Office of Naval Research Contract Nonr-6lO i06)

Task Order NR OÖ4-476

i L ■ J :

'. ■: ■■■■■■ .-: v./

Technical Report No. I

ON THE WESTERGAARD METHOD OF CRACK ANALYSIS

CLEARINGHOUSE FOR FEDWRAL SOBNTIT'ir AND

TECHNICAL INFORMATION " HardoopyT lüöfof id»

%/,>c\%*6>"v\J/fim

AUS LAV 911 Uü ?»

3^1 Ff/ by G. C. Sih

L K HIGH

UNIVERSITY

U •IF

iu c

-^

March I966

Department of Applied Mechanics Lehigh University, Bethlehem, Pennsylvania

Offtee of Naval Research

Contrast Nonr-610 (06)

Task Order NR 061-476

Technical Report No. 1

On the Westergaard Method of Crack Analysis

by

0. C. Slh

Department of Applied Mechanics

Lehigh University

Pethlehem, Pennsylvania

March 1966

Reproduction in whole or in part is permitted by the United

States Government, Distribution of this document is unlimited.

On the Weatergaard Method of Crack Analysis

by

G. C. Slh 2

A survey of the literature on the analysis of crack

problems shows that the Westergaard method [1] has been

most frequently quoted and used by the practitioners In

fracture mechanics for nearly thirty years. Surprisingly

enough it has yet to be pointed out that the method in [1]

suffers severe restrictions for a class of problems dealing

with the infinite medium with a crack (or cracks) subjected

to external loads at infinity. These restrictions will be

derived in the work to follow from the more general con-

sideration of complex potentials originated by Muskelish-

vili [2],

In the theory of two-dimensional Isotropie elasticity,

the stresses and displacements may be expressed in terms of

two complex functions <fr(z) and \|>(z) of the variable z-x+iy.

They are

1 The results presented in this paper were obtained in the

course of an investigation carried out under Contract

Nonr-6l0(06) with the Office of Naval Research in

Washington, D.C.

2 Professor of Mechanics, Lehigh University, Bethlehem, Pa,

3 Numbers in brackets designate References at end of Note.

- 1 -

a + a m l\ Re [♦«(B)]

oy - ox + 21 xxy - 2[z ♦M(z)+*'(z)]

(1)

and

2y(u+iv) » ie $(z) - z ♦' (z) - JTzT (2)

where K takes the value 3-^v for plane strain and (3-v)/(l+v)

for generalized plane stress and v is the Poisson's ratio.

The shear modulus is denoted by y. Eqs. (1) and (2) may be

simplified by introducing symmetry conditions as follows:

Symmetric Problems. If the external loads are placed

symmetrically with respect to the x-axls along which the cracks

are situated, then the shearing stress T must vanish at y»0,

i.e.,

Im [z ♦"(z)+*»(z)] - 0 , for y - 0 (3)

Eq. (3) can be satisfied by taking

i|i'(z) + z 4>"(z) + A - 0 (4)

where A is a real constant depending upon the applied load.

Making use of eq. (4) and letting K ■ 3 - **v for plane strain,

the stresses take the form

- 2 -

ox - 2 Re IV(z)] - 2y Im [♦"(z)3 + A

a - 2 Re [♦•(«)] + 2y Ira [♦"(z)] - A (7)

T - -2y Re [♦"(«)]

and the displacements are

2M U - 2(l-2v) Re [*(z)] - 2y Im [♦'(z)] + Ax

2y v ■ 4(l-v) Im C*(z)] - 2y Re [♦»(z)] - Ay

(8)

Hence, the problem Is reduced to the determination of a single

complex function $(z) satisfying the necessary boundary con-

ditions. Eqs. (7) and (8) agree with eqs. (4-6) and (9-10)

in [1], respectively, only if

2 *'(z) . z , A - 0

In general, the constant A cannot be neglected arbitrarily.

To illustrate this point, consider the problem of an infinite

medium with a central crack of length 2a along the x-axis.

The boundary conditions are

a "T "0, y ■ 0 , -a<x<a y xy * * '

a "60,0 ■» <J , T xy 0, as (xZ+y2)-1'^« 1/2

(9)

- 3 -

(10)

The solution to this problem Is given by [2]

♦'(z) = (o"/2)[z/(Z2-a2)1/23-(l-£)(a°yiO

*'(z) - (a2o"/2)[z/(z2-a2)3/2]+(l-c)(o"/2)

Inserting eq, (10) into (H), A is found to be

A - - (l-e)(o°,/2)

Note that A vanishes only in the special case of e»l corre-

sponding to the case of uniform tension at infinity, The same

applies to the problem of an infinite row of collinear cracks

spaced periodically in an infinite medium [3].

