Elementary Mechanisms of Deformation in Amorphous Solids Anal - PowerPoint PPT Presentation

Elementary Mechanisms of Deformation in Amorphous Solids Anaël Lemaître Rhéophysique

Amorphous materials are, well... disordered In crystals In disordered materials defects = dislocations No topological order => defects? (Volterra, 1930; SEM, 1960) Interaction and motion understoon (Peierls, Nabarro, Friedel, 1950's) Dislocation dynamics in computer What are the elementary mechanisms codes since the 1980's of deformation? How can we up-scale the dynamics?

Length and energy scales in amorphous materials Hard glasses Soft glasses Polymers Colloids Foams Metallic/oxyde glasses Length scales 0.1 µ m's   Length scales nm's Energies ~ kT = 1/40 eV Energies ~ 0.1—1 eV Stresses ~ Pa—kPa Stresses ~ GPa Can we identify some mechanisms of deformation, at least for broad classes of materials, or time-, energy-, length-scales?

Deformation map for a metallic glass Schuh et al , Acta Mat. 55, 4067 (2007)

Deformation map for a metallic glass   ˙ Schuh et al , Acta Mat. 55, 4067 (2007)

Deformation map for a metallic glass   ˙ Schuh et al , Acta Mat. 55, 4067 (2007)

What are the elementary mechanisms of deformation in amorphous solids? Argon (1979): stress-induced hopping among inherent states local shear transformations = flips In real space In PEL

log ˙  The AQS limit Glass Liquid Low temperature: 0 T / T g 1 T < T g γ≪ 1 / τ α ˙ AQS: Neglect any thermally activated process Athermal quasi-static − 1 ≪ ˙ − 1 τ α γ≪τ irr.

Plasticity in a low-T (finite-sized) glass: The system resides at all times in local energy minima L

Plasticity in a low-T (finite-sized) glass: The system resides at all times in local energy minima It track reversibly strain-induced changes in minima L

Plasticity in a low-T (finite-sized) glass: The system resides at all times in local energy minima It track reversibly strain-induced changes in minima L

Plasticity in a low-T (finite-sized) glass: The system resides at all times in local energy minima It track reversibly strain-induced changes in minima Occasionally the occupied minimum L becomes unstable: A plastic event then occurs leading to a new local minimum

Athermal, quasi-static protocol: - Minimize energy - Apply a small increment of strain (homogeneously) - Repeat The system resides at all times in local energy minima It track reversibly strain-induced changes in minima Occasionally the occupied minimum L becomes unstable: A plastic event then occurs leading to a new local minimum

Athermal, quasi-static protocol: - Minimize energy Plastic events - Apply a small increment of strain (homogeneously) - Repeat Elastic branches σ L

L = 20 L = 40 σ γ

L = 20,40,80,160 σ 2 Δσ N /Δγ γ

AQS I: Saddle-node bifurcation σ∼− A √ γ c −γ ~− A /   c − 3 / 2 Δ E ∼(γ c −γ) C. Maloney et al, PRL 93, 195501 (2004)

AQS II: Eshelby quadrupolar events Onset of an event

AQS III: Avalanches Full plastic event = avalanche

AQS III: Avalanches In 2D C. Maloney and AL, PRL 93, 016001 (2004);  E ~ L PRE 74, 016118 (2006) E. Lerner and I. Procaccia, PRE 79, 066109 (2009)  , = 0.74  E ~ L In 3D N. Bailey et al PRL 98, 095501 (2007) 1.4  E ~ L

Particle displacement distribution in AQS =∂ y u x −∂ x u y Maloney & Robbins, J. Phys. Cond. Mat. 20, 244128 (2008)

log ˙  Athermal, finite-strain rate Glass Liquid 0 T / T g 1

log ˙  Athermal, finite-strain rate Glass Liquid 0 T / T g 1 − 12 − 2 r − 6  U = k  r Binary Lennard-Jones AL and C. Caroli, PRL 103, 065501 (2009) T = 0 ≠ 0 ˙ Athermal, finite strain-rate simulations: - Standard MD simulation f ij = m   r   v j − v i  - Damping forces

log ˙  Athermal, finite-strain rate Non-affine Glass Liquid velocity v i − ˙   y i  0 e x T / T g 1 AL and C. Caroli, PRL 103, 065501 (2009) T = 0 ≠ 0 ˙ Athermal, finite strain-rate simulations: - Standard MD simulation f ij = m   r   v j − v i  - Damping forces

Athermal, finite strain-rate Non-affine velocity v i − ˙   y i  e x L = 160 − 5 = 5.10 ˙ PRL 103, 065501 (2009) − 4 T  10

Deformation maps  xy  r  = 1% = 5% = 20%

γ= 10 − 4 γ= 10 − 2 ˙ ˙ Δ γ= 1%

How slow should we drive an athermal system to reach the AQS limit?  ˙    t t 〈 t 〉≫ Average interval event duration

What is the noise received by a weak zone? L System size: 2 ˙ R flip = L  Total flip rate: 2   0 a a 1 / R flip ~ a / c s

