Numerical modeling on strain energy evolution in rock system interaction with energy-absorbing prop and rock bolt

Yang Hao *,Chunhui Liu ,Yu Wua, ,Hai Pua, ,Yanlong Chena, ,Lingling Shen ,Guihen Li

a State Key Laboratory of Intelligent Construction and Healthy Operation and Maintenance of Deep Underground Engineering, China University of Mining and Technology,Xuzhou 221116, China

b School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China

c School of Mines, China University of Mining and Technology, Xuzhou 221116, China

Keywords:Strain energy Coal and rock mass Energy-absorbing prop and rock bolt Strain energy evolution

ABSTRACT The interaction mechanism between coal and rock masses with supporting materials is significant in roadway control,especially in deep underground mining situations where dynamic hazards frequently happened due to high geo-stress and strong disturbed effects.This paper is to investigate the strain energy evolution in the interaction between coal and rock masses with self-designed energy-absorbing props and rock bolts by numerical modeling with the finite difference method.The interaction between rock and rock bolt/prop is accomplished by the cables element and the interface between the inner and outer props.Roadway excavation and coal extraction conditions in deep mining are numerically employed to investigate deformation,plastic zone ranges,strain energy input,accumulation,dissipation,and release.The effect on strain energy input,accumulation,dissipation,and release with rock deformation,and the plastic zone is addressed.A ratio of strain energy accumulation,dissipation,and release with energy input α,β,γ is to assess the dynamic hazards.The effects on roadway excavation and coal extraction steps of α,β,γ are discussed.The results show that:(1)In deep high geo-stress roadways,the energyabsorbing support system plays a dual role in resisting deformation and reducing the scope of plastic zones in surrounding rock,as well as absorbing energy release in the surrounding rock,especially in the coal extraction state to mitigate disturbed effects.(2)The strain energy input,accumulation is dependent on roadway deformation,the strain energy dissipation is relied on plastic zone area and disturbed effects,and strain energy release density is the difference among the three.The function of energyabsorbing rock bolts and props play a key role to mitigate strain energy release density and amount,especially in coal extraction condition,with a peak density value from 4×104 to 1×104 J/m3,and amount value from 3.57×108 to 1.90×106 J.(3)When mining is advanced in small steps,the strain energy accumulation is dominated.While in a large step,the released energy is dominant,thus a more dynamic hazards proneness.The energy-absorbing rock bolt and prop can reduce three times strain energy release amount,thus reducing the dynamic hazards.The results suggest that energy-absorbing props and rock bolts can effectively reduce the strain energy in the coal and rock masses,and prevent rock bursts and other hazards.The numerical model developed in this study can also be used to optimize the design of energyabsorbing props and rock bolts for specific mining conditions.

1.Introduction

Deep coal mining poses a range of dynamic hazards,such as rock and gas hazards [1],water inrush [2],and coal bursts [3].However,the mitigation technology still uses outdated methods of support that do not provide enough energy absorption ability,or yieldable capacity [4].In addition,the mechanism of rock supporting is addressed by stress distribution of rock mass,and roadway deformation control [5].While the strain energy related to dynamic hazards is not concerned.Thus,the strain energy evolution principle should be concerned with the problem of dynamic hazards.Assuming that the underground space is a closed system,the strain energy is conserved.When the excavation is done,the outer space does work to the closed system as strain energy input,and the strain energy accumulation is stored as the elastic strain energy in the rock system.The plastic deformation of rock is a result of strain energy dissipation,and the rock fracture,rock ejection is a strain energy release.The four parts obey energy conversion laws.From 1970 to 1990,in deep mining of metal ores in South Africa where rock bursts frequently happened,Walsh introduced an energy changes law,in which gravitational potential energy was highlighted from enlarging an excavation [6].Then,Salamon gave a strain energy input,accumulation,and release model for golden mining [7],while the strain energy distribution in a rock mass can not be analyzed due to the lack of numerical calculation.Napier considers the fault effect of excavation on strain energy by this modeling,and the fault can increase the strain energy dissipation for his research.Mitri et al.[8] used the finite element method to calculate this model,and strain energy release and storage rates were put forward to assess the rock burst proness.While little research on the strain energy evolution model has been published recently due to the decrease in mining at deep levels.Strain energy evolution principle investigations are more focused on experimental studies related to mechanical properties[9,10],considering the loading path [11-13],different water content [14,15],strain rates [16],and temperature [17].Some analytical models and numerical modeling consider the elastic strain energy changes as the energy dissipation and release,and to assess the rockburst [18],while ignored the strain energy input,dissipation,and release,respectively.Thus,a strain energy evolution model of rock systems with strain energy input,accumulation,dissipation,and release should be developed.

