NASA Contractor Report 175049

Fatigue Crack Growth Under GeneralYielding Cyclic-Loading

(SASA-CB-175049) JATIGOE CBACK GROWTH ONDEB

GENEBAL-YIELDIHG,CYCLIC-LOADING (Syracuse

Univ., fi. ,Y.) .. 28 p HC A03/MF AC I CSCL 20K

N86-21951

.Onclas

G3/39_ :0:S.6.5.6_i.

Zheng Minzhong and H.W. Liu

Syracuse University

Syracuse, New York

February 1986

Prepared for

Lewis Research Center

Under Grant NAG 3-348

Nationa

NAS

l Aeronautics^n

A

d

Space Administration
FOREWORD

The work was conducted at the George Sachs Fracture and Fatigue

Research Laboratory at Syracuse University. The financial support

provided by NASA, Grant No. NAG 3-348, and a Visiting Scholarship from

the People's Republic of China for Mr. Zheng Minzhong are acknowledged.

The authors wish to thank the Academic Computing Center of Syracuse

University for making computer time available and Hibbitt, Karlsson,

and Sorensen, Inc., for the privilege of using the ABAQUS finite-element

program.
FATIGUE CRACK GROWTH UNDER GENERAL-YIELDING CYCLIC-LOADING

Zheng Minzhong,* Visiting Scholar

and

K. W. Liu, Professor of Mechanical and Aerospace Engineering

Syracuse University

Syracuse, New York 13210

SUMMARY

In low cycle fatigue, cracks are initiated and propagated under

general-yielding cyclic-loading. For general-yielding cyclic-loading,

Bowling and Begley have shown that fatigue crack growth rate correlates well with the measured AJ. The correlation of da/dN with AJ

has also been studied by a number of other investigators. However,

none of these studies has correlated da/dN with AJ calculated specifically for the test specimens.

Solomon measured fatigue crack growth in specimens in generalyielding cyclic-loading. The crack tip fields for Solomon's specimens are calculated using the finite element method and the J-values

of Solomon's tests are evaluated. The measured crack growth rate

in Solomon's specimens correlates very well with the calculated AJ.

INTRODUCTION

In strain-controlled fatigue or low cycle fatigue, cracks are

initiated and propagated under general-yielding cyclic-loading. There-

*Aircraft Strength Research Institute

Xian, People's Republic of China.
fore, linear elastic fracture mechanics can no longer be used to analyze

•'."-. ' •.:'-.,: . •-"-.'-'. *:';„/: s.•;•-.-' ' -.

fatigue crack growth rate.

Dowling and Begley [1] and Dowling [2] correlated fatigue crack

growth rate with AJ as shown in Figure 1. The value of J was evaluated

as the rate of change of the deformation work density with respect to

crack extension for through-cracks in plates. The data points in the

figure were measured with specimens under general-yielding cyclic-loading,

and the scatter band was obtained with the linear elastic fracture mechanics specimens. They agree very well with each other. Dowling [2] has

developed an equation for calculating AJ for surface cracks by combining

the finite element method calculations of Shih and Hutchinson [3] for

center-cracked and edge-cracked panels. Haddad and Mukherjee [A] and

Tanaka, Hoshide, and Nakata [5] followed the same procedure to evaluate

J and correlated J with da/dN. Kaisand and Mowbray [6] correlated fatigue

crack growth in general yielding with AJ. They divided AJ into two parts:

elastic and plastic.

AJ = AJ + AJ (1)

e P

Using Shih and Hutchinson's calculation and following a procedure similar

to Bowling's, AJ for a surface crack is approximated by

AJ =1.96 /1/n' AW a . (2)

P P

where n' = the cyclic strain hardening exponent and AW = the applied

plastic deformation work density.

Tomkins [7] and Tomkins,-Sumner, and Wareing [8] correlated crack

growth rate with crack tip opening displacement, CTOD, and J. Here, J

consists of two parts
a

E_

where n is the monotonic strain hardening exponent. However, in all of

the above investigations (1, 2, 4, 5, 6, 7 and 8), no attempt was made

to correlate da/dN data with J calculated for the specific test specimens.

