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.     

On the main mechanisms for defeating AP projectiles,long rods and shaped charge jets

Some of the important mechanisms for defeating various projectiles and shaped charge are reviewed in this paper. These mechanisms are based on the compressive strength of the target material (its inherent resistance to penetration) and on the asymmetrical forces which it exerts on the threat, through proper geometrical arrangements. We discuss the basic features of the resistance to penetration, starting with the classical analysis of the cavity expansion process in elasto-plastic solids. This property of the target is responsible for the deceleration of hard cored projectiles and for the erosion of long rods, under normal impact conditions. We then discuss the asymmetrical interaction of armor piercing (AP) projectiles, long rods and shaped charge jets with inclined plates (stationary and moving). These asymmetric forces, exerted on the impacting threat, are responsible for their deflection and breakup. Our work combines experimental observations with numerical simulations and engineering models, which highlight the basic mechanisms behind these complex situations. This understanding is necessary for optimizing the performance of any armor design against a given threat.     

On the hydrodynamic theory of long-rod penetration

The hydrodynamic theory of long-rod penetration is reexamined by applying the modified Bernoulli equation to the forces acting on both sides of the moving rod-target interface. Using a ratio of 2 for the effective cross sectional areas of the mushroomed and rigid parts of the rod, it is shown that analytical expressions can be used to calculate the resistance to target penetration. The analytical expression used to calculate this resistance is the cylindrical cavity expansion, which yields resistance values of 3–4 times the compressive yield strength of the target material. Calculations based on our model show good agreement with experimental data, for steel and tungsten long rods penetrating various steel targets.     

Penetration of tungsten alloy rods into shallow-cavity steel targets

This paper presents a combined numerical and experimental study on penetration of tungsten heavy alloy long rods (length-to-diameter ratio of 10) into thick RHA (rolled homogeneous armor) steel plates. The main objective of this study was to establish the effects of a shallow cavity at the front of the steel plate on the penetration process. Three experiments were performed at 1.5 km/s on target plates with a shallow-cavity of 19 mm diameter. These results were compared to existing penetration data obtained for flat plates over a range of 1.1–1.7 km/s. In the code simulations three target configurations were considered: a flat target surface without a cavity and two target plates with different cavity diameters (19 and 11.54 mm). The effect of the target’s free surface on the characteristic time that the penetrator takes to reach quasi-steady-state penetration into the target was investigated for three configurations. Based on the experimental results the effect of the shallow-cavity wall constraint on the penetration process was found to be insignificant. The code results matched the measured depths of penetration within the limits of the experimental accuracy for all configurations examined.     

有限柱形空腔膨胀理论及其应用

为研究有限平面尺寸金属厚靶侵彻问题,提出了有限柱形空腔膨胀理论.考虑侧面自由边界,将理想弹塑性材料的空腔膨胀过程分为弹塑性阶段和塑性阶段,得到了空腔壁径向压力的解析解.基于Tate磨蚀杆模型,应用有限柱形空腔膨胀理论计算靶的侵彻阻力,建立了长杆弹侵彻有限直径圆柱形金属厚靶工程模型.与现有文献试验比较表明,文中工程模型计算的侵彻深度与弹道试验结果吻合较好.     

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.     

Analytical solution of the spherical cavity expansion process

The dynamic spherical cavity expansion is treated through a complete analytical solution of the equations of motion for an elasto-plastic solid obeying von-Mises yield criterion. The solutions for various metals, with different elastic and plastic properties, are fitted with a third order polynomial which relates the normalized pressure inside the cavity with the normalized velocity of the cavity wall. An extensive search for material similarities is conducted in order to highlight the roles of the elastic properties of the solid, as well as its strength and equation of state parameters. Using the simple terms we derive, for the coefficients in the third order polynomial, one can easily calculate the relation between pressures inside the cavity and their wall velocities for any solid to within 1%.     

On the role of nose profile in long-rod penetration

The superiority of depleted uranium on tungsten-alloy penetrators has recently been assigned to the self-sharpening mechanism, at the tip of the DU rods, due to the adiabatic shear failure which this material experiences. The purpose of the work presented here was to further investigate the role of deformed nose profile on the deep penetrations of long rods into semi-infinite targets. This was achieved through a series of 2-D numerical simulations and several perforation experiments where we recovered and examined the residual penetrators. The simulations were performed for rigid tungsten-alloy rods having five different nose shapes with the density and elastic properties of tungsten alloys. For the normal impact experiments we chose three rod materials: a tungsten alloy, a copper and a titanium alloy. The residual rods (after perforation of finite thickness targets) were imaged by flash X-ray and softly recovered using sand boxes. As expected, the nose shapes of these rods were very different from each other.     

