Mixed mode crack growth in SM45C and Z02-NKDT steel subjected to repeated impact and fatigue

ELSEVIER Theoretical and Applied Fracture Mechanics 21 (1994) 1-8

tkxfl and

mechanics

Mixed mode crack growth in SM45C and Z02-NKDT steel subjected to repeated impact and fatigue

S.B. Lee a,*, S.I. Noh b,1

a Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Taejon 305-701, South Korea b Samsung Heavy Industry, Ltd., Changwon, South Korea

Abstract

The strain energy density criterion first proposed by Sih accounts for both the rate and direction of crack extension subjected to mixed-mode loading regardless of whether it is applied statically, dynamically or cyclically. This criterion is applied to investigate the mixed mode crack growth behavior in SM45C structural steel and Z02-NKDT maraging steel specimens under repeated impact and fatigue. The specimen contains an off-axis crack whose angle of inclination can be varied with reference to the loading axis to simulate different degrees of mixed loading. A Maekawa type impact tension machine is used. The direction and rate of crack growth under repeated impact and fatigue are measured and compared well with those predicted from the strain energy density criterion.

1. Introduction

Structural members and machine elements are frequently subjected to repeated impact loads. Pile hammers, valve springs and high speed printer elements are just some examples. Failure initiation by fracture is a major concern. Opti- mum design requires a knowledge of the crack growth behavior such that the governing parame- ters can be combined to ensure safe operation and minimize unexpected malfunction.

Repeated impact response differs from that of fatigue in that dynamic wave propagation effects come into play [1-3]. The resulting damage could

be more severe than fatigue loading [4-7] for the same number of cycles. Mixed mode crack growth behavior under repeated impact has not been investigated and is undertaken in this study. Use is made of the Maekawa type [1,2] impact tension machine to collect crack growth data where the orientation of the initial crack plane could be altered with reference to the load axis. Analytical predictions on the direction and rate of crack extension are made by application of the strain energy density criterion.

2. Theory of strain energy density factor

* Corresponding author. x Previously at Korea Advanced Institute of Science and

Technology.

Elsevier Science B.V. SSDI 0167-8442(94)00018-V

2.1. Basic assumpt ions

The strain energy density (SED) criterion [8] is applied in this study in an effort to unify the

2 S.B. Lee, S.L Noh / Theoretical and Applied Fracture Mechanics 21 (1994) 1-8

fracture mechanics approach to mixed mode repeated impact and fatigue loading. It focuses attention on the intensity of the local strain energy density field and has the capability to determine the direction of crack growth. Crack extension is assumed to occur in the direction of minimum strain energy density function, the intensity of which can be designated by S for a fixed radial distance r. An equivalent statement would be for S to be a minimum of Smi n. Initiation of crack is assumed to prevail when Smi n attains a critical value, say S c.

According to Sih [8], the intensity of the strain energy density field in the vicinity of the crack tip, S, may be expressed in terms of the Mode I and Mode II stress-intensity factors K~ and K2, respectively, as

S= (allK2 + 2a12K1K2 + a22K2)/qr (1)

where the coefficients a;j (i, j = 1, 2) depend on the angle 0 shown in Fig. 1 and they are given by

1 a,, = ~ [ ( 1 + cos 0 ) ( K - cos 0)]

1 a u = 1--~s in 012 cos 0 - (K - 1)] (2)

1 a 2 2 - 16G [(K + 1)(1 - cos 0)

+ ( 1 +c os 0)(3 cos 0 - 1)]

Here, G is the shear modulus of elasticity and

G

# 4-- W ID

(y

Fig. 1. Uniaxial extension of an inclined crack in a finite width specimen.

K = 3 - 4v for plane strain and (3 - v) / (1 + v) for plane stress with v being the Poisson's ratio.

