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

Avalanche contribution to nonlinear elasticity

• f 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

• scillatory shear.

3.Theory for critical exponents.

• 4. Effect of the friction between particles.

MO and H. Hayakawa, PRE 90, 042202 (2014)

Jamming transition

• C. O’Hern, et al., Phys. Rev. Lett. 88, 075507 (2002)

φ：volume fraction

φJ : Critical fraction

Low density : Liquid High density : Solid

• Granular materials can flow below a critical density φJ.
• Above φJ, the materials have rigidity and behave as

solids.

• This transition is known as the jamming transition.

Scaling of shear modulus

Shear modulus : G volume fraction：Φ

0.02 0.04 0.06 0.64 0.65 0.66 0.67

Shear stress σ Shear stress σ

Strain γ0

G = σ/γ0

Elastic interaction force

Δ=1 (linear force) Δ= 3/2 (Hertzian force)

Fn ∝ r∆ Fn Fn

contact length : r

r

G ∝ (φ − φJ)∆−1/2

Shear modulus : Theory for elastic network : Δ=1

• M. Wyart, Ann. Phys. Fr. (2005)

P ∝ (φ − φJ)∆

Different scaling relations

Okamura & Yoshino 2013, (Replica theory with small temperature)

G ∝ P

Mason et al., PRE 1996, (experiments of emulsions)

G ∝ (φ − φJ)∆

G : shear modulus, P : pressure

G ∝ γ−c

0 (φ − φJ)

Coulais, Seguin, and Dauchot, PRL 2014

Purpose

• C. O’Hern, et al., Phys. Rev. Lett. 88, 075507 (2002)

Okamura & Yoshino 2013, (Replica theory with small temperature)

G ∝ (φ − φJ)∆−1/2 P ∝ (φ − φJ)∆

G ∝ P

Mason et al., PRE 1996, (experiments of emulsions)

G ∝ (φ − φJ)∆

• We would like to clarify the relationship between

different scaling relations.

• For this purpose, we perform a simulation of granular

materials under oscillatory shear.

G : shear modulus, P : pressure

G ∝ γ−c

0 (φ − φJ)

Coulais, Seguin, and Dauchot, PRL 2014

Purpose :

Contents

• 1. Introduction : Jamming transition and shear

modulus.

• 2. Critical behavior of shear modulus for

frictionless granular materials under finite

• scillatory 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

• Oscillatory shear strain
• Frequency：ω
• Strain amplitude：γ0
• Shear stress : σ(t)

γ(t) = γ0 cos(ωt)

G(γ0, φ) = ω π 2π/ω dtS(t) cos(ωt) γ0

Quasi-static limit : ω→0

We investigate the dependence

• f G on γ0 and Φ.

σ Strain γ(t)

.

Strain γ(t)

Normal interaction force

Shear modulus (storage modulus) Fn = kr∆ + η ˙ r

10-4 10-3 10-2 10-1 10-4 10-3 10-2 10-1

G / G0 φ - φJ

γ0 = 10-5 γ0 = 10-3 γ0 = 10-2

Shear modulus under finite strain (Δ=1)

Infinitesimal strain : Finite strain : Origin : Avalanche (correlated bond breakage)

MO and H. Hayakawa, PRE (2014)

finie strain

strain : γ0

Δ=1

(a) (b) (c) (d)

Initial configuration

Broken bonds at γ=0.0001 Broken bonds at γ=0.0004 Broken bonds at γ=0.0008 Initial configuration

Avalanche-like bond breakage

G ∝ (φ − φJ)1/2 G ∝ γ−c

0 (φ − φJ)

G ∝ (φ − φJ)1/2

G ∝ (φ − φJ)

10-2 10-1 100 10-3 100 103

G( φ - φJ)-a / G0 γ0 ( φ - φJ )-b

φ = 0.650 φ = 0.652 φ = 0.655 φ = 0.660 φ = 0.670

Critical scaling of G

→

lim

x→∞ G(x)

∝ x−c

G : shear modulus, φ：volume fraction, γ0 : strain amplitude

lim

x→0 G(x) = const.

