1. The Statistical Mechanics of Strain Localization in Metallic Glasses Michael L. Falk Materials Science and Engineering University of Michigan

2. July 23, 2007PITP @ UBC Vancouver2 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

3. July 23, 2007PITP @ UBC Vancouver3 Metallic Glass Failure via Shear Bands Amorphous Solids Pushed Far From Equilibrium Electron Micrograph of Shear Bands Formed in Bending Metallic Glass Hufnagel, El-Deiry, Vinci (2000) Quasistatic Fracture Specimen Mukai, Nieh, Kawamura, Inoue, Higashi (2002)

4. July 23, 2007PITP @ UBC Vancouver4 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

5. July 23, 2007PITP @ UBC Vancouver5 Steel @ High RateGranular 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

6. July 23, 2007PITP @ UBC Vancouver6 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. July 23, 2007PITP @ UBC Vancouver7 Simulated System: 3D Binary Alloy  Wahnstrom Potential (PRA, 1991)  Rough Approximation of Nb 50 Ni 50  Lennard-Jones Interactions  Equal Interaction Energies  Bond Length Ratios:  a NiNi ~ 5 / 6 a NbNb  a NiNb ~ 11 / 12 a NbNb  T g ~ 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. July 23, 2007PITP @ UBC Vancouver8 Metallic Glass Nanoindentation 100nm 45nm 2.5nm R = 40nm v = 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)

9. July 23, 2007PITP @ UBC Vancouver9 Metallic Glass Nanoindentation 0% color = deviatoric strain 40% Y. Shi, MLF, Acta Materialia, 55, 4317 (2007)

10. July 23, 2007PITP @ UBC Vancouver10 Metallic Glass Nanoindentation Sample II Sample III Sample I Y. Shi, MLF, Acta Materialia, 55, 4317 (2007)

11. July 23, 2007PITP @ UBC Vancouver11 Metallic Glass Nanoindentation Sample II Sample III Sample I Y. Shi, MLF, Acta Materialia, 55, 4317 (2007)

12. July 23, 2007PITP @ UBC Vancouver12 Cumulative strain up to 50% macroscopic shear Simulations in Simple Shear (2D) Y Shi, MB Katz, H Li, MLF, PRL, 98, 185505 (2007)

13. July 23, 2007PITP @ UBC Vancouver13 10%20%50%100% 2D Simple Shear: Broadening Slope=1/2

14. July 23, 2007PITP @ UBC Vancouver14 Development of a Shear Band 10%20%50%100% Y Shi, MB Katz, H Li, MLF, PRL, 98, 185505 (2007)

15. July 23, 2007PITP @ UBC Vancouver15 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. July 23, 2007PITP @ UBC Vancouver16 Current Constitutive Models Spaepen (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. July 23, 2007PITP @ UBC Vancouver17 Current Constitutive Models Spaepen (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. July 23, 2007PITP @ UBC Vancouver18 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)

19. July 23, 2007PITP @ UBC Vancouver19 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. mechanical disordering thermal annealing Free Volume Theory Shear Transformation Zone Theory

20. July 23, 2007PITP @ UBC Vancouver20 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. July 23, 2007PITP @ UBC Vancouver21 Scaling verifies the hypothesis  Assuming,, E Z =1.9  Y Shi, MB Katz, H Li, MLF, PRL, 98, 185505 (2007)

22. July 23, 2007PITP @ UBC Vancouver22 Implications for Constitutive Models  To model the band a length scale must enter the constitutive relations

23. July 23, 2007PITP @ UBC Vancouver23  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 Fisher-Kolmogorov

24. July 23, 2007PITP @ UBC Vancouver24  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

25. July 23, 2007PITP @ UBC Vancouver25 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

26. July 23, 2007PITP @ UBC Vancouver26 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)

27. July 23, 2007PITP @ UBC Vancouver27 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 kT d /E Z, where E Z 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)