Discontinuous Analysis of CohesiveFrictional Blocky Materials and Assemblies

1
ALERT Geomaterials Alliance of Laboratories in
Europe for Research and Technology
Discontinuous Analysis of Cohesive-Frictional
Blocky Materials and Assemblies Nenad
Bicanic Department of Civil Engineering
ALERT DOCTORAL SCHOOL 2008 9-11 October 2008,
Aussois, France, "Discrete Modelling in
Geomechanics"
2
Mustoe/Willams
Paul Langevin (Labyrinth?) Centre
ALERT DOCTORAL SCHOOL 2008 9-11 October 2008,
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3
Block Assemblies and Masonry Structures
Heritage and Contemporary Context
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4
Offertorium (Faures Requiem)
Harmony
Is the whole equal to the sum of its parts?
The whole can be more, but also can be less than
the sum of its parts
Soprano Alto Tenor Bass
Energy of the composite object sum of the
energies of its parts - energy needed to split
the object apart (binding energy)
TOTUM EST QUOD CONSTAT PLURIUM RERUM UNIONE (The
whole is what comes from a union of more things)
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5
Discontinuous Modelling Frameworks discontinuous,
fractured and disjointed media Molecular
Dynamics MD Discrete element method DEM Rigid
block spring method RBSM Lattice modelling LM
Discontinuous deformation analysis DDA
Manifold Method MM Combined discrete/finite
elements DEM/FEM Non smooth contact dynamics
NSCD Applied element method AEM Stress based
discrete element SDEM etc
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6
Discontinuities at different levels of observation
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7
Discontinuous Media Multi body Problem
Systems of rigid or deformable bodies, of
arbitrary shapes, continuously changing
configurations and contacts, possible
fragmentation, or sintering (plus interactions
with other media or fields)
Harmonic base excitation
Detection of contacts, Treatment of contact
constraints, Contact solvers, Deformability of
bodies, Large displacements and large rotations,
Number and/or distribution (loose or dense
packing) of interacting bodies, Model boundaries,
Subsequent fracturing or fragmentation, Time
integration
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8
Peter Cundalls Early Classification (1989)
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9
Contact Detection
Hashing or binning algorithm for simple particle
shapes and clustered particles
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Representation of Bodies
Continuous implicit function representation of
bodies (inside-outside check)
Polar Discrete Functional Representation (DFR)
Superquadrics in 3D
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Principal Computational Modelling Issues for
Discontinuous Media and Processes

• detection of contacts
• treatment of contact constraints
• deformability (constitutive law) of bodies in
  contact (rigid, pseudo rigid, constrained
  continua, continua)
• large displacements and large rotations
• number (small or large) and/or distribution
  (loose or dense packing) of interacting bodies
  considered
• consideration of the model boundaries
• possible subsequent fracturing or fragmentation
• time stepping integration schemes (explicit,
  implicit)
• Global and Local Frames
• Definition of a contact
  (local) plane

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3D RBSM Rigid Body Spring Method Li, Vance
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13
Rigid Blocks - Elastic Interface Contact in RBSM
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14
Discrete Cracking of Masonry Walls (Alfaiate, de
Almeida)
FE with Multi-Surface Plasticity for Interface
?-?u Fracture Energy Control Masonry Units
Elastic
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Molecular Dynamics MD
potential
Basic Verlet Algorithm
Loup Verlet

