Grainsize effect in 3D polycrystalline microstructure including texture evolution

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Grain-size effect in 3D polycrystalline
microstructure including texture evolution
Bin Wen and Nicholas Zabaras Materials Process
Design and Control Laboratory Sibley School of
Mechanical and Aerospace Engineering101 Frank H.
T. Rhodes Hall Cornell University Ithaca, NY
14853-3801 Email zabaras_at_cornell.edu URL
http//mpdc.mae.cornell.edu/
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Motivations
Grain/crystal
Macro
Meso
Twinning
Inter-grain slip
Grain boundary
Mechanical properties of material are extremely
essential to the quality of products. Preference
on material properties requires efficient
modeling and designing in virtual environment.
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Motivations
Adequate description of material properties using
appropriate mathematical and physical models Use
appropriate model to capture the plastic slip in
polycrystals and simulate the mechanical
properties of the material. Couple the
macro-scale finite element simulation with
underlying meso-scale constitutive model.
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Outlines
Constitutive Model based on crystal slip
theory Texture evolution of polycrystalline
material Grain size effect model Geometric
processing techniques Multiscale simulation with
homogenization method Conclusions
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Modeling of realistic 3D polycrystalline
microstructure
Mesh
Mechanical response
Load
Deformed microstructure
Virtual interrogation of microstructure
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Microstructure constitutive model
The meso-scale microstructure is represented with
a polycrystal aggregate. Each microstructure
contains several grains having different
orientations. The initial orientation is
assigned randomly and the constitutive model of
the crystal is using the rate independent
continuum slip theory developed by Anand. Total
Lagrangian algorithm is adopted.
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Microstructure constitutive model
Active slip systems is determined by comparing
trial shear stress with slip resistance.
Where the hardening matrix is
The update of the slip resistance is based on
shear strain increment
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Microstructure constitutive model
The plastic and elastic deformation gradient can
then be updated. Cauchy stress and PK-I stress
are also ready to be calculated as well as
homogenized equivalent value.
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Verification of constitutive model
Example a uniaxial compression virtual test of a
cubic microstructure.
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Texture Evolution in a discrete manner
The texture is represented by various
orientations of grains in microstruture.
Properties of polycrystals (such as strength,
heat conductivity, etc ) are highly dependent on
the texture. Crystals with random texture
generally demonstrate isotropic characters while
those having preferable texture distribution show
anisotropic.
The evolution of texture along with
microstructure deformation can be tracked by the
elastic twist while assuming plastic deformation
causes glide on slip planes only. The change of
slip system is evaluated as
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Rodrigues representation and ODF
The orientation of a grain is described by a
rotation around a specific axis in the real
space. This rotation can be conveniently
expressed using vector (or point) in 3D Rodrigues
space.
Considering the symmetries of a crystal (cubic
structure for FCC), the Rodrigues space is able
to be contracted into a finite fundamental zone.
The texture is represented using discrete
Orientation Distribution Function (ODF) in that
fundamental zone.
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Texture evolution
Initiated with random texture, a microstructure
subjected to different deformation form (boundary
condition) gives distinct evolution of textures.
Initial random texture
simple compression, szz1.0
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Texture evolution
Plane strain in y-z plane, szz1.0
Simple shear, syz0.6
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Grain size effect
Resistance to plastic flow in crystals is
dominated by dislocation density. The presence
and motion of dislocations lead to permanent
deformation and strain-hardening and can cause
incompatibility in crystals. As the lattice
incompatibility can be measured by elastic
deformation gradient, it is reasonable to
quantifies the incompatibility in Fe (Acharya and
Bassani, 2000)
A evaluation form of dislocation density is
considered as
Where slip system lattice incompatiblity
is the unique skew symmetric tensor defined by
slip normal.
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Grain size effect
Bailey-Hirsch relationship
Differentiate both sides with respect to time, a
isotropic single crystal hardening law is obtained
Substitute k1 and k2 with initial
strain-hardening rate
The first term considers hardening by strain, and
the second term considers the effect from strain
gradient, which is affected majorly by grain
size.
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Examples with different grain size
24x24x24 grid with cubic grain
12x12x12 grid with cubic grain
24x24x24 grid with phase field grain
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Examples of different grain size
12x12x12 elements
Stress-strain curves using grain size effect
model. All of the microstructures are subjected
to compression in z direction and stretch in the
other two.
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Realistic grain
64 grains
Domain decomposition
Stress field
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Voronoi Tessellation method and microstructures
(a)
(b)
(c)

• Steps
• Sample a set of points
• Calculate the grain boundaries with V.T.
• Generate the grains.

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Conforming mesh generation
Advantages No restriction on grains Fully
adaptive to microstructure geometries Element
numbers manageable Simulate the real
microstructures without assuming unrealistic
grain boundaries
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Conforming mesh generation
Conforming grids with 4097 elements
Pixel grids with 202020 elements
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Mesh Generation and Domain Decomposition
Mesh the grains
Domain decomposition
Split into brick elements
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Conforming grids example
Microstructure deformation
Mechanical response
Equivalent Stress field
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Multi-scale simulation
In material processing, the mechanical response
of a work piece is highly interested. To get an
reliable prediction of material property in
processing, accurate micro-scale constitutive
model is needed.
Given the previous microstructure constitutive
models, a multi-scale simulation can be naturally
implemented. The microstructure is coupled with
macrostructure through homogenization assumption.
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Multi-scale homogenization
The macro-scale values are obtained from the
microstructure in the form of volume average. In
this work, Cauchy stress and its derivative with
respect to deformation gradient are returned, and
the PK stress in macro-continuum is calculated
using deformation gradient at that point.
where
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Some examples
A cubic macrostructure consisting of 6x6x6
elements is compressed along z direction and
stretched in the other two. The orientations of
grains in all the microstructures are randomly
assigned.
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Examples
Strain field
Stress field
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Examples
All microstructures have the same texture.
Strain field
Stress field