1. The Discrete Element Method Lecture 1: Fundamentals
Carlos Labra
International Center for Numerical Methods in Engineering
Barcelona, Spain
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2. “The term discrete element method (DEM) is a family of numerical methods for computing the motion of a large number of particles like molecules or grains.”
Wikipedia
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3. Overview DEM, Barcelona 2010 grains of sand SAG mill Conveyor Belts
Sand sorting tower at a gravel extraction pit.
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4. Overview DEM, Barcelona 2010 Operation of a jaw crusher Blasting
Cohesionless Soil Slope
Uniaxial compression test
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5. Fundamentals
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6. Materials structure
Solid materials are characterized with a certain internal structure – in some materials, like rocks or sand, it can be seen without any instruments, in some cases the microstructure can be seen under microscope. Observations can be performed at atomistic scale.
Crystals of gold
Sandstone
Steel microstructure
Sand
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7. Continuum vs. discrete material models
Continuous models: Molecular structure, microstructure or granularity of materials disregarded; mathematical functions are continuous except possibly at a finite number of interior surfaces separating regions of continuity
Discrete models: Discontinuities and discrete internal structure taken into account, material treated as discrete medium
Discrete models
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8. Numerical methods based on discrete models
Physical model of discrete medium
Discrete mathematical model
Discrete model is obtained directly applying equations describing discrete medium
Numerical methods based on discrete material models:
Molecular dynamics
Particle-in-cell method
Cellular automata
Lattice models
Discrete element method
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9. Numerical modelling Remark:
It must be remembered that some particle methods, like SPH (Smoothed Particle Methods) are derived from continuous material models.
Solution obtained by Particle Finite Element Method
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10. Development of the discrete element method
P.A. Cundall and Strack, University of Minnesota A discrete numerical model for granular assemblies, Géotechnique, 1979.
O.R. Walton, LLNL Explicit particle dynamics for granular materials, Num. Meth. in Geomechanics, 1982.
Other authors: G. Mustoe, J.P. Bardet, J.R. Williams
Powders and Grains, 1st International Conference on Micromechanics of Granular Media
1st US Conference on Discrete Element Methods, 1989
Initial period, 1980s, small capabilities of computers limit practical applications of the discrete element method
Fast development since late 1980s
Growing interest in the recent years because of possible new applications, for instance, in the design of new materials
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11. Most important centres developing the DEM
Itasca Consulting Group, Inc., University of Minnesota, P. Cundall
(commercial programs PFC2D, PFC3D)
- Colorado School of Mines, Geomechanics Research Center, G.W. Mustoe
- MIT Intelligent Engineering Systems Lab, J.R. Williams
- Lawrence Livermore National Laboratory, O.R. Walton, R.L. Braun
- Sandia National Laboratories, Geomechanics Department, D.S. Preece
- University of Southern California, J.P. Bardet
- University of Florida, Particle Engineering Research Center
- University of Wales, Swansea, Department of Civil Engineering, A. Munjiza, D.R.J. Owen
- Universität Stuttgart, Institute of Structural Mechanics, E.Ramm
University of Grenoble, Discrete Element Group for Hazard Mitigation, France,
F. Donze, YADE code
- DEM Solutions, Scotland, J. Favier, EDEM
- CIMNE, Barcelona, DEMpack software (DEM since 1999)
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12. Basic assumptions of the discrete element method
Material is represented by a large collection of rigid or deformable discrete elements interacting among one another with contact forces
Discrete elements can be of arbitrary shape:
cylinders (discs), spheres (advantage: efficient contact detection)
ellipses, ellipsoides
polygons, polyhedra
other shapes
Discs
Ellipses
Rombes
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13. Equations of rigid body motion
Discrete element motion is described by the rigid body dynamics equations
Rigid body kinematics
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14. Equations of rigid body motion
The principle of virtual work
– mass centre
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15. Equations of rigid body motion
Newton - Euler equations
where
− inertia tensor
− resultant force
− resultant moment
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16. Equations of motion of a discrete element
The principle of virtual work
for a cylindrical or spherical discrete element:
Newton - Euler equations:
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17. Contact constraints Kuhn-Tucker conditions where For two spheres:
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18. Treatment of collisions in the DEM
Hard contact, also called hard particles, event driven approach
contact time short, it is neglected
instantaneous momentum change is assumed
velocity after collision determined using the restitution coefficient in the normal direction r, the Coulomb friction coefficient μ , and the maximum tangential restitution β0
Soft contact, also called soft particles, time driven, molecular dynamics approach
finite time of contact
change of momentum determined by a potential or interaction law
Remark: Hard contact algorithms are inefficient if the time between events (collisions) is very small, which can occur in densely packed particle models. Soft particle model is considered in the formulation presented.