Very small
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Leap Frog Verlet Algorithm
Velocity Verlet Algorithm
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Repulsion
Lennard-Jones Potential
Attraction
John Edward Lennard-Jones Johannes Diderik van
der Waals Fritz Wolfgang London
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Modified Potentials
L-J
Hard Ball
Shifted L-J
Molecular Dynamics (MD) Nodes moved into the
direction of cumulative interaction force.
Displacements proportional to the magnitude of
the force Monte-Carlo (MC) Number of nodes
fixed. Nodes are moved with random selection of
both the direction and the length of the
displacement along the direction Molecular
Dynamics - Monte-Carlo (MDMC) Number of nodes is
fixed. Nodes are moved into the direction of
cumulative interaction force. The displacements
along the direction are selected randomly Grand
Cannonical Monte-Carlo (GCMC) Molecular Dynamics
- Grand Canonical Monte-Carlo (MDGCMC)
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19
Explicit DEM Rigid Bodies -2nd Order Eqns
(Cundall, Soft Ball)
Contact forces caused by particles position
(overlap)
Event-by-event and Momentum exchange
methodologies different integration for the
ballistic part in between collisions
For dense configurations these lead to an
effective solution locking or inelastic
collapse, manifested in critically slow
simulations.
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Explicit Combined DEM/FEM Discrete Finite
Elements, Ghaboussi 1984 Combined
FEM/DEM Munjiza et al 1995
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Explicit DEM Predictor Corrector Format
Contact at predicted configuration, corrective
acceleration
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Granular Silo Flow Patterns Selective Time
Stepping
Easier to to capture with predictor-corrector
format
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Contact Surface LOCAL Frame
(c)
(a)
(b)
Definition of the contact plane a unique
definition for the corner to edge case (a), the
edge to edge case (b) and an ambiguous situation
for the corner-to-corner (c) contact problem
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Various Algorithmic Ideas Exist e.g. Contact
Plane (Feng et al)
Hertz
Contact forces caused by particles position
(overlap)
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SDEM (Egholm)
Smoothing kernels, centered at contact points
Stress based discrete element method
q normalised distance between the contact point
and the particle center
Particle Strain Rate
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Contact force from average traction
Other constitutive laws possible, e.g. plasticity
(stress update)
Note that the estimate of the particle
deformation is based on similar ideas to
interpolation in SPH (Smooth Particle
Hydrodynamics). Particle strain rate in SDEM
constructed from relative velocities of its
contact points Downscaling of macroscopic laws,
rather than upscaling of interface laws
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27
Discontinuous Deformation Analysis DDA
(Gen Hua Shi)
Lowest order DDA, constant strain (stress) field
per block of an arbitrary shape
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Higher Order DDA
Second order DDA, linearly varying stress field
per block of an arbitrary shape
DDA ? Numerical Manifold Method Fracturing of
Blocks (PU XFEM)
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Contact Constraints in DDA
Penalty Format
Lagrange Multipliers
Augmented Lagrangian
Mohr Coulomb with Tension Cut-Off
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Time Stepping Scheme DDA
Shi (DDA) Newmark
Implicit unconditionally stable scheme
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Stability
Numerical Damping
HIGH
Spectral Radius and High Algorithmic Damping for
the DDA Time integration scheme
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DDA
MAYA animation software Contact detection,
collision rules, energy dissipation
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Four Hinges Arch Bridge Failure
Bridgemill Arch Bridge DDA model
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Bridgemill Arch DDA predictions
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HYDRO-DDA Flow through Fractured Media
Fractured Mudrock Seals
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HYDRO-DDA interface

• HYDRO (fixed mesh)

• HYDRO-DDA Interface

DDA
Pressure
Equivalent porosity
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Fixed fluid mesh
Finite Element Mesh Densification along
discontinuities
DDA Mesh Configuration of discontinuities in
solid domain
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Permeability mapping

• Simple permeability mapping,
• directly proportional to
• equivalent porosity
• The essential behaviour can
• often be expressed
• using the simple
• Kozeny-Carman Relation
• (through-going pipes)

Equivalent porosity
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Staggered Approach between Solids Fluid Domains

