1 / 27

The Statistical Mechanics of Strain Localization in Metallic Glasses

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.

lainey
Download Presentation

The Statistical Mechanics of Strain Localization in Metallic Glasses

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. The Statistical Mechanics of Strain Localization in Metallic Glasses Michael L. FalkMaterials Science and EngineeringUniversity of Michigan

  2. Saotome, et. al., “The micro-nanoformability of Pt-based metallic glass and the nanoforming of three-dimensional structures” Intermetallics, 2002 Applications of Bulk Metallic Glasses http://www.liquidmetal.com PITP @ UBC Vancouver

  3. Electron Micrograph of Shear Bands Formed in Bending Metallic GlassHufnagel, El-Deiry, Vinci (2000) Metallic Glass Failure via Shear BandsAmorphous Solids Pushed Far From Equilibrium Quasistatic Fracture SpecimenMukai, Nieh, Kawamura, Inoue, Higashi (2002) 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 PITP @ UBC Vancouver

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

  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) PITP @ UBC Vancouver

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

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

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

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

  13. 2D Simple Shear: Broadening 10% 20% 50% 100% Slope=1/2 PITP @ UBC Vancouver

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

  19. Free Volume Theory Shear Transformation Zone Theory mechanical disordering thermal annealing 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. 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? PITP @ UBC Vancouver

  21. Scaling verifies the hypothesis • Assuming, , EZ=1.9 Y Shi, MB Katz, H Li, MLF, PRL, 98, 185505 (2007) PITP @ UBC Vancouver

  22. Implications for Constitutive Models • To model the band a length scale must enter the constitutive relations PITP @ UBC Vancouver

  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 PITP @ UBC Vancouver

  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 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. 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 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) 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, 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) PITP @ UBC Vancouver

More Related