Introduction to Crystal Plasticity Modelling

Crystal Plasticity Finite Element Modelling
Crystal Plasticity
Finite Element
Modelling
CDT Lecture – Nov 2018
Suchandrima Das
suchandrima90@gmail.com

Recap of
important
concepts

Crystalline solids: Quick Recap
Crystal: Highly regular arrangement of atoms
• how do we describe them ? - lattices
• how can we study their structures ?
many ways, for example x-ray diffraction.
1-D Lattice
Crystal structures repeat in 3D
Motif  Single atom OR groups of
atoms (unit cell)
Space Lattice + Motif = Crystal
Structure
https://www.seas.upenn.edu/~chem101/sschem/solidstatechem.html/

Crystalline solids: Quick Recap
7 Crystal Systems: 7 unique unit-cell shapes
14 Bravais Lattices
https://www.seas.upenn.edu/~chem101/sschem/solidstatechem.html/

Plasticity
Linear elastic – return to original shape upon removal of load
Plastic  Beyond yield point
Permanent deformation
Energy dissipated
Path dependent
Due to dislocation generation
and movement

How does plastic deformation occur?
Not a rigid body slip
phenomenon
Part slipped Part not yet
slipped
Dislocation Line:
Line defect

1 plane of atoms slides over adjacent plane by defect motion (dislocations).
Adapted from Fig. 7.1, Callister 7e.
Note: Process is isochoric
Dislocations: Key Points

Plastic deformation
G. Dieter, Mechanical Metallurgy, 3rd Edition, McGraw-Hill, 1986.
Energy barrier
ΔE minimized  growing slipped area by
advancing DISLOCATION
Dislocations: Key Points
Caterpillar hump  extra half plane
of atoms
Subsequent bond breaking and
bond forming
Note: Process is isochoric

Dislocation Recap
Material displaces by ‘b’
Key points to remember:
• Dislocations are line defects
• Plastic deformation occurs by dislocation glide
• No volume change involved

** https://www.mm.ethz.ch/teaching.html
** Micro Meso Macro
Bulk  Many single
crystals + GB
Multiscale Modelling
CPFE lies in meso-scale region

Polycrystalline Grain Structure
CPFE needs information about the orientation of the crystal
Electron Backscatter Diffraction
Orientation  Sample relative to the crystal
Electrons diffracted from samples
create diffraction patterns.
Positions of bands  orientation

Crystallographic Orientation: EBSD Intro
Important to acquire information about the orientation of each grain in the polycrystal
Electron Backscatter Diffraction
Electrons leaving
sample
Some exit at Bragg
condition
Bragg condition 
spacing of lattice planes
Electrons diffract 
Kikuchi bands
Bands  phase &
orientation

Crystallographic Orientation: EBSD Intro
EBSD gives us the Euler angles
Euler Angles  3 angles through which the crystal must be rotated in order to relate the
crystallographic axes with the sheet reference axes
1 2 3
4

Why do we need
CPFE?

Crystal plasticity
Crystalline
materials
Thermal &
Mechanical
Behavior
Microstructure
Critical Role played by:
 Texture & orientation of Single crystals
 Total area of GBs
EXAMPLE
 Hall-Petch relationship
 Smaller the grain  Stronger the
polycrystal. Why?
 Effect cannot be captured by
macroscopic continuum model.

Hall-Petch effect & Dislocations: Key Points
So how does Hall-Petch effect come by?
GB offer resistance to dislocation motion. Why?
https://en.wikipedia.org/wiki/Grain_bo
undary_strengthening#/
pileup_force>
GB_Repulsive
force ;
deformation
continues
Pile-up
increases,
pileup_force
increases.
Dislocations
pile-up.
Orientation
mismatch :
hinders
dislocation
motion.
GB_Repulsive
force
GB –
boundary
between 2
grains of diff
orientation
Smaller grain Lesser pile-up
More difficult
to move
dislocation
across GB
Stronger
material

