Numerical Study on the Elastic Deformation and the Stress Field of Brittle Rocks Under Harmonic Dynamic Load

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 disk a compressivewhich only hasdisplacement a z-direction load degree of 5 mm of freedom. is applied Finally, to the upper the sand diskstone which undergoes only has athez-direction same shear degree fracture of freedom. as in theFinally, experiment, the sandstone as shown undergoes in Figure the 1b. same shear fracture as in the experiment, as shown in Figure1b.

FigureFigure 1. 1. TheThe 3D modelingmodeling ofof uniaxial uniaxial compression compression test test of sandstone:of sandstone: (a) The(a) ﬁniteThe finite element element (FE) model (FE) modelconsists consists of a cylinder of a cylinder and twoand thintwo thin disks. disks. The Th highlightede highlighted arrow arrow represents represents that that the the upper upper disk disk is subjected to a downward displacement load of 5 mm. (b) Contour plot of uniaxial compression damage of sandstone. Energies 2020, 13, x 3 of 16

Energies 2020is subjected, 13, 851 to a downward displacement load of 5 mm. (b) Contour plot of uniaxial compression3 of 16 damage of sandstone.

TheThe stress–strain stress–strain curve curve of of sandstone sandstone from from the the numerical numerical simulation simulation with with a gooda good match match with with the the experimentalexperimental results results is is obtained obtained (as (as shown shown in in Figure Figure2), 2), through through a seriesa series of of parameter parameter calibrations. calibrations. Thus,Thus, the the property property parameters parameters of of sandstone sandstone adopted adopted in in the the simulation simulation are are determined, determined, as as shown shown in in TableTable1. 1.

70 Experiment 60 Simulation 50 40 30

Stress/MPa 20 10 0 0 0.005 0.01 0.015 Strain

FigureFigure 2. 2.Stress–strain Stress–strain curves curves of of sandstone sandstone obtained obtained from from experiment experiment and and numerical numerical simulation. simulation.

Table 1. Property parameters of sandstone in the simulation. Table 1. Property parameters of sandstone in the simulation. Material Parameter Value Parameter Value Material Parameter Value Parameter Value 3 9 Density (t/mm )3 2.7 10 −9 Angle of Friction (◦) 50 Density (t/mm ) 2.7× ×− 10 Angle of Friction (°) 50 Elastic Modulus (MPa) 7228 Dilation Angle ( ) 8 Elastic Modulus (MPa) 7228 Dilation Angle◦ (°) 8 Sandstone Poisson’s Ratio 0.3 Flow Stress Ratio 0.8 Sandstone Failure DisplacementPoisson’s Ratio (mm) 0.02 0.3 Yield Flow StressStress (MPa) Ratio 67 0.8 FailureFracture Displacement Strain (mm) 5 10 0.024 Yield Stress (MPa) 67 × − Fracture Strain 5 × 10−4 In addition, it is observed through the experimental and numerical results that the sandstone has obviousIn addition, brittleness. it is observed When the through applied the stress experi reachesmental the and fracture numerical strength results of the that formation, the sandstone it is damagedhas obvious directly brittleness. without plasticWhen deformation.the applied stress Therefore, reaches to improvethe fracture the calculationstrength of e ffitheciency, formation, only the it is elasticdamaged parameters directly ofwithout sandstone plastic are deformation. set and its plastic Theref andore, to damaged improve properties the calculation are ignored efficiency, in theonly followingthe elastic simulations. parameters of sandstone are set and its plastic and damaged properties are ignored in the following simulations. 3. Numerical Model of Indenter Intrusion 3. Numerical Model of Indenter Intrusion 3.1. The 3D Finite Element Modeling 3.1. The 3D Finite Element Modeling The deformation displacement and the stress field of the rock under static and static-dynamic loads areThe analyzed deformation through displacement the process and of the the cylindrical stress fiel indenterd of the intrudingrock under the static rock. and The static-dynamic 3D FE model isloads shown are in analyzed Figure3a, through where the the tool process steel indenterof the cylin is adrical cylinder indenter with aintruding diameter the of 5rock. mm, The and 3D the FE rockmodel is a is cube shown of 100in Figure mm 3a,100 where mm the100 tool mm. steel The indenter rock is is mucha cylinder larger with than a diameter the indenter of 5 mm, so that and × × thethe influence rock is a of cube its boundaries of 100 mm on× 100 the mm deformation × 100 mm. displacement The rock is much and thelarger stress than field the can indenter be ignored. so that Sincethe influence the geometric of its shape boundaries and force on the distribution deformation of the displacement model are symmetric, and the stress a quarter field can of thebe ignored. model isSince adopted the asgeometric shown inshape Figure and3b, force in order distribution to reduce of the computational model are symmetric, time. Besides, a quarter the cylindricalof the model indenteris adopted is regarded as shown as in a rigidFigure body, 3b, in on order which to a re referenceduce computational point is set totime. bind Besides, with it, the limiting cylindrical its freedomindenter only is regarded in the Z-direction. as a rigid Meanwhile,body, on which fully a fixed reference constraints point is are set applied to bind to with the rockit, limiting bottom, its andfreedom symmetrical only in constraints the Z-direction. which Meanwhile, are perpendicular fully fixed to X andconstraintsY axes are are applied applied to to the the A rock and Bbottom, side ofand the symmetrical rock, respectively. constraints which are perpendicular to X and Y axes are applied to the A and B side of the rock, respectively.

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Figure 3. TheThe 3D 3D finite ﬁnite element element model: ( (aa)) The The whole whole model. model. ( (bb)) A A quarter quarter of of the the model. model.