The interaction mechanism between the supporting materials and the rock mass involves transferring stresses between the two.The supporting materials act as reinforcement,distributing the load and limiting deformations in the rock mass.The bond between the supporting materials and the rock mass is critical[19],and it depends on factors such as roughness,chemical compatibility,and additives in the rock mass.The effectiveness of the supporting materials in stabilizing the rock mass depends on how well the interaction mechanism is understood and monitored during construction and operation.The load-convergence curve is the result of the interaction of rock and supporting materials to guide the temporal and spatial pre-stress application.However,due to the limitation of the strain energy evolution model,and the unwieldy used energy-supporting materials,the interaction mechanism between supporting materials and rock mass is limited.For example,Wang et al.[20] employ a 2D distinct element method (DEM) model to evaluate the effects of yielding (D-bolt and Roofex) and the traditional rock bolt (fully resin-grouted rebar) on controlling self-initiated strainbursts.Yokota et al.[21]report the performance of the proposed DC-bolt was verified by numerical simulations using discontinuous deformation analysis(DDA).However,it is quite simple,and cannot get much useful data.Vallejos et al.[22] employ FlAC3Dto assess the dynamic mechanical property on the thread bar,this is a lack of interaction mechanism on bolt and rock mass.The supporting materials of hydraulic props and rock bolts are widely employed in coal mines to prevent roof-to-floor convergence,while the energy absorbing capacity is far behind the energy release.Hydraulic props have a load capacity from 200 to 400 kN with a supporting length of 100 mm,and it cannot be employed in large deformation cases and dynamic situations of rock bursts[23].The hydraulic prop typically consists of a hydraulic cylinder,an adjusting valve,a pressure-relief valve,and a support plate.The pressure inside the cylinder of the prop ensures that it remains stable and keeps the roof and walls of the mine from collapsing.When sudden pressure applies to the cylinder of the prop,the prop might be bending due to the sudden pressure increase.Thus,an energy-absorbing prop based on the plastic flow with a working resistance of 500 kN and supporting length from 200 to 1000 mm had been introduced by our research[24].The energy-absorbing rock bolts,also known as energy-dissipating bolts,are a type of rock reinforcement used in underground mines and civil engineering projects [25].These bolts are designed to absorb energy during rock failure,preventing or mitigating damage to the rock structure and the surrounding environment [26-29].Energy-absorbing rock bolts can be classified based on their mechanism of energy absorption [30-32],which includes plastic deformation [33,34],viscoelasticity [21],and friction [35,36].Friction rock bolts rely on the frictional resistance between the bolt and the rock mass to dissipate energy during rock failure[26].These bolts are designed with a rough surface or coating to increase frictional resistance.The energy absorption mechanism of friction rock bolts is based on the sliding friction between the bolt and the rock mass,which causes the bolt to deform and absorb energy.Plastic deformation rock bolts are designed to deform plastically during rock failure,absorbing energy in the process.These bolts are made of a ductile material,such as steel or aluminum,and are designed with a tapered or enlarged section that allows for plastic deformation.The energy absorption mechanism of plastic deformation rock bolts is based on the plastic deformation of the bolt,which absorbs energy and prevents damage to the surrounding rock structure[37].Viscoelastic rock bolts are designed to absorb energy through viscoelastic deformation,which is the time-dependent deformation of a material under stress.These bolts are made of a viscoelastic material,such as rubber or polymer,and are designed with a tapered or enlarged section that allows for viscoelastic deformation [38].We developed a novel energy-absorbing rock bolt,consisting of a sleeve tube with a slope in the inner surface,steel balls,and rebar,which has been experimentally studied with working resistance of 120 kN,supporting length to 1000 mm [39].

In all types of energy-absorbing rock bolts,the rock bolt is activated to counteract rock deformation by absorbing the energy,thereby the bolt’s performance is precisely judged by its mechanical behavior during the procedure.However,a quantitative analysis of the resistance to deformation and energy evolution characteristics of the surrounding rock of the roadway by energy-absorbing props and rock bolts needs to be studied.In this paper,a strain energy evolution mechanical and mathematical model,and its numerical algorithms are introduced.Then the self-designed energy-absorbing prop and energy-absorbing rock bolt are numerically modeled.Roadway excavation and coal extraction of high geo-stress rectangular roadways with energyabsorbing prop and energy-absorbing rock bolt are simulated with results on roadway rock deformation,plastic zone,strain energy input,accumulation,dissipation,and release density.The effect on strain energy input,accumulation,dissipation,and release with rock deformation,and the plastic zone is addressed.The study aims to investigate the significance and role of energy-absorbing props and rock bolts in controlling the surrounding rocks in deep high geo-stress roadways in deformation and dynamic hazards in strain energy aspect.

2.Modeling on an energy-absorbing prop and rock bolt interaction with rock mass in strain energy aspect

2.1.Mechanical and mathematical modeling on strain energy evolution with rock deformation and its numerical algorithms

The strain energy evolution model is based on the law of energy conversion [40].The rock deformation under the excavation state is based on elastic mechanics [41],and the numerical algorithms are achieved by the finite difference method [42].

