Empirical modelling of subcritical crack growth size and profile under continued loading

/

E L S E V I E R Theoretical and Applied Fracture Mechanics 20 (1994) 85-98

and applied tracmm

mechanics

Empirical modelling of subcritical crack growth size and profile under continued loading

S. Bhattacharya a, A.N. Kumar b.,

a Research and Development Centre, Indian Oil Company, Faridabad-121007, India b Department of Applied Mechanics, Indian Institute of Technology, New Delhi-110016, India

Abstract

During continued loading the slow crack growth regions at the crack tip assume different shapes and sizes. Two different crack front profiles may be identified in practice depending on the extent of crack growth. They may be approximated as parabolic and semi-circular in shape. Crack fronts tend to be semi-circular for relatively large extent of crack growth and parabolic during the early stages of crack growth. Modelling of crack growth size and profile for each crack growth increment is carried out by two approaches; namely, offset measurement and area consideration. The variation of average crack growth size A~ as well as the convergence of A~ as a function of measurement number is studied. The difference in the A~ values obtained by the offset method and area method is also investigated in addition to the influence of specimen thickness and maximum crack growth on ~b. Experimental verification of crack growth front profiles and the change in the A~ value with ~b is carried out. It suffices to consider eight to ten measurements for modelling crack growth size and profile in commonly used specimen thickness.

1. Introduction

Modelling of crack size and profile is an important consideration in subcritical crack growth studies. For materials with high toughness, unsta- ble fracture can be preceded by considerable slow and stable crack growth under quasistatic loading [1]. Stable cracking situations may also prevail during cyclic loading [2], creep loading [3], stress corrosion cracking [4,5], etc. Indirect measurement of crack size is nondestructive. The compli-

* Corresponding author.

ance technique, electrical potential methods, etc., are commonly used. The direct method of crack size measurement serves a different purpose; it can assist the development of empirical models and involves the examination of fractured specimens [6,7]. Fig. 1 shows a schematic of the number of measurement points to approximate a thumbnail shaped crack front. Some points of controversy [8-12] were raised on the number of measurement points to be considered which could affect the estimate of crack front profile, influence of thickness, etc.

The present work is primarily concerned with the empirical modelling of the number of offset

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86 S. Bhattacharya, A.N. Kumar / Theoretical and Applied Fracture Mechanics 20 (1994) 85-98

.~Z_____B

- - - - - - B S P o i n t

Aamax ~ A S P o i n t

iOiilll rq

Fig. 1. Illustration of measurement points on an idealized thumbnail crack growth front.

a 1 = a9 = 0.01 B

measurements ~b for two different slow crack growth front profiles which are parabolic and semi-circular. Some experimental data are used to compare with the estimated crack front profiles for varying extent of slow crack growth. Estimation of the representative crack size is obtained from area consideration under the crack front and compared with the average crack growth size obtained by offset measurements.

2. M o d e l l i n g o f c r a c k f r o n t p r o f i l e s

A slow crack growth front is qualitatively described as a " thumbnai l" type, Fig. 1. Fig. 2 depicts the various stages of slow crack growth regions as may be seen in structural materials. The extent of stable cracking may be clearly seen to be varying across the section thickness. In some cases, the slow crack growth region may not even spread out through the thickness and re- main confined only in the central part, Fig. 2. The crack front profile may also be seen to change with the extent of crack growth as illustrated in Fig. 3. During the early stages of crack growth, the tendency to spread out along the thickness direction is more (sidewise growth) than the tendency of the crack to move in the forward direction [13]. Thus, a parabolic crack front appears to form for relatively small crack growth where A a / B = 0 . 2 5 . However, once the crack fully spreads over the thickness, it tends to move more

Fig. 2. Various stages of the slow crack growth region in structural alloys.

S. Bhattacharya, A.N. Kumar / Theoretical and Applied Fracture Mechanics 20 (1994) 85-98 87

in the forward direction (frontal growth). Contin- ued crack growth with loading eventually results in a semi-circular type crack front profile where Aa/B = 0.5. Two situations as discussed are illustrated in Fig. 3. The appearance of different crack front profiles may also be dependent on the section thickness as well as microstructure of the alloy. The present study is confined to two crack front profiles obtained at varying degree of crack growth in a given section thickness.

Parametric equations have also been derived in [14] to characterize the growth of semi-elliptical surface cracks based on the rate at which energy is released locally around the crack bor- der. More elaborate crack growth profile predic- tions can be made by application of the strain

energy density criterion and the incremental the- ory of plasticity [15].

