Influence of random overloads on fatigue crack lifetime and reliability

Engineering Fracture Mechanics Vol. 30. No. 3, pp. 361-371. Printed in Great Britain.

198X

@

INFLUENCE OF RANDOM OVERLOADS FATIGUE CRACK LIFETIME AND RELIABILITY

0013.7Y44/88 $3.00 + .Oll 1988 Pergamon Press pk.

ON

R. ARONE Israel Institute of Metals, Technion R&D Foundation, Haifa, Israel Abstmct-A stochastic model is presented, describing fatigue crack growth under random overloads and incorporating the situations of fracture occurrence either under an overload or under the base-line load. It is a variant of an earlier model, based on presentation of the delay time as a purely discontinuous Markov process, and modified so as to account for the role of overloads in the event of fracture. It was shown that a strongly beneficial effect of overloads on lifetime is achieved under an optimal combination of material retardation response and frequency and magnitude of the overloads. It was also shown that the more stringent the reliability requirements, the less favourable the lifetime predictions.

1. INTRODUCTION IN RECENT period several probabilistic models has been proposed for description of the fatigue crack behaviour under random overloads[l-41. In our earlier works[3,4] the loading process was presented as a superposition of a base-line constant-amplitude cyclic load and random sequences of overload peaks (Fig. la). The proposed model[3,4] deals with the retardation effects caused by overload peaks and with their influence on fatigue lifetime. This model incorporates the fact that each overload peak delays crack propagation: the higher the overload stress, the longer the delay. The sum of all individual delay intervals over all overload events is the total delay time accumulated up to the observation moment t. Understanding of the stochastic behaviour of the total delay time td is of primary importance in lifetime and reliability assessment; actually, the difference between the total time t and total delay time td is the effective time tp of crack propagation, i.e. the time during which the crack growth is undisturbed by overloads. Thus, a simple dependence can be established between crack length and effective time and the condition for fracture occurrence, based on the critical crack length I,,, can be reformulated in these terms: fracture takes place if effective time exceeds (or equals) the critical time tcr necessary for the crack to reach the value 1,, under base-line loading. Since each individual delay interval depends on several factors of stochastic nature (overload magnitude, material properties and so on), it is a random value. The total delay time, being the sum of individual delay intervals over a definite time interval, is a random process. Thus the effective time as defined above is also a stochastic process. (In what follows, designations like ‘delay process’ and ‘effective process’ are also interchangeably used where convenient.) It was shown[l] that an individual delay interval is a function of the average retardation coefficient Cg and effective time. Replacing the latter by the difference t - t& it is seen that the individual delay interval depends on the value of total delay time up to the moment in question, but not on the history of the latter. This imparts the Markov property to the delay process. It was also shown [3,4] that under definite assumptions the total delay time can be simulated by a purely discontinuous Markov process. Numerical solution of the forward KolmagorovFeller equation yields the probability distribution function of the total delay time. This in turn yields the distribution of the effective time, hence the probability of the inequality {Sup tp< tcr; t E (0, T)} (probability of nonfailure, or reliability). In the above works[3,4] critical time tcr was determined under the assumption that fracture occurs under an overload stress, although crack growth is due to base-line cyclic load. Actually, the average overload stress was used for determination of the critical crack length 1,,. This assumption may be too conservative for rare overloads when fracture does not necessarily coincide with the overload in time. 361

362

R. ARONE

1 I ME

Fig. 1. Schematic representation of crack retardation process: Overload moments designated t,, tz t,, Interoverload times designated 7,. r2 TV.Delay intervals designated Ahi. Ardl . A(,,,+, (q-time interval between last overload and observation moments). (a) Base-line constant-amplitude cyclic load with superimposed random overloads. (b) Time dependence of crack length (Dashed line supposed crack growth behaviour in the absence of overloads). (c) Effective (curve I) and delay (curve 2) times.