3.2.3.2. Parameter Parameter Settings Settings TheThe property parametersparameters of of the the rock rock and and the the indenter indenter used used in the in simulations the simulations are shown are shown in Table in2, Tableassuming 2, assuming the indenter the indenter as a rigid as body, a rigid the body, rock onlythe rock with only its elastic with its stage elastic in this stage numerical in this numerical study. study. Table 2. Property parameters of the rock and the indenter in the simulation. Table 2. Property parameters of the rock and the indenter in the simulation. Material Parameter Value Parameter Value Material DensityParameter (t/mm 3) 2.7Value10 9 Elastic Parameter Modulus (MPa) 7228 Value Sandstone × − DensityPoisson’s (t/mm Ratio3) 2.7 × 0.3 10−9 Elastic Modulus (MPa) 7228 Sandstone Poisson’sDensity (t/ mmRatio3) 7.85 0.3 10 9 Elastic Modulus (MPa) 2.06 10 5 Indenter × − × − DensityPoisson’s (t/mm Ratio3) 7.85 × 0.3 10−9 Elastic Modulus (MPa) 2.06 × 10−5 Indenter Poisson’s Ratio 0.3 In the simulation, the static load of 1 kN is applied and the influence of dynamic load on the deformationIn the simulation, displacement the andstatic the load stress of field1 kN of is rock applied are analyzed and the byinfluence changing of itsdynamic frequency load between on the deformation100 Hz and 150 displacement Hz and its amplitudeand the stress between field 0.5 of kN rock and are 1 kN. analyzed by changing its frequency between 100 Hz and 150 Hz and its amplitude between 0.5 kN and 1 kN. 3.3. Mesh Size Sensitivity Analysis 3.3. MeshAs the Size results Sensitivity of the Analysis FE model are highly sensitive to mesh size, mesh size sensitivity analysis is conducted to minimize the influence of the grid size on the results. The model was run with various As the results of the FE model are highly sensitive to mesh size, mesh size sensitivity analysis is numbers of elements and the deformation displacements of the same point on the rock surface under conducted to minimize the influence of the grid size on the results. The model was run with various static load are compared to each other. numbers of elements and the deformation displacements of the same point on the rock surface under As displayed in Figure4, when the number of elements exceeds 400,000, the deformation static load are compared to each other. displacement of rock starts to converge and tends to a constant value. At this point, the corresponding As displayed in Figure 4, when the number of elements exceeds 400,000, the deformation mesh size is 0.25 mm. Meanwhile, based on the interaction area between the indenter and the rock displacement of rock starts to converge and tends to a constant value. At this point, the obtained from the simulation results, in order to maximize the computational efficiency, the FE model corresponding mesh size is 0.25 mm. Meanwhile, based on the interaction area between the indenter is meshed, as shown in Figure5. The grid is 0.2 mm in the area below and adjacent to the indenter, and the rock obtained from the simulation results, in order to maximize the computational 1 mm in the next area and 10 mm in the area away from the indenter. efficiency, the FE model is meshed, as shown in Figure 5. The grid is 0.2 mm in the area below and adjacent to the indenter, 1 mm in the next area and 10 mm in the area away from the indenter.

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0.030.03 0.03 0.0290.029 0.029 0.0280.028 0.028 0.0270.027 0.027 0.0260.026 0.026

Deformation Displacement [mm] 0.025 Deformation Displacement [mm] 0.025 00 200000 200000200,000 400000 400000400,000 600000600,000 600000 800000800,000 800000 Deformation Displacement [mm] 0.025 200,000 400,000 600,000 800,000 ElementElement Number Number 0 200000200,000 400000400,000 600,000 600000800,000 800000 Element Number

FigureFigure 4. 4.Deformation Deformation displacements displacements from from the the FE FE model model for for different different element element numbers. numbers. Figure 4. Deformation displacements from the FE model for diﬀerent element numbers. Figure 4. Deformation displacements from the FE model for different element numbers.

FigureFigureFigure 5. 5. 5.MeshingMeshing Meshing elements elements elements of of ofthe the the rock: rock: rock: (a (a )( )aPart Part) Part view. view. view. ( (bb )() bTop Top) Top view. view. view. 4. Results and AnalysisFigure 5. Meshing elements of the rock: (a) Part view. (b) Top view. 4. 4.Results Results and and Analysis Analysis 4.1.4. Results Inﬂuence and of Analysis the Position on the Rock 4.1.4.1. Influence Influence of ofthe the Position Position on on the the Rock Rock 4.1. InfluenceAA static static loadof load the of Position of 1 kN1 kN and on and the a harmonica Rock harmonic dynamic dynamic load load with with an amplitudean amplitude of 1 of kN 1 kN and and a frequency a frequency of 100 HzA static are applied load of to 1 the kN rock, and anda harmonic the deformation dynamic displacements load with an amplitude of the rock of under 1 kN the and static-dynamic a frequency ofof 100 100A staticHz Hz are loadare applied appliedof 1 kN to and tothe the a rock,harmonic rock, and and dynamicthe the deformation deformation load with displacements andisplacements amplitude ofof of 1the kN the rockand rock aunder frequency under the the loadsstatic-dynamic are compared. loads In are Figure compared.6a, the contact In Figure area 6a, between the contact the cylindrical area between indenter the cylindrical and the rock indenter has static-dynamicbeenof 100 circled Hz are with loadsapplied a dotted are tocompared. red the line. rock, Four In and Figure points the 6a, ondeformation the the contact surface displacementsarea of the between rock have theof been cylindricalthe selected,rock under indenter among the andstatic-dynamicand the the rock rock has has loads been been arecircled circled compared. with with a dotted aIn dotted Figure red red 6a,line. line. the Four Fourcontact points points area on on thebetween the surface surface the of cylindricalof the the rock rock have haveindenter been been whichselected, points among A and which B are points below A the and indenter, B are below point the C isindenter, near the point indenter C is andnear pointthe indenter D is away and from point selected,theand indenter.the rock among has which been circled points withA and a dottedB are below red line. the Fourindenter, points point on theC is surface near the of indenterthe rock andhave point been Dselected, Dis isaway away among from from the which the indenter. indenter. points A and B are below the indenter, point C is near the indenter and point D is away from the indenter. 0.25 Point A Point B 0.25 Point A Point B 0.2 0.250.2 Point A Point B

0.15 0.150.2 [mm]

[mm] 0.1 0.150.1

[mm] 0.05 0.050.1 Deformation Displcement Deformation Displcement 0 0.050 0 0.02 0.04 0.06 0.08 0.1 Deformation Displcement 0 0.02 0.04 0.06 0.08 0.1 Time [s] 0 Time [s] 0 0.02 0.04 0.06 0.08 0.1 (a) Time(b) [s] (a) (b) (a) (b) FigureFigureFigure 6. 6.Numerical Numerical 6. Numerical results results results showing showing showing deformation deformation deformation displacements displacements displacements of ofdifferent of different diﬀerent positions positions on onon the thethe surface surface surface of of of the rock for cylindrical indenter: (a) Top view of the rock surface. (b) The relationships of deformation theFigure rockthe 6. for rockNumerical cylindrical for cylindrical results indenter: indenter:showing (a) Top (deformationa) Topview view of the ofdisplacements the rock rock surface. surface. of ( bdifferent ()b The) The relationships relationshipspositions on of ofthe deformation deformation surface of displacement versus time of points A, B, C and D. displacementthe rockdisplacement for cylindrical versus versus time indenter: of time points of points( aA,) TopB, C A, viewand B, C D. of and the D. rock surface. (b) The relationships of deformation displacement versus time of points A, B, C and D.

Energies 2020, 13, 851 6 of 16

Figure6b depicts the relationship of deformation displacement versus time at di fferent positions on the rock surface. It can be seen from Figure6b that under the combined action of static load and harmonic dynamic load, the deformation displacement of the rock presents a stable harmonic variation after the initial unstable fluctuation (transient response). With the increase of the distance from the center of the indenter, the deformation displacement of the rock decreases. It should be noted that in the contact area between the indenter and the rock, the deformation displacements of points A and B are very close, and the amplitude of the deformation displacement decreases slightly. However, the deformation displacements of point C and D decrease significantly when the area is outside the indenter. Previous research results [23] have given that deformation displacement on the rock surface under static load for a cylindrical indenter can be expressed outside the contact area between the indenter and the rock as   2  r !  4 1 µ qrZ π/2 2 2 Z π/2  −  a 2 a dθ  w =  1 sin θdθ 1 q  (1) πE  − r2 − − r2 2   0 0 1 a sin2 θ − r2 Within the contact area between the indenter and the rock as 2 Z r 4 1 µ qa π/2 r2 w = − 1 sin2 ϕdϕ (2) 2 πE 0 − a where µ and E are the Poisson’s ratio and elastic modulus of the rock, respectively; a is the radius of the indenter, m; r is the distance from the center of the indenter, m; q is the static load per unit area, N/m2. Based on the above-mentioned formulas, it can be also concluded that the deformation displacement on the rock surface under static load decreases with increasing distance from the center of the indenter. Compared with the static load, the static-dynamic loads are equivalent to adding a dynamic load which changes periodically with time. For a certain moment, it can still be regarded as a static load problem with the superposition of two loads. Therefore, Equations (1) and (2) are still applicable to these static-dynamic loads’ problem, just replace the q with q + q(t), where q(t) is the dynamic load per unit area. That is why similar results are obtained from static and static-dynamic loads conditions.

4.2. Static Load vs. Static-Dynamic Loads Static load and static-dynamic loads are applied to the cylindrical indenter separately, both static loads are 1 kN and the dynamic load is a harmonic load with an amplitude of 1 kN and a frequency of 100 Hz. Then, the deformation displacement and the stress field of the rock under two kinds of loading conditions are compared. Figure7 shows the curves of deformation displacement at the same position on the rock surface under two loading conditions. As shown in Figure7, the deformation displacements of the rock under two types of loads are obviously different, which converges to a constant value under static load and varies in a harmonic form under static-dynamic load, respectively. In addition, there is a grey dotted line representing the failure displacement of the rock in the figure as a reference line, of which the value is determined from the uniaxial compression simulation in Section2. Energies 2020, 13, 851 7 of 16 Energies 2020, 13, x 7 of 16

0.05 Static Load Static-Dynamic Loads 0.04

0.03

0.02

0.01 Deformation Displacement [mm] 0 0.05 0.06 0.07 0.08 0.09 0.1 Time [s] FigureFigure 7. Numerical 7. Numerical results results showing showing deformation deformation displacement versus versus time time relationships relationships of the of rockthe rock for cylindricalfor cylindrical indenter indenter under under two two kinds kinds of of loading loading conditions.conditions.

