energies Article Application of the Mohr-Coulomb Yield Criterion for Rocks with Multiple Joint Sets Using Fast Lagrangian Analysis of Continua 2D (FLAC2D) Software Lifu Chang 1,2,* ID and Heinz Konietzky 1 1 Institut für Geotechnik, Technische Universität Bergakademie Freiberg, Freiberg 09599, Germany; [email protected] 2 School of Mechanics & Civil Engineering, China University of Mining & Technology, Beijing 100083, China * Correspondence: [email protected]; Tel.: +86-189-1102-2674 Received: 6 February 2018; Accepted: 6 March 2018; Published: 9 March 2018 Abstract: The anisotropic behavior of a rock mass with persistent and planar joint sets is mainly governed by the geometrical and mechanical characteristics of the joints. The aim of the study is to develop a continuum-based approach for simulation of multi jointed geomaterials. Within the continuum methods, the discontinuities are regarded as smeared cracks in an implicit manner and all the joint parameters are incorporated into the equivalent constitutive equations. A new equivalent continuum model, called multi-joint model, is developed for jointed rock masses which may contain up to three arbitrary persistent joint sets. The Mohr-Coulomb yield criterion is used to check failure of the intact rock and the joints. The proposed model has solved the issue of multiple plasticity surfaces involved in this approach combined with multiple failure mechanisms. The multi-joint model was implemented into Fast Lagrangian Analysis of Continua software (FLAC) and is verified against the strength anisotropy behavior of jointed rock. A case study considering a circular excavation under uniform and non-uniform in-situ stresses is used to illustrate the practical application. The multi-joint model is compared with the ubiquitous joint model. Keywords: jointed rock masses; multi-joint constitutive model; strength anisotropy; tunnel excavation; numerical simulation 1. Introduction The behavior of a jointed rock (within this paper this term includes also jointed rock masses) is anisotropic, non-linear and stress path dependent due to the presence of discontinuities. Compared to intact rock, jointed rock generally exhibit reduced stiffness, higher permeability and lower strength. Various analytical, numerical and empirical methods are suggested in order to take into account the influence of discontinuities on the mechanical behavior of jointed rock [1–6]. In the past several decades, a lot of lab tests with samples containing a single or multiple joint sets under confined and unconfined conditions have been implemented to investigate the joint orientation effect [7–9]. With the ongoing development of computer based simulation techniques, more sophisticated models have been developed in recent years and it is possible to incorporate more aspects into the simulations [10–13]. Two numerical techniques are common in rock mechanics: continuum-based methods and discontinuum based methods. Discontinuum methods consider the rock mass as an assemblage of rigid or deformable blocks connected along discontinuities [14], and investigate the microscopic mechanisms in granular material and crack development in rocks [3,15]. Boon et al. [16] performed 2D discrete element analyses to identify potential failure mechanisms around a circular unsupported tunnel in a rock mass intersected by three independent joint sets. In continuum methods, the discontinuities are considered as smeared planes of weakness (joints) incorporated in each zone. The equivalent Energies 2018, 11, 614; doi:10.3390/en11030614 www.mdpi.com/journal/energies Energies 2018, 11, x FOR PEER REVIEW 2 of 23 tunnelEnergies in2018 a, 11rock, 614 mass intersected by three independent joint sets. In continuum methods,2 ofthe 23 discontinuities are considered as smeared planes of weakness (joints) incorporated in each zone. The equivalentcontinuum continuum method contains method three contains parts: three the equivalent parts: the continuumequivalent compliancecontinuum partcompliance (incremental part (incrementalelastic law), theelastic yield law), criterion the yield and criterion the plastic an correctionsd the plastic part. corrections part. WillWill [ [1717]] implemented implemented a a multi multi-surface-surface plasticity plasticity model model into into a a finite finite element element code code with with internal internal iterationsiterations in in case case that that several several plasticity plasticity surfaces surfaces are are touched touched at at the the same same time. time. Zhang Zhang [ [1818]] developed developed furtherfurther the the Goodman’s Goodman’s joint joint element element model model and and derived derived the the equivalent equivalent method method for for rock rock masses masses with with multimulti-set-set joints joints in in a a global global coordinate coordinate system. system. Rihai Rihai [19 [19,20,20] ]proposed proposed a a method method which which takes takes into into accountaccount the the influence influence of of micro micro moments moments on on the the behavior behavior of of the the equivalent equivalent continuum continuum based based on on the the CosseratCosserat theory. HurleyHurley etet al.al. [ 21[21]] presented presented a numericala numerical method method which which implemented implemented the the method method into intoGEODYN-L GEODYN code.