The Handbook Excerpts

8/6/2019 The Handbook Excerpts

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THE STRESS

ANALYSIS OF CRACKS

HANDBOOK

THIRD EDITION

HIROSHI TADAWashington University

PAUL C. PARISWashington University

GEORGE R. IRWINUniversity of Maryland

" A 3MEP R E S S : ' Professional

EngineeringPublishing

Tile MateriaisInformation Society


2/18

41l Part II 2.1

THE CENTER CRACKED TEST SPECIMEN

A. Stress Intensity Factor

t t t~ t t---t---

!

I hNumerical Values of F(alh)

(Isida 1962, 1965a, b, 1973) "Isida's 36-tenn power series of (alb) (Laurent series

expansion of complex stress potential, 1973) givespractically exact values of F(a lb) up to a lb= 0.9.Numerical values of F(a lb) are shown in the followinggraph and table.

alb F(%)

0.0 1.0000

0.1 1.0060

-~ 0.2 1.0246. . ._ ,LL 0.3 1.0577

~n8 0.4 1.1094

0.5 1.1867

t * 0.6 1.3033M; ..6371.00.7 1.4882

0.3 1.8160

0.9 2.5776

(~

1.0 _2_/ 11 _ a( **';7l'2-4. V tb

#Exact limit (Koiter 19650)

*See Note 2


3/18

Stress Analysis Results for Common Test Specimen Configurations 41.1

Empirical Formulas

a. Accuracyb. Method of derivation, reference

.) ri~F (G ! b = \i~tan~V 7fG lb

a. Better than 5% for a lb : : ;0.5b. Approximation by periodic crack solution (Irwin 1957)

a. 0.5% for a lb : : ;0. 7

b. Least squares fitting to Isida's resul ts (Brown 1966)

a. 0.3% for alb : : ;0.7, l% at a lb = 0.8b. Guess based on Isida's results (Feddersen 1966)

a. 1% for any a lbb. Asymptotic approximation (Koiter 1965b)

a. 0.3% for any a lbb. Modification of Keiter's formula (Tada 1973)

a. 0.1% , for any a lbb. Modification of Feddersen's formula (Tada 1973)


4/18


5/18

2.7 Stress Analysis Results for Common Test Specimen CO!ltigurations 47

Empirical Formulasa. Accuracyb. Method of derivation, reference

i2 b Ir a= \i -t M--

l tr a 2b

a. Better than S% for alb> 0.4b. Approximation by periodic crack solution (Irwin 1957)

,F(%) = 1.12 + 0.203 ({ I/h ) -1.197(%f + 1.930 (il/h)

a, Better than 2% for alb'S 0.7b. Least squares fitting to Bowie's results (Brown 1966)

7 -

1 1")' - 0 "6'1 (0 / I - 0 O l"(a' )"..L O091 (1 1 /' \"()

. -~ .) ,i /)1 .) [b ;. . b JF% =, v r= a 7 b

a, Better than 2% for any a lbb. Asymptotic approximation (Benthem 1972)

r;: :: ;----

{ II\ ( 41W) ('2b ITaF \'ib) = \ 1 + 0.122 cos - \, -tan~. 2b V 710 2h

a, 0.5% for any a lbb. Modification of Irwin's interpolation formula (Tada 1973)

a. O.SS'" to r any ~/bb. Modification of Benthem's formula (Tada 1973)


6/18

52 Part II

THE SINGLE EDGE NOTCH TEST SPECIMEN

2.10


Numerical Values of F(Glb )

The curve in the following figure was drawn based on theresults having better than 0.5% accuracy.

