Conservative estimation of crack growth under random overloads

Engineering Fracture Mechanics Printed in Great Britain.

Vol. 29, No. 1. pp. 19-29.

0013-7944/88 $3.00 + .OO @ 1988 Pergamon Journals Ltd.

1988

CONSERVATIVE ESTIMATION OF CRACK GROWTH UNDER RANDOM OVERLOADS R. ARONE Israel Institute of Metals, Technion R&D Foundation, Haifa, Israel Abstract-A conservative stochastic model describing fatigue crack growth under random overloads is presented. An auxiliary process, simulating delay time due to retardation as a purely discontinuous Markov Process, is introduced. A numerical procedure based on the Kolmagorov-Feller integro-differential equation is used to determine the probability of failure. It is shown that three main factors influence the difference between conservative and nonconservative estimates: the overload frequency, the retardation intensity, and the prescribed reliability level. The lower the two first factors and higher the third, the smaller the disparity.

1. INTRODUCTION the difficulties of life-time estimation under spectrum loading is adequate accounting for interaction effects. In this context, a major role is played by sharp overload peaks, which can significantly prolong the fatigue life of cracked structural components. As a consequence life estimates based on crack-propagation laws which fail to take retardation effects into account are too conservative. With a delay interval defined as the time lost in terms of crack propagation due to overloads, a stochastic model describing crack growth under random sequences of rare overloads superimposed on a base-line cyclic load was developed[l]. This model while realistic in the case of rare overloads, gradually becomes unconservative as the overload frequence increases since one of its premises-absence of other overloads within the delay interval of the overload in question-also fails gradually. According to the original model, two overlapping delay intervals are summed, thereby exaggerating the total delay time; since the actual total delay time of two overlapping delay intervals is as a rule smaller than the arithmetical sum of two distinct intervals (separated in time), this estimation can become unconservative. The remedy lies in dispensity with summation of overlapping intervals, which is achieved by stipulating that an individual delay interval cannot exceed the interoverload time, and when overlapping occurs, than the overlap time be accounted for only once. Numerous investigations having shown[2-71 that the retardation effects of double (or otherwise multiple) overloads are as a rule smaller than the summary effect of an equal number of distinct single overloads: nevertheless the conservatism of the imposed restriction should be moderate. The model presented in this work yields the lower boundary of the life-time estimates. With the upper bound provided by the original model, the two approaches combined yield the range of life-time estimates, thereby, offering basic data for reliability assessment. ONE OF

2. MODEL Consider a body with a growing fatigue crack, under a constant-amplitude cyclic load with super-imposed overloads random in time and magnitude-as shown schematically in Fig. l(a), where overload moments are designated by ti, interoverload times by ri and overload magnitude by uo; the base-line cyclic load is bounded by umin and a,, with Au = u,,,, - u,,,~,. Each overload retards crack propagation as shown in Fig. l(b), where crack length is plotted against time (number of cycles) in the same time scale as in Fig. l(a). Here, the solid line represents the actual crack length, and the dashed lines the growth process that would take place in the absence of an overload. EPI

29:1-B

19

R. ARONE

20

_

0‘.

LI , 'I

r2

T3 I7

q IT l '3

'2 TIME

Fig. I,, tz

1.

Schematic representation of crack retardation process; overload moments designated interoverload times designated by q, r,, . . . , 6; delay intervals designated &,, Adz.. . At,,+,. 7ris time interval between last overload and observation moment t.

. . t,;

by by

We define the individual delay intervals Atdi, corresponding to the i-th overload, as the difference between times d, required for the crack to travers the zone “affected” by overload a(a = af - a(>),and &, required for the crack to travers the same distance in the absence of an overload: A& = til- rf. The total delay time &,up to observation time t is thus the sum of the individual delay intervals preceding t-except perhaps the last of them, which may include the moment of observation. With the aid of the above formalization the total time up to observation moment t may be presented &the sum of an effective time tp and delay time &, during the first of which crack growth can be considered as unaffected by overloads, whereas during the second-no growth is assumed at all.

(1)

t=tp+td.

The effective ( I$ and delay (fd) times are represented in Fig. l(c) by curves 1 and 2 respectively. Note that, when the effective time increases, the delay time remains constant and vice versa; their sum equals the total time t at any moment. Generally, an individual delay interval depends on the degree of overload and crack length at the moment of the former; crack length in turn depends on the effective time. Since the degree of overload, number of overloads and the interoverload times, which determine the relative fractions of the effective and delay times are random-so are the individual delay intervafs. This in turn permits stochastic representation of the effective time as the difference between the total time elapsed up to the moment of observation c, and the total delay time accumulated up to that moment. b(t) = t-

&,(t).

(2)

Conservative

estimation of crack growth

21

Our objective is to find a stochastic representation of the effective time, compare it with the critical time cclrequired for the crack to become unstable, and eventually derive the probability of the inequality (3) which is the probability of nonfailure, or the reliability of the component. In order to have a conservative estimate of the delay (hence also of the effective) time, we introduce an auxiliary process Z(t) as follows: td(fi-O)+ Z(t)=

Zdiif tdi s ~;+i and tE (Zi, t, +Afrli,

fd(&-o)+‘r,+l

if A&i 2 Ti+l and fE (&y&+Afdi),

(da)

&i(f) if fdi c Ti+l and t E (Ii + ACdi,ti+i -0) and z(ti-o)=

fd(&-0).

W-4

This means that immediately prior to the overload, Z(t) equals the total delay time, but at the overload moment acquires a jumpwise increment of Atd or T, whichever the smaller. Figure 2 illustrates the behavior of the Z(r) process. The upper plot (Fig. 2a) present the effective and delay times and is a reproduction of Fig. l(c). In Fig. 2(b) Z(t) is represented by the dot-dashed graph numbered 1. Before the first 0

0

A

‘I

8

I

c

I

‘2

DE

FG

I

‘3

t

T

TIME

Fig. 2. Schematic representation of actual delay and effective times and auxiliary Z(t) and i,,(l) processes derived from delay and effective times respectively. Curves 1 and 2 in (a) refer to actual effective and delay times respectively; curve 1 in(b) (dot-dashed) refers to process “Z(f) (compared with actual delay time-solid line 2), and curve 1 in (c) (dot-dashed) refers to process tr(r) (compared with actual effective time-solid line 2).

R. ARONE

22

overload occurs at time tl (Point A on the upper absciss of Fig. 2), the actual delay time and auxiliary process Z(t) coincide and equals zero. At moment tl (point A), Z(t) increases jumpwise to the value of the first individual delay interval (Zr = A&). The actual delay time increases gradually to the value Atdl, reaching it at point B (solid line 2 in Fig. 2a). Since the interoverload time ~~exceeds A&,,, Z(t) remains constant throughout it and between points B and C coincides with the actual delay time. At moment t2 (point C) the second overload occurs and Z(t) acquires a new increment of A cd2and so on. If for instance, A cd2exceeds the corresponding interoverload time (in this case Q), than the increment Z(t) would equals TV,so that Z(t) would coincide with the actual delay time at least at the moment of the next overload. This is the way in which the auxiliary jump orocess Z(t) is constructed. This process coincides with the actual total delay time part of the time, and always at overload moments before a jump. The interval rf between the last overload and the observation moment t is a backward recurrent time and for a Poisson flow of overloads (see below) tends asymptotically to the same exponential distribution as the interoverload time 7;-[8]; this means that rf can be included among the Ti’s; thus Z(t) is determined up to the observation moment t, which has the same properties as any overload moment. The auxiliary process underestimates the total delay time, since where the individual delay interval exceeds the interoverload time, this excess is disregarded. Further assumptions are: (a) the probability of occurrence of an overload in the time interval At (base-line single cycle time) can be given as

P = qAt + O(At)

At < min( cd, tp, t)

(5)

while that of two or more overloads within this interval is smaller by one order, (b) the probability of occurrence of K overloads is history-independent, and the flow of overloads is stationary. thus creating a stationary Poisson flow. It was mentioned above that the magnitude of the individual delay interval depends on the elapsed effective time and thus in turn, in accordance with eq. (2)-on the delay time. Also, for the process Z(t) this is always true at overload moments. In other words, the magnitude of the Z(I) jump at an overload moment depends only on the value Z(t) at that overload, but not on its previous values. This enables Z(t) to be regarded as a purely discontinuous Markov process governed by the Kolmog~~r~~v-Feller integrodifferential equation[o, IO]

WT,

x1 t,

St

y) --=

I

-qF(T,X,t,y)+q

P( t, z, y) d,F(T, x, t, z),

7-c

7;

2 E

cl,

(6)

fl

where F(r, x, t, y) is the transition probability of the Markov process, i.e. the probability of the auxiliary process Z(t) being less than y at moment t, subject to the condition that at the preceding moment r 6 t Z(t) = x, q is as per eq. (S), and P( t, t, y) is the conditional probability distribution function defining the probability of the total delay time being less than y just after the jump subject to the condition that a jump occurred at moment t when Z( t - 0) = Z The distribution function P( t, 2, y) determines the magnitude of the Z(t) jump and should account for physical dependence of the individual delay intervals on the total delay time and on the magnitude of the overload. It should also reflect the requirement that the jump of the Z(t)-process should equal the smaller of two values: the individual delay interval or the interoverload time, i.e. AZ(t), = min(Atdi, Tii+r)‘

(7)

Thus if P,(t, 2, y) is the conditional probability distribution function for the individual delay interval and P*(t, Z, y&the same for the interoverload time, then

P(f, z

Y) =

Pl(L

z

y)+

P2(t.

z, y) -P*(r, z, y)P2(t, z, y),

09

Conservative

estimation of crack growth

23

The process Z(t) starts at time zero when its value is also zero (no delay time at the beginning the process) and Z(t) is a nondecreasing function of time (Z=S y). Equation (8) thus yields

where

of

for simplicity

F(O, 0; f, y) = F(4 y). The limiting

behavior

of the above

function

(l(N

is as follows:

lim F(0, 0; t, y) = F(0, y) = E(0, y) = (y iz: i z i

Equation (9) can be solved numerically the initial condition. For this purpose

by a simple step-by-step eq. (9) is rewritten as

+

qAt i

j=l

procedure,

(11)

with eq. (11) stating

P*(G,zj, y/c)(F(ti,zj) - F(t;, zj-l)),

(12)

where F( ti, 2,) = 0 for II < 0 and P*( t, z, y) is a properly chosen value in the appropriate F( T, z) interval. It is seen that all values of the transitional probabilities at time ti+i are expressed through their predecessors at time ti. Thus if the initial values of the transitional probability are available, the whole array of its values can be evaluated by the step-by-step procedure embodied in eq. (12). The initial values are obtainable by substitution of condition (11) in (12) which yields

F(t,, y,J = FW, yd = Cl- qAOE(O,yd + qA@(O,0, yd.

(13)

the eqs (12) and (13) provide the numerical procedure for determination of F(t, y). We define the reliability as the probability of nonfailure during a given time interval (0, 7). As was mentioned above. there is no fracture if

Thus

{Sup tp(t) < Lk To find the probability

of inequality

t E (0, 7).

(14) let us consider

another

(14) auxiliary

&= t-Z(t).

process (15)

This process is indicated in Fig. 2(c) by the dot-dashed line 1. As can be seen it coincides with the actual effective time tp during part of the time, but always at the overload and observation moments [compare curves 1 in Fig. 2(a) and (c)l. It differs from tp( t) during the delay intervals as shown by the solid line 2 in Fig. 2(c) but coincides with the actual effective time in the same time intervals where fd( t) = Z(t). The auxiliary process $(t) within the intervals of coincidence with the actual effective time, and especially at overload and observations moments, forms a nondecreasing sequence of values &,= r,,. Thus inequality (14) can be reduced to (6 < &,} at arbitrary or, bearing

in mind eq. (15)

observation

moment

t E (0, T)

(16)

23

R. ARONE {Z(r)

*

t-

t,,}.

(17)

now, using the solution of eq. (9), a reliability assessment can be given as follows

R=

1 - F( t, 8- f,,) 1

l

for t 2 tcr for t < tcr.

