Mechanical twinning in crystal plasticity finite element methods

1. 7th GAMM Seminaron 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

2. Introduction 10% strain 50% strain 30% strain undeformed

3. 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

4. 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 = FeFp F Fe Fp Intermediate stress-free Configuration Reference Configuration

5. 1-Basics of Crystal Plasticity Finite Element Modeling Current Configuration Flow rule: (given here for Fp) Fp= LpFp Constitutive equations: Hooke law - T* = C:E* F Fe Fp Intermediate stress-free Configuration Reference Configuration

6. 1-Basics of Crystal Plasticity Finite Element Modeling Current Configuration Crystal Plasticity: Notion of kinematics, i.e. finite number of possible deformation modes Description of Lp Homogeneization: Taylor-type Hooke law T* = Chom:E* F Fe Fp Intermediate stress-free Configuration Reference Configuration

7. 1-Basics of Crystal Plasticity Finite Element Modeling Slip deformation in the parent region Twin formation from the parent region Model for twin: “Flow rule” and “Hardening rule” give Model for slip: Flow rule and Hardening rule give

8. 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

9. 2-Review of Kalidindi’s phenomenological approach to mechanical twinning • Model initially proposed by S. Kalidindi (Kalidindi 2001) Flow rule for slip: - 12 reduced slip systems - a viscoplastic power-type law - a CRSS-based activation Flow rule for twin: • 12 twin systems • a power-type law • a unidirectional CRSS-based activation assumedanalogy

10. 2-Review of Kalidindi’s phenomenological approach to mechanical twinning Twins contribute to an extra-hardening for non-coplanar slip systems Twins do not contribute to an extra-hardening for coplanar slip systems

11. 2-Review of Kalidindi’s phenomenological approach to mechanical twinning

12. 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 unidirectionaltension

13. 2-Review of Kalidindi’s phenomenological approach to mechanical twinning [MPa]

14. 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

15. 2nd idea: 3rd idea: 1st idea: deeper explore the morphological and topologicalproperties of microstructure consider deformation twinning as nucleation-growth process introduce more physically-based variables, e.g. dislocation densities 3-Introduction to a more physically-based approach to mechanical twinning • Some ideas initially proposed by S. Allain (Phd thesis 2004)

16. 3-Introduction to a more physically-based approach to mechanical twinning • Physically-based state variables: Introduction of , immobile dislocation dentisity per glide system Derivation of 3 populations of dislocations: , and • Flow rule: Description of the shear rates using mobile dislocation densities and corresponding velocities (Orowan equation) • Hardening rule: Evolution of the immobile dislocation densities from multiplication and recovery rates

17. 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 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 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:

18. dislocation lines dA d* slip plane dislocation reactions capture volume 3-Introduction to a more physically-based approach to mechanical twinning

19. system β eβ mfpβ dβ‘ system β‘ 3-Introduction to a more physically-based approach to mechanical twinning • Twin volume fraction evolution: Computation assuming a recrystallisation like behaviour and instantaneous growth of the freshly nucleated twins: with

20. 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

21. 4-Results and discussion

22. 4-Results and discussion Increase of d*

23. 4-Results and discussion decrease of C4

24. 4-Results and discussion Advantages: • Introduction of relevant variables, e.g. grain size, temperature, stacking fault energies • Dislocation-based twin nucleation law Drawbacks: • Lost of computational efficiency, long calculation time, numerical instabilities • Crystal plasticity induced limitation, e.g. use of continuous and derivable equations

25. 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

26. 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

27. 5-Conclusions and outlook Thank you all for your attention!