-L code. This This method method uses uses a series a series of stress of stress and strain-rateand strain- updaterate update algorithms algorithms to rule tojoints rule jointsclosure, closure, slip, and slip, stress and distributionstress distribution inside the inside computational the computational cells which cells contain which multiple contain embedded multiple embeddedjoints. A model joints. with A model multi-sets with ofmulti ubiquitous-sets of jointsubiquitous with anjoints extended with an Barton extended empirical Barton formula empirical was formulaproposed was by proposed Wang [22] by and Wang pre- and[22] post-peakand pre- and deformation post-peak characteristics deformation characteristics of the rock mass of werethe rock also massconsidered. were also Many considered. models haveMany been models formulated have been to simulateformulated the to effect simulate of joint the angle effect on of the joint strength angle onand the deformability strength and of deformability rock masses, butof rock only masses, a few consider but only several a few jointconsider sets several having differentjoint sets strengthhaving differentparameters strength and their parameters effects on and failure their characteristics.effects on failure characteristics. ThisThis paper paper presents presents a a new new equivalent equivalent continuum continuum constitutive constitutive model model named named multi multi-joint-joint model model toto simulate simulate the the behavior behavior of ofjointed jointed rock. rock. This This model model is suited is suited for jointed for jointed rock containing rock containing up to three up to arbitrarythree arbitrary joint sets joint with sets corresponding with corresponding spatial spatialdistribution. distribution. The equivalent The equivalent compliance compliance matrix matrixof the rockof the mass rock is mass establis is establishedhed and the and Mohr the Mohr-Coulomb-Coulomb yield yield criterion criterion is used is used to tocheck check the the failure failure characteristicscharacteristics of thethe intactintact rock rock and and the the joints. joints. A circular A circular tunnel tunnel subjected subjected to uniform to uniform and non-uniform and non- uniformin-situ stresses in-situ isstresses used to is illustrateused to illustrate the application the application of this model. of this model. 2.2. Methodo Methodologylogy for for Equivalent Equivalent Continuum Continuum Multi Multi-Joint-Joint Model Model 2.1.2.1. Ubiquitous Ubiquitous Joint Joint and and Related Related Approaches Approaches TheThe term term ‘ubiquitous ‘ubiquitous joint’ joint’ used used by by Goodman Goodman [ [2323]] implies implies that that the the joint joint sets sets may may occur occur at at any any pointpoint in in the the rock rock mass mass without without a a fixed fixed location. location. The The joints joints are are embedded embedded i inn a a Mohr Mohr-Coulomb-Coulomb solid solid (Figure(Figure 11aa).). FigureFigure 1. 1. SchematicSchematic for for isotropic isotropic and and anisotropic anisotropic models: models: (a) ubiquitous (a) ubiquitous joint model, joint model, (b) multi (b )joint multi model, joint (model,c) anisotropic (c) anisotropic model with model transversely with transversely isotropic isotropicrock matrix rock and matrix representative and representative joint orientation joint orientation.. Failure may occur in either the rock matrix or along the joints, or both, according to the stress Failure may occur in either the rock matrix or along the joints, or both, according to the stress state, the orientation of the joints and the parameters of matrix and joints [24]. More advanced models state, the orientation of the joints and the parameters of matrix and joints [24]. More advanced models based on ubiquitous joint or strain-harding/softening ubiquitous joint (subiquitous) were developed based on ubiquitous joint or strain-harding/softening ubiquitous joint (subiquitous) were developed in recent years [13,22,25–27] as shown in Figure 1b,c. These approaches consider aspects like stiffness Energies 2018, 11, 614 3 of 23 Energies 2018, 11, x FOR PEER REVIEW 3 of 23 in recent years [13,22,25–27]