeed images that were taken during the pressurizationprocess. This characteristic of the setup results in anaxial load that is close to being equal to zero. Thepure internal pressure loading reduces the complexityof the problem of the far field stress resultants whichare now entirely a circumferential hoop stress compo-nent. Closer to the crack, bending of the crack flangesintroduces other stress and moment resultants that pro-duce a departure from the pure hoop stress case. In thesecond set of tests, the axial motion of the top plug isconstrained from moving outward by a plate, which isattached to a loadcell in order to be able to measurethe axial load. This second set of boundary conditionsis more specific to achieving near zero axial elongationduring the pressurization process. That is, the two endsof the shell are restrained from moving axially duringthe pressurization, by a clamping ring.

The base is connected to a pressure transducer and

an amplifier, which are connected to a high speed os-cilloscope. As a result, the pressure is measured withan accuracy of 0.3 psi. Finally, the crack in the shell issealed from inside with a very thin sheet of plastic, toprevent the gas leaking through the crack. The plasticsheet is glued at its corners to the inside of the shell toensure that it remains in place, but at the same timewill not influence the properties of the shell locally atthe crack tip. When conducting a test, the Nitrogengas is released from a gas cylinder into a reservoir athigh pressure after which a high speed electric valve re-leases the pressure into the shell. This approach makesit possible to 'instantly' pressurize the shell. A typicalpressure versus time history plot for sudden pressuriza-tion is as indicated in Figure 2.3. A strain gage placedat one of the crack tips is used to trigger an ultra highspeed variable framing rate digital camera(CORDIN),which is used to capture images of the crack propa-gation process (Figure 2.4 and Figure 2.5). Some ofthese results will be discussed later. One of the ob-jectives of the experimental work is to determine thedynamic crack growth characteristics when the shellsare exploded by "overpressurization". For these tests,the straingage used to monitor the the strain at thecrack tip is also used to trigger the CORDIN camera (8frames at 0.5 //sec. between frames, with a 0.01 jjisec.exposure time) that can acquire images which will showthe events in the vicinity of the crack tip at the onsetof tearing.

2.1 Proposed MethodologyThe main objective of the experiments was to obtain anexperimental correlation between the cracking pressure,the crack angle and the crack length. This was doneby considering different crack lengths, each for variouscrack angles, as defined in Figure 2.6.

In [7], a non-dimensional crack curvature parame-ter is defined for isotropic shells as A = a/V^Rt, where2a is the crack length, and R and t are the radiusand thickness of the shell respectively. We will usethis definition of A = a/Vftt, throughout this paper.The dimensions of the shells as given in [7] are com-pared to the shells that were used in the present exper-iments in Table 2.1. The equivalent crack lengths forthe present experiments are then calculated such thatthe non-dimensional crack curvature parameters, A, forthe shells with 0=0, are similar to those correspondingto the shells used in [7]. The equivalent crack lengthsare presented in Table 2.2. The crack angles that weremachined are displayed in Table 2.1. The method ofproducing the cracks has a large impact on the "qual-ity" of the crack tip. In the present case the cracks areall produced using a rigid fixture in which the shell is


placed over a 'snuggly' fitting mandrel and an exact-oknife is implemented to produce the crack. A magnifiedphotomicrograph of the crack tip is presented in Figure2.7. A typical crack tip radius is indicated as (0.0011inches). This radius is 16% of the shell wall thickness,and 0.16% and 0.12% of the initial crack lengths of0.668 and 0.886 inches respectively. This is representa-tive of the "sharpness" of the cracks that are studiedin the present experiments. Note that the definition ofthe non-dimensional crack curvature does not includethe free length of the shell, and hence the ratios for r/Land a/L for the shells used in the present experimentsare not compared to the corresponding ratios as usedin [7].' It is expected that the effect of these parameterson the cracking pressures is of a lot lower significancethan the other parameters, since ^<Cl.

When studying the preliminary images that were ob-tained using the CORD IN camera, it becomes apparentthat the crack opens significantly before it starts grow-ing in an unstable manner. This indicates that a non-linear phenomenon with respect to the loading, possiblydue to crack tip plasticity is predominant and thus thiswill have to be implemented in modelling. The peakpressures that are reported in the present paper (pointA in Figure 2.3) were read off as is indicated in Figure2.3. However, it is not certain that at this pressure thecrack starts growing in a stable or unstable manner orthat perhaps the crack starts to grow at a lower pres-sure (like point B, as indicated in Figure 2.3). In orderto clarify these uncertainties a more detailed study ispresently underway using high speed photography andusing more refined ways of synchronizing the pressuretransducer output and the image-time history data.

2.2 Experimental ResultsFigure 2.8 shows the results for the peak pressures thatwere obtained for identical crack lengths (equivalent tothe 3 and 4 inch crack lengths as presented in [7]), butfor several crack angles. Note that unlike what onewould expect for a crack in a "flat sheet", the pressureat which the shell catastrophically fails is about 6 timeshigher when the crack is in the hoop direction (90 deg.)than when the crack is in the axial direction (0 deg.). Alinear fracture mechanics based analysis of the exper-imental data indicates that a quadratic crack growthcriterion is able to capture the qualitative trends thatare indicated by the experimental results. This analysisis presented later.

In Figure 2.9, the crack tip velocity is set out againsttime, where the time origin is chosen at the peak wherethe crack tip strain starts to relax, for a shell contain-ing a crack at 0—0 and loaded by a sudden increaseof internal pressure. These measurements were made

with the images obtained via the CORD IN camera.The first frame in Figure 2.4 corresponds to a cracklength of 1.1 inches at t=0. Note that the initial un-pressurized crack length is 0.6 inch. The pictures aresnap shots in a window of time of the crack propagationevent. The crack tip velocity versus time in this windowprovide insight into the non-uniform crack propagationevent. The crack tip velocity increases with a exponen-tial type behavior. This shows that the crack acceler-ates in a nonuniform manner while it is propagating.If the shells were long enough, it would be interestingto determine the terminal (asymptotic) velocity thatthe crack will finally reach. Dynamic crack growth inflat metallic structures have been a hot bed of researchfor sometime now and the interested reader is referredto [10], for more details pertaining to dynamic fracturemechanics.

3 Finite Element AnalysisA Finite Element Analysis to simulate the pressurizedshell experiments was conducted using the commercialFEA package ABAQUS with Hypermesh as the pre-processor. Other packages were investigated for pre-processing, but none were as effective as Hypermesh.A mesh of the entire shell was made using S8R, 8-node quadrilateral doubly curved shell elements, imple-mented using reduced integration. These elements al-low for through the thickness shear deformation, whichcan be significant for high deformation zones. Further,mesh refinement was achieved using 6-node, STRI65,triangular shell elements far away from the high de-formation regions. The STRI65 are 6-node triangularshell elements, using 5 degrees of freedom per node andimplemented using reduced integration. This type ofmesh refinement was chosen over the use of a multiplepoint constraint approach due to the recommendationsprovided in the ABAQUS manual [12]. The principlereason is that MFC's introduce constraints that lockthe response in the finer mesh.

Due to the difficulties associated with meshing acurved surface, the shell was constructed by modellinga flat sheet with the imperfection present (the crack)and then have ABAQUS convert the given coordinatesto a cylindrical coordinate system. This resulted ininserting x,y,z-coordinates where the x coordinate wasthe radius of the shell, the y coordinate was the circum-ferential distance and the z coordinate, the coordinatealong the axial direction of the shell.

3.1 Mesh SensitivityAs part of the finite element work, a mesh sensitiv-ity study was conducted. A shell with a longitudinal

(c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)1 Sponsoring Organization.

crack was modelled using three different mesh densi-ties. Mainly, the area around the imperfection wasmodelled with a finer/coarser mesh, but also the den-sity of elements in the far field was varied in these threecases. Table 3.1 indicates the results of this study. Allfurther finite element analyses were conducted with amesh density as is indicated in table 3.1. Note that allthe models used in the mesh sensitivity study involvedgeometric nonlinearity as well as material nonlinearity(J2 incremental theory of plasticity) to create a worstcase scenario. As indicated in Table 3.1, the normalizedcrack flange deflection (-^r), has recahed a convergedvalue with the second mesh containing 167,748 degreesof freedom. Thus, further studies were carried out withthis less dense mesh.

