On the hydrodynamic approximation for long-rod penetration

Steady-state hydrodynamic theory, or variations thereof, has been applied to long-rod penetration since the 1940s. It is generally believed that projectile strength is of little consequence at high velocities, and that hydrodynamic theory is applicable to long-rod penetration when penetration pressures are much greater than the target flow stress. Substantiating this belief is the observation that at approximately 2.5 km/s, for tungsten alloy projectiles into armor steel, normalized penetration (P/L) nominally saturates to the classical hydrodynamic limit of the square root of the ratio of the projectile to target densities. Experimental data herein, however, show penetration velocities and instantaneous penetration efficiencies fall below that expected from hydrodynamic theory, even at impact velocities as high as 4.0 km/s. Numerical simulations, using appropriate strength values, are in excellent agreement with the experimental data. Parametric studies demonstrate that both projectile and target strength have a measurable effect even at such high impact velocities.     

Penetration of semi-infinite AD995 alumina targets by tungsten long rod penetrators from 1.5 to 3.5 km/s

The performance of confined AD995 Alumina against L/D 20 tungsten long rod penetrators was characterized through reverse ballistic testing. The semi-infinite ceramic target was cylindrical with a diameter approximately 30 times the rod diameter. The target configuration included a titanium confinement tube and a thick, aluminum coverplate. The impact conditions ranged from 1.5 to 3.5 km/s with three or four tests performed at each of six nominal impact velocities. Multiple radiographs obtained during the penetration process allowed measurement of the penetration velocity into the ceramic and the consumption velocity, or erosion rate, of the penetrator. The final depth of penetration was also measured.

Primary penetration approaches 75% of the hydrodynamic limit. Secondary penetration is very small, even at 3.5 km/s. The effective R_t value decreased from 90 kbar to 70 kbar with increasing impact velocity over the range of velocities tested.

In tests in which the ratio of target diameter to penetrator diameter was reduced to 15, R_t, dropped by 30% to 50%. When a steel coverplate was used, total interface defeat occurred at 1.5 km/s.     

A numerical comparison of the ballistic performance of unitary rod and segmented-rods against stationary and moving oblique plates

The purpose of this paper is to investigate the ballistic performance of segmented-rods against stationary or moving oblique plates. To do this, a series of three-dimensional numerical simulations for the impact characteristics of segmented-rods (5 of L/D=1) into stationary or moving oblique thin-plate targets is conduced. To provide a base line data, an L/D=5 unitary rod projectile which has the same mass and kinetic energy is also considered. The ballistic characteristics of the projectiles are evaluated by examining the crater profile in a thick witness target that is placed behind the oblique plate. The impact velocities considered are 1400, 1800 and 2200 m/s. The results for the test range show that the unitary rod projectile shows better performance in the moving oblique target than the stationary one and the segmented-rods always show slightly better performance in the stationary target. From the impact velocity of 2200 m/s, the outstanding penetration performance of the segmented-rods can be observed. This trend is due to the interaction between the reactive plate and projectile. The extent of the interaction relies on the relative velocities of the plate and projectiles, the plate angle and extended total length of the segmented-rods     

The penetration of rigid long rods – revisited

The penetration process of rigid long rods with different nose shapes (ogive, spherical, conical and flat) is analyzed through a series of 2D numerical simulations. Aluminum and steel targets with different strengths (and large dimensions) are used to follow the deceleration process of these rods from impact, at different velocities, to the final penetration point. We find that for low impact velocities the deceleration of these rods is practicably constant, depending only on the strength of the target and the nose shape of the rod. Above a threshold (critical) impact velocity rod deceleration becomes velocity dependent due to the inertial response of the target. These critical velocities depend on the strength of the target and the nose shape of the rod. These observations led us to propose a simple penetration formula which accounts very well for penetration depths data for rigid steel rods with different nose shapes, impacting various aluminum targets at velocities up to about 1.5 km/s. For higher impact velocities, where the dynamic (inertial) contribution to the target resistance is important, we find good agreement between our model predictions and the simulation results for final penetration depths.     

