Collision of an elastic finite-length cylinder with a rigid barrier

Вайсфельд, Н. Д.

doi:10.1007/s10778-007-0100-0

Cited by 5 publications

(6 citation statements)

References 8 publications

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“…B λ , and P = P B λ (here and in what follows, the overbars are omitted). The properties of the kernel and of the right-hand side of the integral equation (11) enable us to find its solution satisfying conditions (5) and (6). This solution has the form [27]…”

Section: Methods Of Volterra Equationsmentioning

confidence: 99%

“…These problems become especially urgent in connection with the intense introduction of units with contact interaction of thin-walled structural elements into the contemporary engineering practice. In this case, contact problems differ from the classical contact problems of the theory of elasticity [6,8,41,43,54,62,65] and their solution requires the application of different mathematical methods [11,20,64,73,77,80,81]. By using different versions of the theory of plates and shells, one can significantly simplify the mathematical model of the problems of contact interaction and wear and, at the same time, take into account the anisotropy of physicomechanical and thermal characteristics of contacting bodies (typical of new modern antifriction composite materials) and the presence of thin protective coatings [9,47,61,70,71,75].…”

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confidence: 99%

“…The extensive introduction of thin-walled structural elements, new composite materials, bodies with coatings, interface interlayers, patches, and stiffening ribs into engineering practice leads to the intense development of the investigations of contact problems for these structures. On the one hand, this class of contact problems is closely related to the classical contact problems of mechanics of deformable bodies and, on the other hand, it is characterized by certain specific features of the transfer of loads in the course of contact of thin-walled structures [6,8,11,19,29,31,39,47,53,54,64]. As the most popular methods used for the solution of contact problems for thin-walled structural elements, we can mention the conjugation method according to which the systems of key equations are solved separately in the contact zone and outside it with subsequent matching of the solutions; the semiinverse method, where, on the basis of some arguments used in the statement of the problem, we choose expressions for contact stresses with a certain collection of unknown parameters determined in the course of solving; the method of influence functions (Green's functions method), which makes it possible to reduce the problem of the evaluation of contact stresses to the solution of Fredholm or Volterra integral equations.…”

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confidence: 99%

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Specific Features of the Contact Interaction and Wear of Thin-Walled Structural Elements

Maksymuk

2023

J Math Sci

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We consider contact problems for thin-walled structural elements and their wear. We propose a unified method for solving the problems based on the reduction to Volterra integral equations. This enables us to detect singularities of solutions depending on the hypotheses characterizing the process of deformation of thin-walled elements. We present the solutions and analyze the problems of wear of the plates by a rigid punch and a hot punch with regard for the friction heating and changes in the thickness of the plate in the process of wear.

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Section: Methods Of Volterra Equationsmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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Specific Features of the Contact Interaction and Wear of Thin-Walled Structural Elements

Maksymuk

2023

J Math Sci

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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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“…The solution found demonstrates the behavior and features of the elastic waves in the layer, taking into account their repeated reflections from the layer boundaries. The problem of impact of an elastic cylinder against a rigid barrier was solved in [14], this problem being similar to that treated with asymptotic approaches in [8]. The solution constructed numerically in [14] has a certain error, especially at the wavefront.…”

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Nonstationary indentation of a blunt rigid body into an elastic layer: An axisymmetric problem

Kubenko

Marchenko

2008

Int Appl Mech

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The nonstationary indentation of a blunt rigid body into an elastic layer is studied. The general formulation of the problem includes different boundary conditions in the contact area and on the free surface of the layer. The simplified nonmixed problem that arises at the early stage of interaction and allows obtaining approximate results for later times is solved exactly. The solution obtained is compared with that for the plane case Keywords: nonstationary indentation, blunt body, elastic layer Introduction. The current state of the art in the research on nonstationary contact interaction and contact problem of elasticity is elucidated in, e.g., [4,6,9]. In the general case, the modern problem of impact of a body against an elastic medium or structural member is formulated as a nonstationary mixed initial-boundary-value problem of elasticity with an unknown time-dependent boundary, which is determined during problem solving. 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 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 (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 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 demonstrated in a number of studies, the supersonic solution is in some cases enough to make assessments.Such an approach was used in [11,12] to solve the plane problem of nonstationary indentation of a smooth blunt body into a layer of perfect fluid and into a layer of elastic material. The solution found demonstrates the behavior and features of the elastic waves in the layer, taking into account their repeated reflections from the layer boundaries. The problem of impact of an elastic cylinder against a rigid barrier was solved in [14], this problem being similar to that treated with asymptotic appro...

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Collision of an elastic finite-length cylinder with a rigid barrier

Cited by 5 publications

References 8 publications

Specific Features of the Contact Interaction and Wear of Thin-Walled Structural Elements

Specific Features of the Contact Interaction and Wear of Thin-Walled Structural Elements

Collision of dissimilar blunt elastic bodies: a plane problem

Nonstationary indentation of a blunt rigid body into an elastic layer: An axisymmetric problem

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