The document provides an introduction to the discrete element method (DEM). DEM is a numerical modeling method for simulating the motion and behavior of large assemblies of discrete particles. It works by computing the motion of individual particles based on Newton's laws of motion and models contact forces via spring and damping models. DEM has various applications in modeling granular materials, aggregates, and other particulate systems. It offers advantages over other methods for capturing the discrete nature of such materials.

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DISCRETE ELEMENT METHOD

2. INTRODUCTION
What is DEM?
 The computation of the motion and effect of a large number of small particles
 A simulation of discreet elements
https://www.youtube.com/watch?v=-j1lCCznSrU
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3.  a way of simulating discrete matter
 a numerical model capable of describing the mechanical behavior of
assemblies of discs and spheres
 a particle-scale numerical method for modeling the bulk behavior of
granular materials and many geomaterials (coal, ores, soil, rocks,
aggregates)
 capture dual nature of materials
CONT’D….
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DEM
9/13/2022 Discrete Element Method (DEM)

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CONT’D….
3
Discretization of Space!
Lagrangian (ex: DEM)
 Discontinuous
 Classical mechanics interaction (general)
 Resolution at particle level
 Computationally expensive
Eulerian (ex: FEM)
 Continuous
 Related stresses/stains via constitutive EQs
 Resolution filled throughout grid
 Computationally cheaper
Track position and velocity of moving particle Track velocity (or flux) at fixed grid locations
http://15462.courses.cs.cmu.edu/fall2018/lecture/pdes/

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CONT’D….
What is unique about DEM?
 Each particle has its own rotational, positional, radial, and momentum vectors that can be
calculated using simple Newtonian physics (Kong, 2019)
 Simulation consists of three parts
 Small timesteps must be used, as solution is only conditionally stable (O’Sullivan & Bray, 2004)
 Ideal for modeling separate, discrete particle situations, like, Colloids, granular mater. Powder,
bulk materials in storage, progressive fracture and failure
Initialization Time Stepping
Post
processing

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Industrial applications of DEM?
CONT’D….
Chemicals
Pharmaceuticals
Ceramics
Metals
Food
Agriculture
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 Modeling movement of individual particles
 Micromechanical level of analysis
 Coupled with FEM, CFD
 Complex particle geometries and arrangements
 Complicated validation process
 Computationally expensive
Advantages and Disadvantages of DEM
CONT’D….
ADVANTAGES DISADVANTAGES
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HISTORY OF DEM
1971
1974
1978
1985
1992
Cundall develops DEM to
assist with modeling rock
mechanics (Cundall 1971)
Cundall translates
the method into an
RBM code
(Cundall 1974)
Cundall translates
the method into a
FORTRAN code
(Cundall et. al 1978)
Williams and Mustoe
generalize the method,
comparing it to FEM
(Williams & Mustoe 1985)
Cundall & Hart develop
codes to perform the
DEM in 3 dimensions
(Cundall & Hart 1985)
Shi develops
Discontinuous
Deformation Analysis
(Shi 1992)
DEM

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General Principles
 Newton’s Second Law of Motion
 Conservation of momentum
 Particle motion
 Force Displacement Law
 Stiffness
 Friction
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Displacement / velocity boundary condition
Force boundary conditions
Force Displacement Law
(e.g. stiffness, friction)
Newton’s Second Law of
Motion
DEM

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CONT’D….
Soft Sphere and Hard Sphere
 Rigid particles but small overlap is allowed
 Evaluates forces accurately
 Simultaneous contacts possible
 Impulsive forces
 Exchange of momentum
 One collision at a time
Soft Sphere Hard Sphere
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CONT’D….
Advantage of Soft Sphere
𝑃1
𝑃2 𝑃1 𝑃2 𝑃1
𝑃2
Pre-contact
Contact
(Instant contact and no overlap) Post-contact
𝑃1 𝑃2 𝑃1 𝑃2
𝑃1 𝑃2
Pre-contact Contact
(Long lasting contact)
Post-contact
𝑉1 𝑉2
𝑉1 𝑉2
𝑉1
𝑉2
𝑉1 𝑉2
Overlap
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CONT’D….
DEM
Main micro-parameters of particle system
Category Name
Intrinsic parameters Poisson’s ratio (ν)
Density (ρ)/kg/m3
Shear modulus (G)/Pa
Contact parameters between
particles
Coefficient of restitution
Coefficient of static friction
Coefficient of rolling friction
Contact parameters between
particles and geometry
Coefficient of restitution
Coefficient of static friction
Coefficient of rolling friction
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General Principles DEM
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Particle motion: Governed by Newton’s equation
Rotation of particle
M = Iα
M = I
𝑑𝜔
𝑑𝑡
M – torque acting on particle
I – moment of inertia
α – angular acceleration
𝜔 – angular momentum

