A laboratory method to simulate seismic waves induced by underground explosions

Yuguo Ji,Mingyng Wng,,Jie Li,,*,Shuxin Deng,Zhiho Li,Tinhn Xu,Fei Go

a School of Mechanical Engineering,Nanjing University of Science and Technology,Nanjing,210094,China

b State Key Laboratory of Disaster Prevention and Mitigation of Explosion and Impact,Army Engineering University of PLA,Nanjing,210007,China

Keywords:Laboratory method Seismic wave Underground explosion Deep rock mass Coupled loading Experimental apparatus

A B S T R A C T The seismic waves induced by underground explosions generate geological hazards affecting deep buried tunnels such as rockbursts and engineering-induced earthquakes.This issue is difficult to study through full-scale testing due to the expense and unpredictable danger.To solve this problem,the authors developed experimental apparatus and presented a laboratory method to simulate seismic waves induced by underground explosions.In this apparatus,a combined structure of a diffusive-shaped water capsule and a special-shaped oil capsule was designed.This structure can provide an applied confining stress and freely transmit the stress wave generated by external impact.Therefore,the coupled loading of in situ stress and seismic waves induced by underground explosions in the deep rock mass was simulated.The positive pressure time and peak value of the stress wave could be adjusted by changing the pulse-shaper and the initial impact energy.The obtained stress waves in the experiments correspond to that generated by 0.15-120 kt of TNT equivalent explosion at a scaled distance of 89.9-207.44 m/kt1/3.? 2022 Institute of Rock and Soil Mechanics,Chinese Academy of Sciences.Production and hosting by Elsevier B.V.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1.Introduction

Underground explosions at a scale of tens of kilotons of TNT equivalents are usually spherical explosions.The rock around the spherical charge chamber is compressed violently due to the severe detonation process,and a strong spherical shock wave is formed(Donzéet al.,1997).The spherical shock wave continuously propagates and attenuates in the rock mass.The motion of the medium at the wave front is radial and irrotational(Favreau,1969),in other words,the stress wave in the rock propagates outwards in the form of longitudinal wave(P-wave).During propagation,the stress peak of the P-wave decreases and the rise time and positive pressure time(the normal phase time of the stress wave)increase.The Pwave cannot break the rock mass when it reaches the elastic zone,and it is considered as a seismic wave induced by the underground explosion in terms of the strength.Due to the distance from the center of the explosion,the curvature of the stress wave front is negligible relative to the size of the structure affected.In terms of geometry,the seismic wave can be approximately regarded as a one-dimensional(1D)plane wave(Wang et al.,2005,2016;Adushkin and Spivak,2015).The seismic wave carries considerable kinetic energy and poses a serious threat to the safety and stability of a deeply buried tunnel(Kocharyan and Spivak,2001;He et al.,2018;Tang et al.,2019;Dong et al.,2020).When the tunnel is at great depth,the surrounding rock stores a large amount of deformation energy due to the in situ stress(Wang et al.,2016).However,the rock mass comprises complex internal structural layers such as cracks,geological faults,and joints(Sadovskii,1979)and is composed of rock blocks of different sizes(Qi et al.,2005).The seismic waves induced by explosions can destroy weak structural surfaces,release the deformation energy stored in the surrounding rock,result in large deformations,collapse and instability of tunnels,and even cause engineering earthquakes(Mitelman and Elmo,2016;Li et al.,2018).In any study of explosion disturbance to the surrounding rock involving numerical simulation,many simplifications were required(Chen et al.,2011;Dong et al.,2020;Deng,2021),so that the real engineering conditions cannot be reflected completely.Besides,the deformation energy induced by the seismic waves is difficult to control because of the complicated geological and engineering conditions,which implies that it is unfeasible to investigate this problem quantitatively through in situ tests.As a result,to reveal the triggering mechanism of geological hazards induced by the seismic waves,it is necessary to develop experimental apparatus that can quantitatively simulate the seismic waves induced by underground explosions.

In the existing apparatus for three-dimensional(3D)geomechanical experiments,the simulated in situ stress of the rock mass is loaded by fully enclosed loading structures,and there is no suitable space to install the impact components(Gu et al.,2014;Li et al.,2014;Zhang et al.,2017;Shi et al.,2021),thus making it difficult to apply impact loads to the model.For this reason,some scholars simulated the explosion disturbance to the elastic zone by adding tiny explosive balls to the model(Liu et al.,2020),but the stress peak and the positive pressure time generated using this method are difficult to be adjusted.In addition,the use of tiny explosive balls has another shortcoming that cannot be ignored.Compared with the plane stress waves generated in the elastic zone by underground explosions,the explosive stress waves of the tiny explosive balls are classified as spherical waves,whose effect is different from that of plane waves in the elastic zone.Other experiments have achieved the failure phenomenon of rock mass induced by impact load(Aleksandrova et al.,2006;He et al.,2012;Deng et al.,2018),but the impact loads that cause the rock mass to fail are quite different from the seismic waves induced by explosions.Besides,the physico-mechanical properties of the model are inconsistent with the geomechanical properties of the surrounding rock of deeply buried tunnels.The experimental results cannot demonstrate the disturbing effect of the seismic waves induced by underground explosions to such tunnels.

The use of non-explosive loading is a reliable method of simulating impact loads.The earlier application of the non-explosive loading involved the split Hopkinson pressure bar developed by Kolsky(1949).Additionally,Li et al.(2008)developed a set of reaction structures to couple the impact load and confining stress on the rock specimen.To ascertain blast-resistant structures and underwater materials,Deshpande and Fleck(2005)and Deshpande et al.(2006)designed a fluid-structure coupling experimental device according to the waveform characteristics of the explosive stress waves underwater.Espinosa et al.(2006)designed a diffusion pressure tube and overcame the dimensional limitation of the flyer-plate size.Also,without the interference of reflected stress waves to the loading process in liquid,plane wave loading is realized by adjusting the length and angle of the pressure tube.However,most of the aforementioned devices simulated strong impact effects of explosions to study the dynamic response of blastresistant structures in close proximity to the explosion(Deshpande and Fleck,2005;Deshpande et al.,2006;Espinosa et al.,2006;Huson et al.,2011;Huang et al.,2017).The stress peak and the positive pressure time of the impact loads are high and short,respectively.Besides,loading components to apply the confining stress are not required.Therefore,the impact loads imposed in existing devices differ greatly from the seismic waves induced by underground explosions.

In order to solve the above problems,the authors developed experimental apparatus to simulate the disturbance of seismic waves induced by underground explosions.Based on the nonexplosive loading method(Deshpande and Fleck,2005;Deshpande et al.,2006;Espinosa et al.,2006;Huson et al.,2011;Huang et al.,2017),a combined structure consisting of a diffusiveshaped water capsule and a special-shaped oil capsule was designed so that the impact load can be transmitted to the model under an applied confining stress.The waveform can be adjusted by changing the material of the pulse-shaper,and the impact energy can be controlled by adjusting the gas pressure and acceleration length.The stress waves obtained from the apparatus correspond to seismic waves of the order of tens of kilotons of TNT equivalents.

2.The laboratory method to simulate seismic waves induced by underground explosions

2.1.Simulation of seismic waves induced by underground explosions

There should be a good corresponding relationship between the action of the stress waves to the model and the action of the seismic waves on the surrounding rock of a deeply buried tunnel(Westine et al.,1991).Taking this principle as a research requirement,the research was conducted according to the process shown in Fig.1.Model parameters and loads are first determined,and the designed models should be able to be used in tests of disturbance of seismic waves induced by underground explosions at different high in situ stresses.In terms of load analysis,for a deep tunnel,when largescale underground explosions occur,the rock mass is loaded by the seismic waves and the in situ stress.Therefore,the characteristics of the seismic waves and the in situ stress are studied for the loading design in these model experiments.The second step is the analysis of the apparatus and techniques to simulate the seismic waves.Combined with the loading techniques used in the existing device to apply impact load,the main techniques of coupled loading and the experimental components to realize these techniques are explored.The third step is the experimental process designed to simulate seismic waves induced by underground explosions using the apparatus developed.The fourth step is to study the experimental methods of adjusting the waveform.The waveform can be adjusted by changing the working conditions,so that different explosion equivalents and scaled distances are simulated.

2.2.Analysis of model parameters and loads

2.2.1.Definition of model parameters

Generally,to establish similarity between geomechanical model and engineering prototype,equilibrium equation,physical equation,geometrical equation,and compatibility equation of elastic theory need to be considered(Jiang,2016).When modeling dynamic loads such as the seismic waves induced by explosions,the similarity of wave propagation in stress wave theory must be considered.The similarity ratio is the ratio of the same physical quantity between the engineering prototype(P)and the geomechanical model(M).The related physical quantities and their corresponding similarity ratios are displayed in Table 1.

Table 1The related physical quantities and their corresponding similarity ratios.

The static equilibrium equation of engineering prototype in tensor form is expressed as(Jiang,2016):

Fig.1.Simulation of seismic waves induced by underground explosions.

where σijis the stress,andWidenotes the body force.

The corresponding equilibrium equation for a geomechanical model that meets the similarity criterion is as follows:

According to the physical equation in elastic theory,the stressstrain relationship in the elastic stage of the engineering prototype can be written as

where εijis the strain,sijrepresents the deviatoric stress,andGis the shear modulus.

Taking the strain in thex-direction as an example,the stressstrain relationship of the corresponding geomechanical model is given by

Eq.(5)can be simplified as

According to the geometrical equation,the strain of the engineering prototype is

The corresponding strain in the geomechanical model can be written as

As for the time in the wave propagation model,when the elastic stress wave passes through a solid material,the elastic stress wave velocityCof the solid material determines the propagation time of the wave.In the study of wave propagation in engineering prototypes,the relationship between distanceDand the propagation timetis calculated as follows(Fan et al.,2014):

The corresponding relationship in the geomechanical model between the distanceDand the propagation timetcan be written as

Thus,we have

As the seismic wave induced by underground explosions is regarded as a 1D plane wave(Wang et al.,2005,2016;Adushkin and Spivak,2015),the elastic wave velocity in the engineering prototype is expressed as follows:

whereELis the elastic modulus under lateral restraint,andEL=

E(1-μ)／［(1+μ)(1-2μ)］.

The corresponding elastic wave velocity in the geomechanical model is calculated as

According to Eqs.(1)-(12),the following set of equations is obtained:

As the similarity ratios of dimensionless physical quantities are 1,i.e.

according to Eqs.(14)and(15),the following relationships between the similarity ratios of different physical quantities are derived:

An important principle invoked to determine the model scale is such that the mechanical characteristics of the scaled model can be measured in convenient methods.From this perspective,the range of tunnel excavation at model scale is set to 200 mm,and the whole model is cube-shaped.According to the existing model tests(He et al.,2012;Zhang et al.,2017;Shi et al.,2021),the dimensions of model are usually 3-10 times those of the simulated tunnel,and they are 6.5 times those of the simulated tunnel in our study.The real tunnel measures about 10 m in diameter,and a grout mortar is adopted to simulate the rock mass.Therefore,the similarity ratio of geometrySlis 50,the similarity ratio of bulk densitySγ is 1,the similarity ratio of stressSσ is 50,and the similarity ratio of strainSεis 7.07.

2.2.2.Analysis of model loads

When an underground explosion occurs,the surrounding rock at the elastic zone is found to be subjected to two types of loads(Wang et al.,2016;Deng et al.,2018;He et al.,2018;Dong et al.,2020):in situ stress and the seismic waves induced by the underground explosion.Therefore,in the model experiment,the confining stress should be loaded to simulate the in situ stress,and the stress wave should be applied to simulate the seismic waves induced by underground explosions,thus coupling the two loads.

2.2.2.1.In situ stress

One of the most important differences between the deep rock mass and the rock mass in the shallow strata is that the in situ stress of the deep rock mass is larger.The in situ stress can be divided into horizontal and vertical in situ stresses according to the stress direction.Based on the statistics pertaining to in situ stress measurement conducted across many countries(Brown and Hoek,1978),the vertical in situ stress increases linearly with depth,which is characterized as

where σhis the vertical in situ stress(MPa),andzrepresents the depth of rock mass(m).

However,the variation of in situ stress in the horizontal direction is relatively complex.With the increase of burial depth,the tectonic stress caused by crustal movement has an obvious effect on the horizontal in situ stress.Therefore,the horizontal in situ stress does not increase linearly.The horizontal in situ stress at the same depth is more discrete than the vertical in situ stress.The relationship between the lateral pressure coefficientkeand the buried depth is usually expressed as

At a burial depth of 1000 m,the vertical in situ stress is about 27 MPa,the lateral pressure coefficient is 0.4-2,and the corresponding horizontal in situ stress is between about 10.8 MPa and 54 MPa.Therefore,the vertical stress can directly reflect the burial depth,and the horizontal stress can reflect prevailing geological conditions.

2.2.2.2.The seismic waves induced by underground explosions

The seismic waves induced by underground explosions in the elastic zone can be approximately regarded as a plane wave,and the wave front can be understood as the surface with the peak particle velocity(Achenbach,1975;Kocharyan and Spivak,2001;Wang et al.,2016):

where σgis the stress peak at the wave front;ρrdenotes the density of the rock mass(kg/m3);Cpis the velocity of the longitudinal wave(m/s);andvris formulated as

Many underground explosions at the scale of tens of kilotons of TNT equivalents have been carried out in Azgir,Novaya Zemlya,Semipalatinsk,and Balapan.The data measured show that whenthe explosion equivalent and the scaled distance are determined,the peak particle velocity in different rocks is related only toAandn.The values ofAandnare summarized in Table 2(Rodinov et al.,1971;Adushkin and Spivak,2015).

Table 2Values of A and n for different rocks.

Besides,much data measured from large-scale underground explosions show that the seismic waves induced by explosions are approximately regarded as triangular waves(Kocharyan and Spivak,2001;Adushkin and Spivak,2015).The relationship between the pressure rise timetriseand the positive pressure timet+satisfies the following formula(Kocharyan and Spivak,2001;Qian and Wang,2010;Adushkin and Spivak,2015):

The positive pressure timet+is written as follows:

The stress curve of the seismic wave is illustrated in Fig.2.According to Eqs.(19)-(22),the stress peak induced by an explosion is written as follows:

The stress peak and the positive pressure time of seismic waves induced by underground explosion can be calculated using Eqs.(19)and(22),respectively.The scaled radial crack radiusthe scaled crushing radiusthe scaled cavity radiusand the real cavity radiusare derived as follows(Wang and Li,2019):

Fig.2.Stress curve of the seismic wave induced by underground explosion.

where β is a coefficient;σ*and σtrepresent the ultimate compressive and tensile stresses of the rock mass(Pa),respectively,and σt≈0.1σ*;andE0represents the elastic modulus of the rock mass,and it is related to the longitudinal wave velocityCpunder the condition of limited compression,which is expressed asE0(1-μ)／［(1+μ)(1-2μ)］The common parameters of granite are summarized in Table 3.

According to the calculation results based on the parameters in Table 3,the scaled radial crack radius of rockin an underground explosion is 84.36 m/kt1/3.Therefore,the scaled distance beyond 84.36 m/kt1/3is classified as the elastic zone.When the explosion equivalents of TNT are set to 1 kt,10 kt,50 kt,and 100 kt,respectively,the positive pressure time of the selected explosion equivalents is 7.52 ms,16.2 ms,27.71 ms,and 34.89 ms,respectively,and the stress peak is about 2.3 MPa.It can also be calculated that the positive pressure time is longer,and the stress peak is lower when the scaled distance is greater than 85 m/kt1/3.

Table 3The parameters of granite used in calculation(Wang and Li,2019).

2.3.Developed apparatus and techniques to simulate the seismic waves induced by underground explosions

2.3.1.Introduction of apparatus

The 3D section diagram of the apparatus developed by the authors is displayed in Fig.3.The main structure of the apparatus is 6 m high and 3 m wide,which is composed of loading components of confining stress and impact disturbance,and components of tunnel excavation.The model is placed in a rigid reaction box composed of rigid reaction walls.In addition,the confining stress applied to the model is supported by the rigid reaction box and flexible rubber capsules.Square rubber capsules are attached to the four lateral sides of the model,and a special-shaped oil capsule and a diffusive-shaped water capsule are affixed on the top.The simulated tunnel is excavated by the components of tunnel excavation.The impact stress is applied by the loading component of impact disturbance,which consists of the high-pressure gas cavity,the solenoid valve for air supply,the solenoid valve for gas emissions,the impact bullet,and the acceleration tube.The acceleration tube passes through the top rigid reaction wall and is fixed to the outer sleeve attached to the top rigid reaction wall.Furthermore,the pulse-shaper in a suitable material is added between the impact bullet and the water-sealing piston to change the loading rate so that the waveform can be adjusted.The diffusive-shaped water capsule transmits the stress wave from the loading components of impact disturbance,and it also transfers the confining stress from the special-shaped oil capsule.

Fig.3.3D section diagram of the developed apparatus.1-High-pressure gas cavity;2-Solenoid valve for gas supply;3-Solenoid valve for gas emissions;4-Acceleration tube;5-Impact projectile;6-Outer sleeve;7-Loading components of impact disturbance;8-Special-shaped oil capsule;9-Diffusive-shaped water capsule;10-Rigid reaction wall;11-Components of tunnel excavation;12-Water sealing piston;13-Oil sealing piston;14-Model.

2.3.2.Coupling of the stress wave and the confining stress

The research mentioned in the Introduction(Kolsky,1949;Li et al.,2008)indicated that the combination of loading system of reaction force with loading components of external impacts can realize the coupled application of the confining stress and the stress wave,and the diffusive pressure tube with liquid in it can transmit 1D plane waves to the large-scale model(Deshpande and Fleck,2005;Deshpande et al.,2006;Espinosa et al.,2006).On this basis,the confining stress and the stress wave resulting from the external impact can be transferred because only pressure can be transmitted by the liquid.Besides,extraneous interference can be avoided and the conflict of the loading boundary between the confining stress and the stress wave is eliminated.In our apparatus,the combined structure of the diffusive-shaped water capsule and the special-shaped oil capsule was designed using the aforementioned loading techniques.

The diffusive-shaped water capsule is composed of a water sealing piston,a conical inner shell and a rubber cushion.The special-shaped oil capsule consists of an oil sealing piston,an outer circle,a sealing ring,and a conical outer shell.The main components for the coupled loading are demonstrated in Fig.4,and the mechanism of coupled loading is illustrated in Fig.5.In the stage of static loading,after the water capsule is filled with water,the oil is pumped into the special-shaped oil capsule.The reaction wall provides support for the oil sealing piston,which together with the outer circle remains quasi-stationary.After that the pressure is acted directly on the conical outer shell and the conical inner shell.

Fig.4.Components to ensure coupled loading:(a)The rigid reaction wall;(b)The outer sleeve;(c)The water sealing piston;(d)The outer limiting sleeve;(e)The limiting sleeve;(f)Oil sealing piston;(g)The outer circle;(h)The conical outer shell;and(i)The rubber cushion.

Fig.5.Method for coupled loading of the confining stress and the stress wave.1-The outer sleeve;2-The limiting sleeve;3-The rigid reaction wall;4-The pulseshaper;5-The outer limiting sleeve;6-The inner limiting sleeve;7-The water sealing piston;8-The oil sealing piston;9-The conical outer shell;10-The conical inner shell;11-The rubber cushion.

The reaction wall provides a reaction force to the outer sleeve,which supports the limiting sleeve propping the water sealing piston,so that the confining stress is transmitted to the model.Then the impact loads are applied:the impact bullet is driven by the high-pressure gas from the gas cavity,and the impact bullet impacts the pulse-shaper and the water sealing piston,generating a stress wave.

According to the results in Section 2.2.2.2,the positive pressure time of the stress wave at the elastic zone is 7.52 ms(at least).If the impact bullet strikes the sealing piston directly,the positive pressure timetpof the whole stress wave is approximated as(Achenbach,1975):

whereLiis length of the impact bullet,andLi=1.1 m;andCidenotes the wave velocity induced by the impact bullet,andCi=6000 m/s.The calculatedtpis 0.36 ms.

Fig.6.The deformation of the pulse-shaper corresponding to the generation of a stress wave.

Fig.7.Main components and the overall structure after installation of the apparatus:(a)The outer sleeve was installed on the top reaction wall;(b)The structure comprising the diffusive-shaped water capsule and the special-shaped oil capsule was set on the top of the concrete blocks;(c)The overall structure after installation of the apparatus;(d)The impact bullet was settled in the acceleration tube;and(e)The limiting sleeve was set in the outer sleeve.

The length of the water sealing piston is 0.3 m,thus the process in which the impact bullet strikes the sealing piston can be roughly considered as the process whereby the long rigid bar impacts the short rigid bar axially.The long bar can be regarded as a long striker bar,and the short bar as a short incident bar.When the stress wave reaches the side without the long striker bar,reflection and transmission of stress wave occur,but the complete stress wave does not propagate into the short incident bar,and at this moment,the internal impact emerges in the incident bar,thus complicating the wave form and rendering Eq.(25)unsuitable for calculating the positive pressure time of the stress wave.It also can be seen that the rise time of the stress wave is very short,which differs from the seismic wave induced by underground explosions.The waveform can be affected by the lengths of impact bullet and the sealing piston under the condition of axial impact between rigid bars.Therefore,a pulse-shaper made of a soft material is selected and can be regarded as a soft spring with low loading rate.When the soft pulse-shaper is added,the impact between rigid bars is eliminated,the length and shape of the impact bullet will not affect the waveform to any significant extent,and it contributes to providing kinetic energy.As the impact bullet strikes the pulse-shaper,the water sealing piston and water capsule are quasi-rigid,causing the pulse-shaper to deform.The deformation process of pulse-shaper corresponds to the generation process of stress wave,as shown in Fig.6.Taking the moment when the impact bullet begins to strike the pulse-shaper as the starting time(t=0),Fig.6a corresponds to the stage before the impact bullet strikes the pulse-shaper.In this process,t＜0,and the stress is 0.When 0＜t＜trise,the pulseshaper is being compressed corresponding to the stress increase process(Fig.6b).Whentrise＜t＜t+,the pulse-shaper is rebounding,and the stress decreases with time(Fig.6c).According to Eq.(21),the falling time(i.e.the time when the stress decreases from the stress peak to zero)of the stress wave is twice the rise time,therefore,the selected pulse-shaper should be viscoelastic,which can help deceleration of unloading process,so as to prolong the falling of the stress wave amplitude.

