13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- that lie parallel to the crystallographic slip planes and have a width of one element. These bands seek to reflect domains in which cyclic plastic deformation localizes via dislocation dipole structures and in experimentally observed persistent slip bands. By averaging the local FIPs over bands [3], a methodology was developed that predicts the path of a fatigue crack along bands through multiple grains, considering grain size effects [4]. The fact that FIPs can be averaged over multiple volumes (sizes and shapes) raises questions about the role of volume domains for averaging on the variability of nonlocal FIPs and the influence of grain size effects on the variance of the distribution of fatigue life. This work employs a crystal plasticity finite element model for RR1000 Ni-base superalloy to compare the variability of the Fatemi-Socie FIPs averaged over grains or bands, and its local magnitudes for a number of realizations of ostensibly the same microstructure. The influence of grain size effects on normalized FIP distributions is also considered. 2 Modeling and simulation 2.1 Constitutive model At the scale of individual grains we employ a physically-based crystal plasticity constitutive model for RR1000 superalloy adapted from the work of Lin et al. [5]. The crystallographic shearing rate is given by ( ) ( ) ( ) 0 ( ) ( ) ( ) 0 0 0 0 / = exp 1 , / q p b B S F sgn B k T (1) in which ( ) is the shearing rate of slip system , ( ) is the resolved shear stress, T is the absolute temperature, Fo, p, q, 0 , τ0, µ, and µ0 are material parameters that may differ for octahedral and cube slip systems, as listed in Table 1 for 650°C, and kb is Boltzmann‘s constant. This formulation considers 12 octahedral and 6 cube slip systems, the latter representing a zigzag deformation mechanism [6] along octahedral planes, but producing net slip along cube planes. The model was implemented as a user-material subroutine (UMAT) in ABAQUS 6.9 [7] using an implicit integration scheme based on the Newton-Raphson and the backward-Euler methods. Discussion of model parameters and their estimation can be found in Ref. [8]. The slip system shearing rate relation includes a directional slip resistance ( ) S that functions as a theshold stress below which no plastic flow occurs and a back stress ( ) B that accounts for directional hardening (Bauschinger effects) on the slip system. The evolution laws for slip resistance and back stress are written as ( ) ( ) ( ) ( ) 0 = S D S h d S S (2)
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