Conversely, in cases of frequent overloading fracture behaviour is governed by the maximum values rather than by the average of the overload stress, and lifetime estimates may be nonconservative. The object of the present work is development of an adequate stochastic model incorporating the situations of fracture occurrence either under an overload or under the base-line cyclic load. 2. MODEL As in the earlier works[3,4] a superposition of random overload sequences on a base-line constant-amplitude cyclic load is considered (Fig. 1a), overload peaks being generally random in time and magnitude. It is assumed that each individual overload contributes little to crack length but delays crack growth; the higher the overload stress, the longer the delay. The individual delay interval A& associated with the given overload is determined as the difference of the times r, required for the crack to traverse the zone affected by the overload, and tr required for it to traverse the same distance in the absence of an overload (Ah = ra- tt, see Fig. lb). The individual delay interval is a function of the overload stress a0 crack length 1, the maximum (o,& and minimum (u,,,~.) base-line stresses, material parameters and so on. With the base-line stresses and material characteristics as parameters, the delay interval as a function of aa and ‘I can be given in the general form:

Ah = f~(uoo, 4. We assume that the expression for the stress intensity factor and the crack propagation known

(1) law are

Random overload influence on lifetime

363 (24

K1 = &CO

(2b) where K is the stress intensity factor, u the applied stress, I crack length, and AK and Au the stress intensity and stress ranges of the base-line load respectively. Integration of equation (2b) yields the time-dependence of crack length

1= f‘%(t).

(3)

Accordingly, the individual delay interval can be presented in terms of the crack growth time, namely, eq. (1) can be written as

At., = fi(uo, I) = fduo, fdf)l=

(4)

fs(O.

We subdivide the time interval of crack growth (0, t) into two sub-intervals: total delay time td (see Fig. lc).

effective time tp and

(5)

t= tp+ td.

The total delay time, being the sum of the individual delay intervals, represents that part of the total time which is ‘lost’ in terms of crack growth; thus, effective time is that of active crack growth under the base-line load as if no overloads occur. Combination of eqs (4) and (5) yields

Ah = fs(%, tp)= fs[Uo, (t - cd)]

(6)

which indicates that at an arbitrary moment of observation l the individual delay interval depends on the total delay time at that moment, irrespective of the history of the process. Considering the total delay time as a random process due to the random nature of the overload moments, material parameters and so on, and using the Markov property implied by eq. (6), stochastic model representing the delay process as a purely discontinuous Markov process was developed[3,4]. It was assumed that individual delay intervals can be considered as jumps in’ the delay process, for which transition probabilities are obtainable from the Kolmogorov-Feller integro-differential equations[5]:

@(to,x; --4 Y) = t

-qF(t&

x; t, y) + q

I P(to, n

x; t, 2) 4eto;

p = qAt+O(At)

x; t; z)

(74

(7b)

where F(&,, x; t, y) is the transition probability, (i.e. the probability of the total delay time td being less than Y at moment t, provided that at moment to < t td = x); p-the probability of overload occurrence during time interval At (approximately equal to the base-line cycle time), q-the overload intensity and P(t, z, y)-the conditional probability distribution function, defining the probability of & being less than Y subject to the overload occurring at moment t and just before the overload & = Z. In the first version of the model[3] it was assumed, by virtue of infrequency of the overloads, that the individual delay intervals are smaller than the interoverload times. In such a case, bearing in mind eq. (6)

P(t, z, y) = p{z+fS(uO>

t -

cd)<

y}.

(8)

R.ARONE

364

At the starting moment to = 0 and x = 0, and the initial condition can be formulated

lim F(0, 0; t, y) = F(0, y) = ~(0, lq =

as

(9)

t+O

where for simplicity

w,

0; 4 y) = F(t, y).

(I())

Equation (7a) can be solved numerically by a simple step-by-step procedure, with eq. (9) stating the initial condition[3]. The first version of the model may be nonconservative at high values of the overload intensity parameter q (high frequency of occurrence) since in that case interoverload times may be comparable to or even shorter than the individual delay intervals. The second conservative version of the model[4] preserves conservativeness of the reliability assessment even at high values of q. According to this version each individual delay interval equals Aid as per eq. (6) only when it is less than the interoverload time T. Otherwise, the value of Ard is truncated so as to equate the individual delay interval to 7. By this means overlapping of individual intervals is eliminated. This leads in turn to some underestimation of the total delay time and imparts conservativeness to the reliability assessment. The dissimilarity between the two versions manifests itself in the structure of the conditional probability function P(t, z, y). Since in the second version individual delay interval Ai is identified with the smaller of two random values (A&,.T), the function has the structure P(t,z,y)=P,+P*-P*P;?

