13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- alloys in engineering applications from airframe components to compressor blades applications. The experimental data used in this work were obtained from Haritos et al [1], Lanning et al [13] and Naik et al. [15] on Ti-6Al-4V for various notch root geometries and stress ratios. The Ti-6Al-4V specimens used in these papers were obtained from forged bar, which were initially heat treated to 705oC for 2 h, and then followed by static argon cooling to below 1490C. The material was subsequently annealed in vacuum at 5490C for 2 h. This is then followed by static argon cooling to below 1490C. The microstructure of the resulting bar is as shown Figure 2. Figure 2. Ti-6Al-4V forged plate microstructure [1] 3.0. Crystal Plasticity model of Ti-6Al-4V Crystal plasticity models are more suited for studying heterogeneity and interaction across grains in the notch root field as they relate grain scale stress to crystallographic slip response [16 - 18]. The use of crystal plasticity is thus relevant for the accurate prediction of the stress-strain field response at the notch root. The crystal plasticity model used in this work is based on 3D crystal plasticity models developed by Mayeur and McDowell (2007)[19]. Thus, only the summary of the crystal plasticity constitutive models is presented in this section. The deformation in the material is based on a standard two term multiplicative decomposition of the deformation gradient into elastic and plastic parts, i.e., F=FeFp (4) Here, F is the total deformation gradient, F p captures the dislocation glide through the lattice while Fe captures the rigid body rotation and elastic stretching of the lattice. The plastic velocity gradient ˆLp defined as the sum of the crystalline shear displacement rates over all slip systems k is given in the isoclinic, lattice invariant intermediate configuration as [20]: ˆ ( ) ( ) 1 0 0 1 sys L F F S n N p p p k k k k γ = ⋅ = ⊗ − = ∑ ɺ ɺ (5) Here, k is the slip system shearing rate, and S0 k and n 0 k are fixed unit vectors in the slip direction and slip plane normal direction, respectively. The slip vectors S0 k and n0 k remain unchanged through deformation Fp from the reference to the intermediate configuration and maintain orthogonality through Fe. The relationship between the slip system shearing rate and the resolved shear stress of the kth slip system is described by the power law flow rule given by McGinty [21] as: 0 sgn( ) M k k k k k k k D τ χ κ γ γ τ χ − − = − ɺ ɺ (6)
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