Avalanche contribution to nonlinear elasticity of granular materials - PowerPoint PPT Presentation

Avalanche contribution to nonlinear elasticity of granular materials Michio Otsuki (Shimane Univ.), Hisao Hayakawa (Kyoto Univ.)

Contents 1. Introduction : Jamming transition and shear modulus. 2. Critical behavior of shear modulus for frictionless granular materials under finite oscillatory shear. 3.Theory for critical exponents. 4. Effect of the friction between particles. MO and H. Hayakawa, PRE 90, 042202 (2014)

Jamming transition φ：volume fraction φ J : Critical fraction Low density : Liquid High density : Solid •Granular materials can flow below a critical density φ J . •Above φ J , the materials have rigidity and behave as solids. •This transition is known as the jamming transition. C. O’Hern, et al., Phys. Rev. Lett. 88, 075507 (2002)

Scaling of shear modulus Elastic interaction force Δ=1 Theory for elastic network : Shear modulus : r contact length : r F n F n Δ= 3/2 (Hertzian force) Shear stress σ Shear stress σ Δ=1 (linear force) M. Wyart, Ann. Phys. Fr. (2005) G = σ / γ 0 F n ∝ r ∆ G ∝ ( φ − φ J ) ∆ − 1 / 2 P ∝ ( φ − φ J ) ∆ 0.06 Shear modulus : G 0.04 Strain γ 0 0.02 0 0.64 0.65 0.66 0.67 volume fraction ：Φ

G : shear modulus, P : pressure Different scaling relations G ∝ γ − c G ∝ P 0 ( φ − φ J ) Mason et al., PRE 1996, (experiments of emulsions) G ∝ ( φ − φ J ) ∆ Coulais, Seguin, and Dauchot, PRL 2014 Okamura & Yoshino 2013, (Replica theory with small temperature)

Purpose different scaling relations. G : shear modulus, P : pressure materials under oscillatory shear. •For this purpose, we perform a simulation of granular •We would like to clarify the relationship between Purpose : G ∝ ( φ − φ J ) ∆ − 1 / 2 G ∝ P Mason et al., PRE 1996, (experiments of emulsions) G ∝ ( φ − φ J ) ∆ P ∝ ( φ − φ J ) ∆ Okamura & Yoshino 2013, (Replica theory with small temperature) C. O’Hern, et al., Phys. Rev. Lett. 88, 075507 (2002) G ∝ γ − c 0 ( φ − φ J ) Coulais, Seguin, and Dauchot, PRL 2014

Contents 1. Introduction : Jamming transition and shear modulus. 2. Critical behavior of shear modulus for frictionless granular materials under finite oscillatory shear. 3.Theory for critical exponents. 4. Effect of the friction between particles. MO and H. Hayakawa, PRE 90, 042202 (2014)

Model of frictionless particles Normal interaction force Strain γ(t) σ of G on γ 0 and Φ. We investigate the dependence Quasi-static limit : ω→0 Shear modulus (storage modulus) Strain γ(t) • Oscillatory shear strain γ ( t ) = γ 0 cos( ω t ) • Frequency：ω • Strain amplitude：γ 0 • Shear stress : σ(t) . � 2 π / ω dtS ( t ) cos( ω t ) G ( γ 0 , φ ) = ω π γ 0 0 F n = kr ∆ + η ˙ r

Initial configuration Broken bonds at γ=0.0004 Δ=1 Origin : Avalanche (correlated bond breakage) Finite strain : Infinitesimal strain : Broken bonds at γ=0.0001 Shear modulus under finite strain (Δ=1) Broken bonds at γ=0.0008 Initial configuration Avalanche-like bond breakage MO and H. Hayakawa, PRE (2014) 10 -1 G ∝ ( φ − φ J ) 1 / 2 (a) (b) 10 -2 G / G 0 G ∝ ( φ − φ J ) strain : γ 0 10 -3 (c) (d) γ 0 = 10 -5 γ 0 = 10 -3 γ 0 = 10 -2 finie strain 10 -4 10 -4 10 -3 10 -2 10 -1 φ - φ J G ∝ ( φ − φ J ) 1 / 2 G ∝ γ − c 0 ( φ − φ J )

φ：volume fraction, G : shear modulus, r contact length : r F n F n Δ= 3/2 (Hertzian force) Δ=1 (linear force) Elastic interaction force Δ = 1 critical exponents a =Δ - 1/2, b = 1, c = ? γ 0 : strain amplitude Critical scaling of G MO and H. Hayakawa, PRE (2014) G ( γ 0 , φ ) = ( φ − φ J ) a G ( γ 0 ( φ − φ J ) − b ) → x →∞ G ( x ) lim x − c x → 0 G ( x ) = const. lim ∝ γ 0 → 0 G ( γ 0 , φ ) ∝ ( φ − φ J ) a lim 10 0 G ∝ ( φ − φ J ) ∆ − 1 / 2 G ( φ - φ J ) - a / G 0 M. Wyart, Ann. Phys. Fr. (2005) 10 -1 φ = 0.650 F n ∝ r ∆ φ = 0.652 φ = 0.655 φ = 0.660 φ = 0.670 10 -2 10 -3 10 0 10 3 γ 0 ( φ - φ J ) - b

Exponent c G : shear modulus, critical exponents a =Δ - 1/2, b = 1, c = ? Δ-dependence? from the strain dependence of G. We theoretically estimate the value of c Numerical estimation : C = 1/2 ? Finite strain : φ：volume fraction, γ 0 : strain amplitude MO and H. Hayakawa, PRE (2014) G ( γ 0 , φ ) = ( φ − φ J ) a G ( γ 0 ( φ − φ J ) − b ) → x →∞ G ( x ) lim x − c x → 0 G ( x ) = const. lim ∝ 10 0 G ∝ γ − c 0 ( φ − φ J ) a + bc G ∝ γ − c G ( φ - φ J ) - a / G 0 0 10 -1 φ = 0.650 φ = 0.652 φ = 0.655 φ = 0.660 φ = 0.670 10 -2 10 -3 10 0 10 3 γ 0 ( φ - φ J ) - b

Contents 1. Introduction : Jamming transition and shear modulus. 2. Critical behavior of shear modulus for frictionless granular materials under finite oscillatory shear. 3.Theory for critical exponents. 4. Effect of the friction between particles. MO and H. Hayakawa, PRE 90, 042202 (2014)

There exist stress drops due to avalanches appear. Non-linear and hysteric relation. stress σ strainγ Effect of avalanches stressσ 0.0013 0.0002 S ( t ) / ( k σ 0 -1 ) 0 0.0012 (c) -0.0002 0 0.1 0.2 0.0011 γ ( t ) 0.14 0.16 0.18

slip size δs Size distribution Size distribution of avalanches Avalanches Size of stress drop earthquakes Dahmen et al., (2010) Analysis of a mean field lattice model: ρ ( s ) ∝ s − 3 / 2 ρ ( s ) ∝ s − τ τ = 3 / 2

s：yield stress = stress drop each elements have different yield stress σ σ σ Elastic-plastic model σ MO and H. Hayakawa, PRE (2014) � ∞ ds ρ ( s ) ˜ ˜ S ( t ) = S ( s, t ) S ：stress of element 0 ρ ( s ) : size distribution G 0 s 1 G 0 s 2 γ γ ... ... G 0 s n ... ...

Δ= 3/2 Phenomenological result C . Coulais, et al., PRL. 113, 198001 (2014) critical scaling small γ 0 no Δ-dependence σ This is consistent with the previous talk. σ σ a = Δ - 1/2, b = 1, c = 1/2 Dahmen et al., (2010) large γ 0 MO and H. Hayakawa, PRE (2014) � 2 π / ω G ∝ γ − ( τ − 1) dtS ( t ) cos( ω t ) G ( γ 0 , φ ) = ω τ = 3 / 2 0 π γ 0 0 G ∝ γ − c � ∞ 0 ds ρ ( s ) ˜ S ( t ) = S ( s, t ) c = τ − 1 = 1 / 2 0 ρ ( s ) ∝ s − τ G ( γ 0 , φ ) = ( φ − φ S ) a G ( γ 0 ( φ − φ S ) − b ) G ∝ ( φ − φ J ) ∆ − 1 / 2 G ( φ - φ J ) - a / G 0 10 0 G ∝ γ − 1 / 2 ( φ − φ J ) ∆ φ = 0.6500 0 φ = 0.6550 10 -1 φ = 0.6600 φ = 0.6650 φ = 0.6700 10 -3 10 0 10 3 γ 0 ( φ - φ J ) - b

