On the penetration of nonaxisymmetric bodies into a deformable solid medium and their shape optimization

We consider the problem of penetration of rigid pyramidal bodies (impactors) into a strained medium in the case of large speeds of penetration and estimate the depth of the impactor penetration. To this end, we use the two-stage penetration model proposed by Forrestall. We state the shape optimization problem for the penetrating body, which is based on the consideration of a set of bodies of pyramidal external shape with given fixed mass. We study both solid and hollow (shell-shaped) bodies. For the optimization functional we take the penetration depth of the penetrating body, and for the projection variable we take the number of faces of the pyramidal body. We present the results of computations of the penetration depth for different shapes of the impactor and show that, both for shells and solid impactors, the bodies of the shape of a circular cone are optimal. The problems of high-speed penetration of rigid bodies into a deformable medium are nowadays very topical problems [1] which have been studied by Russian and foreign authors [2–8].     

Possibility of generating strong magnetic fields in conducting materials by the action of high-velocity penetrators

The possibility of increasing the intensity of a magnetic field produced previously in a conducting medium moving under the action of a high-velocity penetrating body is analyzed. A simplified model of the interaction of an impactor and a conducting target with a transverse magnetic field is constructed within the framework of a one-dimensional scheme. It is shown that the degree of increase in the field intensity is determined by the relation between the magnetic-field compressibility and diffusion factors, and the corresponding dimensionless determining parameters is determined. Magnetic-field compression is estimated for a perfectly conducting medium and media with real conductivity. The significance of the thermal and mechanical effects accompanying the penetration of an impactor into a target with a transverse magnetic field is assessed. Bauman Moscow State Technical University, Moscow 107005. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 3, pp. 13–18, May–June, 2000.     

Method of calculating the resistance force upon an impact on a composite target

A kinematically possible velocity field allowing calculation of all the necessary integrals in quadratures and obtaining an analytical solution for the resistance force induced by impactor penetration into the target is constructed. The Saint-Venant model of a rigid-plastic body and the theorem on the upper bound of the limit load are used in solving the problem. The essence of the method applied is using the equilibrium equation in the form of the Lagrange equation. The kinematically possible velocity field allows obtaining an upper bound of the limit load, i.e., estimating the resistance force to impactor penetration.     

Experimental‐theoretical analysis of nonstationary interaction of deformable impactors with soil

An experimental method for determining the force of resistance to penetration of a deformable impactor into soft soil was developed in the inverted formulation: the impactor and the target exchange roles and the necessary parameters of contact interaction are recorded in an immovable measuring rod (impactor). To verify the basic principles of this experimental technique, wave processes were analyzed numerically using a modified Godunov's scheme. The applicability of various models of soil deformation was studied, and the calculation results obtained were compared with experimental data.     

Shape optimization of impactor penetrating into concrete or limestone targets

The shape of the normally striking impactor that attains the maximum depth of penetration into a concrete or a limestone semi-infinite target for a given impact velocity is found. It is shown that the optimum shape is close to a blunt (in general case) cone and it is independent on the properties of the material of the target in the framework of the employed penetration model. The performance of some other typical shapes of the nose of the impactors (spherical-conic impactors, sharp-conic impactors, truncated-ogive impactors) are analyzed and compared with the optimal impactor.     

On the penetration of a rotating impactor into an elastic-plastic medium

This research is devoted to the modeling of high-speed rectilinear penetration of a rigid axisymmetric body (impactor with a flat bluntness) into an elastic–plastic media with account for its rotation about the axis of symmetry. The body has an arbitrary shape of the meridian. The resistance to the motion is represented as the sum of the body drag and the contribution of friction. The dynamic system governing the body motion is derived and the qualitative and numerical analysis of the projectile movement and perforation of a slab are performed. The problem of shape optimization of impactor with a flat bluntness is studied using evolutionary algorithm.     

Penetration of a copper rod into a sandy target

This paper presents the results of experimental and theoretical studies of high-velocity penetration of cylindrical copper rods into sand. The hydrodynamic Alekseevskii-Tate theory is modified to determine the penetration depth and wear velocity of the material of the rod penetrating into soil target in the plastic and hydrodynamic stages of penetration. The case where the target material is significantly less strong than the rod (impactor) material is considered.     

