A study on the fatigue crack growth in 9% Ni steel plate

.E~i~~inmhtg Fmctun

Mduuics.

1975. Vol. 7. pp. 4&4’W.

Pcrpm

Press.

Primed in Great Britain

A STUDY ON THE FATIGUE CRACK GROWTH IN 9% Ni STEEL PLATE GROWTH RATE OF SURFACE CRACK IN A PLATE UNDER ARBITRARILY COMBINED TENSION AND BENDING A. NAGAI, M. TOYOSADA and T. OKAMOTO Tc&ical

ResearchInstitute,Hitachi ShiiuildingandEngineer@ Co., Ltd., Osaka,Japan

Abstract-Cyclic growth rate of a surface crack is complicatedby the three dimensionalstressfield around the crack and also by an interactionexpectedbetween the growth rate at every positionon the contour. An analysis, based on the total energy around a smfoce crack, derived a correlation between strain energy rekase rate and cyclic increase in cracked area. And the correlation was confirmed by experiments conducted,with 9% Ni steel plate, under tension, bendingand combinedtension and bending.

1. INTRODUCTION IN THEpast two decades an enormous effort has been devoted to the studies on the fatigue crack growth. In 1%3, Paris and Erdogan[l] found a correlation between stress intensity factor and crack growth rate in aluminum ahoys. A number of metallic materials has since been tested and the similar correlation has been recognized. Most of the preceding studies were fundamental. Their major purpose was related with mechanism of the fatigue crack growth or the fundamental description of the growth rate. Therefore, theoretical models or specimens used in the studies were generally simple. A major part of these studies treated a through-thickness crack in a plate under cycling of uniform tension and the crack growth was analysed uni-axially. In application of the fatigue crack growth analysis to usual structures, further development is necessary. Variation in shape of crack and type of load shall be considered. In considering crack growth in tanks or pressure vessels, it is important to know the crack behaviour until it penetrates through the plate thickness. In this paper we will discuss the growth rate of a surface crack in a plate which is subjected to cyclic loadings arbitrarily combined tension and bending. 2. BASIS OF CONSIDERATION Since Paris and Erdogan’s analysisHI, crack growth rate of a number of metals and alloys has been investigated and following correlation has been ascertained;

where u = a half length of crack, n = number of stress cycles, K = range in stress intensity factor, and C, and m , = constants. In the case of surface crack, however, unified correlation between stress intensity factor and crack growth rate in the uni-axial analysis cannot be obtained, as stated later. The constants m , and C, in eqn (1) are not same between the growth rate under different types of load. Therefore we cannot use the eqn (1) for prediction of behaviour of surface crack in usual structures subjected to combined loads of tension and bending. One of the probable cause of the difference in ml and C, is an interaction between the crack growth rate at different positions on the crack contour. Therefore, bi-axial treatment will be required for the growth rate of surface crack. In the bi-axial treatment, it will be reasonably considered that the cyclic increment of cracked area is correlated to the total amount of contributory energy for crack growth. As the contributory energy is to be strain energy release rate G, growth rate in area, dA/dn, or mean 481

482

A. NAGAI. M. TOYOSADA and T. OKAMOTO

crack growth rate, d(Als)ldn, will be expressed as (2) or 1 dA 31y sdn

3 C,(AG,,)“J,

(2’)

where A = cracked area,

s = tine element of crack contour, and C2,C1,m2 and m3 = constants. In the following chapters, experimental results are analysed with this consideration and usual method by qn (1). 3. -AL PROCRDURJI The growth rate of surface crack was investigated by tests under three types of load cycles; tension, bending and combined tension and bending. Materialused is !@ Ni steel plate of which chemical composition and mechanicalproperties are shown in Table 1. The shape and size of the specimens are shown in Fig. 1. Table I. !3eeipiateuacdforcxpriment dlemial compoah

(%)

U.S. Wmm’

75

Y.S.

)

er.

Wrndf

(%)

68

30

Fii 1. Shape of specimen (dimensions in mm).

The cyclic bending test was conducted with a 860 kg-m bending fatigue machine. The generation of axial force in the specimen is prevented by steel-leaf springs in the supporting system (Fig. 2b). Cyclic tension test was conducted with a lOO-tonservo pulser. Fatigue test under combined tension and bending was carried out by setting Ron lateral bending equipment on the specimen put on the IO&tonservo pulser (Fig. 2a). The lateral force was synchronized with the axial force.