The stress intensity factors K~ and K 2 for an inclined crack of length 2a in a panel with finite width W can be written as [9,10]:

g 2 =F 2 ---~-, /3 o-af~ (3)

in which cr is the uniaxial tensile stress and /3 is the angle of inclination as shown in Fig. 1. Ex- pressions for the functions Fj(2a/W,/3) ( j = 1, 2) in Eqs. (3) are given in [9]. For /3 = 90 °, F 2 = 0 and F 1 takes the form [11]:

FI(_~) = { 1 [2a~ 2 2a --0.025~--~) +0.06(-~) 4}

× see - -~ (4)

2.2. Direction of crack growth

Let 0 o denote the direction of crack initiation. Application of the strain energy density criterion requires S to be a minimum, i.e.,

dS d2S d---if=0, d0-----5->0 for 0 = 0 o (5)

For an infinite sheet with a crack inclined at an angle /3 with the direction of uniaxial load, F 1 and F 2 are given by [8]

F 1 = sin 2 fl, F 2 = sin /3 cos /3 (6)

Substituting Eqs. (6) into Eqs. (3) and the results into Eq. (1) yield S. The first condition in Eq. (5) can be applied to yield

(K - 1) sin(0 o - 2/3) - 2 sin[2(0 o - / 3 ) ]

- sin 200 = 0 (7)

The critical value of S or S c can be obtained from a Mode I test as [8]:

S c = W ( 1 - 2 v ) for 0 =0 ~ (8) ,+t1~

S.B. Lee, S.L Noh / Theoretical and Applied Fracture Mechanics 21 (1994) 1-8 3

in which Kic is the ASTM valid plane strain fracture toughness value.

2.3. Crack growth under repeated loading

A crack growth relation for repeated loading based on the accumulation of the energy density hysteresis can be found in [12]. The crack growth rate d a / d N is given by

da d'-N = A A S (9)

in which AS must be computed from a contin- uum mechanics theory that accounts for damage accumulation and energy dissipation. In Eq. (9), only a single parameter A is needed which can be obtained from a uniaxial repeated loading test. I f AS in Eq. (9) is to be related to the stress intensity factors K 1 and K 2 where no damage accumulation effects are considered, then the approximation requires the addition of another parameter. That is, d a / d N would be expressed by

da d U = B ( A S ) m (10)

In this case, the range of AS becomes

AS = Smmin ax - Smmifn n (11)

with Smin and rain max Smi n being, respectively, the maximum and minimum of the minimum strain energy density factor, i.e.,

Smm~n --- S( O o, O'max), Smmii~ = S( O o, O'min) (12)

Note that the superscripts max and min on S correspond, respectively, to O'ma x and ~rmi n while the subscript min refers to the direction of crack growth 0 o as determined by Smi n as in the case of static loading.

Actual C r a c k - Shape

Assu s%edeCrac k ~ .

13 c

9" Aft

Fig. 2. Incremental crack growth of an inclined crack.

2.4. Equivalent crack length

A method for computing approximate expressions of K 1 and K 2 for a kinked crack has been developed in [13]. A brief account of this approach is given. Once the crack grows non-self similarly by an amount Aa, oriented at angle 0 o with respect to the original crack plane O A (Fig. 2), an equivalent crack OB may be defined. It attains a new angle /31 = /30 + A/3 and half crack length a 1. The expressions for /31 and a I are given in [13] as

Aa I sin 0 o /31 =/30 +

a 0 + Aa I cos 0 o '

Aa 1 + a 0 cos 0 o a I = a 0 + Aa 1 (13)

a 0 + Aa 1 cos 0 o

For each crack growth increment Aaj ( j = 1, 2, etc.) a new set of stress intensity factors K 1 and K 2 and hence strain energy density factor S can be calculated. A new direction of crack initiation is found each time by application of the first condition in Eqs. (5).

