Penetration of rigid bodies into loose and layered media

Penetration and motion of rigid bodies in ground media attracts the researchers’ attention because of various problems arising as the technology evolves. In fact, there are two independent directions of studies in this field: (1) the problem of earth excavation when a rigid body of a definite shape slowly moves along a given trajectory in the ground; (2) an impact of a rapidly flying free rigid or deformable body against the ground. In the latter case, to which the proposed studies pertain, it is sometimes of interest to study the medium behavior and the motion of the free body, which moves in the medium after the impact owing to the kinetic energy of itself. In this field, a majority of studies deal with collision and penetration of bodies of various shapes into clay media. An extensive survey of these studies is given in [1]. After this survey appeared, numerous paper dealing with complicated collision conditions have been published [2]. Penetration in loose media has been studied much more rarely. The direct collision with fractured rock was studied in connection with the expected landing of spacecraft on other planets [3, 4]. In this case, the influence of grain dimensions and the density of the filling and vacuum on the penetration was studied for the initial velocities in the range of 1.7–10 m/s. On the other hand, in [5], the results of investigating the penetration of conic bodies in sand at entry velocities of 700–900 m/s are given; these velocities significantly exceed the speed of sound in this medium, which lies in the range of 100–200 m/s for dry sand. Analyzing the experimental results, the author came to the conclusion that it is necessary to use different representations of the drag force in the supersonic and subsonic modes. In the present paper, we do not consider the influence of the grain distribution, sand density, and filling methods on the penetration. But, as follows from the experiments whose results are described in [6] and [7], to represent the results of penetration of rigid bodies at velocities up to several hundreds of meters per second, in addition to the characteristics listed above, it is also required to describe the technology of the experiment preparation, because such media have the property of shape “memory.”     

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].     

Stability of spatial motion of a body of revolution in an elastoplastic medium

A previously constructed model that describes the spatial motion of a body of revolution in an elastoplastic medium (without flow separation and with nonsymmetric separation of the medium flow taken into account) is used to study the Lyapunov stability of rectilinear motion of a body in the case of frozen axial velocity on a half-infinite time interval. Some stability criteria are obtained and the influence of tangential stresses is analyzed.     

弹体侵彻混凝土的临界跳弹

为了保证钻地战斗部打击防护层目标时不发生跳弹,需要对弹体侵彻目标的临界跳弹角度进行分析和估算。开展了一定大长径比弹体斜侵彻混凝土的跳弹实验,分析了在250~430 m/s速度下弹体侵彻30和60 MPa钢筋混凝土的临界跳弹角度,给出了弹体临界跳弹角度包络线。当靶板强度相同时,随着侵彻速度的增加,弹体的临界跳弹倾角增大,增大的趋势逐渐变缓;在相同侵彻速度下,随着靶板强度的增加,弹体的临界跳弹倾角减小;经验公式分析得到的弹体临界跳弹倾角偏低于实验,但偏差基本在3°以内。     

Elasticoplastic Model of Sludge Injection into a Weak Water-Pressurized Reservoir

Sludge injection, i. e., injection of a mixture of water and solid particles (rock waste or drilling cuttings), can be interpreted as the penetration of a plastic mass into an elastoplastic water-pressurized stratum under the action of high gradients of the pore pressure. A system of elastoplastic deformation of a saturated porous medium describing the injection processes in both the sand formation and the zone occupied by sludge is formulated. An axisymmetric self-similar problem implying the frontal stable displacement of sand by sludge is considered. The chosen interval of variables and the results obtained correspond to the field experimental data.     

弹体斜侵彻混凝土靶的实验研究及其数值模拟

以弹体斜侵彻混凝土的弹道特性为研究内容,通过侵彻实验与数值模拟得到了不同速度下的侵彻深度、开坑尺寸、偏转角等参数,实验结果与模拟结果吻合较好。研究结果表明:倾角对开坑深度和开坑形状影响很大;倾角越大,对侵彻深度和偏转角的影响越明显,弹体偏转角随着速度的增大呈现减小的趋势;当倾角增至一定角度后发生跳弹现象,据此得到跳弹极限角与倾角、侵彻速度的关系。     

