A new approach to the DEM, with applications to brittle, jointed rock

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A new approach to the DEM, with applications to
brittle, jointed rock
Peter A Cundall Itasca Consulting Group, USA
Lecture, ALERT School Aussois
9 October 2008
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Overview
In this talk, we propose an alternative
formulation for the DEM simulations of blocky
systems. The complete scheme has not been
implemented, although some components exist and
have been tested.
The objective is to increase the calculation
efficiency of a 3D DEM model of interacting
angular blocks, at the expense of accuracy. There
is an added advantage (over, say 3DEC), in that
the blocks may fracture.
Thus, the subject discussed here is speculative,
and may or may not lead to a viable method (e.g.,
one in which the accuracy is acceptable).
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Topics covered here

• Bonded assembly of particles
• Smooth Joint Model
• Review of 3DEC code true polyhedral blocks
• Equivalence between true blocky system and SJM
  overlay.
• Lattice scheme for improving efficiency
• Proposed scheme

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Bonded particle assembly for brittle rock
The distinct element method (DEM) may be used to
model brittle rock, using an assembly of bonded
particles. Each bond-break represents a
micro-crack, and a contiguous chain of
micro-cracks represents a macro-crack.
We use circular particles (in 2D and 3D), with
bonding at each contact, using the code PFC. It
is possible to relate the behaviour of such a
bonded assembly to classical fracture mechanics
concepts
Recently, the inclusion of joints (or
pre-existing discontinuities) has been added to
the PFC bonded-particle representation of brittle
rock. This is the Smooth Joint Model (SJM).
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The Smooth-Joint Model (SJM)
A joint in a PFC bonded assembly consists of
modified properties of contacts whose 2
host-particle centroids span the desired joint
plane.
joint plane
To avoid the bumpy-road effect, a new smooth
joint model is employed that allows continuous
slip
SJM in 3D -
Illustration of smooth joint mechanics
Note that the SJM also represents normal joint
opening
Example with several sliding joints
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We have performed an extensive series of
validation comparisons with laboratory
experiments, in 2D (Wong et al, 2001) and 3D
(Germanovich and Dyskin, 2000)
Initial crack (joint)
Laboratory
Numerical
R.H.C. Wong, K.T. Chau, C.A. Tang, P. Lin.
Analysis of crack coalescence in rock-like
materials containing three flaws Part I
experimental approach. Int. J. Rock Mech. Min.
Sci. 38 (2001).
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SJM in 2D
SJM in 3D
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True polygonal polyhedral block DEM models
For many years, DEM codes UDEC and 3DEC have been
available to model angular blocks of rock. Both
codes are computationally intensive, using
detailed interaction logic e.g., edge-to-edge,
edge-to-corner, face-to-corner, etc). We review
briefly the formulation for 3DEC, and then
propose a simpler alternative. Note that 3DEC
and UDEC do not include block fracture, although
each block may contain a nonlinear constitutive
model (e.g., Mohr Coulomb), which accounts for
smeared Example of a 3DEC simulation
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Summary of equations used in 3DEC for the contact
and motion of arbitrary polyhedra (from Cundall,
Lemos Hart papers, 1988).
Relative contact velocity -
shear
normal
In all DEM codes (and especially 3DEC) there is
also a great deal of housekeeping logic to detect
and manage contacts efficiently.
(similarly for block B)
similarly for rotation
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Block assemblies with bonded spheres SJM
We may form angular blocks with assemblies of
spheres, separated by smooth joint planes. This
is an approximate representation of
polyhedra. To illustrate the approach, we
compare the same model of grain structure
simulated with both UDEC and PFC2D. Note that 2D
is used for clarity. An identical approach
operates in 3D, using 3DEC and PFC3D,
respectively.
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An alternative to UDEC (or 3DEC)
bonds
joint planes
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UDEC model
Stress-measurement patch
block with different modulus
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Stress-measurement patch
PFC2D model
smooth joint plane
bonds
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Force distribution in PFC assembly when loaded
axially (in Y direction)
(blue compression red tension)
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Comparison of stress in a circular patch UDEC
and PFC2D
(Note the measurement schemes within the
patches are not identical)
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Lattice model version of packed particle assembly
In a further simplification, we replace balls and
contacts by nodes and springs, where a node is a
point mass. The advantage of this formulation is
a great increase in efficiency. For example, in
3D, the computational speed is increased by a
factor of 10 and the memory requirement decreased
by a factor of 7. The node/spring representation
is called the lattice model, and as an example
it is used in the code BLO-UP which simulates
the fragmentation of a rock mass due to blasting.
The Smooth Joint Model (already discussed) may be
used to overlay joints on the lattice model.
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For example, we make 2 joint continuous sets
Each dot is a spring that is intersected by a
joint plane
Each such joint element obeys the SJM formulation
(ie, angle of joint, not the spring, is respected)
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Thus, we may create a system composed of
polyhedral blocks with -

• A lattice network to represent the interior
  material of each block.
• The Smooth Joint Model (SJM) to represent the
  boundary of each block. The boundary description
  is stored as a separate data structure that is
  tied to (rotated with) the block material.
• Interaction between blocks determined by the mean
  of the SJM planes of the two contacting blocks.

Note that simple point-to-point interaction is
used for contact detection, eliminating the
time-consuming polyhedral interaction logic of
3DEC.
Further, interior springs may break, resulting in
possible splitting of blocks. The new crack is
assigned an SJM normal vector.
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Visualization of proposed scheme in two dimensions
Lattice nodes
mean SJM plane
(slip normal closure resolved in SJM direction)
Block boundary polygon
interaction lattice spring
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The detection and interaction error is related
to the resolution (mean spacing between lattice
nodes). Thus, the error may be reduced by making
the resolution finer.
The lattice scheme acts as a meshless method
resolution may be improved locally at any time,
if required. If this is done, the block boundary
geometry remains the same.
Lattice springs are calibrated to reproduce the
required elastic and strength properties of the
block material. (This is fundamentally different
from, say, the finite element method, which is
based on a volumetric formulation).
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Conclusions
The proposed scheme for simulating assemblies of
polyhedral blocks promises to be very efficient,
at the expense of less accuracy in contact
conditions, compared to, say, 3DEC (which uses
exact polyhedral contact laws). However, the
accuracy is related to the lattice resolution,
which may be refined as necessary (even
locally). Splitting of blocks is an integral
part of the scheme. The location and angle of
such splits is not constrained by a grid. (New
SJMs may be placed arbitrarily).