A Dynamic Relation Model Analysis of High-speed Impact and Sabot Strength

To understand how a projectile penetrates a sabot during a high-speed impact testing, which led to the failure of the impacting experiment, a dynamic model was built based on the interactions between the projectile and the sabot; and the bearing strengths of the sabot with varying impact speeds were analyzed. The correctness of the model was verified by an experiment. It shows that, with all the other conditions kept the same, the maximum stress between the sabot and the projectile increases with the decrease of the sabot mass, the increase of the projectile and impact speed.


Introduction
In earlier days, high-speed impact problems mainly appear in the military field, and now they emerges in civilian areas [1]. Air gun is a useful experiment tool for high-speed launching and high pressure loading. It is commonly used for the study of mechanics of materials, penetration, and high-speed impact [2]. The launching tube of the gun normally has a large diameter to meet the broader size experiment requirements. In this condition, a small-size projectile should be supported by a sabot, which traps the propellant gas and keeps the projectile centered in the launching tube while steadily moving forward. The sabot is not an indispensable component, the mass of the sabot is required to be designed as low as possible [3]. When firing, the high pressure gas forces the sabot to carry the projectile down the lauching tube, the contact surface of the sabot suffers huge stress. The material of the sabot should have high strength and good toughness to ensure its structure integrity [4]. Earlier sabots were made of metallic materials, such as red copper, copper alloy, pure iron, aluminum, aluminium alloy, etc. A large number of new materials who are light weight yet have high mechanical strength and good toughness have been appeared, such as various resin materials, fiber glass, carbon fiber, etc. [5]. The majority of the research on the sabot has been focused on the sabot separation process [6,7,8], the structural analysis of the composite sabot [9,10], the motion analysis during sabot opening process [11], sabot opening lift force [12], etc. While this paper focuses on how to keep the projectile and sabot always be a whole to ensure the high impact experiment doing successfully. Polyurethane foam is a good candidate for normal impact experiment which can fulfill the loading requirements with low cost. The strength of polyurethane foam depends on the foaming compactness. If the strength is insufficient, the projectile may penetrate it and causes the experiment failure. The relationships among the required strength of the sabot, the projectile mass and the experimental velocity should be analyzed and built to ensure a successful impact test.

The phenomenon of sabot penetrates projectile
The first-class air gun is commonly used during an impact testing. The diameter of the sabot is approximately equal to the diameter of the launching tube, so that it can carry various projectiles with different sizes (Fig. 1).

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2nd International Conference on Manufacturing Technologies (ICMT 2018) IOP Publishing IOP Conf. Series: Materials Science and Engineering 398 (2018) 012030 doi:10.1088/1757-899X/398/1/012030 Figure 1. Sketch of projectile and sabot in air gun When firing, the sabot and the projectile are accelerated instantly by the compressed air, a significant pressure is applied to their contact area. Fig. 2 shows a home-made sabot with density of 47kg/m 3 . When the velocity was higher than 145m/s, the projectile penetrated its support sabot and caused the failure of the high speed impact experiment (Fig. 3).   3 showed the moment of the projectile penetrated through the sabot, which was captured by a high speed camera. After recording the sabot at t 1 , the projectile rolled into the shot range at t 2 . In this condition, there was no way for projectile to hit the target plate with scheduled trajectory and speed. It hit the sabot separator and was rebounded. This led to a complete failure of the experiment. The free flying projectile would easily hit on the sabot separator, and both of them were damaged (Fig. 4), the projectile may not pass through the sabot separator successfully. And the irregular bouncing of projectile may cause severe safety issues. This penetration problem must be solved to ensure the high speed impact examination can be carried out normally.  It is necessary to find out the relationship between the shear strength of the sabot, the projectile velocity and their mass to ensure a successful high speed impact experiment.

Dynamic analysis of interaction force between sabot and projectile
Define m 1 and m 2 as the mass of the sabot and the projectile respectively, 1  and 2  as the density of the them, f D , f S and f L as the diameter, sectional area and effective length of the launching tube, p as the gas pressure at some moment, x as the distance between the projectile and its initial position. Sabot velocity v is an important parameter in air gun test. It can be estimated by the following formulas. When the gas pushes the projectile, the gas expansion process can be regarded as the ideal gas adiabatic expansion process [13].
In the formula: p -gas pressure； V -space volume behind projectile； g  --the mocular number of gas； M -injected gas mass； R -gas constant； T -gas temperature；  --gas adiabatic index； f S --cross section area of launching tube， The stress analysis of the sabot is carried out when the high pressure air is suddenly released (Fig. 5). The sabot is subjected to the force exerted by compressed air( 01 F )and projectile( 21 F ).