The first step is the illustration of the stress state on roadway excavation.Fig.1 is a schematic diagram of the stress state in roadway excavation and surrounding rocks,in which the outermost black wireframe area is the underground semi-infinite space,the red wireframe area is the roadway surrounding rock,the yellow area is the roadway excavation area,and the area enclosed by the solid black line is the excavated part of the roadway.Any unit of rock and semi-infinite space is suffered from a one-point stress state.

Fig.1.Mechanical modeling of rock during roadway excavation and one-point stress.

It is known that in rock and coal mass under loading and unloading state,the failure process is undergo strain energy input,elastic strain energy accumulation,dissipation as plastic deformation,and strain energy release as an over-bearing capacity.Assuming that in underground space,the rock system is strain energy balanced,which means the whole space and each zone of coal and rock mass obey energy conservation law.Thus,a strain energy inputW,accumulationUa,dissipationUd,and the releaseUrmathematical relationship is analyzed as follows:

The strain energy input can be described as:

whereWis the total strain energy input;Wpthe potential energy changing due to excavation;andWoutthe external force acting on the rock system.The strain energy accumulation in the elastic zones is:

whereUIIandUIare the two-state of elastic strain energy before and after excavation.

The strain energy dissipation is as follows:

whereUdis the strain energy dissipation;Upthe rock plastic strain energy;andUBthe energy-absorbing supporting strain energy.

Finally,the strain energy release is the difference between input,accumulation,and dissipation,as follows:

The potential energy changesWpis as follows:

whereVis the volume of rocks;Xithe gravity of rocks,anduithe displacement vector on each zone.

The external force on the rock system is obey Gauss divergence theorem [43]:

whereSis the surface area of the rock system;Tithe traction vector in surfaceS;and the superscript k the excavation state.

The elastic strain energy is as follows:

where φ is the elastic strain energy of the zone;τijthe stress vector in the zone;eijthe strain vector in the zone;and superscript e the elastic state.

The plastic deformation on strain energy dissipation is as follows:

where superscript p is the plastic state.

The strain energy absorbed by rock bolts and other supporting materials can be concluded as:

whereF(l) is the function of the load-displacement curve.

Finally,substituting Eqs.(6)-(10)into Eq.(5),the strain energy release is obtained.

Based on Eq.(11),the total strain energy release amount is related to the following parameters:V,Ti,τij,eij,F(l),andl.The amount of strain energy released can be solved combined with boundary conditions.

For underground mining projects,the main objects are to obtain rock deformation,failure,stress redistribution,strain energy,etc,of surrounding rock mass on a roadway,thus providing supporting methods,and optimizing mining methods and processes.FLAC3Dis a numerical modeling software that has many advantages in mining engineering,including complex geologic structure construction,accurate and fast stress analysis,a wide range of constitutive models,and time-dependent modeling.

Elastic strain energy and dissipated plastic energy can be tracked for zones containing a mechanical model.FLAC3Duses an incremental solution procedure: the equations of motion (at the grid points) and the stress-strain calculations (at the zones) are solved every timestep.In the stress-strain equations,the incremental change in energy components is determined and accumulated as the system attempts to reach equilibrium.While the strain energy input of the zone,and the release are not clear.

The Eqs.(2),(6),and(7)can be achieved in FLAC3Delastic strain energy input obeying the Gauss divergence theorem to the tetrahedron,

where the integrals are taken over the volumeVzand the surfaceSzof the tetrahedron zone,respectively;the subscriptzrepresents the zone;andnjthe exterior unit vector normal to the surface.

For a constant strain-rate tetrahedron,the velocity field is linear,andnjis constant over the surface of each face.Hence,after integration,Eq.(12) yields:

where the superscriptfrelates to the value of the associated variable on facef;and v-ithe average value of velocity componenti.

The external work rateEmay be expressed as:

whereEbis the external work-rate contributions of the body forces;andEIthe external work-rate contributions of the inertial forces.

TheEbandEIare expressed as:

where a constant-body force ρbiconstant is assumed.

whereNn(withn=1,...,4) are linear functions of the form and expressed as:

where δnjis the Kronecker delta,the superscriptjrelates to a nodal value of a variable.

By definition of the centroid,all integrals of the formvanish,and the substitution of Eq.(16) into Eq.(15),theEbandEIare expressed as:

Using Cramer’s rule to solve Eq.(18)fortaking advantage of the properties of the centroid:

From Eq.(20) into Eq.(19),Ebyields as:

Thus,the external work rate is expressed by Eqs.(14),(19),and(21):

With the calculation step Δt,the total strain energy input with calculation step can be achieved as follows:

The strain energy accumulationWein a specific step can be decomposed into deviatoric and volumetric component,WesandWev,the numerical implementation can be referred to website as:

where the shear and volumetric components are corresponding toGandKof part,superscriptdof theis deviatoric stress in FLAC3D,theis the mean value of zone stress.

The numerical implementation of Eq.(3) is

wherethe previous elastic strain energy of any step.

The elastic shear and volumetric components are as:

where Δ is the difference between the current and previous steps.

