Nonlinear visco-elastic properties of granular materials near - PowerPoint PPT Presentation

Nonlinear visco-elastic properties of granular materials near jamming transition Michio Otsuki (Shimane Univ.) Hisao Hayakawa (Kyoto Univ.)

Granular materials (Assemblies of particles with dissipation ) Sand Saturn ring mustard seed Ginkaku-ji temple

Shear stress Shear stress Shear stress Shear stress Shear stress Shear stress Sheared granular materials packing fraction : Φ Homogeneous flow No flow . Inhomogeneous flow (Shear rate γ ) Gas Dense liquid Amorphous solid ( Φ = 0.12) ( Φ = 0.8) ( Φ = 0.85)

Shear stress Shear stress Shear stress Shear stress Jamming transition Below Φ J Above Φ J Transition point Φ J Homogeneous flow No flow . (Shear rate γ ) Onset of rigidity Dense liquid Amorphous solid ( Φ = 0.8) ( Φ = 0.85)

Rheology under steady shear frictionless case Shear stress σ T. Hatano, M. Otsuki, S. Sasa, J. Phys. Soc. Jpn, 76, 023001 (2007) non-linear rheological property . For Φ < Φ J , σ ∝ γ 2 (liquid) For Φ > Φ J , σ ≃ const (solid) . For Φ ≃ Φ J , σ ∝ γ y γ . Shear rate γ

α, y Φ : Critical exponents Rheology under steady shear frictionless case Shear stress σ σ / | Φ - Φ J | β T. Hatano, J. Phys. Soc. Jpn, 77, 123002, (2008) T. Hatano, M. Otsuki, S. Sasa, J. Phys. Soc. Jpn, 76, 023001 (2007) scaling plot . γ | Φ - Φ J | - α . Shear rate γ non-linear rheological property . Critical scaling law . . For Φ < Φ J , σ ∝ γ 2 (liquid) σ ( γ , Φ ) = | Φ - Φ J | y Φ S ± ( γ | Φ - Φ J | - α ) For Φ > Φ J , σ ≃ const (solid) . For Φ ≃ Φ J , σ ∝ γ y γ

Fn=kδ Shear stress Kinetic energy Pressure δ Theory for exponents M. Otsuki and H. Hayakawa, PRE, 80, 011308, (2009) Three Critical scaling laws Four Assumptions . . T( γ , Φ ) = | Φ - Φ J | x Φ τ ± ( γ | Φ - Φ J | - α ) • S / P is constant. Coulomb’s friction : Hatano (2007) . . • P in high density region : σ ( γ , Φ ) = | Φ - Φ J | y Φ S ± ( γ | Φ - Φ J | - α ) P ~ Φ O’Hern, et al., (2003) . . • Characteristic time : P -1/2 P( γ , Φ ) = | Φ - Φ J | y Φ ’ p ± ( γ | Φ - Φ J | - α ) Wyart, et al. (2005) • Low density region : collision frequency ∝ T 1/2 Kinetic theory Theoretical prediction for critical exponents x Φ = 3, y Φ = 1, y Φ ’ = 1, α = 5/2 (for disks) Linear repulsive force

Rheology under steady shear frictionless case M. Otsuki and H. Hayakawa, PRE, 80, 011308, (2009) . . σ / | ΔΦ | σ ( γ , Φ ) = | Φ - Φ J | y Φ S ± ( γ | Φ - Φ J | - α ) T. Hatano, J. Phys. Soc. Jpn, 77, 123002, (2008) Theoretical prediction : α = 1, y Φ = 2/5 (for disk) linear repulsive force . γ | ΔΦ | - α . . σ ( γ , Φ ) = ΔΦ S ± ( γ / ΔΦ 5/2 ) ΔΦ = Φ - Φ J

Problem • The system under steady shear is not suitable to study the rigidity near the jamming transition. • In experiments, the steady shear is hard to realize. We numerically investigate the rheological properties under oscillatory shear (OS)

Previous study on the system under OS B. Tighe, PRL 107,158303 (2011) Complex shear modulus ： G* G’ : real part close to the critical point close to the critical point G’’ : imaginary part frequency ： ω • System ： no mass, fixed contact networks, tangential friction • Complex shear modulus exhibits critical scalings.