Solomon [9] measured fatigue crack growth in specimens in generalyielding cyclic-loading. In this paper, the crack tip field of Solomon's

specimens will be calculated with FEM. The J values will be evaluated,

and the measured crack growth rate will be correlated with the calculated

J values.

ANALYSIS OF SOLOMON'S EXPERIMENT

Solomon measured fatigue crack growth in 1018 steel under generalyielding cyclic-loading. The chemical composition of the steel is given

in Table 1. The test section of the cylindrical specimen was reduced

by two semi-curcular notches (Figure 2a). The gross cross-sectional area

2

of the test section of the specimens was 1.24 x 12.6 mm . A sharp notch

was used to initiate the fatigue crack. Both ends of a specimen were

rigidly attached to the loading frame of the test machine. The fatigue

crack growth rates were measured at six different applied cyclic plastic

strain ranges. The applied strain range was controlled by the extensometer located 7.62 mm from the edge of the specimen as shown in Figure 2a.

The total applied strain range consists of two parts

AL = At + Ac (4)

e p

and

Ao . , _ •. •

Ar = -ET (5)
Solomon [9] plotted his crack growth data in terms of pseudo stress

intensity factor defined as

A(PK) = E(Ac) /a (6)

Subsequently, the data were analyzed by Haigh and Skelton [10] in

terms of a strain-intensity factor defined as

AK_ = (- Acg + Ac ) Aa (7)

For comparison with (7), (6) can be written in the form

+ Aep) (8)

Solomon correlated his data with the total strain range AE. In Haigh

and Skelton' s analysis, the elastic strain amplitude was used instead of

the elastic strain range because of the consideration of crack closure

when the applied stress became compression. The data and the correlations

proposed by Solomon [9] and Haigh and Skelton [10] are shown in Figure 3.

The scatter band of Solomon's correlation is a factor of slightly more

than 3, and the scatter band is reduced to a factor of 2.5 by the Haigh

and Skelton correlation.

Brown, et al. [11] calculated the plane stress crack tip field of

Solomon's specimen with FEM. In their calculation, a constant stress

boundary condition was used. The crack growth rate was correlated with

the size of the severe strain zone, r . The constant stress boundary

S

condition will introduce a bending moment at the test section, and the

bending moment could be more than that experienced by the specimen.

Solomon's specimens were tested in the strain controlled fatigue test

fixture. Both ends of the specimen were firmly attached to a rigid test
frame. Therefore, the test condition can best be simulated by a constantdisplacement boundary condition, which was used in the present FEM calculation.

For a specimen under a general-yielding cyclic-load, the entire test

section of the specimen experiences cyclic plastic deformation. The applied

cyclic stress and cyclic strain as well as the crack tip cyclic stress and

strain fields are related to each other through the cyclic stress-strain

curve.

The cyclic yield stress (AoY/2 = 170 MPa) was obtained as the intersection of two straight line segments in the log Ao/2 versus log Ac/ 2 plot,

one in the elastic region and one in the plastic region (Figure 2b). The

cyclic-stress cyclic-plastic-strain relation derived from the

data in Figure 2b is

0.26

Ae

-y (MPa) = 1360 (-f) (9)

where Ao/2 and Ae /2 are cyclic stress amplitude and cyclic plastic strain

amplitude, respectively.

The complete cyclic stress-strain history experienced by the material

in a crack tip region is rather complicated. As a crack tip advances

ahead toward a point, the material at the point experiences increasing

stress and strain ranges. As the cyclic loading goes on, the material at

the point either cyclically hardens or softens. The detailed

cyclic constitutive relation of the steel is unknown. Crack

closure even complicates the matter further. In view of the

complications of the crack tip field, it is necessary to make a

few simplifying assumptions in order to calculate crack tip

field for fatigue crack growth analysis.
6

During the loading half cycle, the material near'the crack tip in

the plane stress region near the specimen surface will be stretched.

Upon unloading, the crack front in the plane stress region will close.

\ :

Crack closure will reduce the effective stress intensity range. It is

well known that the compressive residual stress and crack closure due to

a tensile overload will cause crack growth retardation. Normally fatigue

crack growth is measured under a tension-tension load. If a tensile

overload is followed by a compressive overload, the crack retardation is

greatly reduced because the crack tip compressive residual stress field

and crack closure are washed out by the compressive overload [13]. For

the completely reversed loading of low cycle fatigue, the large compressive strain will "flatten" the crack surface and will even up the residual

stresses throughout the specimen. Therefore it is reasonable to assume

that at the beginning of the tensile half cycle, the specimen is stress

free, the crack tip will open as the applied stress becomes tensile as

suggested by Haigh and Skelton [10], and the fatigue crack growth mechanism

will become operational.

