PSI - Issue 39

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

243

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Fig. 4. Probabilistic description of the kinetics of cracks and fatigue failure taking into account randomness of the parameters a 0 , m and ω

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

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

(14)

} 1 / 

P

M

n

( ) {

j tr a a 

Table 1. Parameters of the numerical study.

Parameter type (deterministic / probabilistic))

Distribution / Value

Parameter

The average value of the internal pressure in the loading cycle , p 0 , MPa

Deterministic Deterministic Deterministic Deterministic Deterministic Deterministic Probabilistic Probabilistic Deterministic Deterministic Deterministic Deterministic

8,0 0,9 0,8 1.6

Stress ratio

Rang of the internal pressure in the loading cycle, Δ p , MPa

Overload factor

Number of the overload cycle, i OL

5000

Initial crack depth, a 0 , m

Probabilistic, Normal N (1∙10

-3 , 1∙10 -4 ) 2,9 ·10 -11

Parameter C of the Paris equation, m/cycle (Pa√m) -m

Parameter m of the Paris equation

Uniform U (2,8; 3,2) Uniform U (2,0; 2,2)

Retardation exponent, ω

Radius of the middle surface r 0 ,m Thickness of the wall δ , m

0,63 0,025 201,6 61.0

Hoop stress S , МПа

Fracture toughness K Ic , MPa√m

Thus, an estimate of the failure probability can be obtained using the Monte Carlo method by repeatedly solving Eq. (6) and comparing the value of the crack depth a N ( j ) achieved after N loading cycles with the corresponding value of the critical crack depth a C for various combinations of random parameters a 0 , m and ω. Since within the framework of a priori estimate, all combinations of parameters are equally probable the probability of the component failure P' F ( N ) after N loading cycles will be equal to the ratio of the number of M' C crack growth curves from the sample j = 1,2, ..., M n , for which the failure condition (11) is satisfied, to the total number of crack growth curves M n : ( ) ( ) C F n M N P N M    . (15)