• DDDDA parallels

Fluid flow calculation
Transform pressures into Nodal forces Fw(n)
DDA mesh
Permeability mapping
Fluid boundary conditions
New DDA calculation
Transform pressures into Nodal forces Fw(n1)
New permeability mapping
New fluid flow calculation
New DDA calculation
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Model Problem - SHEAR Changes in Flow Pattern
though Fractured Seal Layer as a Result of
Blocks Movement and Deformation (Changes in
Mechanical Boundary Conditions)
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Effective Permeability for Problems A and B
Steady State Fluid Fluxes and Effective
Permeability for Different Overpressures
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Concepts in Non Smooth Contact Dynamics (NSCD)
and Applications Jumps in velocities is 2nd
order equation of motion appropriate?
Non Smooth Mechanics accounting roughly for
the real behaviour, discarding details, either
because there is insufficient data for this to be
meaningful or consideration of details would not
bring about significant change, hence should be
ignored (Moreau, Jean)
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Non smooth treatment and regularised treatment of
normal contact (linear and nonlinear penalty
term) and non smooth and regularised treatment of
frictional contact
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NSCD Non Smooth Contact Dynamics Method (Moreau,
Jean)
Signorini Coulomb Problem Nonpenetrability
Average Contact Force
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45
Inelastic Shock Law - Volume Exclusion in
1D Signorini Condition
(Brendel, Unger, Wolf)
Global to Local Frame of Reference
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Volume Exclusion in 1D Signorini Condition
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NSCD - Bouncing Ball Newton Law (e0.7)
Discontinuous Velocity
Velocity
Time
Contact Force
Time
Average Force during Impact Step
Position
Time
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NSCD Elementary 2 Rigid Bodies Problem Volume
Exclusion
Global and local frame of reference
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NSCD Global and Local Frame
Relative velocity and forces at contact point
Global (centroid) and Local (contact)
frame
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Equations of motion transformation from global to
local frame
power conjugacy between velocities and forces
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Time Integration, transformation from global to
local frame
Global
Local
Generalised Mass Inverse
Contact Force
Free Velocity (without interaction forces)
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Signorini_Coulomb Problem in Local Frame 1/3
(1) Check what happens to the gap without
interaction
Gap positive, no contact force
Gap negative, Contact Sticking or Sliding
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Signorini_Coulomb Problem in Local Frame 2/3
no tangential relative velocity
(2) Sticking Condition
free velocity
Contact Force computed as a result of its effect
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Signorini_Coulomb Problem in Local Frame 3/3
(3) Sliding, if Sticking Condition does not hold
normal component remains the same as with
sticking tangential
component changes
colinear with
with opposite sign
Again, contact Force computed as a result of its
effect
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NSCD Pseudo Rigid Bodies (Cohen/Murnaghan)
Constrained Continuum Constant deformation
gradient per body (recall low order
DDA, restricted to small displacements)
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NSCD Pseudo Rigid Bodies Algorithm
(1) For each body define intermediate
configuration
(2) Contact Detection (Define Contact Set) (3)
Loop over Overlaps within the contact set
iterate
Gauss-Seidel
(3.1) compute free velocities
(3.2) solve Signorini-Coulomb problem for
iterative contact forces
(3.3) check
(Other contact forces temporarily frozen during
iteration)
(4) For each body evaluate the
velocity at the end of the step
(5) Define position at the end of the step

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Local Contact Frame based on an Intersection
Volume
Normal with the smallest variance defines the
local frame Computation of intersection I done
efficiently for polyhedral geometries
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NSCD - illustration of the concept
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Adhesion/Cohesion
L. Brendel, T. Unger, D. E. Wolf
Extensions of Signorinis graph to include
Adhesion/cohesion. Maximal attractive force FC
at zero distance only and within finite range
dC. Energy criterion similar to cohesive crack
model
Graphs describing rolling friction (left) and
torsion friction (right) in contact dynamics
Local problem becomes more involved
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General Multibody Problems
Generalised Mass Inverse
Gauss Seidel

• Global Iteration Loop
• Iterate locally at a contact, keeping
  neighbour forces frozen at previous state
• Move to another contact, update status

Random Sweep
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NSCD Pseudo Rigid Body Issues re Computational
Robustness
H between 12 component space of generalised
velocities u and 3 component space of local
velocities U
Generalised Mass Inverse
Singularity of W
Non smoothness of Signorini-Coulomb
ECCOMAS MJS - New Computational Challenges in
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62

