1. Continuous and discrete models for simulating granular assemblies
Akke S.J. Suiker
Delft University of Technology
Faculty of Aerospace Engineering
Chair of Engineering Mechanics
January 1, 2019

2. Configuration of Lattice
Graphical representation of 9-cell square lattice model
Suiker, Metrikine, de Borst, Int. J. Sol. Struct, 38, , 2001
Suiker & de Borst, Phil. Trans. Roy. Soc. A., 363, , 2005
January 1, 2019

3. Equations of motion lattice
January 1, 2019

4. Long-wave approximation of EOM (I.A. Kunin, 1983)
Replace discrete kinematic d.o.f.’s by continuous field variables:
Replace discrete d.o.f.’s of neighbouring cells by second-order Taylor approximations of continuous field variables:
January 1, 2019

5. Equations of motion in long wave-approximation
January 1, 2019

6. Equations of motion for Cosserat continuum (Cosserat E. , Cosserat F
Equations of motion for Cosserat continuum (Cosserat E., Cosserat F., 1909; Günther, W., 1958; Schaefer, H., 1962; Mindlin, R.D., 1964; Eringen, A.C., 1968; Mühlhaus, H.-B., 1989; de Borst, R., 1991)
The Cosserat continuum model is useful for studying:
Localised failure problems, where rotation of grains is important
High-frequency wave propagation, with deformation patterns of short wavelengths
January 1, 2019

7. Mapping long-wave approximation on Cosserat model
Relation between continuum material parameters and lattice parameters:
Constraints that have to be satisfied to match the anisotropic lattice
model with the isotropic Cosserat continuum model:
January 1, 2019

8. Configuration reduced lattice
Graphical representation of reduced 9-cell square lattice model
January 1, 2019

9. Dispersion relations for plane harmonic waves
Lattice
Continuum
Substitution into equations of motion yields:
Dispersion relations:
January 1, 2019

10. Direction of propagation (kx,kz) = (0,k)
Dispersion curves for 9-cell square lattice and Cosserat continuum
January 1, 2019

11. Second-gradient micro-polar model (microstructural approach)
Constitutive coefficients are of the form (using )
Suiker, de Borst, Chang, Acta Mech., 149, , 2001
Suiker, de Borst, Chang, Acta Mech., 149, , 2001
Suiker & de Borst, Phil. Trans. Roy. Soc. A., 363, , 2005
January 1, 2019

12. Reduced forms of the second-gradient micro-polar model
Linear elastic model, C(1) to C(6) = 0,:
Second-gradient model, C(3) to C(6) = 0, (Chang & Gao, 1995):
Cosserat model, C(1), C(2) and C(4) = 0, (Chang & Ma, 1992):
January 1, 2019

13. Dispersion curves for various models
Dispersion curves for compression wave, shear wave and micro-rotational wave
January 1, 2019

14. Boundary value problem
Layer of thickness H, consisting of equi-sized particles of diameter d
- forced vibration under moving load
Suiker, Metrikine, de Borst, J. Sound Vibr., 240, 1-18, 2001
Suiker, Metrikine, de Borst, J. Sound Vibr., 240, 19-39, 2001
January 1, 2019

15. Cells of square lattice
Inner cell
Boundary cell
January 1, 2019

16. Response of layer to a moving load
4 boundary conditions (in Fourier domain):
- boundary cells at top of layer subjected to moving load in z-direction
and free of loading in x-direction
- displacements at bottom of layer are zero (in x- and z-directions)
Substituting harmonic displacements into these 4 boundary conditions gives:
Solve above system, and transform solution to time domain by Inverse Fourier Transform (numerical).
January 1, 2019

17. Displacement profile (H=300mm) (uz taken at 0.2H below layer surface)
Case 1
Case 2
Velocity dependence
Case 3
Harmonic load
January 1, 2019

18. Model Configuration
Cuboidal volume of randomly packed, equi-sized, cohesionless spheres
(initial porosity is 0.382).
Suiker & Fleck, J. Appl. Mech., 71, , 2004
January 1, 2019

19. Stress-strain Response at various Contact Friction
Stress-strain response for various contact friction angles
January 1, 2019

20. Effect of Contact Friction on Sample Strength
Macroscopic friction angle versus contact friction angle
January 1, 2019

21. Effect of Particle Redistribution
Three different kinematic conditions:
Particle sliding and particle rotation are allowed
Particle sliding is allowed, particle rotation is prevented
Particle sliding is allowed in correspondence with an affine deformation field, particle rotation is prevented.
January 1, 2019

22. Stress-strain Responses
left: Volumetric strain versus hydrostatic stress (volumetric deformation path )
right: Deviatoric strain versus deviatoric stress
(deviatoric deformation path )
January 1, 2019

23. Collapse Contour in the Deviatoric Plane
Left: Collapse contour for unconstrained and constrained particle rotation ( )
Right: Collapse contour for DEM model (unconstrained particle rotation) and various continuum models
January 1, 2019

24. Points of discussion
Higher-order continuum models approach discrete models accurately up to a certain wavelength of deformation
Higher-order continuum models may be unstable for small wavelengths;
 remedy: inclusion of higher-order time derivatives
(and coupled time-space derivatives)
Deformations with wavelengths < few times the particle diameter can not be decribed accurately with continuum models
The number of constitutive coefficients increases drastically when continuum models are further kinematically enhanced (i.e., 4th-order, 6th-order etc.)
January 1, 2019