Polycrystal plasticity
Single crystal
plasticity
Homogenization
Polycrystal
plasticity
Assumption: GB do not play an active role in plastic deformation other than providing a
constraint on plastic strain of neighboring grains (G.I.Taylor 1938).
Assumption confirmed by exp at ambient temp. At high temp. GB sliding & diffusion 
creep
Each single crystal  elastically & plastically
anisotropic
Aggregation of many grains  Isotropic
Assumption: lengthscale ‘l’ over which fields of
interest (e.g. strain) varies is << than macroscopic
lengthscale ‘L’.
‘Effective homogeneous
material’  relaxed
continuity requirements

Crystallographic
Slip: Theory

Single Crystal Slip systems
Stretching single crystal  Slip on crystallographic planes along specific directionsStretching single crystal  Slip on crystallographic planes along specific directions
Remember: There is an energy
barrier to dislocation slip.
Inherent resistance to
deformation by crystal lattice
Peierls stress is the shear stress required to move a dislocation through
a crystal lattice at 0 K. Peierls found
𝜏 = 𝐺exp −
2𝜋𝑤
𝑏
Large w and small b will reduce 𝜏

Width  measure of degree of disruption a dislocation creates wrt perfect lattice
𝜏 = 𝐺exp −
2𝜋𝑤
𝑏
Large w and small b will reduce 𝜏
Therefore, slip most likely to occur in close-packed planes in close-packed directions.

Stretching single crystal  Slip on crystallographic planes along specific directions
Slip Directions: Shortest Lattice repeat vectors
Slip Planes: Planes with highest in-plane density
Stretching single crystal  Slip on crystallographic planes along specific directions
X
Y
Z
(111)
½[𝟏ഥ𝟏 𝟎]
FCC
Z
X
Y
(110)
½[ഥ𝟏𝟏𝟏]
BCC
Q. How many slip systems in BCC & FCC?

Schmid Factor
https://www.doitpoms.ac.uk/tlplib/slip/slip_geometry.php
Angle between Ԧ𝒍 and 𝑺 = λ
Angle between Ԧ𝒍 and 𝒏 = Φ
cos λ = Ԧ𝒍. 𝑺
cos Φ = Ԧ𝒍. 𝒏
τ  Shear stress on plane 𝒏 along 𝑺
γ  Shear strain on plane 𝒏 along 𝑺
𝜏 = 𝑷. 𝑚
𝒎 = 𝒄𝒐𝒔 𝝀 𝒄𝒐𝒔 𝜱 = Schmid Factor
How much normal stress can be
effectively transferred on to the slip
plane as shear stress?

Slip occurs when the shear stress acting in the slip direction on the slip plane reaches
some critical value  CRSS
Critically Resolved Shear Stress
𝜏 𝐶 = 𝝈 𝑦. 𝑚
𝝈 𝑦 =
𝜏 𝐶
𝑚
𝑚  Depends on relative orientation of loading axis w.r.t slip
systems
Hence, we expect 𝝈 𝑦 of single crystal to be orientation-
dependent (but not on tension vs. compression)
FCC obeys this law well
BCC shows tension compression asymmetry
𝝉 𝑹 > 𝝉 𝑪
For deformation
Yield stress and Schmid Factor

Taylor Factor: Quick Look
Further Info: https://nptel.ac.in/courses/113108054/module4/lecture17.pdf
https://inis.iaea.org/collection/NCLCollectionStore/_Public/29/000/29000976.pdf
𝝈 𝑦 =
𝜏 𝐶
𝑚
Macroscopic yield stress  Schmid factor of single grain
Different yield stress for applied stress along different directions
Active slip system  one with smallest m
Single Crystals
• Crystal cannot change its shape freely  Constraints from surrounding crystals
• Grains must fit together without voids after deformation
• Several slip systems should be activated in a grain, or in a part of grain
• Slip from 5 independent slip systems is generally required to accommodate the five
independent strain components for plastic deformation (Taylor 1938)
• Active combination  combination of 5 systems which minimizes accumulated slip
• Uniaxial tension in X direction 𝑀 =
σ 𝑖 ሶ𝛾 𝑖
ሶ𝜀11
𝑃 ; 𝝈 𝑦 =
𝜏 𝐶
𝑀
• Assumption: Strain and strain rate in any grain = Average strain
Poly Crystals

