A P P L I C A T I O N S

Use of Three-dimensional Photoelasticity in Fracture Mechanics

An approach ut i l iz ing stress-freezing photoelast ic i ty to obtain stress- intensi ty factors for three-dimensional problems is described which does not require stress separation

by C. William Smith

ABSTRACT The philosophy of fracture mechanics is re- viewed and utilized to formulate a simplified approach to the determination of the stress-intensity factor photoelas- tically for three-dimensional problems. The method involves a Taylor Series correction for the maximum in-plane shear stress (TSCM) and does nat involve stress separation. The results are illustrated by applying the TSCM to surface flaws in bending fields. Other three-dimensional problems solved by the TSCM are cited.

Nomenclature Kz ---- Mode I stress-intensity factor [ lb / ( in . ) 3/~]

Kic ---- critical Mode I s tress- intensi ty factor [ lb / (in.) 3/2]

r,o = polar coordinates (in., tad) a ---- flaw depth (in.) p ---- radius of curvature of crack or notch root

(in.) 2c ---- flaw length in plate surface (in.)

t ---- plate thickness (in.) n = fringe order 5 ---- mater ial-fr inge value ( lb / in . /o rder )

"~max, "t:m ~ max imum shearing stress in piane perpen- dicular to crack border (psi)

Tmo = max imum remote shearing stress in plane perpendicular to crack border at plate surface (psi)

Kap = apparent stress-intensity factor [ lb / (in.)3/~]

Kth = theoretical stress-intensity factor [ lb / (in.) 3 IS]

KTSCM = approximate s t ress- intensi ty factor [ lb / (in.) ~/2]

Introduction Fracture mechanics is based upon the concept that a crack tip is surrounded by a singular elastic-stress field which controls catastrophic crack extension. The field s trength of this stress field is measured by KI, the Mode I stress-intensity factor (SIF), 1 and when KI reaches a critical value Kit , the crack is expected to propagate. Values of Kzc are determined

C. William Smith is P~ofessor of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA. Paper was presented at Third SESA International Congress on Ex- perimental Mechanics held in Los Angeles, CA on May 13-18, 1973.

from "fracture toughness" tests, 24 where the loads and crack length at the onset of fast crack growth are measured. In order to convert these measured quanti t ies into Kxc values, a functional relationship between these quanti t ies and KI is needed and may be derived from l inear elastic theory for many two- dimensional problems.~ However, for three-dimen- sional problems, usual ly characterized by a variat ion of K along the crack border, exact analyt ical expres- sions have been provided for only a few problems, such as embedded circular 5 and elliptical fiaws. 6

Because of the absence of complete analytical solu- tions for many three-dimensional configurations, and the prevalence of such geometries in real engineer- ing problems, considerable effort has been devoted to s tudying problems in this class. An excellent sur- vey of recent approximate analytical results is due to Shah and Kobayashi r for surface flaws, a practical geometry of this class of problems.

Since stress-freezing photoelasticity is well known as an exper imental method for studying three-dimen- sional stress distributions, it is na tura l to consider this technique as a candidate method for determining the SIF for three-dimensional problems. The use of this technique to study crack-tip stress fields was ap- parent ly introduced by Post s and by Wells and Post 9 in the early 1950's. In a discussion of the lat ter paper, I rwin 10 presented an idea for characterizing the local elastic stress field by two parameters for obtaining the stress-intensity factor. Since that time, a substan- tial use has been made of photoelasticity in studying crack-tip stress fields, n-16 and other techniques~-20 have been recommended for extracting approximate K~ values from photoelastic data. However, the origi- nal idea of Irwin, after successful use in a number of investigations 1t,~4,16,I7 has recent ly been refined by Kobayashi and his associates 21-24 and, more recently, by the author and his associates, 25-26 and extended to a mult idegree of freedom system, 27-2s so as to make stress-freezing photoelasticity a very useful tool for SIF determination. The purpose of this paper is to describe the latter technique with emphasis on the applied aspects of the method.