Table 1 Mechanical properties of the steels used in tests

Steel Young's Yield Ultimate Area Rockwell type modulus strength strength reduction hardness

E (GPa) % (MPa) tr u (MPa) RA (%) R A

SM45C 202 447 662 33.2 54.6 Z02-NKDT 380 2045 2165 8.5 76.3

t:3 f ~12

I :'h I i

200 . . . . . . . . . ~'-

3. Repeated load tests

3.1. Specimens and materials

Two materials are selected for the tests; they correspond to SM45C structural steel and Z02- NKDT maraging steel. The mechanical properties of these steels are summarized in Table 1. Two types of specimen are used: one for the regular fatigue tests with sinusoidal and trapezoidal wave form as shown in Fig. 3 and one for repeated impact as shown in Fig. 4. In both cases, the inclined crack is 6 mm long and is made by using a 0.3 mm diameter wire. A MTS servohydraulic machine is used for precracking. The specimen surface is polished with emery paper no. 1200 and diamond compound such that the crack length and direction of growth could be accurately determined from the optical microscope. Test data are obtained for/3 = 30 °, 45 ° and 90 ° .

Fig. 4. Inclined crack specimen for repeated impact load (dimension in mm).

pulses recorded by gages pasted on the specimen center line 30 mm from the mid-section. A strain amplifier and an oscilloscope are used for this purpose. Fig. 6 shows a typical stress wave form for the impact load while Fig. 7 shows the trapezoidal wave generated by the MTS servohydraulic machine.

4. Di scuss ion o f results

3.2. Test machine

t=3

4.1. Crack growth data

A special repeated load test machine is con- structed; it is similar to that of the Maekawa type [1,2]. Modifications are made to enhance the con- tinuous and stable application of impact tension. A schematic of the machine is given in Fig. 5. The magnitude of the impact force can be varied by adjusting the spring constant a n d / o r the spring compression length with the cam mechanism which is driven by a 3 hp variable speed motor. As the cam passes through the maximum compression stroke, the block is knocked back by a compressed spring and strikes the impact block. The dynamic stresses are obtained via the strain

Impact Block Specimen Impact Bar Guide Impact Bar Sliding Guide

I] 24

F,,--- 100 .._1 I

i4 200 - I

Fig. 3. Inclined crack specimen for fatigue loading: sinusoidal and trapezoidal wave form (dimensions in mm).

Since the crack changes its direction of growth, each increment of extension must be recorded in terms of two coordinates, say x and y. A travel- ling microscope is used with accuracy of +0.01 mm. Plotted in Fig. 8 are the crack growth fatigue data for the SM45C steel. The letters S and T denote, respectively, the sinusoidal and trapezoidal wave form while the numbers 30, 45 and 90 correspond to the crack angle /3. Slower crack

/

S.B. Lee, S.I. Noh / Theoretical and Applied Fracture Mechanics 21 (1994) 1-8

Grip Spring Impact Block

Fig. 5. Schematic of repeated impact test machine.

Impact Head

S.B. Lee, S.I. Noh / Theoretical and Applied Fracture Mechanics 21 (1994) 1-8 5

B0

6O

0 .

b 40

8

20

0 2 4 6 8 10

Time , t (msec )

Fig. 6. Stress wave form generated by repeated impact machine.

8

E E 7

v

O

6

c 5 o

_J

P O

3

2

$45 T30 +

T45 + $30

$90 + o o ,

÷ T90~ •

-" * + ~ o

:/ . .° "~ * o °

e . I ,~ ooooo °

~,o .o ,~8-oo. o . o . o ° o ° ° * °

i , l l l l l l l i i i l l l l l l l i i i i i i l l l l , l l l l l l l l l l l l t l l l l I

0 5 10 15 20 25

Number o f Cycles ( x 1 0 4 ) , N

Fig. 8. Crack growth data for SM45C steel under fatigue (S, sinusoidal and T, trapezoidal) for different crack angles.

growth is seen for the sinusoidal fatigue load. The difference decreases with increasing /3 such that at /3 = 90 ° the difference in the wave form has negligible effect on crack growth. Fig. 9 dis- plays the variations of crack growth segments with the number of impact load for the SM45C steel. The letter I stands for impact load. A similar trend is observed. That is, crack growth

decreases as the initial crack aligns closer to the axis of impact load, i.e. for decreasing angle/3.