(114

for A i, = min{A td, T} where PI is determined

(lib)

by eq. (8), and P*=P{7
(12)

F,(e) being the probability distribution function of the inter over load time. Thus, using P(t, Z, y) from eq. (8), or for high q, eq. (lla)-the whole array of transitional probabilities F(t, y) can be determined by simple step-by-step procedure embodied in eqs (7a) and (9). We define reliability as the probability of nonfailure within a given time interval (0, T). Failure occurs if one of the following two events takes place during (0, T): (a) a crack growing under the base-line load reaches critical length for maximum base-line stress a,,,,,, and (b) at least one of the overloads has stress magnitude u. sufficient for the current crack length to become critical, thus causing unstable fracture. According to eq. (2a), the critical crack length for a given stress magnitude is obtainable from equation (13)

where f;‘(a) is theinverse of f2( m), KIc fracture toughness of material the index 0 refers to an overload and l-is the base-line load. Recalling eq. (4) the critical time required for the crack to reach critical length is obtainable as follows ti = fil(

li);

i=O,

1.

Failure does not take place when the following two events coincide

(14)

Random

overload

{Sup tp <

influence t1;

t E (0,

{for none of the overloads As was shown previously[3,4]

365

on lifetime TN

(154

tp 2 to; f E (0, T)}.

(1%)

the probability of the first event is

Let us now consider inequality (15b). An overload causes failure when the following two events coincide: (1) an overload occurs, and (2) its random magnitude a0 is critical for the current crack length. The probability of the first event according to eq. (7b) is qbt. Combination of eqs (13) and (14) yields the following expression for critical time as function of the overload stress co: (17) If the probability density function ol(ao) of a0 is known its counterpart for to [02(fo)] is obtainable with the aid of eq. (17). Accordingly, the probability of a0 being critical can be given in terms of the effective and critical times P{f$

to}=

‘omx

J

P{ t -

td 2~ X/to =

X)0&X) dx.

(18)

&lli”

Equation

(18) can be rewritten as

FCC,

t-

toh2(to)

dto

(19)

and the probability of an overload causing fracture (or, to coin a term, of its being ‘damaging’) can be found as the product

(204 or Gob) where ql( t) is the time-dependent

intensity of damaging overloads:

Thus, for Poisson flow of overloads, during the time interval (0, T) is

the probability of nonoccurrence

of a damaging overload

(21) Reverting now to eqs (15a, b) and bearing in mind eqs (16) and (21) the probability of nonfailure during the time interval (0, T), or in other words the reliability can be formulated as follows R=[l-F(T,

T-t,)]exp

and if the overloads do not cause retardation. EPM 30:3-p

( - 6’ 41(t) d+

(22a)

366

R. ARONE

R =[l-E(t,,

T)]exp [ - loT q,(t) df]

where E( tl, T) =

0 1

(2%)

if T < t, if T 2 t,.

Here it should be borne in mind that for Y I 0 F( t, Y) = 0.

3. EXAMPLE Except for the procedure of critical time determination, all conditions remain the same as in the example of [4]. An edge crack, propagating in a half-space under constant-amplitude zero-to-tension cyclic load (base-line load) with superimposed random overloads, is considered. In this case eqs (2a), (2b) and (6) are as follows[4]: K = dz(l) = (T1.12JGl

(23)

; = f3(AK) = C(l.l2A&$“’

(24)

where C and m equal 10V9 and 3 respectively

and

Atd=fs(ao,(t-fd)=D[fm-(t-td)]

(25)

where D = (SZ - l)( 1 - l/I+-*)‘*)

(26)

L = 2/(C(1.12 aJ+%+-*)‘*, ai is the initial crack length, r the crack ratio, and Sg the inverse of C&the latter being the average retardation coefficient dependent on ao. It should be noted that D depends on a0 (so that the higher moo,the greater Sx) but not on the delay and effective times. Since cro is random, so is SZ, and consequently also D. Substitution of eq. (25) in (8) yields

(274 or, D being random,

Wb) where FD(*) is the probability distribution function of D. Different materials differ in their retardation response to the same overload. In order to illustrate the possible influence of random overloads on fatigue life, several materials with different retardation responses (Cg = 0.8; 0.5 and 0.25) under the same overload conditions are compared. Two types of overload behaviour are considered: type l-characterized by maximum and minimum overload stresses uo,max= 450 MPa, and mO,min= 225 MPa, respectively (the latter being the maximum value of the base-line cyclic load), and type 2-characterized by ~o,max= 90 MPa and u~,,,~~= 225 MPa. The functional dependence of Cg on the overloads being rather complicated and some of the relevant parameters unknown, the triangular distribution of u. is assumed to convert at first approximation into an exponential distribution of D with mean value fi dependent on that of Cg as per eqs (25) and (26).

367

Random overload influence on lifetime

Thus, eq. (27b) yields if zy

P(r,z.y)=[~-cxp{-hI_yctfZ)}

(2W

where A =

l/D.

(2Sb)

The interoverload time is also distributed exponentially (12) to be rewritten as NY

-

z) =

l-exp{-q(y-z)} (()

with parameter

q. This permits eq.

if zy’

(29)

In this example a conservative estimation is applied[4], namely eq. (11) is used for determination of P(t, z, y) with eqs (28a) and (29) replacing PI and PZ, respectively. With a triangular overload distribution eq. (19) can be rewritten (see eq. (1 la) in Appendix) as -_2 ( toAt;

fO,min)

dto

(30)

where A to = tO.max- t().mine This enables the reliability to be calculated from the expression

which was embodied in an assessment procedure supplementing the computer programme developed for determination of F( t, y)[4]. Some of the results (lifetime vs q) are summarized in Figs 2-5 in normalized ordinates, the left-hand axis representing the ratio of actual to base-line loading lifetime, and the right-hand axis the ratio of actual lifetime to its counterpart determined under the assumption that fracture takes place under the average overload magnitude. Figure 2

2.5 -

0.5 -

0 -5

I

-45

I

-4

I

-3.5

I

-3

I

-2.5

-2

Logorithmof overbad pommeterg

Fig. 2. Dependence of lifetime on overload intensity parameter 9 for 0.9 reliability level. Type 1 overload behaviour ((T~,,,~~= 450 MPa, u”,,,~~ = 225 MPa). Curves 1, 2, 3 refer to Ci = 0.25, 0.5 and 0.8 respectively (both ordinates in normalized form).

368

R. ARONE

25

E 3

-

2

f x .N 6 E b 2

1.5 -

I-

01 -5

I

I - 4.5

-4 Logarithm

-3

of overlood

1

I

I

I -3.5

-2 5 parameter

-2

g

Fig. 3. Dependence of lifetime on overload intensity parameter q for Cz = 0.5. Type 1 overload behaviour ((T~,,_~ = 450 MPa, (T,,,,,~” = 225 MPa). Curves 1. 2, 3 and 4 correspond to 0.9, 0.95. 0.99 and 0.999 reliability level respectively (both ordinates in normalized form).

refers to type 1 overload behaviour and to the three levels of Cx mentioned above. The lifetime values are given for the 0.9 reliability level. As can be seen, the three materials have similar lifetime curves in spite of their differing retardation response. At low overload intensities the actual lifetime practically equals its base-line counterpart; as the overload intensity increases the graphs acquire a slight slope (even descends for the highest value of Ci), and finally ascend steeply, the steeper the lower Cg. Figure 3 shows that the lifetime estimates depend significantly on prescribed reliability level; the higher the latter, the lower the estimate. From Fig. 4 it can be seen that this effect becomes more prominent with decreasing C&in other words with increase of retardation response. As regards the influence of increase of maximum overload stress, calculation results for the two above types of overload behaviour were compared, assuming that a twofold increase in (+O.max (transition from type 1 to type 2 behaviour) leads to a similar increase in the retardation response (decrease of C;l from 0.5 to 0.25). The results of this comparison are shown in Fig. 5.

2.5 -

ol-LuL-J -5

-4.5

-4

Logorithm

-35

of overload

-3

parameter

-2.5

-2

g

Fig. 4. Dependence of lifetime on overload intensity parameter q for Cg = 0.25. Type 1 overload to 0.9, 0.95. 0.99 behaviour (u,,~*~ = 450 MPa, u~,,,,~ = 225 MPa). Curves 1, 2, 3 and 4 correspond and 0.999 reliability level respectively (both ordinates in normalized form).

