Axisymmetric Collision Problem for Two Identical Elastic Solids of Revolution

2004

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The impact interaction between a solid body and a liquid or elastic half-space and between two elastic bodies is analyzed. The main approaches and representative numerical results are set out.Introduction. The present review analyzes and generalizes the results from studies of the interaction between a body and a liquid, a body and an elastic medium, and between two deformable bodies, which are originally undeformed and come into contact at some initial moment. We will consider an impact of blunted bodies, namely, the initial stage of penetration of bodies with a smoothly curved surface into a medium. By penetration is usually meant a set of mechanical phenomena accompanying an impact of a moving body on the surface of a medium, followed by immersion into it. The forces of resistance exerted by the medium on the body are, as a rule, much greater than the forces that move the body prior to the impact. Penetrating bodies, especially those with the frontal part blunted, usually suffer maximum loads at the early stage of penetration. In this connection, it is important to analyze the impact interaction of penetrating (colliding) bodies and structural members.Solid-fluid impact theory came to be developed intensively in the 1930s of the 20th century in connection with water-landing studies. Later, similar studies were conducted in shipbuilding (slamming of high-speed vehicles), in new branches of mechanical engineering (impact of solids or liquid particles on structural members), in space engineering (splash-down (landing) of descent modules), etc. The general problem of impact of a body on a liquid (medium) is extremely complex, since the penetration process is accompanied by a variety of phenomena such as large strains of the body and medium, breaking of the medium surface, splash and jet formation, discontinuous flows, entrainment of the air space, cavitation, moving unknown boundaries, etc. The complete nonlinear analysis of the penetration of a structural member into a liquid is hardly feasible. Therefore, common practice is to isolate the most significant aspects from the general formulation of the problem, leaving less important factors aside. Note that we do not address numerical methods here (they are discussed, e.g., in [2]).Historically, there are several approaches to the problem in question. The earliest approach is to consider the medium as a perfect incompressible liquid. The first approaches due to Von Carman and Sedov, which helped to solve many applied problems, described the process globally. However, a researcher or an engineer often needs to know not only integral but also other characteristics such as contact pressure, stress-strain state at some (frontal) points or in the whole body, liquid velocity fields, etc. In this case, the problem needs a more rigorous formulation. One was proposed by Wagner [98] (modeling penetration as an impact on a floating rigid plate) and used as the basis for many research projects. Logvinovich [67] established the applicability limits for Wagner's theory, solved t...

Section: Impact On An Elastic Medium (Impact Of Elastic Bodies)mentioning

confidence: 99%

Impact of blunted bodies on a liquid or elastic medium

2004

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“…Eventually, the velocity of the contact boundary reduces down to transonic and then subsonic magnitudes because of the geometry of the bodies and the deceleration of the impactor, which causes the waves to reach (and interact with) the surface of the body outside the contact region. However, as shown in, e.g., [9,11,13], the supersonic solution is in some cases enough to make assessments.As follows from the cited publications, most studies on the topic address a rigid or elastic body colliding with an elastic half-space. There are much fewer studies on the interaction of a blunt impactor with an elastic body of finite dimensions.…”

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Nonstationary indentation of a rigid blunt indenter into an elastic layer: A plane problem

