Msoud Rhmni ,Alirez Nddf Oskouei ,Amin Moslemi Petrudi

a Department of Mechanical Engineering,IHU University,Tehran,Iran

b Department of Mechanical Engineering,Tehran University,Tehran,Iran

Keywords: Granular materials Shock tube Blast wave Numerical simulations

ABSTRACT Among the intrinsic properties of some materials,e.g.,foams,porous materials,and granular materials,are their ability to mitigate shock waves.This paper investigated shock wave mitigation by a sandwich panel with a granular core.Numerical simulations and experimental tests were performed using Autodyn hydro-code software and a shock tube,respectively.The smoothed particle hydrodynamics (SPH)method was used to model granular materials.Sawdust and pumice,whose properties were determined by several compression tests,were used as granular materials in the sandwich panel core.These granular materials possess many mechanisms,including compacting (e.g.,sawdust) and crushing (e.g.,pumice)that mitigate shock/blast wave.The results indicated the ineffectiveness of using a core with low thickness,yet it was demonstrated to be effective with high thickness.Low-thickness pumice yielded better results for wave mitigation.The use of these materials with a core with appropriate core reduces up to 88%of the shock wave.The results of the experiments and numerical simulations were compared,suggesting a good agreement between the two.This indicates the accuracy of simulation and the ability of the SPH method to modeling granular material under shock loading.The effects of grain size and the coefficient of friction between grains have also been investigated using simulation,implying that increasing the grain size and coefficient of friction between grains both reduce overpressure.

1.Introduction

Shock waves in two or multi-phase environments include several phenomena,which have been considered by mathematicians,physicists,and mechanical and aerospace engineers.In some engineering designs,there is a need to reduce the power of waves.Wave dampers are required to build structures and armor to mitigate the blast wave in the military industry and the damping of the sound wave in the construction industry.The use of explosive wave dampers is not limited to these cases.Moreover,it is applicable in the explosive shaping industry,the construction of tunnels,and explosion test tanks.Foams,porous granular materials,and layered materials have been commonly used to reduce the impact and damping of the blast wave.These capabilities are due to their high energy absorption,among the energy-absorbing materials,relatively inexpensive grain materials,such as pumice,iron filings,and wood chips can be considered.Several studies have been carried out on the use of granular materials.Nesterenko[1]examined the shock wave damping of soft condensed materials,in which the effect of using sawdust in a metal tank was investigated in the explosion test and observed that the deformation of the tank was greatly reduced and the subsequent degradation was reduced.Serna et al.[2]investigated the reduction of the blast wave in grain flow.Nguyen [3] studied the shock wave dynamics in a chain of granular materials,in which this chain was a simplified specimen of the granular material used to study the dynamics of wave propagation.Gueders et al.[4]simulated the explosion-wave damping in granular materials of alumina ceramic by examining the pressure and impulse of the wave.Moreover,Levy[5]investigated the shock wave contact with granular materials.It is necessary to create the criteria for the properties and geometry of materials,which will ensure the mitigation for specific conditions of blast loading.The fact that such materials are appropriate energy absorbers does not guarantee their mitigation performance.For instance,the application of a porous layer with a small thickness equal to an effective radius of explosive charge did not result in the damping of strains in the explosive chamber [1].In the past two decades,granular materials have received significant attention due to their military and civil applications,such as impulse reduction,freeze plumbing,silos,and air purification [5].Liu et al.[6] presented a numerical simulation of the shock-powder interaction based on the particle kinetic theory and gave a close detection to the particle phase response behavior to the passage of shock wave within the dense particle powder.The simulations were carried out using a developed code compiled in FORTRAN.They used particle kinematic theory to model the powder and compared the results with the experimental results.According to the results,the propagation of waves in the particle powder showed a decreasing trend in the gas phase pressure.Burgoyne et al.[7] employed numerical simulation to investigate the two-and three-dimensional shock absorption among grain crystals.Their experimental and computational study showed that the behavior of three-dimensional grain crystals could be simulated.The rules of normal contact between components are well described,and it also turns out that the properties of a onedimensional granular chain can also be generalized and applied to two-and three-dimensional conditions.Luo et al.[8] investigated the shock wave contact with granular materials.In this study,a miniature shock tube was used.Furthermore,they used a highspeed camera to record wave propagation in the granular material and used metallic spheres as the geometry of granular material.Duan et al.[9] studied the shock wave in a one-dimensional granular chain by contacting Hertz between the particles.They found that the shock wave velocity increases with increasing piston velocity and initial overlap.Kandan et al.[10]studied the instability of the granular position surface for shock loading.In their study,the growth and instability of granular materials loaded by air shock waves were studied using the shock tube experiments and numerical simulation.The Eulerian-Lagrangian solver and the Druker-Prager model were used for granular materials.In a study,Fan et al.[11]examined the computational study on Peridynamics modeling and simulation of soil fragmentation under explosive loads.In their study,the SPH method was used.The results of the simulation were compared with experimental data,which are relatively close to each other.Moreover,the study showed that employing the Druker-Prager model,which describes steady-state and quasistatic loads behavior of granular soil material,may not be suitable for describing soil behavior at high strain rates.Mirova et al.[12]investigated the reduction of shock wave passing through a sand layer by changing the position of the sand layer installed at the end of the shack tube.Liang et al.[13]studied the shock waves in a onedimensional bead produced by a constant velocity impact in a short period using the numerical simulation.In their study,a cylinder and piston were used to generate the wave.It was found that the attenuation strength depends on the plasticity,the piston velocity,and the bead radius (see Tables 3 and 4).