By comparing the three curves in Figure7, it can be seen that at the same position on the rock surface, By comparing the three curves in Figure 7, it can be seen that at the same position on the rock the deformation displacement of the rock under static load does not exceed its failure displacement surface, the deformation displacement of the rock under static load does not exceed its failure which would result in no fracture occurring, while the fracture can occur under static-dynamic loads displacementbecause of itswhich larger would deformation result displacement. in no fracture This alsooccurring, explains whywhile static-dynamic the fracture load can is beneficialoccur under static-dynamicto rock fracture loads from because the perspective of its oflarger deformation. deformation In addition, displacement. we know from This Section also 4.1 explainsthat as the why static-dynamicdistance from load the indenteris beneficial increases, to rock the deformationfracture from displacement the perspective of the rock of deformation. decreases, which In is addition, also we knowapplicable from to staticSection load 4.1 conditions. that as Wethe can distance draw aconclusion from the from indenter the results increases, in this section the deformation that the displacementdeformation of displacementthe rock decreases, of the rock which under is static-dynamicalso applicable loads to static is greater load than conditions. that under We static can load draw a conclusionat the same from position the results of the rock,in this which section means that that the although deformation the area underdisplacement the indenter of maythe fracturerock under static-dynamicunder both kindsloads of is loading greater conditions, than that the under static-dynamic static load loads at canthe worksame on position a wider rangeof the than rock, static which meansload that in the although rock area the which area is away under from the the indenter indenter. may fracture under both kinds of loading conditions,The the 3D static-dynamic stress contour plots loads of can the rockwork under on a twowider loading range conditions than static are load given in in the Figure rock8. area As shown in Figure8, the distribution of the stress field of the rock under static load is stable, and the which is away from the indenter. maximum stress is generated at the boundary of the indenter and a certain depth below the rock. FiveThe representative3D stress contour stress plots contour of the plots rock of the under rock tw ino an loading impact periodconditions under are static-dynamic given in Figure loads 8. As shownare in shown. Figure It 8, can the be distribution seen that the of stress the fieldstress of fi theeld rock of the also rock presents under periodic static changesload is stable, due to and the the maximumexistence stress of harmonic is generated dynamic at the load. boundary When the of direction the indenter of dynamic and a load certain is opposite depth tobelow static load,the rock. Five therepresentative stress field of stress the rock contour gradually plots decreases of the rock and eventuallyin an impact becomes period lower under than static-dynamic that under static loads are shown.load. While It can when be seen the direction that the of stress dynamic field load of the is consistent rock also with presents the static periodic load, the changes stress fielddue ofto the existencerock increasesof harmonic and exceeds dynamic that load. under When static load.the direct In addition,ion of dynamic the response load range is opposite of the stress to fieldstatic of load, the stressthe rock field caused of the by rock static-dynamic gradually loadsdecreases is larger and than eventually that under becomes static load. lower Therefore, than that it alsounder can static load.be While concluded when from the direction the perspective of dynamic of thestress load fieldis consistent that when with the fracturethe static strength load, the of thestress rock field is of rock betweenincreases the and stress exceeds strength that under under static static and load. static-dynamic In addition, loads, the theresponse latter is range more advantageousof the stress field to of rock fracture. the rock caused by static-dynamic loads is larger than that under static load. Therefore, it also can be concluded from the perspective of the stress field that when the fracture strength of the rock is between the stress strength under static and static-dynamic loads, the latter is more advantageous to rock fracture.

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Figure 8. The comparison of 3D Mises stress contour plots under static and static-dynamic loads for Figure 8. The comparison of 3D Mises stress contour plots under static and static-dynamic loads for cylindrical indenter during an impact period. cylindrical indenter duri duringng an impact period.

4.3. Influence Inﬂuence of of the the F Frequencyrequency of Dynamic Load The frequency of dynamic load is one major pa parameterrameter that could influence the deformation displacement andand thethe stress stress field field of of the the rock rock under under the the cylindrical cylindrical indenter. indenter. Therefore, Therefore, a constant a constant static staticload of load 1 kN of and 1 kN a dynamicand a dynamic load of 1load kN withof 1 kN the with frequencies the frequencies of 100 Hz, of 125 100 Hz Hz, and 125 150 Hz Hz, and are 150 applied Hz, areto the applied rock in to the the three rock simulations.in the three simulations. Figure 99 depictsdepicts thethe relationshiprelationship betweenbetween deformationdeformation displacementdisplacement versusversus timetime atat thethe samesame position onon the the rock rock surface surface with with diﬀerent different frequencies frequencies of dynamic of dynamic load. It isload. obvious It is that obvious the variation that the in variationthe frequency in the of dynamicfrequency load of dynamic only changes load the only period changes of rock the deformation period of rock response deformation without changingresponse withoutthe magnitude changing of itsthe deformation magnitude of displacement. its deformation Within displacement. the same time,Within the the higher same thetime, frequency the higher of thedynamic frequency load of is, dynamic the more load the reciprocatingis, the more the times reciprocating of deformation times are.of deformation Based on the are. above Based analysis, on the aboveit can beanalysis, further it found can be that further even found if the that deformation even if the displacement deformation of displacementdisplacement the rock under ofof the static-dynamicthe rockrock underunder static-dynamicloads does not loads reach does its failure not reach level, its high-frequency failure level,l, high-frequencyhigh-frequency dynamic load dynamicdynamic also promotes loadload alsoalso the promotespromotes rock fracture, thethe rockwhich fracture, is due to which the high-frequency is due to the reciprocatinghigh-frequency motion reciprocating will aggravate motion the will fatigue aggravate damage the ofthe fatigue rock. damageActually, of the the selection rock. Actually, of the frequency the selection of dynamic of the frequency load should of alsodynamic take intoload accountshould thealso resonance take into accountfrequency the range resonance of rocks frequency and the ra allowablenge of rocks range and of the practical allowable conditions. range of practical conditions.

0.15 100Hz 125Hz 150Hz

0.12

0.09

0.06

0.03 Deformation Displacement [mm] Deformation Displacement [mm] 0 0 0.02 0.04 0.06 0.08 0.1 Time [s]

Figure 9. NumericalNumerical results results showing showing deformation deformation displaceme displacementnt versus versus time time relationships relationships of of the the rock rock forfor cylindricalcylindrical indenter indenter under under static-dynamic static-dynamic lo loadsloads with didifferentﬀerent frequenciesfrequencies ofof dynamicdynamic load.load.