The total shear energy change ΔWTscan be found from the total deviatoric strain and deviatoric stress:

wheree′is the strain vector in FLAC3D.

The total volumetric change ΔWTvcan be found from the total mean straine-and mean stress σ-:

The total shear plastic work dissipated during a timestep is the difference between the total shear energy and the elastic shear energy change at any timestep:

Therefore,the strain energy releaseWris the difference among Eqs.(23),(27) and (28),as follows:

The flow chart of strain energy evolution numerical implementation is shown in Fig.2.

Fig.2.Numerical model of new energy-absorbing rock bolt interaction with rock mass.

2.2.Numerical implementation on an energy-absorbing prop and rock bolt interaction with rock system

In FLAC3D,rock bolt implementation can be achieved by either cable elements or pile structure elements.The difference between the two lies in that cable elements are connected through nodes,while pile elements are similar to foundation piles and are considered as a whole part.Cable elements cannot withstand shear loads,while pile elements can.This part mainly focuses on the axial force and energy absorption performance of energy-absorbing rock bolts;therefore,the shear resistance capacity is ignored.The cable elements are selected to simulate the energy-absorbing rock bolt.

Fig.3 shows the numerical implementation of an energyabsorbing rock bolt.The force exerted by the rock bolt on the surrounding rock is realized by setting the shear stiffness of the sliding component.When the rebar is subjected to axial tensile load,each cable element is connected by a spring,and the support reaction force on the surrounding rock is achieved through the sliding component.The anchored section and the free section are realized by setting different shear stiffness of the grout.The setting of the face plate and the surrounding rock is achieved by rigidly connecting the outer cables element of the rebar to the nearby rock nodes.The constitutive model of cables is ideal elastic-plastic,that is,its tensile strength limit is set,and when it exceeds the limit,the axial force of the rebar will remain constant with displacement.The stiffnessKof the anchor is calculated as follows:

Fig.3.Numerical model of new energy-absorbing rock bolt interaction with rock mass [39].

whereA1represents the cross-sectional area of the rebar;E1is the elastic modulus of two nodes,andLthe length of‘‘cables”.It should be noted that the real energy-absorbing rock bolt is contact type,while the numerical modeling is cablesel element.

Fig.4a shows the experimental results of conventional rock bolts and energy-absorbing rock bolts under the pull-out test.The geometric of a conventional rock bolt with a strength of 220 kN is 22 mm in diameter,and 1000 mm in length,and the energy-absorbing rock bolt with 120 kN is with a sleeve tube.A numerical model of the interaction of rock mass and rock bolt is established as shown in Fig.4b.The parameters of energyabsorbing rock bolts in laboratory experiments and numerical simulations are shown in Table 1.The constitutive model of energy-absorbing rock bolts is the ideal elastic-plastic constitutive model in FLAC3D.While the constitutive model of conventional rebar is a typical elastic-plastic-hardening-plastic breaking three stages.Thus,the three stages should be implemented into FLAC3Dby the FISH function.The rock mass with fixed boundaries was used,and two sets of cables were established.The bottom of the rock bolt was fixed,and a velocity of 2 cm/s was applied at the ends.The implementation of the conventional rock bolt is as follows: First,the positions of all nodes in the cable structure unit in the conventional rock bolt are monitored using the FISH function,and the displacement differences between each node are calculated.Then,the axial force and node displacement difference corresponding to the elastic limit and strength limit were set.When it reaches the maximum strength limit,the axial force of the rock bolt becomes zero as the nodes continue to move.The implementation of the energy-absorbing rock bolt is similar to calculating the relative displacement between nodes.The constant working resistance in the experimental test as the threshold value is set based on the constitutive model of cables of ideal elastic-plastic.

Fig.4.Load-displacement curves of rebar and new energy-absorbing rock bolt.

The experimental and numerical implementation of the energyabsorbing prop is shown in Fig.5.The experimental compressive tests were conducted at the China University of Mining and Science Technology as shown in Fig.5a.The bottom plate is lifted as a speed control at 1 cm/min,and the top plate is fixed.The maximum shrinkage is 600 mm determined by the length of the inner prop.The structure of a prop is similar to a rock bolt,in that the energy-absorbing mechanism is achieved through the axial force of the steel ball on the prop.There are three stages as shown in Fig.5b.In the numerical model of this prop as shown in Fig.5c,it is created for the roof,floor,and prop,with displacement constraints applied to the floor and displacement loading applied to the top roof surface on thez-axis.The prop consists of an inner prop and an outer prop,each equivalent to a cube with the same surface area as the top surface of the prop.The steel ball connection part at the inclined surface of the outer and inner props is implemented using interface elements.Note that the steel prop model corresponds to real conditions because the interface link between the inner and outer cubes can achieve large deformation by sliding components.

Fig.5.Numerical modeling of high energy-absorbing prop and Mechanical modeling of interface.