We also assume that the material at a crack tip experiences "stabilized" hysteresis loop.This assumption is reasonable if crack growth rate

:

is slow enough and the material has experienced enough number of high

strain cycles. In the small scale yielding case, the material will have

to experience several thousand fatigue cycles in the monotonic plastic

zone and, in addition, it will have to experience several thousand cycles

of cyclic plastic deformation before!it reaches the crack tip. Therefore,

the hysteresis loop of the material close to the crack tip is fully

stabilized. In the case of low cycle fatigue, the number of fatigue cycles

is much less, but the cyclic strain range is much higher. Because of the
high strain range, it is also reasonable to expect stabilized hysteresis

loop. If hysteresis loops are stabilized, the cyclic stress-strain relation can be used for the finite element calculation. One will be able

to calculate the cyclic stress and cyclic strain fields by making a

static calculation using the cyclic stress-strain relation, if the hysteresis loop is stabilized everywhere, and if the residual stress is

negligible at the beginning of the tensile half cycle.

For Solomon's specimens, the size of the plastic zone is much larger

than the plate thickness, therefore, plane stress finite element should

be used. The mesh is shown in Figure 4a. The lateral curvature of the

specimen was simulated by layers of different thicknesses, as illustrated

in Figure 4b. The detailed mesh near the crack tip is in Figure 4c. The

solid lines in 4c delineate the meshes, and the dashed curves are the

paths for J-integral evaluation. All the elements in the layer closest

to the crack line are of the same thickness, and the values of J-integral

are evaluated along the paths within this layer.

The ABAQUS FEM program was used. The plane eight noded quadrilateral

quadratic isoparametric element was used. The crack tip triangular elements

were formed by collapsing one side of the quadrilateral element.

According to Hutchinson [14], and Rice and Rosengren [15], the crack

tip stress and strain fields for non-linear elastic solids obeying the

power-law stress-strain relation, (o/o ) = a(e/e ) , are

o. .11

l+n

'o'. . (6,n)

= a

ae I {r/(J/o )}

on o

l+n

T±j- (6,n)

(10a)

(10b)
8

According to Equation 10, J is capable of characterizing the entire

crack tip stress and strain fields. However, the recent' plane-strain

finite element calculations (16, 17, 18) indicate that crack tip field

characteristics are strongly dependent on specimen geometry as well, as

load level. In other words, at the same J-value, the crack tip field may

vary widely from one specimen geometry to another. For the same specimen

shape, at the same J-value, the crack tip field in a small specimen in

deep general-yielding may differ considerably from that of a large specimen

at a limited amount of plastic deformation. According to Equation 10, if

J is capable of characterizing crack tip field, the plot of (a../a ) or

(C../E ) versus r/(J/o ) should fall on the same curve regardless of the

specimen geometry and load level..

Crack tip stresses and strains for three crack lengths, a = 1.27,

2.54 and 5.08 mm, were calculated. The data on effective stress, (a /cO

and effective plastic strain (E /E ) for a = 1.27 mm at various load levels

are plotted against r/(J/aY) in Figure 5a. All the data in the crack tip

region fall on the same curve. The slopes of the lines close to the crack

tip are 0.85 and 0.23 which are slightly higher than the values given by

Hutchinson [13] and Rice and Rosengren [14]'; 1/(1 + n') = 0.8 and

n'/(l + n') = 0.2 respectively, n' is the cyclic strain hardening exponent.

Similar plots for o and e are shown in Figure 5b. Figure 6 shows the

normalized strain distribution, (r. /EY) versus r/(J/Oy) , near the crack

tip for three different crack lengths. In the crack tip region, the data

for all three crack lengths fall on the same straight line as suggested by

Equation (10). The data in figures 5a, 5b and 6 indicate that J-integral

is capable of characterizing the crack tip field. In other words, at
. 9.

the same J-value, the crack tip stress and strain fields are the same

regardless of the length of the crack. Since crack growth rate is controlled by crack tip field, J will be able to correlate well with Solomon's

crack growth data.

