nasa technical note nasa tn d-8325 - dtic

NASA TECHNICAL NOTE

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NASA TN D-8325

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^ FRACTURE ANALYSIS OF SURFACE AND THROUGH CRACKS

IN CYLINDRICAL PRESSURE VESSELS

J. C. Newman, Jr.

Langley Research Center

Hampton, Va. 23665

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON, 0. C. • DECEMBER 1976

19960314 049 DISTRIBUTION STATEMENTlT"?

Approved for public release; Distribution Unlimited

1. Report No.

NASA TN D-8325 2. Government Accession No. 3. Recipient's Catalog No.

4. Title and Subtitle FRACTURE ANALYSIS OF SURFACE AND THROUGH CRACKS IN CYLINDRICAL PRESSURE VESSELS

5. Report DatP

December 1976 6. Performing Organization Code

7. Author(s)

J. C. Newman, Jr.

8. Performing Organization Report No.

L-11118

9. Performing Organization Name and Address

NASA Langley Research Center Hampton, VA 23665

10. Work Unit No. 505-02-31-01

11. Contract or Grant No.

12. Sponsoring Agency Name and Address National Aeronautics and Space Administration Washington, DC 20546

13. Type of Report and Period Covered Technical Note

14. Sponsoring Agency Code

15. Supplementary Notes

16. Abstract

A previously developed fracture criterion was applied to fracture data for surface- and through-cracked cylindrical pressure vessels to see how well the criterion can correlate fracture data. Fracture data from the literature on surface cracks in aluminum alloy, steel, and epoxy vessels, and on through cracks in aluminum alloy, titanium alloy, steel, and brass vessels were analyzed by using the fracture criterion. The criterion correlated the failure stresses to within ±10 percent for either surface or through cracks over a wide range of crack size and vessel diameter. The fracture criterion was also found to correlate failure stresses to within ±10 percent for flat plates (center-crack or double-edge-crack tension specimens) and cylindrical pressure vessels containing through cracks.

17. Key Words (Suggested by Author(s))

Fracture Stress-intensity factor Structural mechanics Materials (metallic)

18. Distribution Statement

Unclassified - Unlimited

Subject Category 39

19. Security Classif. (of this report)

Unclassified

20. Security Classif. (of this page)

Unclassified

21. No. of Pages

33

22. Price*

$3.75

* For sale by the National Technical Information Service, Springfield. Virginia 22161

FRACTURE ANALYSIS OF SURFACE AND THROUGH CRACKS

IN CYLINDRICAL PRESSURE VESSELS

J. C. Newman, Jr. Langley Research Center

SUMMARY

A previously developed fracture criterion was applied to fracture data for surface- and through-cracked cylindrical pressure vessels to see how well the criterion can correlate fracture data. Fracture data from the literature on surface cracks in aluminum alloy, steel, and epoxy vessels, and on through cracks in aluminum alloy, titanium alloy, steel, and brass vessels were analyzed by using the fracture criterion. The criterion correlated the failure stresses to within ±10 percent for either surface or through cracks over a wide range of crack size and vessel diameter. The fracture criterion was also found to correlate failure stresses to within ±10 percent for flat plates (center-crack or double-edge-crack tension specimens) and cylindrical pressure vessels containing through cracks.

INTRODUCTION

Failures of many pressure vessels have been traced to surface cracks or to through cracks. These cracks initiate at structural discontinuities such as holes, material defects, or other abrupt changes in configuration and may propagate to failure under operating stress levels. To prevent such failures, the designer must be able to predict the effects of crack size on structural strength. Linear elastic fracture mechanics (LEFM), which uti- lizes the concept of the elastic stress-intensity factor, has been used to correlate fracture data and predict failure for cracked plates and structural components when the crack-tip plastic deformations are constrained to small regions (plane-strain fracture, ref. 1). However, when plastic deformations near the crack tip are large (plastic zone greater than plate thickness) the elastic stress-intensity factor at failure Kje varies with crack size and structural dimensions. (See refs. 2 to 4.) To account for the variation in K,-e with structural dimensions, the elastic-plastic stress- strain behavior at the crack tip must be considered.