Optimal 3D impactors penetrating into layered targets

Optimization of 3D sharp high speed impactors with given form of a longitudinal contour, length, and volume, penetrating into layered ductile targets, both for conical and thin non-conical strikers using approximate models is studied. It is found that the impactor with the minimum drag moving in a homogenous target with a constant velocity penetrates to the maximal depth into a semi-infinite target and has the minimal ballistic limit when it penetrates into a finite thickness target, regardless of the distribution of the material properties of the target along its depth, the number of the layers, etc. Using the analogy with the hypersonic flow over the flying projectiles it is predicted that the optimal impactor should have a star-shaped form of the cross section. If an impactor has a polygonal cross sections allowing the inscribed circles, the ballistic limit and maximum depth of penetration are independent not only of the properties of the target but also of the form of the polygon in the cross section and equal to the corresponding values for the inscribed body of revolution.     

不同重力场下颗粒冲击过程的离散元分析

开展不同重力场下颗粒材料冲击动力学研究有助于加深对颗粒运动机制的理解和深空探测任务的实施。本文采用离散元模拟对颗粒介质受球形冲击物的冲击过程进行了数值模拟,并与地球重力下冲击的试验结果进行对比验证。在此基础上,进一步研究了重力加速度对冲击物动力学的影响规律。计算结果表明,在所有重力加速度下,冲击物的穿透深度d与冲击速度v₀的关系可以用Poncelet模型表达;d与冲击物下落的总高度H表现为d～Hⁿ的幂律关系,当H<10 m时,d与H的幂率标度为0.322,而H>10 m时,d与H的幂率标度下降到0.211。此外,穿透深度小于冲击物半径时,重力加速度对冲击物减速过程无影响。在所有的重力加速度下,当冲击速度大于5 m/s时,冲击物的持续碰撞时间t_c是恒定的,且与重力的-1/2次方呈正比。     

A model of high speed penetration into ductile targets

A model developed by Mileiko et al. [J. Appl. Mech. Tech. Phys. 5 (1981) 711–713; Theor. Appl. Fracture Mech. 21 (1994) 9–16] describing a high speed penetration of an impactor into a ductile target is generalized.     

Generation of a magnetic field in the shear deformation region of a conducting material upon high–velocity penetration

It is shown that when a high–velocity impactor penetrates into a conducting target with a transverse magnetic field, conditions for considerable field amplification are produced in the shear deformation region on the lateral surface of the impactor. Field generation in a conducting medium deformed in shear is considered within the framework of a plane one–dimensional problem of magnetohydrodynamics. The results obtained indicate that along the boundary of the cavity produced by the impactor in the target with a magnetic field, a thin layer with a very high field intensity (about 100 T) is formed. The possibility of explosion of this layer due to the magnetic pressure acting in it is analyzed.     

Localized interaction models with non-constant friction for rigid penetrating impactors

We suggest approximate penetration models for rigid body penetration that take into account sliding velocity (SV) and pressure dependence of the friction coefficient (FC). It is showed that introducing variable FC in a localized interaction model (LIM) yields a model that belongs to the class of LIM. We developed a general method for determining the depth of penetration (DOP) using the piecewise linear approximation of the impactor’s generatrix. For some classes of SV dependent friction models we obtained analytical formulas for calculating the DOP. Using the experimental data available in the literature, we determined the dependencies of FC vs. pressure and SV. We conducted numerical modeling of penetration of a metal striker into metal and concrete shields employing models with variable and constant FC. Numerical simulations showed that taking into account variable FC strongly effects the DOP when FC changes appreciably for large velocities that are characteristic for the high-speed penetration.     

Effect of air gaps on ballistic resistance of targets for conical impactors

High velocity penetration of a rigid conical impactor into a ductile target with air gaps between the plates is studied using the cylindrical cavity expansion approximation describing impactor–target interaction. It is showed that the latter model predicts improvement of the ballistic performance of the target with the increase of air gaps. It is found analytically that the ballistic limit velocity of the target consisting of N plates with a fixed total thickness with large air gaps increases with the increase of N. The conditions are discussed when the predicted effects can be most pronounced.     