Purpose of this work • In the previous work, the attention is restricted to the small shear limit and the change of the contact network is not considered. • However, the change of the network dominates the rheological property near the jamming transition point. ★ We investigate the rheological properties under OS in a wide range of shear amplitude.

Model of granular materials (frictionless) Shear strain γ(t) δ:contact length Contact force . Shear strain γ(t) Fn Fn . F n = k δ - η δ Elastic part Dissipative part

Shear strain γ(t) Oscillatory shear Shear strain γ(t) • Shear strain : γ (t) = γ 0 cos ( ω t) • Amplitude : γ 0 , Frequency : ω • Shear stress : σ (t) • Volume fraction : Φ • Shear modulus : G* = G’ + i G” • G’ ∝ ∫ dt σ (t) cos ( ω t) / γ 0 . Real part ： Storage modulus • G’’ ∝ - ∫ dt σ (t) sin ( ω t) / γ 0 Imaginary part ： Loss modulus We numerically investigate G * ( γ 0 , ω , Φ ).

Shear strain γ(t) Oscillatory shear Shear strain γ(t) Force network • Shear strain : γ (t) = γ 0 cos ( ω t) • Amplitude : γ 0 , Frequency : ω • Shear stress : σ (t) • Volume fraction : Φ • Shear modulus : G* = G’ + i G” • G’ ∝ ∫ dt σ (t) cos ( ω t) / γ 0 . Real part ： Storage modulus • G’’ ∝ - ∫ dt σ (t) sin ( ω t) / γ 0 Imaginary part ： Loss modulus We numerically investigate G * ( γ 0 , ω , Φ ).

. G * for the Voigt model Model of typical visco-elastic materials σ = σ E + σ K, σ E = E γ , σ K = η γ G’ = E, G” = η ω Complex shear modulus ： G* G’ ： storage modulus red ： ω = 10 -4 Shear stress ： σ (t) green ： ω = 10 -3 ω→ 0 blue ： ω = 10 -2 slope ： G’ G” ： loss modulus width ： G” G” ∝ ω Shear strain ： γ (t) frequency ： ω

Critical scalings of G * • We find three critical behaviors. 1. G * ( γ 0 , ω , Φ ) for γ 0 ≧ 1 . (Large amplitude region) 2. G * ( γ 0 , ω , Φ ) for γ 0 < 1 . (Small amplitude region) 3. G * ( γ 0 , ω , Φ ) for ω→ 0 . (Quasi static limit)

⇒Energy dissipation in the quasi-static limit. relation remains in ω→0. The width in the plot of the σ-γ ω-dependence • c.f. the Voigt model : G” ∝ ω G*( γ 0 , ω , Φ ) for γ 0 ≧ 1 Complex shear modulus ： G* Φ = 0.67, γ 0 =1 Φ = 0.67, γ 0 =1 Shear stress ： σ (t) G” ： loss modulus ω→ 0 G’ ： storage modulus frequency ： ω Shear strain ： γ (t) • G” remains for ω→0.

relation remains in ω→0. ⇒Energy dissipation in the quasi-static limit. The width in the plot of the σ-γ • c.f. the Voigt model : G” ∝ ω ω-dependence G*( γ 0 , ω , Φ ) for γ 0 ≧ 1 Complex shear modulus ： G* Schematic model for ω→ 0 Φ = 0.67, γ 0 =1 elastic plastic � G” ： loss modulus � G’ ： storage modulus plastic elastic { σ = E γ , ( γ < γ c ) σ Y , ( γ c < γ ) frequency ： ω • G” remains for ω→0.