(18)

Example

In our example we consider an edge crack propagating in a half-space under a constant amplitude base-line cyclic ~zero-tag-tensions load with surer-imposed random overloads. The stress intensity factor range is in this case AK = l.l’LA&%a. We assume that crack propagation

(19)

under a base-line cyclic load is governed

$ =C(AK)“’ and within the overload affected zone-by

by the power law

Gw

the expression

where C and m are the power-law parameters and Cn s 1 is the retardation factor dependent on the magnitude of the overload. It was shown in earlier work [ 1 l] that under the above assumptions the individual delay interval A&i can be expressed directly through the effective time tp Atdi=

J

uO+dda/CnC(AK)” %

-

Jz‘”cczi,,,,=B(Sg

- l)(l - l/y’“~2”2)(1/a~m~2~‘2), (22)

where a~ is crack length at overload, d is the dimension of the overload-affected zone, ~=2/C(l.l2Aa~%)‘“(m-2); y is the crack ratio, i.e. r=(a~+d)laoc 1, and S*, is the reciprocal of C&--the latter being the “mean ” value of CR within the overload-affected zone found from the condition

J

a,,++

a,,+d

dalC~C(AK~m

= I/C;

a,,

J

dalC(AK)“.

%

Similarlyt

tp=

da/C(AK)”

1

1

-=-(La&m-2ttZ B

= L- Blai;“-2”2

(23)

tp),

(24)

where ai is initial crack length and t, = Blai (m-21’2. Combination Whe

argument

t in processes

+,(t) aAd i,(t) is omitted

of eqs (22) and (24) yields

in what follows for simplicity.

Conservative

estimation

of crack

Atdi = @(L-

growth

25

CJ,

(25)

where CD= (SX- l)(l - l/y (m-2)‘2) depends on the magnitude of the overload (through on the crack ratio y (assumed to be unsensitive to crack length), but not on the effective time. Using eq. (2), eq. (25) can be rewritten as follows: A~di=qtm-(t-

Cdi)].

Si) and or delay

(26)

It is thus seen that the individual delay interval is dependent on the total delay time fd at the overload moment, but not on td values at preceding times. Since the auxiliary process Z(f) coincides with the actual delay time at least at the overload and observation moments, we may write (27)

AZ=@{&,-[t-Z(t)]}. Denoting by Z,, the value of the process we can rewrite eq. (27) as follows

after the jump,

and bearing

in mind that AZ = Z, - %

Z,=Z+@[t,-((t-z)].

(28)

If the magnitude of the overloads is random, then parameter @ in eq. (27) is also random. In principle, if the distribution function of the overload peaks is known, its counterpart for the parameter CDcan be derived. In practice, however, this is hardly possible since the functional dependence of @ on the overload is rather complicated with some of the relevant material parameters still unknown. A more practical way would be to assign to @ some model distribution and compare it with experimental results. The assigned distribution should obviously allow for the fact that increase of the overload leads to increase of @ and thus to increase of the delay interval. Now we seek the probability Z, < Y at moment t, given that Z(t) for that moment is known and equals, say, Z. Bearing in mind eq. (28), this probability can be given as follows:

~l(t,~,Y)=PiZ~y/z(r)=z}=P(Z+~[L-_(t-_Z)1
Assuming

for @ an exponential

distribution

with parameter

(29)

A, we obtain

the following

expression

for pr(t, 2, y)

l-exp

P,(r, 2, y) =

I

Y-Z L-(t-Z)

-A

0

\

for y > Z (30) for y C Z.

To find the probability distribution function P(t, Z, y) as per eq. (8) the distribution function for the interoverload time P2(f, Z, y) should be known. As mentioned above, if the individual delay interval Afdi exceeds the interoverload time 7, then AZ=r

(314

or

z,=

7+z.

@lb)

R.ARONE

26

Thus P2(~,Z,Y)=PKl~yIZ(t)=Z)=P{T+Z
For a Poisson flow of overloads, the interoverioad distributed (with parameter q) random variables. Thus

(32)

times are independent

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

if y>Z if y G Z.