弹体在内摩擦介质中的斜侵彻深度计算

考虑介质处于内摩擦状态下,利用球腔膨胀理论及内摩擦介质侵彻近区运动方程求得刚性弹体在侵彻过程中的阻力。在此基础上,考虑自由表面对弹体斜侵彻的影响,将斜侵彻分为3个阶段,推导了斜向侵入弹体表面上任一点的应力公式,进而得出了新的刚性弹体的斜侵彻解析解。该解与别列赞公式在形式上一致,计算结果也符合较好,从而在理论上对别列赞经验公式进行了解释。     

Kinetic Dissipation in Ballistic Tests of Soft Body Armors

The objective of this study was to assess the amount of energy transferred to the backing material of oil-based clay (i.e. kinetic dissipation) in ballistic tests of soft body armor panels. To determine the relationship between penetration depth (or dent volume) and impact velocity (or energy), weight dropping test with a series of steel spheres was carried out at low impact velocities, and direct shooting with a 5.56 mm NATO fragment simulating projectile (FSP) was carried out at high impact velocities. At high and low impact velocities, the volume of the dent made in the oil-based clay was proportional to the velocity of the impactor. The change in dent volume per the change in impact velocity was found to be proportional to the 1.5th power of the mass of the impactor, while the energy absorption per unit dent volume increased linearly with the impact velocity. The relationship between trauma depth (or dent volume) and kinetic dissipation of a soft body armor panel subjected to a 9 mm bullet at 436 m/s is presented, where the trauma diameter approached that of a 1.043 kg steel ball.     

Measuring yield stress: a new, practical, and precise technique derived from detailed penetrometry analysis

The displacement of an object through a yield stress is a complex process which involves the continuous deformation and transition of new regions from the solid to the liquid regime. We studied the force vs depth variations during the progressive penetration of a plate or a cylinder in a bath of simple yield stress fluids with negligible thixotropic character (Carbopol solutions, emulsions, and foams). Three regimes could be distinguished: elastic deformation, penetration (partially immersed object), and displacement through the fluid (fully immersed object). A detailed analysis of the force vs depth curves makes it possible to show that in the partially immersed regime the force is the sum of the critical force before penetration and a term associated with a uniform shear stress along the main plate surface, which is independent on the object geometry (plate dimensions and cylinder radius). This understanding can be used to precisely determine the yield stress as the critical shear stress along the plate at vanishing velocities. We also show that it is possible to measure accurately the yield stress from relaxation tests (stress vs time curve for motion stoppage): it indeed appears that the additional force term associated with penetration is negligible in that case so that the asymptotic average shear stress after stoppage is equal to the yield stress.     

Non-linear behavior in a discretely forced oscillator

The simulated and experimental responses of a rigid-arm pendulum driven by an external impactor are considered. Here, impact occurs if the trajectory of a rotating impactor intersects that of the pendulum. Using the rotation rate of the impactor as the control parameter, experimental trials have demonstrated much of the dynamic behavior predicted by numerical simulations. The system exhibits chatter (i.e., multiple impacts within a single forcing period), sticking (i.e., contact between the pendulum and the impactor for non-negligible amounts of time), high-order periodicity, and behavior suggestive of chaos. A new convention for classifying periodic motions as well as insights regarding the nature of the coefficient of restitution (COR) in an experimental impacting system are also presented.     

Stability, Paths, and Dynamic bending of a Blunt Body of Revolution Penetrating into an Elastoplastic Medium

The deep penetration of a thin body with a blunt nose and rear into a lowstrength medium is explored. The motion of the body is described by a system of autonomous integrodifferential equations using the physical model of a separated asymmetric flow over the body and the localinteraction method. An analytical calculation of the Lyapunov stability boundary for straightline motion is performed for bodies with a parabolic meridian. The dependences of the dynamic stability of the body on various parameters are studied numerically. Curved motion paths are constructed in the region of instability, and the classification of paths proposed in previous studies of the motion of pointed bodies is confirmed. It is shown that an reverse ejection is possible when a blunt impactor enters a semiinfinite target. It is established that there is a fundamental possibility of attaining a path close to a specified one and that there is a weak dependence of motion characteristics with a developed separation on the separation angle. Examples are given of calculations of the evolution of the lateral load, the transverse force and moment, and the strength margin of the body using the theory of dynamic bending of a nonuniform rod.     

Shock compression of some porous media in conical targets: numerical study

Axially symmetric flows in conical solid targets filled by porous aluminum, graphite or polytetrafluoroethylene under impact of an aluminum plate with the velocity of 2.5 km/s are simulated numerically within the framework of the model of the hypoelastic ideal-plastic solid. The porosity of the samples is taken into account by conservation laws at the leading shock wave; the medium behind that is supposed to be nonporous. Equations of state for all materials in question are used to describe thermodynamic properties of the impactor and target over a wide range of pressures and temperatures taking into account phase transformations. The maximal over space and time pressure as a function of the initial relative density is presented and discussed.