Contents 1. Introduction : Jamming transition and shear modulus. 2. Critical behavior of shear modulus for frictionless granular materials under finite oscillatory shear. 3.Theory for critical exponents. 4. Effect of the friction between particles. MO and H. Hayakawa, PRE 90, 042202 (2014)

Effect of friction F n : Normal force time t γ 0 Protocol : Discontinuous transition φ C : True critical density Two critical densities : φ S : (fictitious) critical density for scaling μ = 2.0 P : Pressure μ：friction coefficient F t : Tangential force Δ = 1 P ∝ ( φ − φ S ) Discontinuous change φ S φ C P ∝ ( φ − φ S )

Frictional grains (μ=0.1) 0.843 elastic network with friction μ：friction coefficient F t : Tangential force F n : Normal force μ = 0 : This scaling law may be superficial. 0.900 0.870 0.850 0.845 0.844 Δ = 1 G ( γ 0 , φ ) = ( φ − φ S ) a G ( γ 0 ( φ − φ S ) − b ) → a = 1 / 2 , b = 1 , c = 1 / 2 x →∞ G ( x ) lim x − c ∝ a = 0 . 13 , b = 1 , c = 0 . 87 1 G/ ( φ − φ S ) a G ∝ γ − c 0 0.1 φ 0.0001 0.0001 0.0001 0.0001 0.0001 0.01 0.0001 γ 0 / ( φ − φ S ) b 0.01 1 100

μ-dependence of exponents a μ = 0 frictionless particles. The exponents do not converge to that of the μ = 0.01 μ = 0 Exponent for strain dependence Exponent for linear elasticity c μ μ μ = 0.01 0.7 1 0.6 0.5 0.4 µ → +0 0.5 0.3 G ∝ γ − c 0.2 0 G ∝ ( φ − φ S ) a 0.1 0 0 0 0.5 1 0 0.2 0.4 0.6 0.8 1

Discussion (Linear elasticity) cannot be verified in our system Result in our system Origin : discontinuous transition Z J : coordination number at jamming Z iso : coordination number at isostatic state Z : coordination number, Δ = 1 P : Pressure 1.0 0.5 0.1 0.01 Somfai, et al., PRE (2007) : μ Our estimation : Z J (μ) elastic network with friction Z - Z iso G Z iso infinitesimal strain G ∝ ( φ − φ J ) a G = G 0 ( µ ) + A ( φ − φ J ) 1 / 2 µ → +0 G 0 ( µ ) = const. lim G ∝ Z − Z iso Z = Z J ( µ ) + β ( φ − φ J ) 1 / 2 G ∝ Z − Z iso µ → +0 { Z J ( µ ) − Z iso } > 0 lim G ∝ Z − Z iso 1 0.0001 0.1 0.0001 0.0001 0.0001 0.1 1

Discussion (Protocol dependence) G Discontinuous Previous protocol Coutinuous Somfai, et al., PRE (2007) : New protocol Z - Z iso μ = 2.0 φ P New protocol : time t γ 0 time t γ 0 Previous protocol : is plausible in new protocol. elastic network with friction New Protocol G = G 0 + A ( φ − φ J ) 1 / 2 G ∝ Z − Z iso 0.02 0.3 G ∝ Z − Z iso 0.2 0.01 0.1 0 0 0 0.5 1 1.5 0.81 0.84 0.87 G = G 0 + A ( φ − φ J ) 1 / 2