3. Observed crack profiles

A comparison is made between the modelled and observed crack front profiles. The experimental average crack size values A~ are plotted against the corresponding maximum crack size Aama x for various stages of crack growth. The experimental points as obtained for various structural alloys are shown in Figs. 4 and 5. The average crack size and maximum crack size values are normalized by dividing by specimen ligament size, b. The estimated lines between the two

,eb------- B -~b. m

2

%

L

I W

I

9=------ B --~---~

2

ao i ( a ) (b) (c)

/" amax /b = 0.I02 Aamax/b = 0.25 Aamax/b = 0.375

B --I

I oi (d)

Aama x / b = 0 . 3 2 5

. . . . Expe r imen ta l l y measured Crack f ront p ro f i les for specimen with b = l O m m Theo re t i ca l l y obta ined c r a c k f ron t p ro f i l es

1 - Parabo l i c c rack f ront

2 - - Semi c i r c u l a r c rack f ront

Fig. 3. Experimental and theoretical crack front enclosing the crack growth region.


t .£1

IO <J

0-3

0.2

0.1

E x p e r i m e n t a l po in t s

o AP[ 5L x 60 7, API 5L x 46

• API 5L x 42 4- Free c u t t i n g s t e e l ( d e e p c r a c k )

• - - t ~ - - ( S h a l l o w c r a c k ) ,Ix A IS I 4140

Jr ~

4-

r y,

A

T h e o . c r a c k f r o n t M e t h o d

A P a r a b o l i c - - S e v e n p o i n t B P a r a b o l i c A r e a C S e m i - c i r c u l a r - - S e v e n p o i n t D S e m i - c i r c u l a r - A r e a

-T I I i 0.1 0.2 0.3 0.4

~ a m a x / b

Fig. 4. Variation of A~/b with Aamax/b.

0 .3~

I 0 . 2 -

JD

Io ,<1

0.(

i ( 0 . 5 Mo steel at 30°C ) v ( ,~ 2oo°c )

( ~ 400°C ) o ( C r - M o s t e e l a t 30°C) 0 ( ~',' 200°C ) • ( ,~ 400°C ) ÷ ( A P [ 5 L x 6 0 w e l d z o n e )

•E3

Q

I 0.1

+

Theo. c r a c k f r on t Method

A. P a r a b o l i c S e v e n p o i n t B. ---~' Area

C. Semi c i r c u l a r ~ S e v e n po in t D. ~ ~ Y ' - - A r e a

I 0.2

A a m a x / b

Fig. 5. Variation of A~/b with Aamax/b.

I I 0.3 0.4


parameters, i.e., A~/b and Aamax/b are also shown on the plots for the two crack front profiles, i.e., parabolic and semi-circular [13]. The A~/b values used to get the estimated lines are based on two considerations. Firstly, the area measurement under the crack front profiles and divide by the thickness to obtain A~/b and sec- ondly the seven point average estimation A~ following the British Standard BS5762. The estimated lines are shown in Figs. 4 and 5 for each of the crack front profiles considered.

It may be noticed from Figs. 4 and 5 that the experimental points between A~/b and Aamax/b match closely with the estimated lines for a parabolic front in the regime of small crack growth A~/b <_ 0.25. The experimental points, however, tends to deviate gradually and match with the estimated lines drawn for semi-circular crack front for larger crack growth. The intermediate (transition) stages of the crack growth, however, exhibit points that lie between the two theoretical lines. Refer to Figs. 4 and 5. Therefore, the estimated crack front profiles appear to represent the observed profiles. The number of offset measurements required for a representative average crack size in two crack front profiles is considered in the work to follow.

4. Parabolic crack front

The parabolic crack front surrounding the small crack growth Aa region as depicted in Fig. 6 may be analytically represented by

x 2 = 4ky (1)

where k is a constant. For x = B/2, the maxi-

Aai = A a m a x - 4 & a m a x ( }2

J(0,0 )

Fig. 6. Parahofic crack growth front represented by x 2 = 4ky.

mum crack growth Aama x is given at point (0, 0), i.e., y = Aama x. This gives

B 2 k (2)

16Aam~,

The general expressions for individual crack length Aa; along the y direction may be written a s

where i refers to any arbitrary offset measurement at regular intervals along x.