The normal and tangential stiffness on the interface unit is set to achieve outer and inner prop connection and exert shear force on the prop unit,thereby generating working resistance.The contact mode is realized by the sliding component between the interface node and the face.In each calculation step,the normal displacement and relative shear velocity on each node of the interface and its acting surface are calculated and then substituted into the mechanical components to calculate the normal and tangential contact forces.The constitutive model of the contact surface is employed with the linear coulomb shear law,which means that relative displacement occurs after the shear force on the acting surface reaches its maximum value.The normal and tangential contact forces exerted by the nodes on the contact surface on the target surface are as follows:

whereFnis the normal contact forces;Fsithe tangential contact forces computed;(t+Δt)the relative displacement.They represent the relative displacement between the nodes on the contact surface and the target surface,which is the incremental relative shear displacement.They also represent the normal and tangential stresses that the nodes on the contact surface apply to the target surface.The normal and shear stiffness are denoted byKnandKs,respectively.A2denotes the contact surface area between the interface and the target surface.

The interaction between the interface and the target surface can be classified into three types: (1) Bonding mode: When the bonding force between the node and the target surface is within the shear and tensile strength limit.(2) Slip and bond: This stage is a transitional phase between bonding and slipping.(3) Slip mode:When the shear force on the contact surface exceeds the maximum shear force,the structural surface will undergo slip.The calculation of the maximum shear force is:

wherecis the bonding strength on the structural surface;ϕ the internal friction angle on the structural surface;andpthe pore pressure on the target surface.

By monitoring the displacement of the element on the top of the inner prop and the support force on the top plate,the numerical implementation of the energy-absorbing prop can be realized.The numerical calculation parameters for the prop and interface unit are listed in Table 2,and their load-displacement curves obtained from numerical calculations and experiments are shown in Fig.5e.

Table 2 Parameters of energy-absorbing prop in numerical simulation.

3.Strain energy control effect on energy-absorbing rock bolt and prop

3.1.A coal burst case and its numerical model with energy-absorbing rock bolt and prop

The Bingchang coal mining base is located in Bin County and Changwu County,Xianyang City,Shaanxi Province.The coal resources are extracted from Huanglong Jurassic Coal Field,where coal is in depths from 600 to 1000 m.Due to the high tectonic geo-stress,the coal bursts frequently happened in roadway excavation and coal extraction stages.In this section,a case study of a coal burst in the roadway excavation stage from Mencun coal mine is studied in strain energy aspect,and exam the supporting effect on energy absorbing supporting method.

The No.4 coal strata with a thickness of 24 m are in a depth of 720 m.The lithology of the roof and floor are sandstone of 10 m thickness and mudstone of 12 m thickness.The geometric model is a cube with a length of 100 m.The coal in the roadway is in the center of the cube as shown in Fig.6,with a width and height of 3 and 2.5 m.The geo-stress obtained by the hollow inclusion stress meter is σzz,σxx,σyy(17.5,31.0,and 26.0 MPa),respectively,and they are set to initial stress in this model.The displacement axially symmetric boundary condition is restricted atx=100 m in thex-direction andy-direction.The other boundary condition is fixed in all directions.The working conditions are considered with roadway excavation and coal extraction,with advances of a total of 50 m,divided into 5 stages of excavation and coal extraction.The working face thickness is 3 m.The physical and mechanical parameters of the surrounding rock by triaxial tests are shown in Table 3.To compare the supporting effect of the energy-absorbing rock bolt and prop with the conventional supporting type,the three support methods are divided into three types as no support,conventional rock bolt,and energy-absorbing support.By comparing the manifestation rules of roadway surrounding rock pressure and the evolution of energy inside the surrounding rock under different support forms,the role of the energy-absorbing support system is demonstrated.The support strength parameters and supporting length for schemes 2 and 3 are obtained through simulation in Section 2.2.Note that the energy-absorbing prop is installed near the coal extraction face to better resist roof-to-floor deformation.The numerical calculation ends in default with an unbalanced force ratio of 1×10-5.The monitoring points are employed with 1# to 20#,in the rib side and the middle of the roof and floor.

Table 3 Physical and mechanical parameters of high geo-stress deep roadway.

Table 4 Different supporting methods on deformation,plastic zone area,strain energy input,accumulation,dissipation,and release during roadway excavation steps of 50 m.

Table 5 Different supporting methods on deformation,plastic zone area,strain energy input,accumulation,dissipation,and release during coal extraction steps of 50 m.

Fig.6.Supporting method of the roadway and monitoring station.