Plasticity
So we know now that dislocations gliding bring about plastic deformation.
Then how do they cause hardening?
Due to dislocation generation
and movement

Dislocation Hardening
material
hardens
slip rate
reduced
more
obstacles
created
Dislocation
density
increases
progressive
plastic
deformation
Dislocations multiply and entangle with one another
Distance of separation between them reduces.
They restrict each others motion. Dislocation motion becomes more difficult

Dislocations
SSD
Inherently
present. No
material is perfect
GND
To accommodate
strain gradient

SSDs & GNDs
Extra half plane of atoms. So there is lattice distortion
around the dislocation; Bonds are bent or strained.
Dislocations  Generate strain field around them
No effective strain
field  SSDs
Effective strain field
 GNDs
If you zoom in and just see one dislocation, is it a SSD or GND?

Direction of dislocation slip
Non-uniform shear applied

What do we know so far?
• Plastic deformation  Permanent deformation & path dependent
• Dislocations glide on slip systems when the 𝝉 𝑹 > 𝝉 𝑪
• Dislocations bring about hardening.
Jiang J, Zhang T, Dunne FPE, Britton TB. 2016 Deformation compatibility in a single crystalline Ni superalloy. Proc.R.Soc.A 472: 20150690

What do we want to know from CPFE?
Remember  Continuum macroscopic model does not consider
dislocation slip. Primarily we want CPFE to consider this.
Given
• Slip systems; Crystal
• Elastic properties
• Orientation
• Apply macroscopic
loading
CPFE
• Resolve load into assigned slip
systems
• Calculate slip if there is any
• Resulting plastic strain
• Lattice rotation
• Dislocation density
UMAT

Remember: Schmid Factor
Angle between Ԧ𝒍 and 𝑺 = λ
Angle between Ԧ𝒍 and 𝒏 = Φ
cos λ = Ԧ𝒍. 𝑺
cos Φ = Ԧ𝒍. 𝒏
τ  Shear stress on plane 𝒏 along 𝑺
γ  Shear strain on plane 𝒏 along 𝑺
𝜏 = 𝑷. 𝑚
𝒎 = 𝒄𝒐𝒔 𝝀 𝒄𝒐𝒔 𝜱 = Schmid Factor
How much normal stress can be
effectively transferred on to the slip
plane as shear stress?

Slip occurs when the shear stress acting in the slip direction on the slip plane reaches
some critical value  CRSS
Critically Resolved Shear Stress
𝝉 𝑹 > 𝝉 𝑪
For deformation

Crystallographic Slip 𝜷 𝒑
Crystallographic slip rate ሶ𝛽 𝑝
𝜆
Crystallographic slip
ሶ𝛽 𝑝
𝜆
∆𝑡 = ∆𝛽 𝑝
𝜆
Summed over all slip systems
෍
λ=1
𝑛
ሶ𝛽 𝑝
𝜆
∆𝑡 = ∆𝛽 𝑝
Summed over time
𝛽 𝑝
𝑡+∆𝑡
= 𝛽 𝑝
𝑡
+ ෍
λ=1
𝑛
ሶ𝛽 𝑝
𝜆
∆𝑡
In FE We progress from time 𝑡 to 𝑡 +
∆𝑡
For each time increment ∆𝑡, it is useful
to develop the formulation in terms of
rate. Some formulations like
viscoplasticity are rate dependent.
Plasticity is an incremental process.
Instead of dealing with increments in
slip, we consider slip rate.
Rate of slip is calculated
So we need to find ሶ𝜷 𝒑
𝝀
to find 𝛽 𝑝

