Evaluation of Finite-Element Calculations in aPart-Circular Crack by Coherent Optics Techniques

byG.H. Kaufmann, A.M. Lopergolo, S.R. Idelsohn and E.J. Barbero

ABSTRACT-Two techniques, speckle photography andholographic interferometry, were used to test three-dimensionalfinite-element calculations in an internally pressurized cylinderwith an external part-circular crack. Ope'ning displacementsalong the crack line were rTleasured by speckle photography.Radial displacements Vv'ere obtained from holographic fringepatterns. Good agreement between experimental and numericaldata is obtained. Stress-intensity factor variations along thecrack front are calculated from numerical results.

Introduction

Engineering estimates for stress-intensity factors incracked cylinders under internal pressure have been trans­formed in a very important problem in fracture mechanics.Failure of nuclear piping· systems subjected to a cyclicload condition has been traced to fatigue growth ofsurface cracks. The stress-intensity factor is required topredict the rate of fatigue crack growth fora crack shapewhich changes continuously because of its growth. Forthis reason extc!nsive work has been recently done tocalculate stress-intensity factors for various shapes ofsemi-elliptical and semicircular surface cracks in pressurizedcylinders. Three-dimensional numerical analysis using theboundary-integral equation method is reported by Reliotet al. I Atluri and Kathiresan,2 Miyazaki et af. 3 and Rajuand Newman4 use three-dimensional finite-elementmethods. Variations of the stress-intensity factor alongthe crack front have been obtained for a range of crackshapes and sizes for external and internal cracks. In spiteof the considerable importance of the experimentalevaluation of numerical methods, only a few works areavailable in the literature.

In the past few years, coherent optics techniques. suchas holographic interferometry and speckle photographyhave been used to determine the displacement fields ad­jacent to the crack tip. 5--9 Several authors use thesemeasurements to calculate stress-intensity factors fordifferent types of cracks. The use 9f both techniques infracture mechanics offers several advantages. As otheroptical methods, they are noncontact techniques and themodel surface needs no preparation. They do not require

G.H. Kaufmann is Professor, and A.M. Lopergolo is Research Fellow,lnstituto de Fisico de Rosario (COlvICET-UNRj, A vda, Pellegrini 250,2000 Rosario, Argentina. S.R. Idelsohn is Professor, and E.J. Barbero isResearch Fellow, Instituto de Desarrollo TecnolOgico para la IndustriaQuimica (CONICET-UNL), P.O. Box 91,3000 Santa Fe, Argentina.

Original manuscript submitted: June 26, 1985. rI1101 manuscript received:September 2, 1986.

materials with special optical characteristics, so they canbe directly applied to ~ny opaque materials such as lnetalsor composites.

The purpose of this paper is to show the usefulness ofboth techniques in fracture mechanics to evaluate calcula­tions obtained by means of numerical methods. .As anexample, the case of an external surface part-circularcrack in a thick cylinder is analyzed. Finite-elementcalculations are performed; good agreement is foundbetween experimental and numerical results. Variations ofthe stress-intensity factor along the crack front calculatedby the finite-element idealization are compared withresults by Raju and Newman. 4

Experimen·tal Test

Specim~n

The circular cylinder used in this study was made ofPMMA. This material was chosen because it is a brittleone; so no plastic now was expected around the cracktip. Hence, experimental results can be compared withthose calculated using linear-elastic fracture mechanics.The cylinder had a wall thickness t of 0.5 cm. Its internalradius R i was 2.98 cm. The thickness to radius ratio t/R i

was 0.168. Its length 2b was 20 cm, with a I-cm clampingon both edges, as shown in Fig. 1.

After the cylinder was cut into size, a central axial part­circular slot was inserted on the external surface of thespecimen with a circular saw. The crack lengt~ 2 e was1.36 cm. The crack depth a was 0.23 cm, which gave anale ratio of 0.338 and an alt ratio of 0.46. The slot was0.02-cm wide and terminated in a vee notch with a verysmall radius.

Next the specimen was heat treated to relieve stresses.Afterwards, the area surrouading the cracks was sprayedwith a thin coat of matt-white paint.

Two square-end steel plates were made to clamp theedges of the cylinder. On each end-plate there was acircular groove, I-cm deep, which fitted nicely with thecylinder. Before the t\VO ends of the specimen were slidinto the groove, a cyanoacrylate cement was poured, thusclamping the ends of the cylinder to the end-plates. Fourrods parallel to the cylinder and fastened to both end­plates made sure the clamping would not become unglued.Pressure was applied by means of an hydraulic systemmeasured with a manometer. A valve ,vas used to main­tain the pressure at a constant value during the exposuretime.