This paper is an experimental and numerical study on the blast wave decrease using a sandwich panel by granular materials core.In this study,mineral pumice and sawdust were used as granular materials.Several compression tests were performed on them to determine the behavior of these materials under pressure and to extract the required data of numerical simulation,to achieve the properties of these materials.The simulation results were validated using several experimental results of the shock tube.

2.Numerical simulation

The solid materials in regular loading often have elastic behavior,but in the shock loading,they are usually entered into the plastic phase.The equation of state is responsible for explaining the hydrodynamic behavior of the material.Hydrodynamic behavior in most materials is best described by high-rate dynamic loading of the material.For example,the response of gases and liquids that do not tolerate shear is hydrodynamic and the pressure in them is a function of density and internal energy.Also,the behavior of solids is influenced by high hydrodynamic strain rates,where the applied pressure to the solids is much higher than that of the yield stress,and the strength of the material against the applied stress can be neglected.A material model in general requires equations to relate stress to deformation and internal energy to temperature.The equation of state can be determined from knowledge of the thermodynamic properties of the material and ideally,should not require dynamic data to build up the relationship.However,in practice,the only practical way of obtaining data on the behavior of the material at high strain rates is to carry out well-characterized dynamic experiments [14].To define a material in Autodyn Hydro code software,it is necessary to define the Equation of state(EOS),the Strength model,and the fracture model,each of which is defined by the type of material and its behavior.Once the matter is properly defined,the problem solver must be solved,the appropriate solver for the problem is defined according to the type of matter and the physics of the problem.Eulerian solver is generally used for gases and the Lagrangian Solver is used for solids.But the granular material is out of these two states of matter.Some researchers have used discrete element method (DEM) codes for granular materials,a problem of the DEM method that has made it less developed in particle mechanics is that it requires a large number of coefficients to be able to use the coefficients of hardness the springs and estimate the viscosity between the particles and the friction.Some of these coefficients do not even have a physical meaning,and some of them can only be obtained from highprecision bi-axial tests.Also,the numerical computation of the DEM method requires a computer with high computational capacity and time-consuming.Arbitrary Lagrangian-Eulerian Equation(ALE)solvers also can be used in Autodyn software for granular materials,which have been used successfully to model these materials in some studies [15,16].In this study,using Smoothed Particle Hydrodynamics(SPH)has been investigated.This method is a mesh-independent method such as the DEM method,but it is also broader in scope and its algorithm and computer program are further developed.So far,very few articles have used this method,but the results of using this method can be promising.The equations of state,the Strength model,and the fracture model used for the materials in the simulation are presented in Table 1(see Table 2).

Table 1 The equations of state,the strength model,and the fracture model used for materials.

Table 2 Changes in density and speed of sound at different pressures for sawdust.

Table 3 Changes in density and speed of sound at different pressures for the mineral pumice.

Table 4 Sawdust and mineral pumice yield stress at different densities.

Table 5 Shear Modulus of sawdust and mineral pumice at different densities.

To perform numerical simulation,it is necessary to introduce the properties of matter to the software,Properties required for simulation of granular material in Autodyn software include changes in density to pressure,changes in yield stress to pressure,changes in shear modulus to pressure,and changes in the speed of sound in granular material in different densities.Using axial and non-axial compression tests other than the speed of sound at different densities,the remaining required parameters can be obtained[17].In Fig.1 the compression test on the specimen is shown.

In this study,a press machine with a capacity of 30 tons and a compression rate of 2 mm/min as shown in Fig.1 has been used.The internal diameter of the matrix used in this experiment is 7 mm and the cavity depth was 100 mm.The matrix and punch used in the test are shown in Fig.1.The pumice grains are poured into the matrix cavity up to a height of 75 mm.The pressure to density changes can be obtained using an axial press.To obtain the most accurate results,three specimens of each material were tested and the data obtained from the three simulations were averaged using the data obtained from the three experiments.Two of the diagrams obtained from the compression test for sawdust and mineral pumice are shown in Figs.2 and 3.

Using the mean graph data,the density-pressure changes can be introduced into software,which is needed to define the compression equations of state.Considering some simplification assumptions,other properties needed for numerical simulation can be obtained from the compression test.In compression test specimens only the volume changes with the length of the cylinder,to define the granular material model a nonlinear elastic-plastic model can be assumed,Properties are assumed isotropic (homogeneous).In the elastic portion,the bulk modulus k is a function of the material density and the Poisson ratio can also be constant and is approximately 0.29.Then the relation between the elastic constants is as Eq.(1).

Fig.1.Compression test on granular material specimens.

In this relation,G is the shear modulus.The speed of sound in the elastic state in a grain material is obtained by Eq.(2) [18].

The bulk modulus is obtained by Eq.(3).

Where P is the pressure and εvis the volume strain[18].Therefore,using Eqs.(1)-(3),the speed of sound and the shear modulus at different densities are calculated.To calculate the yield stress of granular materials,a simplified relation is used by Coulomb [19],which for non-adhesive granular materials,Eq.(4) is presented:

Fig.2.The results of the sawdust compression test.

Fig.3.The results of the mineral pumice compression test.

Fig.4.The Mohr-Coulomb failure criterion,for the principal stresses and the angle of internal friction [19].

Fig.5.Nonlinear loading pattern in compression equation of state.

Fig.6.Simulation of the problem in the Autodyn software.

The Coulomb yield criterion for the granular material can be written as Eq.(5) from the principal stresses[20]:

In these relations p is the axial pressure,Cand φ represent the cohesion and internal friction angle of the granular material.The friction angle is about 43°for the sawdust and about 37°to 45°[21,22] for the mineral pumice.C is the coefficient of adhesion,which for non-adhesive granular materials such as pumice is a very small(C ≈0 kPa)[23,24]and for sawdust with a moisture content of less than 10%,it is about 7 kPa[24].The coefficient of friction for mineral pumice is between 0.3 and 0.7 and for sawdust is between 0.15 and 0.25.In this study,the average values,0.5 for mineral pumice,and 0.2 for sawdust according to Refs.[22,24] are considered.In both cases,the humidity is less than 10%.The principal stresses and internal friction angles are by Mohr-Coulomb yield as shown in Fig.4.

Eq.(6) is provided for soil shear strength and can be used for other granular materials.

In this relation σ is the normal stress[19].Using Eqs.(1)-(6)and laboratory results,the yield stress and shear modulus at different pressures and densities for the studied granular materials(soil and mineral pumice) are calculated.The calculation results are presented in Tables (2) to(5).

For the granular materials,the compression equation of state is used in this study.Using this model,there is more control over the porous materials during the loading and unloading phase.In this equation of state,the speed of sound is considered as a function of density.This model is widely used in shock simulation in granular materials such as modeling of soil,sand,etc in Autodyn software.The compression model in the unloading state gives more accurate results for nonlinear behaviors.The pressure can be calculated using the bulk modulus and instantaneous density:

Compression Equation of State loading and unloading is shown in Fig.5,at the point(Pn,ρn)the material reaches full compression.As can be seen in Fig.5,the unloading is done in a non-linear way,which is in the form of a dotted line,and their approximation is in the form of a solid line.The CTMDspeed of sound in the fully compressed state of matter [25].Because of the symmetry of the geometry and the loading of the problem,there is no need to simulate the problem in 3D and to reduce the cost of time,the problem is solved in two-dimensional axial symmetry,which shows in Fig.6 [25].

3.Experiment results

The explosive shock tube is used for experimental testing.Fig.7 shows a schematic diagram of the location of the specimens,explosive charge,and detonator in the test.In the first specimen,two sheets without core are adhesive together as shown in Fig.8.In subsequent experiments,the core of the specimens is 3,6,and 8 cm thick(see Table 5).

Table 7 Specifications of experimental test samples.

Fig.7.Schematic image of shock tube used in the experimental test.

Fig.8.A specimen without core and measure specimen deformation.

Fig.9.Specimens preparation.

Table 6 Mechanical properties of AA3105 aluminum alloy[26].

Fig.12.Explosion simulation for double layer aluminum foil.

The dimensions of the used shock tube are shown in Fig.7.The inside of all samples has circularly meshed for ease of measurement.Fig.9 shows the prepare the specimens.Granular materials are put into cloth bags and then into cardboard cylinders.The aluminum plate used on both sides is AA3105,the mechanical properties of which are listed in Table 6 [26].The cardboard cylinder is adhesive to the aluminum sheets by adhesive.

The explosive charge is of type C4 which is pasty-shaped and is shown in Fig.10.This charge is inserted into Teflon containers shown in Fig.11 left side and placed in the back of the shock tube as shown in Fig.11 in the middle.Teflon container back hole is for exit electric detonator wire ejection,and at the front is the location of the C4 explosive charge.In this study,an electric detonator with a score of 8(8 mm in diameter)is used,which has a power of 0.6 g of C4 explosive which should be considered in the simulation.The shape of the explosives is cylindrical.The diameter of the cylinder is twice its height and has been shaved inside the C4 and electric detonator holder due to the mass of the explosive charge.The sandwich panel specimens are fastened to the open end of the shock tube by 12 screws as shown in Fig.11 on the right side.The support is assumed to be completely clamped due to the high number of screws.After the test,the deformation of the outer sheet of the specimens at the mesh points(the mesh points on each line were 1 cm apart) is measured by a caliper as shown in Fig.8.The specifications of the experimental test samples are presented in Table 7.

Fig.10.Type C4 explosive charge.

Fig.11.Different views of the shock tube and explosive material holder.

4.Results and discussion

The simulation image for the non-core specimens with 4 and 5 g of the explosive charge is shown in Fig.12.

Fig.13.Pressure wave before impacting the panel in a 1 mm gap.

Fig.14.Explosion test of non-core specimens for 5 g of C4 explosive charge.

The shock wave at 1 mm before impacting the sheet is visible for 4 and 5 g of C4 explosive in Fig.13.

The result of the blast test by shock tube for the specimen with double layer aluminum foil without core is shown in Fig.14.

Fig.15.Simulation result for the core specimen.

In the second experiment,30 mm of sawdust was used as the core of the sandwich panel between the two sheets.The experiment was performed with 5 g of C4 explosive charge.In this case,the outer sheet (screwed to the open end of the shock tube) is completely torn and separated.The test result can be seen in Fig.21 on the left side.Fig.15 shows the simulation results of the Autodyn software for second specimen.It is evident that in the simulation,the specimen was also destroyed and the outer aluminum sheet was destroyed both from the stand and the center of the specimen.

In the third simulation,a specimen with a core thickness of 30 mm mineral pumice and an explosive charge of 5 g was used,and the simulation results shown in Fig.16.In this case,the sheet is not torn and only deformed.The test result is shown in Fig.22 on the left side.

Fig.16.Simulation result of a 30 mm thickness mineral pumice.

Fig.17.Simulation result of an 80 mm thickness mineral pumice.

Fig.18.Simulation result of a 60 mm-thickness sawdust.

In the fourth test,80 mm of mineral pumice was used as the core of the sandwich panel and the experiment was performed with 5 g of the explosive charge.The simulation result and the amount of deformation of the center of the outer sheet are shown in Fig.17.According to Fig.17,the center of the sheet is about 15.5 mm deformed.The result of the empirical test is shown in Fig.22 on the right side.

Fig.19.Is the result of simulating a specimen with an 80 mm thickness sawdust core.

In the fifth test,60 mm of sawdust was used as the core of the sandwich panel and the experiment was performed with 3.5 g of the explosive charge.The result of the simulation is shown in Fig.18.According to Fig.18,the center of the outer sheet has displaced about 20 mm.The experimental test result is also shown in Fig.21 in the middle section.

In the sixth simulation,80 mm of sawdust in the core and 3.5 g of explosive charge were used,according to the simulation result in Fig.19,the displacement of the center of the outer sheet is about 19 mm.Specimen of the empirical test shown in Fig.21 on the right side.

The simulation result of the seventh sample of a sandwich panel with 60 mm of mineral pumice in the core and 5 g of the explosive charge is shown in Fig.20.According to Fig.20,the center of the sheet has been deformed about 18 mm.The seventh test specimen is shown in Fig.22.The results of the measured values from the empirical test and numerical simulation are presented and compared in Tables 8 and 9.The results of the experiments and numerical simulation were compared and there was a good agreement between these two modes of investigations,indicating the validity and accuracy of simulation assumptions.

Fig.20.Simulation result of a 60 mm thickness core of mineral pumice.

Fig.21.Status of specimens with sawdust core after explosion test.

Fig.22.Status of specimens with mineral pumice core after the explosion test.

The shape of the profile created by the experimental test and numerical simulation can also be compared.For this purpose,it is necessary to measure the displacement of all nodes of a line from a circular mesh in the samples.In the numerical simulation,instead of each of these nodes(mesh nodes),a gauge is placed in the same coordinates that report the displacement of that point on the sheet.Figs.23 and 24 show the comparison of sheet deformation profiles in simulation and test for different specimens with mineral pumice and sawdust,respectively.The main issue in the explosion problems is the pressure of the blast wave which hits the plate as shown in Fig.13 for 5 and 4 g of C4 charge.Each of these sandwich panel reduces the percentage of the wave.Overpressure and impulse are among the basic parameters that are considered in the issue of wave mitigation.Fig.25 shows the percentage of overpressure depreciation for the gauge on the center of the outer sheet in numerical simulation compared to the wave hitting the panel in shock tube (Fig.13).According to the diagrams,it is clear that increasing the core thickness of the specimens did not have much effect on increasing the damping of the blast wave overpressure;but it was observed in Tables 7 and 8 that the rate of deformation of the center of the plate decreased significantly with increasing core thickness(see Fig.4).

Table 8 Comparison of Sheet Center Displacement in experimental testing and sawdust simulation (mm).

Table 9 Comparison of Sheet Center Displacement in experimental testing and Mineral Pumice Simulation (mm).

5.Investigating the effect of particle size and friction coefficient on shock wave depreciation

After observing the results for experimental and numerical testing and comparing them,using numerical simulation,the effect of grain size parameters,and the coefficient of friction between grains were investigated.The particle size of granular material as a factor that is directly related to sample porosity can be one of the factors affecting the propagation speed of the wave.In this section,the collision of the blast wave with four samples with the properties of mineral pumice with a thickness of 60 mm and a grain diameter of 5,7,9,and 10 mm has been simulated.The pressure wave that hit the samples for 5 g of C4 is according to Fig.13,which was measured 1 mm before it hit the sample.The pressure diagrams compared to the time in the gauge on the back of the aluminum plate after passing the samples are observed for different grain sizes in Fig.26.The particle size of the granular material is directly related to porosity.By increasing the particle size,which means more porosity,the overpressure is further reduced.

Fig.23.Comparison of deformation profile of numerical simulation and experimental panel testing with mineral pumice core.

Fig.24.Comparison of the numerical simulation deformation profiles and the experimental test panel with sawdust core.

Fig.25.Percent wave damping for different sandwich panel specimens.

Fig.26.Effect of grain size on wave mitigation.

The following are samples with mineral pumice properties and only with different coefficients of friction,the core thickness of the samples is 60 mm and the mass of C4 is 5 g.As mentioned earlier,the coefficient of friction for mineral pumice is between 0.3 and 0.7[20] Fig.27 shows changes in overpressure with a change in the coefficient of friction between the grains.

According to Fig.27,it is clear that with increasing the friction coefficient,the amount of overpressure is slightly reduced,so that for friction coefficients of 0.3,0.5,and 0.7,the overpressure is equal to 1290 kPa,1480 kPa,and 1500 kPa,respectively is observed.

Fig.27.The effect of the friction coefficient between grains in reducing shock wave.

6.Conclusions

This paper explored the ability of granular materials,such as sawdust and pumice to absorb blast energy was investigated.Pumice had a more significant effect at the same core thickness on blast energy absorption compared to sawdust for stronger waves,and its effect was noticeable even at lower thicknesses.Using these materials also had a great effect on reducing the sound of the blast.However,if the weight factor is also considered,sawdust had better performance,leading to more damped wave energy at a higher thickness,which can be highly effective.In the case of using a core with appropriate thickness,overpressure decreased to 88%,which indicates the ability of these materials (sawdust and pumice) to mitigate the blast wave.The deformation of the plate center varies significantly with the change in core thickness.Nevertheless,it is observed that the percentage of reduction in [shock/blast] wave overpressure has not changed significantly.The SPH method was used for simulation.The results revealed the appropriateness of this method in modeling granular materials under shock loading.An examination of the effects of grain size indicated a decrease in blast wave mitigation with a reduction in grain size.Investigating the coefficient of friction between particles exhibited that increasing the coefficient of friction between grains also led to a slight reduction in overpressure.

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.