13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- steady state cyclic stress strain solutions are then used to calculate the number of cycles to failure due to the creep fatigue damage, where the numerical lifetime results are validated by the existing experimental solutions [11, 12] for the Type 2 cruciform weldment. 2. Numerical Procedures 2.1. Asymptotic cyclic solution We consider a structure subjected to a cyclic history of varying temperature λθθ(xk,t) within the volume of the structure and varying surface loads λpP(xk,t) acting over part of the structure’s surface ST. The variation is considered over a typical cycle t t 0 in a cyclic state. Here λθ and λp denote the load parameters, allowing a whole class of loading histories to be considered. On the remainder of the surface S, denoted by Su, the displacement uk=0. Corresponding to these loading histories there exists a linear elastic stress history: ˆij e(x k,t) ˆij (x k,t) p ˆij p(x k,t) (1) where ij ˆ and p ijˆ denotes the varying elastic stresses due to θ(xk,t) and P(xk,t), respectively. The asymptotic cyclic solution may be expressed in terms of three components, the elastic solution, a transient solution accumulated up to the beginning of the cycle and a residual solution that represents the remaining changes within the cycle. The general form of the stress solution for the cyclic problems involving changing and constant residual stress fields is given by ij (xk,t) ˆij e(x k,t) ij (xi ) ij r (x k,t) (2) where ij denotes a constant residual stress field in equilibrium with zero surface traction on ST and corresponds to the residual state of stress at the beginning and end of the cycle. The history ij r is the change in the residual stress during the cycle and satisfies: ij r (x k,0) ij r (x k, t) 0 (3) For the cyclic problem defined above, the stresses and strain rates will become asymptotic to a cyclic state where: ij (xk, t) ij (xk, t t) ij (xk, t) ij (xk, t t) (4) It is worth noting that the above asymptotic cyclic solutions are common to all cyclic states associated with inelastic material behaviour including both the plasticity and creep. 2.2. Numerical procedure for the varying residual stress, creep strain and plastic strain range Adopting the same minimum theorem for cyclic steady state solution and the same Linear Matching condition as described in [5, 6] for each iteration, we assume that plastic or creep strains occur at N instants, 1t , t2......tN, where tn corresponds to a sequence of points in the cyclic history. Hence the accumulation of inelastic strain over the cycle is 1 ( ) N T ij ij n n t where ij (tn) is the increment of plastic or creep strain that occurs at time nt . Define the shear modulus by linear matching 0 2 ( ( )) n ij nt (5) where σ0 is the von Mises yield stress or creep flow stress and n is the iterative shear modulus. The von Mises yield stress σ0 will be replaced by creep flow stress if the creep relaxation occurs at the load instance. The Linear Matching Method procedure for the assessment of residual stress history and the
RkJQdWJsaXNoZXIy MjM0NDE=