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19. Equations of motion of a discrete element
Newton - Euler equations
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20. Enforcement of contact constraints
Lagrange multipliers method
Penalty function – this method is adopted in the soft particle formulation of the discrete element method, a certain particle penetration is allowed
− penalty parameter
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21. Contact model Decomposition of the contact force
– normal contact force
– tangential contact force
Pair of interacting particles
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22. Frictional contact model
Normal contact force
Elastic part of the contact force
- Elastic part
- Contact damping
Rheologic model
- contact stiffness in the normal direction
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23. Frictional contact model
Hertz model for the elastic normal contact force
Contact stiffness in the normal direction
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24. Frictional contact model
Normal contact force
Contact damping
Rheologic model
– relative normal velocity at contact point ,
– damping coefficient ,
– critical damping for the system of two masses with a spring
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25. Frictional contact model
Tangential contact force – Coulomb friction model
Relative tangential velocity
Slip function
Slip law
Stick/slip conditions:
Coulomb friction model
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26. Frictional contact model
Tangential contact force – Regularized Coulomb friction model
Decomposition of the relative tangential velocity
Slip law
Slip function
Stick/slip conditions: if if
Regularized Coulomb friction model
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27. Frictional contact model
Tangential contact force – Numerical algorithm
Given: - old tangential force: Fsold
- normal force: Fn
- relative velocity: Vrs
Find: - new tangential velocity: Fsnew
Trial friction force
Stick/slip condition
- stick
- slip
Regularized Coulomb friction model
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28. Contact models with cohesion
Discrete element models of rocks, cohesive soils and other cohesive materials require contact models with cohesion (tensile resistance)
Cohesive bonds are applied to pairs of elements being in contact
Cohesive bonds can be broken – initiation and propagation of cracks can be modelled
After debonding frictional contact model is assumed
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29. Elastic perfectly brittle contact model
Normal contact force (elastic part)
Failure (decohesion) criterion Fn
kn – contact stiffness in the normal direction
g – gap or penetration
compression
tension
Elastic perfectly brittle model
In the absence of cohesion (or after decohesion) no tensile interaction is allowed
– bond strength in the normal direction
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30. Elastic perfectly brittle contact model
Tangential contact force (cohesive):
Failure (decohesion) criterion:
In the absence of cohesion:
g < 0 us Fs Rs ks
ks - contact stiffness in the tangential direction
us - relative tangential displacement
g ≥ 0 FS
Rs - bond strength in the normal direction
Elastic perfectly brittle model
In the absence of cohesion it is either zero (for separattion) or friction force RS us ks
g = 0
Coulomb friction
g ≥ 0
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31. Other contact models with cohesion
Elasto-plastic contact models with softening
Elastic damage models
Normal contact force
Tangential contact force
Elastic perfectly brittle model
In the absence of cohesion it is either zero (for separattion) or friction force
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32. Rotational interaction
Interaction moment
Relative
Slip law
Slip function
Stick/slip conditions: if if
Regularized Coulomb friction model
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33. Rotational interaction
Interaction moment
Relative angular velocity
Decomposition of the relative angular velocity
– normal (torsional) component
– tangential component
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34. Rotational interaction without cohesion
Torsion
Regularized slip law
Slip function
Stick/slip conditions: t
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35. Rotational interaction without cohesion
Rotation about the tangential axis (rolling friction)
Regularized slip law
Slip function
Stick/slip conditions: t
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36. Rolling friction
Similarly to interaction between particles, respective models for the interaction particle-plane can be defined