Finding ሶ𝜷 𝑝
Constitutive Slip Law is used to define how much slip will take place
ሶ𝛽 𝑝 𝜆
∝ 𝑓 𝜏, 𝜏 𝐶
Can be a simple power law
1. Phenomenological: Commonly Used Power Law
ሶ𝛽 𝑝 𝜆
= K
𝜏
𝜏 𝐶
𝑛
for 𝜏 > 𝜏 𝐶

Finding ሶ𝜷 𝑝
2. Physically Based Slip Law Thermally activated process of movement of
dislocations, overcoming pinning obstacles
ሶ𝛽 𝑝 𝜆
= 𝑞
𝑣 𝜆
𝑏 𝜆
𝐿ℎ
= 𝜌 𝑚
𝜆
𝑏 𝜆
𝑣 𝜆
𝑣 = 𝑑𝛤 = 𝑑
𝜈𝑏 𝜆
2𝑙
exp −
∆𝐺
𝑘𝑇
∆𝐺 = ∆𝐹 − 𝜏𝑉
ሶ𝛽 𝑝
𝜆
= 𝜌 𝑔 𝜈 𝑏 𝜆 2
exp −
∆𝐹 𝜆
𝑘𝑇
sinh
sgn(𝜏 𝜆)( 𝜏 𝜆 − 𝜏 𝑐
𝜆)𝑉 𝜆
𝑘𝑇
ሶ𝛽 𝑝
𝜆
= α sinh(β( 𝜏 𝜆
− 𝜏 𝑐
𝜆
))

How to find strain, rotation and dislo density?
ሶ𝜷 𝑝 or 𝜷 𝒑 F L
Strain
Rotation
Total lattice
deformation 
Gradient GND

Deformation Gradient
U determines deformation:
• Translation
• Rigid Body rotation
• Stretch/Shape change
• Or a combination
Second-order tensor
Maps the undeformed state to the deformed state of a sample
𝑭 =
𝜕𝒙
𝜕𝑿
= 𝑰 +
𝜕𝒖
𝜕𝑿
= 𝑰 + 𝜷
Representation Analogous to
strain for small deformations
Displacement gradient
Infinitesimal line element dX becomes dx
𝑭 Contains all the information about
stretch, rotation, translation.

F.P.E.Dunne: Lecture – The Crystal Approach 2014
Examples of Deformation

Deformation Gradient Calculation
𝜕𝒙 = 𝑭 𝜕𝑿
𝑑𝒙
𝑑𝒚
𝑑𝒛
=
𝑭 𝑥𝑋 𝑭 𝑥𝑌 𝑭 𝑥𝑍
𝑭 𝑦𝑋 𝑭 𝑦𝑌 𝑭 𝑦𝑍
𝑭 𝑧𝑋 𝑭 𝑧𝑌 𝑭 𝑧𝑍
𝑑𝑿
𝑑𝒀
𝑑𝒁
=
𝜕𝒙
𝜕𝑿
𝜕𝒙
𝜕𝒀
𝜕𝒙
𝜕𝒁
𝜕𝒚
𝜕𝑿
𝜕𝒚
𝜕𝒀
𝜕𝒚
𝜕𝒁
𝜕𝒛
𝜕𝑿
𝜕𝒛
𝜕𝒀
𝜕𝒛
𝜕𝒁
𝑑𝑿
𝑑𝒀
𝑑𝒁
𝑭

Deformation Gradient Calculation: Examples

Deformation Gradient Calculation: Examples
𝑭 Contains all the information about stretch, rotation, translation.
How can we extract strain and rotation from it?

Strain
Rotation
Total lattice
deformation 
Gradient GND
So we know what is 𝑭
Is this only computed for CPFE?
Any FE implementation undergoing plastic deformation will compute 𝑭
So what does CPFE do in this context?

