On the penetration of semi-infinite targets by long rods

The one-dimensional eroding-rod penetration theory proposed by A. Tate (J. Mech. Phys. Solids15, 387, 1967;17, 141, 1969) is modified in two ways. In the equation of motion of the rigid end of the rod, a proper accounting is made of mass transfer into the plastic region. Also, the mushroom strain at the deforming end of the rod is incorporated into the analysis. It is shown that this latter factor has a very substantial effect on calculated penetrations.     

Analytical models for the penetration of semi-infinite targets by rigid, deformable and erosive long rods

A theoretical study is presented herein on the pen- etration of a semi-infinite target by a spherical-headed long rod for Yp 〉 S, where Yp is the penetrator strength and S is the static target resistance. For Yp 〉 S, depending upon initial impact velocity, there exist three types of penetration, namely, penetration by a rigid long rod, penetration by a deforming non-erosive long rod and penetration by an erosive long rod. If the impact velocity of the penetrator is higher than the hydrodynamic velocity （VH）, it will penetrate the target in an erosive mode; if the impact velocity lies between the hydrodynamic velocity （VH） and the rigid body velocity （VR）, it will penetrate the target in a deformable mode; if the impact velocity is less than the rigid body velocity （VR）, it will penetrate the target in a rigid mode. The critical conditions for the transition among these three penetration modes are proposed. It is demonstrated that the present model predictions correlate well with the experimental observations in terms of depth of penetration （DOP） and the critical transition conditions.     

On the deep penetration and plate perforation by rigid projectiles

We present results of a large number of 2D numerical simulations in which we investigated various aspects in the deep penetration of rigid short projectiles into semi-infinite targets, as well as their perforation through thin metallic plates. In particular, we analyze the effect of the entrance phase on the penetration characteristics of short ogive and spherical nosed projectiles. The second issue which we investigate here concerns the perforation of metallic plates by sharp nosed projectiles. Our simulation results show that a simple model, which is based on energy conservation, accounts for the residual velocities when the target is penetrated by the ductile hole enlargement process. In addition, we define a new concept, the effective resisting stress which the plate exerts on the projectile during perforation. We show that it has some valuable insights for the process of perforation and we perform a parametric study to understand its dependence on various parameters. This effective stress, which determines the ballistic limit velocity of the projectile, depends on the strength of the plate, as well as on its thickness, as we show here.     

On finitely deforming rigid-plastic materials

In the context of a Lagrangian strain-space formulation of finitely deforming elastic-plastic materials, a theory of rigid-plastic materials is first derived as a limiting case. Hardening, softening, and perfectly plastic responses are discussed. Consequences of a physically plausible work assumption are examined, and special classes of materials are studied. An a priori theory of finitely deforming rigid-plastic materials is then proposed.     

A theory for the deceleration of long rods after impact

A modified hydrodynamic theory which takes some account of strength effects is used to predict the deceleration of a long rod after striking a target. The results are then compared with experimental data from X-ray observations.     

Numerical simulation of penetration of aluminum targets by spherical-nose steel rods

Numerical simulations of the penetration processes in aluminium blocks by spherical-nose steel rods were performed in this study. The specific impact configuration of this study involves 152-mm diameter 6061-T651 aluminum bars impacted by spherical-nose projectiles machined from T-200 maraging steel rods at nominal impact velocities between 300 and 1000 m/s. The transient dynamic finite element code LS-DYNA2D was used for the numerical analysis. The erosion capability in LS-DYNA2D was exercised in conjunction with the maximum equivalent plastic strain criterion to carry out failure simulations in the target. Calculated results were compared to the experimental data. Good correlation was obtained.     

Further results in the theory of long rod penetration1

The theory of long rod penetration as given in a previous paper by the author is extended to take account of the deformation of a soft rod against a rigid target and the penetration of a rigid projectile into a soft target. It is shown that it is theoretically possible to have a decrease in depth of penetration with increasing impact velocity, and a method for deducing the average radius of the hole is given. The theory is compared with experimental results.     

On the free shapes of elastic rods

The problem of free shape consists in finding the form that an elastic body must have in a natural state in order that it shall assume a given form in an equilibrium configuration under the action of assigned loads. The problem, that is of interest in itself, arises in some practical applications and can constitute a preliminary step in the study of some mechanical properties of classes of equilibrium configurations that are not natural states. This paper examines the problem of free shape for inextensible elastic rods which in equilibrium are subject only to the action of forces and couples applied to the ends, and whose deformations can be described by the theory of finite displacements of thin rods due to Kirchhoff. After the general equations governing the problem have been deduced, they are employed to give a classification of the free shapes of rods that in equilibrium are circular rings.     

On the theory of porous elastic rods

We consider the direct approach to the theory of rods, in which the thin body is modelled as a deformable curve with a triad of rigidly rotating orthonormal vectors attached to every material point. In this context, we employ the theory of elastic materials with voids to describe the mechanical behavior of porous rods. First, we derive the dynamical nonlinear field equations of the model. Then, in the framework of linear theory, we prove the uniqueness of the solution to the associated boundary-initial-value problem. We identify the relevant field quantities from the theory of directed curves by comparison with the three-dimensional equations of straight porous rods. Finally, for orthotropic and homogeneous rods, we determine the constitutive coefficients in terms of the three-dimensional elasticity constants by solving several problems in the two different approaches.     

On the stability of elastic annular rods

The stability of equilibrium of non-linearly elastic rods, whose deformations obey the classical Kirchhoff’s equations, is considered. A variational formulation of the equilibrium problem is given, and the equilibrium equations for infinitesimal deformations superimposed to a finite transformation of a rod are deduced. The stability of annular rings, in which the twisting strain is non-null, is investigated by study of the second variation of the energy functional.     