4.2. Crack growth rate

Crack growth rates d a / d N can be obtained from the curves in Figs. 8 and 9 for the SM45C steel by using the seven point incremental poly-

0_

v

200 -

150 '

I O0

50

0 0

,,,,,,,,e '''''''t'''''''''e '''''''r'l'''''''''e

200 4-00 600 800 1000

7 A

E E

v 6

o

5

. J

C.)

3

0

190 0

J • • • AAAAA AA~A A • •

~ L l l l e e l l l l l , l L l e t e l l e e e l l l t t t J t l t l t l l l l

0 2 4 6

I 4 5 I 3 0

A A

Time , t ( m s e c )

Fig. 7. Trapezoidal wave form for fatigue generated by servohydraulic machine.

Number o f Cycles ( x l 0 4 ) , N

Fig. 9. Crack growth data for SM45C steel under repeated impact for different crack angles.

6 S.B. Lee, S.1. Noh /Theoretical and Applied Fracture Mechanics 21 (1994) 1-8

"6

E E

z

1 0 -~

10 -~

10 4

1 0 - i

1 0 -~

-=i

, h i

t •

v

ii w

.J I I . " - ; nu - !

E L - - :

TF IJ

21 1 0 -~

1 I i i i r ] l l l l l U

i i iiii! I I I I I l l I I I l l l l

1 0 -~

1 0 -3

b" E

E

g

J=

[_)

E~

1 0 -4

! / 10-5 : i ~ i i : : . . : : . ,

z~ Impact

0 Fatigue

1 0 - 6

10-5 10-4 10 -3

Range of Strain Energy Density Factor , &S (MPa.m)

Fig. 10. Log-log plot of crack growth rate versus strain energy density range for SM45C steel under fatigue and repeated impact with /3 = 30 °, 45 ° and 90 °.

Range of Strain Energy Density Factor , AS (MPa.m)

Fig. 11. Log-log plot of crack growth rate versus strain energy density range for Z02-NKDT steel under fatigue and repeated impact with /3 = 90 °.

nomial technique [14]. A log-log plot of da/dN versus AS is given in Fig. 10 for SM45C steel. The two sets of data correspond to S (sinusoidal) and T (trapezoidal) fatigue load and I impact load for/3 = 30 °, 45 ° and 90 °. The crack growth rate data for repeated impact is more scattered than those for fatigue. The former appears to be higher than the latter. Similar results are observed for the Z02-NKDT maraging steel in Fig. 11. These results are caJnsistent with the observa- tion in [15] that the striation spacing for repeated impact is larger than that for fatigue. The argu- ment in [4] is that a smaller crack tip plastic zone is developed under repeated impact than under fatigue. A higher crack opening would prevail for repeated impact and hence a higher crack growth rate would be predicted when correcting for crack closure effects. The work in [16] has used an effective strain energy density factor range ASef f to correct for crack closure effects.

The data in Figs. 10 and 11 can be used to determine the coefficients B and m in Eq. (10). They are summarized in Table 2 in which B has the dimensions of mm/cycle × (MPa.m)-m.

4.3. Direction of crack path

The x- and y-coordinates of the crack growth increments for the SM45C steel for fl = 30 ° and 45 ° are shown in Fig. 12. One set of the data applies to sinusoidal fatigue and the other to impact. They are closely grouped together. The two solid curves are the analytical predictions based on the strain energy density criterion. Good agreement is obtained for the early stages of crack growth up to about 2 mm of length. The discrepancies tend to increase with increasing crack growth. This is somewhat expected because Eqs. (13) do not account for the specimen edge

