Threshold stress intensity factor of fatigue cracks

Engineering Fracture Mechonic.~Vol. 49, No. I, pp.29-33. 1994 Copyright I> 1994 Elsevier Science Ltd Prin1ed.k &eat Britain. All rights reserved 0013-7944(94)EOO94-W 0013-7944194 $7.00 + 0.00

Pergamon

THRESHOLD

STRESS INTENSITY CRACKS

FACTOR

OF FATIGUE

C. R. CHIANG Department

of Power

Mechanical

Engineering, National Taiwan, R.O.C.

Tsing Hua University,

Hsinchu

30043.

Abstract-Based on a micromechanical model, the threshold stress intensity factor is predicted. It is shown that the threshold stress intensity factor is given by M$E, where E is the Young’s modulus; b is the Burgers vector; M is a microstructure parameter depending on the crack tip geometry and its surrounding microstructure. Although M is structure sensitive, the values of M for most polycrystalline metals are found to lie within a quite narrow range. The model leads to an explanation of the dependence of the threshold stress intensity factor on the stress ratio. It is concluded that the random stress variation due to the stress sources other that the applied cyclic stress is responsible for its nonlinearity. The grain size effect on the threshold stress intensity factor is also investigated. It is shown that M = a + bD’ where a, b and c are material constants and D is the grain size. The theoretical predictions are in agreement with the experimental data available.

1. INTRODUCTION BASED

ON

developed

the extreme value theory of statistics, [l]. It was concluded that the growth

g

=

a theory of fatigue rate can be written

crack propagation as

C{(Km,,- Ki)” - (Kmin- Kl)“},

has been

(1)

where C, K, and x are the material constants; Km,, and K,,,,, are the maximum and minimum stress intensity factors during each loading cycle, respectively. It was pointed out that K, can be considered as the threshold value for the stress intensity factor, i.e. AK,,, . The purpose of the present paper is to provide a micromechanical model for the estimation of Ki. In order to make contact with current literature, we shall review some theoretical models concerning the prediction of AKth. Weiss and La1 [2] proposed that once the stress at the crack tip falls below the theoretical cohesive The threshold stress intensity factor range is thus strength o,,, the crack stops growing. given by (2)

AKth = J(~~cP 1210,~~

where p is the Neuber’s micro support constant. Davidson [3] assumed that the threshold condition is determined by a slip band just emanating ahead of the crack tip. Accordingly, he derived that AK,h = J(27-w) Y, where Y is the flow stress of the material at the end of the slip band, band. Another model proposed by Fine [4] indicates that

A&, = where ~~ is the stress to activate a dislocation depending on how much the stress intensity models [S, 61, of course.

q,,h, ,

(3) r is the length

of the slip

(4)

source a distance s from the crack tip and q is a factor near the tip is reduced by plasticity. There are other

29

30

C. R. CHIANG

2. THEORETICAL

MODEL

It is conceivable that near the threshold value of the stress intensity factor the actual stress dislocations, inhomogenities, etc. The intensity factor KtlP is screened by the surrounding relationship between K, and Krlpr in general, can be written as Kllp = K, - 6K,

(5)

where 6K accounts for the shielding effect. A precise specification of 6K requires very detailed information on the microstructures. A simple form of 6K is suggested here by considering the following simple mode III configuration. For a series of dislocations with the Burgers vector h lying ahead of the crack tip, 6K can be written [7, 81 as

6K =

c pb I&T

(6)

1,

where x, is the position of the ith dislocation. For other fracture be derived. It is convenient to write 6K in the following form

modes

similar

expressions

6K = mEblJ(2rrL),

can

(7)

where m accounts for the number of the dislocations; E is the Young’s modulus and L represents the effective distance between the crack tip and the dislocations. Similar to the suggestion of Weiss and Lal, it is proposed here that the crack growth is possible only if the theoretical cohesive stress is attained at the crack tip. The stress at the tip can be evaluated by

wK,,, ,:JPq ),

CJ =

63)

where p is the radius of the crack tip. o is a numerical number shape [9-l l] and it is reasonable to let w = 3. Since it is known q Ih then by defining

--

depending on the actual that approximately

(9)

E/10,

p = kh, thus from eq. (8) with TC= 3.14 we have, K,,, = O.O84J(kb)E.

Combining

crack

(10)

eqs (5), (7) and (10) we find that K,= M&E

(11)

where M = 0.0844

+ mdlJ’(2xL).

(12)

It is noted that M is composed of two factors. The first one reflects the crack tip geometry; the second reflects the microstructure surrounding the crack tip. The values of M of three representative materials (steels, Al-alloys, Ti-alloys) have been calculated and listed in Table 1. It is interesting to note that for these materials the values of M are within some specific range despite the fact that they belong to the different crystalline structures. This phenomenon seems to indicate that the material behavior near threshold conditions has its universal characteristics. Table

1. The values of M for some representative

materials M

E (GPa)

K, (MPafi)

Steel

2.49

195

3&l

(BCC) Al-alloy

2.86

72

3-7

2.5-5.7

2.95

117

6.5-1.5

3.2-3.7

h (lO~“m)

I

2.8-3.6

@XI k-alloy

(HCP)

Threshold

stress intensity

factor

of fatigue

31

cracks

P= 1.0

0.8

.- 0.6 ?a

0.2

0.2

0.4

0.6

0.8

1 .o

R Fig. 1. Influence

of the stress ratio

3. INFLUENCE It has been experimentally suggested [ 121 that

R on the threshold

stress intensity

factor

AK,,,.

