flow trough adjustable crystal lattice Jan Kratochvl Czech - - PowerPoint PPT Presentation

Outline

• Motivation: observed SPD substructures (high pressure torsion)
• Flow model of crystal plasticity
• Example: high pressure torsion

Crystal plasticity treated as quasi-static material flow trough adjustable crystal lattice

Jan Kratochvíl Czech Technical University and Charles University, Prague

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• After an initial adjustment to a tool, specimens twisted under high axial compression

do not change their shape and they withstand unlimited amount of plastic deformation

• An initial non-steady material flow (up to strain >~ 20) is followed by a steady flow

(saturation) where no further work hardening and structural changes are observed

• The observations excluded grain boundary sliding as the main mechanism

in explored HPT; the deformation was achieved by intergranular glide

• Strain can be defined approximately as simple shear

high pressure torsion

Austrian school: Habesberger, Pippan, Schafler, Stüve, Vorhauer, Wetscher, Zehetbauer

Motivation

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HPT copper: Hebesberger et al.: Acta Mat. (2004) axial viewpoint radial viewpoint

• preferred alignment of structural elements in radial direction inclined with respect to the

torsion axis the alignment is changing with reverse of twist no alignment is observed in axial direction

Motivation

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Motivated by severe plastic deformation experiments it seems that crystalline materials at yield behave as a special kind of incompressible, anisotropic, highly viscous fluids.

material flow though adjustable crystal lattice space

analogously as a riverbed adjusts to a water flow.

The material flow through the crystal lattice has been regarded by Asaro as crystal plasticity "basic tenet“ Advances in Applied Mechanics,1983.

Crystal plasticity can be interpreted as:

HPT copper: Hebesberger et al.: Acta Mat. (2004)

Motivation

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crystal plasticity framework:

rate dependent (rigid-viscous-plastic) principal variables of the model:

misoriented cells (subgrains): scheme of corresponding lattice space: velocity slip rates rotation of lattice stress yield stresses

governing equations:

• flow rule
• GND density
• stress equilibrium
• dissipation inequality
• yield condition
• hardening law

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Flow model of crystal plasticity

kinematics: dynamics: constitutive relations:

Kinematics

• flow rule:
• GND density:

seen in micrographs

material flow rate of lattice adjustment

material stretching: evolution of lattice adjustment:

Flow model of crystal plasticity

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Dynamics

• stress equilibrium:
• dissipation inequality:
• resolved shear stress:

Flow model of crystal plasticity

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Constitutive relations

• yield condition:
• hardening law:

local hardening non-local hardening (close range dislocation interaction)

Grama et al Acta Materialia 2003, Kratochvil et al Physical Review 2007

Flow model of crystal plasticity

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Boundary conditions: • periodic

• surface layer

Kuroda & Tvergaard Int. J. Mechanics and Physics of Solids 2008

radial direction ~ simple shear axial direction ~ plastic strain gradient

rb b h

G G

γ ρ ρ θ γ = = = ∇ ,

14 13

10 10 − =

G

ρ m

2 −

h rθ

γ =

Example: high pressure torsion

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double slip rigid-plastic model

α

) 2 ( ) 1 ( , κ

κ

lattice rotation v velocity field slip rates

∑

= −

⊗ + = ∇

2 1 ) ( ) ( ) ( 1 i i i i

m s R R v κ 

φ φ α φ α κ 4 sin ) ( 2 sin ) ( ) ( 2 cos 2

) 2 (

+ ∂ ∂ + ∂ ∂ + + ∂ ∂ − = x v y v x v

y x x

φ φ α φ α φ α φ α α α 2 cos ) sin( ) sin( ) cos( ) cos( 2 sin − + ∂ ∂ − − + ∂ ∂ + ∂ ∂ − = x v y v x v

y x x



in components: rate of lattice rotation material flow

φ φ α φ α κ 4 sin ) ( 2 sin ) ( ) ( 2 cos 2

) 1 (

− ∂ ∂ − ∂ ∂ − − ∂ ∂ − = x v y v x v

y x x

Kratochvíl, Kružík and Sedláček: Acta Materialia (2009) 10/14

Example: high pressure torsion

rate of rotation double slip activity steady state single slip is reached asymptotically (saturation)

steady state single slip

• 1. rotation of slip systems: homogeneous solution

! homogenous states are unstable !

y v L

x xy

∂ ∂ =

• Kratochvíl, Kružík and Sedláček: Acta Materialia (2009)

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• 2. fragmentation = spontaneous structuralization

formation and reconstruction of structural elements (subgrains)

Kratochvíl, Kružík and Sedláček: Rev.Adv.Mater.Sci. (2010)

fragmentation of crystal or polycrystalline grains into a pattern of structural elements (misoriented cells) is a result of a trend to reduce energetically costly multislip.

• rientation of the pattern follows the orientation of the slip systems

rotation of the slip systems causes a permanent subgrains reconstruction

Scheme of misoriented fragmented structural elements.

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cell size:

shear modulus G hardening h G/h = 100 dislocation density in boundaries ρ = m width of non-equilibrium boundaries δ = 10 nm

Kratochvíl, Kružík and Sedláček: Phys. Rev. B (2007)

16

10

2

−

R ~ 1μm

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Example: high pressure torsion

Summary: interpretation of HPT observations

key assumption: glide is carried by a double slip along rotating slip systems

• bservations model

preferred alignment structural misorientation

“3D meanders”

saturation size

?

Kratochvíl, Kružík and Sedláček: Acta Materialia (2009), Rev.Adv.Mater.Sci. (2010)

modeling: material flow through adjustable crystal lattice

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