Analytical Philosophy In order to design a rat ional experiment, the under-

Experimental Mechanics I 539

r

2c

A P P L I C A T I O N S

Fig. 1--Typical flaw geometry and notation

Fig. 2--Typical fringe pattern near a crack tip

lying theory or philosophy should be clear ly recog- nized. For the problem at hand, it is sufficient to recognize the influence of the fol lowing factors:

(1) The nature and distribution of the local elastic stress field

(2) The form of the raw data (3) The zone of valid data

In f racture mechanics, it is convenient to describe the singular stress field in a plane normal to the crack border, such as the nz plane in Fig. 1. The functional forms of the stress components which dominate in the singular region in this plane are well known 1,26,27 and will not be repeated here. The max i m um in- plane shearing stress computed from these stress components also contains singular terms. However,

the size of the singular zone varies widely f rom prob- lem to problem depending upon the geometry, and it is clear from an observation of a typical photo- elastic-fringe pat tern near a crack tip (Fig. 2) that fr inge orders can be discriminated most readily along lines approximate ly normal to and passing through the crack tip due to the separat ion of the fringes in that direction. Thus, it is convenient to evaluate the max imum in-plane shear stress along e ---- ~/2 (Fig. 1) which simplifies the necessary form of the maxi- mum in-plane shear stress to:

Tmax ~-~ Tmax(~')

On the other hand, there are certain complicating factors which have to do with the location of the data zone along e ---- n/2. Figure 3 shows the results of an analysis comparing results for an ell iptical hole to those for an internal crack which indicates that if one measures too close to the crack tip, invalid data may result. Moreover, if one measures too far away from the crack tip, effects other than the f ree-crack surfaces will requi re terms in addition to those usu- ally associated with the field dominated by the singu- lar stresses. In the author 's experience with a var ie ty of three-dimensional problems, the size of the blunted zone is so small that it is difficult to obtain data close enough in to find the effect. On the other hand, the photoelastic-data zone is often influenced by boundary effects other than the crack surfaces. In order to ac- count for this effect, the author and his associates have chosen to express the quant i ty being measured photoelastically (i.e., the m ax im um in-plane shearing stress) in the standard singular form with additional terms provided by a Taylor series expansion which serve to account for or correct the data so as to yield

540 l December 197,3

A P P L I C A T I O N S

valid values for the SIF. This procedure avoids the necessity for stress separation. The equation is:

A M

N=O

where A is proportional to the SIF (A = KI/X/8~). By combining eq (1) with the stress-optic law

nf Tma= = - - (2)

2t

exper imental values of (n, r) may be employed through a least-squares procedure to determine A and, thus, the SIF. The function of the Taylor series correction method (TSCM) is to provide a fitted curve through the experimental data along e = n/2. This curve is then used to extrapolate back to the crack tip in order to determine the SIF. Experience to date indicates that eq (1) may be t runcated at about five terms or less, leading to a computer - run t ime (in- cluding compiling) of less than two seconds for the double precision program. Details of the program are described in Refs. 27 and 28.

The TSCM, using eq (1), will yield the same re- sults as the two parameter method of I rwin if the t runcat ion order is the same for each program. More- over, it can be shown that, for two-dimensional prob- lems, TSCM is equivalent to the Williams stress- function approach along e ---- =/2.

Experimental Considerations In this section, the experimental technique will be

described. Litt le emphasis will be placed on the equipment since no special equipment outside of that normal ly available in a photoelastic laboratory is required.

Materials and Models

Any t ransparent mater ia l exhibi t ing the diphase characteristics of a stress-freezing photoelastic mate- rial may be used. The author has used Hysol 4290 (Hysol Corp.) and PLM-4B (Photolastic Inc.) ex- tensively. Both are satisfactory in general but Hysol specimens are sometimes marred by mott l ing and usual ly require annealing, and PLM-4B requires the removal of a surface layer of stressed mater ia l but no annealing. Cracks can be made either by tapping a sharp edge against the mater ia l surface to produce a starter crack to be grown later above critical tem- perature or by sawing in sharp slits.

