Deformation Twinning in Crystal Plasticity ModelsDeformation Twinning in Crystal Plasticity Models

Su Leen WongMSE 610

4/27/2006

Su Leen WongMSE 610

4/27/2006

OutlineOutline

1. Brief introduction to twinning

2. Brief introduction to continuum mechanics- Kinematics of deformation- Deformation gradient

3. Constitutive Equations

4. Twinning in crystal plasticity models- Taylor model- Problems and solutions- Recent additions- Deformation twinning during impact- Results and Simulations

1. Brief introduction to twinning

2. Brief introduction to continuum mechanics- Kinematics of deformation- Deformation gradient

3. Constitutive Equations

4. Twinning in crystal plasticity models- Taylor model- Problems and solutions- Recent additions- Deformation twinning during impact- Results and Simulations

Brief Introduction to TwinningBrief Introduction to TwinningDefinition 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

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

Introduction to Continuum Mechanics

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)

• 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 configurationat time t

Ω0 - initial configuration (reference configuration)at t = 0

Ω - current configurationat time t

Introduction to Continuum Mechanics (cont.)

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.

• 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.

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F =∂x

∂X

Twinning in Crystal Plasticity ModelsTwinning in Crystal Plasticity Models

Constitutive Equations (Tensor Algebra)

Constitutive Equations (Tensor Algebra)

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F = F * ⋅F p

Total deformation

gradient

Total deformation

gradient

Elastic deformation

gradient

+

Lattice rotation

Elastic deformation

gradient

+

Lattice rotation

Plastic deformation

gradient

Plastic deformation

gradient

Flow rule (viscoplastic model):

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˙ F p = Lp ⋅F p

Sum of the shearing

rates on all slip and twin

systems

Evolution of the plastic

deformation gradient

Plastic deformation

gradient

Constitutive Equations and Conditions

Constitutive Equations and Conditions

Constitutive equation for stress: Constitutive equation for stress:

Slip and twinning conditionsOccurs when the critical resolved shear stress on the slip or twin plane is reached

Slip and twinning conditionsOccurs when the critical resolved shear stress on the slip or twin plane is reached

Evolution equations for slip and twin resistancesIncreasing number of twins produces a hardening of all slip systems and twin systems.Slip and twinning resistance increases.

Evolution equations for slip and twin resistancesIncreasing number of twins produces a hardening of all slip systems and twin systems.Slip and twinning resistance increases.

Evolution equations for twin volume fractionsDescribes rate of increase of twin volume contentTwin volume fraction saturates, twin formation decreases

Evolution equations for twin volume fractionsDescribes rate of increase of twin volume contentTwin volume fraction saturates, twin formation decreases

Lattice reorientation equationsNew orientations of twins have to be kept track of.Lattice reorientation equationsNew orientations of twins have to be kept track of.

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T* = [C]E e

Taylor model of crystal plasticityTaylor 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.

• 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.

Beforedeformation

Afterdeformation

• Over predicts the responses for fcc polycrystals deforming by slip and twinning

• Over predicts the responses for fcc polycrystals deforming by slip and twinning

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 toinclude deformation by twinning.

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 toinclude deformation by twinning.

Twinning in Crystal Plasticity ModelsTwinning 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)

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)

Twinning in Crystal Plasticity ModelsTwinning in Crystal Plasticity Models

Twinning in Crystal Plasticity ModelsTwinning 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)

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)

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)

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)

Twinning in Crystal Plasticity ModelsTwinning in Crystal Plasticity Models

Deformation Twinning during ImpactDeformation Twinning during Impact

Taylor impact testTaylor impact test

QuickTime™ and aBMP decompressor

are needed to see this picture.

Taylor Impact TestTaylor 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.

• 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)

Results and SimulationsResults and Simulations

Although simulated textures agree quantitatively withexperimentally measured textures, no quantitative comparisons

has been made yet.

Although simulated textures agree quantitatively withexperimentally measured textures, no quantitative comparisons

has been made yet.

Strain pole figures of rolling textures in α-titanium

Cubic metals• α-brass• MP35N • Copper

HCP metals:• α-titanium• Ti-Al alloys• magnesium alloy

AZ31B

Cubic metals• α-brass• MP35N • Copper

HCP metals:• α-titanium• Ti-Al alloys• magnesium alloy

AZ31B

ReferencesReferencesLapczyk I, Rajagopal KR, Srinivasa AR (1998) Deformation twinning during impact - numerical calculations using a constitutive theory based on multiple natural configurations. Computational Mechanics 21, 20-27Kalidindi, SR (1998) Incorporation of Deformation Twinning in Crystal Plasticity Models. Journal of the Mechanics and Physics of Solids, Vol. 46, No. 2, 267-290Staroselsky 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, 671-696Myagchilov 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, 975-1004Staroselsky 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, 1834-1864Salem 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:http://www.eng.utah.edu/~banerjee/curr_proj.htmlhttp://scholar.lib.vt.edu/theses/available/etd-07212004-215953/http://www.tms.org/pubs/journals/JOM/0109/Holm-0109.htmlhttp://zh.wikipedia.org/wiki/Image:Crystruc-hcp.jpghttp://home.hiroshima-u.ac.jp/fpc/oguchi/graphics/fcc.gif

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, 20-27Kalidindi, SR (1998) Incorporation of Deformation Twinning in Crystal Plasticity Models. Journal of the Mechanics and Physics of Solids, Vol. 46, No. 2, 267-290Staroselsky 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, 671-696Myagchilov 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, 975-1004Staroselsky 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, 1834-1864Salem 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:http://www.eng.utah.edu/~banerjee/curr_proj.htmlhttp://scholar.lib.vt.edu/theses/available/etd-07212004-215953/http://www.tms.org/pubs/journals/JOM/0109/Holm-0109.htmlhttp://zh.wikipedia.org/wiki/Image:Crystruc-hcp.jpghttp://home.hiroshima-u.ac.jp/fpc/oguchi/graphics/fcc.gif