PSI - Issue 39

Dmitry O. Reznikov / Procedia Structural Integrity 39 (2022) 236–246 Author name ./ Structural Integrity Procedia 00 (2019) 000–000

240

5

It should be borne in mind that estimates of crack growth obtained using these models, which do not take into account the interaction of loads and are based on the linear damage accumulation rule, often turn out to be excessively conservative. In particular, if in the process of constant amplitude loading of the component a single tensile load with increased amplitude (overload) occurs, an effect of crack growth retardation is observed while the crack propagates in the zone of the plastically deformed material, that was induced by the overload (Fig.1,range III ). In this regard, a number of models have been proposed that take into account the effect of crack retardation after overloads (Wheeler, O.E., 1972; Willenborg, J, et al, 1971; Newman, J.R., 1982; Noroozi, 2008; Yamada, Y., et al , 2011). In particular, in Wheeler's model (Wheeler, O.E., 1972) , the effect of crack retardation is taken into account by introducing the crack retardation function � � in the crack growth equation (1): ( ) ( , ) da i f K R dN     , (4) , p OL (5) Here i is the number of the loading cycle , a OL is the crack depth at the instant of the overload, a i is the current crack depth after i loading cycles, r p , i is the size of the plastic zone after the i -th loading cycles, R p,OL is the size of the plastic zone induced by the overload. The crack retardation function ( ) i  changes from cycle to cycle from its minimum value al at the beginning of the retardation period after the overload cycle i = i OL   1 , 1 , / OL p OL p OL r R      to the value * 1 i   that is reached at the cycle i = i * at which the current plastic zone r p* induced by the regular i * loading cycle reaches the boundary of the plastic zone R p,OL , caused by the overload. ω is an empirical exponent depending on the material and loading scheme, which is usually in the range 1 <ω <4 (Sheu, B.C., et al, 1995). The formulated Cauchy problem, which consists of integrating the differential equation (4) with the initial condition (2), can be solved in finite differences (assuming dN = 1, da =Δ a = a i +1 - a i ). For the considered loading mode by cycles of constant amplitude with a single overload in the cycle with the number i OL , the kinetics of the crack will have three sections (according to Fig. 1): , p i ( ) i 1 OL i OL OL i r if a R a r    if a R a r                     , p i , p OL , p i , p OL i a R a  

(6)

1 i i a a i f K R       ( ) ( , ) i i



1

, p i если i i r 

OL

     



   

  





( ) i

if

i i 

, p OL a R a r    OL i



 

, p i

OL

 a R a if i i    , p OL i OL

 

  



1

, p OL a R a r    OL i

, p i

OL

m

1 R        i i K

,

(7)

( , ) f K R C   

i

i

Then the kinetic expression for the crack depth after N loading cycles can be written as: