Yoon Keon Kim,Woo Chun Choi

Department of Mechanical Engineering,Korea University,145 Anam-ro,Seongbuk-gu,Seoul,02841,Republic of Korea

Keywords: Ricochet Exploded debirs Environmental change Sand Temperature Water content Internal friction angle Travel distance

ABSTRACT The debris from exploded buildings can ricochet after colliding with the ground,thus increasing the debris travel distance and danger from any associated impacts or collisions.To reduce this danger,the travel distance of ricocheted debris must be accurately predicted.This study analyzed the change in the travel distance of ricocheted concrete debris relative to changes in the properties of a sand medium.Direct shear tests were conducted to measure the change in internal friction angle as a function of temperature and water content of the sand.Finite element analysis(FEA)was then applied to these variables to predict the speed and angle of the debris after ricochet.The FEA results were compared with results of low-speed ricochet experiments,which employed variable temperature and water content.The travel distance of the debris was calculated using MATLAB,via trajectory equations considering the drag coef ficient.As the internal friction angle decreased,the shear stress decreased,leading to deeper penetration of the debris into the sand.As the loss of kinetic energy increased,the velocity and travel distance of the ricocheted debris decreased.Changes in the ricochet velocity and travel distance of the debris,according to changes in the internal friction angle,indicated that the debris was affected by the environment.

1.Introduction

When ammunition or a building explodes,a large amount of debris is generated and scattered in all directions.When the debris collides with the ground,it is buried,stopped,rolled,or bounced.The phenomenon in which an object bounces after colliding with a medium is called a ricochet.Ricocheting increases the travel distance of the debris,as well as the risk of collision with people or objects.Because ricocheted debris possesses high kinetic energy,it is extremely dangerous when it impacts other objects.Therefore,it is necessary to have an accurate understanding of the ricochet phenomenon so that safety distances can be set accurately.

The first country to study and use ricochets was the United Kingdom.Ricochets were used to increase the range of projectiles in naval battles in the 16th and 17th centuries.According to World War II records,dams that were dif ficult to reach with airplanes were destroyed by using ricocheting projectiles[1,2].

Since then,research has been conducted on ricochets off water and solid mediums.A solid medium comprising numerous small grains is not as homogeneous as water;therefore,associated ricochet behavior changes with the variable types and conditions of each medium[3-8].In forensic science,ricochet is also an important factor.This is because it is often necessary to reconstruct the circumstances of a shooting by tracking the range and path of a ricocheted bullet[9,10].

The travel distance of debris from building explosions has been studied through experiments and simulations[11-14].The importance of ricochets is increasing in the research on building explosions[15].Some software is being developed to predict the extent of damage around exploded buildings due to ricocheted debris,blast waves,heat,etc.[16,17].Ricochets of debris vary depending on the structure of the exploding building,quantity of explosive,and type and characteristics of the medium.Changes in physical properties,such as temperature,water content,internal friction angle,and cohesion,affect ricochets from debris[18-20].However,in the study of ricochets,many factors are not considered and are expressed in simple equations[19].The safety distance-meaning that a person’s life is not lost,but there is a risk of injury from exploded debris-is assumed as 400 m[21].However,depending on the net explosive quantity(NEQ),the final travel distance of the ricocheted debris might exceed 400 m.Although it is dif ficult,with uncertain information,to predict the travel distance of debris from accidental blasting of buildings,it is possible to predict the incident speed and angle of debris when the structure of the building and NEQ are known.In order to minimize the risk of injury,when calculating the travel distance of the debris in the building blasting software,the initial conditions of the debris,as well as the type and conditions of the medium,should be considered.However,simple ricochet equations that do not take into account the type and condition of the medium are not suitable for calculating the travel distance of the ricocheted debris.