4.1. Average crack size by offset method (A~)

If there are ~b number of offset measurement points (including the mid thickness point at x i = B/2) for crack length, then the number of measurements points falling in one-half of the thickness is (~b- 1)/2. The interval x at which the intermediate measurements are to be taken is given by

B/2 B

The individual crack length may be obtained from Eq. (3) at x i --x, 2x, 3x . . . 4~x by substituting x, 2x, etc., for xi. The first offset length, for instance, is given by

Aalatx,=x=Aamax[l--4(n) 2] ( 5 )

Considering the offsets in one-half of the thickness (x <_ x i < (~b - 1)/2 x) the sum of Aa i values may be obtained as

(• - 1 ) /2 (tb - 1 ) /2 4 A a m a x (4' - 1 ) /2

E Aai= E Aamax n 2 E X2 i = l i = l i = l

=Aam~x[~b-1 4 { fiE x2 + 4x2

;)l + 9 x 2 + . . . + x


05- 1 4 { 2 2 = Aamax 2 B"2 x2 12+

0 5 - 1 [ 4 2054-1 2 Aam~ x[1 -- B-Tx 2

Substituting for x and on simplification

)'Aai 05-1 [ 05 ] 2 Aamax 1 3(05 + 1)

(6)

(7)

The sum of all the offset measurements across the full thickness may now be obtained as

Aa i = Aama x + 2 05 - 1 4' i=] 2 Aama x 1 3(05 4- 1)

05(05 + 2) = ~-Aama× (05 + 1) (8)

The average crack growth size (AK) may be obtained dividing the sum by 05

1 4, 2 , [(/)4-2 t Agt= 7 E Aa i= glaamax~ ~-~-~

I i=1 (9)

AK may also be expressed in terms of thickness (B) for a fully grown parabolic crack growth

E E

IO

7 b

6

5

3

1 -

)¢ E x p e r i m e n t a l l y o b s e r v e d A'~

in o s p e c i m e n of 7ram

t h i c k n e s s

20ram

15ram

10ram

X X 7 r a m

5mm

B = 2 m m

I I I I I I I I I I 2 4 6 8 10 12 14 16 18 20

Fig. 7. Variation of A~ with offset measurement number for parabolic crack front.


region where B is equal to 4Aama x as shown in Fig. 6. Equation (9) then reduces to

-~ B ( q ) + 2 1 _ A~ = ~ - ~ y (10)

Fig. 7 shows the variation of A~ with the offset measurement number ~ for some arbitrary crack growth size as well as thickness. A~ may be seen to decrease with decreasing slope as ~ increases. The nature of the variation of A~ as shown in Fig. 7, however, suggest that the effect of ~ is more significant for small ~ values. The point of inflection may be seen to lie for 5 _< ~ < 10 de-

pending upon the section thickness. This is also evident from Eqs. (9) and (10) where the factor (~b + 2)/((/) + 1) tends to decrease gradually and may reduce to 1 for large ~b. As, for instance, the factor turns out to be 1.125 and 1.11 for ~b equals 7 and 8 respectively. The best estimation of A~ may thus be obtained considering ($ + 2)/((/) + 1) equals 1 when ~b tends to infinity, i.e.,

B A ~ = ~-Aama x = ~ (11)

Fig. 7 displays some experimentally observed A~ values from a structural steel of 7 mm thickness

E E

7 0

x

O0

9G

80

7C

BC

5C

4C

3 0 -

2O

lC

z /

X~ B =20mm.ama x =Smm

Y ~ J'B =10ram. amax=2-Smm

t =20ram. a mox=2.Smm

Z - - B = 1 0 r a m . O m o x = 1 , 5 r a m

I I I I 5 10 15 20 25

N o . o f o f f s e t m e a s u r e m e n t (qb)

Fig. 8. Variation of dA~/d(k as a function of offset measurement for parabolic crack growth front profile.


and A a m a x equal to 1.75 mm. The points may be seen to follow the modelled line between A~ versus 4) closely.

Modelling of crack size estimation is also dependent on how the ~b values converge with A~. The difference in the Aft value may be large enough for relatively small ~b values. Differentiat- ing Eqs. (9) and (10), we have (omitting minus signs)

dA~ 2 1 1 B = ~ A a m a x (12)

d~b ( 6 + 1 ) 2 6 ( 6 + 1 ) z

Fig. 8 shows the variation of dAK/d~b as a function of 4) for different Aama x and B values. As 4) increases initially, there is a very rapid fall in dA~/d~b which tends to converge gradually. Complete convergence can only be obtained as tends to infinity.