Φ-dependence dependence on ω with a non-trivial exponent. As Φ approaches Φ J , G * shows a power-law G*( γ 0 , ω , Φ ) for γ 0 ≧ 1 γ 0 =1 γ 0 =1 Storage modulus ： G’ Loss modulus ： G” close to Φ J close to Φ J red ： Φ = 0.67 red ： Φ = 0.67 green ： Φ = 0.66 green ： Φ = 0.66 blue ： Φ = 0.65 blue ： Φ = 0.65 magenta ： Φ = 0.648 magenta ： Φ = 0.648 frequency ： ω frequency ： ω

We assume the rheological property under OS with Critical scaling . G*( γ 0 , ω , Φ ) for γ 0 ≧ 1 G*( ω , Φ ) = ΔΦ g( ω / ΔΦ 5/2 ) ΔΦ = Φ - Φ J γ 0 =1 γ 0 =1 G’’ / Δφ G’ / Δφ red ： Φ = 0.67 red ： Φ = 0.67 green ： Φ = 0.66 green ： Φ = 0.66 blue ： Φ = 0.65 blue ： Φ = 0.65 magenta ： Φ = 0.648 magenta ： Φ = 0.648 ω / Δφ 5 / 2 ω / Δφ 5 / 2 a large γ 0 is dominated by that under steady shear. σ ( γ , Φ ) = ΔΦ F ± ( γ / ΔΦ 5/2 )

Critical scalings of G * • We find three critical behaviors. 1. G * ( γ 0 , ω , Φ ) for γ 0 > 1 . (Large amplitude region) 2. G * ( γ 0 , ω , Φ ) for γ 0 < 1 . (Small amplitude region) 3. G * ( γ 0 , ω , Φ ) for ω→ 0 . (Quasi static limit)

Loss modulus : G” ∝ ω Storage modulus : G’ ∝ (Φ - Φ J ) 1/2 (small ω-dependence) The behavior of G* is consistent with the Voigt model. G*( γ 0 , ω , Φ ) for γ 0 ≪ 1 Complex shear modulus ： G* 0.06 γ 0 =10 -3 γ 0 =10 -3 , ω =10 -4 G’ : storage modulus 0.04 G’ ~ ΔΦ 1/2 G’ close to Φ J red ： Φ = 0.67 0.02 green ： Φ = 0.66 blue ： Φ = 0.65 G” ： loss modulus 0 0.64 0.65 0.66 0.67 Volume fraction ： Φ frequency ： ω C. O’Hern, et al., Phys. Rev. Lett. 88, 075507 (2002)

Critical scalings of G * • We find three critical behaviors. 1. G * ( γ 0 , ω , Φ ) for γ 0 > 1 . (Large amplitude region) 2. G * ( γ 0 , ω , Φ ) for γ 0 < 1 . (Small amplitude region) 3. G * ( γ 0 , ω , Φ ) for ω→ 0 . (Quasi static limit)

Quasi-static limit G*( γ 0 , ω , Φ ) for ω→ 0 G 0 ’ ( γ 0 , Φ ) ≡ lim G’( γ 0 , ω , Φ ) ω → 0 γ c ( Φ ) : yield strain • G 0 ’ = const. for γ 0 < γ c ( Φ ). G 0 ’ close to Φ J • G 0 ’ decreases as γ 0 increases for γ 0 > γ c ( Φ ). red ： Φ = 0.650, green ： Φ = 0.652 blue ： Φ = 0.655, magenta ： Φ = 0.660 cyan : Φ = 0.670 • G 0 ’ decreases as Φ approaches Φ J . Shear amplitude : γ 0

Theoretical prediction Critical scaling Three Assumptions G*( γ 0 , ω , Φ ) for ω→ 0 G 0 ’( γ 0 , Φ ) = ΔΦ 1/2 h( γ 0 / ΔΦ ) lim h(x) ∝ x -1/2 x →∞ G’ ~ ΔΦ 1/2 , for γ 0 → 0 C. O’Hern, et al., Phys. Rev. Lett. 88, 075507 (2002) G ~ γ 0-1/2 The yield strain γ c is G 0 ’ / ΔΦ 1/2 proportional to the contact length. red ： Φ = 0.650, green ： Φ = 0.652 γ c ( Φ ) ~ ΔΦ blue ： Φ = 0.655, magenta ： Φ = 0.660 cyan : Φ = 0.670 B. Tighe, et al., Phys. Rev. Lett. 105, 088303 (2010) G 0 ’ is independent of Φ γ 0 / ΔΦ for ΔΦ→ 0.