3.2 Modelling ApproachThe finite element analysis was conducted using theNLGEOM function in ABAQUS which incorporates ge-ometric nonlinearity. This is needed because of the rel-atively large rotations near the crak flanges of the shell.The pressure loading is gradually built up in a fashionsimilar to the pressure buildup during the tests, allow-ing ABAQUS to reduce the step size for it to reachconvergence. For purposes of comparison, a geometri-cally and materially linear analysis was also carried out.Note that symmetry with respect to crack position, inthe case of the 0° degree cracks was not implementedsince future work will contemplate the use of cohesivezone modeling [11], to provide crack propagation trajec-tories. In such an approach, the crack growth directionis determined as a part of the solution and not asumeda priori.

3.3 Material propertiesThe material properties of the shell wall material weredetermined by doing several tensile tests on dogbonespecimens cut out of the shell wall. These tests resultedin a stress strain curve which is represented in Figure3.1. The test data were fitted with a Ramberg-Osgoodformula:

The two unknowns in the above relation are found byobtaining the best fit to the test data. Note that theabove representation is convenient since it splits the to-tal starin into an elastic and plastic part. This decom-position is conveneient for the implementation of J2 in-ceremental flow theory of plasticity for an isotropic solidas provided in built-in subroutines within ABAQUS.The Young's modulus, Poisson's ratio and yield stressfor the shell material were found to be, E=24.43 GPa,

v — 0.30, and cryieid—51 MPa), respectively. Note thatrecycled aluminum has degraded properties comparedto virgin aluminum.

3.4 ResultsTo compare the results for the different model charac-teristics and different crack configurations, the normal-ized crack face bulging displacement, , is plotted as afunction of normalized hoop stress in Figure 3.2. Thisfigure indicates the results separated by crack configu-ration. As it can be observed for the cases of pure elas-tic material (without any plasticity included), all thecrack configurations for which finite element analysiswas performed, the relation between the central crackflange deflection and the hoop stress is approximatelylinear. The nonlinear geometry function in ABAQUSdoesn't initiate a nonlinear response (yet) for the pres-sures at which the shells fail in the experiments. Thisis consistent for shells of 'small' A as was discussed byStarnes and Rose in [7]. The value for the present shellis A — 5. Figure 3.3 shows the deformed mesh of the0°, crack case.

The picture is very different for models which includematerial nonlinearity. Here the nonlinearity effect (ofthe material this time) kicks in at the point where theplastic curve departs from the elastic curve. The imageof rapidly increasing ^ is a result of the decreasingstiffness of the crack flange material loaded into theplastic range as can be observed from the stress-straincurve in Figure 3.1. The stiffness of the material is rep-resented by the slope of the curve, which clearly reducessignificantly well into the plastic regime of the mate-rial. The stiffness reduces so much that at a specificloading, the rotational increments that are a result ofthe ramp loading (that is inherent in any geometricallynonlinear model)are too large for ABAQUS to handle,even when time increments that are automatically re-duced by ABAQUS to values that would make a singleanalysis run for a long time. ABAQUS then judgesconvergence unlikely and discontinues the analysis.

Further, it is interesting to note that the effect ofaxially constraining the boundaries of the shell has avery small stiffening effect on the shell as can be ob-served in Figure 3.2 for the fully elastic material cases.However, this effect is larger when the material is mod-elled with plastic stress-strain characteristics included.The reason for doing analyses including the axial con-straints are related to the test setup, which does notallow for the boundaries to move in the negative di-rection (outward, axially). However, the clamping ringin the experiment may allow the shell to contract dur-ing the pressurization of the shell. This aspect is notmodelled here.


