G := γ. Introduction. σxy: shear stress. γ: shear strain. γ = yvx. Characteristic properties of granular matters They have a liquid and solid region.

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2 Introduction Characteristic properties of granular matters They have a liquid and solid region. " In the liquid region we use the viscosity η. σxy η := γ σxy: shear stress shear rate γ: y Shear Velocity In the solid region we use the shear modulus G. G := σxy γ γ: shear strain Ly!2 γ = yvx x δx Shear γ = δx/ly

3 G = G + ig. σ G = lim γ γ G G = ω lim γ 0 (σ 0) σ γ, η = G ω G (, μ) G (ϕ ϕ J ) 1/2 ( 0) ΔZ = Z ΔZ 3 = Z 3 C. S. O Hern, et al., Phys. Rev. Lett. 88, 7 (2002). E. Somfai, et al., Phys. Rev. E 75, (2007). M. Otsuki & H. Hayakawa, Phys. Rev. E 95, (2017). G ΔZ ( 0) G/G (lin) = ( μ b 1(ϕ ϕ c ) b 2 )!3

4 f ij = ( f ij,n + f ij,n = k n ξ ij f ij,t = { k t ζ ij n ij η n μ c f ij,n f ij,t ) Θ(r ij x ij ) t ij η t t ij v ij,n r i v ij,t ( f ij,t < μ c f ij,n ) (otherwise) ω i t ij ξ ij ω j n ij r j S. Luding, Granular Matter 10, 235 (2008). η n / mk n = 1 (restitution coefficient 0.043) k t /k n = 0.25 η t /η n = 0.5 μ c = 0 1!4

5 Oscillatory shear system We explain how to apply oscillatory shear. Lx P A sin(ωt) In 2 dimensional N = 4000 (tetradisperse) P/kn γ0 1 A γ0 := (A : amplitude) Lx /2 A sin(ωt) y x P We press the walls with a pressure P and they move according to ± A sin(ωt)!5

6 = , P/k n = = , P/k n = !6

7 G = lim γ σ γ. ( σ = σ σ ) μ = 1 μ = 1 G = ω lim γ 0( σ 0) σ γ. ( σ = σ σ ) G /G res P= G res = lim 0 /P β G > 0, β 1 = 1. M. Otsuki & H. Hayakawa, Phys. Rev. E 95, (2017). G P (ϕ ϕ J )? P M. Otsuki & H. Hayakawa, Phys. Rev. E 80, (2009).!7 G /P α P= /P β α 1 = 0.2, β 1 = G

8 G G = lim γ σ γ. ( σ = σ σ ) G = ω lim γ 0( σ 0) P = µ=0 µ=0.1 µ=0.25 µ=0.5 µ=0.75 µ= G G G μ=0 μ=0 μ G μ α μ σ γ. ( σ = σ σ ) P = /μ β μ γ 1 0 µ=0 µ= µ=0.25 µ=0.5 µ= µ= α μ = 0.3, β μ = μ G G μ 0.1 μ 0.1!8

9 G γ 1 0 σ σ / lim P= σ(t) σ(t) P = γ(t)/ P = γ(t)/ max σ(t) const ( >,c ). G = lim γ σ γ 1!9

10 G < 0 P = σ(t) P = γ(t)/ω G eff /Pα P= σ(t) γ /P β 0 1 γ(t)/ω G < 0 G G eff = ω α 1 = 0.2, β 1 = lim γ 0( σ 0) eff σ γ. ( σ = σ σ )!10

11 G G G (Z) G /k n G eff /k n Z P= P= P= !11

12 G ϕ<ϕj > 0.! UJ DI P= ϕ J

13 G G /G res /P β G eff /Pα P= P=2 10-5!13 /P β

14 !14

15 G /P α G = lim γ σ γ. /P β 0 α 0 = 0.5, β 0 = 1. G ( σ = σ σ ) μ = 0 μ = 0 P= / G /P α 0 M. Otsuki, H. Hayakawa, Phys. Rev. E 90, (2014). M. Otsuki, H. Hayakawa, Phys. Rev. E 95, (2017). P! P= γ /P β 0 0 α 0 = 0.35, β 0 = 1.2. P (ϕ ϕ J )? M. Otsuki, H. Hayakawa, Phys. Rev. E 80, (2009). 1

16 SJ J SJ J UJ P= ϕ UJ P= ϕ G ϕ<ϕj > 0.!16

17 G res G res := lim 0 G G res a log P + b, G res a = 0.04 ± 0.01, b = 0.62 ± P G res!17