An equation which accounts for the effects of plastic deformation on fracture was derived and is presented in references 4 and 5. This equation relates K^Q to the elastic nominal failure stress and two material fracture parameters and is designated the two-parameter fracture criterion (TPFC). The TPFC has correlated fracture data for surface and through cracks, for different specimen types, and for a wide range of materials (refs. 4 to 7).

The purpose of this paper is to apply the TPFC to surface- and through- cracked cylindrical shells subjected to internal pressure (fig. 1) to see how well the TPFC can correlate such fracture data. Fracture data for pressurized cylinders made of various materials (aluminum alloy, titanium alloy, steel, brass, and epoxy) were taken from the literature. (See table I and refs. 8 to 17.) Some of the literature sources reported fracture data on flat plates (fig. 2) in addition to the data for cylinders. To determine whether or not the failure stresses in the cylinders may be predicted using fracture data for the flat plates, experimental and predicted failure stresses in the flat plates were compared. The predicted failure stresses were computed using the two material fracture parameters determined from the cylinder data. In order to apply the TPFC to fracture data, the elastic stress-intensity factors for these crack configurations must be known.

The elastic stress-intensity factors for through cracks in pressurized cylinders have been obtained theoretically by Folias (ref. 18), and Erdogan and Kibler (ref. 19) for crack lengths less than about five times the square root of the product of vessel radius and thickness. In this paper an empirical equation giving elastic stress-intensity factors for through cracks has been obtained using the numerical results from reference 19 and some experimental fracture tests on brass vessels from reference 16. The resulting equation applies over a range of crack lengths about twice as large as those

considered in references 18 and 19.

The elastic stress-intensity factors for surface cracks in pressurized cylinders have not been obtained theoretically, but some experimental stress- intensity factors have been determined from surface cracks in brittle epoxy vessels (ref. 17). In this paper an empirical equation giving elastic stress-intensity factors for surface cracks has been developed using the results from references 4 and 19. The results from this equation are compared with the experimental stress-intensity factors from reference 17.

SYMBOLS

a initial depth of surface crack, m

c initial length of surface or through crack (see figs. 1 and 2), m

F complete boundary-correction factor on the stress intensity

f, shell-curvature correction factor for a through crack

f shell-curvature correction factor for a surface crack s

H height of uniformly stressed specimens, m

KF fracture toughness , N/m^'2

Kj elastic stress-intensity factor, N/nP'2

elastic stress-intensity factor at failure, N/m3/2 Kle

M combined front-face and back-face correction on the stress- intensity factor for the surface crack

M-i front-face correction on the stress-intensity factor for the surface crack

m fracture-toughness parameter

p internal pressure, Pa

Q elastic surface-crack shape factor

R internal radius of cylindrical pressure vessel, m

S gross section stress at failure, Pa

S nominal (net section) stress at failure, Pa

S nominal stress required to produce a plastic hinge on net section (S = a for center-crack and double-edge-crack tension specimens and S = 1.15a for pressurized cylinders), Pa

T temperature, K

t shell thickness, m

W specimen width, m

c A|. through-crack shell parameter,

\/it c /a\

\_ surface-crack shell parameter, 's

v Poisson's ratio

au ultimate tensile strength (uniaxial), Pa

a yield stress (uniaxial), Pa

(j) ratio of KIe to Kp

TWO-PARAMETER FRACTURE CRITERION

The two-parameter fracture criterion (TPFC) was developed and success- fully applied to plane fracture specimens containing either surface or through cracks in metallic materials (refs. 4 to 7). The TPFC accounts for the effects of plastic deformation on fracture properties. The equation is