Fig.8.Positions of sensors:(a)Side view;and(b)Plan view.

The stress wave generated from the impact is diffused when passing through water and acting on the top surface of the model.At this time,the model has already been subjected to the confining stress for a long time.Consequently,the coupling of the stress wave and the confining stress is achieved.

2.4.Experiments to simulate seismic waves induced by underground explosions

2.4.1.Establishment of the experimental system

Fig.9.Control and measurement systems.

Fig.10.The pulse-shapers in A3-A6:(a)80-mm thick PU rubber;and(b)25-mm thick PU rubber.

Fig.11.The stress curve of A1.

The coupled loading method of the confining stress and the stress wave was investigated by model tests without considering quasi-static stress unloading because of tunnel excavation and lateral quasi-static loading.A layer of concrete blocks consecutively placed on a massive concrete platform was selected to apply the loading.There were 13 blocks in each row and each column,and the side length of each block was 0.1 m.To ensure the overall stability of the concrete model during the experiment,the massive concrete platform was directly placed in the steel box.The main components and the overall structure after installation of the apparatus are shown in Fig.7.The establishment of the experimental system is also demonstrated in Fig.7.The top reaction wall and the combined structure of the diffusive-shaped water capsule and the specialshaped oil capsule were set on the top of the concrete blocks.To transmit the reaction force from the special-shaped oil capsule and the diffusive-shaped water capsule,a U-shaped steel piece was welded between the massive concrete platform and the top reaction wall,and then the outer sleeve was installed on the top reaction wall to fix the water sealing piston,the limiting sleeve,and the acceleration tube.After the impact bullet was settled in the acceleration tube,the high-pressure gas cavity and solenoid valve for gas supply were installed.

Piezoresistive sensors were used to measure the dynamic stresses in the experiments(Jordana and Pallas-Areny,2006).The corresponding relationship between the measured voltage signalVoleand the stress signalPgwas obtained by

wherecois the sensitivity coefficient of the sensor,which is 8.18 mV/(V MPa),8.12 mV/(V MPa),8.07 mV/(V MPa),8.41 mV/(V MPa)for sensors 1-4,respectively;modenotes the magnification of the voltage signal;andVolbis the voltage applied to the bridge,andVolb=2 V.

The positions of the four sensors are shown in Fig.8.As shown in Fig.8a,the top sides of each of the four concrete blocks were first grooved,and then the stress sensors were placed in the concrete grooves.The positions of the grooved concrete blocks with stress sensors are shown in the plan view(Fig.8b).The distance from each sensor to the adjacent model boundary is 350-450 mm,therefore,the four sensors can be used to verify the planeness of the stress wave.The thickness of the sensor is 6 mm,the diameter is 25 mm,the response frequency is 50 kHz,and the maximum range is 10 MPa.

The control and measurement systems of the experiment are depicted in Fig.9.The working conditions were controlled by the control board,which consists of the control systems of gas and oil.The gas was used to control the solenoid valves for gas supply and emission.The oil was employed to control the confining stress.The load signals of the stress wave and the confining stress were converted to electrical signals by the stress sensors.Then the electrical signal was amplified and stored in the data-acquisition device.Finally,the data were post-processed.

2.4.2.Experimental process to simulate seismic waves induced by underground explosions

Six experiments were designed to verify the laboratory method for simulating the seismic waves induced by underground explosions,numbered A1-A6,respectively.The pulse-shaper was not used in A1 and A2.The polyurethane(PU)rubber(Somarathna et al.,2018;Wang et al.,2020)was used as the pulse-shaper in A3-A6(Fig.10).The elastic modulus of PU rubber was about 300 MPa,and its thickness was 80 mm,80 mm,25 mm,and 25 mm in A3-A6,respectively.The pressure on the water capsule was about 0.5 MPa,and the gas pressure was 1 MPa,1 MPa,0.5 MPa,0.5 MPa,0.5 MPa,and 0.25 MPa,respectively.The working time of the solenoid valve was 30 ms.Taking the sensor 3 as an example,the stresses are demonstrated in Figs.11-16.

According to Figs.11 and 12,when there is no pulse-shaper,the impact bullet directly strikes the water sealing piston,and the pressure rise time is 0.5 ms and 0.6 ms for A1 and A2,respectively.After the stress reaches the peak,it begins to decrease.After 5 ms,the stress increases again,and then multiple fluctuations occur,with decreasing amplitude.Finally,a static equilibrium is achieved.The overall waveform is not consistent with the seismic waves induced by underground explosions.In Figs.13 and 14,the corresponding pressure rise times of the stress waves are 13.31 ms and 13.33 ms,the positive pressure times are 37.13 ms and 36.05 ms,and the peak stresses are 1.05 MPa and 1.03 MPa,respectively.In Figs.15 and 16,the corresponding pressure rise times of the stress waves are 9.67 ms and 9.43 ms,the positive pressure times are 24.39 ms and 26.89 ms,and the peak stresses are 1.06 MPa and 1.42 MPa,respectively.In Figs.13-16,the fluctuation amplitude of the stress curves is very low,and the whole waveform is relatively smooth.The ratiotrise/t+of each stress wave is about 1/3,which is consistent with the data captured from seismic waves induced by underground explosions(Kocharyan and Spivak 2001;Adushkin and Spivak,2015).