Values of the elastic constant, E = 2920 MPa and v =0.33, were obtained from a calibration of a bar of thesame material using electrical strain gages.

Speckle Photography

The speckle-photography technique lO was used tomeasure the crack opening (in-plane) displacements u()along the slot line. They were obtained by measuring thecrack aperture along t\VO lines parallel to the slot direction,one at each side of the crack and at a distance of 0.025cnl from it. The u() displacements were calculated as onehalf of the difference between the cracks apertures corres­ponding to a pair of points at each side of the slot.

The sensitivity of this technique is limited by the averagespeckle size. It can be shown that displacements to bemeasured must be greater than lO

where A is the wavelength of the laser, F is the apertureratio of the recording optical system and m is its magnifi­cation. Nevertheless displacements observed along thecrack line were very small (about a few microns) whichwere outside the sensitivity of the technique given byeq (1). This could be increased, however, by superimposinga small known artificial displacement. For OUf application,a rigid-body displacement of about three times the specklediameter was produced by mounting the photographiccamera on a vernier table. This displacement was cancelledwhen the difference between the crack apertures corres­ponding to a pair of opposite points along the slot wasdone.

When the cylinder was pressurized between bothexposures, its surface moved in the radial direction. It iswell known that out-of-plane displacements producespeckle decorrelation which causes a decrease in fringecontrast and introduces errors in the measurements. 10

The Rayleigh criterion establishes the movement whichcan be tolerated without any appreciable degradation ofthe fringes. This movement must be less than

(3)

Ri = 2.98 em

t =0.5 em

2b= 20 em

a = 0.23 em

2e=1.36em

t /Rj = 0.168

a Ie = 0.338

alt =0.46

U r = NA/2

-4-Fig. 1-Circula.r cylinder with apart-circular crack

where N takes on integer values at the center of each

double-exposure holograms \vere recorded on Agfa­Gevaert 8E75 plates for different pressures. After pro­cessing the hologram, the image showing deformationfringes was reconstructed and photographed at norrnalincidence. Under these conditions, any point on the axialcrack line has a radial displacenlent given byll

(1)00 :::: 1.22 AF(l + m)/m

(2)

For the experiments presented in this paper ~ -- 500 /-tm.l'aking into account that the radial expansion for themaximum pressure utilized 'was about 30 ,urn, fringes withgood contrast were obtained without using any com­pensation procedure. Figure 2 shows a typical fringepattern.

To simplify data reduction, the cylinder surface wasimaged on to a Kodak SO-253 photographic plate mountedparallel to the specimen by means of a well-correctedcamera lens. A 21-cm focal length of F = 8 aperture wasused at m = 2.8 magnification. The specimen wasilluminated by an He-Ne laser. Double-exposure speckle­grams were analyzed using the pointwise technique. Bymeasuring the spacing and inclination of the fringes,aperture displacements along the crack line were calculated.

Holographic Interferometry

Holographic interferometry was used to measure theradial (out-of-plane) displacement U r along the crack line.The usual holographic setup to measure out-of-planedisplacements was utilized~ It The cylinder was normallyilluminated with a collimated beam of He-Ne laser light.It was mounted parallel to the holographic plate. Several Fig. 2-Typical s,peckle fring.e pattern

(xperimentalll.Manlcs • 155

bright fringe. Figure 3 shows a typical fringe pattern.From each pattern, the change in radial displacementalong the crack line was obtained and was comparedwith numerical predictions.

Numerical Analysis

Three-dimensional finite-element calculations wereperformed so as to be compared with the experimentalresults. Figure 4 shows the discretization used. It employs275 elements with 4107 degrees of freedom. Two types ofelements were used to model the cylindrical vessel: a layerof collapsed singular elements 12 around the crack frontand 20-node isoparametric elements with 60 degrees offreedom elsewhere. The curvature of the model wasrepresented through second-grade" parabolas. Three­dinlensional elements were used because the model was athick cylinder. The dimensions of the elements in theneighborhood of the crack were determined by means of aconvergence study on a known plane-strain crack prob­lem. As shown by Ref. 12, collapsed singular elements arethe pest choice among the singular isoparametric three­dimensional elements to model the singular crack problem.

Fig. 3-Typical interference fringes

The model idealizes one-eighth of the vessel, so that itsimulates a vessel with two diametrically located surfacecracks, ISO-degrees apart. As is shown by Ref. 4, thisdoes not introduce appreciable error.

Symmetry boundary conditions were applied on thex = 0, y = 0, Z = 0 planes, and clamping restrictions onthe z = b plane. Internal pressure was introduced bymeans of a system of energetically equivalent radialforces. The cOlnputer program SAMCEF13 was used forthe analysis.

