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

105

2

polycrystalline metals, various crystal plasticity models have been developed. In this work, we have focused on integration algorithm developed by Maniatty et al. (1992) to model crystal plasticity of single crystals. This algorithm has been modified by Sarma and Zacharia (1999) by using evolution equation for elastic deformation gradient as it can be directly used to calculate the elastic strain and stress and rigid rotation which eventually helps in getting the texture evolution. These time integration models use Taylor (1938) hypothesis for stress-strain behavior of polycrystalline metals. To get the full field solution, mechanical equilibrium needs to be solved over the whole polycrystal for which several techniques like finite element (FEM) or fast Fourier transform (FFT) can be used. For linearly elastic inhomogeneous composites, Moulinec and Suquet (1998) had developed a FFT based scheme. For multiphase metals or polycrystals, the microstructure evolution drives the mechanical behavior. The microstructural evolution is driven by composition gradients, thermal energy and strain fields etc. To model the microstructure evolution, phase field method (PFM) based on diffused interface formulation has evolved as one of the most versatile tool. In the current work, a code has been developed for elasto-viscoplastic (EVP) stress-strain response using Sarma and Zacharia (1999) time integration algorithm for single crystal. This code is then extended for full field solution of polycrystal and multiphase metals by applying the modified FFT scheme for EVP response. The crystal plasticity formulation is then coupled with the phase field formulation to get the effect of microstructure evolution on the stress-strain response and vice-versa. 2. Elasto-viscoplastic crystal plasticity modeling In this section we briefly describe the integration of a phenomenological single crystal plasticity model by using the integration scheme described by Sarma and Zacharia (1999). 2.1. Crystal kinematics Consider a single crystal as a continuum body with initial (undeformed) configuration B 0 , deformed to the current configuration B . By considering a stress-free intermediate configuration  B , the deformation gradient F can be written by using multiplicative decomposition (Lee (1969)) as:

 F F F

F

F

;

det

1, det

0

e p

p

e

(1)





where, F e is the elastic part of deformation gradient that represents the rigid rotation and stretching of lattice and F p is the plastic part of deformation gradient that represents shearing along the crystallographic slip planes which is assumed to be isochoric. Similarly in a crystal, the velocity gradient can also be multiplicatively decomposed where the plastic velocity gradient is defined as a linear combination of shearing rates    on all the active slip systems, N .

N

N

(    L P s m      p        

(2)

)



1

1









where,  P  is schmid tensor, a dyadic product of slip direction  s  and slip plane normal vector  m  .

2.2. Constitutive model Constitutive equations provide the relation between stress and strain for both elastic and plastic deformation. From kinematics, the Cauchy-Lagrange strain during elastic deformation can be defined as

1 2

1 2

e E C I

eT e F F I 

 

   

 

(3)

e   