3.2.Deformation characteristics and plastic zone of roadway

The deformation and plastic zone (excavation damaged zone)are two most concerned characteristics in field application.These are also related to strain energy evolution.Therefore,the two characteristics are analyzed first.The deformation of the surrounding rock at the monitoring stations is shown in Fig.7.The results can be drawn:(1)The roof-to-floor convergence of three different supporting methods are 338,145,and 68 mm respectively.The rib deformation of three different supporting methods are 195,101,and 63 mm,respectively.The deformation feature shows that the roof-to-floor convergence is larger than it is in the rib.This is corresponding to site observation,thus the numerical model is validated.(2) By comparing the deformation of the surrounding rock of the roadway,it can be seen that the energy-absorbing support system can significantly reduce the deformation of the roof,floor,and two sides of the roadway.(3) The use of conventional rock bolts can also reduce the deformation of the surrounding rock to a certain extent,but due to its low elongation rate,in this case,the rock bolts are broken after the deformation of the roadway reaches 100 mm.Therefore,it cannot resist the large deformation of the surrounding rock over 100 mm.(4) In contrast,the energyabsorbing rock bolt can coordinate with the roadway deformation up to 1000 mm,thus providing continuous resistance to the deformation of the surrounding rock.The energy-absorbing prop can coordinate with the roof and provide support to the surrounding rock under a high and constant strength of 500 kN.Therefore,it can continuously support the roof.

Fig.7.Roadway surface deformation under three different supporting methods.

Fig.8 is a displacement contour of the cross in the roadway and coal extraction face by no supporting and energy-absorbing supporting system,from which it can be seen: (1) The energyabsorbing prop can significantly reduce the displacement of the roof and floor of the roadway,from 180 to around 45 mm.Therefore,the energy-absorbing supporting system has significant importance in reducing the subsidence of the floor and the settlement of the roof at the end support.(2) Comparing Fig.8a and b,the range of the contour lines for roof and floor displacement is reduced on the side of the coal pillar using the energy-absorbing prop.This indicates that the energy-absorbing prop can play the role of an equivalent coal pillar in controlling deformation at the cross in the roadway and coal extraction face.

Fig.8.Roadway deformation cross-section in no supporting and energy absorbing supporting system.

Fig.9 shows the plastic zone range of the surrounding rock in a roadway excavation under three different support conditions.As no support,the plastic range is 550 mm in length and 460 mm in height.While the supporting method on conventional rock bolts and energy-absorbing rock bolts and prop shows the same plastic range,with 385 and 352 mm in length,372 and 369 mm in height.This phenomenon indicates that in the roadway excavation state,the rock mass deformation caused by stress distribution is not severe compared with that suffered from the disturbance effect.The plastic range is more relied on rock bolts,as a function to mitigate the stress distribution on rock mass,so it is an active support.The function of the prop is more on deformation control,as passive support.Comparing the deformation of the surrounding rock during excavation,which is about 80 mm,with the elongation of conventional rock bolts when they break,which is around 100 mm,itcan be concluded that the strengthening effect on the surrounding rocks is the same for conventional rock bolts and energy-absorbing rock bolts.

Fig.9.Supporting methods of the roadway under three different support conditions after excavation.

Fig.10 shows the plastic zones in the surrounding rocks under three different support conditions when the roadway suffered from mining pressures.It can be seen that the area of the plastic zone in the rocks is reduced when a conventional rock bolt is used compared to the unsupported state.This is because,in FLAC3D,the rock bolt transmits the force to the adjacent elements through the nodes on the cable element,thereby reducing the damage to the plastic zone.This is consistent with the fact that the rock bolt element can actively enhance the bearing capacity of the roadway in the field.However,when the deformation of the roadway exceeds 100 mm,especially under the mining pressure,rock bolts are no longer able to resist further deformation,thus to be failed.Comparing the results of the plastic zone ranges between the energyabsorbing rock bolt,prop,and conventional rock bolt,the plastic zone in the former is significantly smaller than that in the latter.This is because the energy-absorbing rock bolt has the characteristic of supporting 1000 mm,so it can maintain a stable working resistance and long-distance support during the mining process,effectively enhancing the bearing capacity of the surrounding rock and reducing the plastic zone ranges.

Fig.10.Plastic zone evolution in advanced roadway step 30 m under three different supporting methods.

Therefore,by comparing the deformation and plastic zone ranges of the surrounding rock in three different support conditions,it can be concluded that the energy-absorbing rock bolt and prop can reduce the deformation of the surrounding rock and the plastic zone in the best supporting effect.The surrounding rock deformation mainly depends on the energy-absorbing supporting force,and the reduction of the plastic zone ranges mainly depends on the long-distance coordinated movement between the energy-absorbing rock bolt with the surrounding rock.In conclusion,considering the resistance to deformation and the range of the surrounding rock plastic zone,the energy-absorbing rock bolt and prop are significant for controlling the surrounding rock in deep high-stress roadways.

3.3.Strain energy evolution of rock mass under three supporting type

Fig.11 shows the contours of strain energy input density,energy accumulation density,energy dissipation density,and energy release density in the roadway excavation stage.The following conclusions can be concluded:

Fig.11.Contour of the energy density of deep roadway after excavation.

(1) The distribution patterns of energy input and energy accumulation density are consistent,and their maximum values are both located above the roof and floor.This is due to the deformation results that the roof to the floor is larger than its rib deformation.This phenomenon indicates the deformation is positively related to strain energy input and energy accumulation location.Compared with the three supporting types,the energy input and energy accumulation density of no support is higher than those of conventional rock bolts,while the corresponding values under conventional rock bolts are higher than those under energyabsorbing rock bolts and props.This indicates that the energy-absorbing rock bolt can mitigate the strain energy input and accumulation with deformation control.