• Efficient Large Scale Solvers for Multibody
  Contacts
• Variational Inequality converted into Nonlinear
  Complementarity Problem
• Non Linear Gauss Seidel, PSOR, Conjugate
  Projected Gradient, Iterative Splits into Normal
  and Frictional Contact, Primal-Dual Active Set,
  Newton-like
• Sources of Convergence
• Difficulties
• potential singularity of Z, due to the
  configuration of contact points
• non smoothness
• lack of normality with Coulomb Law

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Application Context 1 - Failure of Masonry Arches
and Walls
Edinburgh Waverley Station
Need for Benchmark Problems
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Couplet/Heyman Minimum Thickness Arch NSCD
Benchmark Problem

Classical Assumptions - No Tension, Infinite
Friction, Rigid Blocks Failure mode based purely
on geometry arguments t/a 0.098 lt 0.10598, arch
unstable under self weight
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Departure from Couplet/Heyman Interface Conditions
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Couplet/Heyman Minimum Thickness Arch NSCD
Benchmark Problem
Location of Contact Points - Block Subdivision
with Thin Edge Layers
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Couplet/Heyman Minimum Thickness Arch NSCD
Benchmark Problem
unstable
unstable
stable
stable
Kinetic energy histories for µ 0.311, four
different ratios h/r
Kinetic energy histories for µ 0.4, four
different ratios h/r
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Couplet/Heyman Minimum Thickness Arch NSCD
Benchmark Problem
Departure from Classical Assumptions Extension to
Finite Friction
h/r 0.15 and µ 0.35
h/r 0.1 and µ 0.5
h/r 0.25 and µ 0.25
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NSCD study of the Couplet-Heyman Minimum Arch
Thickness Problem for a Range of Friction and
Cohesion Values
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Arch Failure Modes
h/r 0.1 and µ 0.5
h/r 0.15 and µ 0.35
h/r 0.25 and µ 0.25
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Application Context 2 - AGR Graphite Core Bricks
Safety Cases Core Integrity, Irradiation
Damage, Seismic Events, Cracking, Lowering of
Control Rods etc
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Industrial Relevance Advanced Gas-Cooled Reactor
AGR Graphite Core
Graphite core (moderator). Fuel channels (tubes
containing uranium) run vertically through the
core. Graphite core slows down the neutrons
released by the fuel to sustain the chain
reaction.
Graphite core also contains channels for boron
steel control rods. These can be raised and
lowered to control the reactor power by absorbing
neutrons and stopping them splitting atoms.
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Graphite Core Complex Brick Geometry
AGR Core 2000-3000 blocks
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1/8 Scale Rig Plastic Bricks
Box-kite Rig Perspex Aluminium
1/4 Scale Rig Aluminium Bricks
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NSCD - Seismic Excitation of AGR Graphite Core
Bricks
Direct contact
Direct Shear Rocking
Full Scale Model - Rigid Bodies Links (No
explicit Friction)
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NSCD Example Dynamic Excitation of AGR Graphite
Core Bricks
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Physical and Computational Models
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Graphite core boxkite assembly with
pre-cracked bricks scenarios NSCD Simulation
and Comparison with Slow Mode I (Separation) and
Mode II (Shear) Locking Experiment
Mode II (Exp)
Mode I (Exp)
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G-Core Boxkite Separation Simulation
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GCore Boxkite Shear Simulation
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AGR Core 2000-3000 blocks
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NSCD Pseudo Rigid Bodies Representation
(Koziara)
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Friction Coefficient 0.0
Friction Coefficient 0.3
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Collapse of Masonry Stack under Sinusoidal Base
Motion
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Similarities and Differences in Various Frameworks
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Range of Possibilities for Modelling Deformability
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Is the whole equal to the sum of its
parts? Simulation of discontinua allows
imagination to thrive
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