细长尖头刚性弹对金属靶板的斜侵彻/穿甲分析

给出细长尖头刚性弹（如尖卵、尖锥形）斜侵彻/穿甲金属靶的一个分析模型。在细长尖头弹对中厚度金属靶的斜穿甲中,韧性孔洞扩张为主要的穿甲机理;着靶初期,发生方向角的改变。研究表明,金属靶的斜穿甲仅由4个量纲一参数控制,即冲击函数I、弹体几何函数N、量纲一靶厚和撞击斜角。分析得到显式的侵彻深度、终点弹道极限、剩余速度和撞击方向改变角表达式。该模型可预期跳飞发生的临界条件。理论预期与实验结果吻合较好。     

The stability of the interface between two bodies compressed along interface cracks. 2. Exact solutions for the case of equal roots

The problem of buckling of the interface between two bodies is considered for the case where several plane cracks are located in the interface and the bodies are compressed along the cracks (along the interface of two different materials). The studies were carried out for a plane problem using the three-dimensional linearized theory of stability of deformable bodies. The complex variables and potentials of the above-mentioned linearized theory are used. This problem is reduced to the problem of linear conjugation of two analytical functions of a complex variable. The exact solution of the above-mentioned buckling problem is obtained for the case where the roots of the basic equation are equal. Some mechanical effects are analyzed under general conditions (elastic, elastoplastic, compressible, incompresible, isotropic, and orthotropic bodies). S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 5, pp. 66–73, May, 2000.     

圆柱形钨破片撞击铝靶时的跳飞临界角

采用理论计算、数值模拟与实验相结合的方法,研究了直径5.7 mm、长6.7 mm的圆柱形破片以800~1 200 m/s的速度撞击2~10 mm厚铝靶时的跳飞特性。建立了破片斜侵彻有限厚靶板的跳飞临界角理论模型,计算得到破片跳飞临界角与破片入射速度、靶板厚度的关系,并与模拟值、实验值对比,三者吻合较好。结果表明:破片撞靶速度相同时,随着靶板厚度的增加,破片的跳飞临界角减小。靶板厚度相同的情况下,在所计算的速度范围内,入射速度越大,破片跳飞临界角越大。速度在800~1 200 m/s时,破片撞击2 mm厚靶板的跳飞临界角为81°~81.25°;撞击4 mm厚靶板的跳飞临界角为72.5°~76.25°。     

Finite-element simulation of contact interaction with the use of concepts of contact pseudomedium mechanics

In the present paper, we analyze an algorithm for solving problems of elastic and elastoplastic contact between rigid deformable bodies taking into account the presence of additional media with various properties in the contact region. These may be nonlinear properties of surface rough layers and films. To take into account the influence of such layers, the finite-element scheme uses contact finite elements with nonlinear properties. We describe an algorithm developed for solving problems with various types of nonlinearities. Using the example of experimental studies described in detail in [1, 2], we identify the nonlinear properties of contact finite elements. Several examples are used to demonstrate the algorithm operating capability.     

Determination of the ultimate states of elastoplastic bodies

The equations of quasistatic deformation of elastoplastic bodies are considered in a geometrical linear formulation. After discretization of the equations with respect to spatial variables by the finite-element method, the problem of determining equilibrium onfigurations reduces to integration of a system of nonlinear ordinary differential equations. In the ultimate state of a body of an ideal elastoplastic material, the matrix of the system degenerates and the problem becomes singular. A regularization algorithm for determining solutions of the problems for the ultimate states of bodies is proposed. Numerical solutions of test problems of determining the ultimate loads and equilibrium configurations for ideal elastoplastic bodies confirm the reliability of the regularization algorithm proposed. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 5, pp. 196–204, September–October, 2000.     