Skew-Symmetric Problems. For loads applied skew-

symmetrically with respect to the crack line, say along the x-

axis, the normal stress o is required to vanish at y»0, or

Re [2<j»*(z)+z ♦ l,(z)+*»(z)] ■ 0, at y - 0 (11)

It follows that

I|I'(Z) + 2 4>'(z) + z ♦"(z) + IB » 0 (12)

where B is a real constant. Substituting eq. (12) into eqs,

(1) and (2) and separating the real and imaginary parts give

ox - *J Re [*'(z)3 - 2y Im [♦"(*)]

av - 2y Im [♦"(z)] y - 4 -

(13)

T • - 2 Im [#•(«) * >y Re U"(z)] - B (13) xy

and

2y u w 4(l«v) Re [♦(z)J - 2y Im [+'(z)] - By

(1M)

2W v « 2(l-2v) Im U(z)] - 2y Re [♦»(«)] - Bx

The Westergaard version of eqa. (13) and (14) may be obtained

by selecting an Airy stress function of the form y Im Z.

The results are the same as those given above if

2 $»(z) - 2, B ■ 0

The restriction of B»0 leads to a trivial solution for

the problem of uniform in-plane shear applied to an infinite

medium containing a crack. For this problem, the conditions

are

O«T "0, y ■ 0 , -a<x<a y xy • J '

(15)

x y "xy o ■ a ■ 0 , T * T , as (x2+y2)w ■*■«»

Prom [2], the complex functions are

♦ •(z) - -(iT°/2)[z/(z2-a2)1/2] + i r/2

#*(s) - iTaB[z/(z2-a2)1/2]-i(a2T°'/2)[z/(z2.a2)3/2]

- 5 -

(16)

«»

The constant B may thus be found from eqs. (12) as

B - - T

Hence, B cannot vanish for a non-trivial solution.

It should be mentioned that the Westergaard method is

valid for loads applied to the crack surfaces since in such

cases the constants A and B have no contribution.

„ 6 -

References

1. H.M. Westergaard, "Bearing Pressure and Cracks", Journal

of Applied Mechanics, Vol. 6, 1937, pp. A 49-53.

2. N.I. Muskhelishvili, "Some Basic Problems of Mathematical

Theory of Elasticity", P. Noordhoff Ltd., Groningen,

Holland, 1953»

3. Private communication with I, N„ Sneddon.

- 7 -

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Department of Mechanics

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tu. MWRT MCUMTTT c UMWMtHII

Unclassified tft. «neu«»

■- ». IMPORT TITLE

On the Westerraard Method of Crack Analysis

4. oiaemrrivt MOTH (Typ* m —pm mm Research Rro.lect

I. AuTMOAO) (U»t MM Bmtt

Sih, G. C.

kMtm)

•■ ACRORT DATS

March 1966 »•- »OTA». MO. «r PAOM TA, MO. OP mmn

Nonr-6l0(06) A MNOJOCTNO.

NR-06^-l76

MT MMMORAffi)

No. 1

• A tfHRRRfMOMT MOfQ <*«r <

10. A VA H.AOU.ITV/LMMTATICM MOTKU

Reproduction in whole or in part is permitted by the United States Government. Distribution of this document is unlimited.

n. TARY NOT» It- IROMSORIMO MILITARY ACTIVITY

Office of Naval Research

II. ABSTRACT

The Westerpaard method of crack analysis, published

almost thirty years ago, is shown to be invalid for a class

of crack problems dealing with the infinite medium with

cracks under applied loads at infinity. The necessary

modifications of the Westereaard method are derived from

the complex potential formulation of Muskhelishvili. The

examples of a single line crack in an infinite plate owin?

to biaxial tension and pure shear are discussed.

DD I JAN «4 14/W Unclassified Security ClaooJficatloa

Unclassi^ied Security Classification

14 KEY WOStDS

LINK A

KOLI

LINK •

NOLI

LINK C

^racture mechanics

Cracks

Plane extension

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2a, REPORT SECUWTY CLAÄHFICATSON: Eater tha ovar- all security clasaiffccette« of the report- indicate wNethwr "Restricted Data" la kndnded, Barking la to be 1M accord- anc« with appropriate »acsrtty ragwletlotsn.

