PSI - Issue 14

B V S S Bharadwaja et al. / Procedia Structural Integrity 14 (2019) 612–618 B V S S Bharadwaja, A.Alankar/ Structural Integrity Procedia 00 (2018) 000–000

613

2

trapping process and are responsible for work hardening in crystalline materials. GNDs are formed to accommodate the inhomogeneous deformation and are formed to preserve the lattice continuity during deformation. In this work, total dislocation density is related to strength via Taylor’s approximation of mean square root density of SSDs and GNDs. The evolution of SSDs and GNDs is implemented in the framework of UMAT developed by Huang (1991) and modified by Kysar (1997) for single crystal plasticity. Plastic shear on each slip system is tracked as separate state variables. SSDs are modeled using Kocks-Mecking (KM) equation that consists of dislocation generation and annihilation mechanisms during plastic deformation. GNDs are modeled using the gradient of shear on each slip system. Objective of the present work is to show the distribution of SSDs and GNDs during pure bending of a Cu single crystal.

Nomenclature F

total deformation gradient elastic deformation gradient plastic deformation gradient

F e F p

ρ SSD statistically stored dislocations (SSDs) ρ GND geometrically necessary dislocations (GNDs) ρ T total dislocation density τ resolved shear stress g slip resistance µ shear modulus b magnitude of Burger’s vector k a dislocation generation parameter k b dislocation annihilation parameter m ɑ slip direction of α slip system n ɑ slip normal of α slip system p ɑ

direction perpendicular to both slip and normal direction

screw GNDs

   s e,    ,   

edge GNDs in normal direction

n

edge GNDs in p

ɑ direction

e p

slip rate on each α slip system

  

P ɑ

Schmid tensor

1.1. Single crystal model A rate dependent model of crystal plasticity based on Asaro (1983) is implemented in UMAT as described in the previous section. In the present model, we consider only 2 slip systems in FCC copper single crystal. A brief summary of the framework utilized in the present model is given in the following. For large deformations, in crystal plasticity, the kinematical theory starts with the multiplicative decomposition of total deformation gradient. Plastic deformation gradient accommodates shear on slip systems and elastic deformation gradient accommodates lattice rotation and stretch.

 e p F F F

(1)

The evolution of

p F is related to the plastic velocity gradient in the intermediate configuration and is given as

n

 



 p p

α

 p L F F

P

(2)







1