G(γ0, φ) = (φ − φJ)aG(γ0(φ − φJ)−b)

• M. Wyart, Ann. Phys. Fr. (2005)

a =Δ - 1/2, b = 1, c = ?

critical exponents

lim

γ0→0 G(γ0, φ) ∝ (φ − φJ)a

G ∝ (φ − φJ)∆−1/2 Δ = 1

Elastic interaction force

Δ=1 (linear force) Δ= 3/2 (Hertzian force)

Fn ∝ r∆ Fn Fn

contact length : r

r

MO and H. Hayakawa, PRE (2014)

Exponent c

G : shear modulus, φ：volume fraction, γ0 : strain amplitude

G(γ0, φ) = (φ − φJ)aG(γ0(φ − φJ)−b)

10-2 10-1 100 10-3 100 103

G( φ - φJ)-a / G0 γ0 ( φ - φJ )-b

φ = 0.650 φ = 0.652 φ = 0.655 φ = 0.660 φ = 0.670

G ∝ γ−c

C = 1/2 ? Numerical estimation :

We theoretically estimate the value of c from the strain dependence of G.

Δ-dependence? a =Δ - 1/2, b = 1, c = ?

critical exponents

→

lim

x→∞ G(x)

∝ x−c lim

x→0 G(x) = const.

G ∝ γ−c

0 (φ − φJ)a+bc

MO and H. Hayakawa, PRE (2014)

Finite strain :

Contents

• 1. Introduction : Jamming transition and shear

modulus.

• 2. Critical behavior of shear modulus for

frictionless granular materials under finite

• scillatory shear.

3.Theory for critical exponents.

• 4. Effect of the friction between particles.

MO and H. Hayakawa, PRE 90, 042202 (2014)

• 0.0002

0.0002 0.1 0.2

S(t) / ( kσ0-1) γ(t)

Effect of avalanches

0.0011 0.0012 0.0013 0.14 0.16 0.18

stressσ strainγ stress σ Non-linear and hysteric relation. There exist stress drops due to avalanches appear.

(c)

slip size δs Size distribution

Size distribution of avalanches

Avalanches

ρ(s) ∝ s−3/2

Size of stress drop

earthquakes

Dahmen et al., (2010)

Analysis of a mean field lattice model:

ρ(s) ∝ s−τ

τ = 3/2

... ...

G0 s1 G0 s2 G0 sn

γ γ ... ...

each elements have different yield stress

Elastic-plastic model

σ

MO and H. Hayakawa, PRE (2014)

S(t) = ∞ ds ρ(s) ˜ S(s, t) ρ(s) : size distribution ˜ S ：stress of element

σ σ σ

s：yield stress = stress drop

S(t) = ∞ ds ρ(s) ˜ S(s, t)

G(γ0, φ) = ω π 2π/ω dtS(t) cos(ωt) γ0

Phenomenological result

σ σ σ

no Δ-dependence

MO and H. Hayakawa, PRE (2014)

ρ(s) ∝ s−τ

G ∝ γ−c

τ = 3/2

G ∝ γ−(τ−1)

critical scaling

c = τ − 1 = 1/2

G(γ0, φ) = (φ − φS)aG(γ0(φ − φS)−b)

a = Δ - 1/2, b = 1, c = 1/2

This is consistent with the previous talk. C . Coulais, et al., PRL. 113, 198001 (2014)

small γ0 large γ0

G ∝ (φ − φJ)∆−1/2

G ∝ γ−1/2 (φ − φJ)∆

Dahmen et al., (2010)

10-1 100 10-3 100 103

G( φ - φJ)-a / G0 γ0 ( φ - φJ )-b

φ = 0.6500 φ = 0.6550 φ = 0.6600 φ = 0.6650 φ = 0.6700

Δ= 3/2

Contents

• 1. Introduction : Jamming transition and shear

modulus.

• 2. Critical behavior of shear modulus for

frictionless granular materials under finite

• scillatory shear.

3.Theory for critical exponents.

• 4. Effect of the friction between particles.