Generally,the scattering and ricocheting behaviors of the debris are considered to be most affected by its initial conditions due to the building structure and NEQ.However,the ricocheting of debris is also affected by the type and condition of the medium;additionally,the environment affects ground media.This implies that the ricocheting and travel distance of the debris can vary depending on the environment.Therefore,in this research,the effects of property variations due to changing temperatures and water content of sand on the behavior and travel distance of ricocheted debris were studied.An analysis was conducted on sand temperature and water content through a low-speed steel sphere ricochet test.To calculate the flight trajectory of the debris more accurately,a flight trajectory equation was used,considering the change in drag with the Reynolds number.Using the curve fitting tool in MATLAB,the speed and angle of the debris were fitted to a three-dimensional(3D)surface,thus allowing the speed and angle of the debris after the ricochet to be predicted.Finally,the travel distances of the ricocheted debris,considering the change in sand properties with the environment,were calculated using the fitted curve and the flight trajectory equations with varying drag.

2.Change in sand properties according to environment

Changes in the surrounding environment must be considered while studying the causes for any phenomenon or while designing an object.Depending on the environment,the properties of sand and soil change.This is one of the reasons that many ricochet studies have been conducted on solid media such as concrete and tarmac.Even if the environment changes,their properties do not change as much as in the case of water and sand.In this study,changes in temperature and water content were selected for the examination of variations in the sand properties.To investigate the changes in the sand properties due to these two factors,a direct shear test,which is used to measure the shear resistance of sand,was performed.The direct shear test was required in order to perform the finite element analysis(FEA)because the change in the physical properties of the sand,which affects the shear stress of the sand,depends on its temperature and water content.Fig.1 illustrates a direct shear test arrangement[22].The sand sample to be measured is placed into a brass box divided into two sections,and then,a constant vertical load is delivered to the sand.Avertical load of 1 kg/cm2was used to measure the shear stress as a function of temperature.Vertical loads of 0.5,1,and 1.5 kg/cm2were used to measure the internal friction angle and cohesive strength as functions of water content.Under a vertical load,the lower box could move horizontally.The experiment was repeated three times under the same conditions,and a graph of the shear stress according to the displacement and internal friction angle of the sand was obtained.

Fig.1.Direct shear test arrangement.

2.1.Changes in sand properties as a function of temperature

Fig.2 depicts the change in the shear stress of the sand according to the measured temperature,during the direct shear test[20].As the temperature of the sand increased from 20 to 100°C,the shear stress decreased,and the shape of the graph changed,indicating a change in the internal friction angle(φ)of the sand.Fig.3 depicts the graph of shear stress according to the internal friction angle of the sand.This indicates that the internal friction angle of the sand changes according to the shape of the graph of shear stress measured during the direct shear test[22].A comparison of Figs.2 and 3 demonstrated that the internal friction angle decreased with increasing sand temperature[20].

Fig.2.Sand direct shear test at 20,50,and 100°C.

2.2.Changes in sand properties as a function of water content

The internal friction angle and cohesion according to the water content of the sand were also measured through a direct shear test.Using the Mohr-Coulomb failure criteria,the internal friction and cohesion of the sand could then be obtained.Eq.(1)is the Mohr-Coulomb failure criterion[22,23].

whereτis the shear stress,cis the cohesive strength,σnis the normal stress on the failure plane,andφis the internal friction angle.The vertical loads used in this test were 0.5 kg/cm2,1 kg/cm2,and 1.5 kg/cm2.The water content of the sand was set as 3,6,9,12,15,18,21,and 24%.Fig.4 depicts the change in the internal friction angle and cohesion according to the water content of the sand,obtained through the direct shear test and Mohr-Coulomb’s failure criteria.The internal friction angle increased to 38°until the water content of the soil reached 15%,and then,the internal friction angle decreased to approximately 25°.The cohesion continued to decrease as the water content of the sand increased.

Fig.3.Results of the direct shear test in loose,medium,and dense sand and the corresponding internal friction angles.

Fig.4.Internal friction angle and cohesive strength for a variable water content.

Fig.5.Sand properties.

2.3.Sand properties used in FEA

Thus,changes in the sand temperature and water content signi ficantly affected the sand properties,indicating the effect of the environment on the sand properties.A change in the internal friction angle was common to these two factors.The internal friction angle was an essential property of sand in the FEA.ANSYS Explicit Dynamics was the analysis software used in this study and the sand model used was the Drucker-Prager model.It is expressed as in Eq.(2).

whereσYis the total yield stress,σpis the pressure yield stress,σρis the density yield stress,andFis the variation of shear modulus.Since sand is a granular material,the Drucker-Prager model was determined as the most suitable model to be used for the FEA.Three tabular data are required for Eq.(2)and are depicted in Fig.5.These were the sand properties used in ANSYS explicit dynamics analysis.Fig.5(a)presents MO granular pressure hardening.The yield stress varied depending on the pressure and the internal friction angle of the sand.Fig.5(b)presents the MO Granular variable shear modulus and compaction path.Shear modulus and pressure changed depending on the density of the sand.These three graphs are interrelated,that is,when the internal friction angle of the sand changed,the yield stress changed despite an equal sand pressure;despite the pressure changing according to the internal friction angle even at the same yield stress,the shear modulus and density of the sand changed.In other words,when environmental factors such as temperature and water content varied,the internal friction angle of the sand changed,thereby affecting the yield and shear stresses of the sand.Shear modulus and pressure change depended on the density of sand.

In this study,FEA was conducted by changing the MO granular pressure hardening,which affects the ricochet of debris,in the Drucker-Prager strength model.The MO granular pressure hardening represents the internal friction angle of the sand,which is an important parameter in this analysis.The focus of this ricochet analysis was to express the appropriate internal friction angle according to the variable temperature and water content.Thus,among the Rankine,Mohr-Coulomb,and Tresca theories,the Mohr-Coulomb failure criteria and Tresca failure criteria were selected for this analysis.The Mohr-Coulomb and Tresca theories were suitable the sand depending on the internal friction angle and cohesion[24].

3.Ricochet and flight trajectory equations based on drag coef ficient

In this study,the effects of changes in sand properties on the travel distance of the debris generated by the explosion of a building were examined.To obtain the travel distance of the ricocheted debris,it is necessary to understand the ricochet and trajectory of the flying debris after the building explodes.The ricochet and flight trajectory equations of the debris considering the drag coef ficient are explained in the following section.

3.1.Ricochet equations

Fig.6 illustrates the process when an object collides with the medium.When an object collides with a medium,a repulsive force(FL)is generated,which can be expressed as in Eq.(3)[25,26].

Fig.6.Schematic view of the impact process of a spherical projectile.

whereCLis the lift coef ficient,ρmdis the density of the medium,Uis the velocity of the object,Scis the contact area between the object and the medium,andf(α,β)is a dimensionless function that contains the angular dependence of the repulsive force.The most relevant factor in the presence or absence of a ricochet of an object is the repulsive force in the direction perpendicular to the surface of the medium.Eq.(4)de fines the repulsive force in thez-axis direction of an object.It can be simply expressed as Eq.(5),whereεis the penetration depth of the object,andVzis the velocity in thezaxis direction.It can also be represented by Eqs.(6)and(7)[25,26].

Eq.(6)de fines the repulsive force,and Eq.(7)serves as the effective friction coef ficient.In other words,if the repulsive force in thez-axis direction is greater than the energy loss due to friction between the debris and the medium,the object will ricochet.Therefore,both penetration depth of the object and contact area with the medium are important factors in the ricochet process because they affect both repulsion and energy loss.

3.2.Flight trajectory equations for an object with air drag

In this study,an object’s trajectory equations were used to obtain the exact trajectory with the drag coef ficients according to the Reynolds number of the debris.The drag(D),Reynolds number(Re),and time constant(ψ)can be expressed as Eqs.(8)-(10),respectively[27]:

whereCDis the drag coef ficient,ρa(bǔ)is the air density,Ais the area of the object projected perpendicular to the air flow,Uis the velocity of the object,dis the radius of the object,μis the kinematic viscosity of the air,andmis the mass of the object.Applying Eq.(10)to Newton’s second law,the acceleration can be determined from the Reynolds number and time constant,as in Eq.(11).

Fig.7.Logarithmic plot of the sphere’s drag coef ficient as a function of the Reynolds number.

Substituting Eq.(10)into Eq.(11),the acceleration of the object with respect to thexandz-axes can be obtained simply as Eq.(12).

Using the modi fied central difference method,the velocity of an object considering the Reynolds number and drag coef ficient can be expressed as Eqs.(13)and(14),respectively.

Eqs.(15)and(16)indicate the location of the debris.

The sphere’s Reynolds number is used in this study,and Fig.7 depicts the drag coef ficient as a function of the sphere’s Reynolds number.

4.Ricochet pilot analyses

The process of a steel sphere colliding with sand and ricocheting was studied using FEA.The most important aim of the analysis was to set the physical properties of the model as close as possible to reality.However,the sand used in this study exhibited a signi ficantly wide range of physical properties depending on the type of surrounding environment;therefore,it was necessary to determine the similarity of the sand used in the analysis to real sand.In this study,to improve the reliability of the properties of the sand used in the FEA,ricochet experiments at low velocities were conducted to compare the results with those of the FEA.When the error of the comparison was low,the FEA of the spherical debris could be performed under various initial conditions.In this analysis,the sand was assumed to be a continuum with small grains and flow.For the sand in the FEA,it was appropriate to use the smooth particle hydrodynamics technique that models the sand as a grain itself.However,because the time required for the analysis was considerably long,the arbitrary Lagrangian-Eulerian method was selected to solve the behavior of the ricochet off the sand.This model is suitable for the collision between solid steel and continuum sand[18].The constitutive equations of the sand material models are founded on isotropic and anisotropic elasto-plastic flow functions[28].In most FEAs,sand uses the elasto-plastic theory,which exhibits elastic behavior under small loads and plastic behavior under further increased loads,and the elasto-plastic model is used with Euler[29].Therefore,the Lagrange model was used for the steel sphere,whereas the Euler model was used for the sand.

4.1.Low-speed ricochet experiments considering the temperature and water content of sand

Low-speed ricochet experiments were performed using an air tacker and air compressor to launch a steel sphere.Some previous ricochet studies have launched projectiles using an air pressure device[18,30].In this study,an air tacker was used to launch steel projectiles using air pressure.Fig.8 illustrates a schematic diagram of the steel sphere ricochet off sand experiments.The pressure conversion of the air compressor allowed for various changes in the initial velocity of the steel sphere.When a long cylinder operating at high pressure struck the steel sphere,it would be fired.The air tacker was equipped with a device for changing the angle of incidence.The incident angles used in this experiment were 15,20,and 25°.The process was recorded using a 600 fps high-speed camera.After extracting still-images from the video,the speed and angle of the steel spheres after the ricochet were calculated.The diameter of the steel sphere was 10 mm,and the density of the steel was 7850 kg/m3.The angle of incidence of the steel sphere was 15°,and the initial speed of the steel sphere was 24 m/s in the variable temperature experiment and 32 m/s in the variable water content experiment.The experiment was repeated three times under identical conditions.The density of sand used in the experiment was 1500 kg/m3.

Fig.8.Experimental setup for ricochet off sand.

The sand temperature was changed by heating the sand on an iron plate.The heated sand was then moved to a ricochet sandbox.The sand temperature was measured using a thermometer by selecting five places from the path and depth of the steel sphere predicted by the ricochet pilot experiments,and the ricochet experiment was conducted when the average temperature was±3°C of the desired value.

The sand temperatures used in this experiment were 20,50,and 100°C.The internal friction angle according to the sand temperature was assumed to be 30,35,and 40°,based on the direct shear test.The low-speed ricochet experiments for the water content used sands with water content of 3,6,9,12,15,18,21,and 24%by accurately mixing the appropriate amounts of sand and water.The internal friction angle of the sand was applied in the FEA,as depicted in Fig.4.

4.2.Low-speed ricochet analysis considering the temperature and water content of sand

The initial conditions for the FEA were the same in both experiments.Table 1 presents the physical properties of steel.Table 2 presents the physical properties of sand.Other sand properties were the same as depicted in Fig.5.The steel sphere was set as a rigid body.The meshes representing the steel sphere and the sand were hexahedral.The mesh representing the sand was set to a cube,and the size of the mesh was 1 mm.The body interaction of the two objects was set to frictional.The contact between the solid projectile and the continuum sand was analyzed by applying the friction coef ficient between the two objects.The friction coef ficient between steel and sand is 0.4[31].A problem with this FEA is that the varying environmental factors could not be applied to the sand model in the analysis.If the temperature and water content are input into the analysis,the change in physical properties must also be input.Therefore,the analysis was conducted assuming that the temperature and water content were replaced with the internal friction angle.

Table 1 Steel properties in FEA.

Table 2 Sand properties in FEA.

4.3.Comparison of ricochet analysis and experimental results

Fig.9 depicts the kinetic energy loss factor for the experiments and FEA,as well as the error factor according to the temperature,when the incident angle and speed were 15°and 24 m/s,respectively.The sphere diameter was 10 mm.The most in fluential factor in the ricochet is the kinetic energy loss due to the loss of repulsive force owing to friction between the medium and the projectile;therefore,the experiment and analysis are compared with respect to the kinetic energy loss factor.Even with varied debris and initial conditions,the variations in debris ricochet behavior with variations in the properties of the medium can be compared with respect to the kinetic energy loss factor.Despite replacing the temperature change with the internal friction angle,the error in all cases did not exceed 4%.Fig.10 depicts the kinetic energy loss and error factors for the experiments and the FEA for the water content when the incident angle and speed were 15°and 32 m/s,respectively.The sphere diameter was 10 mm.As with the temperature changes,in all cases,the error did not exceed 4%.This indicates that the change in temperature and water content is sufficiently applicable even if it is assumed to be the change in the internal friction angle.

Fig.9.Kinetic energy loss and error factors for FEM and experiments with variable temperature.

Fig.10.Kinetic energy loss and error factors of FEM and experiments with variable water content.

5.Ricochet analysis and fitting of concrete spheres

Because the change in the temperature and water content could be replaced by the change in the internal friction angle,the ricocheted speed and angle of a steel sphere off sand were studied according to various sphere diameters,incident speeds,and incident angles through FEA.As all initial conditions could not be analyzed,several were selected for fitting to predict the ricocheted speed and angle.

5.1.Ricochet analysis of a concrete sphere

Because analyses could not be performed on a large number of debris sizes,only representative debris were analyzed.When collecting exploded debris from a building,the debris was classi fied according to its weight and size.Table 3 presents the classi fication according to the weight and size of concrete debris[14].The debris most often collected after blast tests belonged to Mass Bins 8,9,and 10.For the debris generated in the blasting experiment of Ref.[13],Mass Bins 8-10 accounted for 81.98%of the total debris[14].Therefore,in this analysis,the diameters were selected within the size distribution found in Mass Bins 8,9,and 10.The diameters selected were 20,30,and 40 mm,and the debris material was concrete.The initial speeds of the spheres were 10,50,100,and 150 m/s,and the initial incidence angles were 1,5,10,15,20,30,40,and 50°.ANSYS explicit dynamics was used for the analysis.The sand density was 1500 kg/m3,and the internal friction angles were 30,35,and 40°.The friction coef ficient between concrete and sand is 0.6[32].Table 4 presents the physical properties of concrete.The concrete sphere was set as a rigid body.The other properties were the same as those used in the pilot analysis.

Table 3 Concrete Mass Bin characteristics.

Table 4 Concrete properties in FEA.

5.2.Fitting for ricocheted speed and angle

The results obtained under each condition were fitted to a linear or curved line according to the same diameter and incident angle.Then,the fitted lines at various incident angles and speeds were collected to construct a 3D surface using the curve fitting tool in MATLAB.Fig.11 depicts a 3D curved surface that is fitted to the ricocheted concrete debris for various incident speeds and angles for a sand medium.The diameter of the concrete sphere was 40 mm,corresponding to Mass Bin 7.

6.Distribution of debris fragments with variable conditions

MATLAB was used in this study to calculate the travel distance of the debris after a building explosion.To calculate the trajectory of the debris,Eqs.(8)-(16)in Section 3.2 were used.To calculate the exact flight trajectory of the debris,the drag coef ficients were considered according to the change in the Reynolds number of the sphere with a time interval of 0.01 s.The next trajectory,which occurred after the debris ricocheted off the ground,was calculated by obtaining the speed and angle after the ricochet from the 3D curved surface that was made by the curve fitting tool in MATLAB.This process was repeated to set the movement of the debris and was stopped when the speed of the debris became zero or the ricocheted angle became 1°or less.The total distance of the debris was considered up to the point where the motion of the debris stopped.

