13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- ( 1) ( ) 1 1 i i i d d t (14) For example, a value of =2 was enough to increase d1 up to 0.99 by the end of the 1-second half loading cycle; of course, can be varied to accelerate crack stiffness reduction and to allow for convergence while computing stress redistribution effects in the polycrystal over a few computational cycles. Of course, this represents evolution over a much larger number of applied fatigue loading cycles. The plus sign corresponds to the case in which the stress normal to the band is positive and the crack plane is in tension. On the contrary, if the stress normal to the band is in compression, the crack is assumed to be closed and d1 is decreased to zero at a rate proportional to the time step, as described by the minus sign in Equation (14). When roughness- or plasticity-induced closure conditions are detected by virtue of compressive traction normal to the crack face on individual elements, the value of d1 is decreased to 0 such that the initial elastic stiffness is restored. Hence, the degradation of the stiffness tensor is performed on a grain-by-grain basis by increasing the parameter d1 in all the elements after predicting the path of the crack in the following grain. Such a prediction is performed every two computational loading cycles to allow for stiffness degradation and the update of the stress and strain fields. In summary, the fatigue algorithm starts by calculating the nonlocal FIP values on every band in every grain over the third computational cycle, and proceeds by calculating the number of expected cycles to nucleate crack on all bands for all grains using Equation (5). The elements within the band with the lowest nucleate life are marked as “cracked,” and the model applies again a couple of computational loading cycles to update the FIP values and to degraded (d1 increased) the stiffness tensor as necessary to represent crack growth. Thereafter, the algorithm computes the MSC life of all FIP averaging bands that intersect the crack perimeter and renders the elements in the band with minimum life as cracked. The simulation proceeds by applying further loading cycles, in which the stiffness tensor is degraded on the cracked elements to redistribute stress and plastic strain, while checking for grain level closure effects of cracked grains, and the MSC life is evaluated again on the remaining grains. Since we seek to describe the dominant crack, only one crack nucleates per realization. 3 Simulation results 3.1 Specimens and loading conditions Figure 2 depicts the C3D8R-element meshes employed for modeling smooth and notched specimens, each color representing a different crystallographic orientation. The grain size follows a lognormal distribution based on the algorithm by Musinski [13] with a mean value of 18µm. The straining sequence consisted of triangular relative displacement of the upper and lower boundary planes at a 0.05%/s strain rate under shear or tensile mode loading to achieve an overall nominal strain range of 0.8%; lateral faces are free of traction. Simulations with different strain ratios employed a similar strain range to assess only the effect of applied strain ratios (strain/displacement conditions). To achieve equivalence with tensile straining, the magnitude of the displacement vector in shear straining was computed by assuming an elastic model with cubic symmetry. The value of Poisson‟s
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