Heterogeneous deformation and dislocation dynamics in Cu single crystal micropillars under compressi

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Heterogeneous deformation and dislocation
dynamics in Cu single crystal micropillars under
compression

• Sreekanth Akarapu, Hussein Zbib and David Bahr

The support from the Division of Materials
Science and Engineering, Office of Basic Energy
Science at the DOE under grant number
DE-FG02-07ER46435 is gratefully acknowledged.
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Background

• Polycrystalline metals
• Bending
• Torsion
• Indentation
• Nanometallic Multi-layered Composites

• Inhomogeneous loading
• Confined plasticity

Storage of geometrically necessary dislocations
(GNDs)
SIZE EFFECTS
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Experimental Observations
Constant strain rate
Greer et al, Physical Review B, 73, 245410,2006
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Multi-Scale approach FE-DD
Background

• Finite domain problem
• Free Surface effects
• Heterogeneous deformation!!!
• Stress concentrations!!!

Tang et.al Acta Materialia 55 (2007) 16071616
(3D DD only) Benzerga et.al Scripta Materialia 54
(2006) 19371941 (2D)

• Volkert et. al, Phil. Mag., V. 86, PP 5567-5579
  (2006)

Objectives

• Be able to capture the influence of macroscopic
  boundary conditions (fixed bottom compression w
  and w/o friction between indenter and top
  surface)
• Be able to account for image stress effects on
  dynamics of dislocations
• Be able to capture the macroscopic heterogeneous
  shape changes, stress concentrations and their
  influence on dislocation dynamics

• Dislocation mechanisms responsible for observed
  bursts and hardening
• Effect of image stresses on the stress-strain
  behavior
• Effect of boundary conditions (no friction vs.
  rigid)
• Sensitivity of response to dislocation
  distribution

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Multi-scale Dislocation Dynamics Plasticity (MDDP)
Continuum
Discrete Dislocation Dynamics
Equation of Motion
Momentum
Hookes Law
Zbib et. al, IJP, V.18, 1133 (2002)
Discrete systems
Image stress effects Superposition principle
(FEM)
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Problem Setup

• Initial Dislocation Density
• 1013 (1/m2)
• Loading Constant Strain rate compression (100
  /s) along z-direction with bottom fixed.
• Top surface (a) NO FRICTION between the top
  surface and the indenter (b) STICKING FRICTION
  (no sliding).
• Material Cu
• Specimen thickness 0.2 to 2.5 microns
• The two type of defects viz. frank-read sources
  and arms extending from one surface to another
  are distributed on randomly selected slip planes

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Stress-Strain response and size dependence

• The qualitative behavior of the stress-strain
  response is comparable with experiments

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Dislocation burst Dislocation mechanisms
Spiral Source
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• The dislocation density of 1.0 to 2.5 microns
  thick specimens is gradually increasing with
  deformation.
• The dislocation density of 0.2 to 0.5 microns
  thick specimens is almost constant
• The observed hardening in 0.2 to 0.5 micron thick
  specimens is different from the classical work
  hardening observed in 1.0 to 2.5 micron thick
  specimens

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Hardening Dislocation Configurations
Effective Mean Free Path is reduced
Increase in stress to mobilize
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Deformed Contour plots
szz
Effective Plastic Strain
Magnified 10 times
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Effect of boundary conditions
With friction between indenter and the specimen
top surface
Double slip deformation
Without friction between indenter and the
specimen top surface
The thickness of the specimen is 0.5 microns
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Sensitivity of dislocation distribution

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Effect of image stresses
Dislocation density 1014 1/m2
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Conclusions

• The qualitative behavior of the stress strain
  response is comparable to experimental
  observations
• The operation of dislocation arms ending on the
  free surface are identified as the prominent
  contributors to plastic strain
• Hardening is mainly caused due dislocation
  stagnation and effective reduction of mean free
  path due to the formation of entangled
  dislocation configurations
• The influence of stress concentrations as a
  result of heterogeneous deformation is studied
  using a multi-scale approach
• The effects of image stresses and boundary
  conditions are investigated
• The sensitivity of distribution of dislocation
  despite the same density on the stress-strain
  response is presented

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Thank you
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Multi-Scale Discrete Dislocation Plasticity (MDDP)

• Local Stress distribution from applied and image
  stresses
• Shape changes
• /lattice rotation

• FEA3d
• (Macro-Scale)
• Macroscopically applied loading (Homogeneous or
  inhomogeneous) with problem specific kinematical
  constraints

• FEA3d
• Auxiliary boundary value problem for image
  stress effects due to surfaces/interfaces and
  inhomogeneous/ anisotropic properties

• Micro3d
• Discrete Dislocation Dynamics
• (anisotropic)

• Plastic Strain increment

• Internal dislocation stresses