Solomon's crack growth data are shown in Figure 3. At each of the

data points, one can obtain the values of da/dN, Ae and crack length, a.

AK = {(Ae /2) + Ae } /na . With Ae and Ae known, the value of the

e e p e p .

crack length can be found. With both crack length and the plastic strain

at x = 7.62 mm given, the value of J can be calculated. The results of

the calculations for three different crack lengths, a = 1.27, 2.54, and

5.08 mm, are shown in Figure 7. The values of J are plotted against

both c and e^ at x = 7.62 mm, the location of the extensometer.

yy yy

The crack growth data were measured by Solomon at six different

plastic strain levels, Ae = 0.001, 0.002, 0.005, 0.01, 0.02 and 0.05.

The J-values at these six strain levels were, calculated for the three

crack, lengths. At these eighteen combinations of Ac and a, both the

crack growth rates and J-values are obtained. The results are shown in

Figure 8.

For a strain-controlled fatigue test, the stabilized hysteresis loop

is symmetrical, as shown in Figure 9. When the applied compression is

high enough, a crack will close. The exact value of the stress or strain

at which a crack will close is unknown. It will be assumed that during

the lower half of the cycle when the applied stress is negative, a crack

will close and lose its effectiveness as suggested by Haigh and Skelton [10]. Only the applied J corresponds to the positive.part of the

loading cycle (ABC in Figure 9) are used for crack growth data correlation.
10

In this part of the loading cycle, the relation between stress and

strain follows the curve-; ABC. This: segment of the stress-strain curve

consists of the elastic and- plastic parts. The plastic part is Ae and

the plastic part is Ae /2. The value of AJ for Solomon's specimen was

evaluated at the plastic strain range Ae tested, and it corresponds to

the AJ along the loading curve ABC in.Figure 9.

The stress-strain relation along ABC differs slightly from the

cyclic stress-strain curve. However, the cyclic stress-strain curve was

used to calculate the crack tip field for Solomon's test. The values

of J are strongly dependent on strain and much less on stress, therefore

this approximation will introduce a rather small error. Furthermore,

the stress-strain relation along ABC depends on the applied strain range.

It varied from one specimen to another and it is unknown.

The correlation of da/dN with AJ for Solomon's test is shown in

Figure 8. The correlation is better than with either A(PK) or AK

as shown in Figure 3. The data in Figure 8 give the empirical relation

^f (mm/cycle) = 0.7 x AJ1'

7

 (11)

ON

for Solomon's 1018 steel. AJ is in MPa-m. Equation (11) is also plotted

in Figure 1 as the dashed line. The data for 1018 steel are within the

t

data band of A533B steel. It is well known that fatigue crack growth

data of low and medium carbon steels are all close together.

However, in order to use the correlation for engineering applications, it is desirable to compare the measured and calculated AJ.

CONCLUSIONS

In general-yielding cyclic-loading, fatigue crack growth rate correlates very well .with the calculated AJ values for specimens tested at

six different cyclic strain ranges. In order to predict fatigue

crack growth rates in engineering structures, it is necessary

to establish the correlation between da/dN and calculated AJ.
11 .

REFERENCES

[1] Dowling, N.E. and Begley, J.A., "Fatigue Crack Growth During Gross

Plasticity and the J-integral," Mechanics of Crack Growth, ASTM STP,

Vol. 590, 1976, pp. 80-103.

[2] Dowling, N.E., "Crack Growth During Low-Cycle Fatigue of Smooth

Axial Specimens," Cyclic Stress-Strain and Plastic Deformation

Aspects of Fatigue Crack Growth, ASTM STP, Vol. 637, 1977, pp. 97-121.

[3] Shih, C.F. and Hutchinson, J.W., "Fully Plastic Solutions and Large

Scale Yielding Estimations for Plane Stress Crack Problems," J.

Engineering Materials arid Technology, 1976, pp. 298-295.

[4] El Haddad, M.E. and Mukherjee, B., "Elastic-Plastic Fracture Mechanics Analysis of Fatigue Crack Growth," Elastic-Plastic Fracture:

Second Symposium, Vol. II - Fracture Resistance Curves and Engineering

Applications, ASTM STP Vol. 803, 1983, pp. 11-689 - 11-707.

