Athermal, Quasi-static Deformation of Amorphous Materials

Transcription

1 Athermal, Quasi-static Deformation of Amorphous Materials Anaël Lemaître 1 and Craig Maloney 2 1 Institut Navier France 2 Department of Physics UC Santa Barbara Phys. Rev. Lett. 93, (2004) Phys. Rev. Lett. 93, (2004) cond-mat/ Acknowledgements: Jim Langer, Vasily Bulatov

2 Deformation of Small Systems Numerical or Experimental observations (granular materials, foams, nanoindentation): Very intermittent response σ σ γ γ Intermittency smears out with: High temperature T High strain rate γ Large system size N Suggest that deformation is heterogeneous at a small (mesoscopic) scale

3 Spatial Organization of Deformation [Maeda & Takeuchi, (1978)] Quasi-static Deformation (Numerics) [Argon & Kuo (1979)] Disordered bubble raft [Falk & Langer (1998)] Shear occurs in localized spots = Shear Transformation Zones Elementary shear analogous to the nucleation of a dislocation loop (Argon (1979))

4 Avalanche Mechanism at Mesoscopic Scale Elementary shear associated with quadrupolar transformations Transfer of constraint between elementary shear transformations lead to localization [Bulatov & Argon (1994), Langer (2001), Baret et al (2002)] b a

5 Two Types of Theories Hydrodynamic Approaches: [Spaepen(1977)] [Argon (1979)] [Falk et Langer (1998)] Mesoscopic Models: [Bulatov et Argon (1994)] [Baret, Vandembroucq et Roux (2002)] [Picard et al(2004)] Question: Does localization originates from a hydrodynamic instability or from mesoscopic transfer of constraints? How can we test theories? No a priori identification of Shear Transformation Zones Few observations of elementary rearrangements (In dry foams [Kabla & Debreageas (2003)])

6 Two Types of Theories Hydrodynamic Approaches: [Spaepen(1977)] [Argon (1979)] [Falk et Langer (1998)] Mesoscopic Models: [Bulatov et Argon (1994)] [Baret, Vandembroucq et Roux (2002)] [Picard et al(2004)] Question: Does localization originates from a hydrodynamic instability or from mesoscopic transfer of constraints? How can we test theories? No a priori identification of Shear Transformation Zones Few observations of elementary rearrangements (In dry foams [Kabla & Debreageas (2003)])

7 Two Types of Theories Hydrodynamic Approaches: [Spaepen(1977)] [Argon (1979)] [Falk et Langer (1998)] Mesoscopic Models: [Bulatov et Argon (1994)] [Baret, Vandembroucq et Roux (2002)] [Picard et al(2004)] Question: Does localization originates from a hydrodynamic instability or from mesoscopic transfer of constraints? How can we test theories? No a priori identification of Shear Transformation Zones Few observations of elementary rearrangements (In dry foams [Kabla & Debreageas (2003)])

8 Recent Numerical works MD simulations Yamamoto, Onuki (1997) Falk, Langer (PRE 1998) Rottler, Robbins (PRE 2002, PRE 2003) Varnik, Bocquet, Barrat, Berthier (PRL 2003, JCP 2004) Athermal, Quasi-static simulations Utz, Debenedetti, Stillinger (PRL 2000) Malandro, Lacks (PRL 1998, JCP 2000) Schuh, Lund (PRL 2003, Intermetallics 2004)

9 Athermal, Quasi-Static Limit Problem: intrinsic limitation of MD simulations: short timescales hence high shear rates Mimic experiments by taking the limits: T << Tg : Limited role of Thermal Fluctuations Small strain rate: γ << 1/τaging Protocol: Minimize energy Strain simulation cell Start again

10 Shear-induced changes of PEL Indicate elementary transitions associated to localized rearrangements: δσ 0.

11 The PEL Picture Strain biases potential energy landscape Increasing Strain

12 The PEL Picture Strain biases potential energy landscape Elastic Step Elastic segment: System follows an energy minimum Reversible.

13 The PEL Picture Strain biases potential energy landscape Elastic Step Elastic segment: System follows an energy minimum Reversible.

14 The PEL Picture Strain biases potential energy landscape Plastic Step Elastic segment: System follows an energy minimum Reversible.??? Plastic event: Local minimum annhiliates. Irreversible.

