PSI - Issue 14

Ritesh Dadhich et al. / Procedia Structural Integrity 14 (2019) 104–111 R. Dadhich, A.Alankar / Structural Integrity Procedia 00 (2018) 000–000

106

3

where, C e is the elastic part of right Cauchy-Green deformation tensor and I is the second order identity matrix. For elastic part of deformation, the basic constitutive law defines stress in terms of Cauchy-Lagrange elastic strain E e and fourth order stiffness tensor L in the intermediate configuration as shown in Eq. (4).

1

T





e   T E L

e e F F σ F

(det ) [ ] e

e

(4)

where, T e , second Piola-Kirchhoff stress, is the driving stress for dislocation glide due to elastic stretching of crystal lattice in the reference frame. We can relate it to stress state at the deformed configuration, Cauchy (True)

stress, σ , by using above transformation. 2.3. Phenomenological constitutive model

Plastic constitutive equations are used to provide relation between stress on crystal and shearing rate on each slip system. Resolved shear stress, τ α , is the shear component of an applied tensile or compressive stress resolved on slip system α . It is calculated by projecting the second Piola-Kirchhoff stress, T e to the slip system α using the Schmid tensor.

) (    C T s m   ( e e 

)

 



(5)

Resolved shear stress and shear strain rate during plastic deformation can be related by a phenomenological equation as shown below.

1/

m



 

(6)

sign( )  

γ γ    

0

ˆ



* ˆ ( ) ˆ ( )          * ˆ ˆ s

 

    

ˆ 

*

(7)





H  





0



s

i

where, m is strain rate sensitivity parameter, 0 γ  is reference strain rate, H 0 is hardening rate and ˆ i   is initial hardness on the slip system α . The strain hardening, ˆ   , evolves with time using Eq. (7). 2.4. Time integration algorithm The integration algorithm is used for coupling consecutive equations of single crystal plasticity with crystal kinematics to get the EVP response for a cubic single crystal. Further, it can be easily incorporated into the polycrystalline model using mean-field or full-field approach as done by Lebensohn (2001) for their viscoplastic self-consistent (VPSC) model. The time integration algorithm uses the following equation for describing elastic deformation gradient.

e e e p   F LF F L 

(8)

Integration can be done numerically by discretizing the deformation history in time intervals and solving it at each time step to determine the elastic deformation gradient by using the implicit scheme for Eq. (8).

 

  

  

  

N 

e F F L F F e     e t t t t   t t  t t  t

t t  P   

e

 

(9)

t



1



