A primary cause of plastic deformation of crystalline metals is dislocation slip. The crystal plasticity material model is widely used to capture slip-based plastic deformation on the microscale, accounting for the underlying crystal type and microstructure of the material.
Crystal plasticity simulation can predict the structure-property relation of additively manufactured parts to show how the underlying microstructure affects the macroscopic mechanical response. A common microstructure change leading to significant alteration of properties is the change in grain size, which significantly affects the yield stress (known as the Hall-Petch effect) and ductility.
The Hall-Petch effect can be captured in crystal plasticity simulations via a uniform but grain-size dependent initial hardness, or a spatially nonuniform initial hardness distribution based on the closest distance to neighboring grain boundaries.
Theory Overview
Under finite deformation with thermal strain, the total deformation gradient can be
decomposed into the mechanical part 
and the thermal part 
:
The mechanical deformation gradient 
can be further decomposed multiplicatively into its elastic 
and plastic 
parts:
The current crystal plasticity model considers plasticity due to dislocation slip. The
plastic velocity gradient 
can therefore be related to the slip rate 
on a slip system 
:
where:
 = number of slip systems |
 = Schmid tensor, where  ( being the slip direction and  the slip plane normal) |
For face-centered cubic (FCC) materials, the slip rate 
can be related to the resolved shear stress 
as:
where:
 = pre-exponential factor (attack frequency for FCC materials) |
 = activation energy for crossing the hardness without external shear
stress |
 = Boltzmann constant |
 = temperature |
 = hardness (slip resistance) |
 = athermal barrier to slip-comprised immobile dislocation
groups |
The Hall-Petch effect due to grain boundary strengthening can be simulated using a
spatially nonuniform distribution of the initial value of 
(the initial hardness), which is related to the closest distance to
neighboring grain boundaries 
:
where:
 = initial hardness value for a point infinitely far away from the grain
boundary |
 = initial hardness value for a point on the grain boundary |
 = characteristic length that determines how quickly the hardness values
change with  |
| This relation is phenomenological and is not based on any first principles. Other functional forms have been proposed and can be used as well.[3] |
The resolved shear stress 
can be related to the second Piola Kirchoff stress (in the intermediate
configuration) 
as:
where 
is the right Cauchy-Green elastic deformation tensor. 
can be determined from 
as:
where 
is the fourth-order material Jacobian.