13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- Here, 0 is the reference shearing rate, M is the inverse strain-rate sensitivity exponent which controls the rate sensitivity of flow, τ k is the resolved shear stress, χ k is the back stress, к k is the length scale-dependent threshold stress and Dk is the drag stress. As developed by Zhang et al. [22], the drag stress is taken as a non-evolving constant, while the back stress evolves according to an Armstrong-Frederick direct hardening/dynamic recovery type of equation, i.e., k k k k D h h χ γ χ γ = − ɺ ɺ ɺ (7) With χ k(0)=0. The threshold stress is expressed as y k k s k d κ κ κ = + (8) 3.1 Simulation of Notched Components The crystal plasticity constitutive model was coded into ABAQUS 2006 UMAT, based on previous work by Zhang et al [22]; Mayeur and McDowell, 2007[19]. For textured Ti-6Al-4V alloy, some of the material parameters in the crystal plasticity are obtained from Bridier et al [23]. Finite element simulation was performed on three different geometries, meshed using 3D stress four-node linear tetrahedron element type (C3D4) and consisting of approximately 218940 elements to estimate the stress distribution and possible plastic straining that occur in the notched specimens. The dimensions of the specimens used and the different test cases are as given in Table 1. A diagram of the gage section of the specimen is provided in Figure 3. Figure 3. Gage section of the cylindrical specimen with a circumferential V-notch.[1] To reduce computational time, the notched specimen geometries are decomposed into three different regions: an outermost region, far from the notch root, where isotropic linear elasticity is used; an intermediate transition region where macroscopic J2 cyclic plasticity theory is used; and finally the notch root region where crystal plasticity theory is used. The element size at the crystal plasticity region was chosen to coincide with the average grain size of Ti-6Al-4V which is 45 µm. The domain decomposition is as shown in Figure 4. Also, One quarter of the cylindrical notched specimen was modeled because of the symmetry in loading and geometry of the specimen as shown in Figure 5. The bottom of the notched specimen is encastre while symmetry boundary conditions are applied to the two planes of symmetry. The notched specimens were tested at four different load ratios; R=0.1, R=0.5 and R=-1. Average alternating HCF strength at 106 cycles, as determined by Naik et al [15] and as contained in Table 1 for different load ratios, are applied to the top of the specimen.
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