7th GAMM Seminar
on Microstructures
25-26 January
Bochum, Germany
Mechanical twinning in crystal plasticity
finite element methods
Collaboration between:
and
Luc Hantcherli - Philip Eisenlohr - Franz Roters – Dierk Raabe
Introduction
10% strain
50% strain
30% strain
undeformed
Slide 2
Outline of the presentation
• Part 1: Basics of Crystal Plasticity Finite Element Modeling
(CPFEM)
• Part 2: Review of Kalidindi’s phenomenological approach to
mechanical twinning
• Part 3: Introduction to a more physically-based approach to
mechanical twinning
• Part 4: Results and discussion
• Part 5: Conclusions and outlook
Slide 3
1-Basics of Crystal Plasticity Finite Element
Modeling
Current Configuration
Continuum mechnics / FEM:
Space and time discretization
Notion of integration point
Continuum mechanics:
Notion of tensors
Multiplicative decomposition
F = Fe Fp
F
Fe
Fp
Intermediate stress-free
Configuration
Reference Configuration
Slide 4
1-Basics of Crystal Plasticity Finite Element
Modeling
Current Configuration
Flow rule:
(given here for Fp)
Constitutive equations:
F p = L pF p
Hooke law -
F
Fe
Fp
Intermediate stress-free
Configuration
Reference Configuration
Slide 5
T* = C:E*
1-Basics of Crystal Plasticity Finite Element
Modeling
Crystal Plasticity:
Notion of kinematics, i.e. finite
number of possible deformation
modes
Current Configuration
Homogeneization:
Taylor-type Hooke law
Description of Lp
T* = Chom:E*
F
Fe
Fp
Intermediate stress-free
Configuration
Reference Configuration
Slide 6
1-Basics of Crystal Plasticity Finite Element
Modeling
N


Lp  1   f i  α S αslip    c fβ Sβtwin
i 1
β

 α
Slip deformation in
the parent region
Twin formation from
the parent region
Model for slip:
Flow rule and Hardening rule
give
Model for twin:
“Flow rule” and “Hardening rule”
give
α
fβ
Slide 7
Outline of the presentation
• Part 1: Basics of Crystal Plasticity Finite Element Modeling
(CPFEM)
• Part 2: Review of Kalidindi’s phenomenological approach to
mechanical twinning
• Part 3: Introduction to a more physically-based approach to
mechanical twinning
• Part 4: Discussions on the proposed models
• Part 5: Conclusions and outlook
Slide 8
2-Review of Kalidindi’s phenomenological
approach to mechanical twinning
•
Model initially proposed by S. Kalidindi (Kalidindi 2001)
Flow rule for slip:
Flow rule for twin:
- 12 reduced slip systems
- a viscoplastic power-type law
- a CRSS-based activation
- 12 twin systems
- a power-type law
- a unidirectional CRSS-based activation
1
m
  βtwin 
fβ  f0  c 
  
 β 
α
α  0 c sign ( α )
α
assumed analogy
Slide 9
1
m
if
 βtwin  0
2-Review of Kalidindi’s phenomenological
approach to mechanical twinning
Twins contribute to an extrahardening for non-coplanar
slip systems
   H ααα
c
α
α
Twins do not contribute to an
extra-hardening for coplanar
slip systems
  αc 
H αα  f (q)  h0 1  
 sα  
a
Slide 10



sα  s0  stwin 
f j 

 noncoplanar to α 
1
2
2-Review of Kalidindi’s phenomenological
approach to mechanical twinning

