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Workshop on Mean Field Modelling for Discontinuous Dynamic Recrystallization

Workshop on Mean Field Modelling for Discontinuous Dynamic Recrystallization. Fréjus Summer School Recrystallization Mechanisms in Materials. Workshop on Mean-Field Modelling Introduction. Motivation

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Workshop on Mean Field Modelling for Discontinuous Dynamic Recrystallization

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  1. Workshop onMean Field Modelling forDiscontinuous Dynamic Recrystallization Fréjus Summer School Recrystallization Mechanisms in Materials David PIOT

  2. Workshop on Mean-Field Modelling Introduction • Motivation • Illustration of mean-field modelling dedicated to discontinuous dynamic recrystallization (DDRX) • Theoretical derivations related to ergodicity • 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 David PIOT

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

  4. Abstract 2/3Structure of a mean-field model for DDRX • 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 t) distributions (for non initial grains) David PIOT

  5. Abstract 3/3Structure of a mean-field model for DDRX • 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 David PIOT

  6. 1. Strain hardening and dynamic recovery • Constitutive model for • Strain hardening • Dynamic recovery • In the absence of recrystallization • 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 David PIOT

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

  8. 3. Nucleation equation • Various nucleation models available • “Simplest” equation tentative • Nucleation of new grains (t = 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 David PIOT

  9. Exercise 1 1/3Mean dislocation-density • Discrete description of grains (Di) David PIOT

  10. Exercise 1 1/3Mean dislocation-density • Discrete description of grains (Di) David PIOT

  11. Exercise 1 1/3Mean dislocation-density • Discrete description of grains (Di) David PIOT

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

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

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

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

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

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

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

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

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

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

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

  23. Exercise 2 1/2Ergodicity and averages • 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) David PIOT

  24. Exercise 2 1/2Ergodicity and averages • 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 David PIOT

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

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

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

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

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

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

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

  32. Exercise 3 3/3Strain-hardening law influence • 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: example • For PW: example • Grain-boundary mobility and nucleation parameter obtained (direct closed form for PW) from steady-state flow-stress and steady-state average grain-size David PIOT

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

  34. Exercise 4 1/1Impact of the initial microstructure • 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 1/1Test of models for parameters • Mean field models • Relevant tools to test assumptions for modelling the dependence of parameters with straining conditions • 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

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