PSI - Issue 75

Thomas Constant et al. / Procedia Structural Integrity 75 (2025) 660–676 Author name / Structural Integrity Procedia 00 (2025) 000–000

664

5

and Manson, ∆ ϵ p

c may be use to map the plastic strain amplitude and fatigue life in the low-cycle fatigue

2 = ϵ ′ f (2 N f )

(LCF) regime. By combining these two parts, the total strain-life relationship becomes:

σ ′ f E

∆ ϵ 2

b

+ ϵ ′ f (2 N f ) c

(2 N f )

(8)

=

with ∆ ϵ =∆ ϵ e +∆ ϵ p . In practical applications, systems are subjected to multiaxial loadings, requiring the use of appropriate methods. A comprehensive review of multiaxial fatigue models can be found in e.g. Socie and Marquis (2000). Here, a strain critical plane based approach is selected, and aims at identifying the critical planes within the material where fatigue damage accumulation is most likely to occur. Brown and Miller (1973) proposed a fatigue criterion tailored to multi axial low-cycle fatigue (LCF), with particular emphasis on the initiation and early growth of cracks. They concluded that two strain parameters are essential to characterize the fatigue process, namely the cyclic shear strain and the normal strain identified on the plane of maximum shear. From a physical standpoint, the cyclic shear strain primarily contributes to crack nucleation, while the cyclic normal strain assists in their growth. Ultimately, Brown and Miller suggested to adapt the total strain-life equation 8 to multiaxial loading such as:

σ ′ f − σ n , mean E

∆ γ max 2

∆ ϵ n 2

b

+ C 2 ϵ ′ f (2 N f ) c

(2 N f )

= C 1

(9)

+

with ∆ γ max the maximum range of experienced cyclic shear strain, ∆ ϵ n the range of cyclic normal strain and σ n , mean the mean normal stress of the current cycle n . C 1 and C 2 are empirically determined coe ffi cients. Algorithm 1 in Appendix A outlines the critical plane fatigue assessment procedure based on the Brown–Miller criterion. It takes as input the output of the model m A , see Equation 1, along with a set of structural and material parameters. The stress concentration factor k t , is used to convert global stresses into local stresses to take into account the geometrical characteristics of the studied zone. The material parameters include the Young’s modulus E , Poisson’s ratio ν , and fatigue properties σ ′ f and ϵ ′ f , which are required inputs to the Brown–Miller criterion, see Equation 8. In addition, the 0.2% yield strength R p 0 . 2 and the plasticity parameters K ′ and n ′ from the Ramberg–Osgood model are also required for the damage assessment. In this study, k t , σ ′ f , ϵ ′ f and R p 0 . 2 are considered random and gathered in the random vector X B . So far, we have described one common approach to evaluate the damage in critical locations of a system subjected to cyclic loading. Practically, this damage is the outcome of a complex simulation chain, here denoted as d = m B ◦ m A ( x A , x B ), with x A the load parameters which here inform on a given operational condition, i.e. which describe a particular loading configuration. Then, x B gathers the most influential geometrical and material parameters . At this stage, we introduce a performance function g associated with the failure event whose probability is to be estimated. In general, an industrial system will not operate under the same conditions throughout its lifetime, and is more likely to experience a variety of load conditions. To take this into account, several elementary life situations, let say n c , are usually identified and mixed according to the statistical use of the system. The total damage D is therefore computed by: 2.3. Reliability problem

n c c = 1

n c c = 1

p c m B ◦ m A ( x A c , x B )

p c d c =

(10)

D =