What is the noise received by a weak zone? L System size: 2 ˙ R flip = L  Total flip rate: 2   0 l a 2 ˙  R near = l 2  0 a a 1 / R near ~ a / c s l Now isolate a nearby region of size Near field signals are separated iff: 1 / R near ≫τ ⇔ l ≪ √ a 2 Δϵ 0 / ˙ γ τ flip

What is the noise received by a weak zone? L System size: 2 ˙ R flip = L  Total flip rate: 2   0 l a 2 ˙  R near = l Background noise: 2 − l 2  ˙ R back =  L  2  0 a 2   0 a τ During time a Local stress diffuses by:  2 a 2   0 / l 2  〈 2 〉~ ˙ 1 / R near ~ a / c s l Now isolate a nearby region of size Near field signals are separated iff: 1 / R near ≫τ ⇔ l ≪ √ a 2 Δϵ 0 / ˙ γ τ flip

What is the noise received by a weak zone? L System size: 2 ˙ R flip = L  Total flip rate: 2   0 l a 2 ˙  R near = l Background noise: 2 − l 2  ˙ R back =  L  2  0 a 2   0 a τ During time a Local stress diffuses by:  2 a 2   0 / l 2  〈 2 〉~ ˙ 1 / R near ~ a / c s l Now isolate a nearby region of size Near field signals are separated iff: 1 / R near ≫τ ⇔ l ≪ √ a 2 Δϵ 0 / ˙ γ τ flip  〈  2 〉≪ a 2   0 / l 2 

How to characterize avalanches? Transverse diffusion coefficient 〈 y 2 〉 y L=160   L=80 L=40 L=20 L=10   〈 y 2 〉  D = D / ˙    with L

Plasticity-induced diffusion e   y i = ∑ f u y 2 〉= N e   〈 u y 2 〉 e 〈 y r i −  r f  ⇒ Over a large strain interval: Events = single flips Events = linear avalanches y 2   N f  = L 2  0 a l 2  0 u = 2 a x y  4  r Eshelby:  r N a  = N f  / l 4  0 2  L  2 〉 f = a 4   0 2  2 2 2 〉 a = a l 〈 u y ln  L / a  〈 u y ln  L / l  4  2  2   0 2  0   = a 2 〉   = a 2 〉 〈 y 〈 y  l ln  L / l  ln  L / a  4  4  AL and C. Caroli, PRL 103, 065501 (2009) Chattoraj et al, PRE 011501 (2011)

Athermal, finite strain rate: transverse diffusion  2   0   = a 2 〉 D D ≡〈 y   l ln  L / l  4  Large ˙   ⇒ l ~ a D ~ ln L  ⇒   0 l ~ L D ~ L ˙ QS regime L ∝ 1 /  ˙ D / L = f  L  ˙  ⇒ l  ˙   Using

Athermal, finite strain rate: transverse diffusion 2   0   = a 2 〉 D ≡〈 y   D / L  l ln  L / l  4  Large ˙   ⇒ l ~ a D ~ ln L  ⇒   0 l ~ L D ~ L ˙ QS regime L  ˙  ∝ 1 /  ˙ D / L = f  L  ˙  ⇒ l  ˙   Using

Relevance of avalanche size 1 / 3 Extension to 3D l  ˙ ~ a   0 / ˙  flip  − 13 sec ⇒ For atomic glass, with  LJ ~ 10   0 ~ 5% a ∼ 1 nm  ≤ 10 − 3 s − 1 l ≥ 1  m For ˙ (see: Nieh et al (2002)) 2D flow curve   ˙  = 0.74  4.87  ˙  − y ≈ ˙   av  guess:  av ~ l / c s event duration: (domino-like avalanches) ⇒ = y  C  ˙   ˙ 2   0 C =  c s a  ≈ 13

log ˙  Athermal, finite-strain rate Glass Liquid  xy  r  0 T / T g L 1 Avalanche size = 10% = 1% = 5% − 1 / D l  ˙ ∝ ˙  2  c ∝ 1 / L ˙ D  L ˙ l  ˙ ∝ L L  ˙ 

log ˙  At finite temperature Glass Liquid 0 T / T g L 1 Avalanche size − 1 / D l  ˙ ∝ ˙  2  c ∝ 1 / L ˙ l  ˙ ∝ L

log ˙  At finite temperature  = 1%  = 5%  = 10% Glass Liquid T = 0.3 0 T / T g L 1 Avalanche size − 1 / D l  ˙ ∝ ˙  T = 0.2 2  c ∝ 1 / L ˙ l  ˙ ∝ L T = 0.025 Chattoraj et al PRL 105, 266001 (2010) ≠ 0 T ≠ 0 ˙ Finite T, finite strain-rate simulations: - Standard MD simulation - Velocity rescaling

At finite T For independent  D ~ ln L events: Stronger than log Chattoraj et al PRE (2011)

At finite T ∝ 1 /  ˙ l  ˙  Chattoraj et al PRE (2011)

〈 y 2 〉 At finite T D = lim    ∞  t   ˙ ≠ 0 T ≠ 0 ˙ Finite T, finite strain-rate simulations: - Standard MD simulation Chattoraj et al, PRE 2011 - Velocity rescaling