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. Kuhn 1 Ching S. Chang 2 1 University of Portland 2 University of Massachusetts McMAT Mechanics and Materials Conference 2005 Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction Quantifying high-gradient behavior Summary Outline Introduction 1 Quantifying high-gradient behavior 2 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 �✁�✁�✁� ✂✁✂✁✂✁✂ Small (but finite!) Continuum Continuum granular sub-region point Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction Quantifying high-gradient behavior Summary Classical vs. Generalized Continua Continuum representations. . . Classical continuum or Generalized continua 1) Micro-polar 2) Strain gradient dependent 3) Non-local Uniform deformation High-gradient deformation ∂ ǫ /∂ x ≪ 1 ∂ ǫ /∂ x ≈ 1 D D ǫ ǫ Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction DEM “bending” experiments Quantifying high-gradient behavior Questions about granular behavior Summary Experiment results Outline Introduction 1 Quantifying high-gradient behavior 2 DEM “bending” experiments Questions about granular behavior Experiment results Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction DEM “bending” experiments Quantifying high-gradient behavior Questions about granular behavior Summary Experiment results DEM “Bending” Experiments — 2D x 2 x 2 x 1 x 1 “Uniform” deformation “Bending” deformation Horiz. strain Vert. gradient Strain: � d ǫ 11 → | | ← ǫ 11 dx 2 Rotation Horiz. gradient Rotation: ⇔ d θ � | | � dx 1 Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction DEM “bending” experiments Quantifying high-gradient behavior Questions about granular behavior Summary Experiment results Generalized Continuum Stresses Continuum representation of stress . . . δ W Internal σ ji δ u i , j T ji δ θ i , j σ jki δ u i , jk = + + σ 22 σ 11 σ 12 Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction DEM “bending” experiments Quantifying high-gradient behavior Questions about granular behavior Summary Experiment results Generalized Continuum Stresses Continuum representation of stress . . . δ W Internal σ ji δ u i , j T ji δ θ i , j σ jki δ u i , jk = + + σ 22 σ 11 σ 12 Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction DEM “bending” experiments Quantifying high-gradient behavior Questions about granular behavior Summary Experiment results Generalized Continuum Stresses Continuum representation of stress . . . δ W Internal σ ji δ u i , j T ji δ θ i , j σ jki δ u i , jk = + + σ 22 σ 11 σ 12 T 13 σ 121 Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction DEM “bending” experiments Quantifying high-gradient behavior Questions about granular behavior Summary Experiment results Discrete Region DEM Simulations — 256 Particles — Circles or Ovals Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction DEM “bending” experiments Quantifying high-gradient behavior Questions about granular behavior Summary Experiment results Bending Resistance in a Discrete Region Boundary Moments: Boundary Forces: x 2 x 2 x 1 x 1 ⇓ ⇓ T 13 σ 121 T 13 (+) σ 121 Bending Moment = Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction DEM “bending” experiments Quantifying high-gradient behavior Questions about granular behavior Summary Experiment results Outline Introduction 1 Quantifying high-gradient behavior 2 DEM “bending” experiments Questions about granular behavior Experiment results Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction DEM “bending” experiments Quantifying high-gradient behavior Questions about granular behavior Summary Experiment results Granular Behavior ⇔ Questions Questions: Are the boundary moments significant? 1 ¿ | T 13 | > 0 ? Are boundary forces consistent with classical beam 2 theory? d 2 u 1 ¿ σ 121 → E I ? dx 1 dx 2 Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction DEM “bending” experiments Quantifying high-gradient behavior Questions about granular behavior Summary Experiment results Granular Behavior ⇔ Questions Questions: Are the boundary moments significant? 1 ¿ | T 13 | > 0 ? Are boundary forces consistent with classical beam 2 theory? d 2 u 1 ¿ σ 121 → E I ? dx 1 dx 2 Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction DEM “bending” experiments Quantifying high-gradient behavior Questions about granular behavior Summary Experiment results Outline Introduction 1 Quantifying high-gradient behavior 2 DEM “bending” experiments Questions about granular behavior Experiment results Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction DEM “bending” experiments Quantifying high-gradient behavior Questions about granular behavior Summary Experiment results Results Summary Experiment results — incremental response: Question Small strain Large strain 1) | T 13 | > 0 ? No No 2) σ 121 → EI d 2 u 1 dx 1 dx 2 ? Yes No 5 Deviator stress, ( σ 11 − σ 22 ) /p o 4 Ovals 3 Large strain 2 Circles Small strain 1 0 0 0.01 0.02 0.03 0.04 Compressive strain, − ε 11 Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction DEM “bending” experiments Quantifying high-gradient behavior Questions about granular behavior Summary Experiment results Results Summary Boundary Moments: Boundary Forces: x 2 x 2 x 1 x 1 ⇓ ⇓ T 13 σ 121 T 13 (+) σ 121 Bending Moment = Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction DEM “bending” experiments Quantifying high-gradient behavior Questions about granular behavior Summary Experiment results Results Summary Experiment results — incremental response: Question Small strain Large strain 1) | T 13 | > 0 ? No No 2) σ 121 → EI d 2 u 1 dx 1 dx 2 ? Yes No 5 Deviator stress, ( σ 11 − σ 22 ) /p o 4 Ovals 3 Large strain 2 Circles Small strain 1 0 0 0.01 0.02 0.03 0.04 Compressive strain, − ε 11 Kuhn & Chang, McMAT2005 Continuum-Discrete Granular Models with Bending

Introduction DEM “bending” experiments Quantifying high-gradient behavior Questions about granular behavior Summary Experiment results Results Details DEM Simulation Results Dimensionless Bending Stiffnesses 256 particles — 50 assemblies Large Strain Circles Ovals Boundary | T 13 | -0.01 -0.01 moments Boundary 0.60 1.16 σ 121 forces 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 of 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