PSI - Issue 75

Benjamin Causse et al. / Procedia Structural Integrity 75 (2025) 205–218 Author name / Structural Integrity Procedia (2025)

207

3

m × 2 × 10 6 with m = 3 for N C )/(  )]

E ≤ 5 × 10 6 m × 5 × 10 6 with m = 5 for 5 × 10 6 ≤ N

N E = (  C / ) N E = [(2/5)

(1a)

(1/3)  (1b) This method is highly appropriate for components subjected to uniaxial normal stress variation. However, this method does not take into account eventual multiaxial stress states neither average stress. It should be noted that Eurocodes will very soon be updated to take multiaxial fatigue into account (project PR EN 1993-1-9 (CEN/TC 250, 2023) submitted to the CEN enquiry in 2023, revised publication expected in 2025). After publication, to go further in our work, it will be interesting to compare the multiaxial fatigue results of the new Eurocodes, FKM and Dang Van. 1.3. FKM FKM Method (Rennert et al. 2024; Hobbacher, A., 2008) allows fatigue study taking multiaxial stress state into account by using the Von Mises equivalent stress, which has been criticised theoretically in the literature (Dang Van and Papadopoulos, 2014) but nevertheless gives consistent results, particularly on proportional signals, because it is adjusted on numerous tests. FKM uses the S-N curves of Eurocode 3 detail category references (and can therefore be applied not only to extruded tubes, or reconstituted profiles, but also to welded and bolted connections according to EN 1993-1-9:2005 detail categories). Finally, the FKM method takes into account the average stress using a method very similar to the Haigh diagram. Although recognised in the industry ( Kœchlin S., 2015 ), this method remains difficult to re-encode step by step because it is highly parameterised (numerous partial coefficients to minimise or maximise effects depending on experimental trials), and there are limits to its use on complex non-proportional signals. 1.4. Dang Van criterion method Finally, another type of method is based on the study of two physical fatigue phenomena. Firstly, crack initiation, and secondly, crack propagation to failure. On the one hand, experience as well as the theory of continuum mechanics show that crack initiation takes place in the critical plane, i.e. for a tension/compression test, in a plane at 45° to σ I⃗⃗⃗ , where the local shear stress amplitude || || is at its maximum. Davoli et al. (2003) even experimentally verified in the elastic domain that crack initiation does not depend on the mean shear stress, but only on the variable part of shear stress, which is the local (or re-centered) shear stress amplitude || ||. On the other hand, once the crack is initiated, experience and theory converge on the fact that a state of tension (with hydrostatic pressure p > 0) will contribute to opening cracks to failure, whereas a state of compression (p < 0) will keep them closed or at least slow down their opening to failure. The Dang Van criterion takes into account these two phenomena in the material at each time and in each node with a facet of normal vector ⃗ (see Dang Van et al. (1989, 2003), Dang Van and Papadopoulos (2014)). The Dang Van criterion is considered verified locally if inequality (2) is verified: ‖τ( ⃗, ) ‖ +  (N) .p( t ) ≤  (N)  ‖τ( ⃗, )‖ ≤  (N) −  (N) .p( t )) (2) With: • p( t ) : hydrostatic pressure over time (one-third of the stress tensor trace). • ‖τ( ⃗, ) ‖ re-centered shear stress amplitude over time and space orientation. •  (N) and  (N) : the characteristic material parameters, which depend on the number of cycles N and are determined with S-N curves obtained at different ratio R=  min /  max . It can be demonstrated (Schmidt 2024) that the coefficients  and  are a function of the stress amplitude at failure  R and the average stress  m see eq. (3): { = 2( 2√−3 √) 3 ) × 3 + = = √3 ( ℎ ) (3) ≈ 0.232 for R=-1. ≈ 0.116 for R=0 which is the case of S-N curves from EN 1993-1-9:2005, (because of art. 7.2.1)), see Causse et al. (2024) and Poyet (2022). These are theoretical relatively low values compared to those found in literature for real test on steel materials and welded details where  ≈ 0.3 (see Dang Van et al. (2003)). E ≤ 10 8