Test Procedure

Test specimens are assembled in a loading rig in- side the stress-freezing oven. It is highly desirable to use a s t r ing-loaded dead-weight system such as is shown for the plate in bending in Fig. 4. Clamping with dissimilar materials for a stress-freezing cycle may lead to unwanted thermal stresses or even pre- mature fracture in the grips for large specimen masses. The stress-freezing cycle consists of heating the model to critical tempera ture (usually 250 ~ to 275~ at a moderate rate, soaking above critical for

1.4-

1 . 2 -

-L-MAX (NOTCH) I ,O

"C MAX (CRACK)

0.8

I

i = .001

r

CI r I

[ = .01

= .05

0.6

0.4

0.2 ~'MAX(NOTCH)(FROM KOLOSOFF-INGLIS SOLUTION)

"CMAX(CRACK)(FROM MODE E SINGULAR STRESSES)

O 0 02 .04 .06 .08 .lo

r / o

Fig. 3--Elastic assessment of crack-tip blunting effect (Ref. 25)

Fig. 4 - - T y p i c a l tes t se tup fo r cy l i nd r i ca l b e n d i n g

Experimental Mechanics I 541

A P P L I C A T I O N S

"UM

UM

(7 cr

P= POINTS OF KI DETERMINATION

Fig. lO---Other problems solved by the TSCM

flaw in bending. Accuracy is discussed and a fair ly broad class of three-dimensional problems which have been solved using the method are noted.

Any experienced experimental is t can appreciate the author's reluctance to claim that the method described herein will serve as the master key for unlocking the door to all three-dimensional problems. However, the author believes that values of SIF which reflect three-dimensional effects to within reasonable engi- neering accuracy can be obtained by this method for a fairly broad class of three-dimensional problems. Studies utilizing the method on the configurations of Fig. I0 and other problems are continuing.

A c k n o w l e d g m e n t s

The author is indebted to M. A. Schroedl, J. 3". McGowan, and A. E. Harms for their contributions to the studies ment ioned herein. He also wishes to acknowledge the advice of D. Post and H. F. Brinson on optical systems and the work of A. S. Kobayashi and his associates, whose work provided incent ive and encouragement for these studies. The staff and facilities of the VPI&SU ESM Depar tment headed by D. Freder ick is also acknowledged. Parts of the work reported herein were performed under the sponsor- ship of the Depar tment of the Army, Watervl ie t Arsenal, Watervliet , NY, NASA Langley Research Center, Hampton, VA, and US Air Force Materials Laboratory, Wright Field, Ohio.

R e f e r e n c e s

1. Paris, P. and Sih, G. C., "Stress Analysis of Cracks," Fracture Toughness Testing and its Applications, ASTM STP 381, 30-91 (April 1965).

2. Anon., Fracture Toughness Testing and its Applications, ASTM STP 381 (April 1965).

3. Brown, W. F., Jr. and Srawley, J. E., Plane Strain Crack Toughness Testing oJ High Strength Metallic Materials, ASTM STP 410 (Dec. 1967).

4. Review of Developments in Plane Strain Fracture Toughness Testing, ed by W. E. Brown, ASTM STP 463 (Sept. 2970).

5. Sneddon, I. N., "'The Distribution of Stress in the Neighbor- hood of a Crack in an Elastic Solid," Proe. of the Royal Society, Series H, 187, 229-260 (1946),

6. Green, A. E. and Sneddon, I. N., "'The Distribution of Stress in the Neighborhood of a Flat Elliptical Crack in an Elastic Solid," Proc. of the Cambridge Phil. Sac., 46, 159-163 (i950).

7. Shah, R. C. and Kobayashi, A. S., "'On the Surface Flaw Problem" (In Press), Proc. of Cam-CAM Syrup. on the Surface Flaw, Applied Mechanics Die. of ASME (Winter 1972).

8. Post. D., "'Photoelastic Stress Analysis for an Edge Crack in a Tensile Field," Proe. SESA, 12 (1), 99-116 (1954).

9. Wells, A. A. and Post, D., "'The Dynamic Stress Distribution Surrounding a Running Crack-A Photoelastic Analysis," Proc. SESA, 16 (1), 69-92 (1958).

10. Irwin, G. R., Discussion of Ref. 9, Proc. SESA, 16 (1), 93-96 (1958)

11. Kerley. B., "'Photoelastie Investigation of Crack Tip Stress Dis- tributions," GT-5 Test Report Document No. 685D 597, The Gen- eral Electric Co. (March 15, 1965).

