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.     

Stationary contact between a cylindrical metallic projectile and a flat target surface under conditions of dwell

Armour systems capable of defeating an incoming projectile on the surface of a ceramic target have been reported. This capability, called interface defeat or dwell, signifies that the projectile material is forced to flow radially on the surface of the target without penetrating significantly. Under such flow conditions, the hydrodynamic pressure is normally the most important part of the normal load on the target surface. Therefore, projectile properties such as yield strength and compressibility are commonly ignored or assumed to contribute only marginally. In order to investigate the effects of these properties, an analytical expression was derived for the normal load from a cylindrical metallic projectile impacting on a flat, rigid and friction-free surface, which includes the contributions from yield strength and compressibility in addition to that of inertia. At an impact velocity representative of today's ordinance velocities, the contributions to load intensity on the axis from yield strength and compressibility were found to be 15% and 3.4%, respectively, of that of inertia. The analytical results and Autodyn-2D numerical simulations show good agreement within a projectile radius from the axis.     

弹体冲击混凝土半无限靶的侵彻阻力与深度计算

假设弹体是刚性的,将弹体侵彻冲击作用下的混凝土靶体划分为粉碎区、径向裂缝弹性区与原始弹性区,并认为侵彻粉碎区域内的混凝土材料处于类似于流体动力学状态,可采用水动力模型.根据伯努利方程推得了细长杆弹侵彻阻抗力的公式,进而求出锥形弹头侵彻阻抗力的等效平面解,得到了弹体的侵彻深度.利用实弹进行了以不同速度侵彻不同强度的混凝土靶体的验证计算,并与多个经验公式做了对比.结果表明,该模型是合理的,与经验公式相比计算结果趋于保守.     

Mesoscale simulations of a dart penetrating sand

Historically, hydrodynamic calculations have utilized continuum constitutive models to simulate the coupled dynamic response of a solid projectile penetrating a heterogeneous target system such as concrete, foam or a granular porous medium. Continuum models fail to capture the complicated grain level response within the heterogeneous target which can result in asymmetric loading of the projectile leading to variations in projectile performance. These grain level effects can be crucial to predicting the penetration depth or overall effectiveness of the projectile. In order to assess the possibility of using mesoscale simulations to resolve the grain level dynamics, hydrodynamic simulations were performed for an 11.4 cm long, 0.9 cm diameter dart penetrating a bed of porous granular dry sand with an initial velocity of 366 m/s. Simulations were performed using the Eulerian hydrocode CTH in a two-dimensional planar configuration. The goal of the mesoscale simulations is to determine the viability of using these techniques as an alternative to continuum models and to assess the effects of grain level variability such as anisotropic material distributions and variations in the dynamic yield and fracture strength. The results indicate that variations in the size distribution of aggregate added and the fracture strength of the sand system can have a significant effect on penetration performance of the dart; whereas variations in the dynamic strength of the sand had little effect on the dart penetration.     

强度失配对铝合金板焊接接头抗弹性能影响的有限元分析

通过建立7.62mm穿甲弹侵彻铝合金板的模型,采用Johnson-Cook材料模型模拟研究了焊接接头的强度失配对铝合金板抗弹性能的影响。研究结果表明：当子弹侵彻铝合金焊接接头附近时,由于材料的不均匀变形,子弹会改变侵彻方向,其方向角的改变大小与子弹速度、侵入位置、弹头形状、强度失配比、靶板厚度有关;当子弹以低速侵彻铝合金板中间的软焊缝材料时,由于变形局部化,出现材料的抗弹性能低于纯焊缝材料的情况;由于子弹侵彻方向的改变,出现部分区域的抗弹性能高于铝合金基体材料的情况。研究结果为不均匀材料的抗弹性能研究提供参考。     

Normal penetration of an eroding projectile into an elastic–plastic target

The main objective of the present work is to describe normal penetration of a deformable projectile into an elastic–plastic target. The force imposed on the projectile by the target is generally a complex function of the strength of the target material, the projectile velocity, its diameter and shape, as well as the instantaneous penetration depth. When this force exceeds a certain critical value the projectile begins to deform. At moderate-to-high values of the impact velocity, the projectile's tip material flows plastically with large deformations causing the formation of a mushroom-like configuration. This process is accompanied by erosion of the projectile material. In the rear (“elastic”) part of the projectile the deformations remain small and the region can be approximated as a rigid body being decelerated by the projectile's yield stress. The general model allows one to predict the penetration depth, the projectile's eroded length and the crater diameter. It has been shown that in the limit of very high impact velocities the present model reduces to the well-known form of the hydrodynamic theory of shaped-charge jets. Also, a simplified asymptotic formula for the crater radius has been derived which includes the effect of the target's yield stress and compares well with experimental data for very high impact velocities.     