KF = —%T\ (Sn^V (1)

where KIe is the elastic stress-intensity factor at failure, Sfi is the nominal (net-section) failure stress, Su is the nominal stress required to produce a plastic hinge on the net section using the ultimate tensile strength, and Kp and m are the two material fracture parameters. The fracture parameters Kp and m are assumed to be constant for a given com- bination of material, thickness, temperature, and rate of loading. Three conditions are necessary to obtain fracture parameters that are representa- tive for a given material and test temperature: (1) the nominal failure

stress must be less than a , (2) the fracture data must all be from the same specimen thickness, and (3) the test data must encompass a wide range of crack lengths or specimen sizes. Reference 4 shows how the fracture parameters are determined by a least-squares procedure for a given set of fracture data. To define the complete fracture behavior for a material, Kp and m must be determined as functions of thickness, temperature, and load

rate.

If m equals zero in equation (1) , Kp equals the elastic stress- intensity factor at failure and the equation applies to low-toughness (low Kp) materials (plane-strain fracture). However, if m equals unity, the equation applies to extremely ductile or hi'gh-toughness (high Kp) materials. Thus, the fracture parameters, Kp and m, jointly describe the crack sensitivity of the material.

The denominator in equation (1) reflects the influence of the nominal failure stress on fracture toughness. The variation of the denominator with nominal stress for a typical material is shown in figure 3- When the nominal stress is less than the uniaxial yield stress a , the function <t> (ratio of KIe to Kp) is a linear function of nominal stress. However, when the nominal failure stress is greater than the yield stress, the function <|> becomes nonlinear and is dependent upon the stress-strain curve of the material and the' state of stress in the crack-tip region (refs. 4 and 6). For thin materials, where the state of stress in the crack-tip region is biaxial, the expected behavior is estimated by the dash-dot curve. An equation approximating the dash-dot curve is given in reference 6 by

!lf = JLU . m _LM (ffv < Sn < S) (2) Kp S_ I S„ I y n u

and is shown in figure 3 as the dashed curve. For thick materials, where the state of stress in the crack-tip region is triaxial, the fracture behavior for S > a is expected to lie closer to the solid line. The solid vertical line truncates the nominal stress at S . In this paper, equation (2) was used whenever S was greater than o

In order to apply equation (1) to surface and through cracks in pressurized cylinders, the nominal stress required to fail the uncracked vessel S and the elastic stress-intensity factor KIe for these configurations must be determined. For pressurized cylinders, S was 1.15 times the ultimate tensile strength o it was determined by calculating the nominal (hoop) stress required to satisfy the Mises yield criterion (where a was replaced by o ) assuming a 2:1 biaxial stress ratio. The elastic stress-intensity factor equations for these configurations are presented in the next section.

ELASTIC STRESS-INTENSITY FACTORS

The form of the elastic stress distribution near a crack tip that con- tains the stress-intensity factor K-p and the square-root singularity is well known (ref. 2). (The determination of K,. is the basis for linear elastic fracture mechanics.) The stress-intensity factor is a function of load, structural configuration, and the size, shape, and location of the crack. In general, the elastic stress-intensity factor can be expressed as

K Ie = Sn^c" F (3)

for any Mode I crack configuration where S is the nominal stress, c is the initial crack length (defined in figs. T and 2), and F is the boundary- correction factor. The boundary-correction factor accounts for the influence of various boundaries and crack shape on stress intensity. The following sections give the nominal stress equation and the boundary-correction factor equations for the surface- and through-cracked cylindrical pressure vessel.

Through Crack in a Thin Pressurized Cylinder

For the through crack (axial) in a cylindrical shell subjected to internal pressure (fig. 1(a)), the elastic stress-intensity factor at failure is given by equation (3) where the nominal stress is

S = E? (4) n t and

F 1/2

ft = (l + 0.52^t + 1.29\2 - 0.07^t

3) (5)

for 0 ^ *t i 10 where *t = c/yRt. Equation (5) accounts for the effects of shell curvature (refs. 18 and 19) on stress intensity. Poisson's ratio was assumed to be 1/3. The details on the development of equation (5) are given in the appendix.

