A discrete numerical method for brittle rocks using mathematical programming

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.

[1]  Baolin. Wang A block-spring model for jointed rocks. , 1992 .

[2]  Ping Cao,et al.  Mechanical Behavior of Brittle Rock-Like Specimens with Pre-existing Fissures Under Uniaxial Loading: Experimental Studies and Particle Mechanics Approach , 2016, Rock Mechanics and Rock Engineering.

[3]  Peter Eberhard,et al.  A discrete element model to describe failure of strong rock in uniaxial compression , 2011 .

[4]  G. Maier,et al.  On multiplicity of solutions in quasi-brittle fracture computations , 1997 .

[5]  John A. Hudson,et al.  Numerical methods in rock mechanics , 2002 .

[6]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[7]  P. A. Cundall,et al.  A DISCONTINUOUS FUTURE FOR NUMERICAL MODELLING IN GEOMECHANICS , 2001 .

[8]  Dwayne D. Tannant,et al.  Numerical analysis of the stability of heavily jointed rock slopes using PFC2D , 2003 .

[9]  Chun’an Tang,et al.  Micromechanical Model for Simulating the Fracture Process of Rock , 2004 .

[10]  Tadahiko Kawai,et al.  NEW ELEMENT MODELS IN DISCRETE STRUCTURAL ANALYSIS , 1977 .

[11]  Mettupalayam V. Sivaselvan,et al.  Complementarity framework for non‐linear dynamic analysis of skeletal structures with softening plastic hinges , 2011 .

[12]  Chung Yee Kwok,et al.  Discrete element method modeling of inherently anisotropic rocks under uniaxial compression loading , 2016 .

[13]  John E. Bolander,et al.  Voronoi-based discretizations for fracture analysis of particulate materials , 2011 .

[14]  Ming Cai,et al.  Numerical Simulation Of The Brazilian Test And The Tensile Strength Of Anisotropic Rocks And Rocks With Pre-Existing Cracks , 2004 .

[15]  J. Bolander,et al.  Fracture analyses using spring networks with random geometry , 1998 .

[16]  Navid Bahrani,et al.  Distinct element method simulation of an analogue for a highly interlocked, non-persistently jointed rockmass , 2014 .

[17]  Scott W. Sloan,et al.  Lagrangian modelling of large deformation induced by progressive failure of sensitive clays with elastoviscoplasticity , 2017 .

[18]  Chung Yee Kwok,et al.  Micromechanical analysis of the failure process of brittle rock , 2015 .

[19]  Kohei Nagai,et al.  Mesoscopic Simulation of Failure of Mortar and Concrete by 2D RBSM , 2004 .

[20]  G. Paulino,et al.  PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab , 2012 .

[21]  P. Cundall,et al.  A bonded-particle model for rock , 2004 .

[22]  Jian-Fu Shao,et al.  Modelling of induced anisotropic damage in granites , 1999 .

[23]  Tadahiko Kawai,et al.  New discrete models and their application to seismic response analysis of structures , 1978 .

[24]  Hehua Zhu,et al.  Centroid sliding pyramid method for removability and stability analysis of fractured hard rock , 2017 .

[25]  Minoru Kunieda,et al.  Analysis of crack propagation due to rebar corrosion using RBSM , 2011 .

[26]  Qinghui Jiang,et al.  Numerical simulation of damage and failure in brittle rocks using a modified rigid block spring method , 2015 .

[27]  Richard E. Goodman,et al.  Block theory and its application to rock engineering , 1985 .

[28]  Tadahiko Kawai,et al.  The rigid bodies—spring models and their applications to three-dimensional crack problems , 1992 .

[29]  P. Cundall,et al.  A discrete numerical model for granular assemblies , 1979 .

[30]  Conrad Childs,et al.  The orientation and dilatancy of shear bands in a bonded particle model for rock , 2013 .

[31]  Franz Aurenhammer,et al.  Voronoi Diagrams , 2000, Handbook of Computational Geometry.

[32]  Kourosh Shahriar,et al.  Experimental and numerical study of crack propagation and coalescence in pre-cracked rock-like disks , 2014 .

[33]  Vahab Sarfarazi,et al.  Mixed mode crack propagation in low brittle rock-like materials , 2013, Arabian Journal of Geosciences.

[34]  Giovanni Grasselli,et al.  A review of discrete modeling techniques for fracturing processes in discontinuous rock masses , 2014 .

[35]  Andrei V. Lyamin,et al.  Granular contact dynamics using mathematical programming methods , 2012 .

[36]  Yun-Teng Wang,et al.  Numerical simulation of crack propagation and coalescence in pre-cracked rock-like Brazilian disks using the non-ordinary state-based peridynamics , 2016 .

[37]  N. Takeuchi,et al.  An Explicit Dynamic Method of Rigid Bodies-Spring Model , 2015 .

[38]  Chuangbing Zhou,et al.  A discrete approach for modeling damage and failure in anisotropic cohesive brittle materials , 2016 .

[39]  Tayfun Babadagli,et al.  A grain based modeling study of fracture branching during compression tests in granites , 2015 .

[40]  Lianyang Zhang,et al.  A new contact model to improve the simulated ratio of unconfined compressive strength to tensile strength in bonded particle models , 2014 .

[41]  A. Zang,et al.  A grain based modeling study of mineralogical factors affecting strength, elastic behavior and micro fracture development during compression tests in granites , 2015 .

[42]  C. Martin,et al.  A clumped particle model for rock , 2007 .

[43]  Steven Fortune,et al.  A sweepline algorithm for Voronoi diagrams , 1986, SCG '86.

[44]  F. Donze,et al.  Rock slope stability analysis using photogrammetric data and DFN–DEM modelling , 2015 .

[45]  Stefano Berton,et al.  Crack band model of fracture in irregular lattices , 2006 .

[46]  Yoshiaki Okui,et al.  A micromechanical model for brittle failure of rock and its relation to crack growth observed in triaxial compression tests of granite , 2006 .

[47]  Eurípedes do Amaral Vargas,et al.  Application of the discrete element method for modeling of rock crack propagation and coalescence in the step-path failure mechanism , 2013 .