Table 2 Numerical values of B and rn for SM45C and Z02-NKDT steel

Steel Load Coefficient

type type B m

SM45C Fatigue (S,T) 3193 Impact (I) 1131

Z02-NKDT Fatigue (S) 94.46 Impact (I) 3.875

2.17 1.714

1.608 1.134

S.B. Lee, S.L Noh / Theoretical and Applied Fracture Mechanics 21 (1994) 1-8 7

E E

v

E 0

0 Q .

4

F ~- S i m u l a t e d Crack D i r e c t i o n

3

0 0 2 4 6

1 0 ~ , , , , , ,

~ o

1 0 ~

L:l ~°~

.,!f, - - 1

1 0 ~ -

1 0

I I 1 I I i i i i I I I I I I I I I I I 114

H o S30 • s45 II o $ 9 0 I I • T 3 0 B • T 4 5 • T 9 0 [1 - t 3 0 14

1 O 0

X p o s i t i o n ( m m )

Fig. 12. Comparison of theoretical prediction with test data of crack path for SM45C steel under fatigue and impact with /3 = 30 ° and 45 °.

R a n g e o f S t r e s s I n t e n s i t y F a c t o r , A K e q ( M P o 4 " m ' )

Fig. 13. Log-log plot of crack growth rate versus equivalent stress intensity factor range for SM45C steel under fatigue and repeated impact with/3 = 30 °, 45 ° and 90 °.

effect that becomes more pronounced as the crack grows longer. Despite the neglect of specimen width effects, predictions from the equivalent crack length approximation are reasonable.

5. Concluding remarks

The crack growth rates and path for the SM45C and Z02-NKDT steel were determined under mixed mode fatigue and repeated impact loading. Good correlation was obtained by application of the strain energy density criterion with the exper- imental data. Higher crack growth rates were found for the repeated impact load as compared with fatigue load.

An empirical relation between d a / d N and equivalent AKeq correcting for mixed mode has also been suggested [17]:

da = C(AKeq)" (14)

d N

in which

g e q = ( g 4 +8K4) 1/4 (15)

The coefficients C and n are determined by tests. The data for SM45C and Z02-NKDT steel are

replotted in terms of d a / d N versus AKeq on the log - log scale; they are shown in Figs. 13 and 14. Determined values of C and n are summarized in Table 3. The dimension of C is m m / c y c l e x (MPa.v~- ) -" . However, it is not clear how the directions of crack growth are determined in the application of Eqs. (14) and (15).

o

E

z

.=

o

P t J

10 -3 ........ i - i - - i - - i - - i . . . . . . . . . . . . . . . . . . . . . . . . ~

i0 -4

10-5

10-6

::::::::::::::::::::::::::::::: ....... T 1 ' [ ! ! " ' T T [ F i ........ F T - T i - T i ] . . . . . . . .

================================ ........ r - - r - - - r ........ r .-.-i -~ .-..~ • .~ ...... , . < , . , ,

" [ ~ [ i i

" i i T i : i r ! i

-~!~!~!!4!!:!!!!q!q

........ r r r r i i i i i i

........ F T i i i :. ~ i i i i I

i i i ; i

! ! ! !

i i i i i

i i i i i i

: : : : ,

! . . . . . . . . . . . . . . . . . . . . i .... z~ I m p a c t !

! 0 F a t i g u e ~i

L i J i i i i ,

10 100

R a n g e o f S t r e s s I n t e n s i t y F a c t o r , A K e q ( M P o v r ~ )

Fig. 14. Log-log plot of crack growth rate versus equivalent stress intensity factor range for Z02-NKDT steel under fatigue and impact with/3 = 90 °.

8 S.B. Lee, S.L Noh / Theoretical and Applied Fracture Mechanics 21 (1994) 1-8

Table 3 Numerical values of C and n for SM45C and Z02-NKDT steel

Steel Load Coefficient

type type C n

SM45C Fatigue (S,T) 7.74 X 10-12 5.189 Impact (I) 8.69 x 10 -9 3.642

Z02-NKDT Fatigue (S) 3.85 x 10 -9 3.412 Impact (I) 1.69 x 10 -7 2.435

Even though the forms of Eqs. (9) or (10) and (14) are similar, fundamental differences prevail which should be emphasized. Eq. (9) is derived from the hypothesis that elements ahead of the crack break down owing to the accumulation of the energy density hysteresis and the coefficient A could be obtained independent of the fatigue crack growth test. Eq. (10) prevails because of the approximate nature of AS. Eqs. (14) and (15) are strictly empirical; they are not intended for mak- ing predictions. Crack path predictions such as those given in Fig. 12 for the SM45C steel cannot be obtained from Eqs. (14) and (15).