In Figs.11 and 12,there are multiple fluctuations,which correspond to multiple loading/unloading processes over the stress history.These loading/unloading processes mainly come from the axial impact from the impact bullet to the sealing piston,and the reflection of the transmitted wave generated in the water capsule(Achenbach,1975),as shown in Fig.17a and b.When there is no pulse-shaper,the primary downward compression wave(wave 1 in Fig.17)is generated at the interface between the water sealing piston and the impact bullet.When it propagates to the interface between the water sealing piston and the water,the first downward transmitted wave(wave 3)and the first upward reflected wave(wave 2)are generated.Due to the length of the impact bullet,the primary downward compression wave is still being applied to the impact bullet.Therefore,continuous internal impact,at the end face of the water sealing piston,between the upward reflected wave and the primary downward compression wave,is on-going.During this process,the upward and downward travelling waves in the sealing piston are generated.After the upward travelling wave reaches the top surface of the impact bullet,it will evolve into a downward unloading tensile wave(wave 7)and continue to propagate in the impact bullet.When the downward transmitted wave reaches the interface between the water capsule and the concrete blocks,the transmitted wave will be reflected upwards(wave 4).The wave impedance of the water sealing piston is much greater than that of the water and is considered as a rigid surface.Therefore,weak transmission(wave 6)and strong reflection(wave 5)will occur at the interface between the water and the water sealing piston.A complicated reflection ensues in the water capsule,so that the stress history incorporates complex loading/unloading processes.

Fig.12.The stress curve of A2.

Fig.13.The stress curve of A3.

Fig.14.The stress curve of A4.

Fig.15.The stress curve of A5.

Fig.16.The stress curve of A6.

Fig.17.The main loading/unloading process in the impact bullet and the water capsule(the pulse-shaper is not added):(a)Axial impact from the impact bullet to the sealing piston;and(b)The reflection and transmission of the transmitted wave generated in the water capsule.

Fig.18.Reflection and transmission of the transmitted wave generated in the water capsule(the pulse-shaper has been added).

Fig.19.Stress curves of A5.

When the pulse-shaper is added,the stress histories(Figs.13-16)are extremely similar to the seismic waves induced by underground explosions,suggesting that the function of the shaper shown in the experiments agrees with the analysis in Section 2.3.2.However,in Figs.13-16,a local maximum occurs as the pressure rises.Under the condition that extraneous interference is eliminated,a possibility based on the theory of stress wave propagation is considered(Achenbach,1975):when the shaper is compressed,the stress increases with time;however,before the compression limit is reached,as shown in Fig.18,wave reflection and transmission occur when it reaches the interface between the water capsule and the concrete blocks.The reflected wave(wave 4)propagates upwards and will reach the water sealing piston.Because the shaper is compressible,it can be regarded as a flexible boundary,so that the reflected wave(wave 6)can almost transmit through the water sealing piston and the pulse-shaper.When the reflected wave reaches the water sealing piston,it will provide an upward load,and temporarily offset the downward load applied by the impact bullet,causing the local maximum in the pressure-time plot.The shaper continues to be compressed and begins to rebound after being compressed to the limit.

Fig.20.Confining stress applied in the test.

Fig.21.Coupled confining stress and dynamic stress.

Comparing the results obtained from two groups of experiments with different pulse-shaper thicknesses,it is shown that,when the pulse-shaper is thinner,the positive pressure time is shorter,which indicates that,within a certain range,different thicknesses represent different elastic stiffnesses.The thinner the material,the stiffer the spring,and the shorter the corresponding positive pressure time.

The changes of stresses measured by the four stress sensors in each experiment are relatively consistent.Limited by the length of this article,only the stress curves of A5 are demonstrated in Fig.19.In this experiment,the stresses start to rise at about 8 ms,with the similar fluctuations,and they decrease to 0 MPa at 43 ms.The maximum difference of the four stress curves at the same time is no more than 0.2 MPa,indicating the planar nature of the stress waves.

2.4.3.Stability of the coupled loading of the stress wave and the confining stress

The confining stress was set to 0.54 MPa,and the pressure was held for up to 9000 s in the experiment for testing the device’s capacity of maintaining the pressure.The confining pressure applied in the test is plotted in Fig.20.The coupled loading curve of the confining stress and the stress waves is shown in Fig.21.In the initial stage of the experiment,a confining stress of 0.54 MPa was added to the diffusive-shaped water capsule.After about 1088 s,the first impact experiment(C1)was conducted using the peak value of the stress wave of 1.33 MPa and a positive pressure time of 35.9 ms.The second impact test(C2)was performed about 573 s later with the peak value of the stress wave of 1.17 MPa and the positive pressure time of 33.4 ms.The third impact test(C3)was undertaken about 153 s later.In this test,the peak value of the stress wave was 1.63 MPa and the positive pressure time was 36.7 ms.The value shown on the water pressure gage remained unchanged for a long time after the third impact experiment,indicating that there was no obvious liquid leakage.

