ISSN 1318-0010
KZLTET 32(3-503)165(1998)
PROBABILITY FUNCTION OF FATIGUE CRACK GROWTH IN CASE OF MATERIAL OVERLOADING
VERJETNOSTNA FUNKCIJA UTRUJENOSTNEGA [IRJENJA RAZPOKE V PRIMERU PREOBREMENITVE MATERIALA
NENAD GUBELJAK, J. LEGAT
Fakulteta za strojni{tvo, Maribor, Slovenija
Prejem rokopisa - received: 1998-12-06; sprejem za objavo - accepted for publication: 1998-12-14
Studies of structural components under a fatigue environment including overloads are rather interesting in engineering practice today. One important task is to determine the probability distribution function in consideration of two failure modes by a selected fatigue load and overload level. The probability density function permits to evaluate of overloading under fatigue environment on the residual life or to predict the failure of a component. The fatigue crack propagation in case of overloading can be modeled by the Markov's process. The obtained probability functions describe the probability of retarded crack growth propagation as a consequence of the overload cycle.
Key words: fatigue crack propagation, probability function, Markov's process
Utrujenostno {irjenje razpoke na konstrukcijskih materialih ali elementih v primeru preobremenitve je danes deležno intezivnega raziskovanja, kajti z lokalno preobremenitvijo je mogoče hitrost utrujenostnega {irjenja razpoke upočasniti ter s tem delno podalj{ati dobo funkcionalnega obratovanja konstrukcije. Za kvalitativno ocenitev je potrebno določiti verjetnostno porazdelitveno funkcijo s katero lahko primerjamo zanesljivost utrujenostnega {irjenja razpoke pod monotonim dinamičnim obremenjevanjem brez in z enim preobremenitvenim nihajem. Če z verjenostno funkcijo zajamemo učinek probremenitve, takrat lahko napovemo preostalo življenjsko dobo oziroma poru{itev konstrukcijskega elementa. Za izpeljavo verjetostne funkcije je opisano utrujenostno {irjenje razpoke s procesom Markova. Dobljena verjetnostna funkcija verjetnostno opisuje upočasnitev utrujenostnega {irjenja razpoke kot posledico preobremenitvenega cikla.
Ključne besede: utrujenostno {irjenje razpoke, verjetnostna funkcija, Markov proces
1 INTRODUCTION
The crack life time, which practically means the time of fatigue crack propagation up to the critical length, can
Crack legth, a
Figure 1: Fatigue crack growth behaviour in a single overload test Slika 1: Utrujenostno {irjenje razpoke pri enem preobremenitvenem preizkusu
be stretched by overload tests1. If the specimen is overloaded on the tip of the fatigue crack, a overload plastic zone is formed, Figure 1.a. Let us assume that after pre-cracking of a number identical specimens up to a certain crack extension one part (group B) of specimens was submitted to one overload cycle (state 1) and the other part of specimens (group A) was not overloaded (state 0). After that both groups of specimens were fatigued at a constant ÀK. As a consequence, the crack propagation rate of the group B specimens is retarded as long as the tip of the fatigue crack is located within the plastic zone of the overload cycle. The aim of this work is to find the probability function of all the specimens (group A and B) to separate statistically the fatigue behaviour of the two groups of specimens, and to determine the probability that after certain number of cycles the crack tip of group B specimens is still within overload plastic zone.
2 THE MODEL
As stated by J. Tang and B. F. Spencer Jr.2, the fatigue crack propagation is a lognormal random process. To determine experimentally the probability function
da dN
, N
1
1flnN-|iö
V-N-Ö2P
eL 2| v
(1)
of the crack growth rate da/dN after a determined number of cycles N at a given value of stress intensity range ÀK, at least 12 specimens are needed3,4.
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N. GUBELJAK, J. LEGAT: PROBABILITY FUNCTION OF FATIGUE CRACK..
