13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- Table 1. Tension ( aσ ) and torsion ( aτ ) stress amplitude used for different load conditions loading Tension Torsion Combined loading ] ] [ [ MPa MPa a a τ σ Phase shift biaxiality 0= k =∞ k 0.5 = k 1= k 2= k = °0 φ 56/0 0/36 43.5/22 30/30 17/34 = ° 45 φ 47/23 31/31 17/34.5 = ° 90 φ 56/28 34/34 17.5/35 3. Mesoscopic fatigue indicator parameters The studied fatigue indicator parameters (FIPs) were selected from stress criteria widely used in the literature. The multiaxial HCF criteria considered here are Crossland [9], Matake [10] and Dang Van [11]. These fatigue criteria are generally defined in the context of continuum mechanics. In order to evaluate the fatigue criterion on each computed microstructures, the usual HCF criteria are projected on the slip systems of the crystals. This procedure is repeated for each crystal considering its local orientation ( 2 1, , φ ϕ φ ) and local stress state computed by FE for each loading case. For instance, the shear stress vector in a given plane is transformed into a resolved shear stress vector over a slip system. The rotation of the crystal in space (defined by the Euler angles ( 2 1, , φ ϕ φ )) covers all the planes and directions of space, which enables to find the same critical planes and directions (planes and directions maximizing the criterion) than those obtained by the original criterion (with continuous formulation). Table 4 gives FIP expressions adapted to the crystal scale. Table 2. Expression of Fatigue Indicator Parameters (FIPs) of the studied criteria Criterion iI iα Crossland c c hyd s oct a cI β σα τ ≤ = + ,max , ( ) ( )3 3 1 1 1 − − − − = s t s cα Matake m m s s a s mI β σ α τ ≤ + = = ,max 1,12 max( ) 2 1 1 1 = − − − s t cα Dang Van dv dv hyd s t s dv t s t I β σ α τ ≤ + = = ( )]) max(max[ ˆ ( , ) 1,12 ( ) ( )3 2 1 1 1 − − − − = s t s dvα Finally, the parameters iα and iβ describing the median macroscopic threshold of the considered criteria are identified from two median fatigue limits for 107 cycles of the considered material on smooth specimens under fully reversed loadings: tension ( MPa s 56 1 = − ) and torsion ( MPa t 36 1 = − )
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