The Statistical Mechanics of Strain Localization in Metallic Glasses

1
The Statistical Mechanics of Strain Localization
in Metallic Glasses

• Michael L. FalkMaterials Science and
  EngineeringUniversity of Michigan

2
Applications of Bulk Metallic Glasses
http//www.liquidmetal.com
3
Metallic Glass Failure via Shear BandsAmorphous
Solids Pushed Far From Equilibrium

• Electron Micrograph of Shear Bands Formed in
  Bending Metallic GlassHufnagel, El-Deiry, Vinci
  (2000)

Quasistatic Fracture SpecimenMukai, Nieh,
Kawamura, Inoue, Higashi (2002)
4
Indentation Testing of Metallic Glass
Nanoindentation studies of shear banding in
fully amorphous and partially devitrified
metallic alloys Mat. Sci. Eng. A (2005) A.L.
Greer., A. Castellero, S.V. Madge, I.T. Walker,
J.R. Wilde

• Hardness and plastic deformation in a bulk
  metallic glass
• Acta Materialia (2005)
• U. Ramamurty, S. Jana, Y. Kawamura, K.
  Chattopadhyay

5
Examples of Strain Localization
Polymer Crazing
Mild Steel
Nanograined Metal
Young and Lovell (1991)
Van Rooyen (1970)
Wei, Jia, Ramesh and Ma (2002)
Steel _at_ High Rate
Granular Materials
Bulk Metallic Glasses
Xue, Meyers and Nesterenko (1991)
Mueth, Debregeas and et. al. (2000)
Hufnagel, El-Deiry and Vinci (2000)
6
Physics of Plasticity in Amorphous Solids

• How do we understand plastic deformation in these
  materials?
• no crystalline lattice no dislocations
• Can we use inspiration from Molecular Dynamics
  simulation and new concepts in statistical
  physics?
• How do we count shear transformation zones?
• How do these processes lead to localization?

MLF, JS Langer, PRE 1998 MLF, JS Langer, L
Pechenik, PRE 2004 Y Shi, MLF, cond-mat/0609392
7
Simulated System 3D Binary Alloy

• Wahnstrom Potential (PRA, 1991)
• Rough Approximation of Nb50Ni50
• Lennard-Jones Interactions
• Equal Interaction Energies
• Bond Length Ratios
• aNiNi 5/6 aNbNb
• aNiNb 11/12 aNbNb
• Tg 1000K
• Studied previously in the context of the glass
  transition (Lacevic, et. al. PRB 2002)

• Unlike the simulation of crystalline systems, it
  is not possible to skip simulating the processing
  step
• Glasses were created by quenching at 3 different
  rates 50K/ps, 1K/ps and 0.02 K/ps

8
Metallic Glass Nanoindentation
Simulations performed using parallelized
molecular dynamics code on 64 nodes of a parallel
cluster
R 40nmv 0.54m/s
2.5nm
45nm
600,000 atoms
100nm
Y. Shi, MLF, Acta Materialia, 55, 4317 (2007)
9
Metallic Glass Nanoindentation
0
color deviatoric strain
40
Y. Shi, MLF, Acta Materialia, 55, 4317 (2007)
10
Metallic Glass Nanoindentation
Y. Shi, MLF, Acta Materialia, 55, 4317 (2007)
11
Metallic Glass Nanoindentation
Y. Shi, MLF, Acta Materialia, 55, 4317 (2007)
12
Simulations in Simple Shear (2D)
Cumulative strain up to 50 macroscopic shear
Y Shi, MB Katz, H Li, MLF, PRL, 98, 185505 (2007)
13
2D Simple Shear Broadening
10
20
50
100
Slope1/2
14
Development of a Shear Band
10
20
50
100
Y Shi, MB Katz, H Li, MLF, PRL, 98, 185505 (2007)
15
Incorporating Structural Evolution into the Theory

• The established theories of plastic deformation
  in these materials are history independent
  because they did not include structural
  information.
• Clearly to understand this plastic localization
  process and plasticity in general, structure is
  crucial.
• How do we incorporate structure into our
  constitutive theory?