483

A studyon the fatigue crack growthin !J%Ni steelplate

Fig. 2. Schematicillustrationof testingmethod.

Loading condit~ns are shown in Table 2. The surface stress of tbe test under the combined tension and bending was measured by strain gages put on tbe both surfaces at IOmm inwardfrom the edges of the specimen. The stress value of the other specimens was measured by load-cellof the fatigue machines. Table 2. Testingconditions rype of w

kmd

wrrrce rtlsa Gr&nrn~)

r&o or*

cacksidc

back side

bend@ stras

frequency

ntu. min.

max. min.

AS1

tlmsion

26

0

26

0

0

8

ES1

bendin@

43.2

0

-43.2

0

i

8

30

0

Il.5

0

0.206

3

33.1

0

0

-2.16

0.535

2.5

CSI cS2

tctutoa & kndii tension h bending

fHtt

note: Bach mukr were formed by incmait@ the ntinimum stress to I half of tbc m&mum *tress in a per&d of 2000 cyeler after every 6tXtO(5000 for BSI f of the testing cycles. ‘; bmdimJSbBsrm rtresa+tmmLmue stress)

To measure the crack growth rate in the depth direction, beach marks were formed by increasing the minimum load of the cycle to a half of tbe maximum for a period of 2000cycles periodically. The crack length on the surface of specimens was measured by crack gages or photography as shown schematically in Fig. 3.

Fig. 3. Measuringdevicefor crack kngth.

4a4

A. NAGAI. M. TOYOSADA and T. OKAMOTO

4. EXPERIMENTAL DATA The crack growth rate in the width direction measured on the surface is shown in Fig. 4 with relation to the range in stress intensity factor on the surface. Those data in the depth direction measured at the center of crack are shown in Fig. 5 in the same manner. The stress intensity factor was calculated using Shah and Kobayashi’s analysis for semi-elliptical crack[2]. In the calculation of stress intensity factor on the surface, the magnification factor for the longer axis of buried elliptical crack in a semi-infinite body was applied with the correction for free surfaces(31. In case of the combined tension and bending, the sum of stress intensity factor for tensile component and bending component was used. In Fii. 5 linear correlation between AK and dbldn is lost in the range where the plastic zone formed in front of the crack penetrates through the plate thickness. For this range elastically calculated K cannot be applied. The range is shown by semi-solid plottings for each specimen in Figs. 4 and 5, and later in’Figs. 7 and 8. This limit was approximately calculated (Appendix) for each specimen with the formula for edge cracked strip, presented by Gross and Srawley[4] and the corresponding crack depth is shown in Table 3. From the calculation and also experimental data the maximum crack depth where the crack growth rate can be correiated to the elastic stress intensity factor is about a half of the plate thickness, in these experiments. As the prediction of the growth behaviour of the deeper crack is also important, further discussion is given in the following chapters. Besides, in Figs. 4 and 5, some difference is found in the crack growth rate of each specimen even in the range where the plastic zone does not reach the back surface. These results, as they are, are not satisfactory for the basis of the prediction of fatigue crack growth in usual structures. 5. APPROXIMATE ANALYa FOR DEEP SURFACE CRACK An analysis was attempted for the growth of a deep surface crack of which plastic zone penetrates through the plate thickness. When the substantial part under- surface crack has

0 0

b.:crack depth b, : crack depth wh.81 th piatlc am ranch bacfl surhx

OA

1W

mnge in stress intensity tutor, AK Ikgjmm “*) Fii. 4. Crack growth rate at surface.

No0

485

A study on the fatigue crack growth in 9% Ni steel plate apaclmn

b(b,

b,
AS1

0

a

BSl Chl cm

0

0

A 0

A 0

1.0

ttttk (1)

sacFig.4

(2) Solidlines show the 0.6

same range with Fii. 4

loo

1000

nn#e ht ltar )ll&dtyfactor,AKNmm)") Fig.5. Crack growth rate in depth direction.

