Plane Collision Problem for Two Identical Elastic Parabolic Bodies—Direct Central Impact

Kubenko, V. D.; Marchenko, T. A.

doi:10.1023/a:1026273623303

Cited by 6 publications

(6 citation statements)

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“…According to Goldsmith's experiments [18], when V c p 0 2 10 / < − (c p is the velocity of longitudinal waves), the energy of impact went to develop plastic strains is much less than the initial kinetic energy, and the elastic model quite accurately describes the process. A plane impact problem for a rigid blunted body was addressed in [36,37,63,64,70], axially symmetric impact problems in [36,65,71], impact problems for an elastic shell in [45][46][47]66], and collision problems for blunted elastic bodies in [39,41,61,62,69,92].…”

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

confidence: 99%

Impact of blunted bodies on a liquid or elastic medium

Kubenko

2004

Int Appl Mech

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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...

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

confidence: 99%

Impact of blunted bodies on a liquid or elastic medium

Kubenko

2004

Int Appl Mech

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“…The modern formulation of the impact problem leads to a mixed boundary problem with initial conditions and an unknown time-dependent contact boundary determined during the solution. Some recent approaches to and results on the subject can be found in [4,6,[8][9][10]. A variable contact boundary and associates difficulties of its description are also encountered in nonclassical fracture mechanics [7].…”

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confidence: 99%

Collision of dissimilar blunt elastic bodies: a plane problem

Kubenko

Marchenko

2009

Int Appl Mech

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The paper deals with a direct central impact of two infinite cylindrical bodies having differently shaped cross sections and made of different materials. A nonstationary plane problem of elasticity is solved. The contact boundary is moving and determined during the solution. A mixed boundary-value problem is formulated. Its solution has the form of Fourier series. Satisfying mixed boundary conditions gives an infinite system of Volterra equations of the second kind for the unknown coefficients of the series. The basic characteristics of the impact process and their dependence on the physical and mechanical properties of the bodies are determined numerically Introduction. Impact of elastic bodies is a complicated and difficult-to-solve problem in continuum mechanics. The modern formulation of the impact problem leads to a mixed boundary problem with initial conditions and an unknown time-dependent contact boundary determined during the solution. Some recent approaches to and results on the subject can be found in [4,6,[8][9][10]. A variable contact boundary and associates difficulties of its description are also encountered in nonclassical fracture mechanics [7].Here we present a method to solve the central impact problem for elastic cylindrical bodies with different physical and mechanical properties (masses, shapes of frontal surface, materials). This approach was earlier used to study the direct central impact of identical elastic bodies [4,6]. As before, we assume that there is no friction between the bodies, the impact velocities are small, and the frontal surfaces are smooth and have small curvature. At the initial instant of impact, the bodies interact along the common generatrix. In this case, the deformation pattern is the same in all cross sections of the cylinders; therefore, a plane problem can be formulated to study the process. We also assume that the depth of interpenetration of the bodies is small compared with the contact region, which makes it possible to prescribe boundary conditions on the undisturbed surfaces of the bodies. We will examine a period during which the perturbations reflected from the rear and lateral surfaces of the bodies do not yet reach the contact region, which permit formulating somewhat arbitrary boundary conditions on the lateral and rear surfaces.There may be different problem formulations depending on the method used to determine the contact boundary. We will present specific numerical results and analyze the dependence of the stress-strain state of the bodies on their physical and mechanical properties. A central-impact problem is formulated in Sec. 1. The governing system of equations for determining the stress state of and the force of interaction between the bodies is derived in Sec. 2. A simplified governing system of equations for determining the characteristics of the process at its initial stage is derived in Sec. 3. Calculated results are presented and conclusions are drawn in Sec. 4.1. Problem Formulation. Consider two elastic cylindrical bodies having parallel...

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“…Though various approaches were considered in [1-3, 5, 6, 11, 13, 14, etc.] to solve this problem, no exact and complete results have been obtained yet.The findings in nonstationary elasticity and hydroelasticity are discussed in [14,16,18]. We will solve the problem on the basis of linear theory.…”

mentioning

confidence: 99%

“…The findings in nonstationary elasticity and hydroelasticity are discussed in [14,16,18]. We will solve the problem on the basis of linear theory.…”

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confidence: 99%

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Interaction of a spherical shockwave and an elastic circular cylindrical shell

2004

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The paper studies the interaction of a spherical shock wave with an elastic circular cylindrical shell immersed in an infinite acoustic medium. The shell is assumed infinitely long. The wave source is quite close to the shell, causing deformation of just a small portion of the shell, which makes it possible to represent the solution by a double Fourier series. The method allows the exact determination of the hydrodynamic forces acting on the shell and analysis of its stress state. Some characteristic features of the stress state are described for different distances to the wave source. Formulas are proposed for establishing the safety conditions of the shell.Consider an infinitely long elastic circular cylindrical shell immersed in an infinite fluid. A point source of shock waves (SWs) is located at a distance R 0 from the shell axis. A spherical SW is diffracted by the shell, and while deforming, the shell generates radiation waves. Therefore, the stress analysis of the shell must involve the simultaneous solution of the equations of motion of the fluid and shell coupled by boundary conditions at the shell surface. Though various approaches were considered in [1-3, 5, 6, 11, 13, 14, etc.] to solve this problem, no exact and complete results have been obtained yet.The findings in nonstationary elasticity and hydroelasticity are discussed in [14,16,18]. We will solve the problem on the basis of linear theory. Let the shell radius r 0 , the density of the fluid ρ 0 , and the sonic velocity in the fluid c 0 be units of measurement. Then all other quantities are measured in terms of fractions of the power complex r c 0 0 0 α β γ ρ that has the same dimension as a given quantity. With such an approach, all dependences will be dimensionless, which is convenient for theoretical analysis. We will describe the deformation of the shell using the linear theory of shells based on the Kirchhoff-Love hypothesis. Let the displacements of the shell's median surface, u, ν, and w, be the basic variables. Then, written in a cylindrical coordinate system x, r, θ whose axis coincides with the shell axis, the equations of motion of the shell take the following form [11]:

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Plane Collision Problem for Two Identical Elastic Parabolic Bodies—Direct Central Impact

Cited by 6 publications

References 5 publications

Impact of blunted bodies on a liquid or elastic medium

Impact of blunted bodies on a liquid or elastic medium

Collision of dissimilar blunt elastic bodies: a plane problem

Interaction of a spherical shockwave and an elastic circular cylindrical shell

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