Local Anisotropy In Globally Isotropic Granular Packings. Kamran Karimi Craig E Maloney

Transcription

1 Local Anisotropy In Globally Isotropic Granular Packings Kamran Karimi Craig E Maloney

2 Granular Materials 2 A Granular Material Is A Conglomeration Of Discrete Solid, Natural Macroscopic Particles Characterized By A Loss Of Energy Phenomena: Whenever The Particles Interact (Wikipedia) Failure Flows Industry: Tablet s Chemicals

3 Forces In Granular Materials 3 Walk On The Beach Without Sinking: Forces in Granular Materials Veje, Howell, and Behringer, Phys.Rev.E, 59, 739 (1999) Transmission Of Loads : Along Force Chains Forces Are Highly Heterogeneous

4 Forces In Granular Materials 4 2d Shear Experiment liu et al. (1995) Mueth et al. (1998). (2003) Majmudar et al. (2005) Zhou et al. (2006) Biaxial Tester And Photoelastic Particles Howell et al (1999) Macroscopic Particles Forces Are Measurable Experiments Done To Study Forces

5 Main Questions 5 Disorder Anisotropy And Heterogenity Question? Apply Globally Isotropic Compression How To Quantify These Properties In Geometrically Complex Forces? Specifically: Local Anisotropy Despite Isotropic Loading Question? Impact Of Forces In Elasticity And Mechanical Response How Anisotropic Are Elastic Moduli?

6 Previous Work: Study Forces 6 Question? Behringer et al (1999) How To Quantify These Geometrically Complex Forces? Henkes et al (2009) Ostojicet al (2006)

7 Previous Work: Mechanical Response 7 Homogeneous Displacements Inhomogeneous Response, Non- Affine Displacements Elastic Moduli 2D Jennard-Jones Scale Dependent Corrections Convergence To Homogeneous And Isotropic Elasticity eff, eff a, a Tanguy et al. (2002,2004,2005) Lemaître and Maloney (2004,2006) Williams et al. (1997) Debrégeas et al. (2001) Roux et al. (2002) Kolb et coll. (2003) Weeks et al. (2006) Tsamados et al (2009)

8 Model And Protocol 8 Discrete Particle Model (LAMMPS) Generate Configurations Start With Random Positions Quench At Fixed Φ Equations Of Motion m d 2 r i dt 2 f i Overlap 0.5 d i d j r ij Normal Viscous Damping: t 0.1 md 2 r n i r j ij r i r j Bi-Disperse Mixture: O Hern et al (2003) N A /N B = 1, D A /D B = 1.4 V (r ij ) Energy Periodic Boundaries 0 0

9 Stress 9 Stress Tensor: τ Mohr Circle xx yx ( xx, xy ) xy yy Microscopic Expression 1 V i j f ij r ij For Each Bond j σ P ( max min )/2 σ min ( max min )/2 σ max ( yy, xy ) S P Globally : S 1 isotropic i r ij Locally : S 1 anisotropic

10 Coarse-grained Stress 10 Coarse-grained Quantity S P s( R ) 1 N N i 1 i s ( R ) R i N: Number Of Squares ε S Is NOT Linear! On Each Square Region Of Length R

11 Coarse-grained Stress 11 Locally ε S Is Large But Globally ε S <<1! Local Anisotropy Despite Global Isotropy ε S Is NOT Self-Averaging!

12 Pressure 12 Compact φ=0.89 Chain-Like Clusters φ=0.86 Extended

13 No Spatial Correlations: ε s ~ 1/R 13 More Anisotropic Closer to Jamming Power-Law Regime: ε s ~ R The Characteristic Length Grows With φ-φc 0 Scaled by R -1 Random

14 Average Stress 14 Rescale R by ξ and ε s by A Diverging Length Scale!

15 Elastic Moduli 15 2D Case (Voigt Notation): xx yy C xxxx C xxyy C xxxy C xxyx C yyxx C yyyy C yyxy C yyyx xx yy xy yx C xyxx C xyyy C xyxy C xyyx C yxxx C yxyy C yxxy C yxyx Ιndependent Moduli From Symetries C =C 10 Moduli σ =σ 6 Moduli An Isotropic Material: Lame-Navier Form Only 2 Constants C L xy yx Pure Rotation: C =C 7 Moduli

16 3 Nontrivial Eigenvalues/vectors For C L : Isotropic Expansion Axial Pure Shear Elastic Moduli 1 2( ), 1 2 2, Form orthogonal basis [ 1, 2, 3, 4 ] K C 12 C 13 0 C C 21 1 C 23 0 C 31 C Or Pure Shear 3 2, K C 12 C 13 C 21 1 C 23 C 31 C 32 2

17 Anisotropy in Elastic Moduli 17 C ( 1, 2, 3 ) K C 12 C 13 C 21 1 C Κ: Bulk Modulus 23 μ C 31 C 1, μ 2 : Shear Moduli 32 2 ε 1 C (,C ) 2θ ( 1,C 23 ) μ ( max min )/2 ( max min )/2 θ ε θ ε 3 ε 2 μ min m. C. μ max ( 2, C 23 ) Globally : m 1 Isotropic Locally : m 1 Anisotropic cos sin T 1 C 23 cos C 32 2 sin

18 Microscopic Elastic Moduli 18 Born-Huang Approximation: C Born 1 c ij V (r ijc t )r n n n n ij ij ij ij ij ij ij ij 2 V V and t r 2 ij ij r ij Valid For The Ordered Case For The Disrdered Case C C Born 1 V. H 1. Assemble Local Affine Force and Hessian to get full matrices i, (r ij c ij t ij )n ij j n ij n ij H i (c ij t ij j r ij )n ij n ij

19 Elastic Moduli 19 PureShear. C. PureShear ( max min )/2 ( max min )/2 m θ

20 Coarse-graining 20 Coarse-grained Quantities 1 N i m N i 1 m C i R N: Number Of Squares C αβγδ and ε m Are NOT Linear! On Each Square Region Of Length R

21 Average Moduli 21 Small R: μ ~ μ Born Large R: μ μ g No Simple Scaling Behavior Large R: δμ 0 Lame-like Εlasticity Power-Law Regime: ε m ~ R ΝΟ Characteristic Length!

22 Conclusion 22 In Isotropically Prepared, Frictionless Granular Packings Lengthscale Is exhibited By The Anisotropy In The Stress Anisotropy In The Shear Modulus Shows No Characteristic Lengthscale Thank You!