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

109

6

  

   

c

chem U U  



    M

(17)

= i

last

e

ij

t

c

c







j

j

where, the chemical and elastic energy can be calculated by

1 2

1 2

f



  

  

2





σ ε

e ij ij

(18)

;

U

c dv

U

dv



  



chem

elast

c



V

V

where, f is a function of concentration defined by a double well potential and  is the gradient coefficient. The stress is determined using Hooke's law by assuming that all phases show linear elasticity. Apart from the above evolution equation, mechanical equilibrium also needs to be solved which is handled by FFT elastic scheme discussed in the previous section. To include plasticity in the current model, the CPFFT model can be used to calculate stress and strain at every grid point which will help in calculation elastic strain energy. 5. Results A FORTRAN code has been developed for the EVP model using the time integration algorithm as described in section 2.4. It was implemented to simulate the stress-strain response for commercial purity aluminum single crystals. Material parameters are taken from Mathur and Dawson (1989). The single crystal was deformed under tension up to a strain of 2.0 with the velocity gradient given as

0.5 0.0 0.0 0.0 0.5 0.0 0.0 0.0 1.0 



    

   

  L



The variation of von Mises stress (normalized using initial hardness) with the effective strain is shown in Fig. 1(a). This result from the simulation is compared with the stress-strain variation from Sarma and Zacharia (1999) as shown in Fig. 1(b). The code gives similar predictions of the stress-strain response as in the literature. We have not performed one-on-one calibration of the model yet.

Fig.1. Effective stress vs. effective strain plot from (a) present model and (b) Sarma & Zacharia (1999) model.

A code for basic FFT elastic scheme and FFT-EVP scheme has also been developed following the algorithm as described in section 3.1 and 3.2. Numerical simulations were performed for a homogenous matrix of pure aluminum with two hard inclusions deformed in tension up to a strain of 0.001. In Fig. 2, the stress field on a mid-plane of the