1. Centre Sciences des matériaux et des structures Département Rhéologie, Microstructures, Thermomécanique FR CNRS 3410 – CIMReV UMR CNRS 5307 Laboratoire Georges F RIEDEL 11&12 Sept. 2014 (v1)David PIOT 1 Workshop on Mean Field Modelling for Discontinuous Dynamic Recrystallization Fréjus Summer School Recrystallization Mechanisms in Materials

2. David PIOT 2 Workshop on Mean-Field Modelling Introduction n Motivation + Illustration of mean-field modelling dedicated to discontinuous dynamic recrystallization (DDRX) + Theoretical derivations related to ergodicity n Outline + How to average dislocation densities? How to keep constant the volume? + How to test an assumption about the dependency of parameters? + Impact of the constitutive equation choice

3. David PIOT 3 Abstract 1/3 Structure of a mean-field model for DDRX n Mean-field = mesoscopic description + Description at the grain scale + Inhomogeneities at microscopic scale are averaged + Dislocation density homogeneous within each grain + Localization / Homogenization n Assumptions to simplify (but not mandatory) + No topological features + Distribution of spherical grains of various diameters + Localization: Taylor assumption

4. David PIOT 4 Abstract 2/3 Structure of a mean-field model for DDRX n Variables for describing microstrcurure + As no stochastic is considered, all grains of a given age have the same diameter and dislocation density because they have undergone identical evolution → one-parameter (nucleation time  ) distributions (for non initial grains) +

5. David PIOT 5 Abstract 3/3 Structure of a mean-field model for DDRX n Evolution of grain-property distributions + 1. Equation for strain hardening and dynamic recovery giving the evolution of dislocation densities + 2. Equation for the grain-boundary migration governing grain growth or shrinkage + 3. A nucleation model predicting the rate of new grains + 4. Disappearance of the oldest grains included in (2) when their diameter vanishes

6. David PIOT 6 1. Strain hardening and dynamic recovery n Constitutive model for + Strain hardening + Dynamic recovery + In the absence of recrystallization n General equation + Each grain behaviour is described by the same equation + Several laws can be used, e. g.: + The parameters are temperature and strain-rate dependent

7. David PIOT 7 2. Grain-boundary migration n Mean-field model + Each grain is inter- acting with an equiv- alent homogeneous matrix n Migration equation + + M grain-boundary mobility, T line energy of dislocations matrix D

8. David PIOT 8 3. Nucleation equation n Various nucleation models available n “Simplest” equation tentative + Nucleation of new grains (  = t ) is assumed to be proportional to the grain-boundary surface + + Here, p = 3 is assumed  It is the unique integer value for p compatible with experimental Derby exponent d in the relationship between grain size and stress at steady state using the closed-form equation between p and d in the power law case

9. David PIOT 9 Exercise 1 1/3 Mean dislocation-density n Discrete description of grains ( D i )

10. David PIOT 10 Exercise 1 1/3 Mean dislocation-density n Discrete description of grains ( D i )

11. David PIOT 11 Exercise 1 1/3 Mean dislocation-density n Discrete description of grains ( D i )

12. David PIOT 12 Exercise 1 1/3 Mean dislocation-density n Discrete description of grains ( D i ) + I. e. average weighted by the grain-boundary area

13. Annex: On the rush… n What about grain growth? + Hillert ( Acta Metall. 1965)

14. Annex: On the rush… n What about grain growth? + Hillert ( Acta Metall. 1965)

15. Annex: On the rush… n What about grain growth? + Hillert ( Acta Metall. 1965) n Mixed formulation + With stored energy: average dislocation-density + With surface energy: average grain-size

16. David PIOT 16 Exercise 1 2/3 Mean dislocation-density n Continuous description for a volume unit + After vanishing of the initial grains

17. David PIOT 17 Exercise 1 2/3 Mean dislocation-density n Continuous description for a volume unit + After vanishing of the initial grains

18. David PIOT 18 Exercise 1 2/3 Mean dislocation-density n Continuous description for a volume unit + After vanishing of the initial grains  Nucleation is ocurring ( t =  ) and D = 0  Disappearance of old grains ( t =  + t end ) and also D = 0

19. David PIOT 19 Exercise 1 3/3 Mean dislocation-density n Volume constancy

20. David PIOT 20 Exercise 1 3/3 Mean dislocation-density n Volume constancy

21. David PIOT 21 Exercise 1 3/3 Mean dislocation-density n Volume constancy

22. David PIOT 22 Exercise 1 3/3 Mean dislocation-density n Volume constancy

23. David PIOT 23 Exercise 2 1/2 Ergodicity and averages n Steady state = dynamic equilibrium + Ergodicity postulate when S. S. is established + Averages over the system = averages over time for a typical element of the system + All characteristic and their distribution does not depend on time and the only variable to label grains is their strain/age (current – nucleation time)

24. David PIOT 24 Exercise 2 1/2 Ergodicity and averages n Steady state = dynamic equilibrium + Ergodicity postulate when S. S. is established + Averages over the system (constant) = averages over time for a typical element of the system

25. 2014David PIOT 25 Exercise 2 2/2 Ergodicity and averages n  n : average dislocation-density weighted by D n + + Steady-state case

26. David PIOT 26 Exercise 2 2/2 Ergodicity and averages n  n : average dislocation-density weighted by D n + + Steady-state case

27. David PIOT 27 Exercise 2 2/2 Ergodicity and averages n  n : average dislocation-density weighted by D n + + Steady-state case

28. David PIOT 28 Exercise 2 2/2 Ergodicity and averages n  n : average dislocation-density weighted by D n + + Steady-state case

29. David PIOT 29 Exercise 2 2/2 Ergodicity and averages n  n : average dislocation-density weighted by D n + + Steady-state case

30. David PIOT 30 Exercise 3 1/3 Strain-hardening law influence n Comparison YLJ / PW (/KM) + PW tractable with closed forms + Physically still questionable + Easy to switch data from one to another law  M ONTHEILLET et al. (Metall. and Mater. Trans. A, 2014)

31. David PIOT 31 n Exercise 3 2/3 Strain-hardening law influence

32. David PIOT 32 Exercise 3 3/3 Strain-hardening law influence n Alternative codes, both for nickel + DDRX_YLJ + DDRX_PW + Parameters in drx.par  Pure nickel strained at 900 °C and 0.1 s –1  For YLJ: exampleexample  For PW: exampleexample  Grain-boundary mobility and nucleation parameter obtained (direct closed form for PW) from steady-state flow-stress and steady-state average grain-size

33. Comparison ReX Frac. / Soft. Frac. n It depends on… Nb content and what else?

34. Exercise 4 1/1 Impact of the initial microstructure n Comparison quasi Dirac / lognormal + Initial average grain-size : 500 µm + Flag 0  Initial grain-size distribution: Gaussian  “Standard deviation”: Variation coefficient (SD/mean)  Quasi Dirac : variation coefficient 0.05 (already done) + Flag 1  Initial grain-size distribution: lognormal  “Standard deviation”: ln-of- D SD (usual definition, dimensionless)  Parametric study ( e. g. 0.1, 0.25, 0.5, 1)

35. Exercise 5 1/1 Test of models for parameters n Mean field models + Relevant tools to test assumptions for modelling the dependence of parameters with straining conditions n Exemple : strain-rate sensitivity + Rough trial  GB mobility, nucleation, recovery, only depend on temperature  Strain hardening: power law  + Screening by comparing 0.1 with 0.01 and 1 s –1