PSI - Issue 39

Dmitry O. Reznikov / Procedia Structural Integrity 39 (2022) 256–265 Reznikov D.O./ Structural Integrity Procedia 00 (2019) 000–000

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In accordance with the selected probabilistic distributions of stochastic parameters, random combinations of these parameters { a 0 ( j ) , C ( j ) , m ( j ) , Σ ( j ) } ( j =1,2,…, M n ) are generated. Further, for each combination of parameters, the following procedure is performed: - integration of equations (1) and (5) to obtain the crack growth curve Cv ( j ) corresponding to the j th combination of random parameters { a 0 ( j ) , C ( j ) , m ( j ) , Σ ( j ) } (Fig. 4) and representation of the dependence of the crack depth on the number of regular constant amplitude cycles N r and the number of overloads N k :     ( ) , j j Nr Nk r k a a N N   ( j =1,2,…, M n ); - verification of the fulfillment of the failure condition (11) (in the form ( ) j Nr Nk C a a   ).

Fig.4. Statistical modeling of fatigue crack growth according to a block loading simulation scheme.

It is assumed that the events “ Sampled j-th crack growth curve Cv ( j ) c oincides with the true crack growth curve Cv tr ” { Cv ( j ) = Cv tr }( j = 1,2, ..., M n ) form a set of collectively exhaustive events. In this case the assumption is made that all combinations j = 1,2, ..., M n of parameters { a 0 ( j ) , C ( j ) , m ( j ) , Σ ( j ) } are equally probable. It means that the probability of the event { Cv ( j ) = Cv tr } is equal to:

, ( j = 1,2, ..., M n ).

(20)

} 1/ 

P

M

n

( ) {

j tr Cv C 

Thus, an estimate of the fracture probability can be obtained using the Monte Carlo method by repeatedly integrating Eq. (1) and Eq.(5) and comparing the value of the crack depth ( ) j Nr Nk a  achieved after N r regular loading cycles and N k overloads with the corresponding value of the critical crack depth a C for various combinations of random parameters { a 0 ( j ) , C ( j ) , m ( j ) , Σ ( j ) }. Since according to the assumption that all combinations of parameters are equally probable the probability of the component failure P F ( N r , N k ) after N r constant amplitude loading cycles and N k overloads will be equal to the ratio of the number of crack growth curves M C from the sample j = 1,2, ..., M n , for which the failure condition ( ) j Nr Nk C a a   is satisfied, to the total number of crack growth curves in the sample M n : ( , ) ( , ) C r k F r k n M N N P N N M  (21) For the numerical implementation of the described procedure, a file program was written in the Matlab environment, which allows forming samples of values of random parameters from the most frequently used probabilistic distributions, including exponential, uniform, normal, logarithmical, Weibull, etc. Applying the Matlab algorithms for generating random numbers from the selected probabilistic distributions of the parameters a 0 , C , m , and Σ, the procedure of multiple integration of equations (1) and (5) and estimation of failure probability by Monte Carlo method according to expression (21) is implemented.