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DEM
Particle motion: Governed by Newton’s equation
CONT’D….
Translation of particle
F= mα
𝐹
𝑔 + 𝐹𝑐 + 𝐹𝑛𝑐= m
𝑑𝜗
𝑑𝑡
𝐹
𝑔 − gravitational force (mg)
𝐹𝑐 – contact force
𝐹𝑛𝑐 – not contact force
m – mass of the particle
𝜗– translational velocity
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DEM
CONT’D….
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Contact forces: Normal force
𝐹𝑐𝑛 = −𝑘𝑛𝛿𝑛 − η𝑛𝑉
𝑛
Tangential force
𝐹𝑐𝑡 = −𝑘𝑡𝛿𝑡 − η𝑡𝑉𝑡
𝑘𝑛, 𝑘𝑡 − 𝑠𝑡𝑖𝑓𝑓𝑛𝑒𝑠𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑝𝑟𝑖𝑛𝑔𝑠
𝜂𝑛, 𝜂𝑡 - damping coefficients
𝛿𝑛, 𝛿𝑡 - displacement
𝑉
𝑛, 𝑉𝑡 - relative velocities

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DEM
CONT’D….
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Non - contact forces:  Gravitational force
𝑭𝒈 = 𝑮
𝒎𝟏𝒎𝟐
𝒓
𝑚1𝑚2 - mass of the particle
G – gravitational constant
R – distance
 Molecular forces

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CONT’D….
Particle positions:
x (t + ∆t) = x(t) + ν(t)∆t
Particle velocity:
ν (t + ∆t) = ν(t) + a(t)∆t
Numerical Integration:

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Classification of particle interaction force models
by contact force and non-contact force.
Non-Contact Force
Van der Waals force
Liquid bridge force
Electrostatic force
Linear spring model Non – linear spring
Hertz-Mindlin
Hertz-Mindlin + JKR
DMT Model
Linear Spring-Dashpot
Hysteretic Model
Thornton Model
Particle interaction
Contact Non- Contact
Elastic Inelastic
Linear model Non- linear model

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CONTACTMODELS
 Contact between two particles occurs in the finite area
 Area consists of the normal and tangential plane
 Contact force - normal and tangential
 Overlap (δ) = 𝑅1 + 𝑅1-d
 Damping forces - friction forces and cohesive forces
 Determine the acceleration of particles

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ELASTICCONTACT MODELS
LINEAR SPRING MODEL
 Two particles in contact are both normally and tangentially
connected by linear spring
 Energy is not consumed and the contact is considered
completely elastic
 Linear relationship b/w force and displacement
 Limitation: kinetic energy is dissipated by plastic
deformation
Normal force
𝐹𝑛 = −𝑘𝑛𝛿𝑛
Tangential force
𝐹𝑡 = −𝑘𝑡𝛿𝑡

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DEM
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HERTZ-MINDLIN MODEL
 Nonlinear elastic model
 Contact between two particles in the normal direction – Hertz
 Contact between two particles in the tangential direction – Mindlin
 Hertz-Mindlin model – complexity, time-consuming
 Simplification: no slip – Hertz and Mindlin
 Accuracy - pharmaceutical industry
Eeq, Req and Geq are the equivalent Young’s modulus,
equivalent radius and equivalent shear modulus

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 Model the contact of cohesive particles
 The adhesive theory using a balance between stored elastic
energy and loss of surface energy
 Opposite force owing to the pulling force
HERTZ-MINDLIN + JKR MODEL
a - contact area
γ - surface energy

