content Recollection Part I & II

Transcription

1 content Recollection Part I & II

2 recollection of part I & II deformation/velocity gradient characterizes local at material point) change/rate of change of shape and size material point model connects stress response) to strain boundary cond.) finite strain plasticity introduces multiplicative decomposition with intermediate configuration Reference configuration xx Intermediate or relaxed) configuration F p Infinitesimal neighborhood of x Sτ) S = 1 2 C : E Fx) n i m i material constitutive law F e L p τ) Ḟ p = L p F p Current configuration yx) solution of elastic/plastic strain partitioning requires L p from constitutive model F τ) F = FF p 1 F p τ)

3 recollection of part I & II check out the source code... crystallite.f90

4 content PART III Monocrystal Plasticity Models

5 dislocation slip from

6 dislocation slip reason for plane rotation? from

7 slip systems in face centered cubic structure n b

8 shear from slip dγ b h dl dγ = b dl hl = b da V l

9 plastic velocity gradient L p = L p ) ij = ẋ i x j = α γ α) b α) n α)

10 resolved shear stress uniaxial τ α) = σ cos φ α) cos λ α) from

11 resolved shear stress uniaxial τ α) = σ cos φ α) cos λ α) general τ α) = S :b α) n α) ) from

12 room-temperature deformation resistance in fcc metals ɛ low strain-rate sensitivity σ largely monotonic decrease in strain hardening coefficient ɛ

13 phenomenological description deformation kinetics γ α) = γ 0 τ α) τ α) c n sign τ α) microstructure evolution τ α) c = β q αβ h 0 1 τ c β) τ s ) a γ β)

14 phenomenological description drawbacks independent of temperature independent of strain path independent of grain size

15 dislocation density-based description basis γ = b A V = b l V ẇ = b ϱ m v issues parameterization of microstructure velocity of dislocations evolution of microstructure

16 dislocation structure parameterization following Ma & Roters 2004) density on each system ϱ α) with α =1,..., 12

17 dislocation structure parameterization following Ma & Roters 2004) projected perpendicular density ϱ α) = 1 2 β [ ] ϱ β) n α) n β) b β) ) + n α) b β)

18 dislocation structure parameterization following Ma & Roters 2004) projected parallel density ϱ α) = 1 2 β [ ] ϱ β) n α) n β) b β) ) + n α) b β)

19 dislocation structure parameterization following Ma & Roters 2004) derived mobile density τ α) ϱ α) m ϱ α) m = 0 ϱ α) ϱα)

20 dislocation structure parameterization following Eisenlohr & Blum 2005) specific) dipole density

21 dislocation structure parameterization following Eisenlohr & Blum 2005) specific) dipole density

22 dislocation structure parameterization following Eisenlohr & Blum 2005) specific) dipole density

23 dislocation structure parameterization following Eisenlohr & Blum 2005) specific) dipole density ϱ α) sgl with α =1,..., 12 ϱ α) dip with α =1,..., 12 χ α) ϱα) dip h with α =1,..., 12 h spon < h < ĥα) 1 8π1 ν) Gb τ α)

24 dislocation forest interaction from zig.onera.fr

25 dislocation forest interaction from zig.onera.fr

26 dislocation velocity thermally activated forest cutting v α) = λ α) ν attack sign exp Q slip k B T ) τ α)) τ α) sinh eff V α) k B T ) x l

27 dislocation velocity thermally activated forest cutting v α) = λ α) ν attack sign exp Q slip k B T ) τ α)) τ α) sinh eff V α) k B T ) x l V α) = b l x

28 dislocation velocity thermally activated forest cutting v α) = λ α) ν attack sign exp Q slip k B T ) τ α)) τ α) sinh eff V α) k B T ) x l V α) = b l x l ϱ α) ) 0.5

29 dislocation velocity thermally activated forest cutting v α) = λ α) ν attack sign exp Q slip k B T ) τ α)) τ α) sinh eff V α) k B T ) x l

30 dislocation velocity thermally activated forest cutting v α) = λ α) ν attack sign exp Q slip k B T ) τ α)) τ α) sinh eff V α) k B T ) x l τ α) = τ α) τ α) eff pass { τ α) c = 3 Gb ϱ α) + ϱ α) m if τ α) > τ α) pass 0 otherwise

31 dislocation structure evolution χ = 2 γ bn g exchange of constituents dipole decompostion {}}{ {{}}{ ĥ }} ){ ϱ sgl + χ dh 1 χh χh h 1 0.9h/ĥ 1 + 2v ) cχ ln χ) {}}{ { }} ){ h ln h) 1 {}}{}{{} climb of constituents } dipole formation ϱ sgl = 2 γ b N g dislocation generation {}}{ N g 2Λ dipole formation {}}{ ĥϱ sgl + dipole decomposition {}}{ ĥ ) 1 χh h spon 1 0.9h/ĥ 1 dh

32 recollection of part III check out the source code... constitutive.f90