Workshop on Mean Field Modelling for Discontinuous Dynamic Recrystallization

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Workshop onMean Field Modelling
forDiscontinuous Dynamic Recrystallization

• Fréjus Summer School
• Recrystallization Mechanisms in Materials

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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

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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

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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)

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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

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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

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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

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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

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Exercise 1 1/3Mean dislocation-density

• Discrete description of grains (Di)

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Exercise 1 1/3Mean dislocation-density

• Discrete description of grains (Di)

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Exercise 1 1/3Mean dislocation-density

• Discrete description of grains (Di)

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Exercise 1 1/3Mean dislocation-density

• Discrete description of grains (Di)
• I.e. average weighted by the grain-boundary area

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Annex On the rush

• What about grain growth?
• Hillert (Acta Metall. 1965)

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Annex On the rush

• What about grain growth?
• Hillert (Acta Metall. 1965)

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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

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Exercise 1 2/3Mean dislocation-density

• Continuous description for a volume unit
• After vanishing of the initial grains

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Exercise 1 2/3Mean dislocation-density

• Continuous description for a volume unit
• After vanishing of the initial grains

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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

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Exercise 1 3/3Mean dislocation-density

• Volume constancy

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Exercise 1 3/3Mean dislocation-density

• Volume constancy

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Exercise 1 3/3Mean dislocation-density

• Volume constancy

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Exercise 1 3/3Mean dislocation-density

• Volume constancy

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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)

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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

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Exercise 2 2/2Ergodicity and averages

• ?n average dislocation-density weighted by Dn
• Steady-state case

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Exercise 2 2/2Ergodicity and averages

• ?n average dislocation-density weighted by Dn
• Steady-state case

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Exercise 2 2/2Ergodicity and averages

• ?n average dislocation-density weighted by Dn
• Steady-state case

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Exercise 2 2/2Ergodicity and averages

• ?n average dislocation-density weighted by Dn
• Steady-state case

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Exercise 2 2/2Ergodicity and averages

• ?n average dislocation-density weighted by Dn
• Steady-state case

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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
• MONTHEILLET et al. (Metall. and Mater. Trans. A,
  2014)

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Exercise 3 2/3Strain-hardening law influence

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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 s1
• 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

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Comparison ReX Frac. / Soft. Frac.

• It depends on Nb content and what else?

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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)

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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 s1