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
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DEM
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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….
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
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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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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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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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DEM
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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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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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DEM
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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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2. U.S. Food and Drug Administration. Guidance for industry: Q8 (R2) pharmaceutical development; Food
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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….