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

107

4

This integration scheme can be used for a polycrystal by using Taylor (1938) hypothesis. In this approach, the velocity gradient applied on polycrystal can be assumed to be same on each crystal. Then the stress on the whole polycrystal can be determined by averaging on each crystal. 3. Crystal plasticity fast Fourier transform (CPFFT) modeling The integration algorithm with continuum framework explained in previous section is used to determine the stress-strain behavior and texture evolution on the time scale applied on single crystal. To get the full EVP response of polycrystalline material with a full field approach, the whole polycrystal is discretized with grid points and the integration algorithm is implemented at each grid point by using spectral method. 3.1. FFT : elastic case In crystal elasticity or plasticity, FFT approach is used for solving the system of partial differential equations resulting from compatibility and static equilibrium. A basic FFT scheme by Moulinec and Suquet (1998) for an inhomogeneous elastic medium with stiffness tensor L ( x ) and prescribed strain E is shown below. This algorithm is used because it is simple to implement and can be adapted to materials that show non-linear behavior. ( ) = ( ) : ( ) x x x σ ε L (10)

0 * * ( ) = ( ) : (u ( )) + ( ) x x x x σ ε p L

(11)

where, σ ( x ) and ε ( x ) are the local stress and strain field, p * ( x ) is the polarization stress field and L 0 is the fourth order stiffness tensor for homogeneous reference material. The local strain field is assumed to be divided into its average E and a perturbation strain * (u ( )) x ε or * ε . This perturbation strain is a fluctuating term which corresponds to the heterogeneity in the material. The problem to be solved for a heterogeneous elastic material using FFT is

* * ( ) = ( ) : ( (u ( ) + ) div ( ) = 0 u ( ) , . x x x x σ ε E σ σ n L

  

(12)

x  

where, the displacement boundary conditions are periodic and traction is non-periodic. An iterative scheme to solve the above problem by Moulinec and Suquet (1998) is shown below: Initialization:

* ( ) = ; ( ) = 0 ; ( ) = ( ) : ( ) x x x x x ε E ε σ ε L

(13)

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 x x x   σ ε L 5. Convergence test in stress