PSI - Issue 75

Sgamma M. et al. / Procedia Structural Integrity 75 (2025) 709–718 Author name / Structural Integrity Procedia 00 (2025) 000–000

3

711

σ ′ f

σ ′ f 2 σ y

P FS = 1 + ν E

c · 1

b

b + 1 + ν P ε ′ f 2 N f

2 N f

2 N f

E

+ k

(3)

,

where E is the elastic modulus, ν E and ν P are Poisson’s ratios, and σ ′ f , ε ′ f represent axial fatigue strength and ductility coe ffi cients, respectively. Equation 3 allows engineers to evaluate multiaxial fatigue life using axial strain–life data rather than torsional test results. Although the original FSC, shown in equation 1, was formulated under constant-amplitude multiaxial loading, real world service conditions frequently involve variable-amplitude or random loading. Consequently, significant research was devoted to extend P FS to account for these non-constant loadings. For example, Shamsaei et al. [38] investigated block and amplitude load scenarios with di ff erent multiaxial load paths, applying Palmgren–Miner’s linear damage rule to sum partial fatigue damages across multiple loading segments. Similarly, once each pair of strain amplitude and maximum normal stress is extracted by the multiaxial rainflow procedure [2], engineers can accumulate damage across the strain–stress pairs to address the entire load history in a consistent manner. A key hurdle in formulating multiaxial fatigue criteria for frequency-domain applications is the shift from de terministic inputs (e.g., explicit time histories of stress and strain) to probabilistic approaches built upon statistical descriptors of random processes. In other words, instead of relying on fixed values from measured signals, one must use statistical estimates derived from the moments of the power spectral density (PSD) and corresponding probability distributions. Moreover, the classic Fatemi–Socie criterion has been traditionally defined for constant-amplitude loading sce narios with potential phase shifts. When extending this criterion to variable-amplitude conditions in the frequency domain, one must consider later developments that incorporate non-constant loading states. Bannantine and Socie [1] proposed such an approach employing the critical-plane concept alongside rainflow cycle counting. In their frame work, each cycle of shear strain on the critical plane is first identified from the time history. Subsequently, the maxi mum normal stress over the duration of that same cycle is determined. This procedure yields multiple pairs of strain amplitudes and maximum normal stresses, facilitating a final damage calculation by summing partial damages via the Palmgren–Miner rule. Extending the Fatemi–Socie parameter from a single constant-amplitude pair to many amplitude–stress pairs cap tures the essence of variable loading conditions without sacrificing theoretical consistency. However, utilizing only the PSD matrices of stress and strain for such an adaptation introduces several additional challenges: • determining the orientation of the critical plane by either the maximum-damage or maximum-variance criteria; • substituting the time-domain rainflow counting process with spectral methods that predict rainflow amplitude distributions; • replacing the maximum stress value found in each rainflow cycle with a distribution of maximum stresses for the corresponding strain amplitude interval. Within the context of frequency-domain (spectral) analysis, similar strategies as those used in the time domain can be employed to identify the critical plane. Two prominent methods are the maximum damage approach and the maximum variance approach. In the first method, the fatigue damage is computed over numerous possible critical plane orientations; the orientation that yields the highest fatigue parameter is then chosen. Though accurate, this procedure actually is computationally prohibitive and is thus not frequently adopted in practice. Alternatively, the maximum variance technique focuses on maximizing the chosen fatigue parameter directly. In the case of the Fatemi–Socie parameter ( P FS ), this requires expressing P FS in terms of the variances of the pertinent 3. Frequency-Domain Formulation 3.1. Determining the critical plane orientation