Penetration analysis for geo-material based on unified strength criterion

A unified strength criterion is applied for penetration analysis of geo-materials. Based on the cylindrical cavity-expansion theory the relation between the radial traction on the cavity surface and the impact velocity of a rigid projectile is derived. The finial penetration depth of the projectile is analytically obtained and the effect of strength criterion on the penetration depth is investigated. By comparing with existing test results, it is found that the proposed penetration model is effective in the analysis of a rigid projectile penetrating into a semi-infinite geo-material target.     

More on the ricochet of eroding long rods—Validating the analytical model with 3D simulations

The ricochet of eroding long rods, from inclined steel targets, is investigated by a series of three-dimensional (3D) numerical simulations. These are compared with the predictions of our, previously published, analytical model for ricochet. The agreement between simulation results and model predictions is excellent, strongly enhancing our simple ricochet model. We also highlight several aspects of our model which are derived from its simple closed form. One of these is the fact that ricochet of long rods can take place only at velocities and obliquities which are higher than certain threshold values. Otherwise, the process involves rod bending and sliding along the target impact face.     

The impact of a thermoelastic rod against a rigid heated barrier

The impact of a thermoelastic rod with a heat-insulated lateral surface against a rigid heated barrier is considered. The heat exchange between the rod and the wall occurs at one of its ends contacting with the wall, while the other end is heat-insulated and free from external forces. The behaviour of the rod during the impact process is described by the Green-Naghdy theory which allows one to take finite speed of heat propagation into account, neglecting therewith thermal relaxation. The Laplace integral transform with the subsequent expansion of the found images in terms of the natural functions of the problem is used as a method of solution, which is found in explicit exact closed form. The analytical time-dependence of displacements, stresses, and temperature at each rod particle is obtained. The emphasis is on the analysis of the contact stress, the temperature of the colliding bodies during their contact interaction, and on the detection of the duration of contact of the rod with the rigid wall. It is shown that the contact time essentially depends on the relationship between the mechanical and thermal values.     

Energy partitioning and microstructural observations related to perforation of titanium and steel targets

This paper presents an analysis of two target materials and the associated energetics related to the initial penetration into the target and perforation as the penetrator exits the target. Impact tests were conducted for tungsten alloy rods striking rolled homogeneous armor (RHA) and titanium alloy plates. Rod impact velocities were nominal 1,500 and 2,000 m/s. Target thicknesses were chosen so that the rods would overmatch the targets and lose some 200 m/s during penetration. The tests utilized flash x-rays to determine rod residual lengths and velocities and target plug features to include thicknesses and velocities. From these observables, experimental determination of the corresponding kinetic energies (KEs) and estimates for the fracture energies were obtained. Also, in each case, target material adjacent to penetration channel walls was examined by optical and electron microscopy and x-ray diffraction to gain further insight into deformation processes (cavity expansion) during penetration. The analytic penetration model gave results that were in good agreement with the experimental observables. In addition, it was observed that the RHA follows traditional plastic flow of cavity expansion while titanium alloy shows deformation features that deviate significantly. The paper discusses possible causes for these differences.     

Effect of the static strength of a metal on long-rod penetration at low velocities

The effect of the static strength (yield stress) of a material on major penetration parameters (specific work of cavity formation and pressure at the contact surface) was examined. The analytical equations and the quantitative evaluation of the specific work spent for the formation of a spherical or cylindrical cavity and of the pressure at the surface of such cavities upon their static expansion were obtained for rigid-plastic and elastoplastic materials. The evaluation of the specific work of deformation upon the formation of the spherical cavity in the plate and the cylindrical one upon the penetration of a rigid rod are shown to be adequate. The influence of strain hardening on maximum radial stresses on the cavity surface was evaluated. The resistance to penetration and the expansion of a cavity are shown to be influenced by the elastic compressibility of a material in the inelastic region. The kinetics of the stress-strain state of the material upon the formation of the cavity was analyzed for the case of considerable deformation of a rod at penetration. Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev Ukraine. Translated from Problemy Prochnosti, No. 1, pp. 93–110, January–February, 2000.     

砂土中刚性挡墙不同主动变位模式任意位移土压力计算

已有模型实验及现场实测表明,刚性挡墙随着变位模式和位移量的变化,主动土压力合力和分布均发生改变,有时甚至与经典理论的线性分布有很大不同。采用中间状态系数定义非极限状态,提出了砂土中刚性挡墙不同主动位移模式下非极限状态土压力合力系数的计算公式;将墙后土体简化为连续非线性弹簧和刚塑性体的组合体作用在挡墙上,得到了不同位移模式任意位移的土压力分布和合力作用点高度。与已有理论方法和实验结果对比表明,该文方法在三种典型位移模式下与实验数据吻合更好。研究还发现,平动模式土压力呈线性分布,其合力随挡墙位移量的增大易趋于稳定并到达极限状态;绕墙底和绕墙顶转动模式下土压力合力随着位移增大只能接近极限状态且呈非线性分布。绕底转动时,土压力分布曲线逐渐向上凹,合力作用点高度趋于降低;绕顶转动时,分布曲线则逐渐向上凸,合力作用点高度趋于升高,墙顶附近表现出明显的土拱效应。     