P0(N+AN) =
N).AN].P0(N)+h( N).An.P1(N)
and for the state 1 process, i.e., the group B specimens with the tip located inside of the overload plastic zone:
P1(N+AN) =
= [1-h(dN N) AN] P1(N)+1(d!NN, N) An P0(N) (3)
Both I and h denote the intensity of the crack growth level (da/dN), I for state 0 and h for state 1. The expressions l-lAN and l-hAN represents the probability for maintaining state n=0 or n=1. The equations (2) and (3) can be expressed as differential equations:
dP°(N) = -KdN, N).P0(N)+h(^a, NMN-PXN)
dN
dPt(N) dN
,da
W
= -h(|N, N)-P1(N)+1(^da, N)-AN-P0(N)
(4)
(5)
The initial conditions are:
P0(0) = 1	(6)
Pj(0) = 0	(7)
The parameters I and h depend on the crack growth rate and the number of cycles N. The intensities I and h of each crack growth state, are expressed as3:
da f0 (dN N
for n = 0:	N) = —dN-
dN R0(N)
^ f1 da N)
da	1 dN
for n = 1:	, N) =-
dN	Rj(N)
(8)
(9)
Rn(N) = Jf(jN, N)dN for n = 0,1
(10)
Figure 2: Markov's procesess in case of overload Slika 2: Markov process v primeru preobremenitve
m is the expected value of baseline crack growth rate and V is the variance of the lognormal distribution function.
The fatigue process of the specimens can be modeled by the Markov's process shown in Figure 2.
The probability for the state 0 process, i.e., for the specimens of group A or for the specimens of group B, if the crack tip is located outside the overload plastic zone, is given by:
The solution off the differential equations (4 and 5) were obtained by the Euler method. The results belong to specified numbers of cycles Nj(j=1,...,i) with computed values of parameters I and hj by eq. (8) and eq. (9), (step h=Nj+1-Nj).
P0J+! = P0,j + h(-1j.P0,j + hj-Pj	(11)
Pj = Pu + h(-hj-P1,j + 1j-P0,j)	(12)
The probability functions P0(N) and P1(N) are mirror symmetrical around the values P0(N)=P1(N)=0.5. The in-crese of variance V1 makes the confindence interval (N1,min, N1,max) wider, as shown in Figure 3, but has only small effects on the probability functions P0(N) and P1(N). More influence has the increase of the expected value h1, as shown in Figure 4.
(2) >
Figure 3: The influence of standard deviation V1/V0 on the probability function P0(N)
Slika 3: Vpliv spremembe standardne deviacije (v razmerju V1/V2) na verjetnostno funkcijo P0(N)
where the reliability Rn(N) is given by:
Figure 4: The influence of expected values m1/m0 on the probability function P0(N)
Slika 4: Vpliv spremembe pričakovane srednje vrednosti (v razmerju m1/m0) na verjetnostno funkcijo P0(N)
n
504
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N. GUBELJAK, J. LEGAT: PROBABILITY FUNCTION OF FATIGUE CRACK...
The solution of the differential equations (4) and (5) by applying the convolution integral is given as source of the probability density function
fj(x) = ■
. •e"(i+m)x
(13)
(2j-1)!
where x = N/Nj.
The obtained probability density function can be used to predict the interval of the number of cycles until the postoverload fatigue crack grows out of the overload affected zone.
3 CONCLUSION
The obtained probability function determines the moment (number of cycles) when the postoverload fatigue
crack grows out of the overload affected zone. The purpose of the probability function is to estimate the effect of overloading on the extension of the residual life of a structural component without risk of final failure.
4 REFERENCES
1M. Darvish, S. Johansson: Engineering Fracture Mechanics, 52 (1995) 2, 295-319 2 J. Tang, B. F. Spencer Jr.: Engineering Fracture Mechanics, 34 (1989) 2, 419-433
3N. Gubeljak, A. Kuhelj: Strojniški vestnik, 39 (1993) 1/2, 62-64 4H. Ghonem, S. Dore: Engineering Fracture Mechanics, 27 (1987) 1, 1-25
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