Deformation Twinning in Crystal Plasticity Models

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

5. Introduction to Continuum Mechanics (cont.) • The deformation gradient Fallows 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

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

8. Constitutive Equations and Conditions Slip and twinning conditions Occurs when the critical resolved shear stress on the slip or twin plane is reached Constitutive equation for stress: 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. 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.

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)

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 Although simulated textures agree quantitatively with experimentally measured textures, no quantitative comparisons has been made yet. Strain pole figures of rolling textures in α-titanium

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, 267-290 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, 671-696 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, 975-1004 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, 1834-1864 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: http://www.eng.utah.edu/~banerjee/curr_proj.html http://scholar.lib.vt.edu/theses/available/etd-07212004-215953/ http://www.tms.org/pubs/journals/JOM/0109/Holm-0109.html http://zh.wikipedia.org/wiki/Image:Crystruc-hcp.jpg http://home.hiroshima-u.ac.jp/fpc/oguchi/graphics/fcc.gif