Stability Problems for Plates with Short-Term Damageability

The problem on stability of plates with microdamages simulated by hollow randomly dispersed micropores is considered. Two approaches are proposed to investigate the stability of plates weakened by microdamages. These approaches are based on models well known from the theory of stability of elastoplastic bodies — the concepts of tangent-modulus loading and continuous loading     

Study of unsteady flow past bodies behind shock waves

Several theoretical and experimental studies have been devoted to the problem of the nonstationary action of the stream behind a shock wave on bodies of varied shape. In particular, in [1], the pressure and density are calculated for flow about bodies of the more typical shapes in the initial stage of the process. The basic relations which accompany the interaction of shock waves are considered in [2, 3]. The analysis of the phenomena of diffraction of shock waves on the sphere, cylinder, and cone is presented in [4]. Problems of unsteady flow about a wing are examined in [5, 6]. A detailed review of the foreign studies on unsteady flow is given in [7]. Of great practical interest is the question of the time for flow formation and the magnitudes of the unsteady loads during this period. Experimental investigations have been made recently [8, 9] in which some criteria are presented for estimating the bow shock formation time for supersonic flow about the sphere and cylinder with flat blunting. However the question of the formation time of the stationary pressure on the body surface is not referred to in these studies and no relationship is shown between the transient position of the reflected wave and the corresponding unsteady pressure on the surface. Moreover, in [8] the dimensionless time criterion is determined very approximately, independently of the Mach number of the shock wave. The present study was undertaken with the object of determining the basic criteria which characterize unsteady flow about bodies behind a plane shock wave which has time-independent parameters, and clarification of the shock wave reflected from the body and the pressure on the surface of the body during the transient period. The most typical body shapes were studied: 1) a cylinder with flat face aligned with the stream; 2) a spherically-blunted cylinder; and 3) a cylinder transverse to the stream. The experiments were conducted in a conventional shock tube using the single-diaphragm scheme. The measurements of the pressure on the models and the velocity of the incident shock wave were made using the technique analogous to that of [10, 11]. A highspeed movie camera was used to record the pattern of the wave diffraction on the body. The Mach number of the incident shock wave varied in the range from M=1.5 to M≈6.0, which corresponded to a range of Mach numbers M_∞ of the stream behind the shock wave from 0.6 to 2.1. The calculations of the required gas dynamic parameters for high temperatures were made with account for equilibrium dissociation of the air on the basis of the data of [10, 12, 13]. The magnitude of the relative maximal shock wave standoff Δ at the stagnation point obtained in the present experiments was compared with the values of Δ from other studies. In the case of the flat-blunted cylinder it was in good agreement with the results of [8–14], and in the case of the spherically-blunted cylinder and the transverse cylinder it was in agreement with the results of [15].     

Conservation of the resultant force vector in the plane of the angle of attack for slender star-shaped bodies in supersonic flows

At high supersonic flight speeds bodies with a star-shaped transverse and power-law longitudinal contour are optimal from the standpoint of wave drag [1–3]. In most of the subsequent experimental [4–6] and theoretical [6–9] studies only conical star-shaped bodies have been considered. For these bodies in certain flow regimes ascent of the Ferri point has been noted [10]. In [11] the boundary-value problem for elongated star-shaped bodies with a power-law longitudinal contour was solved for the case of supersonic flow. The present paper deals with the flow past these bodies at an angle of attack. It is found that for arbitrary star-shaped bodies with any longitudinal (in particular, conical) profile the aerodynamic forces can be reduced to a wave drag and a lift force, the lateral force on these bodies being equal to zero for any position of the transverse contour.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 135–141, November–December, 1989.     

Unexpected ricochet of spheres off water

A sphere was observed to apparently ricochet off the free surface of water at incident angles as large as 45° while the expected (empirical/analytical) maximum angle to the horizontal for ricochet was 6°. Closer examination of the process revealed that the cavitating sphere penetrated the liquid to depths as great as 35 sphere diameters. Under certain circumstances the sphere was also observed to leave the liquid in a direction close to the incoming direction; that is, the sphere ricocheted backwards! This peculiar behavior was found to be a result of an unintentional spin applied to the sphere upon launching. By crudely modelling the process, the sphere path is qualitatively predicted. It was found that the drag and lift coefficients required to model the trajectory data were several times smaller than those obtained for the non-cavitating case or for the non-spinning case. If more precise sphere trajectory data were available, this experiment could be used to measure the lift and drag coefficients of a spinning and cavitating sphere.The author would like to thank T. Miloh for suggesting an experimental investigation into the ricochet problem and for his help and discussion of this work. Preliminary experiments were performed by Jim Oelberg as a special undergraduate project. Randy Hostetler, our Laboratory Supervisor, designed and built the final apparatus and John Leonard our Instrument Repair Technician, built and improved the associated electronic equipment for the timing device.     

