Guo-qiang Deng,Xiao Yu

Institute of Defense Engineering,AMS,PLA,Beijing,100850,China

Keywords: Explosion Cased bomb Bare explosive Numerical simulation MK84

ABSTRACT When considering the bomb explosion damage effect,the air shock wave and high-speed fragments of the bomb case are two major threats.In experiments,the air shock wave was studied by the bare explosives superseding the real cased bomb;in contrast,the bomb case in fluence was ignored to reduce risk.The air explosion simulations of the MK84 warhead with and without the case were conducted.The numerical simulation results showed that the bomb case signi ficantly in fluenced the shock wave generated by the bomb:the spatial distribution of shock wave in the near field changed,and the peak value of shock wave was reduced.Breakage of the case and kinetic energy of the fragmentation consumed 3 and 38%of the explosion energy,respectively.The increasing factors of the peak overpressure induced by the bare explosive on the ground and in the air were 1.43-3.04 and 1.37-1.57,respectively.Four typical stages of case breakage were defined.The mass distribution of the fragments follows the Mott distribution.The initial velocity distribution of the fragments agreed well with the Gurney equation.

1.Introduction

As an indispensable part of a bomb,the case carries all the explosives inside[1-3].As the bomb detonates,the case is broken by the detonation wave and the detonation products[4].High-speed fragment clusters and air shock waves,the major killing factors of the bomb[5],are subsequently generated.Bare explosives were commonly used in studies without considering the case[6,7].No case breaking process was considered when studying the generation process of the air shock wave.However,a signi ficant effect can be induced by the bomb case[8-10]:generally,the breakage of a case and accelerated flight of the fragments consume part of the explosion energy.This energy consumption generally increases with the thickness of the case,i.e.,it is closely related to the chargeweight ratio.For ordinary blasting bombs,the fragments consume 30-40%of the explosion energy.

For safety considerations,bare explosives are usually used in experiments without considering the effect of the bomb case[11,12],resulting in most of the existing data being obtained from a bare explosive.However,applications of bare explosive data usually encounter dif ficulties in evaluating the in fluence induced by the bomb case because related studies are considerably limited.Moreover,when studying the fragment effect,the flight process,distribution on the target,and the single killing effect of the fragments have been of greatest concern[13-15].Neither the formation of the fragments nor the speci fic impact of the fragments on the air shock wave has been considered.

The MK84 low-impedance aerial bomb is one of the earliest developed,wide-ranging,and large-scale bombs in current US military equipment.Many countries have purchased and equipped themselves with MK84 bombs[16].It is of great practical signi ficance to investigate the explosive damage capability of MK84 bombs.

The MK84 low-resistance aerial bomb is the heaviest in the MK series.The streamlined shape of the MK84 bomb conforms to the principle of high-speed flight aerodynamics,which makes the MK84 bomb suitable for modern aircraft.The MK84 warhead is also an improved prototype of the ground-drilling bomb.For instance,the BLU-109 warhead was improved based on an enhanced type of MK84 bomb.Both warheads have the same pound level and can use universal loading pendants.In the early days of the MK84 bomb,auxiliary or deceleration devices were used to obtain better precision guidance capabilities.Now,precision guidance devices are added to create a warhead for missile-based munitions,such as the GBU-31 Joint Direct Attack Ammunition(JDAM).The MK84 is mainly used to attack ground or shallow buried targets.

The damage ability of the MK84 aerial bomb mainly comes from the air shock wave and high-speed fragments formed by the highly explosive charge.The combined effect of the air shock wave and high-speed fragments on the target is much higher than the single effect.However,due to the lack of fragment-related research,whether it is weapon effect damage research or engineering protection design,a single factor of air shock waves is considered,which is obviously inconsistent with the actual weapon killing effect.In particular,the attenuation of the shrapnel flight speed with distance is much slower than the air shock wave,and the impact on the area where the shrapnel can directly act will be more obvious[5].

In the present study,the in fluence of the bomb case for the MK84 was investigated by numerical simulation.Different explosion processes,energy transition processes,explosion flow fields,and overpressure time histories of the cased explosive and bare explosive were compared.The fragment generation process was described in detail.The distribution of the mass and the initial velocity of the fragments were also discussed.

2.Numerical model

2.1.Basic parameters of the MK84 bomb

The MK84 bomb is a 2000-pound level blasting bomb.It is filled with H-6 explosives and has a charge of 429 kg.The shape of the bomb is streamlined with a thin case,as shown in Fig.1.The length and diameter of the bomb are 2748 and 457 mm,respectively.

Fig.1.Geometric parameters and photo of the MK84 bomb.(a)Geometric parameters of MK84 and(b)unexploded MK84 in soil.

2.2.Material model

H6 is a widely used high-performance aluminum-containing explosive,and its explosion process can be described by the JWL equation of state[17]:

whereA,B,R1,R2,andωare the material constants;Pis the pressure;Vis the relative volume;Eis the initial speci fic internal energy.The parameters of the above equation are shown in Table 1.

Table 1 Parameter table of the JWL equation for the H6 explosive[18].

The case material is 4340 steel,which is modelled by the impact state equation and Steinberg-Guinan strength equation[19,20].

The shock equation of state is a simpli fied form of the Grüneisen equation of state[21].According to the impact Hugoniot condition[21],the form of equations similar to the Grüneisen equation of state can be established as follows:

wherePis pressure;eis internal energy;ρis density;ρ0is initial density;the Grüneisen coef ficientΓis 1.67;c0is 4580 m/s;sis 1.33.

For the Steinberg-Guinan strength equation [19,20],the constitutive relationship between the shear modulusGand yield strengthYat a high strain rate can be expressed as follows:

whereεis the effective plastic strain;Tis the temperature;η=v0/v is the compression ratio.After the melting temperature is exceeded,the yield strength and shear modulus are set to zero.

For 4340 steel,the density is 7810 kg/m3;initial shear modulusG0is 80.1 GPa;initial yield stressY0is 1200 MPa;maximum yield strengthYmaxis 2500 MPa;hardening coef ficients 2;hardening indexnis 0.5;dG/dPis 1.479;dG/dTis-2.262e7 Pa/K;dY/dPis 0.03214;melting temperature is 2310 K;temperature indexnis 1.2.

The air uses the ideal gas equation with a density of 1.225 kg/m3,andγis 1.4.

2.3.Numerical simulation model

The FE model was developed on the LS-DYNA platform,which has multi-material Euler and Lagrange algorithms.It uses explicit algorithms to analyze transient shock dynamics,which are mainly used for shock wave simulation.

The axisymmetric model was built,the symmetric boundary condition was applied to the centreline of the model.The nonre flecting boundary conditions were applied to the boundary of the model except for the ground.The fixed boundary condition was set for the ground.The keyword*INITIAL_DETONATION was defined to detonate the explosive;the detonation point was the charging center.

The calculation was divided into two steps.In the first step,a fine mesh size of 2.5 mm was used to mesh the near-field model,which was 2.5 m in width and 5.5 m in height.The warhead is 2 m above the ground.The Arbitrary Lagrange-Euler(ALE)algorithm,which combines the merits of the Lagrange and Euler algorithms that are often applied to eliminate numerical dif ficulties induced by severe distortion of solid meshes,was adopted to simulate the explosive and air.The mesh type of case was Lagrange.The interaction between the fluid field and solid fragments was simulated by the coupled Eulerian-Lagrangian(CEL)method.In the second step,a coarse mesh size of 20 mm was used to mesh the far-field model,which was 20 m in width and 10 m in height.The initial conditions of the second step were obtained by mapping the simulation result of the first step,as shown in Fig.2.

Fig.2.The two-step numerical model.(a)The near-field model and(b)the far-field model.

The Euler hydrodynamic equations,which consisted of the continuity equation,momentum equation,and energy equation,was used to solve the simulation.It can be expressed as follows:

whereρis the density of the fluid;pis the pressure;e is the internal energy;u,v,andware the velocity components in the directions ofx,y,andz.

3.Numerical simulation results

3.1.Explosion process of the cased explosive

The explosion process of the cased explosive in the near field is shown in Fig.3.After the MK84 warhead detonated in the center,the detonation wave propagated towards the case.When encountering the case,a reflection occurred,and the case bulged outward.At 181μs,the detonation wave reached both ends of the warhead.At this time,the bomb cavity was filled with high-temperature and high-pressure detonation gas,which were uniformly distributed.The warhead was deformed into a spindle shape.The case was swelled outwards and gradually accelerated due to the hightemperature and high-pressure detonation gas.Then,the case began to rupture from the weak points.The high-temperature and high-pressure gas began to flow out of the broken case,and the overall shape became a drum.

The breaking process of the case is shown in Fig.4.Because the thickness of the MK84 case is relatively thin,the strength of the case can be neglected compared with the extremely high-pressure denotation wave.The case expands outwards as the detonation wave propagates,as shown in Fig.4b.The case deforms quickly and gradually deforms into a spindle shape,with obvious weaknesses,as shown in Fig.4c.Attributed to the largest deformation,cracks are first generated in the center portion of the case.Then,the cracks are connected with the expansion of the case,as shown in Fig.4d.The connected cracks continuously spread towards the two ends of the case,as shown in Fig.4e.After the detonation process,the highpressure gas in the case continuously accelerates the fragments outwards.The velocities of the fragments reach the maximum with an increase in the gas leakage and expansion of the case.Then,the fragments decelerated due to air resistance.Finally,the connected fragments are separated and fly outwards independently,as shown in Fig.4f.

The complete formation process of the case fragments can be divided into four stages:(1)the detonation process of the highly explosive charge;(2)the interaction of the detonation waves with the case;(3)the breaking process of the case;and(4)the acceleration and deceleration of the fragments in the detonation products and air.

In the far-field,as shown in Fig.5,the air shock wave and detonation product gradually separated.The shock wave re flects on the ground and forms a Mach shock wave that propagates along the ground.

3.2.Explosion process of the bare explosive

Although the explosion process of the bare explosive exhibited a similar pattern to that of the cased explosive,as shown in Fig.6,the detonation product of the bare explosive spread outwards faster,which was attributed to the absence of the case.The detonation wave had a spherical shape at the beginning.Then,the wave quickly reached the top and bottom ends,forming a fatter spindle shape.

In the far-field,as shown in Fig.7,the shock wave propagation process and the detonation product of the bare explosive showed a similar trend to that of the cased explosive.When the shock wave reached the ground,the superstition of the re flected wave and the incident wave formed the Mach wave,which then propagated along the ground.

Comparing the bare explosive with the cased explosive,it could be concluded that the presence of the case changed the spatial distribution of the detonation wave,limiting the expansion of the detonation product.This phenomenon is more signi ficant in the near field.For a real bomb,the shock wave is generated by the hightemperature and high-pressure detonation gas escaping from a rupture in the case.The case plays a role in hindering the formation of the shock wave.In contrast,in the bare explosion scenario,the shock wave is directly generated from the compression of the detonation product.The shock wave can travel faster and carry more energy.

Fig.3.The explosion process of the cased explosive in the near-field.

Fig.4.Breaking process of the case.(a)Before detonation;(b)expansion of the case;(c)weakness generation;(d)cracks connection;(e)spread of the cracks towards the two ends;and(f)fragments formed.

Fig.5.The propagation process of the blast wave generated by the cased explosive in the far-field.

4.Discussion

4.1.The explosion energy transition process

4.1.1.The cased explosive

After detonation of the cased explosive,the explosive energy of the charge is instantly released.This energy is converted into the internal energy of the explosion product,which is accompanied by detonation wave propagation.When the detonation wave propagates to the top and bottom ends,the explosion energy is completely released.At this time,the internal energy of the explosion product reached the maximum value.With the expansion of the detonation product,the internal energy of the explosion product was gradually converted into the kinetic energy of the explosive product and the kinetic and internal energy of the fragments.The internal energy of the detonation product gradually decreased,while the kinetic energy of the detonation product and the kinetic energy of the fragments gradually increased.During this period,the air shock wave was not yet formed,and this wave carries a very small portion of energy.

Fig.8.The energy transition process of the cased explosive.

The energy distribution data of several typical moments,which are taken from Fig.8,are shown in Table 2.At 0.15 ms,the detonation process had not yet ended.The internal energy of the detonation product was dominant,accounting for 56%of the whole explosion energy.The kinetic energy of the detonation product and the kinetic energy of the case accounted for 17%and 22%,respectively.At 0.5 ms,the fragments formed.The detonation product continuously drove the fragments to accelerate.The energy consumed by the shattering of the warhead case accounted for 3%of the explosion energy.The kinetic energy of the fragment group accounted for 38%of the explosion energy.The total energy of the detonation product accounted for 56%of the explosion energy,of which the kinetic energy and internal energy each account for 28%.The air shock wave was in the initial formation stage,and the sum of its internal energy and kinetic energy accounted for approximately 3%of the explosion energy.

Table 2 Energy distribution of typical moments of the cased explosive.

4.1.2.The bare explosive

Compared with the cased explosive,as shown in Fig.9,the explosive product expanded faster after the detonation of the bare explosive,which led to a faster increase in its kinetic energy.At 0.16 ms,the internal energy of the detonation product reached the maximum value,which was slightly smaller than that of the cased explosive.However,the kinetic energy of the detonation product,which did not yet reach the maximum,was greater than that of the cased explosive.Thereafter,the internal energy of the detonation product gradually decreased,and the kinetic energy of the detonation product gradually increased.At 0.20 ms,the kinetic energy of the explosion product approached the maximum,which was higher than that of the cased explosive.At 0.39 ms,the energy of the detonation product still occupied the main part of the explosive energy.The air shock wave was still in the formation stage.

The energy distribution data of several typical moments,which are taken from Fig.9,are shown in Table 3.At 0.15 ms,the energy of the detonation product was dominant,and the internal energy and the kinetic energy each accounted for approximately half.At 0.30 ms,the kinetic energy of the detonation product was already dominant,accounting for 62%of the total explosive energy.The air shock wave was in the formation stage,and the sum of its internal energy and kinetic energy accounted for approximately 12%of the total energy.

Fig.9.The energy transition process of the bare explosive.

Table 3 Energy distribution of typical moments of the bare explosive.

4.2.Spatial pro file of the explosion flow field

The explosion flow field is cut by the latitudinal surface and the longitudinal surface,as shown in Fig.10 and Fig.11,respectively.The latitudinal surface is the surface through the axis of the explosive and parallel to the paper;the longitudinal surface is the surface through the axis of the explosive and vertical to the paper.The explosion flow field of the bare explosive at 387μs is compared with that of the cased explosive at 587μs?The reason for this comparison is that the explosion wave of the bare explosive propagates faster than that of the cased explosive.Although the time of the bare explosive is less than that of the cased explosive,the propagation distance of the explosion wave of the bare explosive is larger than that of the cased explosive.

As shown in Fig.10,although the case of the cased bomb explosion was already broken,it can still provide a strong constraint on the detonation product.The fragment velocity was higher than the velocity of the air shock wave front.The velocity of the fragment in the middle of the case was 2252 m/s,and the shock wave front speed was 1961 m/s.The velocity of the detonation product in the cavity was roughly linearly related to the distance,which is plotted as a red dashed line in Fig.10a.The velocity of the detonation product in the center was close to zero,and the outer side was 1920 m/s.The velocities of the case at the top end and the bottom end were 730 and 190 m/s,respectively,which were much lower than those in the middle,as shown by the red dashed line in Fig.11a.In the latitudinal section,as shown by the red dashed line in Fig.10b,the overpressure showed a trend of high in the center(approximately 21 MPa)and low on both sides(approximately 8 MPa).In contrast,the overpressure in the longitudinal section changed more drastically,as shown by the red dashed line in Fig.11b.

For the bare explosive,as shown by the green line in Fig.10a,the velocity of the detonation product in the middle was as high as 2800 m/s,the shock wavefront velocity was 1929 m/s,the velocity of the detonation product in the cavity was roughly linear with the distance,and the center was close to zero.The velocities of the detonation product at the top and bottomwere 2018 and 4290 m/s,respectively,as shown by the green line in Fig.11a.In the latitudinal section,a high overpressure of 26 MPa was observed in the center.As shown by the green line in Fig.10b,the overpressure first decreased and then increased as the distance increased.An air shock wave formed at a distance of 1.75 m.The overpressure in the longitudinal section changed drastically,as shown by the green line in Fig.11b.

Fig.10.Distribution of the explosion flow field in the latitudinal section.(a)Velocity and(b)overpressure.

Fig.11.Distribution of the explosion flow field in the longitudinal section.(a)Velocity and(b)overpressure.

4.3.Overpressure time history of the shock wave

Two sets of recording points were set,as shown in Fig.12.The first set of recording points,numbered from 1 to 9,were laid on the ground.Point 1 was 3 m away from the central axis of the warhead.The spacing between points was 2 m.The second set of recording points,numbered from 11 to 19,were at the same height of 3.5 m above the ground,which was identical to the detonation point of the explosive.Point 11 was 1 m away from the central axis of the warhead.The spacing between points was 2 m.

Fig.12.The layout of the recording points of the blast wave.

The overpressure time histories on the ground,recorded by points from 1 to 9,are compared in Fig.13.At the same distance,compared with the bare explosive,the cased explosive had a much lower peak overpressure and a later shock wave arrival time.The distribution of the shock wave and its attenuation effect in the near field were severely disturbed by the case and its fragments.However,the case and its fragments did not cause a signi ficant in fluence of the shock wave in the far field,as shown in Figs.13 and 15a.

The overpressure time histories in the air,recorded by the points from 11 to 19,are compared in Fig.14.At the same distance,compared with the bare explosive,the cased explosive had a lower peak overpressure and a later shock wave arrival time.Compared with the shock wave on the ground,the shock wave in the air and its attenuation effect exhibited a slighter in fluence from the case and its fragments,as shown in Figs.14 and 15b.

Fig.14.Overpressure time histories of the shock wave in the air.(a)Cased explosive and(b)bare explosive.

Fig.15.The attenuation of the peak overpressure with propagating distance.(a)Ground and(b)air.

The data in Fig.15 are tabulated in Table 4,an increasing factor of the bare explosive was defined as follows:

Table 4 Comparison of peak overpressures of the cased explosive and the bare explosive.

wherePbis the peak overpressure of the bare explosive;Pcis the peak overpressure of the cased explosive.

For the points on the ground,the maximum value ofIfis 3.04,which reaches the local area within a diameter of 3 m on the ground.Outside the local area,Ifranges from 1.75 to 1.43.The air recording points show a variation trend,which ranges from 1.57 to 1.37.

4.4.Distribution of fragments

4.4.1.Mass distribution

Fragment mass distribution is usually described by a cumulative distribution function,rather than a probability density function,which is more sensitive to the scatter of the fragment masses data.Mott[22,23]had formulated the well-known fragment distribution law in the form

whereN(mf)is the cumulative number of fragments,which is the total number of fragments with the mass greater thanmf;Mis the total mass of the case;is the average mass of the fragments;iis the dimension(1,2,or 3);μiis the parameter related to,which can be calculated by

Fig.16.Mass distribution of the fragments.

In total,520 fragments were generated in the present simulation.The mass distribution of the fragments shows a roughly uniform trend,as shown in Fig.16.The Mott distribution[2]is followed by the distribution curve of the mass.However,an obvious difference could be observed:compared with the Mott curve,the curve of the simulation results exhibits a lower amplitude in the lower mass but a higher amplitude in the higher mass.This phenomenon may account for a certain mesh size,which cannot be in finitely reduced in the simulation.The size of fragments obtained from the simulation is certainly larger than the mesh size.It is impossible to generate fragments finer than the mesh.

4.4.2.Initial velocity distribution

For a cylindrical charge with a uniform case thickness,the initial velocity of the fragments can be calculated using the Gurney equation[5]:

whereGis the Gurney constant of the charge;Wqis the mass of the charge;Wcis the mass of the case.For the MK84 bomb,Gof H6 is 2316 m/s;Wqis 429 kg;Wcis 421 kg.Then,the initial velocity of the fragments is determined as 1903 m/s by Eq.(13).

The velocity distributions of the fragments obtained from the simulation are shown in Fig.17.Every individual fragment with a certain initial velocity is plotted as a scatter.By dividing the masses of the fragments with initial velocities of different levels to the total mass of the case,the mass percentages of fragments are calculated,as shown the bars in Fig.17.The initial velocity of the fragments is mostly distributed in the range from 1800 to 2100 m/s,which dominates 95.1%of the total mass of the case.It could be concluded that the simulation result is reliable.

5.Conclusions

This paper presents a numerical simulation to evaluate the case effect of an MK84 explosion in the air.The case can signi ficantly in fluence the explosion process.The case changes the spatial distribution of the shock wave in the near field and greatly reduces the peak overpressure of the shock wave.The increasing factors calculated byPb/Pcon the ground and in the air are 1.43-3.04 and 1.37-1.57,respectively.Four typical stages of case breaking are observed.The mass distribution of the fragments follows the Mott distribution,but an obvious difference exists due to a certain mesh size that could not be in finitely reduced.The initial velocity distribution of the fragments is well predicted by the Gurney equation.

The results of the present research can promote the accuracy of evaluating the blast scenario of the real bomb.A further understanding of the case effect of the air-explosion bomb makes it possible to predict the potential load of the structures,weaponry,personnel,and facilities that may confront.Consequently,modi fications can be made for the available standards and guidelines on designing the protective structure or developing the weapons.With the improvement of precision guidance technology and penetration ability of the bomb,underground explosions of the bombs frequently occur in modern warfare.It is well known that the circumstances in fluence the blast propagation and damage effect signi ficantly,which indicates the results obtained from the airexplosion cannot apply to the underground-explosion.In our subsequent research,the cased bomb detonated underground will be investigated.

Fig.17.Mass distribution of the fragments:(a)linear scale,and(b)semi-logarithmic scale.

Declaration of competing interest

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