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

108

5

where, ˆ σ and ˆ ε are the stress and strain and 0 ˆ Γ is the green’s operator in Fourier space derived by Mura (1987). In the above algorithm, the Fourier transform and inverse Fourier transform is referred as  and  -1 respectively. 3.2. FFT : elasto-viscoplastic case The FFT scheme shown above is extended to implement single crystal plasticity to a polycrystal microstructure. To get the full-field response of the microstructure, the EVP algorithm is applied on each material point on the Fourier grid. The output of the EVP algorithm iteration loop is the elastic deformation gradient, from which Cauchy Lagrange elastic strain E e and Cauchy stress σ can be calculated at each grid point. This strain and stress data calculated at every grid point is then used to initialize the perturbation strain and stress before each iterative loop of the FFT scheme shown above.

ε

( ) = ( ) ; e x x E

* ( ) = ( ) : ( ) x x x ε σ L

*

(14)

Iterations: at ( i + 1) with stress ( ) i x σ and strain ( ) i x ε known at every grid point x . 1.     ˆ ˆ ( ) = ( ) ; ( ) = ( ) x x   σ σ ε ε   2. 1 0 ˆ ˆ ˆ ˆ ( ) ( ) : ( ) i i       ε ε Γ σ 3.   * 1 1 1 ˆ ( ) = ( ) i i x     ε ε  4. 1 * 1 ( ) = ( ) : ( ( ) ( )) i i e x x x x    σ ε E L 5. Convergence test in stress where, E e ( x ) = E e calculated at every grid point. 4. Phase field model

To model the microstructure evolution and its effect on overall stress-strain behavior, PFM can be coupled with the CPFFT model explained in previous section. PFM uses conserved variable like concentration, c , and non conserved field variables like order parameter ( η ) etc. for describing microstructure. Two most commonly used models are Cahn and Hilliard (1958) and Allen and Cahn (1979) model for evolution of conserved and non conserved field variables respectively as shown in Eq. (15) and (16).

c



U

 

  M

= i

(15)

ij

t 

c

j

t  

U



= - p

(16)

Ω

pq

q 

where, M is the diffusivity of phases and concentrations and Ω is the mobility of order parameters. The total free energy U in the derivative term can have many sources like chemical energy due to composition gradient, mechanical strain energy, thermal energy, etc. During microstructure evolution, mismatch or eigen-strain occurs due to the variation in lattice parameter of different phases in multiphase metals. This strain causes stress field in addition to the applied stress. The conserved variable based Cahn-Hilliard equation can be modified by considering the total free energy as addition of both chemical and elastic strain energy.