The Statistical Mechanics of Strain Localization in Metallic Glasses

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

http://www.liquidmetal.com/applications/dsp.medical.asp

http://www.liquidmetal.com/applications/dsp.sporting.asp

http://www.liquidmetal.com/applications/dsp.space.asp

http://www.liquidmetal.com/applications/dsp.hinge.asp


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)


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


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


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


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


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)



0%

colo

r =

dev

iato

ric s

trai

n

40%

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are needed to see this picture.




Sam

ple

II

Sam

ple

III

Sam

ple

I



Cumulative strain up to 50% macroscopic shear

Simulations in Simple Shear (2D)

Y Shi, MB Katz, H Li, MLF, PRL, 98, 185505 (2007)


10% 20% 50% 100%

2D Simple Shear: Broadening

Slope=1/2

&εnet =2×10−5t0−1


Development of a Shear Band

10% 20% 50% 100%



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?



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( )


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( )


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)


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


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


Scaling verifies the hypothesis

Assuming, , EZ=1.9ε ∞ =

kTg

EZ

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are needed to see this picture.



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


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.


∂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 ∞ −( )


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.


∂tu=∂x

2u+u 1−u( ) 0 =∂x

2u−v∂xu+u 1−u( )

∂t =D∂x

2 +2sf s( )

c0e−1 ∞ −( )


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


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)


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)



The Statistical Mechanics of Strain Localization in Metallic Glasses

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