(2) The distribution of energy dissipation density is mainly concentrated within the plastic zone of the surrounding rock,and the energy dissipation in the roof and floor ranges is greater than those in the side walls.This is due to the broken range on the roof and floor,which indicate that in engineering application,the broken rock ranges have played the role of strain energy dissipation.After using the energyabsorbing rock bolts and props,the range of energy dissipation and the energy dissipation density is significantly reduced.The energy-absorbing rock bolts play the role of broken ranges mitigation,thus reducing the range of energy dissipation.

(3) The source and amount of strain energy release are directly related to the magnitude and location of dynamic hazards while it is complex to predict.This is due to the strain energy release being determined by strain energy input,accumulation,and dissipation.In this study,the energy release density is mainly concentrated in the roof and floor of a rock mass.After using a conventional rock bolt,the energy release concentration range is reduced.When using the energy-absorbing rock bolt and prop,the range of energy release concentration is further reduced,but its peak density is increased.And the increased peak density area is within the range of energy-absorbing rock bolts.This phenomenon indicates that the energy-absorbing rock bolt and prop can release the dynamic rock failure range and the absorbing strain energy interaction with rock mass as enhancement.The maximum density peak is concentrated at the corner of the rectangular roadway,which requires special attention in field application.

Fig.12 is a column chart that shows the distribution of energy input,energy accumulation,energy dissipation,and energy release after excavation for three different support methods.From the chart,the following conclusions can be drawn:(1)The maximum energy input density is around 1.4×104J/m3,and the relationship between the maximum energy input density and support strength is not significant.(2)As the support strength increases,the maximum energy accumulation and energy dissipation densities decrease.The maximum energy accumulation density decreases from around 1.4×104J/m3without support to around 8.5×103J/m3,and the maximum energy release density decreases from 2×103to 1×103J/m3.(3)The maximum energy release density shows little change in its peak value as the support strength increases,ranging from 4×103to 5×103J/m3.

Fig.12.Maximum elastic energy density of column statistic.

During the coal extraction process,the emphasis of roadway support is on the range of cross on the roadway and working face due to the influence of advanced mining pressure and mining disturbance.Therefore,the energy evolution in areas ahead,behind,and across the roadway and working face are analyzed.The strain energy accumulation,input,dissipation,and release density are shown in Fig.13,from which the following conclusions can be drawn: (1) The maximum strain energy accumulation,input,energy dissipation,and energy release is in the cross-section of roadway and coal extraction face.The dynamic hazards and supporting location should be focused here.(2)The strain energy accumulation,input is larger in the location on the coal pillar for all types of support.The strain energy release shows a different trend.In the state of no support and conventional rock bolt,the strain energy release location is in the coal pillar,while it is transferred into energy-absorbing props,thus reducing a coal pillar with a strain energy release density in the magnitude order of 103J.It indicates that energy-absorbing props can take part in the role of coal pillar,and the absorbing strain energy releases during the coal extraction.

Fig.13.Contour of energy evolution of roadway in the advanced coal face,cross-section roadway,and back coal face.

4.Discussion

4.1.Strain energy evolution relationship with roadway deformation,plastic zone

The variation in strain energy density with roadway deformation is shown in Fig.14a,from which the strain energy input and accumulation are more sensitive to roadway deformation,from a total of 1×108to 4×108J,while the strain energy dissipation and release are flat variations with rock deformation in the magnitude order of 106J.This indicates the amount of strain energy input and accumulation have relied on roadway deformation.It can be seen from Eq.(9)the strain energy dissipation is related to plastic deformation.The strain energy dissipation is non-linearly related to the plastic zone area from Fig.14b,with a relationship from 5 to 10 m2to strain energy 0 to 1×106J.While in the range of 10 to 15 m2shows a sharp increase rate and it moves to linear increase from 15 to 30 m2.The range of 5 to 10 m2is in the stage of roadway excavation,while the range of 10 to 30 m2is in the stage of coal extraction.This result indicates that the strain energy dissipation amount is more influenced by mining disturbed effects.In the field application of coal mining,the rock mass of the roadway is more prone to be deformed and broken in the mining stage [44],this is well corresponding to the results on the calculation results on roadway deformation,plastic zone with strain energy input,accumulation,dissipation,and release are shown in Tables 4 and 5,as an example for excavation and extraction on 50 m.

Fig.14.Variation in strain energy density with roadway deformation and plastic zone area.

The zone of strain energy input,accumulation,dissipation,and release is expressed as strain energy density,which is related to the roadway deformation,failure patterns,and magnitude of rock dynamic hazards.Based on Eq.(1),the ratio of strain energy accumulation,dissipation,and release on strain energy input with symbols α,β,γ can be an index to assess roadway deformation,failure patterns,and magnitude of rock dynamic hazards.

