Continuum Models of Discrete Particle Systems with Particle Shape Considered

Transcription

1 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

2 Outline Introduction 1 Introduction 2

3 Introduction Continuum vs. Discrete Frameworks Continuum Small (but finite!) granular sub-region Continuum point

4 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

5 Outline Introduction 1 Introduction 2

6 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

7 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

8 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

9 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

10 Discrete Region Introduction DEM Simulations 256 Particles Circles or Ovals

11 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

12 Outline Introduction 1 Introduction 2

13 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?

14 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?

15 Outline Introduction 1 Introduction 2

16 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 Small strain Ovals Large strain Circles Compressive strain, ε 11

17 Results Introduction Boundary Moments: x 2 Boundary Forces: x 2 x 1 x 1 T 13 σ 121 Bending Moment = T 13 (+) σ 121

18 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 Small strain Ovals Large strain Circles Compressive strain, ε 11

19 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 EI u Beam theory

20 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.

21 Appendix Further Reading Further Reading I M. R. Kuhn Are granular materials simple? An experimental study of strain gradient effects and localization. Mechanics of Materials, 37(5): C. S. Chang and M. R. Kuhn On virtual work and stress in granular media. Int. J. Solids and Structures, 42(13):