Surface Crack in a Thin Pressurized Cylinder

For the internal or external surface crack (axial) in a cylindrical shell subjected to internal pressure (fig. Kb)), the elastic stress-

intensity factor at failure is also given by equation (3), where the nominal stress is given by equation (4) and

F = (6)

The square-root term converts the through-crack expression to that for a surface crack, tor, and f The elastic shape factor Q was given in reference 20 as the square of the elliptic integral of the second kind. An expression was chosen in reference 4 as a simple approximation for Q and is given by

M is the combined front-face and back-face correction fac- is the shell-curvature correction factor for a surface crack.

1 + 1.47

1 + 1.47

J.64

J.64

x>>

< 1.0

> 1 .0

(7)

j

The expression for Mg (ref. 4) is given by

M = e (8)

where q was determined empirically as

- - € (9)

The term M. is the front-face correction, and the a/t term is the back- face correction. The expression for M.. is given by

M1 = 1.13 - 0.1

'1 M<Hi('*0-03a

^,

0.02 S - S. 1 .0

> 1.0

(10)

J

The shell-curvature correction factor for a surface crack f is given by s

1/2 fs = M + 0.52XS + 1.29XS

2 - 0.01k\s3) (11)

c a for 0 < X < 10 where X0 = -. Again, Poisson's ratio was assumed

~ s " s ,f^ t {m, to be 1/3. The form of X was obtained by assuming that the surface crack could be replaced by an "equivalent" through crack of equal area. As a/t approaches unity, X approaches X, and equation (6) reduces to equation (5). In the appendix, equation (6) is compared with some experimentally-determined correction factors for a brittle epoxy.

Through Crack in Axially Loaded Flat Plates

Center-crack tension.- For the center-crack tension specimen (fig. 2(a)), the elastic stress-intensity factor is given by equation (3) where

S S = Ü— (12) n 2c

W~

and

1 - H_clLc !-c (13) w / y w

for 0 1 2c/W < 1.0 and H/W 2 2. The secant term is the finite-width correction on stress-intensity factor and was obtained from reference 1.

Double-edge-crack tension.- For the double-edge-crack tension specimen (fig. 2(b)), the elastic stress-intensity factor is given by equation (3), where S is given by equation (12), and F was obtained by a boundary- collocation analysis of a configuration with H/W = 0.625. (Fracture data analyzed from ref. 15 used this particular configuration.) The stress S was assumed to be uniformly applied. An equation fit to the collocation results gave

F = 1.13 1 - — h/sec - (14) \ W/ V W

for 0 S. 2c/W ii 0.4 and H/W = 0.625. (Stress-intensity factors for H/W =1,3, and °° are given in ref. 2.)

ANALYSIS OF FRACTURE DATA

Fracture data on pressurized cylinders made of various materials and containing either surface or through cracks were taken from the literature and were analyzed using the TPFC (eqs. (1) and (2)). The fracture parameters, KF and m, for a given material, thickness, and test temperature were determined from the fracture data using a best-fit procedure described in reference 4. These values of Kp and m were then used to calculate failure stresses for the same fracture tests to see how well the TPFC correlated the failure stresses. Where fracture data on flat plates (center- crack tension or edge-crack tension) were available, experimental and predicted failure stresses for the flat plates were compared. The failure stresses were computed using the two material fracture parameters determined from the cylinder data. The failure stresses were calculated by substitut- ing equation (3) into equations (1) and (2), and were given by

Sn = mKL (fire F +

<Sn ^ V (15)

and

\/(mY); 2YS, mY (*y < Sn < V (16)

where

Vy 2Su^c"F

(17)

Table I summarizes the materials, thicknesses, test temperatures, and crack configurations that were analyzed. Figures 1 to 15 show the correlation obtained for each set of fracture data. Table I also indicates which fracture data were obtained using sharp saw-cut slits instead of fatigue cracks.