Splitting F
Introduce an intermediate imaginary
configuration dp
𝑑𝒑 = 𝑭 𝑝
𝑑𝑿
𝑑𝒙 = 𝑭 𝑒
𝑑𝒑
𝑑𝒑 = 𝑭 𝑝 𝑑𝑿 = 𝑭 𝑒−1
𝑑𝒙
𝑭 =
𝝏𝒙
𝝏𝑿
= 𝑭 𝒆 𝑭 𝒑
𝑭 can be split into the elastic (𝑭 𝑒
) and the plastic (𝑭 𝑝
) component
CPFE considers plastic deformation by dislocation slip and computes the resulting 𝑭 𝒑

Plastic Deformation Gradient
How can we calculate 𝑭 𝒑
?
𝑭 𝑃
= ෍
𝜆
𝛽 𝑝 𝜆
𝒔 𝜆
⨂𝒎 𝜆
crystallographic slip 𝜷 𝑝
s  Slip direction
M  Slip normal
𝜆  A particular Slip system
Just like we consider slip rate, we will also consider the rate of 𝑭 𝑃
i.e. ሶ𝑭 𝑝
ሶ𝑭 𝑝 = ෍
𝜆
ሶ𝛽 𝑝 𝜆
𝒔 𝜆⨂𝒎 𝜆
Remember  Any FE implementation undergoing plastic deformation will compute 𝑭
CPFE will compute 𝑭 𝑷
& ሶ𝑭 𝒑 by considering dislocation slip.
Knowing 𝑭 and 𝑭 𝑷
, you can then compute 𝑭 𝒆

Strain
Rotation
Total lattice
deformation 
Gradient GND
So we know what is 𝑭 and how we can compute its elastic and plastic components.
Now what is L?
We have discussed so far rate quantities like ሶ𝜷 𝑝
& ሶ𝑭 𝑝. L is a form used to represent
ሶ𝑭 𝑝 .
Simply, 𝑳 =
𝜕𝒗
𝜕𝒙
= ሶ𝑭𝑭−1 (theoretically  spatial rate of change of velocity).
FE implementation will compute L. Once again we need CPFE to compute its plastic
component. So how to get strain and rotation from L?

Velocity Gradient
L
Symmetric
Component
1
2
𝑳 + 𝑳 𝑇
rate of
deformation D
Stretch related
Anti-Symmetric
Component
1
2
𝑳 − 𝑳 𝑇
continuum spin
W
Rotation related
𝑳 𝑒
𝑳 𝑝
1
2
𝑳 𝑝
+ 𝑳 𝑝 𝑇
1
2
𝑳 𝑝
− 𝑳 𝑝 𝑇
𝑫 𝑝
=
𝑾 𝑝 =
1
2
𝑳 𝑒
+ 𝑳 𝑒 𝑇
1
2
𝑳 𝑒
− 𝑳 𝑒 𝑇
𝑫 𝑒 =
𝑾 𝑒
=

Equations to complete the picture
We know the slip rate from the slip law

Crystallographic Slip 𝜷 𝒑
Crystallographic slip rate ሶ𝛽 𝑝
𝜆
Crystallographic slip
ሶ𝛽 𝑝
𝜆
∆𝑡 = ∆𝛽 𝑝
𝜆
Summed over all slip systems
෍
λ=1
𝑛
ሶ𝛽 𝑝
𝜆
∆𝑡 = 𝛽 𝑝
Summed over time
𝛽 𝑝
𝑡+∆𝑡
= 𝛽 𝑝
𝑡
+ ෍
λ=1
𝑛
ሶ𝛽 𝑝
𝜆
∆𝑡
In FE We progress from time 𝑡 to 𝑡 +
∆𝑡
For each time increment ∆𝑡, it is useful
to develop the formulation in terms of
rate. Some formulations like
viscoplasticity are rate dependent.
Plasticity is an incremental process.
Instead of dealing with increments in
slip, we consider slip rate.
Rate of slip is calculated
So we need to find ሶ𝜷 𝒑
𝝀
to find 𝛽 𝑝