Energies 2020, 13, 851 9 of 16 Energies 2020, 13, x 9 of 16

FigureFigure 10 showsshows thethe maximummaximum and and minimum minimum stress stress of theof the rock rock under under static-dynamic static-dynamic loads loads with withdifferent different frequencies, frequencies, and compares and compares them with them the with stress the field stress of the field rock of under the rock static under load, respectively.static load, respectively.In Figure 10 a,In itFigure can be 10a, seen it thatcan be the seen maximum that thestress maximum of the stress rock underof the rock static-dynamic under static-dynamic loads with loadsdifferent with frequencies different frequencies is greater than is thatgreater under than static that load, under but static the maximum load, but stressthe maximum is almost thestress same is almostwith the the change same with of the the frequency change of ofthe dynamic frequency load. of dynamic Similarly, load. it can Similarly, be seen it in can Figure be seen 10 bin that Figure the 10bminimum that the stress minimum of the stress rock is of less the than rock that is less under than static that load,under and static its valueload, and does its not value change does much, not changeeither. Therefore,much, either. we Therefore, can conclude we thatcan theconclude variation that of the the variation frequency of ofthe dynamic frequency load of doesdynamic not aloadffect doesthe amplitude not affect of the the amplitude stress field of thethe rock,stress but field static-dynamic of the rock, loads but withstatic-dynamic any frequency loads have with a better any frequencyeffect on the have rock a better fracture effect than on static the rock load fracture only. than static load only.

FigureFigure 10. 10. TheThe comparison comparison of of Mises Mises stress stress contour contour plots plots under under static static and and static-dynamic static-dynamic loads loads with with differentdiﬀerent frequencies frequencies of dynamic load for cylindrical indenter: ( (aa)) MaxMises. MaxMises. ( (bb)) MinMises. MinMises.

NoteNote that if wewe dodo notnot change change the the amplitude amplitude of of load, load, as as expected expected we we do notdo not get anyget any different different level levelof deformation of deformation and stressand stress as the asmodel the model does does not considernot consider fracture. fracture. Nevertheless, Nevertheless, we we can can still still see see the thesignificance significance of the of changethe change in frequency, in frequency, which changeswhich changes the response the response period and period affects and the affects fatigue the life fatigueof the rock. life of the rock.

4.4.4.4. Influence Influence of of the the Am Amplitudeplitude of of Dynamic Dynamic Load Load TheThe amplitude amplitude of of dynamic dynamic load load is is another another main main factor factor affecting affecting the the intrusion intrusion effect effect under under the the static-dynamicstatic-dynamic loads. loads. Therefore, Therefore, a aconstant constant static static load load of of1 kN 1 kN and and dynamic dynamic load load with with a frequency a frequency of 100of 100 Hz Hzare areadopted, adopted, and and only only the theamplitude amplitude of dynamic of dynamic load load is set is setto 0.5 to 0.5kN, kN, 0.75 0.75 kN kNand and 1 kN, 1 kN, to analyzeto analyze the thedeformation deformation displacement displacement and and stre stressss field field of the of the rock rock in this in this section. section. TheThe deformation displacement isis simulatedsimulated for for the the varying varying amplitude amplitude of of dynamic dynamic load load giving giving the theresults results shown shown in Figure in Figure 11. The11. The magnitude magnitude of deformation of deformation displacement displacement of the of rockthe rock increases increases with withthe increase the increase of the amplitudeof the amplitude of dynamic of dynamic load, but load, the frequency but the frequency of fluctuations of fluctuations of the deformation of the deformationdisplacement displacement does not change. does not The change. greater The the amplitudegreater the ofamplitude dynamic of load dynamic is, the load more is, serious the more the seriousdeformation the deformation of the rockis, of which the rock is more is, which conducive is mo tore rock conducive fracture. to However, rock fracture. it is worth However, noting it that is worththe amplitude noting that of dynamicthe amplitude load cannotof dynamic be increased load cannot optionally, be increased its maximum optionally, value its maximum needs to bevalue less needsthan or to equalbe less to than the staticor equal load. to Thisthe static is because load. This if the is dynamic because if load the isdynamic greater load than is the greater static load,than the the staticindenter load, would the beindenter lifted atwould some timebe lifted instead at ofsome intruding time theinstead rock, of which intruding is unfavorable the rock, for which the rock is unfavorablefragmentation. for Inthe addition, rock fragmentation. it can be also In concluded addition that, it can the rockbe also can concluded be broken that even the under rock a smallercan be brokenstatic load even because under thea smaller dynamic static load load would because aggravate the dynamic the deformation load would of the aggravate rock on the basisdeformation of static ofload, the whichrock on implies the basis that of forstatic drilling load, which application, implies a that smaller for drilling weight application, on the bit can a smaller be allowed weight when on thedrilling bit can under be allowed the static-dynamic when drilling loads. under the static-dynamic loads.

Energies 2020, 13, 851 10 of 16 Energies 2020, 13, x 10 of 16 Energies 2020, 13, x 10 of 16

0.15 0.15 0.5kN 0.75kN 1kN 0.5kN 0.75kN 1kN

0.120.12

0.090.09

0.060.06

0.030.03 Deformation Displacement [mm] Deformation Displacement [mm] 0 0 00 0.02 0.02 0.04 0.04 0.06 0.06 0.08 0.1 0.1 TimeTime [s] [s]

FigureFigureFigure 11. 11.Numerical 11.Numerical Numerical results results results showing showing showing deformation deformation deformation displacement displacement displacement versus versusversus time time relationships relationshipsrelationships ofof of thethe the rock rock for cylindrical indenter under static-dynamic loads with different amplitudes of dynamic load. forrock cylindrical for cylindrical indenter indenter under under static-dynamic static-dynamic loads loads with with diff erentdifferent amplitudes amplitudes of dynamicof dynamic load. load. Figure 12 demonstrates 3D contour plots of the maximum and minimum stress of the rock FigureFigure 12 12 demonstrates demonstrates 3D 3D contour contour plots plots of of the the maximum maximum and an minimumd minimum stress stress of theof the rock rock under under static and static-dynamic loads with different amplitudes of dynamic load, respectively. As staticunder and static static-dynamic and static-dynamic loads with loads di ffwitherent differ amplitudesent amplitudes of dynamic of dynamic load, respectively. load, respectively. As shown As in shown in Figure 12a, the maximum stress of the rock increases with the increase of the amplitude of Figureshown 12 ina, Figure the maximum 12a, the maximum stress of the stress rock of increasesthe rock increases with the with increase the increase of the amplitude of the amplitude of dynamic of dynamic load and is higher than that under static load, while in Figure 12b, it can be seen that the load and is higher than that under static load, while in Figure 12b, it can be seen that the minimum dynamicminimum load stress and is of higher the rock than decreases that under as the static amplitude load, whileof the dynamiin Figurec load 12b, increases, it can be andseen is that lower the stressminimumthan of the that stress rock under of decreases thestatic rock load, asdecreases thewhich amplitude means as the that amplitude of the the amplitude dynamic of the dynamiof load dynamic increases,c load load increases, has and an is effect lowerand ison thanlower the that underthanmagnitude staticthat under load, of static whichthe stress load, means field which thatof the means the rock. amplitude Itthat can the be ofalsoamplitude dynamic concluded of load dynamic that has when an load etheffect rockhas on andoes the effect magnitudenot breakon the of themagnitude stressunder field static of ofthe load, the stress rock. adding field It candynamic of the be alsorock. load concluded It and can adjust be also thating concluded whenthe amplitude the rockthat ofwhen does dynamic not the breakrock load does can under make not static break the load, addingunderstress dynamicstatic of theload, rock load adding exceed and dynamic adjustingits fracture load the limit and amplitude an adjustd breaking ofeven the dynamic ifamplitude its mini loadmum of can dynamicstress make decreases. theload stress can make of the the rock exceedstress itsof the fracture rock exceed limit and its fracture break even limit if an itsd minimumbreak even stress if its mini decreases.mum stress decreases.

Figure 12. The comparison of Mises stress contour plots under static and static-dynamic loads with different amplitudes of dynamic load for cylindrical indenter: (a) MaxMises. (b) MinMises. FigureFigure 12. 12.The The comparison comparison ofof Mises stress contour contour plots plots under under static static and and static-dynamic static-dynamic loads loads with with 4.5.different Influence amplitudes of the Type of ofdynamic Indenter load for cylindrical indenter: (a) MaxMises. (b) MinMises. different amplitudes of dynamic load for cylindrical indenter: (a) MaxMises. (b) MinMises. In addition to the cylindrical indenter, the spherical and conical indenters are also common 4.5.4.5. Influence typesInfluence of of ofindenters. the the TypeType ofofTherefore, IndenterIndenter in order to study the influence of the type of indenter on the IndeformationIn addition addition to displacementto the the cylindrical cylindrical and indenter, indenter,stress field the the sphericalof spherical the rock and andunder conical conical static-dynamic indenters indenters are areloads, also also commonthe common same types numerical simulations are carried out by replacing the indenter in sequence. As shown in Figure 13, oftypes indenters. of indenters. Therefore, Therefore, in order in to order study to the study influence the influence of the type of ofthe indenter type of onindenter the deformation on the these three kinds of indenters have a diameter of 5 mm, and a reference point PR is set at the center displacementdeformation anddisplacement stress field and of the stress rock field under of static-dynamic the rock under loads, static-dynamic the same numerical loads, the simulations same of the upper surface of each indenter, which is also the position applied by static-dynamic loads. arenumerical carried outsimulati by replacingons are carried the indenter out by replacing in sequence. the indenter As shown in sequence. in Figure As 13 shown, these in three Figure kinds 13, of indentersthese three have kinds a diameter of indenters of 5 mm,have anda diameter a reference of 5 mm, point and PR a is reference set at the point center PR of is the set upper at the surfacecenter of of the upper surface of each indenter, which is also the position applied by static-dynamic loads. each indenter, which is also the position applied by static-dynamic loads. Besides, in order to satisfy the conditions for the convergence of the three models, the static load applied is 100 N and the harmonic dynamic load is 100 N, with a frequency of 100 Hz. Energies 2020, 13, x 11 of 16 Energies 2020, 13, x 11 of 16

EnergiesBesides,EnergiesBesides,2020, 13in2020, order 851in, 13 order, x to satisfyto satisfy the the conditions conditions for forthe the convergence convergence of theof the three three models, models, the the static11 static of load1611 ofload 16 appliedapplied is 100 is 100 N andN and the the harmonic harmonic dynamic dynamic load load is 100 is 100 N, N,with with a frequency a frequency of 100of 100 Hz. Hz. Besides, in order to satisfy the conditions for the convergence of the three models, the static load applied is 100 N and the harmonic dynamic load is 100 N, with a frequency of 100 Hz.

Figure 13. 3D finite element models of indenters: (a) cylinder; (b) sphere; (c) cone. RP, Fs and Fd FigureFigure 13. 13.3D 3D ﬁnite finite element element models models ofof indenters:indenters: ( a) cylinder; ( (bb) )sphere; sphere; (c ()c )cone. cone. RP, RP, Fs Fs and and Fd Fd Figurerepresent 13. the3D referencefinite element point, models static ofand indenters: dynamic ( aload,) cylinder; respectively. (b) sphere; (c) cone. RP, Fs and Fd representrepresent the the reference reference point, point, static static and and dynamic dynamicload, load, respectively.respectively. represent the reference point, static and dynamic load, respectively. Firstly, in order to analyze the influence of the type of indenter on the deformation of the rock Firstly,Firstly, in in order order to to analyze analyze the the inﬂuence influence ofof thethe typetype of indenter on on the the deformation deformation of ofthe the rock rock Firstly, in order to analyze the influence of the type of indenter on the deformation of the rock underunder static-dynamic static-dynamic loads, loads, the the deformation deformation displace displacementsments of theof the rock rock in thein the horizontal horizontal and and vertical vertical underunder static-dynamic static-dynamic loads, loads, the the deformation deformation displacements displacements of of the the rock rock in inthe the horizontal horizontal and andvertical vertical directionsdirections are are compared, compared, as shownas shown in Figuresin Figures 14 and14 and 15. 15. directionsdirections are compared, are compared, as shownas shown in in Figures Figures 14 14 and and 1515..