Multiple impact penetration of semi-infinite concrete

An experimental study was performed to gather multiple impact, projectile penetration data into concrete. A vertical firing range was developed that consisted of a 30-06 rifle barrel mounted vertically above a steel containment chamber. 0.41 m cubes of an Air Force G mix concrete were suspended in wet sand and positioned in the steel chamber. The concrete targets were subjected to repeated constant velocity impacts from 6.4 mm diameter steel projectiles with an ogive nose shape and a length to diameter ratio of 10. A laser sight was adapted to the rifle to ensure alignment, and a break screen system measured the projectile velocity. After each impact, the projectile penetration and crater formation parameters were recorded. The penetration and crater formation data were consistent with single impact penetration data from previous studies conducted at Sandia National Laboratories. In addition, an analytic/empirical study was conducted to develop a model that predicted the penetration depth of multiple impacts into concrete targets. Using the multiple impact penetration and crater formation data, a single impact penetration model, developed by Forrestal at Sandia National Laboratories, was extended to account for the degradation of the target strength with each subsequent impact. The degradation of the target was determined empirically and included in the model as a strength-modifying factor. The model requires geometry parameters of the ogive nose projectile, projectile velocity, the number of impacts, and target compressive strength to calculate the overall penetration depth of the projectile.     

刚性弹侵彻有限直径金属厚靶的机理与模型研究

采用统一强度理论,考虑靶板中间主应力效应和靶体侧面自由边界的影响,得到线性硬化靶材在弹塑性阶段和塑性阶段的空腔壁径向应力的表达式,建立线性硬化靶材的统一侵彻模型,求出中低速(v₀≤1000 m/s)刚性弹体侵彻有限直径金属厚靶时侵彻阻力、侵彻深度计算公式,并利用Simpson算法对其进行求解,分析了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侵彻过程中弹靶的动态响应,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、对靶材在不同撞击速度下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、靶体半径和弹头形状对有限直径金属厚靶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,侵彻深度增加了22.45%;随着靶弹半径比的减小,侵彻深度不断增大,当靶弹半径比小于等于16时,侵彻深度增大的程度显著,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。     

A time-dependent model for long-rod penetration

The one-dimensional, quasi-steady-state, modified Bernoulli theory of Tate [J. Mech. Phys. Solids, 15, 287 (1967)] is often used to examine long-rod penetration into semi-infinite targets. In general., the time histories of penetration predicted by the Tate model can be in good agreement with those computed from numerical simulations. However, discrepancies exist between the model and numerical simulations at the beginning and at the end of penetration. From insights provided by numerical simulations, assumptions are made concerning the velocity and stress profiles in the projectile and the target. Using these assumptions, the time-dependent, cylindrically-symmetric, axial momentum equation is explicitly integrated along the centerline of the projectile and target to provide the equation of motion. The model requires the initial interface velocity—which can be found, for example, from the shock jump conditions-and material properties of the projectile and target to compute the time history of penetration. Agreement between the predictions of this one-dimensional, time-dependent penetration model are in good agreement with experimental results and numerical simulations.     

Hypervelocity impact of rod projectiles with L/D from 1 to 32

Beside a short remark on the “hydrodynamic theory of rod projectiles”, the paper deals with the terminal ballistic behaviour of cylindrical projectiles against semi-infinite targets. Experimental data of EMI, completed by results of some other authors, are presented. Crater parameters like depth, diameter and volume and their dependence on projectile velocity (up to 5000 m/s), projectile and target material properties, as well as L/D-ratios (1–32), will be discussed. Mainly the projectile materials steel and tungsten sinter-alloys are considered. Target materials are mild steel and high strength steel, an Al-alloy and a tungsten sinter-alloy. The results show that the influence of material density on the crater dimensions is considerably greater than the influence of strength. The L/D ratio determines the velocity dependence of crater depth, diameter and volume. At high velocities in the hydrodynamic regime, the crater depth of short cylinders (L/D 1) is approximately proportional to v_p^2/3 (V_p=projectile velocity). With increasing L/D-ratio, the slope of the penetration curves decreases and converges for rods (L/D 1) versus a saturation, i. e. becomes nearly independent on v_p. A consequence of this saturation is the existence of a so-called “tangent velocity”, above which an optimal increase of efficiency is only realized by increasing the projectile mass and not the velocity. Furthermore, ballistic limits of real targets like single plates and symmetric double plates meteorite bumper shield) are taken into account. The expected better performance of “segmented rods” is also discussed.     

Numerical–analytical model of penetration of long elastically deformable projectiles into semi-infinite targets

A new numerical–analytical model of penetration of long axisymmetric elastically deformable projectiles in semi-infinite targets is presented. A background of this model is the integral–differential equation of ballistics for non-deformable projectile. This equation is obtained on the basis of the Lagrange–Cauchy integral for non-stationary irrotational motion of an incompressible fluid, as well as the solutions for the quasi-static spherical cavity expansion problem in an infinite medium. The velocity field in a target is defined by actual projectile shape. The functional dependence of penetration velocity is determined for both elastic and rigid projectiles. The effect of forced elastic longitudinal oscillations on penetration velocity is estimated. An estimate is made for the critical impact velocity at which point the projectile plastically deforms causing irreversible changes in its shape, and also leads to instability of its trajectory in the target. This velocity depends on both elastic and strength characteristics of the projectile and target, their densities and projectile shape. Results from our penetration modeling are compared with existing experimental and calculated data.     