The excavation and extraction steps on 10,20,30,40,and 50 m versus α,β,γ under three different supporting methods are shown in Fig.15,from which it can be concluded that:(1)For three supporting types,with the roadway excavation and coal extraction steps increase,the ratio α,β are decreased,while the ratio γ is increased.It suggests that,when mining is advanced in small steps,the strain energy accumulation is dominant.While in large steps,the released energy is dominant,thus a more dynamic hazards proneness.(2)Compared with no support and conventional rock bolt,energyabsorbing rock bolt and prop can reduce the ratio γ to a large extent on 0.1 to 0.3 in the coal extraction stage,from extraction steps 10 to 50 m,while it is 0.55 to 0.80 of no support and conventional rock bolt.It indicates that the energy-absorbing rock bolt and prop can reduce three times strain energy release amount,thus reducing the dynamic hazards in coal extraction stages.

Fig.15.The excavation steps and coal extraction steps versus α,β,γ under three different support.

4.2.Limitation and future investigation

The limitations of numerical modeling for the evolution of strain energy in coal and rock system interactions with energyabsorbing props and rock bolts include the difficulty in accurately determining the material parameters and boundary conditions,as well as the challenges of modeling complex geo-structures on-site.Additionally,the accuracy and reliability of numerical models heavily depend on the validation data,which can be difficult to gather for these types of systems.Future research should focus on improving the accuracy and reliability of numerical models by developing better experimental techniques to obtain validation data,improving the understanding of the physics of coal and rock system interactions,and developing more advanced modeling techniques to capture the complex inner faults that occur.

On-site experiments could provide valuable insights into the complex behavior of coal and rock systems and the role of energy-absorbing props and rock bolts in mitigating strain energy.These experiments could involve monitoring strain energy and deformation in coal and rock systems with and without energyabsorbing props and rock bolts under different loading scenarios.The results of these experiments could be used to validate numerical models and to identify the key parameters that influence the evolution of strain energy in these systems.In addition,field trials of different types of energy-absorbing props and rock bolts could also be conducted to evaluate their performance in real-world conditions and to identify any design flaws or performance limitations.The lessons learned from these experiments could be used to optimize the design and selection of energy-absorbing props and rock bolts for use in coal and rock systems.

5.Conclusions

Numerical modeling is employed to investigate the mitigation strain energy effect of energy-absorbing rock bolts and props on the surrounding rocks in high geo-stress roadways,under excavation and coal extraction conditions.The numerical implementation of the energy-absorbing rock bolt relies on the cables element in FLAC3D.By setting the stiffness and yield strength between the cable elements,the load-displacement curve of the rock bolt in pull-out tests is obtained,and the results are validated against experimental results.The verification results show that the simulation results are in good agreement with the experimental results.The following conclusions on energy-absorbing rock bolt and prop interaction with rock mass in strain energy evolution aspect can be obtained:

(1) In deep high geo-stress roadways excavation stages,the energy-absorbing support system plays a dual role in deformation and strain energy control.It can resist deformation and reduce the scope of plastic zones in surrounding rock.The energy-absorbing rock bolt can mitigate the strain energy input and accumulation with deformation control,and decrease the broken ranges of rock mass,thus reducing the range of energy dissipation.The strain energy release is the difference among the three above.

(2) In the coal extraction stage,the energy-absorbing props can play the role of coal pillar.The spatial distribution of energy release is transferred from the coal wall at the intersection of the coal pillar and the working face to the roof and floor.The peak value is reduced from 104to 102J/m3.It shows that the energy-absorbing rock bolt and prop have a significant absorption effect on the energy release of deep roadway from mining disturbance.

(3) The relationship between roadway excavation and coal extraction steps and strain energy evolution is investigated.With the roadway excavation and coal extraction steps increase,the energy accumulation and dissipation ratio α,β are decreased,while the energy released ratio γ is increased.It suggests that,when mining is advanced in small steps,the strain energy accumulation is dominant.While in large steps,the released energy is dominant,thus a more dynamic hazards proneness.Compared with no support,conventional rock bolt,energy-absorbing rock bolt,and prop,γ can be reduced from a large extent on 0.1 to 0.3 corresponding to a coal extraction step from 10 to 50 m,while it is 0.55 to 0.80 of other two support types.

(4) The numerical modeling and methods can give a numerical tool to assess the energy-absorbing rock bolt and prop interaction with rock and coal mass.In further study,on-site deformation,rock broken ranges,microseismic data,and rock bolt force monitoring results should be employed to be calibration with rock mass deformation and failure and reconstruct to strain energy to predict and prevent coal or rock bursts.

Acknowledgments

This research was funded by the National Natural Science Foundation of China (Nos.52204114,52274145,U22A20165,and 52174089),the Natural Science Foundation of Jiangsu Province(No.BK20210522),the National Key Research and Development Program of China (No.2022YFE0128300),the China Postdoctoral Science Foundation (No.2023M733758),and the Shandong Postdoctoral Science Foundation (No.SDCX-ZG-202302037).