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The Statistical Mechanics of Strain Localization in Metallic Glasses. Michael L. Falk Materials Science and Engineering University of Michigan. Saotome, et. al., “The micro-nanoformability of Pt-based metallic glass and the nanoforming of three-dimensional structures” Intermetallics, 2002. - PowerPoint PPT Presentation
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The Statistical Mechanics of Strain Localization in Metallic Glasses
Michael L. FalkMaterials Science and EngineeringUniversity of Michigan
July 23, 2007 PITP @ UBC Vancouver 2
http://www.liquidmetal.com
Applications of Bulk Metallic Glasses
Saotome, et. al., “The micro-nanoformability of Pt-based metallic glass and the nanoforming of three-
dimensional structures” Intermetallics, 2002
July 23, 2007 PITP @ UBC Vancouver 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)
July 23, 2007 PITP @ UBC Vancouver 4
Indentation Testing of Metallic Glass
“Hardness and plastic deformation in a bulk metallic glass”Acta Materialia (2005)U. Ramamurty, S. Jana, Y. Kawamura, K. Chattopadhyay
“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
July 23, 2007 PITP @ UBC Vancouver 5
Steel @ High Rate Granular Materials
Polymer Crazing
Young and Lovell (1991)
Xue, Meyers and Nesterenko (1991)
Mueth, Debregeas
and et. al. (2000) Hufnagel, El-Deiry
and Vinci (2000)
Bulk Metallic Glasses
Mild Steel
Van Rooyen (1970)
Nanograined Metal
Wei, Jia, Ramesh
and Ma (2002)
Examples of Strain Localization
July 23, 2007 PITP @ UBC Vancouver 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
July 23, 2007 PITP @ UBC Vancouver 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
July 23, 2007 PITP @ UBC Vancouver 8
Metallic Glass Nanoindentation
100nm
45nm
2.5nm
R = 40nmv = 0.54m/s
600,000 atoms
Simulations performed using parallelized molecular dynamics code on 64 nodes of a parallel cluster
Y. Shi, MLF, Acta Materialia, 55, 4317 (2007)
July 23, 2007 PITP @ UBC Vancouver 9
Metallic Glass Nanoindentation
0%
colo
r =
dev
iato
ric s
trai
n
40%
QuickTime™ and a decompressor
are needed to see this picture.
Y. Shi, MLF, Acta Materialia, 55, 4317 (2007)
July 23, 2007 PITP @ UBC Vancouver 10
Metallic Glass Nanoindentation
Sam
ple
II
Sam
ple
III
Sam
ple
I
Y. Shi, MLF, Acta Materialia, 55, 4317 (2007)
July 23, 2007 PITP @ UBC Vancouver 11
Metallic Glass Nanoindentation
Sam
ple
II
Sam
ple
III
Sam
ple
I
Y. Shi, MLF, Acta Materialia, 55, 4317 (2007)
July 23, 2007 PITP @ UBC Vancouver 12
Cumulative strain up to 50% macroscopic shear
Simulations in Simple Shear (2D)
Y Shi, MB Katz, H Li, MLF, PRL, 98, 185505 (2007)
July 23, 2007 PITP @ UBC Vancouver 13
10% 20% 50% 100%
2D Simple Shear: Broadening
Slope=1/2
&εnet =2×10−5t0−1
July 23, 2007 PITP @ UBC Vancouver 14
Development of a Shear Band
10% 20% 50% 100%
Y Shi, MB Katz, H Li, MLF, PRL, 98, 185505 (2007)
July 23, 2007 PITP @ UBC Vancouver 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)
July 23, 2007 PITP @ UBC Vancouver 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.
&ε pl =n R+ s( )−R− s( )⎡⎣ ⎤⎦
R± s( ) =νexp−ΔG ±sΩ
kT⎛
⎝⎜⎞
⎠⎟
&ε pl =2nνexp −ΔGkT
⎛
⎝⎜⎞
⎠⎟sinh
sΩ2kT
⎛
⎝⎜⎞
⎠⎟
&n =−krn n−neq( ) + P &ε pl( )
n =exp −γv* vf( )
July 23, 2007 PITP @ UBC Vancouver 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.
&ε pl =2nνexp −
ΔGkT
⎛
⎝⎜⎞
⎠⎟sinh
sΩ2kT
⎛
⎝⎜⎞
⎠⎟
&n =−krn n−neq( ) + P &ε pl( ) n =exp −γv* vf( )
July 23, 2007 PITP @ UBC Vancouver 18
Relevant Statistical Mechanics Observations
Jamming - shear induced effective temperature in zero T systems (Ono, O’Hearn, 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)
July 23, 2007 PITP @ UBC Vancouver 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.
&ε ij
pl =e−1/ fij skl( )
c
0& =2sij
&ε ijpl ∞ −( )−κ T( )e−β
mechanical disordering
thermal annealing
≡
vf
γv*
Free Volume Theory
≡
kTd
EZ
Shear Transformation Zone Theory
nSTZ∝ e−1
July 23, 2007 PITP @ UBC Vancouver 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?
C1 =PE −PE0
&ε pl =e−1/ f s( )
c
0& =2s&ε pl ∞ −( )−κe−β
&ε pl y( )&εb
=e1/ b−1/ y( )
ln&ε pl y( )&εb
⎡
⎣⎢⎢
⎤
⎦⎥⎥=
1b
−C1
PE −PE0
ln&ε pl y( )&εb
⎡
⎣⎢⎢
⎤
⎦⎥⎥−
1∞ −r&εb
−1=−
C1
PE −PE0 2s&εb ∞ −b( ) =κe−β b
July 23, 2007 PITP @ UBC Vancouver 21
Scaling verifies the hypothesis
Assuming, , EZ=1.9ε ∞ =
kTg
EZ
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Y Shi, MB Katz, H Li, MLF, PRL, 98, 185505 (2007)
July 23, 2007 PITP @ UBC Vancouver 22
Implications for Constitutive Models
∂t −D∂x
2 =2s&ε pl
c0∞ −( )
∂t =
2s&ε pl
c0∞ −( )
To model the band a length scale must enter the constitutive relations
July 23, 2007 PITP @ UBC Vancouver 23
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.
Implications for Constitutive Models
∂tu=∂x
2u+ f u( )Fisher-Kolmogorov
∂tu=∂x
2u+u 1−u( )
f 0( ) =0, f 1( ) =0
f ' 0( ) =0, f ' 1( ) < 0
∂t =D∂x
2 +2sf s( )
c0e−1 ∞ −( )
July 23, 2007 PITP @ UBC Vancouver 24
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 u=0, u=1.
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.
Implications for Constitutive Models
∂tu=∂x
2u+u 1−u( ) 0 =∂x
2u−v∂xu+u 1−u( )
∂t =D∂x
2 +2sf s( )
c0e−1 ∞ −( )
July 23, 2007 PITP @ UBC Vancouver 25
Numerical Results(M Lisa Manning and JS Langer,
UCSB; arXiv:0706.1078) These equations closely
reproduce the details of the strain rate and structural profiles during band formation
July 23, 2007 PITP @ UBC Vancouver 26
Stability Analysis(M Lisa Manning and JS Langer,
UCSB; arXiv:0706.1078) 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)
July 23, 2007 PITP @ UBC Vancouver 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, e1.
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, MB Katz, H Li, MLF, PRL, 98, 185505 (2007)
Y. Shi, MLF, Acta Materialia, 55, 4317 (2007)