MESA 3-D calculations of armor penetration by projectiles with combined obliquity and yaw

We introduce and briefly describe MESA, a new 3-D Eulerian hydrodynamic code, developed specifically for simulations of armor and anti-armor systems. The code's current capabilities and its planned model improvements and additions are discussed. MESA models hydrodynamic flow and the dynamic deformation of solid materials. It uses simple elastic-perfectly plastic material strength models as well as models with thermal softening and strain and strainrate hardening. Future versions will incorporate advanced fracture models. It treats detonations in explosives using a programmed burn. The code's current capabilities are illustrated with simulations of experiments on yawed rods obliquely impacting armor plates at 1.29 km/s. With nominal elastic-perfectly plastic strength parameters MESA predicts well the experiment measurements of rod length, velocity deflection, and emergent yaw rates but underpredicts exit rod velocities. An artificial simulation of fracture indicates that fracture modeling would further improve agreement with experiment. The utility of MESA to treat hypervelocity impacts is demonstrated with simulations of a rod obliquely impacting a thin plate at 5 km/s. At this much higher impact velocity, hole sizes are much larger and material strength plays a minor role in hole size and rod deformation.     

Monolithic and segmented projectile penetration experiments in the 2 to 4 kilometers per second impact velocity regime

A series of experiments was performed to evaluate the performance of projectiles impacting targets at velocities two to three times larger than conventional ordnance velocities. The results were positive, where low L/D ratio projectiles exceeded the theoretical hydrodynamic limit of penetration for the given projectile-target combination. High L/D ratio projectiles did not appreciably exceed the limit.

A second set of experiments was devised to test the hypothesis that a segmented projectile, - consisting of a series of low L/D projectiles, assembled in a long rod configuration, - could penetrate deeper into the target than a monolithic projectile of equivalent mass. The results were again positive, with a gain of about 10% shown in some cases. The balance of the experiments was devoted to developing a set of design rules and to exploring variations in the configuration and materials.     

Penetration dynamics of rods from direct ballistic tests of advanced armor components at 2–3 km/s

The penetration of semi-infinite steel and spaced-plate armors by continuous and segmented rods has been analyzed and measured by direct ballistic tests, hydrocode calculations, and hydrodynamic models at velocities from 2 to 4 km/s. An empirical equation of rod penetration in semi-infinite steel was formulated from hydrodynamic models of rod impact. Penetrations predicted by the equation agreed well with measured values. Increasing the spacing between segments from one to two diameters increased the penetration significantly (20%). Structures to support and align the segments can either increase or decrease the penetration, depending on their design. The relative penetrations of continuous and segmented rods depend on the parameters selected for the comparison: the segmented rod having greater penetration for equal mass and diameter and vice versa for equal mass and length. Tests of segmented rods penetrating spaced-plate armor showed that the armor is defeated by the front segment (or segments) punching a hole in the front plate (or plates) that allows the remaining segmented rod through intact to attack the main armor.     

The dependence of penetration velocity on impact velocity

Recent experimental measurements show that eroding long-rod penetration velocity is a linear function of impact velocity over a very wide range of impact velocities and for an interesting range of rod–target material combinations. These experiments all show that U=a+bV, where U and V are the penetration and impact velocity, respectively, and “a” and “b” are constants for given projectile and target materials. Numerical simulations also show that U=a+bV. The accumulation of these results suggests that a linear relationship between penetration and impact velocity may be fundamental over a very large range of impact velocities. A linear relationship between penetration and impact velocity has a number of implications. Some implications of this result for the Tate–Alekseevskii model are briefly examined in this paper.     

Yaw impact of rod projectiles

The purpose of this numerical study is to investigate the penetration regimes for L/D 30 tungsten-alloy rod projectiles for cases where the impact yaw angle varies from 0 to 90° and for impact velocities from 1.4 to 2.6 km/s. The target is modeled as a semi-infinite or half-space block of rolled homogeneous armor (RHA) at zero obliquity. For cases of mild interference, the penetration channel is still deep and narrow but may be skewed with respect to the original shot-line. While penetration is degraded the efficiency of the rod projectile remains relatively high. With increasing yaw angle the rod may deform due to transverse loading to the extent such that it contacts and produces gouges on the opposite side of the penetration channel. Additionally, lateral loading may induce angular acceleration to the extent such that the tail of the projectile rotates (in the plane of symmetry) and also contacts the opposite side of the penetration channel. In the next discernible penetration regime, the long-rod deforms under transverse load but the tail does not rotate significantly. It is seen that nearly the entire rod length experiences the lateral load with the result that the original shot-line is significantly altered. The deformed rod, again, has multiple contact or loading points (or regions) and the resultant angular acceleration appears to be insufficient to induce rotation of the projectile tail. Thus, rather than ricochet, the projectile cuts a significant slot into the target. Finally, for very large yaw angles the crater becomes indistinguishable from one produced by a side-on or 90° impact even though the impact yaw angle may be significantly less than 90°.     