Mode-I elastic-stress-intensity factors were calculated bymeans of the crack-opening displacement method, assum­ing plane-strain conditions.

Results

Opening displacement along the crack line for anapplied pressure p = 1-.2 MPa are shown in Fig. 5.Radial displacement along the same line for p = 0.2 MPaare plotted in Fig. 6. Satisfactory agreement betweenexperimental and numerical results is obtain~d. The dis­agreement of data with the numerical solution at distancesfar from the crack tip could be due to the fact that theplane-strain condition is lost at those distances. Further­more, it must be pointed out that displacements weremeasured over the free surface and so measurements areneither plane strain nor plane stress.

Figure 7 shows the distribution of nondimensionalstress-intensity factors along the crack front obtainedfrom numerical calculations. The results are normalizedby the stress-intensity factor Ko at ¢ = 90 degrees for anelliJjtical crackemb-eddedin an infinite body· subjected toa uniform tension stress (Jo,

Ko = (10 -.J ~with

where R o is the external radius of the cylinder, and Q isthe square of the complete elliptic integral of second kindapproximated by Q = 1 + 1.464 (a/c)1.65. The results ofRaju and Newman4 for a semielliptical crack of t/R =

156 • June 1987

Fig. 4-Finite-element mesh

0.1, ale = 0.4 and alt = 0.5 and the results from ASMEBoiler & Pressure Vessel Code Section XI are alsopresented for comparison in Fig. 7.

using both coherent-optics techniques to evaluate calcula­tions obtained by means of numerical methods, such asfinite elements.

MUtlEfU CAl RESUl TS

ASf£ SECTlON Xl

EXPERIMENTAL RESUlTS

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o RAJU AND NE\IttAN C4J

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ooo

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References1. Heliot, J., Lahbens, R.C. and Pellissier-Tanon, A., HSemi­

Elliptical Cracks in the Meridional Plane of a Cylinder Subjected toStress Gradients, " Fracture Mechanics, ASTM STP 677, A mer. Soc. Test.and Mat., 341-364 (1979).

2. At/uri, S.AT. and /(athiresan, K., "3 D Analysis of Surface Flawsin Thick- Walled Reactor Pressure Vessels Using a Displacement-HybridFinite Element Afethod," Nucl. Eng. Des., 51, 163-176 (1979).

3. Miyazaki, N., Shibata, K., Watanabe, T. and Tagata, K., "StressIntensity Factor Analysis of Surface Cracks in Three-dimensionalStructures - Comparison of the Finite Element Solutions with the ResultsObtained by the Simplified Estimation Methods," Int. J. Pres. Ves. &Piping, 15, 37-59 (1984).

4. Raju, I.S. and Newman, J.C., "Stress-Intensity Factors for Internaland External Surface Cracks in Cylindrical Vessels," J. Pres. Ves. Tech.,Trans. ASME, 104, 293-298 (1982).

5. Barker, D.B. and Fourney, M.E., "Displacement Measurements inthe Interior of Three-dimensional Bodies Using Scattered-light SpecklePatterns," EXPERIMENTAL MECHANICS, 16, 209-214 (1976).

6. Chiang, F.P. and Asundi, A., HA White Light Speckle MethodApplied to the Determination of Stress-Intensity Factor and DisplacementField Around a Crack Tip, " Eng. Fract. Mech., 15, 115-121 (1981).

7. Kaufmann, G.H., Lopergolo, A.M. and Idelsohn, S., HEvaluationof Finite Element Calculations in a Cracked Cylinder under InternalPressure by Speckle Photography, H J. Appl. Mech., Trans. ASME, 50,896-897 (1983).

8. Hsu, T.R., Lewak~ R. and Wilkins, B.J.S., HMeasurements ofCrack Growth in a Solid at Elevated Temperature by HolographicInterferometry, " EXPERIMENTAL MECHANICS, 18, 297-302 (1978).

9. Ennos, A.E. and Virdee, M.S., HApplication of ReflectionHolography to Deformation Measurement Problems," EXPERIMENTAL

ME<::H~NIC~,~,202-2(1.9 f.!S!8?).10. Erf, R.K., ed., Speckle Metrology, Academic Press (1978).11. Vest, C.M., Holographic Interferometry, John Wiley and Sons

(1979).12. Barsoum, R.S., HTriangular Quarter-Point Elements as Elastic and

Perfectly-Plastic Crack Tip Elements," Int. J. Num. Meth. in Eng., 11,85-98 (1977).