Thermophysical Property Measurements on Mold Materials: Thermal Expansion and Density

In order to optimize casting processes, computational models of solidification have proven to be very valuable to foundrymen. It is experimentally proven that the casting defects are primarily related to mold properties. During the eutectic growth the temperature rises, which is commonly referred to as recalescence. This has a strong effect on the mold walls, and mold wall movement can occur. The huge pressures generated at this time can block voids if mold is rigid. In green sand molds the moisture content will be reduced and mold wall will expand easily. According to previous research results, a distribution of thermophysical properties of the mold in the mold cavity, and the movement (expansion or contraction) of the mold and the metal interface are crucial for formation of many defects. The thermal expansion and bulk density of selected mold materials (olivine sand and zircon sand) and silica sand cores in transient regimes were determined in this study using a computer-controlled dual-pushrod dilatometer.Paper presented at the Fifteenth Symposium on Thermophysical Properties, June 22--27, 2003, Boulder, Colorado, U.S.A.     

Penetration of rigid objects into semi-infinite compressible solids

Penetration of rigid object into semi-infinite compressible solid is investigated in the present study. First a detailed numerical analysis of the penetration process is performed via smooth particle hydrodynamics. Based on the numerical results, estimate for the resistive force that the target exerts on the penetrating object is obtained. It is shown that in the computation of the resistive force the quasi-static coefficient can accurately be obtained through the spherical cavity expansion theory. However for a given target material the coefficient associated with the hydro-dynamic term significantly depends on the impact velocity unlike commonly assumed constant values. Penetration equations for rigid object with arbitrary nose geometries are derived. Developed analytical model is then verified through some experimental and analytical results reported in the literature.     

高速长杆弹对有限直径金属厚靶的侵彻分析EI北大核心CSCD

采用基于统一强度理论的有限柱形空腔膨胀理论,结合Tate磨蚀杆模型,考虑中间主应力、靶体侧面自由边界的影响及高速(1500 m/s~2200 m/s)侵彻弹体的变形和消蚀现象,推导线性硬化有限直径金属厚靶在长杆弹高速侵彻时的空腔壁径向应力,建立侵彻阻力和侵彻深度计算模型,并利用MATLAB软件编程求解,分析包括强度准则差异在内的弹道终点效应的一系列影响因素。结果表明:该文计算方法可以更好地描述弹靶的动态响应,还可以得到一系列基于不同强度准则的侵彻阻力和深度的解析解、对不同靶弹半径比的靶材侵彻深度的区间范围进行有效预测;强度参数、弹体撞击速度和靶体半径对有限直径金属靶体的抗侵彻性能均有较大的影响,其中强度参数值由1减小为0时,侵彻阻力可减小33.33%,侵彻深度可增加15.93%;当靶弹半径比小于等于20时,侵彻深度增大的程度显著,当靶弹半径比由19.88减小至4.9时,侵彻阻力减小了41.30%,侵彻深度增长了32.61%,此时靶体边界尺寸对侵彻性能的影响很大,不能继续按照半无限靶体进行计算。     

Penetration into ductile metal targets with rigid spherical-nose rods

We developed penetration equations for rigid spherical-nose rods that penetrate ductile metal targets. The spherical cavity-expansion approximation and incompressible and compressible elastic-perfectly plastic constitutive idealizations simplified the target analyses, so we obtained closed-form penetration equations. We compared predictions from our models with previously published penetration data and results from Lagrangian and Eulerian wavecodes.     

Penetration of 6061-T6511 aluminum targets by ogive-nosed VAR 4340 steel projectiles at oblique angles: experiments and simulations

In this paper we present the results from a combined experimental, analytical, and computational penetration program. First, we conducted a series of depth-of-penetration experiments using 0.021 kg, 7.11 mm diameter, 71.12 mm long, vacuum-arc-remelted 4340 ogive-nose steel projectiles. These projectiles were launched with striking velocities between 0.5 and 1.3 km/s using a 20 mm powder gun into 254 mm diameter, 6061-T6511 aluminum targets with angles of obliquity of 15°, 30°, and 45°. Next, we employed the initial conditions obtained from the experiments with a new technique that we have developed to calculate permanent projectile deformation without erosion. With this technique we use an explicit, transient dynamic, finite element code to model the projectile and an analytical forcing function derived from the dynamic expansion of a spherical cavity (which accounts for compressibility, strain hardening, strain-rate sensitivity, and a finite boundary) to represent the target. Results from the simulations show the final projectile positions are in good agreement with the positions obtained from post-test radiographs.