For a reasonable convergence of dAa/d~b, a limit of 10 ~m may be set as this is usually the resolution limit of travelling microscopes commonly employed for direct (physical) measurement of Aa values. A horizontal line corresponding to 10 ~m is drawn in Fig. 8 to obtain a realistic number for average crack growth size measurements. As may be seen for a thickness of 10 mm, around 12 measurements may be required for the estimation of crack size. However, 4) is also found to be dependent on the extent of crack growth.

4.2. Average size from area consideration (~1~* )

The best estimation of average crack size may be made from area considerations for the parabolic crack front considered. Referring to

E E

v

I

. d

T h i c k n e s s ( r n m )

A 2 m m

B S r n m

C - - 1 0 r a m

E) - - 15 m m

E ~ 2 0 m m

I I I ~ I I 2 3 4

, I i 5 6 7

I ,1 8 9

Fig. 9. Variation of I A~ -- Aft* I with 4) for parabolic crack front.

s. Bhattacharya, A.N. Kumar / Theoretical and Applied Fracture Mechanics 20 (1994) 85-98 93

Fig. 6, the area under the parabolic crack growth front may be obtained as

Area(A) = 2 f;/2y dx

Substituting Aa for y from Eq. (3), we have

Area(A) :2 f ; /2[Aamax- -4Aamax( -X-~)2 l dx

8 amax IX/I"" =BAamax B 2 [ 3 ]0

1 2 =BAama x - 3BAama x= ~BAama x (13)

(AK*) is obtained dividing by B as

2 _ 1 (14) A~* = ~Aama x -- gB

using the parabolic relation between B and

Aamax. Eqs. (11) and (14) may be seen to be identical

as the number of measurements ~b in the area average consideration is infinite. The equations reveal that a most conservative estimation of A~ may be obtained by simply dividing the Aama x value by 1.5 or thickness by 6 for the situation when the slow crack growth region may be represented by a parabola.

5. Semi-c ircu lar crack front

A semi-circular crack growth region tends to form at the crack tip when substantial slow crack growth takes place under continued loading. The transition from parabolic crack front profile to semi-circular crack front profile is discussed in an earlier section. In this section, modelling average crack growth size is carried out for semi-circular front. The general equation representing the crack front as illustrated in Fig. 10 may be given as

x2 + y2 = a 2 (16)

where a is the radius of the semi-circle. In the case of a fully spread crack growth region, over the thickness (B), we have " a " equals Aama x and Eq. (16) reduces to

Aa i = ~/(Aamax) z - x/2 (17)

where i refers to any arbitrary single offset measurement and x represents the interval for offsets across the thickness.

5.1. Average by offset measurement (A~)

4.3. Comparison of A~ and A~* values

The two average values A~ and Aft* as discussed earlier need to be compared for the optimization of $. The difference between the two average values as given by Eqs. (9) and (14) is

B 1 A~ - A~* = (15)

6 ~ b + l

The difference may be seen to vary inversely with ~b and tends to 0 as ~b approaches infinity. Fig. 9 shows how I Aft - A~* I varies with ~b. The difference between the two decreases with increase in ~b. However, the effect of ~b appears to be quite pronounced when ~b lies in the range of 0 to 5. This implies that when the crack growth region is small so that the crack front profile can be described by a parabola, even a small ~b value (around 5) may give a fairly good approximation of the best average crack size as measured from area considerations.

Referring to Fig. 10, let there be ~b offset measurements including the mid-section one (0, 0) across the thickness. A similar approach for A~ estimation as followed earlier in the case of parabolic crack growth front is followed here. Equation (17) thus gives the offset length as

Aa I x,=x = ~/(Aamax) 2 --X2 (18)

~y Slow crock 9rowth zone

L (0 x r - ' -

J )2_ 2 Aa i = ( A a r n a x (x i )

Fig. 10. Schematic illustration of semi-circular slow crack growth front represented by x 2 + y2 = a 2.