Through Cracks

Pressurized cylinders.- Figures 4 to 7 show fracture data on pressurized cylinders containing through cracks. The figures show the nominal failure stress normalized to Su (1.15cr ) plotted against half-length of crack c. The symbols show the fracture data, and the curves indicate the best fit of the TPFC using the values of K and m determined from these data. A knee occurs in all curves when the nominal stress is equal to the yield stress of the material (transition from eq. (15) to eq. (16)), but some are hardly perceptible. The calculated failure stresses were generally within ±10 percent of the experimental failure stresses.

Pressurized cylinders and flat plates.- Figures 8 to 13 show fracture data on pressurized cylinders and flat plates. In each figure the results are for cylinders and flat plates made of the same material and thickness, and tested at the same temperature. Both the cylinders and flat plates con- tained through cracks. The figures show nominal failure stress normalized to S plotted against half-length of crack c. For the flat plates Su = a and for the cylinders Su = 1.15° The fracture parameters, Kp and m, were determined from an analysis of the fracture data on the pressurized cylinders. The fracture data on the flat plates were insufficient to obtain the two parameters because only one specimen size and crack length was tested. The solid curve or curves show the calculations from the TPFC for various vessel radii. The agreement was considered good. The dashed curve on each figure shows the predicted results for flat plates (R = <*>) using the values of Kp and m determined from the cylinders. The circular symbols show the experimental results for the flat plates. The predicted results were within ±10 percent of the experimental failure stresses.

Surface Cracks

Pierce (ref. 11) conducted cryogenic fracture tests on surface cracks in 2014-T6 material. Figure 14 shows S normalized to S (1.15° ) plotted against cF , where c is the half-length of crack and F is the boundary-correction factor (eq. (6)). The TPFC indicates that this type of plot (Sn/S against cF ) gives a single curve for various a/c and a/t ratios. The open and solid circular symbols denote experimental data on externally or internally located surface cracks, respectively.

The surface-crack data included variations in crack shape 0.07 ^ - =0.9

( a \ \ c * and crack size 0.36 £ - £ 0.98 . The curve shows the calculations from the

TPFC using the values of Kp and m determined from these data. The calculated failure stresses were within ±10 percent of the experimental failure stresses.

Kiefner, Maxey, Eiber, and Duffy (ref. 14) conducted surface-crack fracture tests at room temperature on steel pressure vessels. Figure 15 shows the nominal failure stresses normalized to S plotted against cF . The

surface-crack data included variations in crack shape 0.02 ^ - = 0.14 and

/ a \ \ " c / crack size 0.38 1 - = 0.82J. The symbols denote the fracture data and the

the curve shows the calculations from the TPFC. The agreement was considered good.

CONCLUDING REMARKS

A two-parameter fracture criterion that relates the elastic stress- intensity at failure, the elastic nominal failure stress, and two material

parameters was used to analyze fracture data on surface- and through-cracked cylindrical pressure vessels. Fracture data from the literature on steel, titanium alloy, aluminum alloy, brass, and epoxy vessels tested at either room or cryogenic temperature were analyzed. The two-parameter fracture criterion correlated the data well (generally within ±10 percent of the experimental failure stresses) for a broad range of materials, including some that were extremely ductile. The fracture criterion was also found to correlate fracture data from flat plates and cylindrical pressure vessels within ±10 percent for the same material, thickness, and test temperature.

Langley Research Center National Aeronautics and Space Administration Hampton, VA 23665 September 27, 1976

10

APPENDIX

DEVELOPMENT AND VERIFICATION OF BOUNDARY CORRECTIONS ON STRESS-INTENSITY

FACTORS FOR THIN PRESSURIZED CYLINDERS

Through Cracks

Folias (ref. 18) and Erdogan and Kibler (ref. 19) have obtained the elastic stress-intensity factors for a longitudinal (axial) through crack in a pressurized cylinder (fig. 1(a)) for X^. J 4.5. Figure 16 shows the shell- curvature correction (or boundary-correction factor) on stress intensity as a function of Xfc. The symbols are numerical values obtained from reference 19 for v = 1/3. The shell-curvature correction factor F was determined by

F=FT+2|FB| (18)

where FT is the contribution due to the membrane solution (or normal forces) and Fß is the contribution due to bending. Reference 21 has shown that including one-half of the bending contribution was necessary to correlate crack-growth rates from pressurized cylinders and flat plates. The bending term contributed only about 10 percent to the total correction

factor.