Figure 5. Force analysis of sabot
Without considering the specific forms of various energy loss, the mass of the sabot is increased from m 1 to φm 2 in order to replace various loss factors. So the motion equation of the sabot is as follows: where φ is called virtual mass coefficient, which is calculated as follows in the interior ballistics theory: The factor K is determined by experiment, which is about 1~1.10. The letter m represents the sum mass of sabot and projectile and M is the mass of the gas.
According to the motion consistency, the sabot and projectile can be regarded as one part, the change law of stresses exerted on them are similar to the chamber pressure in launching tube [14], which increases sharply at first then decreases gently (Fig.6). In order to simplify the calculation, it is assumed that the contact stress at the bottom of the projectile increases linearly during the launching process, and k is the linear slope. It is assumed that m v is the maximum speed and m L is the moving distance of sabot when the stress between sabot and projectile reaches the peak value. The relation law between 21 p and x can be expressed as Eq. (7).
Eq. (7) is put into Eq.(6), and Eq.(8) is obtained through integral. The projectile is analyzed without considring other forces (Fig. 7). The force from the sabot acting on the projectile 12 F is equal to its reaction force 21 F in numerical value and opposite in direction. The projectile is embedded in the sabot then move together with the same speed after firing. The mass loss of projectile is set as 2 m  to represent various loss factors. Then Eq.(9) is obtained as follows:

Experimental verification
From Fig.6, when projectile travels m L and the velocity of it reaches m v , the pressure exerted on it becomes the maximum. With the distance increasing, the pressure in launching tube decreases, but the projectile is still accelerated during the whole launching process. Its velocity e v at the end position of gun barrel must be higher than m v . The pressure calculated by e v will be higher than the actual maximum pressure. Replacing m v as e v to analyze the strength of sabot will be more reliable. The experimental device is shown in Fig. 9. The diameter of the metal projectile is 25mm, the air gun's tube diameter is 65mm, the sabot's length l is 100mm, the design impact speed e v is 150m/s, the cs p value is 0.132MPa after calculating by above equations. The calculation parameters of this high speed impact experiment are shown in Table 1.  . In other words, when the sabot and the projectile as a whole run to the middle position of the launch tube, the pressure is expected to reach the maximum of 2.68MPa. Because the damage between projectile and sabot is obviously shear failure, the shear stress is calculated as follows: Where dw d is the diameter of projectile， l is the length of sabot， k l is the depth of hole in projectile.
If we put the related parameters of projectile and sabot ( ). The shear test is carried out, the compressive strength and shear strength of the sabot are 0.15MPa and 0.02MPa. They are less than 0.24MPa, which means that the damage will occur. This calculation result is consistent with the examination phenomenon (Fig. 3), the projectile penetrated through the sabot and formed a hole. The material of sabot was changed to increase its density and shear strength. Three new sabots were made, Fig.10(a) and (b) were polyurethane foams,whose density were 200 3 / kg m and 300 3 / kg m . Fig.10(c) was made by ethylene vinyl acetate (short for EVA) foam. Their shear strength were 0.12 MPa ,0.18 MPa and 0.4MPa respectively and higher than the calculation value. The three material were all used and the experiments were all successful. The density of EVA foam was only 40 kg/m 3 , and it was the best choice for sabot among the three material.

Conclusions
In view of the separation of projectile and sabot in high-speed impact experiment, the dynamics of high speed impact and the bearing strength of the sabot were analyzed and deduced. The model of the interaction between the sabot and the projectile is established, and the correctness of the model is verified. According to the relationship model, the following conclusions can be acquired:  Under the same conditions, the lighter the sabot, the larger the maximum stress between sabot and projectile.  Under the same conditions, the heavier the projectile, the larger the maximum stress between sabot and projectile.  The higher the impact speed, the larger the maximum stress between sabot and projectile. This model is acquired by theoretical derivation, but it still needs more data to verify its validity, and the difference between theoretical data and real data should be researched deeper.