Summation of all the Aa offsets in one-half of the crack growth region is given by

first two terms in the series, Eq. (19) may be reduced to

( 6 - 1) /2 (ok- 1) /2 ,

E A a i = E i (Aamax) 2 - X2 i=1 i=1

+ ~(Aamax)2 -- (3X) 2

(19)

E x p a n d i n g e a c h t e r m a n d c o n s i d e r i n g o n l y t h e

Aa i = Aarnax 1-- "~ i=l

+1

1( 2x )2) 2

1 / , ; b - 1 l 2] x + 1 - + 1 - - ~ x

" ' " 212Aamax ] ]

1(x)2 = Aamax 2 2

E E

1 0 . 0 - ~

9.0-

Experimentally observed A~ for a specimen of thickness (B ) =lOmm

=20ram

=15mm

6,0

5.C

4,0

3.0

2,C I

1.0

O.C

^ ~-, = 10 mm

=5ram

- w

B = 2 m m

l 1 J 1 l t , , I I _, t t 2 z., S a ~o 12 ~/., ~6 18 20

Fig, 11. Variation of A~ with offset measurement number ~b for semi-circular crack front profile.


× 1 2 + 2 2 + 3 2 + . . . +

1(x)2 =Aamax ~" 2 A--'~m ~

4 ) - 1 4 ) + 1 4 ) × - - 2 2 6

amax 1[ 2 = 1-- 2

4) ] (20) x 24(4) + 1)

The average A~ may be given as

1 ~ 1 ( ~ - 1)/2 |

A~ = ~ i=IE Aai = ~- Aamax + 2 i=IE Aai 1 - - - 1 + ( 4 ) - 1 ) 1 -

× 24(4) + 1) (21)

For a semi-circular crack front, B may be re- placed by 2Aama x and therefore

54) + 7 B(54) + 7) A~

Aaron6(4) + 1) 12(4) + 1) (22)

Eq. (22) is used to plot A~ at different 4) for some arbitrary B and Aama x value as may be seen in Fig. 11. The plot may be seen to become flat at 4) values around 7 to 10 for all values of B and Aam~ x considered. However, the slope of the plot may be seen to be steep for 4) values up to 5. Some experimental results are shown in the same figure to indicate the close agreement between the proposed model and the experimental points.

The dA~/d4) may be given by [Aaron,/3(4) + 1) 2] and the variation of dA~/d4) with 4) is shown in Fig. 12. The dA~/d4) values may be seen to drop very fast initially when 4) is very small. However, the plots tend to become flat for relatively higher values of 4) which depends on the thickness and extent of crack growth. This

implies that the scattering in the determination of A~ decreases rapidly with 4). It may be seen that the scattering becomes zero or very small as 4) tends to infinity. As done before, a horizontal line is drawn on the plots at dA~/d4) equal to 10 ixm. As may be seen in Fig. 12, around 8 to 12 point measurements are necessary for reasonable optimization of crack size from the proposed model for a section thickness of 10 mm.

5.2. Area consideration a~*

The area under the semi-circular crack growth front (Fig. 10) may be given as

Area = 1 2 ~4)Aama x (23)

1 arAama x rrB A~ = 2B4)Aa2ax= 4 S (24)

The difference in the average crack size value obtained by the offset method and the area method may be given by

I A ~ - A ~ * I = - ~ - 1 4)+1

[ ° 41 = 5Aama x 1 ~ + ] (25)

The difference may be seen to increase with 4) marginally and may be taken as constant as the second term in Eq. (25) is negligible as shown in Fig. 13. The difference between the two values tends to be constant around 4)= 7, indicating that A~ by the offset method is independent of 4) at 4)> 7. The initial increase in the difference between the two may be attributed to the approximation made by considering the first two terms of the series expansion

~/(Aamax) 2 --X/2

5.3. Effect of thickness and crack growth size

It is clear from Figs. 7 to 9 and 11 to 13 that section thickness as well as extent of crack growth is of great importance in the modelling of average crack growth size considering various crack fronts. Fewer offset measurement points are required


for optimum convergence when either the thickness or extent of crack growth is small. As may be seen from Fig. 8 (Y and Z), when the thickness is kept constant (10 mm), the required number of offset measurements is 9 and 12 for a variation of 0.25 mm in the extent of crack growth. Similarly, for the semi-circular crack front profile, a variation of 3 mm crack growth increases the optimum number of offset measurements points from 7 to 12 (Fig. 12, R and P) for specimens of 10 mm thickness. On the other hand, when the extent of crack growth is the same, the thicker specimen requires a larger number of measurements, e.g.,

13 and 17 for specimens with 10 and 20 mm thickness, respectively. Another important obser- vation is that when the ratio between the two quantities (B and Aama x) for each of the profiles attains the presumed value (4 for parabola and 2 for semi-circular), the required number of offset measurements points are found to be the same for both profiles, Figs. 8 (Y) and 12 (P and S). Earlier work also showed a strong dependence of extent of crack growth on offset measurement numbers [12]. For an ideal thumbnail crack front profile, a difference of two in the number of measurements (as considered, 7 in BS5762 and 9

E

O3

r 0

35

30

25

20

15

t

P B = 10 ~ a m o x = B / 2 , O B = 2 0 J

= max B / 5

I I I I 4[ 1 1 ~ 1 2 4 6 8 10 12 14 16 18

Fig. 12. Variation of dA~/dd~ versus $ for semi-circular crack growth front.