15 The PEL Picture Strain biases potential energy landscape Plastic Step Elastic segment: System follows an energy minimum Reversible.??? Plastic event: Local minimum annhiliates. Irreversible. σ γ

16 Atomistic Model and Algorithm Potentials: Harmonic: U = 1 2 ks2 Hertzian: U = 1 2 ks5/2 Lennard-Jones: U = k(r 12 2r 6 ) Binary mixture. Lees-Edwards boundaries. Quasistatic Shearing: limit T 0, γ 0 Apply uniform shear. Minimize energy at fixed box. Repeat. s

17 Stress-strain Curve Energy Stress Pressure Strain (.1)

18 Small System Small Events Energy d a b c Force Minimization Step

19 Small System Small Events

20 Observations on a Large System Phys. Rev. Lett. 93, (2004) U <F*F> Number of Minimization Steps

21 An Issue about Localization

22 Size of avalanches P(δΕ) x x25 50x δε P(N) x x25 50x50 1e N <δε> <δε>.0056 L L <N> <N> ~ 1.6 L L

23 Zooming on the First Quadrupole L=10 L=10

24 More about the Eschelby Calculation Decay in 2D: σ xy = 2F a cos(4θ) With: π r 2 F aµ ǫ0 ǫ pl = a 2 ǫ 0 Hence: a F a 2 µ ǫ 0 Constant when a 0

25 Fourier Analysis of the First Quadrupole Take radial component of displacement field Take its amplitude In 2D: should decay as, A r step=200 step=210 step=220 step=230 step=240 step=250 step=260 step=270 1/r r

26 Catastrophes What happens when H becomes singular? dr dσ dγ, dγ. 0.9 σ σ µ µ

27 Scaling at the Onset of Failure Recall: dr dγ = H 1.Ξ At incipient failure: λ (x) ax x(γ) 2(γc γ)/a λ (γ) 2a(γ c γ) µ (ξ ) 2 / γ c γ 1e+00 1e-02 1e-04 1/µ λ λ p -λ 0p 1e-06 1e-10 1e-08 1e-06 1e-04 γ c -γ

28 Critical Mode in Real Space ds iα ( γ 10 9 )

29 Summary Elementary quadrupolar transformation can be observed at the onset of a cascade...but they are difficult to separate at later stages A plastic event is a complex process: avalanche Power-law distribution of avalanches sizes Size of largest events determined by system size Results from long-range elastic coupling Speaking of criticality: are power laws a property of the quasi-static limit? Directions: Need for 3D numerical studies Finite strain-rates: departure from quasi-static limit How timescales come into the picture? How to construct theories of plasticity?

30 Summary Elementary quadrupolar transformation can be observed at the onset of a cascade...but they are difficult to separate at later stages A plastic event is a complex process: avalanche Power-law distribution of avalanches sizes Size of largest events determined by system size Results from long-range elastic coupling Speaking of criticality: are power laws a property of the quasi-static limit? Directions: Need for 3D numerical studies Finite strain-rates: departure from quasi-static limit How timescales come into the picture? How to construct theories of plasticity?

31 Summary Elementary quadrupolar transformation can be observed at the onset of a cascade...but they are difficult to separate at later stages A plastic event is a complex process: avalanche Power-law distribution of avalanches sizes Size of largest events determined by system size Results from long-range elastic coupling Speaking of criticality: are power laws a property of the quasi-static limit? Directions: Need for 3D numerical studies Finite strain-rates: departure from quasi-static limit How timescales come into the picture? How to construct theories of plasticity?

32 Summary Elementary quadrupolar transformation can be observed at the onset of a cascade...but they are difficult to separate at later stages A plastic event is a complex process: avalanche Power-law distribution of avalanches sizes Size of largest events determined by system size Results from long-range elastic coupling Speaking of criticality: are power laws a property of the quasi-static limit? Directions: Need for 3D numerical studies Finite strain-rates: departure from quasi-static limit How timescales come into the picture? How to construct theories of plasticity?

33 Summary Elementary quadrupolar transformation can be observed at the onset of a cascade...but they are difficult to separate at later stages A plastic event is a complex process: avalanche Power-law distribution of avalanches sizes Size of largest events determined by system size Results from long-range elastic coupling Speaking of criticality: are power laws a property of the quasi-static limit? Directions: Need for 3D numerical studies Finite strain-rates: departure from quasi-static limit How timescales come into the picture? How to construct theories of plasticity?