Acknowledgements

The Z02-NKDT maraging steel was supplied by Professor G. Pluvinage of the University of Metz in France. Mr. Dong-choon Lee performed the fatigue tests while, Mr. Jin-young Kim performed the computations.

References

[1] I. Maekawa, Y. Tanabe and Z. Jin, Size effect in impact tensile fatigue, Proc. ICF Int. Syrup. on Fracture Mechan- ics, Beijing, China, pp. 757-762, 1983.

[2] I. Maekawa, Y. Tanabe and M. Aiba, Influence of c o r r o -

sive environment on impact fatigue of structural steel, Ins. Mech. Eng. 347-352 (1986).

[3] W.H. Hsi and K.Y. Lee, Repeated impact response of XT-75 20 mm cannon support, J. Theor. Appl. Fract. Mech. 12, 33-43 (1989).

[4] T. Tanaka and H. Nakamura, On the impact fatigue strength of metallic materials, J. Soc. Mater. Sci. Jpn. 23, 678-685 (1974).

[5] T. Tanaka, H. Nakamura and K. Kimura, On the impact fatigue crack growth behavior of metallic materials, Fa- tigue Fract. Eng. Mater. Struct. 8, 13-22 (1985).

[6] M. Birch and J.E. Brocklehurst, The impact endurance of polycrystalline graphite, Carbon 21,497-510 (1983).

[7] M. Futakawa, K. Kikuchi, Y. Muto and H. Shibata, Impact bending fatigue and impact response of a nu- clear-grade graphite beam, Carbon 28, 149-154 (1990).

[8] G.C. Sih, Some basic problems in fracture mechanics and new concepts, Eng. Fract. Mech. 5, 365-377 (1973).

[9] Y. Marakami, Stress Intensity Factors Handbook (Per- gamon Press: Oxford, 1986) 909-911.

[10] H. Kitagawa and R. Yuuki, Analysis of arbitrarily shaped crack in a finite plate using conformal mapping, Trans. Jpn. Soc. Mech. Eng. 43, 4354-4362 (1977).

[11] H. Tada, A note on the finite width correction to the stress intensity factor, Eng. Fract. Mech. 3, 345-347 (1971).

[12] G.C. Sih, Mechanics of Fracture Initiation and Propaga- tion (Kluwer Academic Publishers: The Netherlands, 1991).

[13] G.C. Sih and B.M. Barthelemy, Mixed mode fatigue crack growth predictions, Eng. Fract. Mech. 73, 439-451 (1980).

[14] ASTM Designation E647-91, Standard method for mea- surement of fatigue crack growth rates, pp. 654-681, 1991.

[15] R. Murakami and K. Akizono, The influence of cyclic impact loading and stress ratio on fatigue crack growth rate in aluminum alloy, Fracture Mechanics Technology Applied to Material Evaluation and Structural Design, G.C. Sih, N.E. Ryan and R. Jones, eds. (Martinus Nij- hoff, 1983) 505-516.

[16] Y.C. Lam, Mixed mode fatigue crack growth and the strain energy density factor, J. Theor. Appl. Fract. Mech. 12, 67-72 (1989).

[17] G.C. Sih and C.K. Chao, Fatigue initiation in unnotched specimens subjected to monotonic and cyclic loading, J. Theor. Appl. Fract. Mech. 2, 67-74 (1984).

[18] G.C. Sih and C.K. Chao, Size effect of cylindrical specimens with fatigue cracks, J. Theor. Appl. Fract. Mech. 1, 239-247 (1984).

Documents

Mixed mode crack growth in SM45C and Z02-NKDT steel subjected to repeated impact and fatigue