Methods and Referencesa. Boundary Collocation Method (h lb > 0.8): Gross 1964b. Mapping Function Method (hl

b= 1.53): Bowie 1965

c. Green's Function Method (hlb> 1.5): Emery 1969, 1972d. Weight Function Method: Bueckner 1970, 1971e. Asymptotic Approximation: Benthem 1972f. Finite Element Method (hlb = 2.75. 1.0): Yamamoto 1972

\.2

LL~ 1.1~

'*I \.0-. . . . .,O .q

t t o " 'f......!- ----- 1--,.--

II h,

~a-

Ib hII

f---L-- ~L--

! I 1

\.122-3/2 ~\


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2.11 Stress Analysis Results for Common Test Specimen Configurations 53

Empirical Formulasa. Accuracyb. Method, reference

a. 0.5% for a lb :S 0.6b. Least squares fitting (Gross 1964; Brown 1966)

(a' ) ( 01 )4 0.857 + 0.265 %F Ib = 0.265 1 ~ b + 3/

(1_%)12

a. Better than 1% for a lb < 0.2, 0.5% for alb:: : 0.2b. Tada 1973

3

I /2b 7r a 0.752+2.02(%)+0.37(I~.sin;b)F ( a l b )= v ' -tan-. 7T _

7TG 2b cos2b

a. Better than 0.5% for any a lbb. Tada 1973

r---

I

- - -

TI h+t . . . 0 . ., . ,

I h- r,-- --

B. Displacements

Crack Opening at Edge

Gross' results (Gross 1967, Boundary Colloca-tion Method) are expected to have 0.5% accuracyfor 0.2 :::; a lb < 0.7. An empirical formula with 1%accuracy for any a lb is (Tada 1973)

. . 1.46 + 3.42 (1 ~ cos ; a ) "t r fa! ' _ _bPl\!b}- ".

(cos2g ) -


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2.13 Stress Analysis Results for Common Test Specimen Configurations 55

THE PURE BENDING SPECIMEN


h

Numerical Values of F(a /h)

The curve in the following figure was drawn based on theresults having better than 0.5% accuracy. Also used for four-pointbending.

a I~-b----i

- - l - - -+- - 'h

1 .

\ 221o \!

\ (1- 9> f J kf ( %),

'\-,


9/18

58 Part II 2.16

THE THREE-POINT BEND TEST SPECIMEN

A. Stress Intensity Factorp

_ 6M (M _p s )a- "-b2 4

Numerical Values of F(a lh)

The curves in the following figure have 1 %accuracy. P/z

1.122

.......~ ~ - - ~ - - ~ ~ - - ~ *

,.....,

*_, O.SJ._~tn t0.5 0.6

I -~0.6'-'t

0 .4 -

02. 0.4 0 . 6 O.S 10-- crb

,

Methods and References1. Boundary Collocation Method (S ib= 4,8) (Gross 1965b)2. Green's Function Method (% = 3,8) (Emery 1969)

Empirical Formulas

a. Accuracyb. Method, reference

For si b = 4,

a. 0.5% for any ab. Srawley 1976


10/18

19.1 Two-Dimensional Stress Solutions for Various Configurations with Cracks 289

ecgee''--------------

t l -

--

t t" t 001:~~ --------~--~ 0

--

Q.s - R+dK: ; t= r r / i rO .K(S)1


11/18

290 Part HI 19.2

t

-

t 0- t

-A()-

, l o - +s = --.:__

4 R+a

k:l.= cr.,lifa .F ; . .(S)

F A(S) = ( ,-X) Fo(s)-tl\ F ,(S)3

FoeS) =[ \-\-.'1(1-5)+.3(\-5)') F1(s)3

u : F 't(5) = 2.2tr'3 -:d:04~,. \.'3S2S~-.11t8S 3

.: 2 K11 . .= ll..{iii .F J l r < '5) . sin Y 2t

F II F.m : {5 )=:k( 2- 5 51 2

-

04 0&S-_g_- R+Q

o . " C 10

-Method: Mapping Function MethodAccuracy: F o and FJ Better than 1%

FIJ] Exact

References: Bowie 1956; Yokobori 1972 (or Kamel 1974); Tada 1985


12/18

292 Part In 19.4

t

ih

j

h

4~------~--------------------~----~~! K I = (S - / r r a Fe.fi. R lL)b '" b ) b I

1

0.25

hb =2

O.B 1. 0.2 04 0.&

Method: Boundary Collocation Method.