To determine the effect of the internal friction angle on the fitted 3D surface,the trajectory and travel distance of the debris according to the internal friction angle were analyzed under the same initial conditions.Fig.12 illustrates the trajectories of the debris when a 40-mm concrete sphere had identical initial conditions at three internal friction angles.The initial velocity of the debris was 100 m/s,and the initial launch angle was 10°.All of the travel distances of the debris were between 260 and 280 m.When the internal friction angle was 40°,the debris traveled the farthest,and when the angle was 30°,the debris traveled a lesser distance,con firming the effect of the internal friction angle on the travel distance of the debris.

Fig.11.3D surface plot of 40-mm concrete sphere ricocheted off sand.

Fig.12.Trajectory of 40-mm concrete sphere with variable internal friction angles and equal initial conditions after the first ricochet.

Additionally,the trajectory of the debris,which was represented in two dimensions,was converted to three dimensions,and the travel distance of the debris was analyzed.The initial speed and launch angle of the debris were randomly selected within the speci fied range.The initial launch speed range of the actual debris was signi ficantly wide.Thus,the speed of the debris is affected by many variables,and the initial launch speed of each debris fragment varies in the same exploded building test under different conditions.Many studies on debris obtain the debris launch velocity(DLV)for the explosive test conditions using Eq.(17)[33].

where theNEQis the total mass of explosive material,Vwis the volume of the building or ammunition,ρwis the wall density,andtwis the wall thickness.To compare the results of the actual explosive test with the results of this study,the initial speed of the debris was determined by the blast test conditions of the SciPan 4 report[14].The building volume used in the experiment was 252.57 m3,and the density and thickness of each of the four walls used was 2400 kg/m3and 0.19 m,respectively.The NEQ was 1000 kg.The initial velocity of the debris obtained using Eq.(17)was 122.99 m/s.Therefore,in this study,the initial speed range was set at 30-160 m/s by increasing the range above and below the value obtained through Eq.(17).The initial launch angle range of the debris was 1-15°[34].The number of repetitions of the analysis per condition was 10000.In this study,the travel distance of the debris for each diameter of the concrete sphere and the internal friction angle of the sand were obtained,and the number and percentage of debris in each zone were represented.The sizes of the concrete spheres used were 20,30,and 40 mm.

Fig.13.Travel distance distribution chart of 20-mm concrete sphere on sand with variable internal friction angle.

Fig.14.Travel distance distribution chart of 30-mm concrete sphere on sand with variable internal friction angle.

Figs.13-15 present distribution plots depicting the number and percentage of debris at each travel distance after ricocheting off sand,with three internal friction angles,and when the diameter of the concrete sphere was 20,30,and 40 mm,respectively.As the internal friction angle in the sand increased,more debris could be found to have moved farther.In the top two zones of the concrete spheres of all sizes,the number of debris fragments was the highest when the internal friction angle was 40°,and the number of debris fragments was the lowest when the internal friction angle was 30°.In particular,the 40-mm concrete sphere was the heaviest among the three concrete spheres,and owing to its large kinetic energy,the debris traveled up to 500 m beyond the safety distance of 400 m.The distribution of debris from 400 to 500 m was 21.2%when the internal friction angle was 40°and 17.9%when the internal friction angle was 30°.In summary,as the internal friction angle,which is proportional to the shear stress,increases,the kinetic energy loss of debris decreases after ricocheting.Therefore,the travel distance of debris is longer on the sand of a high internal friction angle.

Fig.15.Travel distance distribution chart of 40-mm concrete sphere on sand with variable internal friction angle.