Table 3. Crack depth when plastic zone ahead of crack reaches to back surface apdmen

crack depth plate thickness

g;

E 0.49

csz

0.51

comparatively small resistance to the applied load, we may neglect the contribution of the part. Then, the deep surface crack can be approximated by a through-thickness crack when we calculate the stress intensity factor on the surface. The stress intensity factor for a through-thickness crack in a finite plate under tension is given by

(3) where u= gross stress, and B = a half width of the plate. For bending stress component, u = 0-50~ (bending) was applied likewise Erdogan and Roberts 151. On the other hand, for the crack growth in depth direction we assumed a variable which is corresponding to stress intensity factor in a range of the small scale yielding. From the trends of crack growth behaviour shown in beach marks in Fig. 9. corresponding K was assumed as Figs. 6(a) and &I). In the assumption the corresponding K for tensile stress is fixed to the value at

A. NAGAI, hf. TOYOSADA and T. OKAMOTO

(a)

(b)

tension

bending

Fig. 6. Assumption of corresponding K after plastic zone reaches back surface.

which the tensile plastic zone penetrates through the plate thickness, and for bending stress the corresponding K decreases linearly to zero with the increase of the crack depth because the crack growth stopped near the back surface as noted in Fig. 9(b). For the combined load of tension and bending, corresponding K was calculated by superposing those for tensile component and for bending component. The modified plottings are shown in Figs. 7 and 8. In Fig. 7 the whole data from initial to linal failure are rearranged. The same correlation in daldn vs AK is found regardless before or after crack penetrating the plate thickness, except for the early stage where the plastic zone does not reach to the back surface. The fact shows the effectiveness of the approximation by eqn (3) for the deep surface cracks. A remarkable improvement is found in the correlation between the crack growth rate and stress intensity factor in the depth direction in Fig. 8. However, it is noticed that the crack growth rate still depends on the type of load. 6. INCREMENT IN CRACKED AREA In the uni-axial analysis in the preceding chapters we obtained no decisive unified correlation in the data of different type of load. One of the reasons will be the existence of aforementioned interaction. I

-note b : crack depth h,,: cnckdepthwhen the pintic ZOM reach= back surface t : plate thickness

Noo

Nu

nye

in strem intensity f&or,

AK (kg/mm’“)

A studyon the fatigue crack growthin 9% Ni steelplate

487

1.0

gg

0.5 E K

See Fig. 7.

i a I # \

0.1

F 0.0:

00 *

0

0" 0

0 0.0:

/

0

I, 10 nqc

in ua

iaMJty

~ ;

100

raeto& hK org’mm’*)

Fig. 8. Creek growthrate in depth direction(AK is calculatedas shownin Fig. 6).

Here, we will show an analysis based on the consideration that the increment in the cracked area is related with the total energy release rate of surface crack. The calculation of G-value for semi-elhptical surface crack seems very complicated. It will be, therefore, not practical to caIculate the value in every stage of crack growth. It is known that G is proportional to K2 for a through thickness crack. The same correlation was shown for buried penney shaped crack[6). Thus, we believe the same correlation for semi-elliptical surface crack;

G=zKK2, E

(4)

where K= 1 for plain stress, (1 - u2) for plain strain, E = Young’s modulas, and Y= Poisson’s ratio. As shown in beach marks in Fig. 9 the crack growth rate along the contour increases or decreases monotonously from the surface to the center of the crack. Therefore, as a primary approximation for the calculation of the mean Gvaluc on the crack contour, we took mean of G-value at the surface and at the center, as G-

3t t[G(surface) + G(depth)l.

(3

K value in eqn (4) was calculated from Shah and Kobayashi’s analysis[2] for the early stage of crack growth and the modified value, as shown in the previous chapter, was taken for the later stage. In calculating eqn (4) K was assumed always to be (1 - u’). Increment of the cracked area was calculated as a semi-ellipse from the increment in depth and length of the crack. The result is shown in Fii. 10, in a relation between mean G vs mean crack growth rate on the crack contour. All data are plotted on a narrow range in the figure. As the eqns (2) and (2’) are

A. NAGAI, M. TOYOSADA and T. OKAMOTO

II 2 -L ‘ ia) As1

10mm

,

(uMion)

(b) 881 NEWT) Fig.9. Exampksofbcachmarks.

