energies Article Numerical Study on the Elastic Deformation and the Stress Field of Brittle Rocks under Harmonic Dynamic Load Siqi Li 1 , Shenglei Tian 1, Wei Li 1,* , Xin Ling 1, Marcin Kapitaniak 2 and Vahid Vaziri 2 1 Institute of Petroleum Engineering, Northeast Petroleum University, Daqing 163318, China; [email protected] (S.L.); [email protected] (S.T.); [email protected] (X.L.) 2 Centre for Applied Dynamics Research, School of Engineering, University of Aberdeen, Aberdeen AB24 3UE, UK; [email protected] (M.K.); [email protected] (V.V.) * Correspondence: [email protected] Received: 30 December 2019; Accepted: 11 February 2020; Published: 15 February 2020 Abstract: In order to study the deformation displacement and the stress field of brittle rocks under harmonic dynamic loading, a series of systematic numerical simulations are conducted in this paper. A 3D uniaxial compression simulation is carried out to calibrate and determine the property parameters of sandstone and a model of the cylindrical indenter intruding the rock is proposed to analyze the process of elastic deformation. Four main parameters are taken into account, namely the position on the rock, the frequency and the amplitude of dynamic load, the type of indenter and the loading conditions (static and static-dynamic). Based on the analysis undertaken, it can be concluded that both of the deformation displacement and stress field of the rock change in a harmonic manner under the static-dynamic loads. The frequency and the amplitude of harmonic dynamic load determine the period and the magnitude of the rock response, respectively. In addition, the existence of harmonic dynamic load can aggravate the fatigue damage of the rock and allow a reduction in static load. Our investigations confirm that the static-dynamic loads are more conducive to rock fracture than static load. Keywords: deformation displacement; stress field; harmonic dynamic load; FE simulation; brittle rock 1. Introduction In order to improve the crushing efficiency of hard rocks or some unconventional resources [1], some auxiliary rock-breaking tools, such as axial hydraulic impactor [2], and few emerging rock-breaking methods, such as Resonance Enhanced Drilling (RED) [3,4] and ultrasonic drilling [5,6], have been proposed. There is one thing in common in terms of the working principle that they all apply an additional periodic dynamic load to the rock based on static load to achieve Rate of Penetration (ROP) improvement. Moreover, compared to other kinds of periodic dynamic loads, the dynamic load in the harmonic form which changes in a cosine (or sine) law with time has been proven to be the most effective [7,8]. In short, the dynamic load plays an increasingly important role in rock fragmentation. The intrusion of indenter into the rock is considered an ideal method to analyze the mechanical problems during excavation, which has been widely used in various engineering fields including oil drilling, mining, tunneling and so on [9–12]. The interaction between the rock and the indenter has been studied mainly focusing on two aspects: deformation and stress field [13–15], and fracture and fragmentation of the rock [16–19]. Both of them are of great significance to the mechanism research, optimization of operation parameters and process of mechanical tools for rock fragmentation. At present, a large number of studies on the indenter intrusion process have been proposed for different geometries of indenters such as spherical, pyramidal, conical and flat-ended cylindrical Energies 2020, 13, 851; doi:10.3390/en13040851 www.mdpi.com/journal/energies Energies 2020, 13, x 2 of 16 fragmentation of the rock [16–19]. Both of them are of great significance to the mechanism research, optimization of operation parameters and process of mechanical tools for rock fragmentation. At present, a large number of studies on the indenter intrusion process have been proposed for Energies 2020, 13, 851 2 of 16 different geometries of indenters such as spherical, pyramidal, conical and flat-ended cylindrical indenters through theoretical, numerical and experimental methods [20–23]. Elastic mechanics is the mostindenters common through theoretical theoretical, approach numerical for studying and experimental the deformation methods and [20 stress–23]. Elasticfield of mechanics rocks [24–27]. is the Themost finite common element theoretical method approach(FEM), the for boundary studying element the deformation method (BEM) and stress and fieldthe discrete of rocks element [24–27]. methodThe finite (DEM) element are three method widely (FEM), used the methods boundary to elementsimulate method the indenter (BEM) intrusion and the process discrete [28–32]. element Manymethod experimental (DEM) are tests three have widely been used carried methods out to to verify simulate thethe theoretical indenter results intrusion and process explore [ 28more–32]. complexMany experimental issues [33–36]. tests Based have on been these carried studies, out toa basic verify understanding the theoretical of results the indenter and explore intrusion more processcomplex has issues been [33 developed.–36]. Based on these studies, a basic understanding of the indenter intrusion process has beenHowever, developed. most existing research on the indenter intrusion process considers only static load. A few worksHowever, are conducted most existing under research dynamic on theload, indenter and th intrusione impact processvelocity considersis the main only factor static to load. be consideredA few works [37–39]. are conducted There are under few dynamicstudies on load, the andmechanical the impact behavior velocity of is rocks the main under factor indenter to be intrusionconsidered with [37 –the39 ].harmonic There are dynamic few studies load. on Besides, the mechanical rock often behavior suffers of rocksbrittle under failure indenter under intrusionexternal forcewith due the harmonicto its internal dynamic defects load. [40–42]. Besides, When rock the often stress su ffiners which brittle the failure brittle under rock is external subjected force exceeds due to theits internalfracture defectsstrength [40 of–42 the]. When rock, thethe stress rock inwill which be instantaneously the brittle rock is damaged subjected without exceeds theexhibiting fracture significantstrength of plastic the rock, strain, the which rock will means be instantaneously that the elastic damageddeformation without is the exhibitingmain stage significant before the plastic rock fracturestrain,. whichAs the meansmechanical that theproperties elastic deformationof the brittle isrock the at main the elastic stage beforestage are the the rock key fracture. to its fracture, As the inmechanical order to investigate properties the of effect the brittle of harmonic rock at dyna the elasticmic load stage on the are mechanical the key to behavior its fracture, of rock in order under to indenterinvestigate intrusion, the effect the of harmonicelastic deformation dynamic load and on thethe mechanical stress field behavior of the of brittle rock under rock indenterunder static-dynamicintrusion, the elasticload are deformation analyzed through and the FE stress numerical field of thesimulation. brittle rock under static-dynamic load are analyzedThe paper through is organized FE numerical as follows. simulation. In Section 2, a 3D uniaxial compression simulation is carried out basedThe paperon the isexperimental organized as data follows. to calibrate In Section the 2rock, a 3D model. uniaxial Then, compression the 3D FE simulationmodel is established is carried andout the based mesh on thesize experimental sensitivity analysis data to calibrateis conducted the rock to ensure model. the Then, accuracy the 3D of FE results model in is establishedSection 3. Finally,and the the mesh effects size sensitivityof loading analysisconditions, is conducted parameters to of ensure the dynamic the accuracy load ofand results the type in Section of indenter3. Finally, on thethe deformation effects of loading displacement conditions, and parameters stress field of of the the dynamic brittle rock load are and analyzed the type ofand indenter discussed on thein Sectiondeformation 4. Finally, displacement the conclusions and stress of this field study of the are brittle presented rock arein Section analyzed 5. and discussed in Section4. Finally, the conclusions of this study are presented in Section5. 2. The 3D Uniaxial Compression Simulation 2. The 3D Uniaxial Compression Simulation The lithology used in the simulation, through this study, is sandstone. The 3D uniaxial compressionThe lithology simulation used inis thefirstly simulation, carried throughout to calibrate this study, and is determine sandstone. the The property 3D uniaxial parameters compression of thesimulation sandstone is firstly adopted carried in outthe to numerical calibrate and simulation determine according the property to parametersthe uniaxial of thecompression sandstone experiment.adopted in theThe numerical FE model simulation consists of accordinga cylinder to in the the uniaxial size of Ø54 compression mm × 116 experiment. mm and two The rigid FE modeldisks consists of a cylinder in the size of Ø54 mm 116 mm and two rigid disks in the size of Ø108 mm in the size of Ø108 mm × 0.5 mm, as shown ×in Figure 1a. A fully fixed constraint is applied to the× lower0.5 mm, disk as and shown a compressive in Figure1a. displacement A fully fixed constraintload of 5 ismm applied is applied to the to lower the upper disk and