0.06 0.06 0.06 Cylinder Sphere Cone Cylinder Sphere Cone 0.05 Cylinder Sphere Cone 0.05 0.05 0.04 0.040.04 0.03 0.030.03 0.02 0.020.02 0.01 0.010.01 0 0 0 Deformation Displacement [mm]

Deformation Displacement [mm] 0 0.02 0.04 0.06 0.08 0.1 Deformation Displacement [mm] 00 0.02 0.02 0.04 0.04 0.06 0.06 0.08 0.08 0.1 0.1 Time [s] TimeTime [s]

FigureFigureFigureFigure 14. 14.Deformation 14. 14.Deformation DeformationDeformation displacement displacement displacement displacement versus versusversus versus time time relationships time rela relationshipstionshipstionships of the ofof rock thetheof rocktheforrock three rockfor for three typesforthree threetypesof types indenters typesof of of underindentersindentersindenters static-dynamic under under under static-dynamic static-dynamic static-dynamic loads. loads. loads. loads.

0.1 0.1 0.1 CylinderCylinder SphereSphere Cone Cone Cylinder Sphere Cone 0.080.08 0.08

0.060.06 0.06 [mm]

0.04 [mm] [mm] 0.04 0.04 0.02 0.02 0.02Deformation Displacement Deformation Displacement

Deformation Displacement 0 0 0 024681012 024681012Distance [mm] 024681012 Distance [mm] Distance [mm] (a) (b) (a) (a) (b) (b) Figure 15. Comparison of deformation displacement of three types of indenters under static-dynamic FigureFigureloads:Figure 15. 15. Comparison(a 15.Comparison) The Comparison selected of of path deformation deformation of indeformation the vertical displacementdisplacement displacementdirection of of the th threeof reerock. th reetypes types (b types) Deformation of of indenters of indenters indenters displacement under under under static-dynamic static-dynamic static-dynamic versus loads:loads:distanceloads: (a )(a The) The(a from) selected The selected the selected rock path path surface. path in in the the in vertical thevertical vertical directiondirection direction ofof the of therock. rock. ( (bb) ) (Deformationb Deformation) Deformation displacement displacement displacement versus versus versus distancedistancedistance from from thefrom the rock therock surface.rock surface. surface. Figure 14 shows the deformation at the same position on the rock surface for three indenters. It FigurecanFigure beFigure 14seen 14shows shows14from shows the Figure the deformation the deformation 14 deformation that the at at theamplitudes theat same the same same position posiof deformationposition ontion on the onthe rock the displacementrock surfacerock surface surface for forof three forthethree three indenters.rock indenters. indenters.for the It It can It becan seensphericalcan be from beseen seen Figure andfrom fromconical 14Figure thatFigure indenters 14 the 14that amplitudes that arethe the closeamplitudes amplitudes ofto deformationeach other,of deformationof deformationand displacement are higher displacement displacement than of thethat rockofof theof for the cylindricalrock the rock sphericalfor forthe the andspherical conical spherical indentersand and conical conical are indenters close indenters to eachare are close other, close to and eachto each are other, higher other, and thanand are thatare higher higher of the than cylindricalthan that that of theof indenter the cylindrical cylindrical under the same static-dynamic loads. Figure 15a displays the selected path in the vertical direction of the rock, and the corresponding deformation displacement at t = 0.1 s is plotted in relation to the distance from the indenter in Figure 15b. As shown in the ﬁgure, all the deformation displacements of the rock Energies 2020, 13, x 12 of 16

Energiesindenter2020 ,under13, 851 the same static-dynamic loads. Figure 15a displays the selected path in the vertical12 of 16 direction of the rock, and the corresponding deformation displacement at t = 0.1 s is plotted in relation to the distance from the indenter in Figure 15b. As shown in the figure, all the deformation for three indenters decrease with the increase of the distance from the rock surface. The deformation displacements of the rock for three indenters decrease with the increase of the distance from the rock displacement of the rock for the spherical and conical indenter near the rock surface is obviously surface. The deformation displacement of the rock for the spherical and conical indenter near the higherrock surface than that is obviously of the cylindrical higher than indenter, that of but the as cylindrical the distance indenter, increases, but the as the deformation distance increases, magnitude ofthe the deformation rock for the magnitude three indenters of the tends rock tofor be the the three same. indenters In summary, tends to the be deformation the same. In displacements summary, the of thedeformation rock under displacements the static-dynamic of the rock loads under for the the spherical static-dynamic and conical loads for indenters the spherical are similar, and conical and the deformationindenters are degree similar, of and the rockthe deformation is significantly degree greater of th thane rock that is significantly of the cylindrical greater indenter. than that of the cylindricalSubsequently, indenter. the influence of the type of indenter on the stress field of the rock under static- dynamicSubsequently, loads is discussed. the influence Figure of16 thedemonstrates type of in thedenter maximum on the andstress minimum field of stressthe rock counterplots under forstatic-dynamic the three types loads of indenters is discussed. under Figure static-dynamic 16 demonstrates loads, respectively. the maximum Please and note minimum the color codestress for eachcounterplots experiment for whichthe three present types maximum of indenters and under minimum static-dynamic stress for loads, each figure.respectively. It can Please be seen note from Figurethe color 16a code that thefor maximumeach experiment stress which of rock presen undert maximum the conical and indenter minimum is the stress largest, for followedeach figure. by It the sphericalcan be seen indenter, from andFigure it is 16a the that smallest the maximum for cylindrical stress indenter. of rock under Similarly, the theconical same indenter conclusion is the can belargest, drawn followed from the by minimum the spherical stress indenter, of the rock and init is Figure the smallest 16b. This for indicatescylindrical that indenter. the average Similarly, stress ofthe the same rock conclusion under the can conical be drawn indenter from is the higher minimum than that stress of of the the other rock two in Figure indenters 16b. underThis indicates the same static-dynamicthat the average loads. stress However, of the rock from under the pointthe coni ofcal view indenter of the responseis higher than range that of theof the stress other field, two the oneindenters under theunder cylindrical the same indenter static-dynamic is the largest, loads. whileHowever, that underfrom the the point conical of indenterview of the is the response smallest, whichrange is of exactly the stress contrary field, tothe the one previous under conclusion.the cylindrical The indenter reason foris the the largest, above resultswhile that is mainly under due the to theconical difference indenter in the is the contact smallest, area betweenwhich is theexactly three co typesntrary of to indenters the previous and conclusion. the rock, which The reason implies for that a smallthe above contact results area createsis mainly a large due stressto the field difference but a small in the response contact rangearea between of the rock. the Thus, threethe types extreme of valueindenters of stress and andthe rock, the response which implies range that of the a small rock co shouldntact area be considered creates a large comprehensively stress field but whena small the response range of the rock. Thus, the extreme value of stress and the response range of the rock type of indenter is optimized for operation. should be considered comprehensively when the type of indenter is optimized for operation.

