Akke%20S.J.%20Suiker

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Continuous and discrete models for simulating
granular assemblies

• Akke S.J. Suiker
• Delft University of Technology
• Faculty of Aerospace Engineering
• Chair of Engineering Mechanics

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Configuration of Lattice
Graphical representation of 9-cell square lattice
model
Suiker, Metrikine, de Borst, Int. J. Sol. Struct,
38, 1563-1583, 2001 Suiker de Borst, Phil.
Trans. Roy. Soc. A., 363, 2543-2580, 2005
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Equations of motion lattice
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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
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Equations of motion in long wave-approximation
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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

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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
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Configuration reduced lattice
Graphical representation of reduced 9-cell square
lattice model
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Dispersion relations for plane harmonic waves
Plane harmonic waves
Lattice
Continuum
Substitution into equations of motion yields
Dispersion relations
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Direction of propagation (kx,kz) (0,k)
Dispersion curves for 9-cell square lattice and
Cosserat continuum
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Second-gradient micro-polar model(microstructural
approach)

• Constitutive coefficients are of the form (using

)
Suiker, de Borst, Chang, Acta Mech., 149,
161-180, 2001 Suiker, de Borst, Chang, Acta
Mech., 149, 181-200, 2001 Suiker de Borst,
Phil. Trans. Roy. Soc. A., 363, 2543-2580, 2005
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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)

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Dispersion curves for various models
Dispersion curves for compression wave, shear
wave and micro-rotational wave
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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
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Cells of square lattice
Inner cell
Boundary cell
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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).
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Displacement profile (H300mm) (uz taken at 0.2H
below layer surface)
Case 1
Case 2
Case 3
Velocity dependence
Harmonic load
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Model Configuration
Cuboidal volume of randomly packed, equi-sized,
cohesionless spheres (initial porosity is 0.382).
Suiker Fleck, J. Appl. Mech., 71, 350-358, 2004
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Stress-strain Response at various Contact
Friction
Stress-strain response for various contact
friction angles
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Effect of Contact Friction on Sample Strength
Macroscopic friction angle versus contact
friction angle
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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.

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Stress-strain Responses
left Volumetric strain versus hydrostatic
stress (volumetric deformation path
) right Deviatoric strain versus
deviatoric stress (deviatoric deformation path
)
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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
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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 lt 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.)