[5] Tanaka, T. Hoshide, and Nakata, M., "Elastic-Plastic Crack Propagation Under High Cyclic Stresses," Elastic-Plastic Fracture: Second

Symposium, Vol. II - Fracture Resistance Curves and Engineering

Applications, ASTM STP Vol. 803, 1983, pp. 11-708 - 11-722.

[6] Kaisand, L.R. and Mowbray, "Relationships Between Low-Cycle Fatigue

and Fatigue Crack Growth Rate Properties," Journal of Testing and

Evaluation, JTEVA, Vol. 7, No. 5,. Sept. 1979, pp. 270-280.

[7] Tomkins, B., "The Development of Fatigue Crack Propagation Models

for Engineering Applications at Elevated Temperatures," Journal

of Engineering Materials and Technology, Trans, of ASME, Series H.,

Vol. 97, 1975, pp. 289-297.
• 12

[8] Tomkins, B., Sumner, G., and Wareing, J., in Inter. Symposium on

Low Cycle Fatigue Strength and Elasti-Plastic Behaviour of Materials,

Deutscher Verband fUr Material PrUfung, Berlin, 1979, pp. 495-507.

[9] Solomon, H.D., "Low Cycle Fatigue Crack Propagation in 1018 Steel,"

J. of Materials, JMLSA, Vol. 7, No. 3, Sept. 1972, pp. 299-306.

[10] Haigh, J.R. and Skelton, R.P., "Fatigue Crack Growth Rates and

Thresholds in Steels under Oxidising Conditions," Materials Science

and Engineering, Vol. 36, 1978, pp. 17-25.

[11] Brown, M.W., Liu, H.W., Kfouri, A.P., and Miller, K.J., "An Analysis

of Fatigue Crack Growth Under General Yielding Conditions," Advances

in Fracture, edited by D. Francois et al., Pergamon Press, Oxford

and New York, 1980.

[12] Zheng, Minzhong and Liu, H.W., "Crack Tip Field and Fatigue Crack

Growth in General Yielding and Low Cycle Fatigue," NASA Contract

Report 174686 to Lewis Research Center, Sept. 1984.

[13] Schijve, J. and Broek, D., "Crack-propagation-test based on a gust

spectrum with variable amplitude loading," Aircraft Engineering,

34 (1962) pp. 314-316.

[14] Hutchinson, J.W., "Singular Behaviour at the End of a Tensile Crack

in a Hardening Material and Plastic Stress and Strain Field at a

Crack Tip," J. of Mech. and Phys. of Solids, Vol. 16, 13, 1968, p. 377.

[15] Rice, J.R. and Rosengren, G.F., "Plane Strain Deformation Near a

Crack Tip in a Power-Law Hardening Material," J. of Mech. and Phys.

of Solids, Vol. 16, 13, 1968, p. 1.
13

(16) McMeeking, R.M. and Parks, D.M., "On criterion for J

dominance of crack tip fields in large scale yielding",

ASTM STP 668, (1979) 175-194.

(17) Shih, C.F. and German, M.D., "Requirement for a one

parameter characterization of crack tip field by the

H.R.R. singularity", Int. J. of Fracture, 17, (1981), 27-43,

(18) Liu, H.W. and Zhuang, Tao, "A dual-parameter elasticplastic fracture criterion", Accepted for publication

in the Int. J. of Fracture, 1985
oemovD

14

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15

AJ, (MPo-m)

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ID'

.-2

10"

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ELASTIC-PLASTIC TESTS

DOWLJNGS TEST, A533B STEEC

A

 CENTER CRACK ED SPECIMENS

W = l"

o COMPACT SPECIMENS

W=2 " -

SOLOMON'S TEST,

1018 STEEL

I I

10

.2

10

-5

E

^o

10"'

.6

10

IO'7

10' I01

in-lb/in2

TO5

Figure 1. Fatigue crack growth rate versus AJ [1]
1

6

Q>

0>

•o

(ft

SuVaCOeooCO

14-1ococo•HCOc<uB

OOCO

(N0) M00
17

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 o

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O o —

bf oh

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18

EAK« EQUIVALENT ELASTIC STRESS INTENSITY

FACTOR, MNm-S/2

20 50 100 ZOO 500 1000

•! 10"

0

>»

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H

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I

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--- OKI6INAL DATA

USIN8 TOTAL STRAIN

CONVCMTfO DATA

USIN6 rOUIVALTNT

CLASTIC STRAIN

A«p+ 1/2 Ace

I i _ I

2 5 10 20 50 100

AK, .STRAIN INTENSITY AMPLITUDE I0'4m'/*

Figure 3. Comparison of equations (7) and (8) using Solomon's

data [10].
19

FRONT VIEW SIDE VIEW

F

Crack—

%

i

'

2.6n

(i

r\m ^

O

•

-'

^•4

N

x

l ,

-.

,>"^

V

v

N , V

\

1

v

\

F=M 1.24mm

n

• /

0

\

\

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3)

Crack

Figure 4. Finite element idealization for Solomon's specimen,

showing (a) top half of the specimen, (b) cross section,

and (c) crack tip elements.
20

10

10'

10

10

o = 1.27mm

10

-2 10' 10'

Figure 5a. Normalized effective plastic strain (t ) and effective

stress (o ) distributions ahead of the crack tip for

Solomon's specimen.
2

1

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22

10'

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1 1

I • I

J I

I I T

A o = 127 mm

o a = 2.54mm

D a « 5.08mm

D

-i

10° 10'

Figure 6. Normalized strain (e ) distribution near the crack

tip for three different crack lengths.
23

a

CL

10

rl

id2

-3

10

10"

0

(mm)

1.27

2.54

5.08

Stroin «vv

PI o Stic

Q

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A

Total

•

•

A

I I I t I

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i

.a

10'

10" 10

r2

*yy,'yy

Figure 7. J-integral dependence on applied strains at x = 7.62 mm,

the location of the monitoring extensometer.
24

w,

Figure 8. Cyclic stress-strain loop and the deformation work

density, W .
25

AJ,MPo-m

10'

10' •icri

10

r3

o>

1

o

T3

10

-5

10

,-6

1 1 1 1 1

J" in MPo -m

DO -127mm

oo =2.54 mm

AO =5.08 mm

/O'

i i i 1 i i i 1 j_ i I i, .•

10

i i i i

D/

/O'

AJ, /b-in/m'

/o10'

I

10

•3

10'

,..

/o1

Figure 9. Correlation of da/dN versus AJ for Solomon's tests.
1. Report No.

NASA CR-175049

2. Government Accession No. 3. Recipient's Catalog No.

4. Title and Subtitle

Fatigue Crack Growth Under General-Yielding

Cyclic-Loading

5. Report Date

February 1986

6. Performing Organization Code

7. Author(s)

Zheng Minzhong and H.W. Liu

8. Performing Organization Report No.

None

10. Work Unit No.

9. Performing Organization Name and Address

Sycrause University

Dept. of Mechanical and Aerospace Engineering

Syracuse, New York 13210

11. Contract or Grant No.

NAG 3-348

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Washington, D.C. 20546

13. Type of Report and Period Covered

Contractor Report

14. Sponsoring Agency Code

533-04-12

15. Supplementary Notes

Project Manager, Jack Telesman, Structures Division, NASA Lewis Research

Center, Cleveland, Ohio 44135. This research was sponsored by the HOST

Program.

16. Abstract

In low cycle fatigue, cracks are Initiated and propagated under general-yielding

cyclic-loading. For general-yielding cyclic-loading, Dowling and Begley have

shown that fatigue crack growth rate correlates well with the measured AJ. The

correlation of da/dN with AJ has also been studied by a number of other

Investigators. However, none of these studies has correlated da/dN with AJ

calculated specifically for the test specimens. Solomon measured fatigue crack

growth 1n specimens in general-yielding cyclic-loading. The crack tip fields

for Solomon's specimens are calculated using the finite element method and the

J-values of Solomon's tests are evaluated. The measured crack growth rate 1n

Solomon's specimens correlates very well with the calculated AJ.

7. Key Words (Suggested by Author(s))

Fatigue crack growth; Elasto-plastlc;

J-1ntegral

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Unclassified

20. Security Classif. (of this page)

Unclassified

21. No. of pages

26

22. Price'

A03

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