Self-similar flow of a conical projectile on a flat target surface under conditions of dwell

In order to investigate the state of stress in a target material under conditions of interface defeat or dwell it is necessary to determine the load intensity at the interface of the flowing projectile material and the target. Previous studies for a cylindrical projectile geometry at normal impact under stationary conditions show that the load can be considered to be composed of three components, viz., those of inertia, compressibility and yield strength of the projectile material. In order to determine the influence of projectile shape, a conical projectile in axi-symmetric impact on a ridged, friction-free surface is studied by use of an analytical model for self-similar flow and numerical Autodyn simulations. It is shown how the maximum load intensity, and the position of the maximum, depends on the apex angle. Both the self-similar model and the Autodyn simulations show that the contribution to the load intensity from compressibility is positive below and negative above apex angles 80°. The influence of yield strength on the load intensity depends only weakly on the apex angle and therefore corresponds to that for a cylindrical projectile.     

Numerical modeling of projectile penetration into compressible rigid viscoplastic media

We develop computational methods for modeling penetration of a rigid projectile into porous media. Compressible rigid viscoplastic models are used to capture the solid–fluid transition in behavior at high strain rates and account for damage/plasticity couplings and viscous effects that are observed in geological and cementitious materials. A hybrid time discretization is used to model the non‐stationary flow of the target material and the projectile–target interaction, i.e. an explicit Euler method for the projectile equation and a forward (implicit) method for the target boundary value problem. At each time step, a mixed finite element and finite‐volume strategy is used to solve the ‘target’ boundary value problem. Specifically, the non‐linear variational inequality for the velocity field is discretized using the finite element method while a finite‐volume method is used for the hyperbolic mass conservation and damage evolution equations. To solve the velocity problem, a decomposition–coordination formulation coupled with the augmented Lagrangian method is adopted. Numerical simulations of penetration into concrete were performed. By conducting a time step sensitivity study, it was shown that the numerical model is robust and computationally inexpensive. For the constants involved in the model (shear and volumetric viscosities, cut‐off yield limit, and exponential weakening parameter for friction) that cannot be determined from data, a parametric study was performed. It is shown that using the material model and numerical algorithms that developed the evolution of the density changes around the penetration tunnel, the shape and location of the rigid/plastic boundary, the compaction zones, and the extent of damage due to air‐void collapse are described accurately. Copyright © 2007 John Wiley & Sons, Ltd.     

A generalized formula for the penetration depth of a deformable projectile

A simple analytical algebraic formula is developed for predicting the penetration depth of a deformable projectile into a semi-infinite target. This formula is a simplified version of more general equations that have been developed to predict the time-dependent penetration process in finite thickness targets. Specifically, the formula generalizes the classical hydrodynamic theory to include dependence on elastic properties of the target and on the yield strengths of both the target and the projectile. Moreover, the formula is limited to the case of long-rod penetration where both the projectile and the target experience significant plastic flow. The limiting values of the location of the elastic–plastic boundary in the target have been determined, and a single empirical constant has been introduced to characterize the transition between these limiting values. A value for this empirical constant has been determined which produces theoretical predictions that are in reasonable agreement with experimental data for moderate to high values of the impact velocity of steel and tungsten projectiles penetrating a steel target.     

长管体斜侵彻半无限均质靶简化模型

长管体斜侵彻半无限靶的研究具有重要意义。描述了长管体在速度范围在1000m/s~1500m/s时斜侵彻半无限均质靶板的物理图像,对长管体在侵彻各个阶段的受力和运动情况进行了分析,发现侵彻时管体中央的“靶芯”对侵彻过程影响很大。在一定假设条件下,基于变质量系统动量守恒定律建立了不同侵彻阶段长管体侵彻的质心运动方程,基于动量矩定理建立了不同侵彻阶段绕质心的摆动方程,并进行了数值计算。进行了长管体侵彻半无限均质靶板的试验,试验与计算结果表明二者的侵彻行程吻合较好,说明所建立的模型能够描述长管体对半无限均质靶板的侵彻过程,对空心弹和异形穿甲弹的研究和设计具有一定参考价值。     

刚性尖头弹侵彻圆柱形金属厚靶分析模型

考虑金属厚靶侧面自由边界的影响,研究了刚性尖头弹侵彻有限平面尺寸金属厚靶问题。基于有限柱形空腔膨胀理论和线性硬化材料模型,得到了空腔壁径向压力的解析式,建立了刚性尖头弹侵彻有限直径圆柱形金属厚靶工程模型。与试验和数值模拟比较表明,该文工程模型计算精度很好。基于所建立的工程模型,研究了靶板半径对侵彻深度和侵彻阻力的影响,结果表明：当靶板与弹丸半径比值小于20时,靶板半径对侵彻阻力和侵彻深度有显著影响,不能按无限尺寸靶板计算;当靶板与弹丸半径比值大于20时,靶板半径对侵彻阻力和侵彻深度影响较小,可近似按无限尺寸靶计算。     

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 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.