Target strength effect on penetration by shaped charge jets

A limit of the applicability of the hydrodynamic theory was determined for penetration into targets of finite strength. At impact velocities below a limiting velocity, the target strength cannot be considered negligible. The limiting velocity is dependent on the target hardness and density. For the high-hardness aluminum and titanium alloys used in this paper, the limiting velocity, about 3.0 km/s, was found to lie in the range of velocities typical of shaped charge jets. Penetration into the aluminum and titanium alloys by portions of the jet with velocities below 3 km/s was studied experimentally; the depths of penetration were found to be significantly smaller as compared to the hydrodynamic predictions. This deviation has been attributed to the target strength effect.     

Brownian dynamics simulations of rigid rod-like macromolecular particles flowing in bounded channels

Brownian dynamics simulations have been carried out of the joint probability distribution functions (PDF), P(ξ,θ), for macromolecular rod-like particles in the limit of infinite dilution in a solution under hydrodynamic linear flow. These PDF are calculated as a function of the orientations of the rod-like particles, θ and of the positions, ξ, of their centres of mass measured from a solid surface boundary. These simulations are developed in the neighbourhood of a solid surface boundary and in a confined space bounded by two such boundaries. They are constructed for a wide range of key quantities depicting the ratio of the hydrodynamic shear rate to the rotational Brownian diffusion coefficient. The notion of restitution is introduced to develop an algorithm for the consequences of the Brownian and hydrodynamic collisions of these macromolecules with impenetrable solid surface boundaries, which approach applies to a wide range of surfaces and macromolecules. The simulation results for the PDF distributions are given for typically low and high hydrodynamic flow conditions, and their properties are discussed. We show, for example, for low shear rates that a phenomenon which we call Brownian restitution enables the macromolecular rods to pass through a channel that is narrower than the rod length.     

Penetration mechanics and performance of segmented rods against metal targets

Terminal ballistic experiments confirm theoretical predictions that a segmented rod will penetrate a semi-infinite metal target deeper than a continuous rod of the same material and having equal mass, diameter and velocity. For copper segmented rods impacting aluminum targets and tantalum segmented rods impacting 4340 (BHN 300) steel, penetration depths of at least 50 percent greater than that for a corresponding continuous rod are measured at impact velocities of 4 to 5 km/s. Spacing between segments of only about 2.5 segment diameters or more are required to achieve these results. Reducing the Li/D of the segments to less than 1 improves the penetration efficiency of a segmented rod. For segmented rods with segment L_i/D < 1, experiments suggest that penetration may increase with impact velocity rate greater than V^2/3.     

考虑攻角的长杆弹斜穿透中厚铝靶机理

攻角对长杆弹斜侵彻有重要影响,该文通过大量数值模拟研究了攻角对长杆弹斜穿透中厚铝板的影响机理。基于实验验证的有限元模型,开展了变速度和攻角的多工况数值模拟,得到了侵彻过程中弹体的减加速度大小、速度方向以及整体弯曲的变化规律,分析了侵彻速度、倾角和攻角对侵彻阻力、弹体弯曲和弹道偏转的影响。结果表明:带攻角斜侵彻时,负攻角对弹体弯曲的影响明显大于正攻角,且弹体弯曲随着侵彻速度的增大而减小;随着斜侵彻速度的增大,攻角引起弹体甩尾和弹道偏转越明显,此时带攻角的斜侵彻过程的能量损耗机理明显不同于正侵彻和无攻角的斜侵彻。     

Penetration equations for ogive-nose rods into aluminum targets

We present penetration equations for rigid, ogive-nose, rod projectiles that penetrate aluminum targets at normal impact. Comparisons of depth of penetration data and predictions from a previously published penetration equation derived from spherical, cavity-expansion methods show excellent agreement for striking velocities to 1800 m/s. We then identify a small parameter in the penetration equation, perform a power-series expansion, and obtain approximate penetration equations. These approximate equations are very accurate for striking velocities to 1300 m/s and display clearly the dominant problem parameters.     