𝑳 𝑝
≅ ሶ𝑭 𝑝
𝑳 𝑒
= 𝑳-𝑭 𝑒
𝑳 𝑝
𝑭 𝑒−1
𝑭 𝑃
= ෍
𝜆
𝛽 𝑝 𝜆
𝒔 𝜆
⨂𝒎 𝜆
ሶ𝑭 𝑝 = ෍
𝜆
ሶ𝛽 𝑝 𝜆
𝒔 𝜆
⨂𝒎 𝜆
We will know F from FE, so we can find 𝑭 𝒆
𝑭 =
𝝏𝒙
𝝏𝑿
= 𝑭 𝒆
𝑭 𝒑
We also know L from FE. We then find 𝑳 𝑝
& 𝑳 𝑒
We then find 𝑫 𝑝
𝑾 𝑝
𝑫 𝑒
𝑾 𝑒

Velocity Gradient
L
Symmetric
Component
1
2
𝑳 + 𝑳 𝑇
rate of
deformation D
Anti-Symmetric
Component
1
2
𝑳 − 𝑳 𝑇
continuum spin
W
𝑳 𝑒
𝑳 𝑝
1
2
𝑳 𝑝
+ 𝑳 𝑝 𝑇
1
2
𝑳 𝑝
− 𝑳 𝑝 𝑇
𝑫 𝑝
=
𝑾 𝑝 =
1
2
𝑳 𝑒
+ 𝑳 𝑒 𝑇
1
2
𝑳 𝑒
− 𝑳 𝑒 𝑇
𝑫 𝑒 =
𝑾 𝑒
=

Strain
Rotation
Total lattice
deformation 
Gradient GND
We know:
𝑭 𝑃
𝑭 𝑒
𝑳 𝑃
𝑳 𝑒
𝑫 𝑃
𝑫 𝑒

∆𝜺 𝑝 = 𝑫 𝑝
∆𝑡
∆𝜺 𝑒 = 𝑫 𝑒
∆𝑡
∆𝝎 𝑒 = sym 𝑳 𝑒 ∆𝑡
Jiang J, Zhang T, Dunne FPE, Britton TB. 2016 Deformation compatibility in a single crystalline Ni superalloy. Proc.R.Soc.A 472: 20150690
Continuum Rotation: Bonds at the
atomic level remain unchanged.
No necessary change in lattice
orientation.

Dislocation Density calculation
GND can be calculated from Lattice deformation.
Lattice
rotation
Lattice
strain
Total lattice
deformation
 Gradient
 GND
SSD value not evolved. But can be.

𝝎 𝑒
+𝜺 𝑑𝑒𝑣
𝑒
Experiments
CPFE
Dislocation
Tensor
𝛼 𝑁𝑦𝑒
GND
Density 𝝆
𝜶 𝑵𝒚𝒆
≅ (− )𝐜𝐮𝐫𝐥(𝜺 𝒅𝒆𝒗
𝒆
+ 𝝎 𝒆
) 𝑻
𝜶 𝑵𝒚𝒆
= ෍
𝝀
𝒃 𝝀
ໆ 𝝆 𝝀
Geometrically Necessary Dislocation Density
Generated to accommodated lattice curvature

Key Points to Remember
• No volume change involved
• Dislocations bring about hardening too
• FE is an incremental process: Deformation computed for every Δt

FE computes
total
deformation
Calls CPFE
UMAT
Compute
∆𝜺 𝑒 ∆𝜺 𝑝 &
∆𝝎 𝑒
Compute
stress
increment
from ∆𝜺 𝑒
Elastic and plastic
deformation
quantities are
sent back to FE.
F & L
Start from time = 0, deformation has occurred for Δt
Consider
dislocation slip,
computes slip
rate, 𝐹 𝑝
& 𝐿 𝑝
Next time
increment
begins
Key Points to Remember

