1. Fracture Toughness of Metallic Glasses: A Ductile-to-Brittle Transition? Eran Bouchbinder Weizmann Institute of Science Work with Chris H. Rycroft University of California, Berkeley Metallic glass (glassy alloy) Lowhaphandu and Lewandowski Scripta Materialia 38, 1811 (1998)

2. Elastic limit [%] Strength [MPa] Metallic Glasses Unfortunately, metallic glasses can be extremely brittle Elastic limit [%] Strength [MPa]

3. Ductile-to-Brittle transition A single glass under annealing Fracture energy [KJ/m 2 ] Poisson’s ratio “Ductile” “Brittle” The ability of a material to resist failure in the presence of a crack

4. Ductile-to-Brittle transition (cont’d) Main questions: Can one calculate the fracture toughness? What is the origin of the ductile-to-brittle transition? Fracture energy [KJ/m 2 ] Poisson’s ratio We need: A continuum theory for transient elasto-plastic response of glasses

5. Weak coupling between these two subsystems, Timescales separation, Quasi-ergodicity due to external driving forces EB & JS Langer, Physical Review E 80, 031131 (2009) EB & JS Langer, Physical Review E 80, 031132 (2009) Basic idea 1: Separable Configurational + Kinetic/Vibrational Subsystems Focus on two configurations Slow, Non-Equilibrated, Configurational rearrangements Fast, Equilibrated, Vibrational motion Mechanically stable configurations Theoretical approach: Non-equilibrium thermodynamics Total internal energy: Total entropy:

6. Basic idea 2: The non-equilibrium state of the system can be characterized by coarse-grained internal variables The elastic part of the deformation A small number of coarse-grained internal variables (order parameters), describe internal degrees of freedom that may be out of equilibrium A constrained measure of the number of configurations EB & JS Langer, Physical Review E 80, 031131 (2009) EB & JS Langer, Physical Review E 80, 031132 (2009) Non-equilibrium entropy When in the thermodynamic limit

7. Basic idea 2: The non-equilibrium state of the system can be (cont’d) characterized by coarse-grained internal variables Define two different temperatures: Ordinary, equilibrium temperature “Effective” temperature, non-equilibrium degrees of freedom EB & JS Langer, Physical Review E 80, 031131 (2009) EB & JS Langer, Physical Review E 80, 031132 (2009) is a true thermodynamic temperature, e.g. it appears in equations of state, it controls the probability of configurational fluctuations etc. Early ideas in the glass/granular materials community: Edwards, Cugliandolo, Kurchan, Coniglio, Barrat, Berthier and more Thermodynamic interpretation of soft glassy rheology models P. Sollich and M. E. Cates, Phys. Rev. E 85, 031127 (2012)

8. Fluctuation-dissipation relation Theoretical approach (cont’d) L. Berthier and J. L. Barrat, J. Chem. Phys. 116, 6228 (2002) MD simulation of a steadily sheared glass (observed also in experiments on colloids and granular materials) Transient dynamics L. Boue, H.G.E. Hentschel, I. Procaccia, I. Regev and J. Zylberg, Phys. Rev. B 81, 100201 (2010)

9. Constitutive Law within a Non-equilibrium Thermodynamic Framework Potential Energy Landscape Shear Transformation Zones (STZ)

10. Constitutive Law (cont’d) Shear Transformation Zones (STZ) model Two steps: Step 1 – Identify internal state variables and associate with them energy and entropy Step 2 – Develop equations of motion consistent with the laws of thermodynamics Assumptions: Falk & JS Langer, Physical Review E 57, 7192 (1998) EB, JS Langer & I Procaccia, Physical Review E 75, 036107 (2007) EB & JS Langer, Physical Review E 80, 031133 (2009) ML Falk & JS Langer, Annu. Rev. Condens. Matter Phys. 2, 353 (2011) Solid-like starting point Dilute distribution of soft spots of two-level nature Mean field

11. Constitutive Law (cont’d) Shear Transformation Zones (STZ) model STZ flips + STZ creation- STZ annihilation

12. The Equations Plastic flow law: Hooke’s law: Kinematics: Force balance:

13. Fracture Toughness High initial Low initial K-fields of Linear Elastic Fracture Mechanics (  Universal r -1/2 stress singularity)

14. Computational challenges Tracking free-boundaries motion Imposing realistic loading rates

15. Results

16. Results (cont’d) An elasto-plastic tip instability

17. Initiation criterion: A cavitation instability Jiang et al., Philosophical Magazine 2008, 88 407 M. L. Falk, PRB 60, 7062 (1999) Huang et al., JMPS 39, 223 (1991) MD simulation Experiment P. Guan, S. Lu, M.J.B. Spector, P.K. Valaval, and M.L. Falk, Phys. Rev. Lett. 110, 185502 (2013) Atomistic aspects Continuum estimate

18. Results (cont’d) An elasto-plastic tip instability Cavitation threshold

19. Does local cavitation lead to global failure? Lowhaphandu and Lewandowski Scripta Materialia 38, 1811 (1998)

20. Fracture energy [KJ/m 2 ] Poisson’s ratio Final comments

21. Summary The effective temperature concept is important for describing the mechanical properties of glassy/disordered materials. We demonstrated the existence of a new elasto-plastic crack tip instability as a function the degree of structural relaxation Our results offer a possible explanation for the observed annealing-induced ductile-to-brittle transition and may open the way for a better understanding of the toughness of metallic glasses Chris H. Rycroft and Eran Bouchbinder, Phys. Rev. Lett. 109, 194301 (2012)

22. exists and controls configurational fluctuations