PSI - Issue 14

Manish Kumar et al. / Procedia Structural Integrity 14 (2019) 839–848 Manish Kumar et. al/ Structural Integrity Procedia 00 (2018) 000–000

843 5

is the enriched nodal degrees of freedom vector associated with crack tip enrichment function (x) j  . Heaviside enrichment function and elasto-plastic tip enrichment functions are taken from Shedbale et al . (2017). The whole procedure to estimate the creep strain is divided into two parts: elasto-plastic and creep. In elasto plastic analysis, the loading is considered as ramp loading and applied in one-unit pseudo time. A pseudo time step is taken and load is applied on the computational model according to that. The set of discrete equations (Eq. (6)) is solved simultaneously to obtain displacement, strain and trial stress field. Plasticity criterion is checked for trial stress and appropriate action is taken. If the solution is converged, then proceed for next pseudo time step else repeat the same process until convergence is achieved. Once the total pseudo time becomes unity, then elasto-plastic analysis gets completed and creep analysis starts. An actual time step is considered and effective creep strain is computed from creep law using the continuum state of the converged elasto-plastic analysis (Kumar et al ., 2018a). Creep force is computed and used to calculate the new displacement, strain and trial stress (using Eq. (8) and Eq. (9)) fields. The yielding criterion is checked and an appropriate action is taken. If the solution is converged, then next time step starts otherwise the process of creep force and displacement evaluation is repeated until convergence is achieved. New obtained stress field is used to estimate the creep strain via creep law for the next iteration. The next time step is increased by 10% of the last time step. If solution starts diverging then time step is reduced to half and process of evaluating the creep strain and all other secondary fields is repeated. 2.3. Creep crack growth The stress field is relaxed and redistributed within the intact portion during small-scale creep and transition creep. In the formulation of stress intensity factor ( K ) and J -integral, creep deformations are not accounted. Therefore, ( ) C t -integral is used to uniquely characterize the crack under creep conditions. For the contour  , ( ) C t -integral is defined as,

   

1    ds          i u x 



(11)

( )

* W dx T 

C t



2

i

0







 

ij

*

(12)

W

ij    ij d

0

where,  is the contour taken counter clockwise from lower crack face to upper crack face as shown in Fig. 2, * W is the strain energy rate density, i T is the traction vector defied by the outward normal j n at path  ( i ij j T n   ), i u  is the displacement rate and ds is the incremental arc length along the contour  and ij   is the strain rate. The rate of strain energy density given in Eq. (12) can be written as,

c eff





 

c

*

W

eff    d

(13)

eff

0

To evaluate the integration in Eq. (13), creep law is used to relate creep strain rate with stress state. Suppose the material’s creep behaviour is described by the Power law as,

c

b eff A 

(14)







eff

where, A is creep constant and b is creep exponent. Using Eq. (14), strain energy density rate is reduced to,