Examples

Y
ZX
[100]
[110]
[1-10]
5 µm
5 µm
• Substantial Increase in Pile Up
• Slip localization
• Higher max load
0
30
60
90
0 200 400 600
LoadonElement(mN)
Depth (nm)
Load vs Depth
Implanted
Unimplanted
Nano-Indentation  SEM
[010]
4.2 µm radius; 500 nm deep
Unimp
He-imp
S. Das et al., Scr. Mater. 146 (2018) 335–339.
http://linkinghub.elsevier.com/retrieve/pii/S1359646217307145

𝛾ሶ 𝛼
= 𝜌 𝑔 𝜈(𝑏 𝛼 )2
exp⁡−
∆𝐹 𝛼
𝑘𝑇
sinh
(𝜏 𝛼
− 𝜏 𝑐
𝛼
)𝛾0∆𝑉 𝛼
𝑘𝑇
Slip-systems – {110}<111>
• 12 edge type
• 4 screw type
𝑙 =
1
)𝛹(𝜌 𝐺𝑁𝐷 + 𝜌 𝑆𝑆𝐷
𝑉  spacing between the pinning
dislocations 𝑙 .
𝑳 𝑝
= ෍
𝜆
ሶ𝛾 𝛼
𝒔 𝛼
ໆ 𝒏 𝛼
≅ ሶ𝑭 𝑝
Fitted to
experiment
Slip Law for slip system α
Example of Nano-indentation in Tungsten
https://www.sciencedirect.com/science/article/pii/S0749641918300068?via%3Dihub
S. Das et al., Int. J. Plast. 109 (2018) 18–42.

Surface Profile Prediction
What happens without CPFE?
S. Das et al., arXiv Prepr. arXiv1901.00745 (2018)
http://arxiv.org/abs/1901.00745

Surface Profile Prediction for different
orientations
https://ora.ox.ac.uk/objects/uuid:ea8aa247-fd6a-4b2d-a4be-04aa229da828

Lattice rotation
Das, S. et al. Int. J. Plast. https://doi.org/10.1016/j.ijplas.2018.05.001

Lattice strain
Das, S. et al. Int. J. Plast. https://doi.org/10.1016/j.ijplas.2018.05.001

Thank You

Back-up

Strain and Rotation from F
𝑭 =
𝜕𝒙
𝜕𝑿
=
𝜕(𝑿 + 𝒖)
𝜕𝑿
= 𝑰 +
𝜕𝒖
𝜕𝑿
= 𝑰 + 𝜷
𝑰 =
𝟏 𝟎 𝟎
𝟎 𝟏 𝟎
𝟎 𝟎 𝟏
𝜷
Symmetric
1
2
𝜷 + 𝜷 𝑇
Strain
Asymmetric
1
2
𝜷 − 𝜷 𝑇
Rotation
For small deformations only
But we want elastic and plastic components of strain. So to get that we decompose
F into elastic and plastic parts.

ሶ𝛽 𝑝
𝜆
= 𝜌 𝑔 𝜈 𝑏 𝜆 2
exp −
∆𝐹 𝜆
𝑘𝑇
sinh
sgn(𝜏 𝜆
)( 𝜏 𝜆
− 𝜏 𝑐
𝜆
)𝑉 𝜆
𝑘𝑇
𝑉 depends on the spacing between the pinning dislocations
𝑙 . Decrease mean free path of slip 𝑙
𝑙 =
1
𝛹(𝜌 𝐺𝑁𝐷 + 𝜌 𝑆𝑆𝐷)
GNDs can also be introduced in the form of increasing CRSS
ሶ𝛽 𝑝 𝜆
= K
𝜏
𝜏 𝐶
𝑛
for 𝜏 > 𝜏 𝐶
𝜏 𝐶
𝑡+∆𝑡
= 𝜏 𝐶
𝑡
+ 𝐺𝑏 𝛹(𝜌 𝐺𝑁𝐷 + 𝜌 𝑆𝑆𝐷)
1. Phenomenological: Commonly Used Power Law
2. Physically Based Slip Law

Introduction to Crystal Plasticity Modelling

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