The solid curve in figure 16 is an equation chosen herein (eq. (5)) to fit the numerical values. The solid curve was within ±5 percent of the numerical values. Reference 21 has shown that the effect of Poisson's ratio (v = 0 to 1/2) on the curvature correction was less than 5 percent from the values given in figure 16. Therefore, equation (5) was assumed to

apply for any material.

For Xfc > 4.5, an experimental technique was used herein to verify the applicability of equation (5) for calculating the shell-curvature correction factors. Figure 17 shows the shell-curvature correction factor plotted against X.. The symbols show the experimentally derived correction factors from through-crack fracture data on brass cylinders (ref. 16) using the two- parameter fracture criterion. The correction factors are given by

F= ft = ^ ^ "' (19) sn^

The fracture parameters Kp and m were determined from an analysis of the fracture data with L < 1. The crack length c and the corresponding failure stresses S were obtained from reference 16. The solid curve shows the correction factors calculated from equation (5), and these factors

11

APPENDIX

are in good agreement with the experimental data. Therefore, equation (5) was assumed to apply for any material with xt £ 10.

Surface Cracks

Derby (ref. 17) has experimentally determined stress-intensity correction factors for surface cracks in pressurized cylinders made of a brittle epoxy. The critical elastic stress-intensity factor for this material (Kj = 1.02 MN/m^/2) was obtained from four-point notch bend fracture tests (ref. 17). This material was brittle (m = 0). The experimental correction factors for the surface cracks were obtained from equation (3) as

F = he_ __ U02_ (.20) Sn)[ic Snp

where S and c were obtained from the fracture tests on the epoxy vessels. Table II shows a comparison between the experimentally determined boundary correction factors and those calculated from equation (6). The theoretical correction factors were within ±10 percent of the experimental values.

12

REFERENCES

1. Brown, William F., Jr.; and Srawley, John E.: Plane Strain Crack Tough- ness Testing of High Strength Metallic Materials. Spec. Tech. Publ. No. 410, American Soc. Testing & Mater., c.1966.

2. Fracture Toughness Testing and Its Applications. ASTM Spec. Tech. Publ.

No. 381 , c.1965.

3. Kuhn, Paul: Residual Tensile Strength in the Presence of Through Cracks or Surface Cracks. NASA TN D-5432, 1970.

4. Newman, J. C, Jr.: Fracture Analysis of Surface- and Through-Cracked Sheets and Plates. Eng. Fract. Mech., vol. 5, no. 3, Sept. 1973,

pp. 667-689.

5. Newman, J. C, Jr.: Plane-Stress Fracture of Compact and Notch-Bend Specimens. NASA TM X-71926, 1974.

6. Newman, J. C, Jr.: Fracture Analysis of Various Cracked Configurations in Sheet and Plate Materials. NASA TM X-72709, 1975.

7. Newman, J. C, Jr.: Predicting "Failure of Specimens With Either Surface Cracks or Corner Cracks at Holes. NASA TN D-8244, 1976.

8. Peters, Roger W.; and Kuhn, Paul: Bursting Strength of Unstiffened Pres-

sure Cylinders With Slits. NACA TN 3993, 1957.

9. Anderson, Robert B.; and Sullivan, Timothy L.: Fracture Mechanics of Through-Cracked Cylindrical Pressure Vessels. NASA TN D-3252, 1966.

