13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- depends on the crack length/size within the ith grain, which varies from an uncracked to a fully cracked grain. To integrate analytically Equation (8), we consider that the mean evolution of FIPinside the grain follows a decreasing law, i.e., 2 1 FIP FIP 1 2 o ia (9) Here, ai is defined here as the fraction of the area of the i th grain to have cracked (nondimensional measure of cracked area of slip plane within the grain), and FIPo represents the FIP value for the ith grain on the αth slip system along which ai is measured, before the grain is cracked. Interestingly, Equation (9) resembles the empirical laws proposed by Hobson et al. [10] [11] and Miller [12] and has been validated using FEM simulations [1]. After integrating, the total life consumed in the MSC regime ( MSC N ) is the sum of the lives for each grain involved in the growth process for a given 3D crack, i.e., | MSC i MSC i N N . (10) where 2 1 1 1 2 1 | = | tanh i i MSC G MSC History st History c N N N D N c cc (11) 1 0 = 2FIP CTD n i i st nd i th ref gr D D c d and 0 2 FIP = n i i ref st nd gr i c D D d (12) Here, History N corresponds to the number of cycles undergone since the band considered has intersected the crack perimeter, and is necessary to include since cracks grow in multiple grains along a 3D crack front. Further details can be found in Refs. [1][2]. 2.3 Crack growth model To compute stress redistribution due to an increment in crack length, we employ an isotropic damage model along the bands having FIP values that lead to minimum fatigue lives. For the elements within these bands, the elastic stiffness tensor C is degraded according to damage parameter d1, i.e., 1 =(1 ) d C C (13) Here, d1 varies from 0 (uncracked) to 0.99, at which point full damage is assumed. The elastic stiffness is degraded for those elements in bands along which the crack is assumed to grow. The degradation of the elastic stiffness requires a gradual increase of parameter d1 to achieve convergence of the FEM solution. Accordingly, the degradation is imposed over several time steps, i.e., for the current time step it we write
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