PSI - Issue 2_A

Dong-Jun Kim et al. / Procedia Structural Integrity 2 (2016) 832–839 Dong-Jun Kim et al. / Structural Integrity Procedia 00 (2016) 00–000

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o P L P 

ref (2) where P L is the plastic limit load and P denotes the applied load. L r is a parameter related to plastic yielding and is used as the relative loading magnitude. HRR field may be written using the reference stress and Eqn. (1) and be following as:   1 1 1 , m ij ij m ref m ref J m I A r                  (3) The crack-tip stress field under elastic-plastic region can also be obtained from the modified boundary layer analysis by Williams, M. L. (1957) and described in the plane strain condition as:   0 0 0 0 0 2 0 0 I ij ij T K f r T                 (4) where K I is stress intensity factor from converting J -integral and T is known as T -stress. 2.2. Steady-state creep region In creep deformation, the material can be assumed to power-law creep and be expressed as: c n B     (5) where B and n are material constants. In creep conditions the stress field around crack-tip depends on time. However, if the applied load is primary loading type then the stress field will converge after a certain time. The converged stress field is characterized by C * -integral, and the certain time is defined as the redistribution time. C * -integral has the analogous relationship with the J -integral and is expressed as: r o   L   where W * is the strain energy rate density related with the stress and the strain rate and is defined as: * 0 ij ij ij W d        (7) In the steady-state creep condition, the stress field will become independent of time and may have the relationship with the C * -integral by Riedel, H. et al. (1980). This relationship is known as the RR equations which can be expressed using the reference stress as: (8) where I n is a dimensionless constant associated with the creep material properties n and whether conditions of plane stress or plane strain. ij   is the function of the creep exponent, n , and the crack tip angle,  . 2.3. Transient creep region For the transition creep condition (non-steady state), the stress field around the crack-tip is characterized by C(t) - integral which may have the function of time. The C(t) -integral is defined from the definition of C * and is following as:   0 i i u C t Wdy T ds x               (9) The relationship between the stress field on the crack-tip under the transition creep condition and C(t) -integral is expressed as: This relationship is from RR-field.   , n    1 * 1 1 I B r   n ij ij n ref n ref C             * u C W dy T ds x              * i i (6)