The erosion transition of tungsten-alloy long rods into aluminum targets

This work extends and refines the phenomenological understanding of ballistic penetration in the vicinity of the erosion-threshold velocity, for the case of hemispherical-nosed tungsten rods striking ductile targets. Analysis, supported by experimentation, indicates a period of noneroding penetration for these configurations, which results from lateral support exerted by the target crater upon the deforming, yet noneroding, penetrator. Experiments indicate that the magnitude of the lateral support, the direct result of an interference fit between rod and crater, must be on the order of the target’s ballistic-penetration resistance, and does not vary with the impact velocity over the range studied. Analysis suggests that the duration of the noneroding portion of the ballistic event is neither governed by a fixed time, nor by a fixed depth of penetration, but rather by a fixed, permissible level of deformation in the penetrator.     

柱形长杆弹侵彻的界面击溃分析

在Alekseevski-Tate模型的基础上,分析了柱形长杆弹的界面击溃过程,给出了弹体速度下降及质 量侵蚀的计算公式;讨论了弹体速度下降及质量侵蚀对动能损失的影响;特别针对柱形长杆弹在界面击溃过 程中弹体速度准定常小量变化的特点,近似给出了弹体速度、弹体质量随时间变化的简化解析表达式,为工程 应用提供便利。     

On the evolution of crystallographic dislocation density in non-homogeneously deforming crystals

A set of evolution equations for dislocation density is developed incorporating the combined evolution of statistically stored and geometrically necessary densities. The statistical density evolves through Burgers vector-conserving reactions based in dislocation mechanics. The geometric density evolves due to the divergence of dislocation fluxes associated with the inhomogeneous nature of plasticity in crystals. Integration of the density-based model requires additional dislocation density/density-flux boundary conditions to complement the standard traction/displacement boundary conditions. The dislocation density evolution equations and the coupling of the dislocation density flux to the slip deformation in a continuum crystal plasticity model are incorporated into a finite element model. Simulations of an idealized crystal with a simplified slip geometry are conducted to demonstrate the length scale-dependence of the mechanical behavior of the constitutive model. The model formulation and simulation results have direct implications on the ability to explicitly model the interaction of dislocation densities with grain boundaries and on the net effect of grain boundaries on the macroscopic mechanical response of polycrystals.     

On the approximation of acceleration waves in rods

This paper employs an approximate form of analysis based on the assumption of plane stress to find the transport equation and corresponding evolution law governing the intensity of acceleration wave propagation in an elastic rod of slowly varying area of cross-section. The result is then extended to include the case of slightly bent rods. In each of these cases it is shown that for a medium in which the strain energy function Σ(p) is such that d³Σ/dp³ ≠ 0, with p the displacement gradient, the acceleration wave intensity is governed by a Bernoulli equation. The work is concluded by showing that the analysis may also be applied to the case of a composite rod comprising an arbitrary number of homogeneous isotropic plane layers normal to the direction of acceleration wave propagation.     

长杆弹撞击陶瓷靶的一种数值模拟方法

陶瓷材料具有高强度和低密度等特点,抗弹性能优越,被广泛用于各类装甲中。长杆弹撞击陶瓷靶时会发生径向流动、质量显著侵蚀而无明显侵彻的界面击溃现象,是陶瓷抗侵彻性能研究中具有重要研究价值的特殊现象。利用有限元软件AUTODYN建立了长杆弹撞击陶瓷靶的二维轴对称计算模型,采用Lagrange和光滑粒子流体动力学（smooth particle hydrodynamics, SPH）算法,模拟了柱形钨合金长杆弹撞击带盖板的碳化硅陶瓷,通过改变长杆弹的撞击速度,得到了界面击溃、驻留转侵彻和直接侵彻3个不同现象。讨论了不同建模算法、边界条件以及材料参数对模拟结果的影响。通过网格收敛性验证和与实验结果进行拟合,综合验证了计算模型中算法、边界条件和参数设定的可靠性。结果表明,在建模中若同时使用SPH算法和Lagrange算法,需要考虑粒子和网格大小对于模拟结果的影响。针对长杆弹撞击陶瓷靶的界面击溃模拟,不建议对陶瓷材料采用SPH粒子建模。相关建模和参数选择方法对后续陶瓷抗侵彻/界面击溃的数值模拟具有重要的指导意义。

     

On the propagation of signals in Kelvin-Voigt viscoelastic rods

Summary The aim of this paper is to develop a general procedure for predicting the way in which mechanical signals propagate through a semi-infinite, thin, Kelvin-Voigt viscoelastic rod. By use of the Laplace transform and the asymptotic method of stationary phase, the boundary-value problem in question is solved.

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Sommario Nel presente lavoro si sviluppa un procedimento generale per individuare il modo nel quale i segnali meccanici si propagano attraverso barre viscoelastiche di Kelvin-Voigt, sottili e semiinfinite. Utilizzando la trasformata di Laplace ed il metodo asintotico di fase stazionaria, si risolve quindi il problema al contorno in questione.

     

On the geometric conservation law in transient flow calculations on deforming domains

This note revisits the derivation of the ALE form of the incompressible Navier–Stokes equations in order to retain insight into the nature of geometric conservation. It is shown that the flow equations can be written such that time derivatives of integrals over moving domains are avoided prior to discretization. The geometric conservation law is introduced into the equations and the resulting formulation is discretized in time and space without loss of stability and accuracy compared to the fixed grid version. There is no need for temporal averaging remaining. The formulation applies equally to different time integration schemes within a finite element context. Copyright © 2005 John Wiley & Sons, Ltd.