Mixed Mode Crack Initiation and Direction in Circumferentially Cracked Pipes

Mixed Mode Crack Initiation and Direction in Circumferentially Cracked Pipes

1) DISTART-Department, Viale Risorgimento 2, 40136 Bologna, Italy ABSTRACT

In this paper, the S-theory is applied to determine crack initiation and direction for circumferentially cracked pipes. It makes use of a parameter called strain-energy-density factor, S, which is a function of the stress intensity factors. A simple method tbr obtai~fing approximate stress intensity factors is also applied. It takes into account the elastic crack tip stress singularity Mille using the elementary beam theory. Two basic loading conditions in piping systems are studied. The results are in reasonable agreement with flae more accurate calculations.

INTRODUCTION

As well known, Linear Elastic Fracture Mechanics is widely used to describe many aspects of crack behavior. Knowledge of the stress intensity factors plays an important role in fracture control. In structural applications, combined standard loading conditions often involve simultaneously Kt, Kn and Kin. Within the framework of brittle fracture, the well- known "Strain energy density factor theotT" [1 ] allows to predict unstable crack growth in mixed mode. It nmkes use of a parameter called strain-energy-density factor, S, which is a function of the stress intensity factors.

The Sih fundamental hypotheses of crack extension are:

1)The crack will spread in the direction of ma;vimum potential eneti~ density;

2)The critical intensity S~,. of this potential field governs the onset of crack propagation.

Stress intensity thctors for many configurations are available [2,3,4] . In most cases the results were obtained by means of analytical and numerical methods. In many cases the results were obtained by finite element methods and boundary element methods. Experimental methods have been applied to simple cases in order to detemaine the fracture toughness Ktc of engineering materials. Solutions tbr many structural configurations are not available in the handbooks. Simple engineering methods which allow a fast but approximate detemaination of the stress intensity factors are highly valued to a design engineering.

Remarkably simple methods tbr close approximation of stress intensity I5ctors in cracked or notched beams were proposed by Gao and Herrmann [5] and by Nobile [6]. The former has been based on elementary beam theory estimation of strain energy release rate as the crack is widened into a fracture band, the latter has been based on elementary beam theory equilibrium condition tbr internal forces evaluated in the cross section passing through the crack tip, taking in account the stress singularity at the tip of an elastic crack. The derived simple formulas [6] for stress intensity t~ctors are in reasonable agreement with the more accm-ate calculations in literature.

In this paper the latter method [6] is applied to compute stress intensity factors and to determine crack initiation and direction tbr circumti~rentially cracked cylindrical pipes. Two basic loading conditions in piping systems are studied. A P P R O X I M A T E EVALUATION OF STRESS INTENSITY FACTORS

Consider a straight beam of constant cross section. The z-axis coincides with the geometrical axis, and the x and y- axes coincide with the principal axes of the cross section. The stress components due to stress resultants are well known. Suppose that the presence of an edge crack of initial length a doesn't alter the stress resultant on the cross section passing through the crack tip. The singular stress distribution at the crack tip takes the form

Ki (1)

° = 42Jr,, S

wifla the condition that a/j acts at a distance r=b ti'om fl~e tip. The nominal stress is evaluated by the known stress distribution on the reduced solid cross section passing through the crack tip (ligament). The stress distribution doesn't take into account the presence of the crack. Then, the equivalent condition between singular stress and nominal stress resultant at the crack tip determines K~ approximately. Note that Krwdues are better approximated for b< a such that the elastic singularity governs stresses at a distance from the tip lower compared to the geometric dimension of crack length.

Consider a cylindrical thin pipe of radius R and thickness t much smaller than R, containing a crack of length 2R~ along the circumference, as shown in Fig, 1.