13. SAA1CEF, Systeme d'Analyse des Milieux Continus par ElementsFinis, Manuel d'Utilisation, Universite de Liege.

~-r--------------------------,N

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V \l

V EXPOtt£HTAl

--.HUMERI CAl

\l EXPER1MEN TAL

-- NUtEftlCAl

~+---,--..,.....---,--..,.....----r-..,.....----r--r----r--r-----r-..,.....----r--"---"----f"'0.00 1.00 2.00 3.00 4.00 5.00 e.oo 7.00 8.00

DISTANCE FROM CRACK TIP (mm)Fig. 5-0pening displacement Ue along the crack line.__, finite-element calculations. V, from specklemeasurements

g.o-r------------

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Conclusion

Concluding, the use of speckle-photography andholographic-interferometry techniques for the measure­ment of displacements surrounding a crack tip in aninternally pressurized cylinder was successfully demon­strated. This work also demonstrates the practicability of

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0.12~ 00o..&..--.....-.L--........--I'"-----...-~~--'--~-"'---I----I:-....i--.J..----'

24>/1rFig. 7-Stress-intensity-factor distributions. 0 ==numerical results (part-circular, t/ Ri = 0.168, alc ::: 0.338,alt = 0.46).0 == Raju and Newman results· (semi-ellipticalcrack, t/Ri == 0.1, ale :::: O~4, alt = 0.5): ~ from ASMEBoiler & Pressure Ves,set Code Section XI

itN+------T--r-_r----r--r---r-..,----.,.-..,----.-.....,.--r--~-.,.----r---l-0.00 1.00 2.00 3.00 4.00 5.00 1.00 '.00 8.00

DISTANCE FROM CRACK TIP (mm)Fig. 6-Radial displacement U r along the crack line.__7 finite-element calculations. V, from holographicmeasurements

g,.;

JUNE 1987

VOLUME 27-NUMBER 2

168 The Effect of a Symmetric Discontinuity of AdjacentMaterial in a Longitudinally Vibrating Uniform BeamW.T. SPRINGER. K.L. LAWRENCE AND T.J.LAWLEY

172 A Unique Elevated Temperature Tension-TorsionFatigue Test RigE.H. JORDAN AND C.T. CHAN

184 Use of Photoelasticity to Determine Orthotropic K,Stress-Intensity FactorM. MOJTAHED AND L.W. ZACHARY

190 Experimental Study of a Metal-Matrix CompositeD. POST. R. CZARNEK. D. JOH. J. JO AND Y. GUO

195 Dependence of Crack Acceleration on the DynamicStress-Intensity Factor in PolymersK. TAKAHASHI AND K. ARAKAWA

200 Dependence of Crack Acceleration on the DynamicStress-Intensity Factor in PolymersDISCUSSION BY TAKAO KOBAYASHI

201 Photoelastic Assessment of Finite-Element Analysisof a Pipe TeeE. SALEHI·BIGLARI. G.R. BUCHANAN AND D.G.SMITH

208 Transmission Photoelasticity of Centrally LoadedGenerally and Specially Onhotropic BeamsJ.L. SULLIVAN. E. BLAIS AND H. VAN OENE

220 Kill Deformation Modes in Internal Oblique CracksUnder Plane-Stress ConditionsPERICLES S. THEOCARIS

A Journal of the Society for Experimental Mechanics, Inc.

113 Further Development of the Iosipescu Shear TestMethodD.F. ADAMS AND D.E. WALRA11i

120 The Angle of Initiation and Propagation of Cracks inDuctile MediaP.S. THEOCARIS. N.P. ANDRIANOPOULOS AND S.KOURKOULIS

126 Experimental Analysis of Squeal of ReadIWriteHeads Upon floppy-Disk MediaD.L. MILLER AND J.K. GOOD

132 Stress Measurement by Copper Electroplating Aidedby·a Personal ComputerAKIRAKATO

138 Measurement of Crossflow Forces on TubesTOM M. MULCAHY

146 Dynamic Crack Curving and Branching Under Biax­ial LoadingJ .S. HAWONG. A.S. KOBAYASHI.. M.S. DADKHAH.8.S.-J. KANG AND M. RAMULU

154 J Evaluation of Finite-Element Calculations in a Part­Circular Crack by Coherent Optics TechniquesG.H. KAUFMANN. A.M. LOPERGOLO. S.R.IDELSOHN AND E.J. BARBERO

158 Fracture Behavior of Brittle Plates and CylindricalShells Subjected to Concentrated Impulse LoadingS. KIDA AND J. aDA

164 Use of Oscillation on PSD-Based Instruments forX-Ray Measurement of Residual StressMICHAEL R. JAMES