Finally,to increase the accuracy and validity of this study,three types of data were compared with these results.First,the effect of a ricochet on the movement of debris is depicted in Fig.16.The distribution distance of debris with and without the ricochet off sand for 40-mm concrete spheres with a density of 1500 kg/m3and an internal friction angle of 40°is depicted.It can be seen that debris that experienced a ricochet traveled farther than the debris that did not.A total of 12%more ricocheted debris traveled beyond the safety distance of 400 m in comparison with the debris that did not experience a ricochet.This suggests that considering ricochets is essential in determining the travel distance of debris.

Fig.16.Distribution ratio of the travel distance of debris according to the presence or absence of a ricochet.

Fig.17.Distribution ratio of the distance of debris using simple ricochet equations.

The second is the difference in prediction of the ricocheted speed and angle after the debris collides with the sand.In this study,the speed and angle after a ricochet were predicted on a 3D surface that was fitted according to the debris size,sand conditions,etc.;however,in Ref.[18],the ricochet equations were defined simply through experimentation and FEA.The ricochet equations are as expressed in Eqs.(18)and(19)[19].

whereθois the angle after ricochet,θiis the angle of incidence,vois the velocity after ricochet,andviis the velocity of incidence.Fig.17 depicts the distribution of the travel distance for 40-mm concrete spheres obtained using Eqs.(18)and(19).Except for predicting the speed and angle after the ricochet,the travel distance of the debris was calculated by the method used in this study.A total of 20%of the debris exceeded the safe distance,and 6.1%of the debris traveled over 550 m.The maximum travel distance of the debris was 1241 m.Most of these occurred at low angles below 3°.

Fig.18.Comparison of debris distribution plot of explosion test and analysis.

Finally,Fig.18 depicts the comparison of the coordinates of the concrete sphere debris in this analysis using ANSYS explicit dynamics,the curve fitting tool in MATLAB,and MATLAB with those obtained from the actual experiment.The initial conditions of the analysis were as follows:a 40-mm concrete sphere and an internal friction angle of sand of 40°.This was the condition under which the debris traveled the farthest.In actual building explosive tests,the distribution of debris typically appears in various shapes,because of the building structure and the varying reactions when explosives detonate.The building explosive experiment in Fig.18 depicts the distribution of debris in the shape of a cross.This shape appears to be frequently produced in reinforced concrete structures with thick walls[16,33,35].This analysis assumes that the debris spreads in all directions because the spreading azimuth of the debris cannot be determined.The debris distribution using the wall thickness and wall properties in this analysis is represented by the area marked with the black border in Fig.18.

7.Discussion

In this study,the ricochet of debris and the change in travel distance was examined according to the conditions of the medium,and not the initial condition of the debris.In particular,the changes in the ricochet characteristics and the travel distance of the spherical debris were studied according to the change in the physical properties of the sand.The temperature and water content were changed to vary the properties of the sand.To change the temperature of the sand,the sand was heated,and to control the water content of the sand,water and sand were combined to produce the desired mix ratio.This changed the internal friction angle and shear stress of the sand,as depicted in Figs.2-4.The reason for this change in the internal friction angle and shear stress is as follows.

First,when the temperature of the sand increases,the shear resistance of the particle contacts decreases.Consequently,there is partial collapse in the sand structure and a decrease in the void ratio,until a sufficient number of additional bonds are formed to enable the sand to carry the stresses at a high temperature[36].Therefore,as the temperature of the sand increases,its shear resistance and internal friction angle decrease.The water content also varies the internal friction angle of the sand.As the water content increases,the internal friction angle increases and then decreases.When sand and water combine,the tensile force of the water in the sand attracts the sand particles,increasing the capillary tension.Therefore,the contact stress between the sand particles also increases,thereby increasing the frictional resistance between the sand particles.The water film surrounding the individual particles increases the strength of the sand by increasing the cohesion between the sand particles due to capillary tension.Therefore,as the water content of the sand increases,the shear stress and internal friction angle of the sand increase.However,as the water content further increases,the adhesion of the sand almost disappears,reducing the internal friction angle of the sand.Thus,it can be con firmed that the shear stress and internal friction angle of the sand are proportional to each other.Therefore,the sand properties according to weather conditions can be represented by the internal friction angle.Changes in the internal friction angle affect the ricochet of debris.As the internal friction angle decreases,the shear stress also decreases,thus reducing the resistance of the sand.Therefore,when debris collides,it penetrates deeply and consumes a large amount of kinetic energy due to the loss of friction energy to the sand.The debris fragments that do not ricochet at a high speed do not move farther or ricochet while colliding with the medium the subsequent time,as can be observed in Fig.12.As the rigidity of the ground increases,the kinetic energy loss of the debris decreases,and thus,the travel distance increases.Regardless of the size of the debris,the higher the internal friction angle,the farther the debris travels.This result is similar to other studies,as can be seen in Figs.13-15.