SMiRT 16, Washington DC, August 2001 Paper # 1841

j l ~ r

crz = " RcosO (2)

Ix

where Mx = M is the bending moment and Ix is file moment o f inertia:

I x = zcRSt

(3)

(4)

Tile distribution of normal stresses on the reduced section is

cy*= " R c o s O +

1 ~ - a

(5)

- t ~ tJ~ c o s O + ~ dO

E -- ()~

T r o u g h

C:rack

T

M.,

t

J

I

Fig.1. A circumferentially cracked cylindrical pipe under bending. Bending

First consider the case when the pipe is subjected to a bending moment Mx, as shown in Fig. 1. The distribution o f nomaal stresses on the uncracked cross section is

The singular stress component is related to the mode I stress intensity ihctor as follows

Z

K t x/ 2~R ~

The stress resultant arising at the crack tip is equal to

2 ol 42~Rot Rd~ =2v ~K K~

(6)

(7)

According to the condition that o'_ = o "s~ at 0=a'j, /~) can be expressed approximately as

K ! = ~[2zcl{a~ cr z

]o---~

Substituting Eq. (8) into Eq.(7), there results

b K t

sin~ )

J" d r = ~ 2a~ c o s ~ + ~

0 42m"

1~

~ - 0~

The distance b call be detemlined from the equivalent condition for tbrces in file direction of the geometrical axis z

at

Ks

o

2 ~o ~21rRO dr= 2~

Now Eq.(9) may be submitted into Eq.(10). The integral may be evaluated to give

4 _-z7-~ ~/ut c o s a +

I~ ' z c - a I x

Eq.(11) may be used to solve for az. The result can be put into Eq.(8) to give the stress intensity factor

1: (

,,,,n° /

Ix ~ - a

o r

where

sinot

)

(9)

(10)

(1 ] )

(12)

(13)

(14)

.

. . . . [ 7 ]

. . .

[5]

Present method

¢ g

,F

/

./'[.."

0.0 0.1 0.2 0.3 0.4 0.5

[ , \

The function .6 % ) can be compared with the expressions given by [5] and [71, as shown in Fig.2. As pointed out in [5], an improvement can be achieved by introducing a correction fgctor. This factor can only be obtained from optimum fitting to the exact elasticity or the finite element solutions. The agreement between the present result and those in [5,7] appears to be good.

Torsion

Consider the case when the pipe is subjected to a torque T, as shown in Fig.l. The stress at any point of the uncracked cross section is

z(~ = T 2 S

2 tcR e t

The maximum stress on the reduced cross section passing through the crack tip is

(15)

Z'zs)max = T t

d where

s =

3

The singular stress component is related to the mode I stress imensiw factor as follows r~,. = Kfl

~/2rcr

The stress resultant arising at the crack tip is equal to

Ku RdO R

o 4 2 nR O

]

(16)

(17)

(18)

(19)

According to the condition that rz, = "r:"~. at O=a+~l, Kn can be expressed as

Substituting Eq. (20) into Eq.(19), there results

eO t 2

K° - - R d O = T ~ o ~/2~RO 3 -- (Z 1

The angle at can be determined from the equivalent condition for moments about the geometrical axis z

czt t 2 Ct 'l " Czj, f 2

[" Kn --3-RdO- ['":~')tR~d'9+ J'{"r~")m~--Rd')

o ~I2*cRO o o 3

Now Eq.(2 I) may be submitted into Eq.(22). The integrals may be evaluated to give

o~ _ T a r~

T ~ - + T ~

(20)

(21)

(22)

(23)

Eq.(23) may be used to solve tbr ~j. The result can be put into Eq.(20) to give the stress intensity factor

o r

K * * = I 2 zrR rz ( 1- a

2t2~R(1 - ~

)--

K n - 21rR2---

7

(24)

(25)

where

R 2n:~

A plot of normalized stress intensity t~tctor as a function of J2 (/~//~) for different values of R/t is shown in Fig.3.