12. Dixon, .L R. and Strannigan, 1. S., "',4 Photoelastic Investiga- tion of the Stress Distribution in Uniaxially Loaded Thick Plates Con- taining Slits," NEL Report No. 288, Nat. Engineering Lab., Glasgow, Scotland (May 1967).

13. Liebowitz, H., Vanderveldt, H. and Sanford, R. l., "'Stress Concentrations Due to Sharp Notches," ~:Xl'~M~:Z~'rAL ~c~ar~xcs, 7 (12), 513-517 (1967).

14. Smith, D. G. and Smith, C. W. , "',4 Photoelastie Evaluation of the lnauence of Closure and Other Effects upon the Local Bend- ing Stresses in Cracked Plates," Int. ]ournaI of Fracture Mechanics, 6 (3), 305-318 (Sept. 1970).

15. Smith, D. G. and Smith, C. W. , "Influence of Precatastrophie Extension and Other Effects on Local Stresses in Cracked Plates under Bending Fields," EXP~Var~ENWAL MECr~ArCXCS, 11 (9), 394-401 (1971).

16. Marrs, C. R. and Smith, C. W. , "'A Study of Local Stresses Near Surface Flaws in Bending Fields," Stress Analysis and Growth of Cracks, ASTM STP 513, 22-36 (Oct. 1972).

17. Smith, D. G. and Smith, C. W., "'Photoelastic Determination of Mixed Mode Stress Intensity Factors," VPI-E-70-16 (June 1970); ]. of Engineering Fract. Mech., 4 (2), 357-366 (June 1972).

18. Eessler, H. and Mansell, D. 0., "'Photoelastie Study of Stresses Near Crack~ in Thick Plates," 1. of Mech, Engineering Sci., 4 (3), 213-225 (1962).

19. Stock, T. A. C., "'Stress Field Intensity Factors for Propagating Brittle Cracks," Int. J. o[ Fract. Mech., 3 (2), 121-129 (1967).

20. Marloff, R. H., Leven, M. M., Ringlet, T. N. and Johnson, R. L., "'PhotoeIastle Determination of Stress-intensity Factors," ~X- VErtXMEN'rAL ~ECHAr~XCS, 11 (12), 529-539 (1971).

21. Bradley, W. B. and Kobayashi, A. S., "'Fracture Dynamics- A Photoelastle Investigation," 1. of Engineering Fract. Mech., 3 (3), 317-332 (Oct. 1971).

22. Bradley, W. B. and Kobayashl, A. S., "'An Investigation o~ Propagating Cracks by Dynamic PhotoeIasticity,'" ~XX, Era~Er~TAL Wt~CHarCXCS, 10 (3), 106-113 (1970).

23. Kobayashl, A. S., Wade, B. G., Bradley, W. B. and Chlu, S. T., "Crack Branching in Homallte-lO0 Sheets," TR-13, Dept. of Mech. Eneineerlng. Coll. of Engineering, Univ. of Washington, Seattle, W A (June 1972).

24. Kobayashi, A. S. and Wade, B. G., "'Crack Propagation and Arrest in Impacted Plates," TR-I4, Dept. of Mech. Engineering, Coll. of Engineering, Uvlv. of Washington, SeaffIe, W A (July t972).

25. Schroedl, M. A., MeGowan, ]. J. and Smith, C. W. , "An As- sessment of Factors Influencing Data Obtained bu the Photoe?aztie Stress F~eezing Technique for Stress Fields Near Crack Tips," VPI- E-72-6, ]. of Ena~ineering Eract. Mech., 4 (4), 801-809.

26. Schroedl, M. A. and Smith, C. W.. "'Local Stresses Near Deep Surface Flaws Under Cyllndrlcal Bending Fields," VPI-E-72-9, Progress In Flaw Growth and Fracture Toughness Testing, ASTM STP 538, 45-57 (Oct. 1973).

27 SchroedI, M. A., McGowan, I. J. and Smith, C. W. , "'De- termination of Stress Intensity Factors from PhofoeT.astlc Data with Apnlication to Surface Flaw Problems," VPI-E-73-1 (in Press) (Feb. 1973).

28. Harms, A. E. and Smith, C. W., "'Stress Intensity Factors in Long Deep Surface Flaws in Plates Under Extensional Fields," VPI- E-73-6 (in Press) (Feb. 1973).

544 I December 1973