Rolling resistance moment can be interpreted as an effect of the off-set of the normal reaction
Rolling friction compensates the deficiency of the model using particles of ideal shape (spheres, cylinders)
− rolling friction coefficient
Friction , Fricción, frotamiento , rozamiento ; friction brake, freno de Prony; friction clutch, embrague de fricción; friction coefficient, coeficiente de rozamiento; friction disc, disco de fricción; friction head, pérdida de carga (hidráulica); frictionless, sin rozamiento; frictionless bearings, rodamientos sin rozamiento; friction of rolling or rolling, rozamiento de rodadura; friction of sliding, rozamiento de deslizamiento; friction (screw) press, prensa a fricción; friction shoe, patínde rozamiento, zapata; friction socket, cono de friccíón; air friction , frotamiento de¡ aire; angle of friction , ángulo de rozamiento; box of a friction coupling, manguito de fricción; magnetic friction, fricción magnética; skidding friction , rozarniento de derrape; skin friction , rozamiento superficial (aviación); sliding friction, rozamiento de deslizamiento. Friction bearing, cojinete de fricción , ( Ingeniería mecánica ) Cojinete enterizo que soporta y está en contacto con el extremo de un eje.Friction brake, freno de fricción , ( Ingeniería Mecánica ) Freno en el que la resistencia o fuerza de frenado se obtiene por rozamiento o fricción. Friction clutch, embrague de fricciónFriction coefficient, coeficiente de fricción , ( Mecánica ) See friction coefficientFriction damping, amortiguamiento por rozamiento, ( Mecánica ) La conversión de la energía vibracional mecánica de sólidos en energía calorífica producida por el desplazamiento de un elemento seco sobre otro elemento.Friction drive, transmisión por fricciónFriction factor, factor de fricción , ( Mecánica de los fluidos  ) Cualquiera de los números adimensionales utilizados para el estudio de la fricción de los fluidos en tuberías, que es igual al factor de fricción de Fanning multiplicado por cierta constante adimensional. Friction flow, flujo de fricción , ( Mecánica de los fluidos ) Flujo de un fluido en el cual una cantidad significativa de energía mecánica se disipa en forma de calor por la acción de la viscosidad.  Friction gear, rueda de fricción Friction head, carga de fricción, ( Mecánica de los fluidos ) Pérdida de carga hidrostática de un flujo, en una corriente o conducto, debido a perturbaciones friccionales establecidas por el fluido en movimiento y por la fricción intermoiecular.Friction horsepower, potencia absorbida por rozamientoFriction loss, pérdida por rozamientoFriction saw, sierra de fricciónFriction sawing ,serrado por fricciónFriction torque, par de rozamientoFrictional , A fricción, de rozamiento; - losses, pérdidas por rozamiento.Frictional grip, adherencia friccional, ( Mecánica ) Adherencia por rozamiento de una locomotora o adhesión que se produce entre las ruedas y las vías en una línea férrea.Frictional secondary flow, flujo friccional secundario , ( Mecánica de los fluidos ) See Secondary flow. Frictionless flow, flujo sin fricción , ( Mecánica de los fluidos  ) See Inviscid flow.
FRICCION de rodamiento
FRICCION ADHERENTE, DESLIZANTE Y MIXTA. TENSION TRASERA Y DELANTERA.
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37. Equations of motion of a discrete element
Newton - Euler equations
background (global) damping
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38. Integration methods for ordinary differential equations
Implicit integration methods – dependent variables are computed in terms of unknown quantities, for equations of motion – new configuration is obtained from the equations taken at the new (unknown) configuration;
implicit solution is almost always iterative (possible problems with convergence),
better control of accuracy of solution,
unconditional numerical stability, large integration steps can be used
Explicit integration methods – direct computation of the dependent variables is made in terms of known quantities, for equations of motion – new configuration is obtained from the equation taken at the known (last) configuration
Non-iterative solution (no problems with convergence),
conditional numerical stability, large number of small steps,
more efficient solution.

39. Integration methods – example
Problem to be solved (initial value problem):
Explicit integration scheme: forward Euler method
Implicit integration scheme: backward Euler method

40. Integration methods for first order
Explicit Implicit Hybrid
Forward Euler Backward Euler Milnov
Runge-Kutta Trapezoidal Numerov
Merson Simpson Gears
Adams-Bashfort Adams-Moulton Others
Others Linninger
Willoughby
Others

41. Integration methods for second order
Numerical methods have been developed to deal with second order ordinary differential equations directly.