To put all the numerical results in perspective, let usdefine a factor,

Wc

™clinear

This factor compares the crack flange deflection from apurely linear elastic (material and geometry) point ofview, against a fully nonlinear (material and geometry)point of view, as a function of loading (pressure). /?*against pressure is plotted in Figure 3.4, for all of thecases (9 = 0°, 45° and 90°). Also, we have marked by a'dashed' line the conjectured trajectories that we wouldhave expected as the pressure is further increased. No-tice that 0* begins to exponentially increase at aboutao/E ~ 0.0025 - 0.0030. This shows that for this rangeof pressure, /3* becomes unbounded. Further, when weexamine the stresses ahead of the crack tip as a functionof distance from the crack tip (Figure 3.5), we noticethat a very large region of the shell is substantially plas-tic even at 43% of the burst pressure for the 9 = 90°case. Thus, the current static Finite Element resultsyield that the maximum hoop stress sustainable by theshells for the cracks studied are ao/E ~ 0.0030. Thisnumber is higher than the hoop stress corresponsing tothe experimental burst pressure for 9 = 0° case, butlower than the experimental case for 9 = 90°.

Some of the differences in the experimental resultsand the finite element results are due to the non-inclusion of dynamic effects associated with the exper-iments and due to the uncertainty in associating themaximum measured pressure as the burst pressure. Asstated earlier, it is quite possible that crack propagationoccurs while the pressure is still increasing, since theloading mechanism is "over pressurization". Further, itis known that metals are strain rate sensitive, with thematerial properties substantially "stiffer" when rate ef-fects are incorporated. The incorporation of these threeeffects, namely, (a) certainty in establishing the exactpressure at first crack propagation, (b) inclusion of dy-namic effects and (c) inclusion of rate dependent mate-rial properties are relegated to future work.

4 Linear Elastic Fracture Me-chanics

4.1 IntroductionFor cracks which are inclined with respect to the appliedstress, as is indicated in Figure 2.6, the opening(tensile)mode I stress intensity factor and the sliding(shear)mode II stress intensity factor are significant. Thesedepend on the angle of the crack with respect to theapplied load. Consider the configuration corresponding

to the flat plate equivalent as shown in Figure 2.6, sub-jected to pure hoop stress. Then, by transforming thestresses to coordinates that are aligned and normal tothe crack, one obtains the following expressions for themode I and mode II stress intensity factors (SIF's);

= a^/Tra sin 9 cos 9.If crack growth can be described by a quadratic failurecriterion, it can be hypothesized that,

/ //where, KIC is the critical mode I stress intensity factorand is a known quantity for a given material. KJIC istaken to be KIC multiplied by a factor a, and is thecritical mode II stress intensity factor. Thus, assumethat,

KIIc=aKIc.Substituting these in the above crack growth criterion,we obtain,

2 J_ 2 ^R2

With the relations for KI and KH given above, thisyields,

cr2 (TTO) cos4 9 + — a2 (?ra) sin2 9 cos2 9 = Kjc,

as the condition describing crack growth. Simplificationof this relation results in,

KI:

where,c = 7ra cos2 0(cos2 0+ — sin2 0)

As a special case one can assume a to be equal to 1.This results in a relation for the constant c:

ca=i = ira cos2 9.

Ic

This yields for cr,

We can now apply the above development to a cylindri-cal shell containing a crack at some angle to the shellgenerators, when subjected to internal pressure. To doso, we shall assume the shell radius R ^> t, where t is


the shell wall thickness. Then, the above 'flat plate'development is applicable. For pure internal pressure,

a = pRt '

where p is the internal pressure. Thus, the critical pres-sure for crack growth for a "flat plate" is obtained as,

PC = ^

With the definition of /J, therefore, for a shell

shell ( JL\

~\RJi

When the experimental results are compared againstthe predicted results, the crack tip bulging can be com-puted and is found to be 0 = 3.39

4.2 ResultsResults obtained using the above formula for criticalpressure are compared against the experimental resultsin Figure 4.1. In this Figure, two curves computed us-ing two different values of KIC are indicated. The firstcurve uses Kfc — 22Ks^^/m while the second curveis computed using the value of Kjc that would matchthe experimental result corresponding to 0 = 0, whichis the case of an axial crack. From this case we ob-tain, fi = 3.39. Note that, the first curve predictionsare much higher than the experimental result, whereasthe prediction value obtained by matching the exper-imental data is much lower. This apparent loweringis associated with the crack face bulging due to local-ized bending deformations. What is encouraging is thatthe predicted trend for critical pressure by the analysismatches what was obtained experimentally. The anal-ysis has several limitations as pointed out earlier, themost important of which are the restriction to a math-ematically sharp crack and the restriction to a 'flatplate'. These effects must be properly accounted forand the need to include geometrically nonlinear effectsdue to crack face bulging as pointed out in [9], is alsoappreciated.

5 Concluding RemarksIn this paper, a set of experimental results for the burstpressure of internally pressurized shells has been pre-sented. Consistent results for 3 sets of shells corre-sponding to A = 3.8 and A = 5.0 were presented as afunction of crack angle 0. The experimental results forburst pressure show that cracks with 0 = 0° are themost critical. Further the experimental results have

Symbol

Rtit2Le

2a

From [7]

9.0 in.0.020 in.0.040 in.36 in.Odeg.

1,2,3,4 in.

From [7]

10 in.0.1 in....20 in.0 deg.

4 in.

AluminumShells1.30 in.0.006 in.

3.5 in.0, 15, 30,45, 60, 75,90 deg.0.668,0.886 in.

Table 2.1: Dimensions of Shells.

Crack lengthsFrom [7]1.0 in.2.0 in.3.0 in.4.0 in.

Crack lengthscurrent shells

0.668 in.0.886 in.

Table 2.2: Crack lengths for equal crack curvature pa-rameters, 0=0 deg. resulting in equal A

been used to obtain /3 = 3.39 for the present testedshells.

The Finite Element Analysis of selected shell configu-rations were presented. Both geometrical and materialnonlinearity were included. It was observed that sub-stantial plasticity around the crack tip region leads tolarge crack flange deflections. Linear elastic analyseswere carried out for all the cases succesfully. A nondimensional factor /3* = was introduced to

Jclinearsummarize the results. The plot of /3* versus normal-ized hoop stress indicates that when the hoop stressreaches a range of <je/E ~ 0.0025 - 0.0030, /?* -» oo.This indicates that, with the given input and given pres-sure controlled (as opposed to volume controlled) load-ing, this is the maximum range of ((?0/E) that is pos-sible. The reasons for the discrepancy between the ex-perimental results and the FEA analysis for the 0 > 0°cases was discussed earlier and it is felt that the in-clusion of these effects will substantially improve thisdiscrepancy.


.Toppto

CrackAngle

00*04590

element type

S8R/STRI65S8R/STRI65S8R/STRI65S8R/STRI65S8R/STRI65

no. d.o.f.

124542167748235152126066105060

Char,el.length1.491.41.261.41.4

wct

0.55800.56850.5686

Table 3.1: Overview of mesh refinement study

Adjuring

AtI"S

»8^

r

/

i\\I. H .n

I

i\ii

yr1

-Ikadednd

Adjuring

.Pressure fine

Figure 2.2: Schematic of shell setup.

Thickness distribution shell in axial direction

0.05

20 30 40 50 60 70

Coordinate along the shell [mm.]

Figure 2.1: Thickness distribution of shell in axial di-rection.

Pressure vs. Time curve

0.01 0.02 0.03 0.04 0.05 0.06

Time [s]

0.07 0.08

Figure 2.3: Typical pressure vs. time history plot forsudden pressurization during pressure build-up phaseand global failure, for a shell with a crack at 9 = 60°.


Frame S: t = 60 asec Frame 6: t = 80 p&ecFigure 2.5: Typical high speed camera images of crackpropagation of a 0.886 inch crack at 30 degree angle,A = 5.0.

