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

Peter A Cundall

Itasca Consulting Group, USA

Lecture, ALERT SchoolAussois

9 October 2008

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.

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).

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.

Overview

1. Bonded assembly of particles

2. Smooth Joint Model

3. Review of 3DEC code – true polyhedral blocks

4. Equivalence between true blocky system and SJM overlay.

5. Lattice scheme for improving efficiency

6. Proposed scheme

Topics covered here

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).

Bonded particle assembly for brittle rock

The Smooth-Joint Model (SJM)

Illustration of smooth joint mechanics:

Example with several sliding joints:

A “joint” in a PFC bonded assembly consists of modified properties of contacts whose 2 host-particle centroids span the desired joint plane.

Note that the SJM also represents normal joint opening

To avoid the “bumpy-road” effect, a new smooth joint model is employed that allows continuous slip

SJM in 3D -

joint plane

We have performed an extensive series of validation comparisons with laboratory experiments, in 2D (Wong et al, 2001) and 3D (Germanovich and Dyskin, 2000)

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).

Laboratory Numerical

Initial crack (joint)

SJM in 2D

SJM in 3D

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:

True polygonal & polyhedral block DEM models

Summary of equations used in 3DEC for the contact and motion of arbitrary polyhedra (from Cundall, Lemos & Hart papers, 1988).

Relative contact velocity -

Law of motion for translation -

where

(damping force)

shear

normal

} (similarly for block B)

CA

B

In all DEM codes (and especially 3DEC) there is also a great deal of housekeeping logic to detect and manage contacts efficiently.

… similarly for rotation

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.

Block assemblies with bonded spheres & SJM

An alternative to UDEC (or 3DEC) …

bonds

joint planes

“block” with different modulus

UDEC model Stress-measurement patch

PFC2D modelStress-measurement patch

bonds

smooth joint plane

Force distribution in PFC assembly when loaded axially (in Y direction)

(blue = compression red = tension)

Comparison of stress in a circular patch – UDEC and PFC2D

(Note – the measurement schemes within the patches are not identical)

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.

Lattice model version of packed particle assembly

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)

Thus, we may create a system composed of polyhedral blocks with -

1. A lattice network to represent the interior material of each block.

2. 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.

3. 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.

mean SJM plane

(slip & normal closure resolved in SJM direction)

Lattice nodes

interaction lattice spring

Visualization of proposed scheme in two dimensions

Block boundary polygon

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).

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).

Conclusions