10. Sullivan, Timothy L.; and Pierce, William S.: Effect of Radius on Bulg- ing and Fracture of Through-Cracked Cylindrical Pressure Vessels at Cryogenic Temperatures. NASA TN D-4951 , 1968.

11. Pierce, William S.: Effects of Surface and Through Cracks on Failure of Pressurized Thin-Walled Cylinders of 2014-T6 Aluminum. NASA TN D-6099, 1970.

12. Sullivan, Timothy L.: Texture Strengthening and Fracture Toughness of Titanium Alloy Sheet at Room and Cryogenic Temperatures. NASA TN D-4444, 1968.

13. Calfo, Frederick D.: Effect of Residual Stress on Fracture Strength of AISI 301 Stainless-Steel and Ti-5Al-2.5Sn ELI Titanium Cracked Thin- Wall Cylinders. NASA TN D-4777, 1968.

14. Kiefner, J. F.; Maxey, W. A.; Eiber, R. J.; and Duffy, A. R.: Failure Stress Levels of Flaws in Pressurized Cylinders. Progress in Flaw Growth and Fracture Toughness Testing, ASTM Spec. Tech. Publ. 536, c.1973, pp. 461-481 .

13

15. Hahn, G. T.; Sarrate, M.; and Rosenfield, A. R.: Criteria for Crack Extension in Cylindrical Pressure Vessels. Int. J. Fract. Mech.,

vol. 5, no. 3, Sept. 1969, pp. 187-210.

16. Sechler, E. E.; and Williams, M. L.: The Critical Crack Length in Pres- surized, Monococque Cylinders. GALCIT 96 (Contract NAw-6525), Grad. Aeronaut. Lab., California Inst. Technol., Sept. 1959.

17. Derby, R. W.: Experimentally Determined Shape Factors for Deep Part- Through Cracks in a Thick-Walled Pressure Vessel. Progress in Flaw Growth and Fracture Toughness Testing, Spec. ASTM Tech. Publ. 536,

c.1973, PP. 482-491.

18. Folias, E. S.: An Axial Crack in a Pressurized Cylindrical Shell. Int. J. Fract. Mech., vol. 1, no. 2, June 1965, pp. 104-113.

19. Erdogan, F.; and Kibler, J. J.: Cylindrical and Spherical Shells With Cracks. Int. J. Fract. Mech., vol. 5, no. 3, Sept. 1969, pp. 229-237,

20. Irwin, G. R.: Crack-Extension Force for a Part-Through Crack in a Plate Trans. ASME, Ser. E: J. Appl. Mech., vol. 29, no. 4, Dec. 1962,

pp. 651-654.

21. Erdogan, F.; and Ratwani, M.: Fracture of Cylindrical and Spherical Shells Containing a Crack. Nucl. Eng. & Des., vol. 20, no. 1, 1972,

pp. 265-286.

14

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■15

TABLE II.- THEORETICALLY AND EXPERIMENTALLY DETERMINED

BOUNDARY-CORRECTION FACTORS FOR SURFACE CRACKS IN

PRESSURIZED CYLINDERS MADE OF A BRITTLE EPOXY

[R = 68.3 mm and t = 15 mm (from ref. 17)]

a, c> Experimental F Theoretical F Theoretical F

mm mm Experimental F

13.2 14.6 0.959 0.932 0.97

12.2 14.2 .878 .863 -98

11 .1 14.5 .846 .855 1.01

10.9 14.5 .846 .823 .97

10.5 15.9 .808 .841 1.04

11 .5 14.4 ,831 .839 1 .01

10.7 14.5 .785 .819 1.04

6.9 9.5 .682 .696 1 .02

6.9 8.3 .708 .689 .97

7.3 8.3 .672 .689 1.03

5.4 8.5 .621 .677 1.09

16

2c 2c

(a) Through crack. (b) Surface crack.

Figure 1.- Surface crack or a through crack in a pressurized cylinder.

17

g

t t t

1 2c

W

T~r^

2H * »>

g

l_JL_t

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