Fig.22 depicts the spalling of the outer ring concrete in the test,which represents material failure after multiple dynamic and static coupling loading experiments.The failure mechanism is illustrated in Fig.23,where a concrete block at the outermost ring is simplified as a two-dimensional(2D)model.When the static pressure is applied,three sides of the concrete block are compressed,and the remaining side is free.Stable static loading induces continuous stress on the concrete,and tensile deformation increases near the free surface.In this state,the top surface of the specimen is impacted by a simulated seismic wave induced by explosion directly.Due to the Poisson effect,the tensile deformation near the free surface is greater,and the concrete is continuously degraded by multiple shocks,finally leading to the splitting of the concrete near the free surface,which demonstrates the successful coupling of the stress wave and the confining stress.

Fig.22.Spalling of the outermost blocks under dynamic and static coupled loading(multiple experiments).

Fig.23.Spalling mechanism of outermost blocks under dynamic and static coupled loading.

2.5.Experimental adjustment of the waveform

2.5.1.Using gas to adjust waveform

Experiments were conducted to determine the influences of gas on the peak value and positive pressure time of the stress wave.The gas pressure ranged from 0.25 MPa to 1 MPa,and the working time of the solenoid valve for gas supply varied from 1 ms to 30 ms.The thickness of the pulse-shaper is 25 mm and 80 mm,respectively.The experiments are named B1-B10(Table 4),respectively.

Table 4Experimental conditions.

According to the relationship between the gas pressure and the stress peak shown in Fig.24,the higher the gas pressure,the larger the peak stress.Statistical positive pressure time corresponding to different working times of the solenoid valve for gas supply and different pressures is plotted in Fig.25.When the thickness of the pulse-shaper is 80 mm,the pressure and the working time of gas have significant influences on the positive pressure time.For a given working time of the gas,the greater the pressure,the longer the positive pressure time.At a given gas pressure,the longer the gas working time,the longer the positive pressure time.When the thickness of the pulse-shaper is 25 mm(B8-B10),with increasing working time of gas at the same pressure,as listed in the legend of Fig.25,the difference of the positive pressure time in the three experiments is within 2 ms.

These experimental results indicate that the driving action of gas influences the applied stress waveform.After actuating the solenoid valve controlling the gas supply,the driving of gas on the impact bullet can be divided into two processes,as illustrated in Fig.26:early driving and subsequent driving,respectively.The early driving mainly corresponds to the acceleration of the impactbullet,while the subsequent driving corresponds to the process after the impact bullet begins to compress the pulse-shaper.Therefore,the larger the pressurePgas,the higher the velocityV0of impact bullet caused by early driving,and the greater the peak amplitude of the stress wave.When the gas pressure is high and the solenoid valve for gas supply undergoes a long working time,the pulse-shaper will undergo a longer compression and rebound process because of the subsequent driving effect,which is conversely limited by the spring stiffness.When the stiffness of the spring is too low,the effect of the subsequent driving on the stress wave is less significant.

Fig.24.Stress peak at different gas pressures Pg.

Fig.25.Positive pressure time for different gas effects.

Fig.26.The process of early driving and subsequent driving.

Fig.27.Stress peak for different acceleration lengths.

2.5.2.Adjusting the acceleration length

Experiments were conducted to evaluate the influences of the acceleration length of impact bullet on the peak value and positive pressure time of the stress wave.The gas pressure is 0.5 MPa,the working time of the solenoid valve for gas supply is 30 ms,the thickness of the pulse-shaper is 25 mm,the actual acceleration lengthLacceis 0.25,0.5 and 1 times the length of the acceleration tubeLwhole,and the experiments are numbered B11,B12,and B13,respectively.The peak value and the positive pressure time of stress waves are shown in Figs.27 and 28,respectively.It can be observed that the stress peak increases continuously with the increase of acceleration length.

Fig.28.Positive pressure time for different acceleration lengths.

Fig.29.Comparison between the experimental stress curve after similarity transformation and the calculated stress curve according to Eq.(23).

The acceleration process corresponds to the early driving of the gas.Under the same gas pressure,the early driving effect is optimal at the full acceleration length,thus the stress peak increases.When the actual acceleration length becomes shorter,the driving action of gas on the impact bullet is limited;therefore,the stress peak decreases.The influence of acceleration length on the positive pressure time of stress wave is shown in Fig.28 when the thickness of the pulse-shaper is 25 mm.When the gas pressure is 0.5 MPa,the longer acceleration length corresponds to the longer positive pressure time.Whether the early driving is sufficient or not will also affect the positive pressure time.The gas pressure will enhance the early driving effect,and a longer positive pressure time ensues.

Fig.30.Explosion equivalents,distances to the center of the explosion,and scaled distances corresponding to(a)different positive pressure times and(b)different stress peaks.

3.Comparative analysis of experimental results with the measured seismic waves induced by underground explosions

Many underground explosion experiments at the scale of tens of kilotons of TNT equivalents were conducted at the Novaya Zemlya test site.Eqs.(19),(22)and(23)are used for calculating the stress peak,positive pressure time,and stress curve of the seismic waves induced by underground explosions,respectively.The values ofAandnare 1×104and 1.6,respectively,according to the statistical data pertaining to the Novaya Zemlya test site in Table 2.These three equations have good application for granite from the boundary between the elastic zone and crack zone to the scaled distance of 450 m/kt1/3at the Novaya Zemlya test site.Since the waveform obtained from our experiments is similar,only the TNT equivalent and the scaled distance of the experiment A4 are calculated and substituted into Eq.(23).The stress curve is thus obtained allowing comparison with the experimental stress curve generated from our apparatus after similarity transformation.The stress curves of experiment A4 after similarity transformation and the calculated stress curve according to Eq.(23)are shown in Fig.29.The explosion equivalent of the two stress curves is 6 kt,and the scaled distance is 141.96 m/kt1/3.The positive pressure times of experimental and calculated stress curves are about 254.87 ms and 254.5 ms,respectively,and the stress peaks are both about 50 MPa.Fig.29 indicates that the positive pressure time and the peak value of the experimental stress curve are consistent with the calculated ones according to Eq.(23),and the trend in the positive pressure time satisfiestrise=t+/3.