Accuracy: Curves were drawn based on the results having better than 0.1% accuracy.Reference: Newman 1971See also page 19.11.

- p


13/18

19.i5 Two-Dimensiona! Stress Solutions for Various Con!1gurations with Cracks 303

~=o(F... \.122.~

I0 H-_---!fl_=___+_

5= _s_b+ C I.

l < : r=t J " ; rm .F(S, t- )

{

S-O(~""O): F =l.I22 Kts..... -%--00): F... 1.122

.fSC/b~CO : F-4.122

where Kt = Kt (% )= = (I+ 2 ~ ). f (% )c- I Sh

fC1b) ~ \ -+ 0122 ( 1+ Yo)

1.122

G~----+- 1;-C/b

i

o ~ - - ~ - - ~ - - ~ - - - - ~ - - ~ - - ~ - - ~ - - - - ~ I - - ~ - - ~ .o 0.2. 0.6 O.'S 10

0--5"" ---

b1-o.

Methods: Stress Relaxation (Superposition) (Nishitani; C b = =1 . 1, 2 and D.2 :S Cl o S 1)~Estimated

. Better [han 2%Tuda 1913


14/18

342 Part IV

a-t t to -

r

(KlJ = KIll = 0)

Volume of Crack:

Crack Opening Shape:

( 2 )1 - V 2 22v (r,O) = aja - r

rSa 1[E

Opening at Center:

8(I-V2)

60 = 2 v{O , 0)= 7[E (J a

Additional Displacement at (0, s) due to Crack:

where1

a = -r-r-r-r-r-r2 (1 - v)

Methods: Integral Transform, Integration of page 24.5 or 24.11, Paris' Equation (see Appendix B),Reciprocity (see page 24.7)

Accuracy: ExactReference: Tada 1985

NOTE: V(O,s) Is the displacement at (O,s) when uniform pressure o i s applied on crack surfaces.

24.1

. I


15/18

390 Part IV 27.1

ponet = lfa2

Kr-=

~ et..r rr a. F, (alb)

= C J n e tInCb-a) F2(a./b)

= c > n e t !Tr b F a(


16/18

4l\1 Part 1\'

~f

(

(

~j

x

Semi-infinite body (y 2 : 0) (y = 0: Free surface)

F {O ) = Ui 1- ,1860:inO (10' < (J < l70')

28.1

Methods: Alternating Method (Smith, Hartranft), Finite Element Method (Tracey, Raju); F (8) is basedon Smith's result (Merkle)

Accuracy: 2%References: Smith 1967; Hartranft 1973: Trace}' 1973; Merkle 1973; Raju 1'179


17/18

478 Part VB 33.6

t

t

I R( J= -P -

2 I

110F('\) = (1 + .3225.\2)!"

= 0.9 + 0.25.\

Crack Opening Area:

(J

A = E' (2 7rR t) G (.\)

Methods: K; Integral Equation; A Paris' Equation (see Appendix B)Accuracy: K, 1%; A 2~{)References: Fnlias 1967; Fama 1972; Tada 1983a


18/18

35.1 Crackis) in a Sheil 485

t

= 0.6 + 0.9 A

Crack Opening Area:

(T

A = 1 ( 21TRt ) . G (.\ )

G(.\) = / + .625.\4

2 3 4= .14 + .36.\ + .72)" + .405.\

Methods: K Integral Transform; A Paris' Equation (see Appendix B)Accuracy: K/ 1%; A 2%References: Folias 1965; Erdogan 1969; Tada 1983a

NOTE: As a ~oc, K, ~ r r JR (,);7;r . f) (Harris 1997).

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