P2(4z y) = {*

exponentially

(33)

The obtained expressions for Pr(t, Z, y) and P2(t, Z, y) permit numerical solution of eq. (12) with Pfr, z, y) as per eq. (8) and the initial conditions-as per (11). A computer program based on the step-by-step procedure embodied in eqs ( 12) and ( 13) was worked out, and the probabilities of nonfailure over a prescribed time (number of cycles) were determined, based on. the following values of the relevant parameters: (a) Propagation law parameters m and c-3 and lo-” respectively (the latter value for crack length in mm and stress intensity factor in kg mm-“‘*). (b) Base-line cyclic zero-to-tension stress range CT= 225 MPa and average overload magnitude u. = 450 MPa. (c) Crack ratio y = 1.l. (d) Initial crack length ai = 0.002 M and critical crack length a,,=O.Oll M (it was assumed conservatively that fracture occurs at an average value of the overload, for a material with Kr, = 95 MPa&) . For the above parameters t, = 15750 cycles and & = 9100 cycles. Two materials were compared one with an average retardation factor of 0.5 (Sg = 2) and the other-with 0.25 (Si = 4) at the same overloading process. Equation (25) yields the following average values of @: cP1= 4.65 - lo-* (for St = 2) and a2 = 0.14 (for Sg = 4). Accordingly two values of parameter h (as per eq. 30) were used for the calculations: h1=$=21.505 I

and

&=$=7.14, 2

Some of the calculation results are shown in Figs 3 to 6 in which crack length is plotted against the number of cycles; both variables are given for convenience in normalized form, namely as ratios of actual-to-critical crack length (Q/U,,) and actual-to-base-line life time (N/N,,) respectively. Figure 3 ilfustrates the influence of the parameter q on crack propagation behavior; All curves corresponding to the 90% quantile and Si = 2 (AI = 21.505); hence curve 1 refers to the base-line cyclic loading, and accordingly calculated by the usual deterministic procedure, using the power law, the solid curves 2, 4,6 refer to conservative estimates for q = 5 x 1V4; 8 x 10B4 and 10 x 10d4 respectively and the dashed curves 3, 5,7-to nonconservative estimates for the same respective vahres of q[l 11. It is seen that at low values of q there is no disparity between conservative and nonconservative estimates. As q increases, the difference grows too. Figure 4 which represents the 99O/oquantile counterparts to Fig. 3, shows that the difference varies inversely with the reliability level; the higher the latter, the smaller the difference. At the same time the difference depends significantly on the retardation coefficient, i.e. on the parameter h, as is seen from the comparison of the 99% quantiles for two values of A plotted in Figs 4 and 5 (h = 21.505 corresponding to S$ = 2 in Fig. 4 and A = 7.L4 corresponding to S$ = 4 in Fig. 5). It turns out that decrease of A (increase of Sa leads not only to a longer life-time at equal q-values but also to a significant increase in the above difference. However, the influence of the reliability level on the difference between the estimates is much more significant for higher values of the retardation coefficient S i. This is seen from comparison of the

Conservative

21

estimation of crack growth

I.0

0.8

8.6

8. 4

6.2

Y.H

.w

l

----6.5

I

.e

I 1

1

.s

NORMALIZED

2.0

LIFE

TIME.

N/NCcr>

Fig. 3. Dependence of 90% quantiles of crack length on overload intensity parameter q (for A = 21.505); curve 1 refers to base-line constant-amplitude cyclic loading (no overloads); solid curves 2,4,6 refers to conservative and dashed curves 3,5,7-to nonconservative estimates. Curves 2.3: 4.5: and 6, 7 refer to q = 5 x lo+; 8 x 10e4 and 10 X 10S4 respectively.

1.0

9.8

0.6

6.4

0.2

6.6 .e

I 6.5

I 1.6

I 1 .s NORIIALIZED

I 2.0 LIFE

TIME.

N/N

Ccr

>

Fig. 4. Dependence of 99% quantiles of crack length on overload intensity parameter 4 (for A = 21.505); curve 1 referring to base-line cyclic loading coincides with both conservative and nonconservative estimates for 4 = 5 X lo-“ solid curves 2,4 refer to conservative and dashed curves 2,5-to nonconservative estimates. Curves 2, 3 and 4, 5 refer to q = 8 x lo+ and 10 x 10m4 respec. . ttvety.

R. ARONE

.a

I

8.S

1 I

.e

----l-v .s 1

NORHALIZED

LIFE

TIME.