I

2O


1 5 r a m

E

I 0

I Io

5 r a m

B = 2 m m

10ram

J I I I 5 10 15 qb

Fig. 13. Variation of I A~ - A~* I with 4, for semi-circular crack front profile.

in ASTM E1290) leads to a difference of 11 to 15% in crack size and around 13% on realistic crack front profile [16].

6. C o n c l u s i o n s

The following salient points emerge from the present work on the empirical modelling of slow crack growth front and size.

1) Depending on the extent of crack growth, two distinct slow crack growth regions are identified. Relatively smaller crack growth may be described by a parabolic front, while the larger crack growth region appears to be enclosed by a semi-circular front. The proposed modelling of crack front appears to have good agreement with the experimental crack growth front for a number of structural alloys.

2) An empirical model is established to obtain the average crack growth size by the offset method as well as area considerations. The convergence rate of average Aa values with 4' is found to lie in the range of 8 to 12 for commonly used section thicknesses both for parabolic and semi-circular

crack fronts. Experimental data on average Aa values for several structural alloys closely match the values predicted from the model.

3) The section thickness as well as maximum crack growth size is found to have a significant influence on the number of measurements for a representable crack size. Higher 4, values should be taken into consideration in such situations. 4, values required for the two types of crack front cases are found to be the same provided the presumed Aama~/b ratio for each profile is satis- fied.

R e f e r e n c e s

[1] A.A. Willoughley, P.L. Pratt and C.E. Turner, The mean- ing of elastic plastic fracture criteria during slow crack growth, Int. J. Fract. 14, 449-466 (1981).

[2] P.S.E. Forsyth, A two stage process of fatigue crack growth, Crack Propagation Syrup., Cranfield, 1, 1961, pp. 76-94.

[3] T. Ohmura, R.M. Pellouse and N.J. Grant, Eng. Fract. Mech. 5, 909-922 (1973).

[4] B.F. Brown, The application of fracture mechanics to SCC, metals and materials, 2, Met. Rev. 13, 171-183 (1968).


[5] H.H. Johnson and P.C. Paris, Subcritical flaw growth, Eng. Fract. Mech. 1 (1968).

[6] BS 5762: Crack opening displacement (COD) (British Standards Institution: London, 1979) 1-12.

[7] ASTM E1290: Standard Test for Crack Tip Opening Displacement Fracture Toughness Measurement, 1989.

[8] D. Hellmann and K.H. Schwalbe, J. Test. Eual. 14, 292- 297 (1986).

[9] A.N. Kumar, Thickness effect on slow crack growth measurement, Int. J. Fract. 36, R29-R32 (1988).

[10] A.N. Kumar, On accuracy of crack size measurement, Int. J. Fract. 38, R27-R30 (1988).

[11] A.N. Kumar, Effect of physical crack growth on CTOD measurement, Int. J. Fract. 35, R3-R8 (1987).

[12] A.N. Kumar, Analytical optimization of crack size mea-

surement number in R-curve study, Eng. Fract. Mech. 29, 599-608 (1988).

[13] S. Bhattacharya and A.N. Kumar, R-curve generation and CTOD evaluation considering maximum crack growth size and parabolic crack front, J. Test. Eval.

[14] R.J. Hartranft and G.C. Sih, Local energy release rate and geometry of growth of semi-elliptical surface cracks, Computational Fracture Mechanics, eds. E.F. Rybicki and S.E. Benzley (ASME, 1975) 145-159.

[15] G.C. Sih and C. Chert, Non-self-similar crack growth in elastic-plastic finite thickness plate, J. Theor. AppL Fract. Mech. 3, 125-139 (1985).

[16] S. Bhattacharya and A.N. Kumar, A comparative study of crack size measurement, Int. Z Fract. 47, RI I -R16 (1991).

Empirical modelling of subcritical crack growth size and profile under continued loading

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Transcript of Empirical modelling of subcritical crack growth size and profile under continued loading