26. GROUP: Automatic flomngi i«11 ag, to specified in DoD Di- rective 5200.10 and Arssad ftrcee Industrie! iOaaoaL Entar tha group number. Alao, whan nppltraato, show that optional marking» hava baan used for Group 3 and Group 4 aa author* Ued.

3. REPORT TITLE.- Katar tha complete report tfeto to all capital letters. Tltla« la all caeec ateald be wwcUawiflaw» II a w*«nl«ao>i tltla caaawt ha selected withe«* dneeiftca- tlon, «now tüle dasaificatien In aU capital» In parenthesis Immediately following tha title.

4. DESCRIPTIVE NOTE» If raport, e.g,, Interim, progress, Give tha ind serve- ewtms when covered.

5. AUTHOR(S): EnteT tha oamo(s) af authoKs) aa ahowa on or In tha report. Entar last name, fitret aam«, mlcVfle Initial. If military, ahow rank and branch of service. Tha name of tha principal «uthor ia an absolute «lauaaai requiinmeat

6. REPORT DATE: Enter tha data of tha raport aa day, month, year, or month, y*ar> If more than one en tha report, ma data of publication.

7a. TOTAL NUMBER Of PAGES: Tha total pea» coal should follow OMUUU paginationprucanmroa. La., lainiiai of p*-=» cooxatotog tafornaCioo.

7b. NUMBER OF RnTBRSNCB* Eater aha total reference» cited t» tha report.

If appropriate, enter under which

Sa. CONTRACT OR GRANT NUMBER: the applicable wmber of tha contract or tha raport waa written.

lb, 8c. a. ftd. PROJECT NUMBER: Enter the appropriate military department idmtttific atiom, amen aa project number, aubprojact number «yatem mamlims, tee* mmxxu, etc. 9a. ORIGINATOR'S REPORT NUwBER(B> Entar the offi- cial raport number by which the document will be identified and controlled by the originating activity. Tbia number must be unique te tula report. 9b. OTHER REPORT NUmBEmYS): SI the report has bann esslgnod any other raport wambara Odffter »r the arf4*n*ror or by the mpottmor), alao entar thia «■befi.i. 10. AVAIL ABlLTTT/LUCTATIOll NOTICE* Enter any lfab- ltstlons on fartbar dissemination of the raport, other than thooe

impoaed by security daaaiflcation, using standard statements sach aa:

(1) "Qualified renunntnrs may obtain copies of this report bom DDC "

(2) " foreign anmnwnrnmsst and Ma a ami nation of this raport by DDC to net

(3) "U. ft. thia

* npnai la* an» obtain eaplec of y teem DDC Other qualified DDC

turoogh

(4) "V. ft. asUstarf- ehtnto oepiee of thto DDC Other qualified a ear a

(5) M AU du«wltioa of this raport la controlled. Qual- ified DDC awara shall request through

cat* thia fact and

furnish«* to tha Office of Technical of reowaarca, for sals to (he public. temV

«hnprlee, Ul

Uae (or additional eapli 1L SUPPLEMENT.ART NOTE«: tory notes.

IX SPONSORING BUXTART ACTOTTT: Enter tha name of the 4*pmtmamtai prefect office or laboratory apoanoriag (par ini for) the raaaantb ami awvdesaMwL Include addreee.

13. ABtTBACT Sntor an abstract giving a brief and foctoal ■—nry of tha document tedieatrre of the raport. even though It matt aJno appear «ieewham in the body of the technical re- pert. ttaaWttwaal space U required, a continuation sheet ahall

k to highly desirable that the ebstract of elessifted reports (nwjlawaifwid Aneh jMmgMpn of the abstract shall «md with

am indication of the military security clanerflcarion of the in- formation to the paragraph, represented aa (TM). (M). (C), ot (V)

There is ne limitation on the length of (he abstract. Bow- ever, the suggested length ia from 150 to 22S words.

14. KEY WORDS: Key words are technically meaningful terma or short phrases that characterise a raport and may be used ss index sowie« for cataloging Ate report Key werde meet be selected »a met no «ecurtty classification to required Identi- fiers, such aa «mapmsnt modal dealenatlen, trade name, military prefect came naaia, awsemMgdc loewtsno, aury he used as key word« hat wiU be feUoerad by an todicatlnn of techoscal con- text. The «sstgnmittt of ltoka, rnlaa, sod wnights la optional.

DD FORM 1 JAN «4 1473 (BACK) ''nclassi^ici

Security Clnuification