Y Shi, MB Katz, H Li, MLF, PRL, 98, 185505 (2007)
16
Current Constitutive ModelsSpaepen (1977)
Steif, Spaepen, Hutchinson (1982) Johnson, Lu,
Demetriou (2002) De Hey, Sietsma, Van den Beukel
(1998) Heggen, Spaepen, Feuerbacher (2005)

• Typically the strain rate is proposed to follow
  from an Eyring form
• Then the deformation dynamics are described via
  an equation for n, e.g.

17
Current Constitutive ModelsSpaepen (1977)
Steif, Spaepen, Hutchinson (1982) Johnson, Lu,
Demetriou (2002) De Hey, Sietsma, Van den Beukel
(1998) Heggen, Spaepen, Feuerbacher (2005)

• Problems with this formalism
• There is no standard accepted way to directly
  measure n in simulation or experiment
• Attempts to infer n by relating it to the density
  of the material result in low signal to noise.

18
Relevant Statistical Mechanics Observations

• Jamming - shear induced effective temperature in
  zero T systems (Ono, OHearn, Durian, Langer,
  Liu, Nagel)
• Effective Temperature via FDT (Berthier, Barrat
  Kurchan, Cugliandolo)
• Soft Glassy Rheology (Sollich and Cates)
• Granular Compactivity (Edwards, Mehta and
  others)
• STZ Theory/ Disorder Temperature (Falk, Langer,
  Lemaitre)

19
Testing Theories of Plastic Deformation via
Simulations of Metallic Glass(Falk and Langer
(1998), Falk, Langer and Pechenik (2004), Heggen,
Spaepen, Feuerbacher (2005), Langer (2004),
Lemaitre and Carlson (2004))

• Is there an intensive thermodynamic property
  (called ? here) that controls the number density
  of deformable regions (STZs)?
• This would be an effective temperature that
  characterizes structural degrees of freedom
  quenched into the glass.

20
Can we relate ? to the microstructure
quantitatively?

• Consider a linear relation between the ?
  parameter and the local internal energy
• Is there an underlying scaling?

21
Scaling verifies the hypothesis

• Assuming, , EZ1.9?

Y Shi, MB Katz, H Li, MLF, PRL, 98, 185505 (2007)
22
Implications for Constitutive Models

• To model the band a length scale must enter the
  constitutive relations

23
Implications for Constitutive Models

• This equation is not so different from the
  Fisher-Kolmogorov equation used to model
  propagating fronts in non-linear PDEs.
• Both exhibit propagating solutions that can be
  excited depending on the size of the
  perturbation to the system.

Fisher-Kolmogorov
24
Implications for Constitutive Models

• The Fisher-Kolmogorov equation can be simplified
  by looking for propagating solutions in a moving
  reference frame
• This is possible because of steady states at u0,
  u1.
• We also have steady states at ?0 and ? ??
• But our shear band is never propagating into a
  material with ?0. So the invaded material is
  never in steady state.
• Translational invariance cannot be achieved.

25
Numerical Results(M Lisa Manning and JS Langer,
UCSB arXiv0706.1078)

• These equations closely reproduce the details of
  the strain rate and structural profiles during
  band formation

26
Stability Analysis(M Lisa Manning and JS Langer,
UCSB arXiv0706.1078)

• Furthermore analysis of these equations allows
  Lisa to produce a stability analysis that
  predicts (R in the figure below) the onset of
  localization in her numerical results (??in the
  figure)

27
Conclusions

• We can quantify the structural state of a glass
  by a disorder temperature, ?? that is linearly
  related to the local potential energy per atom
• This parameter is predictive of the relative
  shear rate via a Boltzmann like factor, e????.
• If interpreted as kTd/EZ, where EZ is the energy
  required for STZ creation, the quantitative value
  is reasonable, 2x the bond energy.
• The stress-strain behavior is consistent with a
  yield stress assumption, not an Arrhenius
  relation between stress and strain rate.
• Numerical results closely resemble the atomistic
  simulations, and are subject to prediction via
  stability analysis (Manning)

Y. Shi, MLF, Acta Materialia, 55, 4317 (2007)
Y Shi, MB Katz, H Li, MLF, PRL, 98, 185505 (2007)