13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- ( ) ( ) ( ) ( ) ( ) = B D B h r B , (3) in which 1 ( ) 0 0 ( ) = B D c h r S f and S 0 , h B , h S , d D , 0 , f c , λ, are constants that differ for octahedral and cube slip planes (see Table 1). Both evolution equations follow a hardening-dynamic recovery format and the initial values are specified as 0S for the slip resistance and zero for the back stress. Table 1. Parameters of the constitutive model at 650°C for octahedral and cube slip systems. F 0 kJ/mol p Q 0 s -1 τ 0 GPa S 0 MPa f c h B GPa h S GPa d D MPa µ’0 GPa Oct. 295 0.31 1.8 120 810 350 0.42 400 10 6024 72.3 Cube 295 0.99 1.6 4 630 48 0.18 100 4.5 24 28.6 Other parameters: λ=0.85, µ0= 192GPa. Elastic constants: C11 = 166.2GPa, C12 = 66.3GPa, C44 = 138.2GPa. 2.2 Simulations Figure 1 depicts a mesh composed of ―brick‖ elements (C3D8R) employed for modeling smooth specimens, containing 6859 elements and 118 grains, with colors representing different crystallographic orientations. The loading sequence consisted of relative displacement of the upper and lower boundary planes at a 0.05%/s strain rate under tensile mode loading to achieve overall nominal peak strains of 0.8% and strain ratio (Rε) equal to zero, typical of the HCF regime. The lateral faces are free of traction and the model has unidirectional periodic boundary conditions along the loading direction (Y-axis) such that the sum of the displacement perturbations of the nodes on top and bottom faces relative to the mean is null, leaving only the imposed net relative displacement. The grain size follows a lognormal distribution based on the algorithm by Musinski [9] and has a mean grain size about 18 µm. 2.3 Fatigue driving force measures Since the local driving force to nucleate and grow fatigue cracks is affected by the microstructure, several FIPs have been proposed to consider these effects. Fatemi and Socie [10] proposed a FIP based on the critical plane approach that plays a role similar to that of the mixed mode ∆CTD or ∆J-integral in correlating with growth of small fatigue cracks [11][12][13]. Subsequently, several investigators have successfully employed approaches akin to such FIP along with crystal plasticity formulations for studying the effects of microstructure on fatigue crack formation and early growth within the first few grains [14][15][16].
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