Fig.31.Positive pressure time and stress peak of experiments and calculated results.

Table 5Experimental results and the corresponding simulated TNT equivalents and scaled distances of seismic waves induced by underground explosions.

As shown in Fig.30a,the data from experiments B9,C1,B6,B12,A3(B2),A6(B10),B4,and C2 are divided into three groups.The stress peaks in each group are similar,but the positive pressure time differs significantly.The longer the positive pressure time,the greater the distance to the center of the explosion,and the greater the explosion equivalent,but the change of scaled distance at this time is insignificant.As shown in Fig.30b,the data from experiments B12,B8,A4,C1,A3,and C3 are divided into three groups.The positive pressure time in each group is similar,but the peak value differs:the larger peak values correspond to greater distances from the center of the explosion and greater explosion equivalent,but the corresponding scaled distance with the larger peak value and greater distance is smaller at that time.The experimental results in Fig.30a and b are consistent with those obtained by Eqs.(19)and(22).When the correlation constant is determined,the peak value is considered to correspond to a certain scaled distance.On this basis,the real distance to the center of each explosion and its explosion equivalent can be determined according to the positive pressure time.

Fig.31 illustrates the experimental results of the positive pressure time and peak value of stress waves after similarity transformation,and calculated stress peak and positive pressure time corresponding to each experiment were obtained according to Eqs.(19)and(22).The differences between the experimental and calculated results are extremely small,suggesting that the experimental results are consistent with the explosion effect of the seismic waves induced by underground explosions.

The experimental and the calculated results of seismic wave induced by underground explosions are summarized in Table 5.The positive pressure time in the experiment B11 is short and the stress peak is of a low amplitude,and the corresponding TNT equivalent in this experiment is only 0.15 kt.Except for that case,the stress waves of the other 12 experiments correspond to the seismic waves induced by explosions of 2.2-120 kt of TNT equivalents at the elastic zone and the largest TNT equivalent simulated reaches 120 kt.The scaled distance is about 89.9-207.44 m/kt1/3.It was found that the developed apparatus can simulate the seismic waves induced by explosions of kilotons of TNT equivalents in the elastic zone.

4.Discussion

It is difficult to conduct large-scale experiments when studying the seismic waves induced by underground explosions.Besides,the induced energy is difficult to be estimated,and the possible geological disasters are potential hazards faced when investigating this problem in situ.Therefore,through the experimental apparatus developed in the present research,under the definite similarity relationship,the explosion equivalent and scaled distance can be adjusted by controlling the experimental conditions,and the disturbance effect of seismic waves induced by underground explosions to the tunnel surrounding rock can be determined.

The stiffness of the pulse-shaper plays an important role in waveform adjustment in the process of optimizing the stress waveform of seismic waves induced by explosions.In addition,the positive pressure time and the stress peak are quite different when the pulse-shaper is altered.In this study,the pulse-shapers of two thicknesses have two kinds of deformation ability.The influence of the pulse-shaper deformation ability on the stress waveform was qualitatively investigated,but not evaluated in any quantitative sense.Therefore,the stress peak and the positive pressure time of the stress waves in experiments could be adjusted to a wider and more uniform range through the quantitative analysis of the pulseshaper.

5.Conclusions

(1)The diffusive-shaped water capsule and the special-shaped oil capsule are the key components to realize the proposed experimental method to simulate seismic waves induced by underground explosions.The confining stress can be applied to the model by the above two components.On this basis,the special-shaped water capsule can transmit the stress wave generated by the external impact.The stress waves thus obtained correspond to the seismic waves induced by explosions of 0.16-120 kt of TNT equivalents at scaled distances ranging from 89.9 m/kt1/3to 207.44 m/kt1/3.

(2)When the static pressure is 0.54 MPa,the pressure holding time is up to 9000 s,and the maximum confining stress is 0.68 MPa,corresponding to the in situ stress at a depth of 1260 m.The confining stress remains unchanged after repeated impacts,indicating that the device can stably simulate the in situ stress state of rock mass at a great depth before and after explosion-induced seismic wave excitation.

(3)In the experimental apparatus,PU rubber is used as the pulse-shaper.On one hand,its stiffness allows smooth loading/unloading with a prolonged rise time.On the other hand,a flexible boundary is provided to absorb the reflected compression wave and eliminate vibration and its effects upon the waveform.The stiffness of thick PU rubber(80 mm)is greater than that of thin PU rubber(25 mm),so that it is more sensitive to those factors that influence the waveform.

(4)To achieve the purpose of adjusting the explosion equivalent and the scaled distance,the stress peak and positive pressure time of the stress waves produced by the apparatus can be adjusted by controlling the gas pressure,working time of the gas,acceleration length,and pulse-shapers.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors acknowledge the financial support from the National Natural Science Foundation of China(Grant Nos.51527810,51679249,12002171 and 51909120)and Postgraduate Research &Practice Innovation Program of Jiangsu Province(Grant No.KYCX20_0312).