Acta Geotechnica

, Volume 13, Issue 2, pp 283–302 | Cite as

A discrete numerical method for brittle rocks using mathematical programming

  • J. Meng
  • J. Huang
  • C. Yao
  • D. Sheng
Research Paper


A computational formulation of discrete simulations of damage and failure in brittle rocks using mathematical programming methods is proposed. The variational formulations are developed in two and three dimensions. These formulations naturally lead to second-order cone programs and can conveniently be solved using off-the-shelf mathematical programming solvers. Pure static formulations are derived so that no artificial damping parameters are required. The rock is represented by rigid blocks, with interfaces between blocks modelled by zero-thickness springs based on the rigid-body–spring network method. A modified Mohr–Coulomb failure criterion is proposed to model the failure of the interfaces. When the interface’ strength limits are reached, a microscopic crack forms and its strength is irreversibly lost. The microscopic elastic properties of the springs are related to the observed elastic behaviour of rocks with the developed empirical equations. The program is first validated with three simple tests. Then, numerical uniaxial and biaxial compression tests and the Brazilian tests are conducted. Furthermore, the proposed approach is employed to study the rock crack propagation and coalescence using cracked Brazilian disc test. The results are in good agreements with reported experimental data, which shows its potential in modelling mechanical behaviour of brittle rocks.


Discrete element method Mathematical programming Rigid-body–spring network Rock failure Second-order cone programming 

List of symbols

la and lb

Dimensions of the rectangle domain

\(A^{I}\) and \(\varvec{A}\)

Area of the interface I and array containing all interfaces’ area, respectively


Micro-parameter, the cohesion of the interface

C1 and C2

Correction factors

\(\varvec{C}_{n}\), \(\varvec{C}_{t}\) and \(\varvec{C}_{\varphi }\)

\({\text{diag}}\left( {1/k_{n}^{1} , \ldots ,1/k_{n}^{N} } \right),\;{\text{diag}}\left( {1/k_{t}^{1} , \ldots ,1/k_{t}^{N} } \right)\;{\text{and}}\;{\text{diag}}\left( {1/k_{\varphi }^{1} , \ldots ,1/k_{\varphi }^{N} } \right)\), respectively


Local displacements in the normal, tangential and rotational sense, \({\mathbf{d}}^{\text{T}} = (\delta_{n} ,\delta_{t} ,\varphi )\)


Diagonal matrix containing springs’ stiffnesses, i.e. \(\varvec{D} = {\text{diag}}(k_{n} ,k_{t} ,k_{\varphi } )\)


Elastic module

\(\varvec{f}_{\text{ext}}^{i}\) and \(\varvec{f}_{\text{ext}}\)

ith block’s external force vector and corresponding matrix for all blocks

\(f_{n}^{I}\) and \(f_{t}^{I}\)

Tensile limit and tangential limit of the interface I, respectively

\(\varvec{f}_{n}^{{}}\) and \(\varvec{f}_{t}^{{}}\)

Arrays containing tensile and tangential strength limits for all interfaces


Array containing the reaction forces, i.e. \({\mathbf{F}}^{\text{T}} = [p ,q , { }\tau ]\))

\(F_{n}\) and \(F_{t}\)

Applied normal force and tangential force, respectively

\(k_{n}\), \(k_{t}\) and \(k_{\varphi }\)

Normal, tangential and rotational spring stiffness, respectively

ln and lt

Normal and tangential length of the interface


Minimum distance between any two random points

\(\it {m}_{\text{ext}}^{i}\)

ith block’s external moment


Applied torque

\(\varvec{n}_{0}^{I}\) and \(\hat{\varvec{n}}_{0}^{I}\)

Unit normal and tangential vectors at the interface I, respectively

\(\varvec{N}_{0}\), \(\hat{\varvec{N}}_{0}\) and \(\bar{\varvec{N}}_{0}\)

Mapping from local level to global level

O, Oi and Oj

Midpoint of the interface for the initial configuration, block i and block j, respectively

pI and p

Normal reaction force at the interface I and the array containing normal reaction forces at all interfaces, respectively

qI and q

Tangential reaction force at the interface I and the array containing tangential reaction forces at all interfaces, respectively

\(R_{i}^{p}\) and \(R_{i}^{q}\)

Moment arms of the reaction forces p I and q I for the block i

\(\varvec{R}^{\text{p}}\) and \(\varvec{R}^{\text{q}}\)

Matrixes containing all moment arms, i.e. \(R_{i}^{p}\) and \(R_{i}^{q}\)

s1, s2 and s3

Slack variables

xi and x

ith block’s position and matrix containing all blocks’ positions

\(\alpha^{i}\) and α

ith block’s angular position and matrix containing all blocks’ angular positions


Pre-existing crack inclination angle

\(\bar{\varvec{N}}_{ 0}^{\text{T}} \Delta\varvec{\alpha}\), \(\Delta \varvec{u}_{n}\) and \(\Delta \varvec{u}_{t}\)

Relative angular displacement, normal displacements and tangential displacements, respectively

λ1, λ2 and λ3

Arrays containing Lagrange multipliers

\(\varepsilon_{1}\) and \(\varepsilon_{3}\)



Internal friction coefficient, \(\mu = \tan \phi\)


Poisson’s ratio

\(\sigma_{1}\), \(\sigma_{3}\), \(\sigma_{n}\) and \(\sigma_{s}\)



Micro-parameter, critical normal stress of the interface


Micro-parameter, tensile strength of the interface

τI and τ

Torque at the interface I and the array containing torques at all interfaces, respectively


Micro-parameter, friction angle of the interface

\({\varvec{\uppsi}}\) and \(\varvec{R}_{{\varvec{\Omega}}} \left( {\varvec{\uppsi}} \right)\)

Random seeds and their reflections by the boundary of Ω


Specimen domain



The authors wish to acknowledge the support from the Australian Research Council Centre of Excellence for Geotechnical Science and Engineering and the National Natural Science Foundation of China (Project No. 51679117). The first author also wishes to acknowledge the help from Erling D. Andersen from MOSEK and support from the China Scholarship Council.


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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.ARC Centre of Excellence for Geotechnical Science and Engineering, Faculty of Engineering and Built EnvironmentThe University of NewcastleCallaghanAustralia
  2. 2.School of Civil Engineering and ArchitectureNanchang UniversityNanchangChina

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