Simulation of dynamic response of projectile and granular target

In the present study, the three-dimensional elastic discrete element method (DEM) and the elastic finite element method (FEM) analysis were, respectively, applied to the randomly distributed identical spheres as a granular target and to the cylindrical projectiles in order to clarify dynamic response during penetration. In the target, it was found that highly densified region was formed just ahead of the projectile and began to propagate spherically at much higher velocity than that of projectile leaving relatively rarefied region. It was also found that the peak resistance during penetration was closely related to the initial formation of densified region and was expressed in terms of momentum change of target particles accelerated by the projectile. Stress wave propagation in the projectiles with various body lengths was investigated during penetration and was discussed in connection with dynamic response of target particles ahead of the projectile. Effect of mechanical properties of projectile on the peak resistance was also investigated and was understood in connection with dynamic behavior of target particles.


Introduction
Collision impact between continuous medium and granular materials such as sand, soil, and heterogeneous brittle aggregates is interesting subject and has been investigated over several decades.Resistance during penetration has been identified in experiments and a various formulations for the resistance versus penetration relations have been proposed [1,2].Direct measurement of resistance has been also tried [3,4].It would be the first and most fundamental approach to clarify dynamic characteristics in the problem based on elastic wave propagation theory.It is also necessary to find out effect of mechanical impedance of materials upon the dynamic behavior of subject.
In the present study the discrete element method together with the finite element method were applied to simulate the dynamic behaviors of penetration into granular target and the characteristics of the problem were fully discussed in connection with dynamic movement of particles together with dynamic response of projectile.

Projectile
Figure 1 shows a projectile and a target of granular medium.A projectile was a cylinder of 42 mm in diameter and three kinds of length with a flat end.Three dimensional elastic FEM model was applied to the projectile having Young' modulus of 69 GPa and density of 3.8 × 10 3 Kg/m 3 with Poisson's ratio of 0.3.Initial velocity of the projectile was in the range of 60 m/s to 240 m/s. a Corresponding author: paddy-og@khaki.plala.or.jp

Target
The target was formed by randomly distributing identical spheres of 2.18 mm in diameter and of 3.76 g/cm 3 in density with Young's modulus of 310 GPa into a spheroidal container of 218 mm in diameter and of 160 mm in depth as shown in the figure.Average packing density was changed from about 60% to 48%.Elastic Discrete Element Method (DEM) was applied for the target.
Reactions between spheres were expressed by springs both in normal and tangential directions, together with a slider to represent static and kinetic frictions as shown in Fig. 2. Numerical simulations were carried out with time increment of 2.94 × 10 −8 s.

Behavior of target
To clarify deformation state in the particulate target we have focus our attention to the change of apparent packing density which may reflects to stress and strain in the sense of continuous mechanics.Figure 3 represents distribution of packing density caused by penetration of projectile at initial impact velocity of 120 m/s and 240 m/s.Just after the impact, higher density region appears just ahead of the projectile, and then, at almost the peak resistance which will be explained in detail, the region spreads spherically and begins to propagate downward at much higher velocity than that of projectile leaving relatively rarefied region.This rarefied region is essentially caused by difference of velocities of the projectile and the high density region and such velocity difference would be decreased with the increase of initial impact velocity.
With increase of initial impact velocity, the high density region grows much clearly and tails up to just ahead of the projectile as seen in the figure.The propagation velocity of such high density region was determined as follows.The location of the most distant particle which began to move at time t in vertical direction is expressed by (Y , r ) and that at time t + t is as (Y + Y, r + r ), and then the propagation velocity, C, can be evaluated as C = ( Y/ t). (1) In the case of granular medium the elastic wave velocity depends on the density.Elastic wave propagation in randomly arranged particles with separations as illustrated in Fig. 4a can be simplified by that in aligned particles with separations as shown in Fig. 4b.When the velocity, V , is applied to the left end particle, then it moves and hits its next neighbor particle and so on.Then we have the following relation for the elastic wave propagation; where C 0 is the elastic wave velocity in the case of s = 0, and n is a number of particles in the length L. Equation ( 2) can be rewritten as follows.
In Fig. 5, the propagation velocities at the peak resistance of projectile are plotted against the velocity of projectile following the Eq. ( 3) for different packing densities.The values of s/d can be obtained from the slope for respective lines.Since the close packing density is given by π √ 2/6 (1 + s/d) −3 , the estimated values are of 63%, 59%, 54% and 44% which are slightly different from the used packing density of 60%, 57%, 53% and 48%, respectively in the present simulation.The values of C 0 can be obtained by using the respective intercepts of lines together with the values of s/d, and are ranged from 500 m/s to 530 m/s almost corresponding to the value of 530 m/s for rectangular lattice arrangement without separation [5].

Wave propagation in a projectile
Elastic wave propagation in the projectile was investigated and was exampled in Fig. 6  Assuming the elastic wave propagation without retardation and free end reflection on the upper end of the projectile, the force at the impact head can be evaluated by superposing the elastic waves detected at the locations A, and B, and is given in Fig. 8(a,b) for the projectiles of 120 mm and 200 mm in length.It should be noted that the force decreases after the peak value with periodical change corresponding to the reflection of stress wave as indicated by an arrow, since the periods of 48 and 80 microseconds for the projectiles of 120 mm and 200 mm in length, respectively.It is also apparent that the force evaluated from acceleration of the projectile well corresponds to the force obtained by stress wave analysis, even though significant undulations appear at around the peak value.The relations between reaction force and time are strongly influenced by impact velocity as shown in Fig. 9. Significant increase of peak value can be understood by change of momentum of particles in target as discussed before [6].

Effect of mechanical properties of projectile
Effect of mechanical properties of projectile on the resistance force during penetration was also investigated.
Based on the one-dimensional elastic wave propagation theory, the reaction force of the projectile can be evaluated as follows.Let's denote ρ, C, and ρ t , C t , are the respective density and wave velocity of the cylindrical projectile and the target.The reaction force F at initial impact velocity of V 0 can be expressed as follows Where, A and A t are the sectional area of projectile and target, and σ R and σ T are the reflected and the transmitted stress wave amplitude in the projectile and the target, respectively.The effective sectional area of target, A, can be estimated as before [6], and is almost constant independent of the impact velocity.On the other hand, elastic wave velocity of the target depends on the impact velocity as shown in Fig. 5. Now, we can see the effect of mechanical properties of projectile upon the resistance force as shown in Fig. 10.For the projectile of initial values of Young's modulus of 69 GPa and of density of 2.8 × 103 Kg/m 3 , the

k 2 Figure 2 .Figure 3 .
Figure 2. Mechanical model of granular medium with elastic springs and sliders to represent static and kinetic friction.
. At the impact onto the 04040

Figure 6 .Figure 7 .Figure 8 Figure 8 Figure
Figure 6.Distribution of axial stress in projectile in time.