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DEM
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DMT MODEL
 Cohesion at the contact periphery
 Hertz-Mindlin + JKR model based on the surface energy
 Suitable for hard materials
 Solids with a small tip radius and low surface energy

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INELASTICCONTACT MODELS
LINEARSPRING-DASHPOTMODEL
 Elastic models – accumulation of energy
 Inelastic models - to model the dissipation of energy
 Plastic deformation between particles occurs
 Composed of linear spring and dashpot components
 Linear spring describes the repulsive forces
 Dashpot dissipates the relative kinetic energy
Normal contact force
𝐹𝑐𝑛 = −𝑘𝑛𝛿𝑛 − η𝑛𝑉
𝑛

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 Uses various spring constant at the loading, Unloading and reloading
stages
 Hysteretic model - linear contact models
 Normal direction - a partially latched spring force-displacement model
 Mindlin and Deresiwicz theory - the constant normal force in
the tangential direction
 Limitation: it describes the plastic deformation only in the normal
direction
where K1 and K2 are the spring
constants in the loading and
unloading stages
HYSTERETIC MODEL

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 Explains plastic deformation
 Proposed for normal contact between two elastic, perfectly
spherical plastic particles
 Based on the Hertz theory
(normal force-displacement relationship during the initial
elastic loading )
 Plastic deformation occurs if the limiting contact pressure is
reached at the center of the contact area
THORNTON MODEL
where Fny and δy denote the
normal contact force and
displacement

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DEM
CONT’D….
DEM Model Workflow
Accelerations Velocities Positions
Forces Contacts
Newton’s Law
Contact mechanics

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Numerical solution software for DEM
Open Source Commercial
 Often written in C, C++, Fortran, python and interacted
with within the terminal
 Many limited to Linux/ Unix, and ran using code-
specific commands and functions, or ran from an input
file
 Often more powerful, with convenient GUI’s and
built-in system coupling
 Very expensive for limited licenses
 ABAQUS: ~ $ 65,000

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Discrete Element Modeling - DEM Software | Altair EDEM
https://www.altair.com › edem

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1. Rogers, A.; Ierapetritou, M. Challenges and opportunities in modeling pharmaceutical manufacturing
processes. Comput. Chem. Eng. 2015, 81, 32–39.
2. U.S. Food and Drug Administration. Guidance for industry: Q8 (R2) pharmaceutical development; Food
and Drug Administration: White Oak, MD, USA, 2006.
3. Suresh, P.; Basu, P.K. Improving pharmaceutical product development and manufacturing: Impact on cost
of drug development and cost of goods sold of pharmaceuticals. J. Pharm. Innov. 2008, 3, 175–187.
4. Ketterhagen,W.R.; am Ende, M.T.; Hancock, B.C. Process modeling in the pharmaceutical industry using
the discrete element method. J. Pharm. Sci. 2009, 98, 442–470.
5. Kremer, D.; Hancock, B. Process simulation in the pharmaceutical industry: A review of some basic
physical models. J. Pharm. Sci. 2006, 95, 517–529.
References

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6. Pandey, P.; Bharadwaj, R. Predictive Modeling of Pharmaceutical Unit Operations; Woodhead
Publishing: Cambridge, UK, 2016.
7. Björn, I.N.; Jansson, A.; Karlsson, M.; Folestad, S.; Rasmuson, A. Empirical to mechanistic modelling in
high shear granulation. Chem. Eng. Sci. 2005, 60, 3795–3803.
8. Reklaitis, G.V.; García-Munoz, S.; Seymour, C. Comprehensive Quality by Design for Pharmaceutical
Product Development and Manufacture; JohnWiley & Sons: Hoboken, NJ, USA, 2017.
9. Wassgren, C.; Curtis, J.S. The application of computational modeling to pharmaceutical materials
science. MRS Bull. 2006, 31, 900–904.
10. Norton, T.; Sun, D.-W. Computational fluid dynamics (cfd)—An eective and effcient design and
analysis tool for the food industry: A review. Trends Food Sci. Technol. 2006, 17, 600–620.
CONT’D….

32. THANK
YOU
9/13/2022 Discrete Element Method (DEM)