369

Random overload influence on lifetime

I 33

I

I

I Log~ikt~

of

I

I

- 3.5

-4.5

uwW.Wd&k~ter

- 2.5

I -2

g

Fig. 5. Dependence of life time on overload intensity parameter q for different overload behaviours. Curves 1 and 2 refer to type 1 behaviour, curves 3 and 4 to type 2 behaviour. Curves 1 and 3 correspond to 0.9 reliability level. curves 2 and 4 to 0.999 reliability level. Normalized lifetime values represent ratio of actual and base-line lifetimes.

the upper and lower pairs of curves referring to type I and type 2 respectively (each pair at 0.9 and 0.999 reliability level respectively). As can be seen, the doubled retardation response in type 2 not only does not make for longer lifetime, but actually reduces it below its base-line level over a considerable interval of q-values. The above examples show that a strong beneficial effect of overloads on lifetime can only be achieved under an optimal combination of retardation response and overload intensity and magnitude. When overloads are rare their effect is weak. As overload intensity increases, two possible behaviours can be expected: under a combination of high overload magnitudes and moderate retardation response the lifetime is most likely to be shorter than its base-line counterpart; by contrast, under a combination of moderate overloads and strong retardation response-the lifetime increases with overload intensity moderately at first and then rapidly, in some cases up to infinity (complete arrest of the crack).

with

4. SUMMARY

AND CONCLUSIONS

(1) A stochastic model is presented describing fatigue crack growth under random overloads and incorporating the situations of fracture occurrence either at overload moments or under base-line cyclic load. (2) The conflicting influence of overloads on lifetime and reliability was demonstrated. (3) It was shown that overloads do not always make for longer lifetime due to the retardation effects, and may even lead to its reduction. (4) A strong beneficial effect of overloads on lifetime can be achieved under an optimal combination of retardation response and of frequency and magnitude of the overloads. (5) Under a combination of high-stress overload sequences and moderate retardation response the lifetime is most likely to be shorter than its base-line counterpart; by contrast under a combination of moderate overloads and strong retardation response, the lifetime increases significantly with overload intensity. In the latter case for different combinations of overload behaviour and material retardation response there are specific ranges of the overload intensity parameter 4 characterized by a steep increase of the lifetime. (6) The predicted lifetime is strongly dependent on the prescribed reliability level: the more stringent the reliability requirement, the less beneficial the lifetime predictions.

R. ARONE

370

REFERENCES cyclic loading. Engag Fracture Mech. 24, 223-232 (1986). [Z] 0. Ditlevsen and K. Sobczyk, Random fatigue crack growth with retardation. Engng Fracture Mech. 24. X61-87X (1986). [3] R. Arone, On retardation effects during fatigue crack growth under random overloads. Engng Fracture Mech. 27, 83-89 (1987). [4] R. Arone, Conservative estimation of crack growth under random overloads. Engng Fracture Meek. 29, I Y-29 ( 19881. [S] B. V. Gnedenko, The Theory of Probability. Chelsea. New York (1968).

[I] R. Arone, Fatigue crack growth under random overloads superimposed on constant-amplitude

APPENDIX ~~r~T~i~a~~o~ of critical time dis~ibu~io~ for fria~gu~ur d~~~b~tio~ of ~~~r~oad stress Consider a one-sided triangular density distribution of the overload stress o,(X) with the positive skewness; this distribution implies that the higher the overload magnitude, the lower its probability. The condition

r

%.max

w,(x) dx = 1

(IA)

leads to

where AS =

utl.max - uo,,,,. This leads to the following expression for o,(x)

W(X) =

2f~O.m - x)AP

if oO,min5 X 5 00,“~~

0

otherwise.

The critical crack length and overload stress are related as follows (See eq. (24) in main text) (3A) where U = &,/1.12-& dl Z=

C(l.l2Ar4&)“.

Integration of the equation [see eqs (2b) and (24)] in main text] yields t,, = t, - B/1:y-2”’ where t,, is the critical time for a given f,,, B = 2/( m - 2)C( 1. 12&&r)m, Substitution of (3A) in (SA) yields t(, = t, -

or-‘B/

f_P-‘.

(.sA) r, = B/f$m-“‘2, and 1, is initial crack length.

(6A)

Since in our example m = 3, (7Al

f, = L - Boa,, where B,, = B/U. Equation (7A) can be rewritten as

(XA) The density probability of tt, can be obtained as follows

I2 ,I

@2(x)= o,lffo(4,)J which leads to

(9A)

Random overload influence on lifetime

( -~)/B~AS~

o*(x) = 2

~0,nl.x

371 (lOA)

0

where AS = u~),,,,~~- (+O,min. Bearing in mind eq. (8A), eq. (10A) can be rewritten in the form

4x) where 64) = to,,,,

2(x -

= 0

4,,d/AG

if ro,min < x < h,max otherwise

- t~.,i, b.max = t, - &~~~rnin to,rnin= L - B~,,.rnax. (Receioed

16 June 1987)

(114