Stochastic Diffusion Models for Fatigue Crack Growth and Reliability Estimation
Crack propagation is a major task in the design and life prediction of fatigue-critical structures such as aircraft, offshore platforms, bridges, etc. Experimental data indicate that fatigue crack propagation involves a large amount of statistical variati
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B.F. Spencer, Jr. University of Notre Dame, Notre Dame, IN, USA
Abstract Crack propagation is a major task in the design and life prediction of fatiguecritical structures such as aircraft, offshore platforms, bridges, etc. Experimental data indicate that fatigue crack propagation involves a large amount of statistical variation and is not adequately modeled deterministically. The lectures presented herein discuss the basic analysis and use of fracture mechanics-based random process fatigue crack growth models that can be represented by Markov diffusion processes. For completeness, the random variable models are presented as a special case of the random process models. The use of the models in fatigue reliability estimation is also discussed.
K. Sobczyk (ed.), Stochastic Approach to Fatigue © Springer-Verlag Wien 1993
B.F. Spencer, Jr.
186
1. Introduction Failures due to fatigue loading, especially of high performance structures such as aircraft and space vehicles, have always been a main concern of structural engineers. By now it is widely accepted that fatigue failure of a structure involves three stages: (a) crack initiation; (b) crack propagation; and (c) unstable crack growth leading to catastrophic failure [1 ), [2). Because crack propagation often constitutes a large portion of the fatigue life of a structure, researches have proposed many fatigue crack growth models. Unfortunately, at this time the knowledge of the behavior of engineering materials is insufficient for fatigue crack growth models to be based solely upon fundamental physical laws. Generally, phenomenological crack growth models based on the principles of fracture mechanics are of the form (e.g., Bluham [3], Miller and Gallagher [4]) da(t)
--cit = Q (J1K, Kmax• Kc, S, a, R) ,
(1.1)
where a ( t) is the deterministic crack siz~ is the time or cycle number, Q ( ·) is a non-negative function, J1K = Y(a) J1S.J'IT'a is the stress intensity factor range, Y(a) is the geometry factor, J1s is the stress range, Kmax is the maximum stress intensity factor, Kc is the fracture toughness, Sis the maximum stress amplitude in the loading spectrum, and R is the stress ratio. Some commonly used fatigue crack growth models, such as the Paris-Erdogan model [5], the Forman model [6], the hyperbolic sine model [7], and the cubic polynomial model [8], are given respectively by da(t)
dt
da(t)
da(t) (it
(1.2)
'
C(J1K) m (1-R)Kc-J1K'
Cit= da(t) dt
= C(J1K)m
(1.3)
= 10 c sinh[C (1ogAK+C )]+C 1
2
3
4
'
= exp [ C0 + C1 (In J1K) + C2 (In J1K) 2 + C3 (In J1K) a] ,
(1.4)
(1.5)
where the constants C, m, C0, C1, C2, C3 and C4 are determined from experimental crack growth data. Data obtained from experimental tests provide the main source of information regarding fatigue of engineering materials. Rgure 1.1 shows the sample functions of crack length versus cycle number resulting from a statistical investigation of fa-
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Stochastic Fatigue Crack Growth
tigue crack growth reported by Virkler, et al. [9]. A more recent collection of fatigue crack grow
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