Introduction 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
Outline Introduction 1 Introduction 2
Introduction Continuum vs. Discrete Frameworks Continuum Small (but finite!) granular sub-region Continuum point
Introduction Classical vs. Generalized Continua Continuum representations... Classical continuum or Generalized continua 1) Micro-polar 2) Strain gradient dependent 3) Non-local Uniform deformation ɛ/ x ɛ 1 D High-gradient deformation ɛ/ x ɛ 1 D
Outline Introduction 1 Introduction 2
Introduction DEM Bending Experiments 2D x 2 x 2 x 1 x 1 Uniform deformation Strain: Rotation: Bending deformation Horiz. strain Vert. gradient ɛ 11 dɛ 11 dx 2 Rotation Horiz. gradient dθ dx 1
Introduction Generalized Continuum Stresses Continuum representation of stress... δw Internal = σ ji δu i, j + T ji δθ i, j + σ jki δu i, jk σ 22 σ 11 σ 12
Introduction Generalized Continuum Stresses Continuum representation of stress... δw Internal = σ ji δu i, j + T ji δθ i, j + σ jki δu i, jk σ 22 σ 11 σ 12
Introduction Generalized Continuum Stresses Continuum representation of stress... δw Internal = σ ji δu i, j + T ji δθ i, j + σ jki δu i, jk σ 22 σ 11 σ 12 T 13 σ 121
Discrete Region Introduction DEM Simulations 256 Particles Circles or Ovals
Introduction Bending Resistance in a Discrete Region Boundary Moments: x 2 Boundary Forces: x 2 x 1 x 1 T 13 σ 121 Bending Moment = T 13 (+) σ 121
Outline Introduction 1 Introduction 2
Introduction Granular Behavior Questions Questions: 1 Are the boundary moments significant? T 13 > 0? 2 Are boundary forces consistent with classical beam theory? σ 121 E I d 2 u 1 dx 1 dx 2?
Introduction Granular Behavior Questions Questions: 1 Are the boundary moments significant? T 13 > 0? 2 Are boundary forces consistent with classical beam theory? σ 121 E I d 2 u 1 dx 1 dx 2?
Outline Introduction 1 Introduction 2
Results Introduction 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 Deviator stress, (σ11 σ22)/po 5 4 3 2 1 0 0 Small strain Ovals Large strain Circles 0.01 0.02 0.03 0.04 Compressive strain, ε 11
Results Introduction Boundary Moments: x 2 Boundary Forces: x 2 x 1 x 1 T 13 σ 121 Bending Moment = T 13 (+) σ 121
Results Introduction 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 Deviator stress, (σ11 σ22)/po 5 4 3 2 1 0 0 Small strain Ovals Large strain Circles 0.01 0.02 0.03 0.04 Compressive strain, ε 11
Results Details Introduction DEM Simulation Results Dimensionless Bending Stiffnesses 256 particles 50 assemblies Large Strain T 13 Boundary moments σ 121 Boundary forces Circles Ovals -0.01-0.01 0.60 1.16 EI u Beam theory 0.25 0.65
Introduction 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.
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.