Acta Geotechnica

, Volume 13, Issue 2, pp 283–302

# A discrete numerical method for brittle rocks using mathematical programming

Research Paper

## Abstract

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.

## Keywords

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

c

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

d

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

D

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

E

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

F

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

lmin

Minimum distance between any two random points

$$\it {m}_{\text{ext}}^{i}$$

ith block’s external moment

$$M_{a}$$

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

β

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 $$\varepsilon_{3}$$

Strain

$$\mu$$

Internal friction coefficient, $$\mu = \tan \phi$$

ν

Poisson’s ratio

$$\sigma_{1}$$, $$\sigma_{3}$$, $$\sigma_{n}$$ and $$\sigma_{s}$$

Stress

$$\sigma_{\text{cr}}$$

Micro-parameter, critical normal stress of the interface

$$\sigma_{t}$$

Micro-parameter, tensile strength of the interface

τI and τ

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

ϕ

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

Ω

Specimen domain

## Notes

### Acknowledgements

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.

## References

1. 1.
Asahina D, Bolander J (2011) Voronoi-based discretizations for fracture analysis of particulate materials. Powder Technol 213(1):92–99
2. 2.
Aurenhammer F, Klein R (2000) Voronoi diagrams. Hand book of computational geometry. Elsevier, Amsterodam, pp 201–290
3. 3.
Bahrani N, Kaiser PK, Valley B (2014) Distinct element method simulation of an analogue for a highly interlocked, non-persistently jointed rockmass. Int J Rock Mech Min Sci 71:117–130Google Scholar
4. 4.
Berton S, Bolander JE (2006) Crack band model of fracture in irregular lattices. Comput Methods Appl Mech Eng 195(52):7172–7181
5. 5.
Bolander JE, Saito S (1998) Fracture analyses using spring networks with random geometry. Eng Fract Mech 61(5):569–591
6. 6.
Bolzon G, Maier G, Tin-Loi F (1997) On multiplicity of solutions in quasi-brittle fracture computations. Comput Mech 19(6):511–516
7. 7.
Bonilla-Sierra V, Scholtes L, Donzé F, Elmouttie M (2015) Rock slope stability analysis using photogrammetric data and DFN–DEM modelling. Acta Geotech 10(4):497–511
8. 8.
Cai M, Kaiser P (2014) Numerical simulation of the Brazilian test and the tensile strength of anisotropic rocks and rocks with pre-existing cracks. Int J Rock Mech Min Sci 41:478–483
9. 9.
Camones LAM, Vargas EdA, de Figueiredo RP, Velloso RQ (2013) Application of the discrete element method for modeling of rock crack propagation and coalescence in the step-path failure mechanism. Eng Geol 153:80–94
10. 10.
Cao RH, Lin H, Pu CZ, Ou K (2016) Mechanical behavior of brittle rock-like specimens with pre-existing fissures under uniaxial loading: experimental studies and particle mechanics approach. Rock Mech Rock Eng 49(3):763–783
11. 11.
Cho N, Martin CD, Sego DC (2007) A clumped particle model for rock. Int J Rock Mech Min Sci 44(7):997–1010
12. 12.
Cundall P (2001) A discontinuous future for numerical modelling in geomechanics? Proc Inst Civil Eng Geotech Eng 149(1):41–47
13. 13.
Cundall PA, Strack OD (1979) A discrete numerical model for granular assemblies. Geotechnique 29(1):47–65
14. 14.
Ding X, Zhang L (2014) A new contact model to improve the simulated ratio of unconfined compressive strength to tensile strength in bonded particle models. Int J Rock Mech Min Sci 69:111–119Google Scholar
15. 15.
Duan K, Kwok CY, Tham LG (2015) Micromechanical analysis of the failure process of brittle rock. Int J Numer Anal Meth Geomech 39(6):618–634
16. 16.
Duan K, Kwok C, Pierce M (2015) Discrete element method modeling of inherently anisotropic rocks under uniaxial compression loading. Int J Numer Anal Methods Geomech 40(8):1150–1183
17. 17.
El Ghaoui L (2017) Optimization models and applications. http://livebooklabs.com/keeppies/c5a5868ce26b8125. Accessed 2 Jan 2017
18. 18.
Ergenzinger C, Seifried R, Eberhard P (2010) A discrete element model to describe failure of strong rock in uniaxial compression. Granular Matter 13(4):341–364
19. 19.
Fortune S (1987) A sweepline algorithm for Voronoi diagrams. Algorithmica 2(1–4):153–174
20. 20.
Ghazvinian A, Nejati HR, Sarfarazi V, Hadei MR (2013) Mixed mode crack propagation in low brittle rock-like materials. Arab J Geosci 6(11):4435–4444
21. 21.
Golshani A, Okui Y, Oda M, Takemura T (2006) A micromechanical model for brittle failure of rock and its relation to crack growth observed in triaxial compression tests of granite. Mech Mater 38(4):287–303
22. 22.