FigureFigure 16. 16.Comparison Comparison ofof thethe stress field for three three types types of of indenters indenters under under static-dynamic static-dynamic loads: loads: (a) (a) MaximumMaximum Mises. Mises. ((bb)) MinimumMinimum Mises. As As there there are are large large differences differences between between the the maximum maximum and and minimumminimum stress stress for for the the three three types types of of indenters, indenters, the the uniform uniform legend legend is notis not applicable applicable here. here. Please Please note the color code for each experiment which present maximum and minimum stress for each figure.

Energies 2020, 13, 851 13 of 16

4.6. Discussion In fact, although the static-dynamic loads have an additional harmonic dynamic load compared to the static load, the average force applied to the indenter is the same under two loading conditions, which means that no extra work is done on the indenter. This point can also be proven from the equilibrium position of the deformation displacement of the rock under static-dynamic loads, whose average value is also the same as that under static load. However, the beneficial effects of static-dynamic loads on rock fracture are obvious due to the existence of harmonic dynamic load. It can not only change the deformation shape of the rock making it easier to reach the failure level, but also cause the rock to vibrate periodically aggravating its fatigue damage, so that fatigue fracture may occur in the rock even if the fracture limit is not reached. In addition, the static-dynamic loads expand the response degree and range of the rock, which is also beneficial for rock fragmentation. As rock fragmentation is a critical component for many field operations, such as drilling and mining, the application of harmonic dynamic load will help improve the efficiency of these field operations. In other words, it is meaningful to study the response of the rock under static-dynamic loads. This paper only focuses on the deformation displacement and the stress field of the brittle rock which is the process before fracture. However, there are still many works, such as the effect of static-dynamic loads on the plastic deformation, fracture and crack propagation of the rock, which need to be done in the future.

5. Conclusions This work has presented a detailed numerical study on the deformation displacement and the stress field of the brittle rock under static-dynamic loads. The 3D uniaxial compression simulation is firstly carried out to calibrate and determine the property parameters of the rock and the failure displacement in the simulation is obtained. Then, the 3D modeling of the cylindrical indenter intrusion the rock is proposed and mesh size sensitivity analysis is performed to ensure that the grid does not affect the results. At last, four main parameters are considered, namely the position on the rock, the frequency and the amplitude of dynamic load and the type of the indenter to discuss the effect of static-dynamic loads on the deformation displacement and stress field of the rock. Besides, the numerical results are compared between the static and static-dynamic loading conditions. Based on the simulation undertaken, three main conclusions can be drawn as follows:

1. The deformation displacement of the rock presents a harmonic variation under static-dynamic loads and decreases as the distance from the center of the indenter increases. The frequency of dynamic load only changes the fluctuation period of the deformation displacement of the rock without changing its amplitude, while the amplitude of dynamic load has the opposite effect on it. Besides, the deformation displacement of the rock increase with the decrease of the contact area between the rock and the indenter. 2. The stress field of the rock also presents periodic changes due to the existence of harmonic dynamic load. The magnitude of the stress field of the rock increases with the increase of the amplitude of dynamic load and does not change with the change of the frequency of dynamic load. In addition, a small contact area between the rock and the indenter creates a large stress field but a small response range of the rock. 3. Compared with the deformation displacement and the stress field of the rock under the same static load, the deformation displacement of the rock is more likely to exceed its failure level and the maximum stress generated is also larger, and the response degree and range of the rock is wider under static-dynamic loads.

In addition, the static-dynamic loads can also aggravate the fatigue damage of the rock and allow less static load, which is more conducive to rock fracture. Therefore, it is meaningful to further study the mechanism of rock fragmentation under static-dynamic loads. Energies 2020, 13, 851 14 of 16

Author Contributions: X.L. and S.T. conducted the 3D uniaxial compression simulation; S.L. and W.L. performed the indenter intrusion simulation; M.K. and V.V. analyzed the simulation results; S.L. wrote the paper. All authors have read and agreed to the published version of the manuscript. Funding: The support of National Natural Science Foundation of China (No. 51704074) and Youth Science Foundation of Heilongjiang Province (No. QC2018049) are gratefully acknowledged. The work is also supported by Talent Cultivation Foundation (No. SCXHB201703; No. ts26180119; No. td26180141) and Youth Science Foundation (No. 2019QNL-07) of Northeast Petroleum University. Conﬂicts of Interest: The authors are aware of the ethical responsibilities and they declare that they have no conﬂict of interest.

List of Symbols

µ Poisson’s ratio E Elastic modulus, Pa a Radius of the indenter, m r Distance from the center of the indenter, m q Static load per unit area, N/m2 q(t) Dynamic load per unit area, N/m2

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