(26)

"8 350

300

250

200

150

100

50

0

0.0

R/t=20

- - - R/t=30

. . . R/t=40

o m • • w J

• j o J • * J

,0

t 6

J , l * t

, P

•O

II,

, u i

0.1 0.2 0.3 0.4

# o

I S '

. S

0.5

M I X E D M O D E C R A C K I N I T I A T I O N A N D D I R E C T I O N

For general loading, the Strain energy density t~ctor is

S - aizK ~ + 2ai2K~Kn + a2:K h + a:~sK;t~t (27.)

where

a l l =

1

16re# [(3-4V -cosO)(1 +cosO)], a12 = ~ 16 rq.z (2sinO) [cos 0 - ( 1 - 2v )]

(28)

a 2 2 :

1

/ 6 rq~

_ 1

[4(1-v)(1 - c o s O ) + (1 +cosO)(3cosO - 1)], a33 - 4~r/J

with v being the Poisson's ratio and/2 the shear modulus of elasticity.

Consider the case where Mand T are present, as shown in Fig.l. Setting K , U 0 in Eq.(27) and putting Eqs.(13) and (25) into Eq.(27), S becomes

S = - - - M 2 (4a1i.. if2 + 4as2m[1f2 + a ,~.m ~" f]" )

4ycR.~ t 2 . . . . (29)

where j) is expressed as in Eq.(14), .~ is expressed as in Eq.(26) and m=T/M. Assume that the crack would initiate in the direction of S,,~i,, i.e., 3S/30 = 0. This corresponds to the direction where dilatation would dominate. A plot of normalized strain energy density factor as a function of 0 for R/t=40, v=0.3, m=0.1 and different values of ~0/~ is shown in Fig.4. Crack instability; is then assumed to take place when Smi,~ equal to a critical w~lue S'¢, which is a material parameter.

0

o 700

, m ,

cn 600

I:1

500

C

I l l

• ~: 4 0 0 -

"o 3 0 0 -

N

E

"- 2 0 0 -

@

z

o o o ... I ... ...

I \

"k

,k

k \

U d l t r - - ~ , I "" 1

- - - c~ rc--0. 2 /

I

i /

I i

g

/

% j ,

v=-0.3 m = O . l

f f 100 % \ \ \ \ \

\

-2 -1 0 1 2

0 (rad)

CONCLUSION

The strain energy density theory is applied to the case of circumferentially cracked pipes. The attractive feature of this theory over the conventional k~-theory is that the single pameter S~ can simultaneously detemaine the fracture toughness of the material and the direction of crack initiation.

The mixed mode crack growth analysis requires a knowledge of the stress intensity factors. A simple method is applied to find approximate stress intensity factor. The method takes into account the elastic singularity and is based on the elementary beam theory. The results show a good approximation when compared to known solutions.

Acknowledgement

The author gratefially acknowledges the financial support rendered by the Italian Ministry of University.

REFERENCES

[ 1] Sih, G.C., "A Special Theo~ of Crack Propagation: Methods of Analysis and Solutions of Crack Problems", in ,~echanics oj'Fraeture I, edited by G.C. Sih, Noordhoff International Publishing, Leyden, 1973, pp. 21-45.

[2] Sih G.C., Handbook of Stress lntensitv Factors jor Researchers and Engineers, Institute for Fracture and Solid Mechanics, Lehigh University, Bethlehem, 1973.

[3] Tada, H, Paris, P.C., Irwin, G.R., The Stress Analysis of Cracks Handbook; 2 "d edn., Paris Production, 1985. [4] Murakami, Y., Handbook of Stress lntensin~ Factotw, Pergamon Press, Oxford, 1987.

[5] Gao H., Hermaann G., "On estimates of stress intensity Factors for cracked beams and pipes", Engineering Fracture Mechanics, Vol. 41, 1992, pp. 695-706.

[6] Nobile L., "Mixed mode crack initiation and direction in beams with edge crack", Theoretical and Applied Fracture Mechanics, Vol. 33, 2000, pp. 107-116.