Goodman RE, Shi GH (1985) Block theory and its application to rock engineering. Prentice-Hall, Englewood CliffsGoogle Scholar
23. 23.
Guide MUs (1998) The mathworks. Inc, Natick, MA vol 5, p 333Google Scholar
24. 24.
Haeri H, Shahriar K, Marji MF, Moarefvand P (2014) Experimental and numerical study of crack propagation and coalescence in pre-cracked rock-like disks. Int J Rock Mech Min Sci 67:20–28Google Scholar
25. 25.
Hofmann H, Babadagli T, Yoon JS, Zang A, Zimmermann G (2015) A grain based modeling study of mineralogical factors affecting strength, elastic behavior and micro fracture development during compression tests in granites. Eng Fract Mech 147:261–275
26. 26.
Hofmann H, Babadagli T, Zimmermann G (2015) A grain based modeling study of fracture branching during compression tests in granites. Int J Rock Mech Min Sci 77:152–162Google Scholar
27. 27.
Jing L, Hudson J (2002) Numerical methods in rock mechanics. Int J Rock Mech Min Sci 39(4):409–427
28. 28.
Kawai T (1977) New element models in discrete structural analysis. J Soc Naval Arch Jpn 1977(141):174–180
29. 29.
Kawai T (1978) New discrete models and their application to seismic response analysis of structures. Nucl Eng Des 48(1):207–229
30. 30.
Kikuchi A, Kawai T, Suzuki N (1992) The rigid bodies—spring models and their applications to three-dimensional crack problems. Comput Struct 44(1):469–480
31. 31.
Krabbenhoft K, Lyamin AV, Huang J, Vicente da Silva M (2012) Granular contact dynamics using mathematical programming methods. Comput Geotech 43:165–176
32. 32.
Lisjak A, Grasselli G (2014) A review of discrete modeling techniques for fracturing processes in discontinuous rock masses. J Rock Mech Geotech Eng 6(4):301–314
33. 33.
Mosek A (2015) The MOSEK optimization toolbox for MATLAB manual. Version 71 (Revision 28)Google Scholar
34. 34.
Nagai K, Sato Y, Ueda T (2004) Mesoscopic simulation of failure of mortar and concrete by 2D RBSM. J Adv Concr Technol 2(3):359–374
35. 35.
Potyondy D (2010) A grain-based model for rock: approaching the true microstructure. In: Proceedings of Bergmekanikk i Norden, pp 225–234Google Scholar
36. 36.
Potyondy DO, Cundall PA (2004) A bonded-particle model for rock. Int J Rock Mech Min Sci 41(8):1329–1364
37. 37.
Schöpfer MPJ, Childs C (2013) The orientation and dilatancy of shear bands in a bonded particle model for rock. Int J Rock Mech Min Sci 57:75–88Google Scholar
38. 38.
Shao J, Hoxha D, Bart M, Homand F, Duveau G, Souley M et al (1999) Modelling of induced anisotropic damage in granites. Int J Rock Mech Min Sci 36(8):1001–1012
39. 39.
Shi G (1988) Discontinuous deformation analysis—a new numerical model for the statics and dynamics of block systems Ph.D. thesis, University of California, Berkeley, USAGoogle Scholar
40. 40.
Sivaselvan M (2011) Complementarity framework for non-linear dynamic analysis of skeletal structures with softening plastic hinges. Int J Numer Meth Eng 86(2):182–223
41. 41.
Sturm JF (1999) Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim Methods Softw 11(1–4):625–653
42. 42.
Talischi C, Paulino GH, Pereira A, Menezes IF (2012) PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab. Struct Multidiscip Optim 45(3):309–328
43. 43.
Tran KK, Nakamura H, Kawamura K, Kunieda M (2011) Analysis of crack propagation due to rebar corrosion using RBSM. Cem Concr Compos 33(9):906–917
44. 44.
Wang B (1992) A block-spring model for jointed rocks. University of Ottawa, OttawaGoogle Scholar
45. 45.
Wang C, Tannant DD, Lilly PA (2003) Numerical analysis of the stability of heavily jointed rock slopes using PFC2D. Int J Rock Mech Min Sci 40(3):415–424
46. 46.
Wu W, Zhuang X, Zhu H, Liu X, Ma G (2017) Centroid sliding pyramid method for removability and stability analysis of fractured hard rock. Acta Geotech 12(3):627–644
47. 47.
Yagi T, Takeuchi N (2015) An explicit dynamic method of rigid bodies-spring model. Int J Comput Methods 12(04):1540014
48. 48.
Yao C, Jiang QH, Shao JF (2015) Numerical simulation of damage and failure in brittle rocks using a modified rigid block spring method. Comput Geotech 2015(64):48–60
49. 49.
Yao C, Jiang Q, Shao J, Zhou C (2016) A discrete approach for modeling damage and failure in anisotropic cohesive brittle materials. Eng Fract Mech 155:102–118
50. 50.
Zhang X, Sheng D, Sloan SW, Bleyer J (2017) Lagrangian modelling of large deformation induced by progressive failure of sensitive clays with elastoviscoplasticity. Int J Numer Methods Eng. doi:
51. 51.
Zhou X-P, Wang Y-T (2016) Numerical simulation of crack propagation and coalescence in pre-cracked rock-like Brazilian disks using the non-ordinary state-based peridynamics. Int J Rock Mech Min Sci 89:235–249Google Scholar
52. 52.
Zhu W, Tang C (2004) Micromechanical model for simulating the fracture process of rock. Rock Mech Rock Eng 37(1):25–56