PSI - Issue 2_B

Yuebao Lei / Procedia Structural Integrity 2 (2016) 2566–2574

2569

4

Author name / Structural Integrity Procedia 00 (2016) 000–000

   

 W W W  e e

p

for

0





    W W W W W    p p e e

(3)

W





p



p



  p

for

0

ij

d





ij

ij

step stress Residual

p

step stress Residual |



ij

where 2 e   ij  represents the plastic strains extracted at the ABAQUS analysis step defined by “ residual stress step= Ω ” in the ABAQUS “* contour integral ” keyword line. The formula for initial strain calculation is not given in the ABAQUS User’s Manual. Numerical investigations in this work (see Section 4 below) show that ABAQUS calculates initial strains using the following equation based on the variables obtained at the Ω th step in the analysis.            0 for 0 for step stress Residual 0        p th ij e ij ij ij (4) Equation (1) is identical to the definition given by Lei et al. (2000), where the thermal strains are treated as part of the initial strains. The correction to the uncracked-body plastic strain energy density defined in Eqn. (3) is very similar to that given in Lei et al. (2000). The initial strain extraction defined by Lei et al. (2000) for the J definition in Eqn. (1) can be expressed as   step stress Residual 0 th ij e ij ij ij        (5) Comparing Eqns. (4) and (5), it is found that the ABAQUS v6.14 initial strain calculation is correct for a plastic deformation induced residual stress field defining Ω ≠ 0 and for a residual stress field due to initial stress input defining Ω = 0. Otherwise, the extracted initial strains are incorrect. For a residual stress field simulated by a self-equilibrated thermal stress field, the parameter “ residual stress step =” should not be included in the J calculation ( Ω =0), therefore, no correction is made to the uncracked-body plastic strain energy density, from Eqn. (3). It is also found from the numerical investigation that ABAQUS evaluates 1 x th ij    for Eqn. (1) using the nodal temperature difference, Δ T , rather than the thermal strains, th ij  . This may cause a problem in J calculation when an uneven temperature distribution is input as initial conditions because the ABAQUS J function does not use temperature data input in Step 0. 3.2. ABAQUS v6.11-14 formulation for C(t) calculation ABAQUS does not give the C ( t ) definition used in the C ( t ) function in its User’s Manual. It is inferred from this investigation that in the old ABAQUS versions (ABAQUS (2010)), the 2D C ( t ) function may be expressed as (refer to Fig. 1) ij ij e W  and W p are the elastic and plastic strain energy density, respectively, and step stress Residual p step stress Residual ij

  

  

x u  i   1 ij  

(6)

 



*

C t Lim W 

n ds



1

j

j



0





where i u  are the components of displacement rate and W

* is the strain energy rate density, which may be defined as

c ij

   0



(7)

*

c ij   

W

ij d

ij   the components of creep strain rate.

with c