MO and H. Hayakawa, PRE 90, 042202 (2014)

Effect of friction

Fn : Normal force Ft : Tangential force

μ：friction coefficient

Discontinuous change

φS φC P : Pressure

μ = 2.0

P ∝ (φ − φS)

φS : (fictitious) critical density for scaling

Two critical densities :

φC : True critical density

Discontinuous transition

P ∝ (φ − φS)

Protocol : γ0 time t Δ = 1

Frictional grains (μ=0.1)

0.01 0.1 1 0.01 1 100 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001

→

lim

x→∞ G(x)

∝ x−c G ∝ γ−c G/(φ − φS)a

γ0/(φ − φS)b

0.843 0.844 0.845 0.850 0.870 0.900

φ

a = 1/2, b = 1, c = 1/2

a = 0.13, b = 1, c = 0.87

This scaling law may be superficial.

μ = 0 :

Fn : Normal force Ft : Tangential force μ：friction coefficient

elastic network with friction

G(γ0, φ) = (φ − φS)aG(γ0(φ − φS)−b)

Δ = 1

μ-dependence of exponents

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.2 0.4 0.6 0.8 1

a μ c μ

0.5 1 0.5 1

G ∝ (φ − φS)a

G ∝ γ−c

Exponent for linear elasticity Exponent for strain dependence

µ → +0

μ = 0

μ = 0.01

The exponents do not converge to that of the frictionless particles. μ = 0

μ = 0.01

Discussion (Linear elasticity)

Somfai, et al., PRE (2007) :

Our estimation :

elastic network with friction infinitesimal strain

G ∝ (φ − φJ)a ZJ(μ) Ziso Z = ZJ(µ) + β(φ − φJ)1/2 G

Z - Ziso

0.1 1 0.1 1 0.0001 0.0001 0.0001 0.0001

cannot be verified in our system

μ

0.01 0.1 0.5 1.0

P : Pressure Δ = 1

Z : coordination number, Ziso : coordination number at isostatic state ZJ : coordination number at jamming

Origin : discontinuous transition

G ∝ Z − Ziso

Result in our system

G = G0(µ) + A(φ − φJ)1/2 lim

µ→+0 G0(µ) = const.

lim

µ→+0{ZJ(µ) − Ziso} > 0

G ∝ Z − Ziso

G ∝ Z − Ziso

Discussion (Protocol dependence)

Somfai, et al., PRE (2007) :

elastic network with friction

G = G0 + A(φ − φJ)1/2

is plausible in new protocol.

Previous protocol :

γ0 time t γ0 time t

New protocol :

G P φ

G = G0 + A(φ − φJ)1/2

0.01 0.02 0.81 0.84 0.87

Previous protocol Discontinuous

0.1 0.2 0.3 0.5 1 1.5

New Protocol Coutinuous New protocol

Z - Ziso

μ = 2.0

G ∝ Z − Ziso G ∝ Z − Ziso

Discussion (Other critical scaling law)

G(γ0, φ) = (φ − φS)aG(γ0(φ − φS)−b)

Scaling A :

Scaling B : G(γ0, φ, µ) = G0(µ) + (φ − φJ)aG(γ0(φ − φJ)−b)

0.01 0.1 1 0.01 1 100 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001

G ∝ γ−c G/(φ − φS)a

γ0/(φ − φS)b

0.843 0.844 0.845 0.850 0.870 0.900

φ

γ0/(φ − φS)b

φ

The critical exponent a depend on μ. G0 depends on μ. a = 1/2

(G(γ0, φ) − Gc)/(φ − φS)a

μ = 0.1 μ = 0.1

Scaling A :

Scaling B :

We don’t have a clear understanding yet. bad result

Summary

• We perform simulations for frictionless grains under
• scillatory shear.
• We found a crossover from the known exponent for

the jamming to the non-trivial behavior.

• Non-trivial exponent can be understood by the mean

field analysis for avalanches.

• We discuss the effect of the friction coefficient.

MO and H. Hayakawa, PRE 90, 042202 (2014) [Frictionless case]