Explicit: Implicit:
Explicit central difference method Implicit central difference method
Verlet method Newmark method
Velocity Verlet method Numerov (Cowell) method
(leapfrog method) Others
Runge-Kutta method for 2nd order
others
Integration algorithm typically used in the DEM

42. Integration of DEM equations of motion
Explicit leapfrog (velocity Verlet, variant of CD) method
Equations of motion for the known configuration at tn
Determination of the new configuration at tn+1
Translational motion
Remark: Velocities are shifted by with respect to displacements
Δt/2

43. Integration of DEM equations of motion
Explicit leapfrog (velocity Verlet, variant of CD) method
Determination of the new configuration at tn+1
Rotational motion
Remarks:
Angular velocity in 3D is not a derivative of any vector – cannot be integrated, the vector of small incremental rotations is evaluated only.
For spherical discrete elements the rotational configuration need not be evaluated
If necessary, the rotational configuration can be determined

44. Stability of the time integration scheme
Explicit time integration schemes are conditionally stable
Time step length Δt must be smaller than a certain critical time step Δtcr
For the leapfrog algorithm:
− highest natural frequency of the whole discrete system
N - number of discrete elements
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45. Stability of the time integration scheme
Natural frequencies for the i-th discrete element can be determined by solution of the following eigenproblem
2D Example
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46. Stability of the time integration scheme
Single element contacting with a rigid plane
Usually in DEM:
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47. Background (global) damping
Viscous damping
Non-Viscous damping (dissipation does not depend on velocity)
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48. Equations of motion of a discrete element
DEM equations – system of ordinary differential equations
Solution – integration with respect to time
Integration methods:
Explicit,
Implicit,
Hybrid (combine explicit and implicit steps)
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49. Stability of the time integration scheme
Single element contacting with a rigid plane
Influence of the damping
damping given by a fraction of the critical damping for the frequency
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50. Influence of damping – numerical test
Model details:
Gravitational load
Normal interaction only (friction not considered)
Two cases:
without damping
with damping (viscous)
Conclusions
Without damping – no energy dissipation in the simulated process
Damping allows us to simulate a quasistatic process
No damping
Damped

51. Rotational motion Rotational degrees of freedom test t = 0.0054 s

52. Search of contacts
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53. Search of contacts Many searches during the simulation
High computational cost involved
Different techniques/algorithms can be selected
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54. Search of contacts
Algorithms:
Brute force
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55. Search of contacts Algorithms: Brute force Quadtree / Octtree
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56. Search of contacts Algorithms: Brute force Quadtree / Octtree KdTree
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57. Search of contacts Algorithms: Brute force Quadtree / Octtree KdTree
Bins (Hash)
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58. Particles generation Katalin Bagi Hungarian Academy of Sciences
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59. Particles generation Assembly preparation
Methods to generate random dense arrangements of particles:
Dynamic techniques
Follow the particles motions through several time steps
Constructive techniques
Assembly preparation with purely geometrical calculations
Application experiences
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60. Particles generation Dynamic techniques Isotropic compression
Particle expansion
Gravitational deposition
Multi-layer Undercompaction Method
Collective rearrangement techniques
Optimization scheme …
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61. Particles generation Much faster!!! Constructive techniques
Regular arrangements (WHY NOT USE THEM?)
Sequential inhibition model
Sedimentation techniques
Closed front technique
Inwards packing algorithm …
Much faster!!!
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62. Dynamic techniques Isotropic compression Particle expansion
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63. Dynamic techniques Gravitational deposition
Multi-layer undercompaction method
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64. Dynamic techniques Collective rearrangement techniques
Optimization scheme (Labra & Oñate 2009)
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65. Optimization scheme
Dense random particle configurations obtained by minimization of porosity of a loose random collection of spheres (Labra & Oñate 2009).
Unstructured irregular mesh
Initial particle configuration (overlapping is eliminated during the optimization)
Final particle configuration
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66. Optimization scheme Objective function where: Constraints
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67. Optimization scheme
Generation of a particle assembly using the optimization scheme
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68. Constructive techniques
Regular arrangements
Very easy to create!!
Unreliable for the simulation of the mechanical behavior of materials with random internal microstructure
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69. Constructive techniques
Sequential inhibition model (PFC2D)
Very easy to create!!
Unreliable for the simulation of the mechanical behavior of materials with random internal microstructure
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70. Constructive techniques
Sedimentation techniques (Jodrey & Tory 1985, Bagi 1993, Feng & Owen 2003)
Stable position of a grain:
Very easy to create!!
Unreliable for the simulation of the mechanical behavior of materials with random internal microstructure
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71. Constructive techniques
Closed front techniques (Feng & Owen, 2003)
Very easy to create!!
