13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- 2.3. Numerical procedure for the creep strain and flow stress When calculating the creep strain during the dwell period, 0in equation (5) equals to creep flow stress c 0 , which is an implicit function of creep strain c and residual stress c during the creep dwell period. Adopting a time hardening creep constitutive relation: c n m B t (13) where c is the effective creep strain rate, is the effective von Mises stress, t is the dwell time, and B, m and n are the creep constants of the material. When m=0, the time hardening constitutive equation becomes the Norton’s law. During the relaxation process there exists an elastic follow up factor Z, i.e. c Z E (14) where , E is the Young's modulus and ( ) ij . Combining (13) and (14) and integrating over the dwell time, we have 1 1 1 1 1 1 ( 1) 1 ( ) ( ) m n n c s BE t Z m n (15) where sis the effective value of the start of dwell stress, c is the effective value of the creep flow stress at dwell time t, and ( ) c sij cij . Integrating (14) gives the effective creep strain during the dwell period t as, ( ) c s Z E (16) Combining (15) and (16) and eliminating gives 1 1 1 ( 1) ( ) 1 1 ( )( 1) m c s c n n c s B n t m (17) For the pure creep where s c , the creep strain becomes: 1 1 n m c s B t m (18) The creep strain rate F at the end of dwell time t is calculated by Eq. (15) and (17): 1 1 ( 1) 1 1 ( ) ( 1) ( ) n c F n m c c n n s c c s m B t t n (19) For the pure creep where s c , the creep strain rate F becomes: n F m s B t (20) Hence in the iterative process, we begin with current estimated i s, i c and use Eq. (17), (19) or (20) to compute a new value of the creep stress c from Eq. (21) to replace 0 in the linear matching condition (5): 3 / 2(1 ) E E Z E/
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