PSI - Issue 28

Ping Zhang et al. / Procedia Structural Integrity 28 (2020) 1176–1183 Author name / Structural Integrity Procedia 00 (2019) 000–000

1178

3

= c d             ,

(2)

where c is the direct hardening modulus and d is the dynamic recovery modulus. The isotropic hardening is built upon the evolution of the critical shear stress as follows

   

   

  

  

g 

h h q h    

,

(3)

    

   

0     h

2

sech

  h h 

0



0

s

where h  is the slip hardening modulus, subscripts  and  are the indices of slip systems, ( )  q     stands for the latent-hardening constant. The self-hardening modulus   h  follows a hyperbolic function, where 0 h is the initial hardening modulus, 0  is yield stress which equals 0 g , s  is the stage I stress and  is the total cumulative shear strain on all slip systems, which is obtained as q     represents the self-hardening constant and 

0 t

    

dt 

.

(4)





The CP model described above was implemented in ABAQUS/Standard finite element software as a user material subroutine (UMAT) (Huang, 1991) and used in this study. 2.2. XFEM procedure XFEM implemented in ABAQUS is used in this study. When combined with the traction-separation cohesive behaviour, the approximate displacement vector u can be expressed as

N

  u

 

 

i   b ,

(5)

i   a 

N x

H x

i

1

i



where i N is the conventional nodal shape function, i a is the standard nodal displacement vector for the continuous part, i b is the nodal enriched displacement vector of the nodes cut by the crack, and the jump function   H x is the Heaviside function used to mark the two sides of a crack (   and   ) as



1,



x x



 

(6)

.

H x

 

1,



  

To control the development of cracks, a user damage subroutine (UDMGINI) is used, where the damage indicator value and the vector normal to the crack growth direction are provided. During each computation increment, the onset of fracture is determined by the fracture value and normal direction vector averaged at the centroid of an element. When the fracture value is greater than 1.0, the traction-separation behaviour is activated and the crack grows along the given direction. In this work, the damage criterion and crack growth direction are proposed at the crystalline level based on the experimental observations (Carroll et al., 2013) and existing numerical studies (Zhao and Tong, 2008). Since the slip controlled crack paths were reported in in-situ SEM experiments, the crack propagation was assumed to be driven by the individual cumulative shear strain (ICSS) to reflect the contribution of each slip system, which is defined as