0.5

i

Fig.10.Meancrackgrowthntc.

equivalent. we can believe the correlation between the total energy release rate and the increment of the cracked area for surface cracked plate under arbitrarily combined tension and bending. In the practical application of the analysis, we can And the approximate growth rate on the surface from Fii. 4. Then using Fig. 10 the growth rate in the depth direction is calculated. After plastic zone ahead of the crack reaches to the back surface, Fig. 4 shall be replaced by Fig. 7.

489

A studyon the fatigue crack growth in 996Ni steelplate Table 4. Comparisonof estimationand experiment for life and aspect ratio of CSl plate

initial

initial

thick.(t)

dcpth(bo)

length(2a.)

12 mm

3.1

mm

6.2

final length (2~)

mm

u,. 29.4

3”

no. of cycles spent

-$-

exp.

Cal.

exp.

Cal.

20.6

0.408

0.420

76OMI

exp. 85ooO

By the method mentioned above the crack growth behaviour of specimen CSI was estimated. In Table 4 calculated number of cycles required for the penetration of crack to the back surface and aspect ratio of the crack shape at that time are compared with experimental values. Good agreement is found in the calculated and experimental values as shown in the table. Thus, the method will be useful in the prediction of the growth of surface crack in a plate subjected to the arbitrarily combined tension and bending. 7. CONCLUSION A study on the growth rate of surface crack in a plate under arbitrarily combined tension and bending was made. The conclusions obtained are as follows: (1) Cyclic increment of cracked area in a semi-elliptical surface crack has an unified correlation with total energy release rate regardless the type of load. (2) After the plastic zone ahead of a semi-elliptical surface crack penetrates through the plate thickness, the crack growth rate on the surface can be approximated by that of through-thickness crack. Ac&nawludgemm~s-To Prof. Emeritus Hiroshi Kihara, Prof. Takeshi Kanaxawa, Prof. Kunihiro Iida, and AssociateProf. SusumuMachida, of the Faculty of Engineer&. Tokyo University, who contributedso much to the executionof the present studywith invaluablepertinentadviceand &dance. we would like to expressour sincerethanksand appreciationhere.

RWERENCES P. C. Paris and F. Erdogan, Trolls.ASIKE Ser. D 85,528 (1%3). I21 R. C. Shah and A. S. Kobayashi, ASZ’MSTP 513, 3 (1971). [3] T. K anamw S. MachidaandT. Miyata, Dot. to Comm. of TestingAf~lredd,&pan WeldingSot. aOr.TM-104(1973) (in

[ll

Japanese).

(41 B. Gross and J. E. Srawky, NASA Tech. No&, D2603 (1%5). 151 F. Erdogan and R. Roberts, Proc. Intem. Conf Fmctun, Sendai, 1, 341 (1965). 161C. R. Irwin, Tmns. ASME Ser. E, 27, 651 (1962). [7] A. A. Wells, Br. Weld. J. 10, 855 (1%3). 181 A. Ando et et., IIW Dot. XIII-69573 (1973).

APPENDIX M’itOXiMATECALCULATION F’LASTIC

OF CRlTlCAL CRACK DEITB REACHES BACK SURPACE

ZONE

WREN

THJI

The critical crack depthwhen the plastic zone reachestbe back surface,was approximatedby the calculationfor an edge crack strip. Stress intensity factor K for an edge crack is shown by Gross and Stuwky[4] as

gn -~~1~99-0~41~+18~7~2-38~~‘+53~8~4) + 1297/3’- 23*17/3’+

9, - <&199-2.478 where

(A.11 24*8fl’),

t = plate thickness,b = crack depth, fl = b/t,

crm= tend

stress component, and ub = bending stress component. The elastic stressfield in front of a crack is given by

(r=KKIVG,

(A.2)

where r = distancefrom crack tip. On the other hand, plastic zone size p is givenf7] as P = 2r, where r, is a distance given by substitutingo = o, (= yield stress) into eqn (A.2).

(A.3)

A. NAGAI,

490

M.

TOYOSADAand T. OKAMOTO

When the plastic zone reaches the beck surface. the critical crack depth b, is given by (A.4)

b,=t-p, From eqns (A.l)

through(A.4), we obtain

whereb in the term g,, and p is qunkud

to b,. Thus. required WAC b, is obtaimd by numericalcalculationof eqa (A.3. In the cakxWon of the critical crack depth b, for bending component,we assumed p=r,

and

2p=t-b,.