PSI - Issue 39

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

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occurs due to its growth through the plastic deformation zone formed by the overload cycle. The retardation rate is proportional to the overload level and, as a consequence, to the ratio of sizes of the regular r p and the overload plastic zones R p,OL . 3. Numerical assessment of the failure probability taking into account the effect of crack retardation Consider a pipeline component with an axial crack on the inner surface (Fig. 3) loaded by an internal pressure varying according to a cyclic law with constant amplitudes with a single tensile overload imposed at the cycle i OL . We will assume that the parameters a 0 , m , and ω of the kinetic equation (6) are probabilistic. The type of distribution can be selected in accordance with available data the specific parameter or level of uncertainty in the problem under consideration. In this example, the parameters m and ω of the kinetic equation are assumed to be distributed according to the law of uniform density, the initial crack depth a 0 according to the normal law: (13) (where α x and β x are, respectively, the left and right boundaries of the interval of values of the quantity x with a nonzero probability density, E { x }, and S { x } are the mathematical expectation and standard deviation of the quantity x ). The values of parameters of the numerical example are given in Table 1. The kinetics of a surface crack in the pipeline component under consideration is described by equation (6) written in finite differences form with the initial condition (2). The brittle fracture criterion (9) is used. Since a cycle-by-cycle approach to describing the kinetics of a fatigue crack has been chosen, the considered Cauchy problem is solved numerically by the finite difference method. 0 0 0 { }) ~ ; ( ~ ; ( U a N E a S a      , ~ { } ( ) ; ), m m C C m U

Fig. 3. The considered structural component In accordance with the selected distribution of random parameters of the model, random combinations of these parameters { a 0 ( j ) , m ( j ) , ω ( j ) } ( j =1,2,…, M n ) are generated. Further, for each combination of parameters, the following procedure is performed: - integration of equation (6) (solution of the Cauchy problem) to obtain the crack growth curve a ( j ) ( N ) corresponding to the j th combination of random parameters { a 0 ( j ) , m ( j ) , ω ( j ) } (Fig. 4) and representing the dependence of the crack depth on the number of loading cycles a N ( j ) = a ( j ) ( N ) ( j = 1,2,…, M n ); - verification of the fulfillment of the failure condition in the form a N ( j ) > a C (within the specified number of loading cycles [0; N ]).