Marchenko

2008

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The nonstationary indentation of a rigid blunt indenter into an elastic layer is studied. An approach to solving a mixed initial-boundary-value problem with an unknown moving boundary is developed. The problem is reduced to an infinite system of integral equations and the equation of motion of the indenter. The system is solved numerically. The analytical solution of the nonmixed problem is found for the initial stage of the indentation process Keywords: impact, elastic layer, wave diffraction, mixed problem Introduction. The nonstationary contact problem of elasticity has intensively been studied for the past two to three decades owing to its practical importance, specific features of the physical process, and interesting ways of solving the associated boundary-value problems. The current state of the art in the field is reviewed in, e.g., [1,5,11]. Methods for solving such problems are outlined in, e.g., [7,8,10,12,17]. The problem of impact of a body against an elastic medium or a structural member is generally formulated as a nonstationary mixed initial-boundary-value problem of elasticity with an unknown time-dependent boundary, which is found in the course of solution. The problem statement includes the equations of elastic deformation of the impacted body, the equation of motion of the impactor, the relationship between the interaction force and the unknown contact boundary, the relationship between the contact area and the displacement (penetration) of the impactor, and the boundary and initial conditions. In the general case, the problem is coupled one with fuzzy input data, which predetermines the difficulties of its solution.Note the following physical factor that has a direct influence on the ways of solving the problem of impact of a body against an elastic medium. During an impact, the contact boundary between the bodies moves over their surfaces with variable velocity dependent on their shapes. If the bodies are blunted, the velocity of the contact boundary can be very high during the initial period (at least, higher than the velocity of elastic waves in the body (supersonic)). During this period, waves in the elastic body do not interact with its free surface and, hence, the boundary conditions on it can be chosen as one sees fit. This would allow us, at least for the early stage of interaction, to formulate a nonmixed boundary-value problem and, thus, to simplify the solution procedure. Eventually, the velocity of the contact boundary reduces down to transonic and then subsonic magnitudes because of the geometry of the bodies and the deceleration of the impactor, which causes the waves to reach (and interact with) the surface of the body outside the contact region. However, as shown in, e.g., [9,11,13], the supersonic solution is in some cases enough to make assessments.As follows from the cited publications, most studies on the topic address a rigid or elastic body colliding with an elastic half-space. There are much fewer studies on the interaction of a blunt impactor with an elastic body of fi...

“…The associated nonstationary mixed boundary-value problem will be reduced to an infinite sequence of integral equations. Its analytic solution can be found for the early stage of interaction using an asymptotic representation of cylindrical functions (see [9,10] for a similar approach for an elastic medium). In the second limiting case, the radius of the cavity will be assumed much greater than that of the body.…”

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confidence: 99%

Nonstationary interaction of a short blunt body with a cavity in a compressible liquid

2006

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The shock-interaction problem for a rigid spherical body and a spherical cavity in a compressible liquid is formulated and solved. Three typical cases of typical dimensions of the body and cavity are examined. An asymptotic solution valid at the earliest stage of interaction is obtained. In the general case, the problem is reduced to an infinite system of integral equations of the second kind. It is numerically solved for the case of a nonsmall air gap Keywords: shock interaction, compressible liquid, rigid body, cavity Introduction. When moving through a liquid with a nearly sonic velocity, a body gives rise to a thin long cavity (so-called supercavity) inside which the body is in effect flying [6][7][8]. Such a motion is usually unstable, resulting in impacts of the body on the walls of the cavity. The longitudinal dimensions of the body and cavity are as a rule much greater than their lateral dimensions. Therefore, a two-dimensional modeling of lateral impacts would be natural. This was done in [7,13] for an incompressible liquid and in [12] for a compressible liquid. For example, the paper [12] suggests expanding the solution for a rigid cylindrical body in a cylindrical cavity into Fourier series to reduce the problem to an infinite system of Volterra equations and the equation of lateral motion.The present paper develops an approach to the study of the shock interaction between a spherical rigid body and a spherical cavity in a compressible liquid. The results discussed below may find application in a mathematical analysis of a method to launch a supercavitating body from within a preliminarily created gas bubble in a liquid [7]. It is generally assumed that there is an air gap between the body and cavity, i.e., their radii are not equal. Depending on the size of the gap, we will examine one general case and two limiting cases. In the general case, we will formulate a nonstationary mixed boundary-value problem with an unknown boundary varying with time. By expanding the unknown variables into Legendre series, the problem will be reduced to an infinite system of integral equations to be solved simultaneously with the equation of translational motion and the equation of the body-liquid contact boundary. In the first limiting case, the gap will be assumed so small that it can be neglected in formulating the boundary conditions [5]. Hence, the contact area is constant, curved, and not small compared with the cavity. The associated nonstationary mixed boundary-value problem will be reduced to an infinite sequence of integral equations. Its analytic solution can be found for the early stage of interaction using an asymptotic representation of cylindrical functions (see [9,10] for a similar approach for an elastic medium). In the second limiting case, the radius of the cavity will be assumed much greater than that of the body. Hence, the contact area is small and its curvature is insignificant. This makes it possible to model the process as an impact of a spherical body on the flat surface of a compressible liquid...