Unreliable for the simulation of the mechanical behavior of materials with random internal microstructure
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72. Constructive techniques
The inwards packing method (Bagi 2005)
Initial front
Interior
Very easy to create!!
Unreliable for the simulation of the mechanical behavior of materials with random internal microstructure
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73. Characterization of mesh/assembly
Contact graph
Average radius
Coordination number
Void ratio
Fabric tensor
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74. Characterization of mesh/assembly
Contact bond graphs
Contact bond graph represents connections in the particle assembly - visual confirmation of irregularity and compactness
DEM assembly
Contact bond graph
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75. Characterization of mesh/assembly
Contact bond graphs
Detail of the DEM assembly and the contact bond graph
DEM assembly
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76. Characterization of mesh/assembly
Radius: Average value & distribution
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77. Characterization of mesh/assembly
Coordination number
Average number of contacts per particle
Void ratio / porosity
The volume of the voids divided by the total volume of the domain
Constructive algorithms ~ 3.5 – 4.5 (2D)
High density packing ~ 5.0 – 5.5 (2D)
Constructive algorithms ~ 0.5 – 0.7 (2D)
High density packing ~ 0.8 – 0.9 (2D)
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78. Characterization of mesh/assembly
Fabric tensor
The distribution of contact orientations in the assembly.
Based on the unit vectors of the contact directions
Isotropic anisotropic
assembly assembly
The sum of its eigenvalues always equals to 1, and the difference between them reflects the strength of anisotropy
(if the two eigenvalues are equal, the contact distribution is perfectly isotropic)
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79. Discrete – continuum relationship
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80. Discrete-continuum relationships
Definition of DEM parameters
Estimation of DEM parameters based on macro behaviour
Continuum equivalence
Stress-strain relationships
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81. Relationship micro – macro parameters?  
Micromechanical constitutive laws
Macroscopic stress-strain relationships
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82. Relationship micro – macro parameters
Simulation of standard laboratory tests (unconfined compression, Brazilian test, direct shear test) using a discrete (micromechanical) model and determination of macroscopic properties corresponding to a micro parameters used in a discrete model
Homogenization and averaging procedures ?
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83. Relationship micro – macro parameters
Microscopic level
Constitutive parameters
kn – contact stiffness in normal direction
ks – tangential contact stiffness
Rn – normal contact strength
Rs – tangential contact strength
μ – Coulomb friction coefficient
αt – translational damping coefficient
αr – rotational damping coefficient
State variables
g – penetration/gap
us – relative tangential displacement
Fn – normal contact force
Fs – tangential contact force
Geometrical and physical parameters
r – average element radius
ρ – density
others …
Macroscopic level
Constitutive parameters
E – Young’s modulus
ν – Poisson ratio
σc – compressive strength
σt – tensile strength
c – viscosity
State variables
ε – strain
σ – stress
Geometrical and physical parameters
L – sample size
n – porosity
ρ – effective density
others …
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84. Relationship micro – macro parameters
Dimensionless numbers
Mesh influence should be considered!!
f(nc,ϕ,…) : function of the mesh characterization parameters
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85. Relationship micro – macro parameters
Mesh influence
Average radius
Coordination number
Fabric tensor
Poisson ratio
(Best fit approach)
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86. Relationship micro – macro parameters
Unconfined compression test
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87. Relationship micro – macro parameters
Brazilian test
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88. Discrete - continuum relationships
Stress and strain
Stress
Forces
Relative displacement
Strain
Averaging
Contact constitutive relation
Kinematic localisation assumption
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89. Discrete - continuum relationships
Stress
Macroscopic stress is generated by contact forces and configuration
Average stress i j l f
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90. Discrete - continuum relationships
Transition from the microscopic discrete to macroscopic continuum description by averaging over RVE
Around each point we define a representative volume element (RVE)
Averaging of the quantity Q over RVE
The average is assigned to the point
, if Q constant for the particle:
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91. Discrete-continuum relationships
Strain (equivalent-continuum strain)
Can’t be obtained in a direct way
Different aproachs
Kruyt & Rothemburg (1996)
Bagi (1993) (1996)
Kuhn (1999)
Cambou et al (2000)
Kruyt (2003)
Differences between the versions
Differences in the applied equivalent continuum
Differences in how to define a translational field in them
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92. Lecture 2: Application and advanced topics
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