PSI - Issue 82

114 Sigbjørn Tveit et al. / Procedia Structural Integrity 82 (2026) 112–118 S. Tveit et al. / Structural Integrity Procedia 00 (2026) 000–000 where is the stress tensor, and D = − ( " ⁄3) − is the back-stress-modified stress deviator. The back-stress tensor dictates the endurance surface’s centre in the deviatoric plane and its rate is defined as ̇ = ̇ D ; ( = 0) = (3) where is the positive dimensionless back-stress parameter. Moreover, the damage rate is defined as ̇ = ̇ ( ) ; ( =0)=0 (4) where and are the positive dimensionless fatigue damage parameters. The material is undamaged ( = 0 ) in the initial state =0 , while the initiation of a macroscopic fatigue crack is assumed to occur at = 1 . The endurance function rate ̇ can be worked out by means of standard calculus, cf. Tveit et al. (2025). The signs of and ̇ define whether loading or unloading currently occurs: > 0 and ̇ > 0 Otherwise Loading ( ̇ ≠ and ̇ > 0 ) Unloading ( ̇ = and ̇ = 0 ) (5) The scenario of loading is schematically illustrated in Fig. 1 where the endurance surface for duraluminum with !" !" ⁄ =0.641 Nishihara and Kawamoto (1945) is depicted in the stress space. As shown in the figure the current stress point is located outside the endurance surface ( > 0 ) and moving away from it ( ̇ > 0 ), producing a nonzero back-stress rate ̇ ≠ and a positive damage rate ̇ > 0 . For loading where the stress point moves periodically along a straight line in the stress space, the back-stress, after a brief transient phase, takes on a periodic trajectory along the same linear path (Ottosen et al., 2008). This can be used to obtain closed-form analytical expressions for the calculation of finite fatigue lives resulting from uniaxial loading, and to express the model as a classical invariant-based fatigue limit criterion for infinite fatigue lifetimes under linear loading. The original formulation by Ottosen et al. (2008) then reduces to the well-known Sines criterion (Sines, 1955), which is restricted to !" !" ⁄ =1⁄√3 , while the formulation in Eq. (1) reduces to a generalized Sines criterion for arbitrary !" !" ⁄ -ratios (Tveit et al., 2025). 3 → →

̇

"

̇ ≠ ̇

= 0

=0

Fig. 1. Schematic representation of endurance surface = 0 in the stress space during loading, cf. Eq. (5).