Continuum Models of Discrete Particle Systems with Particle Shape - - PowerPoint PPT Presentation

Introduction Quantifying high-gradient behavior Summary

Continuum Models of Discrete Particle Systems with Particle Shape Considered

Matthew R. Kuhn1 Ching S. Chang2

1University of Portland 2University of Massachusetts

McMAT Mechanics and Materials Conference 2005

Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction Quantifying high-gradient behavior Summary

Outline

1

Introduction

2

Quantifying high-gradient behavior DEM “bending” experiments Questions about granular behavior Experiment results

Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction Quantifying high-gradient behavior Summary

Continuum vs. Discrete Frameworks

Continuum Small (but finite!) granular sub-region Continuum point

Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction Quantifying high-gradient behavior Summary

Classical vs. Generalized Continua

Continuum representations. . . Classical continuum

• r

Generalized continua 1) Micro-polar 2) Strain gradient dependent 3) Non-local Uniform deformation High-gradient deformation ∂ǫ/∂x ǫ ≪ 1 D ∂ǫ/∂x ǫ ≈ 1 D

Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction Quantifying high-gradient behavior Summary DEM “bending” experiments Questions about granular behavior Experiment results

Outline

1

Introduction

2

Quantifying high-gradient behavior DEM “bending” experiments Questions about granular behavior Experiment results

Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction Quantifying high-gradient behavior Summary DEM “bending” experiments Questions about granular behavior Experiment results

DEM “Bending” Experiments — 2D

x1 x2

“Uniform” deformation

x1 x2

“Bending” deformation Strain:

• Horiz. strain
• Vert. gradient

→ | | ← ǫ11 dǫ11

dx2

Rotation: Rotation

• Horiz. gradient

| | ⇔ dθ

dx1

Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction Quantifying high-gradient behavior Summary DEM “bending” experiments Questions about granular behavior Experiment results

Generalized Continuum Stresses

Continuum representation of stress . . . δWInternal = σji δui, j + Tji δθi, j + σjki δui, jk

σ11 σ12 σ22

Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction Quantifying high-gradient behavior Summary DEM “bending” experiments Questions about granular behavior Experiment results

Generalized Continuum Stresses

Continuum representation of stress . . . δWInternal = σji δui, j + Tji δθi, j + σjki δui, jk

σ11 σ12 σ22

Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction Quantifying high-gradient behavior Summary DEM “bending” experiments Questions about granular behavior Experiment results

Generalized Continuum Stresses

Continuum representation of stress . . . δWInternal = σji δui, j + Tji δθi, j + σjki δui, jk

σ11 σ12 σ22 σ121 T13

Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction Quantifying high-gradient behavior Summary DEM “bending” experiments Questions about granular behavior Experiment results

Discrete Region

DEM Simulations — 256 Particles — Circles or Ovals

Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction Quantifying high-gradient behavior Summary DEM “bending” experiments Questions about granular behavior Experiment results

Bending Resistance in a Discrete Region

Boundary Moments:

x2 x1

⇓ T13 Boundary Forces:

x1 x2

⇓

σ121

Bending Moment =

T13 (+) σ121

Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction Quantifying high-gradient behavior Summary DEM “bending” experiments Questions about granular behavior Experiment results

Outline

1

Introduction

2

Quantifying high-gradient behavior DEM “bending” experiments Questions about granular behavior Experiment results

Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction Quantifying high-gradient behavior Summary DEM “bending” experiments Questions about granular behavior Experiment results

Granular Behavior ⇔ Questions

Questions:

1

Are the boundary moments significant? ¿ |T13| > 0 ?

2

Are boundary forces consistent with classical beam theory? ¿ σ121 → E I

d2u1 dx1dx2

?

Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction Quantifying high-gradient behavior Summary DEM “bending” experiments Questions about granular behavior Experiment results

Granular Behavior ⇔ Questions

Questions:

1

Are the boundary moments significant? ¿ |T13| > 0 ?

2

Are boundary forces consistent with classical beam theory? ¿ σ121 → E I

d2u1 dx1dx2

?

Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction Quantifying high-gradient behavior Summary DEM “bending” experiments Questions about granular behavior Experiment results

Outline

1

Introduction

2

Quantifying high-gradient behavior DEM “bending” experiments Questions about granular behavior Experiment results

Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction Quantifying high-gradient behavior Summary DEM “bending” experiments Questions about granular behavior Experiment results

Results Summary

Experiment results — incremental response: Question Small strain Large strain 1) |T13| > 0 ? No No 2) σ121 → EI d2u1

dx1dx2 ?

Yes No

Circles Ovals Large strain Small strain Compressive strain, −ε11 Deviator stress, (σ11 − σ22)/po 0.04 0.03 0.02 0.01 5 4 3 2 1

Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction Quantifying high-gradient behavior Summary DEM “bending” experiments Questions about granular behavior Experiment results

Results Summary

Boundary Moments:

x2 x1

⇓ T13 Boundary Forces:

x1 x2

⇓

σ121

Bending Moment =

T13 (+) σ121

Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction Quantifying high-gradient behavior Summary DEM “bending” experiments Questions about granular behavior Experiment results

Results Summary

Experiment results — incremental response: Question Small strain Large strain 1) |T13| > 0 ? No No 2) σ121 → EI d2u1

dx1dx2 ?

Yes No

Circles Ovals Large strain Small strain Compressive strain, −ε11 Deviator stress, (σ11 − σ22)/po 0.04 0.03 0.02 0.01 5 4 3 2 1

Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction Quantifying high-gradient behavior Summary DEM “bending” experiments Questions about granular behavior Experiment results

Results Details

DEM Simulation Results Dimensionless Bending Stiffnesses 256 particles — 50 assemblies Large Strain Circles Ovals

|T13|

Boundary moments

• 0.01
• 0.01

σ121

Boundary forces 0.60 1.16 EI u ′′ “Beam theory” 0.25 0.65

Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction Quantifying high-gradient behavior Summary

Summary

DEM simulations can probe the response of small regions to high strain gradients. Cosserat-type torque stress does not contribute to incremental bending stiffness. A generalized stiffness is associated with the 1st gradient

• f strain. Stiffness is larger for oval particles.

Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Appendix Further Reading

Further Reading I

• M. R. Kuhn 2005.

Are granular materials simple? An experimental study of strain gradient effects and localization. Mechanics of Materials, 37(5):607–627.

• C. S. Chang and M. R. Kuhn 2005.

On virtual work and stress in granular media.

• Int. J. Solids and Structures, 42(13):3773–3793.

Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending