1. Deformation Twinning in Crystal Plasticity Models
Su Leen Wong
MSE 610
4/27/2006

2. Outline Brief introduction to twinning
Brief introduction to continuum mechanics
- Kinematics of deformation
- Deformation gradient
Constitutive Equations
Twinning in crystal plasticity models
- Taylor model
- Problems and solutions
- Recent additions
- Deformation twinning during impact
- Results and Simulations

3. Brief Introduction to Twinning
Definition of Twinning:
Occurs as a result of shearing across particular lattice planes
A region of a crystal in
which the orientation of the
lattice is a mirror image of
the rest of the crystal.
Two basic plastic processes:
Slip
Twinning
Twinning compared to slip:
More complicated deformation than slip
Twinning produces a volume fraction of the grain with a very different orientation compared to the rest of the grain

4. Introduction to Continuum Mechanics
The properties and response of solid and fluid models can be characterized by smooth functions of spatial variables.
Provides models for the macroscopic behavior of fluids, solids and structures.
Each particle of mass in the body has a label x which changes for a body in motion.
The motion of a body can be mathematically described by a mapping Φ between the initial and current position.
x = Φ(X ,t)
Ω0 - initial configuration
(reference configuration)
at t = 0
Ω - current configuration
at time t
A configuration of a body refers to the collection of all x values over omega. Choose one instance as a reference X, usually when time t = 0. Allows us to map the current configuration back to the initial configuration. To refer to the same particle in the body as t changes, use X to describe the position of the particle since X is forever the same.

5. Introduction to Continuum Mechanics (cont.)
The deformation gradient F allows the relative position of two particles after deformation to be described in terms of their relative material position before deformation.

6. Twinning in Crystal Plasticity Models
This is an elastic-plastic constitutive relation. The central feature of this theory is that the plastic deformation occurs in metals by flow of material through the crystal lattice without distorting the lattice itself, and then the lattice with the embedded deformed material undergoes elastic deformation and rigid rotation. The intermediate configuration is a hypothetical configuration, in that the material will not recover to this configuration upon unloading from the loaded configuration.

7. Constitutive Equations (Tensor Algebra)
Total deformation gradient
Elastic deformation gradient +
Lattice rotation
Plastic deformation gradient
Flow rule (viscoplastic model):
Evolution of the plastic deformation gradient
Sum of the shearing rates on all slip and twin systems
Plastic deformation gradient

8. Constitutive Equations and Conditions
Constitutive equation for stress:
Slip and twinning conditions
Occurs when the critical resolved shear stress on the slip or twin
plane is reached
Evolution equations for slip and twin resistances
Increasing number of twins produces a hardening of all slip
systems and twin systems.
Slip and twinning resistance increases.
Evolution equations for twin volume fractions
Describes rate of increase of twin volume content
Twin volume fraction saturates, twin formation decreases
Lattice reorientation equations
New orientations of twins have to be kept track of.

9. Taylor model of crystal plasticity
Originally proposed in 1938
The deformation gradient , F in each grain is homogeneous and equal to the macroscopic F.
Equilibrium across grain boundaries is violated.
Compatibility conditions between grains is satisfied.
Provides acceptable description of behavior of fcc polycrystals deforming by slip alone.
Original model of crystal plasticity. Most models are based on the Taylor model.
Before
deformation
After
deformation
Over predicts the responses for fcc polycrystals deforming
by slip and twinning

10. Twinning in Crystal Plasticity Models
Twinning occurs in
metals that do not possess ample slip
systems
HCP crystals where slip is restricted
FCC metals with low SFE
Alloys with low SFE
Interest in incorporating twinning into
existing polycrystal plasticity models.
Efforts in modifying the Taylor model to
include deformation by twinning.
The SFE modifies the ability of a dislocation in a crystal to glide onto an intersecting slip plane. When the SFE is low, the mobility of dislocations in a material decreases. SFE low, less likely to deform by slip.
Often those metals are ones that do not possess ample slip systems for arbitrary deformations. For example, in hexagonal close-packed crystals (HCP) slip may be restricted to a relatively small number of basal or prismatic slip systems, and there may not be any plausible slip systems which allow extension along the crystal c-axis. Twinning serves as an additional mode of deformation which enables extension along the c-axis, and thus accommodates grain-to-grain compatibility.