Reverse impact experiments against tungsten rods and results for aluminum penetration between 1.5 and 4.2 km/s

Reverse impact experiments against 0.76 mm diameter L/D = 20 tungsten rods have been conducted with a 38 mm diameter launch tube, two-stage light-gas gun using four 450 kV flash X-rays to measure penetration rates. Techniques for projectile construction, sample placement, alignment, and radiography are described. Data for penetration rate, consumption velocity, and total penetration were obtained for 28 mm diameter 6061-T651 aluminum cylinders at impact velocities between 1.5 and 4.2 km/s. It was found that penetration velocity was a linear function of impact velocity over this velocity range. Above 2 km/s impact velocity, penetration was completely hydrodynamic. There was substantial secondary penetration, and the total penetration increased linearly with impact velocity over the range 1.5 to 2.5 km/s.     

Hypervelocity jacketed penetrators

The use of steel jackets was found to significantly improve the penetration efficiency of tungsten alloy rods. Experiments and analyses were conducted with L/D=10 projectiles of constant exterior dimensions at a nominal impact velocity of 2.2 km/s. The fraction of jacket material was varied to see which geometry would have the best performance. For a core-to-jacket diameter ratio (μ) of 0.6, the experiments showed the penetration efficiency (P/KE^1/3) increased by 21% relative to an all-tungsten baseline rod of the same exterior dimensions. Experiments and AUTODYN simulations showed the same penetration efficiency trends. The simulations, however, did not show that the tungsten core outran the jacket, contrary to what was observed in the experiments.     

An examination of long-rod penetration

The one-dimensional modified Bernoulli theory of Tate [J. Mech. Phys. Solids 15, 287–399 (1967)] is often used to examine long-rod penetration into semi-infinite targets. The theory is summarized and the origins of the target resistance term examined. Numerical simulations were performed of a tungsten-alloy, long-rod projectile into a semi-infinite hardened steel target at three impact velocities sufficiently high to result in projectile erosion. The constitutive responses of the target and projectile were varied parametrically to assess the effects of strain hardening, strain-rate hardening, and thermal softening on penetration response. The results of one of the numerical simulations were selected to compare and contrast in detail with the predictions of the Tate model.     

More on the secondary penetration of long rods

The secondary penetration of long rods, impacting semi-infinite metallic targets, has been investigated since the early 60's, both experimentally and analytically. Several models have been proposed for the extra penetration which is achieved by these rods at the later stages of the process. However, the models are of limited applicability since they cover only limited regimes of the relevant parameters. In order to further understand the phenomenon of secondary penetration, we performed a large number of numerical simulations using the PISCES 2 DELK code. These simulations dealt with the relevant parameters in large ranges of variability, such as: the rod impact velocity, its aspect ratio (L/D), as well as the densities and strengths of rod and target material. We show that the semi-empirical formulations do not account for the whole range of these parameters. Our simulations show that the strength of the rod has a major influence on the values of the secondary penetrations. In addition, these values are strongly dependent on L/D and target strength.     

The crater formation due to segmented rod penetrators

The efficiency of segmented rods penetrating into semi-infinite steel targets was investigated numerically by hydrocode simulations with impact velocities varying between 2000 and 5000 m/s.

In a second phase segmented elements were integrated in experimental projectiles and these projectiles were accelerated by means of a powder gun to verify the launchability of such projectiles and to confirm the results of the numerical simulations.

As predicted by the numerical simulations, we observe an increase of the penetration depth in the order of 10% with a 4 segments spaced projectile, in the case of an impact velocity of 2100 m/s.     

The effects of a random off-axis velocity component on the penetration achieved by long rod kinetic energy penetrators and shaped charge jets

The impact of long rod kinetic energy penetrators and shaped charge jets on homogeneous targets is investigated. In particular the effects of a random off-axis velocity component on the penetration achieved are analyzed. The aim of this study is to consider the case where the off-axis velocity component takes a uniform value W along the rod or jet.

It is assumed that penetration takes place according to the classical hydrodynamic penetration law, and that it continues either until the projectile material is exhausted or until a side wall collision occurs. The penetration is evaluated as a function of a reference coordinate q defined along the projectile, and the corresponding crater radius distribution R(q) calculated, on the assumption that W is zero. The locus of the penetration stagnation point S corresponding to the true value of W is then determined as a function of q and a revised crater profile is calculated with radius R(q), centred on S. We determine whether a side-wall collision occurs, and calculate the final penetration, P. The probability that P exceeds a given value P_o is found. We then determine the expected value of P and investigate parameter variations to maximize this value.