It is essential to consider the ricochet phenomenon when studying exploded debris.However,the scant research that has been conducted on the ricochet phenomenon of debris uses a simple equation.This causes many errors in calculating the travel distance of the debris.The resultant values of the velocity and angle of the ricocheted debris are not derived linearly.Conversely,nonlinear results are often obtained.The maximum travel distance of the fragments calculated by Eqs.(18)and(19)results in a deviation of approximately 700 m from the actual debris distribution range,as depicted in Fig.17.Calculated under the same initial conditions as the results in Fig.12,the travel distance of the 40-mm concrete sphere is 228 m.The results of this study differ by more than 50 m from the calculated results.In this study,the distribution of the ricocheted debris that travels up to 500 m is nearly that of actual explosion test debris,as depicted in Fig.18.This can lead to large errors when building a magazine or conducting military operations.The ricochet of debris also varies depending on the type and condition of the medium.The simple equations cannot contain many of property factors.Therefore,the study of ricocheted debris according to the medium is recommended,in order to accumulate a large number of databases and to map data,rather than to create equations for predicting the speed and angle after the ricochet.Certainly,a considerable amount of time is required to accumulate a large number of databases;however,they would produce results similar to the actual fragment distribution as depicted in Fig.18.Issues of danger from exploded debris are highly important for human safety;therefore,there is a signi ficant bene fit to spending more time on this valuable research.

8.Conclusions

In this study,changes in the properties of sand related to changes in the environment and the travel distance of debris depending on the variable properties were studied.The resulting conclusions are as follows:

(1)Changes in the environment cause changes in the sand properties.The temperature and internal friction angle of sand are inversely related.Therefore,sand at high temperatures exhibits a decreased internal friction angle and shear stress.As the water content of sand increases,the internal friction angle increases;however,because the cohesive force is decreasing simultaneously,the internal friction angle also starts to decrease eventually.Reduction of the internal friction angle decreases the shear stress of the sand.Additionally,changes in the internal friction angle affect the travel distance of debris.As the internal friction angle decreases,the sand resistance decreases,and the debris penetrates deeper into the sand.This causes an increase in the loss of kinetic energy from the debris.Therefore,in areas of high temperature or rain,the number of debris ricochets decreases,and the debris does not travel as far with the ricochet as it does in cold or dry areas with high internal friction angles.

(2)Consideration of the debris ricochet is essential,and rather than predicting the speed and angle of the debris using simple equations,it is desirable to predict them through numerous databases.This is because ricochets vary depending on the type of ground medium;however,for a given medium,the ricochets vary owing to the changes in the environment as well.To overcome the limitations of simple equations,the speed and angle of the ricocheted debris should be predicted by data mapping according to the type and condition of the medium.

In most building explosion analysis software,ground conditions and properties are simply assumed and applied.Changes in the environment,as the results of this study show,clearly affect the ricochet of debris off the medium.Therefore,in order to calculate debris travel distance in the software more accurately,it is essential to consider the data for the property change of the medium according to the environment.If this is dif ficult to apply,then types of medium with different characteristics,such as sand and concrete should be considered.With the consideration,it is possible to calculate the speed and angle of the debris after the ricochet more accurately than the existing software,which does not consider the variable medium.It is expected that the calculation of more accurate debris travel distances will further reduce human and property damage.

Declaration of competing interest

The authors declared that there is no con flict of interest.

Acknowledgements

This study was financially supported by the Foundation Research Program[grant number UD170027GD]of the Agency for Defense Development and the Defense Acquisition Program Administration of the Republic of Korea.