13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- associated plastic or creep strain range due to the cyclic load history is described below. The entire iterative procedure includes a number of iteration cycles, where each cycle contains N iterations associated with N load instances. The first iteration is to evaluate the changing residual stress 1 1 ij for the action of the elastic solution ˆ ( )1t e ij at the first load instance. We define n ij m as the evaluated changing residual stress for nth load instance at mth cycle of iterations, where n 1, 2, ... N and m 1, 2, ... M. At each iteration, the above changing residual stress n ij m for nth load instance at mth cycle of iteration is calculated for the combined action of applied elastic stress at the nth load instance and previously calculated accumulation of residual stresses 1 1 1 1 1 ˆ ( ) m N l n l e ij n k m k l l t ij ij . When the convergence occurs at the Mth cycle of iterations, the summation of changing residual stresses at N time points must approach to zero ( 0 1 N n n ij M ) according to the condition of the steady state cyclic response. Hence the constant element of the residual stress for the cyclic loading history must be determined by 1 2 1 1 1 1 N N N n n n ij ij ij ijM n n n (6) The corresponding increment of plastic strain occurring at time nt is calculated by 1 ˆ ( ) ( ) ( ) 2 p e ij n ij n ij n n t t t (7) where notation ( ' ) refers to the deviator component of ˆe ij and ij . ( ) ij nt is the converged accumulated residual stress at the time instant nt , i.e. 1 ( ) n k ij n ij ijM k t (8) In this paper, the Ramberg-Osgood type is adopted for the cyclic stress and strain range relationship: 1 2 2 2 E B (9) where is the true stress range, is the true strain range, E is the elastic modulus, B and β are the Ramberg-Osgood plastic hardening constants. The first term on the right-hand side of the above equation represents the elastic strain amplitude and the second term corresponds to the plastic strain amplitude. Then the plastic strain range from Eq. (9) can be written as: 1 2( ) 2 p B (10) If we define yield stress σ0 in Eq. (5) as half stress range: 0 ( ) 2 2 p B (11) then the iterative von-Mises yield stress 0( )nt from Ramberg-Osgood material model can be obtained from Eq. (7) as: 0 ( ( )) ( ) ( ) 2 p ij n n t t B (12) For the calculation of creep strain and stress relaxation during a creep dwell period, the relevant numerical scheme and theoretical formulations are summarized below.
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