2  
2  
 N 
  hnc 
f j   hc   f i   
f j 

 i 1   coplanar to β 2
 noncoplanar toβ 2


c
β
Slide 11
2-Review of Kalidindi’s phenomenological
approach to mechanical twinning
Geometry/Mesh
- 125 linear cubic elements, each with 8
integration points
- periodic boundary conditions
- 10 random orientations per integration
point (Taylor homogenization)
- deformation in unidirectional tension
Slide 12
2-Review of Kalidindi’s phenomenological
approach to mechanical twinning
s0slip s0twin h0
85
150
355
sslip stwin hnc
265
700
10
4
hc
8000
[MPa]
Slide 13
Outline of the presentation
• Part 1: Basics of Crystal Plasticity Finite Element Modeling
(CPFEM)
• Part 2: Review of Kalidindi’s phenomenological approach to
mechanical twinning
• Part 3: Introduction to a more physically-based approach to
mechanical twinning
• Part 4: Results ans discussion
• Part 5: Conclusions and outlook
Slide 14
3-Introduction to a more physically-based
approach to mechanical twinning
•
Some ideas initially proposed by S. Allain (Phd thesis 2004)
1st idea:
2nd idea:
3rd idea:
introduce more
physically-based
variables, e.g.
dislocation densities
consider deformation
twinning as nucleationgrowth process
deeper explore the
morphological and
topological
properties of
microstructure
Slide 15
3-Introduction to a more physically-based
approach to mechanical twinning
•
Physically-based state variables:
Introduction of  αimm , immobile dislocation dentisity per glide system
parallel
forest
mobile
Derivation of 3 populations of dislocations:  α
, α
and  α
•
Flow rule:
Description of the shear rates using mobile dislocation densities and corresponding
velocities (Orowan equation)
α   αmob bvα with  αmob  imm  and vα  α ,  imm 
•
Hardening rule:
Evolution of the immobile dislocation densities from multiplication and recovery
rates
 αimm   αdislocations   αgrain bounbaries   αtwin boundaries   αrecovery
Slide 16
3-Introduction to a more physically-based
approach to mechanical twinning
•
Requirements for the twin nucleation law:
Need of special dislocation configurations, e.g. locks, as preferential sites for
twin nucleation
3
1
2
imm
imm
imm 2

 
 volume density of dislocation reactions  




Need of local stress increase on these configurations, e.g. pile-ups, to trigger the
formation of a twin nucleus
 volume fraction sampled for building pile-ups dV  dA d *  d d *
V
V
b
Need of a Schmid criterion based nucleation, e.g. classical power-law
•
Final expression for twin nucleation law:
Volume density of activated twin nuclei through expressed as:
N β  N 0 
  
imm
α
3

2
*  β
d  c

 β
Slide 17
  α 
 

b

1
m
3-Introduction to a more physically-based
approach to mechanical twinning
dislocation lines
dA
d*
slip plane
dislocation reactions
Slide 18
capture volume
3-Introduction to a more physically-based
approach to mechanical twinning
system β
•
Twin volume fraction evolution:
Computation assuming a recrystallisation like
behaviour and instantaneous growth of the
freshly nucleated twins:
eβ
fβ  (1   f )  N β Vβ
mfpβ
with
4
Vβ 
 eβ  mfp β2
3
N β  N 0 
  
imm
α
3
dβ‘
    α 
2
*  β 
d  c

 
b
 β
1
m
Slide 19
system β‘
Outline of the presentation
• Part 1: Basics of Crystal Plasticity Finite Element Modeling
(CPFEM)
• Part 2: Review of Kalidindi’s phenomenological approach to
mechanical twinning
• Part 3: Introduction to a more physically-based approach to
mechanical twinning
• Part 4: Results and discussion
• Part 5: Conclusions and outlook
Slide 20
4-Results and discussion
Slide 21
4-Results and discussion
Increase of d*
Slide 22
4-Results and discussion
decrease of C4
Slide 23
4-Results and discussion
Advantages:
Drawbacks:
- Introduction of relevant
variables, e.g. grain size,
temperature, stacking fault
energies
- Lost of computational efficiency, long
calculation time, numerical instabilities
- Dislocation-based twin
nucleation law
- Crystal plasticity induced limitation,
e.g. use of continuous and derivable
equations
Slide 24
Outline of the presentation
• Part 1: Basics of Crystal Plasticity Finite Element Modeling
(CPFEM)
• Part 2: Review of Kalidindi’s phenomenological approach to
mechanical twinning
• Part 3: Introduction to a more physically-based approach to
mechanical twinning
• Part 4: Results and discussion
• Part 5: Conclusions and outlook
Slide 25
5-Conclusions and outlook
•
Conclusion:
We proposed a physically-based CPFEM modeling that capture some of the main
physics of mechanical twinning shown in TWIP steels. Advantages and drawbacks
were discussed.
•
2 ways for future works:
–
To try to pursue the modeling of mechanical twinning, including some new
features like no constant twin thickness, new deformation modes that allow
twins to deform plastically
–
To start microstructural investigations of TWIP steels, with particular focus on
the nucleation of mechanical twins
Slide 26
5-Conclusions and outlook
Thank you all for your attention!
Slide 27