11. Twinning in Crystal Plasticity Models
Problem: Keeping track of twin orientations
Computationally intensive
New crystal orientations have to be
generated to reflect the orientations
of the twinned regions.
An update of the crystal orientation
is required at the end of each time
step in the simulation.
Solution: Evolve relaxed configuration
The initial configuration is kept
fixed and the relaxed configuration
is allowed to evolve during the deformation.
The relaxed configuration is continuously updated during the imposed deformation.
Calculations utilize variables in the intermediate configuration
Kalidindi, S R (1998)
Most models of the inelastic processes of projectile impact neglect twinning and assume that slip is the only mechanism of inelastic deformation. This results in a significant discrepancy between the simulated and experimental results for some materials.
Simulation based on the multiple reference configuration theory
Deformation twinning is not observed under normal temperatures and strain rates because the material is more likely to slip rather than twin
Under impact, due to very large strains, the stress in the material can be high enough to initiate twinning especially for materials that have few active slip systems.
High strain rates can be achieved, for example, when a projectile is fired onto a rigid plate. (Taylor impact test)

12. Twinning in Crystal Plasticity Models
Problem: Evolution of twin volume fractions
Twinned regions are treated as one of the
other grains after twinning.
Twinned regions are allowed to further
slip and twin.
Cannot predict increase in twin volume
fraction.
Solution: Introduce an appropriate hardening model
A criterion to arrest twinning is based on that twins are not likely to form if they must intersect existing twins.
The probability that the twin systems will intersect in a grain is computed.
If the probability of intersection is high, the twin systems are inactivated by increasing the CRSS to a value is large in comparison to slip system strength.
Myagchilov S, Dawson P R (1999)

13. Twinning in Crystal Plasticity Models
More recent additions:
Twin volume fraction saturates at some point.
Intense twin-twin and slip-twin interactions.
Increased difficulty of producing twins in the matrix at high strain levels.
Further twinning or slip does not occur inside twinned regions.
Twin volume fraction is always positive.
Twinned regions are not allowed to untwin.
Each grain is modeled as a single finite element.
Grain boundary effects are considered.
- Grain boundary sliding
- Decohesion phenomena
Staroselsky A, Anand L (2003)
Salem AA, Kalidindi SR, Semiatin SL (2005)

14. Deformation Twinning during Impact
Taylor impact test

15. Taylor Impact Test
Most models neglect twinning, assume slip takes place only
Energy released can compensate for energy dissipation due to twinning.
Onset of twinning controlled by an activation criterion.
As soon as twins are nucleated, sufficient energy is available for propagation.
Dynamic equations during impact are solved
Progress of twinning can be tracked
Static problem is solved after impact to find final shape
Agrees with experimental observations
Twinning confined to small area near impact zone and near the rear surface.
Lapczyk I, Rajagopal K R, Srinivasa A R (1998)

16. Results and Simulations
Cubic metals
• α-brass
• MP35N
• Copper
HCP metals:
• α-titanium
• Ti-Al alloys
• magnesium alloy AZ31B
(nickel-cobalt-chromium-molybdenum alloy)
Strain pole figures of rolling textures in α-titanium
Although simulated textures agree quantitatively with
experimentally measured textures, no quantitative comparisons
has been made yet.

17. References
Lapczyk I, Rajagopal KR, Srinivasa AR (1998) Deformation twinning during impact - numerical calculations using a constitutive theory based on multiple natural configurations. Computational Mechanics 21, 20-27
Kalidindi, SR (1998) Incorporation of Deformation Twinning in Crystal Plasticity Models. Journal of the Mechanics and Physics of Solids, Vol. 46, No. 2,
Staroselsky A, Anand L (1998) Inelastic deformation of polycrystalline face centered cubic materials by slip and twinning. Journal of the Mechanics and Physics of Solids. Vol. 46, No. 2,
Myagchilov S, Dawson PR (1999) Evolution of texture in aggregates of crystal exhibiting both slip and twinning. Modeling and Simulation in Materials Science and Engineering 7,
Staroselsky A, Anand L (2003) A constitutive model for hcp materials deforming by slip and twinning: application to magnesium alloy AZ31B. International Journal of Plasticity 19,
Salem AA, Kalidindi SR, Semiatin SL (2005) Strain hardening due to deformation twinning in α-titanium: Constitutive relations and crystal-plasticity modeling.
Hosford, WF (2005) Mechanical Behavior of Materials. Cambridge University Press.
Bonet J, Wood RD (1997) Nonlinear continuum mechanics for finite element analysis. Cambridge University Press.
Taylor GI, (1938) Plastic strain in metals J. Inst. Met., 62, p. 307.
Images: