PSI - Issue 75

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

714

6

ϕ µ =

m σ 0 m γ 0

(9)

γ a + σ g ,

ϕ σ = 1 − | r γσ | + 1 −

γ a √ m γ 0 F ( N ) ,

E 0 E P

2 √

m σ 0 1 −

(10)

where m σ 0 is the zero-th spectral moment of the normal stress PSD, σ g denotes the global mean stress, r γσ is the correlation coe ffi cient between shear strain and normal stress, and E 0 , E P are the expected zero crossings and peaks of the normal stress, respectively. By inserting these terms into equation 8, one obtains the joint probability distribution for amplitude–stress pairs purely from PSD-based parameters. An important computational challenge involves calculating the correlation coe ffi cient r γσ between the shear strain and normal stress on the critical plane. Uncorrelated signals produce a broad range of normal stress values for a given γ a , while highly correlated signals keep them aligned. This coe ffi cient can be derived by examining the cross-power spectral density (CPSD) between the normal and shear stress components on the critical plane. Specifically, it is often assumed that the shear component is proportional to the shear strain, enabling the evaluation of: r γσ = r n τ = corr ⃗ n σ ( t )⃗ n ′ ,⃗ τσ ( t )⃗ τ ′ , (11) where σ ( t ) is the stress tensor, and⃗ n and⃗ τ define the plane’s normal and the direction of maximum shear. In the frequency domain, the correlation coe ffi cient can be estimated by: r n τ = G n τ ( f ) 2 df G nn ( f ) df G ττ ( f ) df , (12) where G n τ ( f ) is the CPSD between the normal and shear stresses, and G nn ( f ), G ττ ( f ) are their respective PSD func tions. Methods to compute these PSDs on the critical plane have been detailed by Mrsˇnik et al. [26] and Gao et al. [20], among others.

4. Results and Conclusions

This section presents a set of preliminary outcomes intended to validate the proposed frequency-domain approach for multiaxial fatigue. Two key aspects were examined: (i) the selection of the critical plane and (ii) the construction and use of a joint probability density function (PDF) to assess fatigue damage under random loading. Special attention was given to two scenarios which represent narrowband, fully correlated loading and broadband, uncorrelated loading.

4.1. Validation of the Critical Plane Selection

Figure 2 illustrates the comparison between the maximum damage and maximum variance criteria in identifying the critical plane over a spherical representation of all potential orientations. In each examined loading scenario, ranging from simpler uniaxial, narrowband conditions to fully multiaxial, wideband, and uncorrelated load scenarios, the maximum variance method successfully pinpointed the same plane as that yielding the highest damage, computed