钨球侵彻LY-12铝合金靶板的有限元分析

介绍了利用LS-DYNA进行球形破片侵彻靶板的网格划分方法，确定了钨球破片和LY-12铝合金的材料模型，利用不同尺寸的球形破片对不同厚度的靶板进行了仿真分析，得出了不同情况下破片对靶板的穿透曲线图、正向跳飞曲线图和破片穿透靶板时的临界入射角和正向跳飞时的临界入射角，并对破片的剩余速度和末角度进行了分析。     

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.     

Experimental investigation of supersonic flow past blunt bodies with strong distributed injection

The entry of bodies into planetary atmospheres at high supersonic velocities is accompanied by intense evaporation of the surface due to radiative heat fluxes. A series of problems involving the conduction of investigations of such kind has been proposed by Petrov. In [1], in particular, the entry of a meteorite into an atmosphere was examined. The gasdynamic aspects of this problem have been approximately simulated by many authors by intense injection of gas in theoretical, e.g., [2–5], and experimental [6, 7] studies. The theoretical studies were based on two-layer [3, 4] or three-layer [5] schemes of gas flow between the shock wave and the surface of the body. The aim of the present work was an experimental investigation of the interaction of injection with a counter supersonic flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 84–95, May–June, 1978.     

The stability of the interface between two bodies compressed along interface cracks. 3. Exact solutions for the combined case of equal and unequal roots

The problem of the stability of the interface between two bodies is considered for the case where several plane cracks are located in the interface, and the bodies are compressed along them (along the interface of two different materials). The study is carried out for a plane problem by using the three-dimensional linearized theory of stability of deformable bodies. Complex variables and potentials of the above-mentioned linearized theory are used. This problem is reduced to the problem of linear conjugation of two analytical functions of complex variable. The exact solution of the above-mentioned problem is derived for the case where the basic equation has unequal roots for the first material and equal roots for the second material. In earlier authors' publications, the exact solutions were obtained for the cases where both materials have either equal or unequal roots. Some mechanical effects are analyzed for the general formulation of the problem (elastic, elastoplastic compressible and incompressible isotropic and orthotropic bodies). It is pointed out that, in accordance with the exact solutions, the main result and conclusions have a general form for the above-mentioned cases of roots. S.P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 6, pp. 67–77, June, 2000     

Motion of a cylinder in a viscous electroconductive medium with a magnetic field

The method of force sources is used to consider the planar problem of the motion of a circular cylinder in a viscous electroconductive medium with a magnetic field. The conventional and magnetic Reynolds numbers are assumed to be small. Expressions are obtained for the hydrodynamic reaction forces of the medium, acting on the moving cylinder. It is shown that as a result of the flow anisotropy in the medium, caused by the magnetic field, in addition to the resistance forces on bodies moving at an angle to the field, there are deflecting forces perpendicular to the velocity vector. The velocity field disturbances at great distances from the moving cylinder are determined.The problems of viscous electroconductive flow about solid bodies in the presence of a magnetic field constitute one of the divisions of magnetohydrodynamics. Motion of an electroconductive medium in a magnetic field gives rise to inductive electromagnetic fields and currents which interact with the velocity and pressure hydrodynamic fields in the medium [1, 2]. Under conditions of sufficiently strong interaction, the number of independent flow similarity parameters in MHD is considerably greater than in conventional hydrodynamics. This circumstance complicates the theoretical analysis of MHD flow about bodies, and therefore we must limit ourselves to consideration of individual particular flow cases.Here we consider the linear problem of the motion of an infinite circular cylinder in a viscous incompressible medium with finite electroconductivity located in a uniform magnetic field.There are many studies devoted to the flow of a viscous electroconductive medium with a magnetic field about solid bodies (see, for example, [3–5]). Because of this, some of the results obtained here include previously known results, which will be indicated below. In contrast to the cited studies, the examination is made by the method of force sources, suggested in [6]. This method permits obtaining integral equations for the distribution of the forces acting on the surface of the moving body. Their solution is obtained for small Reynolds and Hartmann numbers. Then the nature of the velocity disturbances at great distances from the body are determined. These results are compared with conventional viscous flow about a cylinder in the Oseen approximation.