Frame 7,1100 tisec. Frame 8,1200 jjsec.Figure 2.4: Typical high speed camera images (100000Hz) of crack propagation of a 0.6 inch crack at 0 degreeangle, A = 3.40. 2L

Figure 2.6: Schematic of shell parameter definitions.


Figure 2.7: Microscopic image of crack tip.

Stress-Strain Curve

2000 3000

Strain [^e]

4000 5000

6-pressure curve

.--&- Stable crack growth 3 in. [1]-x—Stable crack growth 4 in. [1]

-a—Unstable crack growth 3 in. [1]

-<—Unstable crack growth 4 in. [1]

-£_ Experimental data equivalent to 3 in. crack [1](2 tests)

n Experimental data equivalent to 4 in. crack[1] (2 tests)

Figure 2.8: -pressure curve.

Crack propagation velocity vs. relative time, o = 0

-£- 22.50 mm. Crack length-B-16.97 mm. Crack length

Tir

200 400 BOO

T = Relative time [ ^sec.]

800 1000

Figure 3.1: Stress-strain curve shell material.

0 deg. Angle: Crack Flange Deflection vs. Hoop Stress

J2 3

-Linear [A]- Nonlinear elastic [B]- Nonlinear elastic ax. constrained [C]-Nonlinear plastic [D]- Nonlinear plastic ax. constrained [E]-Starnes'datafrompJF]

0O.OE- •00 5.0E-04 1.0E-03 1.5E-03 2.0E-03 2.5E-03 3.0E-03 3.5E-03 4.0E-03 4.5E-03

Normalized Hoop Stress 0»/E

Figure 3.2: Finite element analysis results for 0 degreeangle crack

Figure 2.9: Crack propagation speed against crackgrowth increment.

10


Von Mises Stress at cracktip

0 Degree angle [22 psi. 100% burst pressur45 Degree angle [27 psi. 75 % Burst pressure]90 Degree angle [43 psi. 30 % Burst Pre

Figure 3.5: Von Mises stress vs. distance away fromthe crack tip

Figure 3.3: Deformed mesh for a shell with crack at 0degree angle

* c

1 E

(

p* vs. normalized hoop stress

— ©— 0 degree crack configuration I ^-Q-45 degree crack configuration I I

t \

'•/[ iI. / I& '$ !

'&-' F

1 D m DOBP>^ -0^2^ —————————————

) 0.001 0.002 0.003 0.004

Normalized hoop stress [o,/E]

| B

urst

pre

ssur

e [p

si]

210 -

190 -

170 -

150 -

L

30 -i

0-pressure curve

-©-Acquired data equivalent to 3 in. crack f T[2](average 2 tests) / /

-B- Acquired data equivalent to 4 In. crack £ 1[2] (average 2 tests) / f

-^-Fracure mechanics solution, generated £ Iwith theoretical Klc of 22 Ksi\jin. £ f

—*- Fracture mechanics solution, based on jr 1experimental data. P=3.39 ^T I A\

JF f //

^ f //

f^,^^^0^^ J^

^^^

T

0 20 40 60 80

angle [deg]

Figure 3.4: Degree of nonlinearity as function of the pigure . Comparison of results obtained from frac.hoop stress ture mechanics and experimental data

11


References1. Hoogkamer, D., Waas, A.M. and Arbocz,

J., "Damage Tolerance of Cylidrical ShellsContaining Cracks.", Proceedings of the 41si

AIAA/ASME/ASC/ASCE Structures, StructuralDynamics and Materials Conference, April 2000,published by the AIAA.

2. Hoogkamer, D., Waas, A.M. and Arbocz, J.,"Bulging Factors for Cracked Thin Shells: AnApproximate Analytical Model.", Int.J.NonlinearMechanics, in review, 2001.

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for Longitudinal Cracks in Pressurized CylindricalShells", NASA Langley Research Center Techni-cal report under Grant NASA-CR-208454, Vol. 3,AIAA, 1997, pp. 2389-2402.

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Damage Tolerance of Cracked Cylindrical Shells under