NfNCcr3

Fig. 5. Dependence of 99% quantiles of crack length on overload intensity parameter 9 (for A = 7.14); curve 1 refers to base-line loading; solid curves 2,4 refer to conservative and dashed curves 3.5-w nonconservative estimates: curves 2,3 and 4.5 refer to 4 = 8 x IOY and 10 x IO-” respectively.

a. e

8.0

8. 4

8.0

I 8.6

I

I

5,s

I .8

I

i

1 .s NORMALIZED

2.e LIFE

TX&E,

N/Ntcr3

Fig. 6. Dependence of 95% quantiles of crack length on overload intensity parameter 4 (for h = 7.14): curve 1 refers to base-line cyclic loading; solid curves 2,4 refer to conservative and dashed curves 3, S-to nonconservative estimates; curves 2,3 and 4,5 refer to 4 = X x IOP and IO x IO-“ respectively.

Conservative

estimation

of crack

growth

29

plots in Figs 5 and 6, representing reliability levels 99% and 95% respectively at A = 7.14 (& = 4). For instance, for q = 10 X 10e4 the difference at 99% equals approximately one base-line lifetime, where as at 95% it tends to infinity (since nonconservative estimation predicts full arrest in this case; see curve 5 in Fig. 6).The above example shows that the most significant difference occurs where high frequencies of the overloads, or high retardation effects, or both, are involved. This circumstance is particularly marked at lower levels of reliability, but much less so at 99% reliability and above. Since important structures normally required high levels of reliability, the difference between the estimates for such structures should be moderate. Thus the procedure yields reasonable gap between the lower and upper life-time bounds and some intermediate value could be taken as the normative life-time for the prescribed reliability level. SUMMARY

AND CONCLUSIONS

(1) A conservative

stochastic model describing fatigue crack growth under random overloads superimposed on a base-line cyclic load is presented. (2) Calculations comparing conservative and nonconservative life-time estimates for the prescribed reliability level were performed, and the differences estimated. (3) It was shown that three main factors influence the difference between conservative and nonconservative estimates: (a) overload intensity (q), (b) retardation intensity (SX or A), and (c) the prescribed reliability level: the higher q and the lower A (higher S;l) and the reliability level-the larger the difference between estimates. (4) For important structures characterized by high reliability requirements (above 0.99) the difference between the estimates is relatively moderate, thus permitting convenient estimation of the lower and upper life-time bounds for the prescribed reliability level. REFERENCES [I] [2] [3] [4] [5]

[6] [7]

[8] [9] [lo] [I I]

R. Arone, On retardation effects during fatigue crack growth under random overloads. Engng Fracture Me&. 27. 83-89 (1987). J. Schijve, Observations on the prediction of fatigue crack growth propagation under variable-amplitude loading. Fatinue Crack Growth Under Swctrum Loads. ASTM STP 595, 3-23 (1976). S. Matsuoka, K. Tanaka and M. Kawahara, The retardation phenomenon of fatigue crack growth in HT 80 steel. Engng Fracture Mech. 8, 507-523 (1976). W. J. Mills, R. W. Herzberg and R. Roberts, Load interaction effects on fatigue crack growth in A5 14F steel alloy. Cyclic Stress-Strain and Plastic Deformation Aspects of Fatigue Crack Growth, ASTM STP 637, 192-208 (1977). R: I. Stephens, E. C. Sheets and G. 0. Njus, Fatigue crack growth and life predictions in man-ten steel subjected to sinale and intermittent tensile overloads. Cyclic Stress-Strain and Plastic Deformation Aspects of Fatigue Crack Grow&, ASTM STP 637, 176-191 (1977). . C. Bathias and M. Vancon, Mechanisms of overload effects on fatigue crack propagation in aluminium alloys. Engng Fracture Mech. 10,409-424 (1978). J. B. Chang, R. M. Engle and J. Stolpestad, Fatigue crack growth behaviour and life predictions for 2219-T851 aluminium subjected to variable amplitude loadings. Fracture Mechanics, Thirteenth Co@., ASTM STP 743,3-27 (1981). D. R. Cox, Renewal Theory. Wiley, New York (1962). A. T. Bharucha-Reid, Elements of the Theory of Markou Processes and Their Applications. McGraw Hill, New York, (1960). B. V. Gnedenko, The Theory of Probability. Chelsea, New York (1968). R. Arone, Fatigue crack growth under random overloads superimposed on constant-amplitude cyclic loading. Engng Fracture Mech. 24, 223-232 (1986). (Received

19 February

1987)