OF THE STRESS RATIO ON AK&

confirmed

that

AKth depends

on the stress

ratio

R. It was

AK,,, = (1 - R)YAK:h

(13)

where AK:,, denotes the threshold value at the condition that the stress ratio R = 0 and y = 0.7-l. On the other hand, La1 and Nambooddhiri [13] developed a more sophisticated model and proposed that AK

th

=

Cl- WK!,

R -=c R,

constant

R>R,’

(14)

where R, is a critical value for R. According to the present theory [I], by definition AKth = (1 - R)K,.

(15)

Although eq. (15) fits the experimental data very well for small values of R, some deviations are observed for R close to 1. Such behavior, in fact, is not surprising. Remember that as R approaches 1, the externally applied stress variation becomes smaller and smaller so that other “random” stress variation becomes important and may effectively reduce the applied stress amplitude. In other words, near the threshold condition for R close to 1, we have the effective maximum stress intensity factor as (Km&r

= ( 1 - 9 Kax 3

(16)

where q denotes the reduction fraction of K,,,,, due to random stress sources and is a function Since at the threshold condition (K,,,),rr must equal Ki, thus Km,, = K/(1 - v).

of R.

(17)

Therefore AKth = (1 - R)K,,,,, = (1 - R)K,/(l

- q).

(18)

Since at R = 0, AK,, must equal Ki, a good approximation for r] is proposed, i.e. r] = R”P with 1 > p > 0. Curves of AK,,/K, against R are plotted for different values of p. All experimental data [12-151 can be fitted by choosing proper values of p.

C. R. CHIANG

32

4. INFLUENCE

OF GRAIN

SIZE

ON Ki

It is known that K, is very sensitive to the microstructure of the material. As far as the grain size dependence is concerned, no general agreement is found among the experimental data [ 166191. Some reports [15, 171 indicate that the threshold stress intensity is increased with grain size, while others indicate the reverse tendency [18, 191. The exact reason for the controversy is unclear. But this must be caused by different microstructure involved in different material systems. A simple micromechanical model is proposed here to explain this phenomenon. When the number of the shielding dislocations is large, it is appropriate to consider the continuized version of eq. (6). i.e. the shielding effect can be written as L ,ubm (x) dx 6K = s II

(19)

4271x)

where L is an appropriate microscale and m(x) represents the density distribution of the dislocations. It is reasonable to assume that L is proportional to the grain size D, i.e. L = fiD, where p is the proportional constant. Now, with different distribution of m(x), we may predict different dependence of the threshold stress intensity on grain size. Consider two extreme cases for m(x): (1) uniform distribution and (2) Dirac delta distribution. By eq. (19), with the uniform distribution we have M = J(2Pla)@mfi, while with the Dirac

delta distribution,

(20)

we have 6K = pbmlJ(2nBD).

According

(21)

to eq. (20) or eq. (21), we have from eq. (11) that K,=a+bD”

(22)

K, = a + bD-“.5.

(23)

or

In conclusion, if the distribution of the dislocation within the shielding zone is uniform, then eq. (22) is valid. On the other hand if most dislocations are concentrated at a distance (proportional to the grain size) away from the tip, then eq. (23) is valid. Thus, in general we have K,=a+bD’ where -0.5 < c < 0.5. The prediction data [16-191.

(24)

of the present

model

is confirmed

by the experimental

5. CONCLUSIONS A micromechanical model has been developed for the estimation stress intensity factor Ki for crack growth. It is shown that K, = M $E,

of the threshold

value of the (25)

where M consists of two terms which reflect the crack tip geometry and the microstructure around the crack tip. It is remarkable that, despite the fact that M is structure sensitive, the value of M is found to lie within a narrow range for most polycrystalline metals. Theoretically, the threshold stress intensity range of fatigue cracks should be related to Ki by AKth=(l

-R)Ki.

(26)

Nevertheless, as K, stands for a system parameter of a stochastic process (fatigue) [I], the influence of random stress variation due to sources other than the applied cyclic stress should be taken into account for cases in which R approaches 1. As a result, it is proposed that I-R AK,,, = ~1 _ R l/P

Ki

’

(27)

Threshold

stress intensity

factor

of fatigue

cracks

33

A wide range of experimental data can be fitted by this simple equation. The effect of the grain size on Ki is also studied based on a simple dislocation shielding mode. It is shown that M can be written as M=a

+bD’

(28)

where a, b and c are material constants. It should be noted that the general present theory allows one to study other factors influencing M.

framework

of the

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33, 965-977

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cracks

through

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Engng

Fracrure

(1989).

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