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

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

doi:10.1007/s10778-008-0044-z

Cited by 14 publications

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“…The analogue pattern was also obtained in plane case [9]. Note that additional results calculated with (16) depending on parameters of layer material and its thickness are presented in [2].…”

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confidence: 74%

“…Analytical solution. This section is based on the problem solving technique presented in [2]. The formulated problem admits an analytical solution on the condition that the indenter contour is a sufficiently smooth, gently changing curve.…”

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confidence: 99%

“…Following to [2] we use the Laplace integral transform in time (parameter s , upper index L ) and the Bessel integral transform along the radial coordinate (parameter x , upper index B ): Our problem is to reverse expression (14) with respect to integral transforms. To this end, we restrict analytical results by obtaining normal stresses in the axis z (r = 0) in the case of a parabolic surface of the indenter moving with a constant velocity V 0 .…”

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confidence: 99%

“…Omitting details of the reversion technique (see [2]) we present the final expression of the above-mentioned stresses linearly increase with time and inversely proportional to the general rigidity of the layer (or, it the same, to the layer thickness). The analogue pattern was also obtained in plane case [9].…”

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Impact indentation of a rigid body into an elastic layer. Axisymmetric problem

2011

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An axisymmetric contact-impact problem is considered for an elastic layer subjected by normal indentation of a rigid body. An exact analytical solution is obtained in the case1. Introduction. This paper is the continuation to the axisymmetric case of the plane problem analyzed in [9]. The introduction presented in [9] could be practically completely replicated herein, as well as sited works within [9]. With analytical approaches in mind, we refer review [1] reflecting the multitude of studies of a body's impact interaction with elastic and liquid media, while numerical approaches (primarily the method of finite elements) can be found in review [17]. A generalizing monograph in the field of contact interaction [3] is devoted to the development of analytical approaches to the solution of problems about the action of impact on an elastic medium. In the common case, the indentation problem is formulated as a unsteady-state mixed initially-boundary elasticity problem with an unknown (temporally varying) boundary, which must be determined in the course of the solution. The problem statement includes:· equations of dynamic deformation of the impacted solid; · the equation of the indenter motion; · the ratio presenting the resistive force (drag) as a function of contact stresses on a priori unknown contact surface; · the equation connecting the contact zone size with the indenter displacement, · the corresponding boundary and initial conditions; The overwhelming majority of publications (at least of those in which analytical methods are used) are devoted to the problem of impact by rigid or deformable indenter against a halfspace that precludes the possibility of analyzing the waves reflected from the boundaries of the impacted solid. Studies of indenter interaction with solids of finite size are much less represented. Positing such a problem appears topical in the practical aspect as well -in particular, in view of the wide use of laminate materials in modern aircraft and shipbuilding. It is noteworthy that scale effect is among the determinant qualitative factors for problems of stresses and fracture in impact interaction (see e.g. [10,11]): the structure element under impact loading is destroyed by stresses whose level is formed due to superposition of waves reflected from boundary surfaces. The classical Hertz theory of collision is known to be applicable in dynamics at large time values, i. е. after the wave processes have faded in the solid. The Saint -Venant wave theory of rod collision is well developed only for one-dimensional or quasi one-dimensional problems and does not take into account energy transfer in directions different from the impact one. The foresaid evaluates the motivation of the presented work devoted to the construction and investigation of more adequate models and methods for dynamic processes of indentation. This paper, in similar to [9], consists of two parts: in the first one a precise analytical solution of the problem is built of the indentation with a constant impact velocity of an smoot...

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Impact indentation of a rigid body into an elastic layer. Axisymmetric problem

2011

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“…The initial stage of the study was to neglect the dependence of the drag torque on the angular velocity, not neglecting its dependence on the angle of attack. All the results found under this elementary assumption lead us to the conclusion that it is impossible to establish conditions in which the systems would have solutions describing angular oscillations of the body with a limited amplitude (see also [16,17]). …”

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Spatial motion of a rigid body in a resisting medium

Шамолин

2010

Int Appl Mech

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The paper studies the spatial motion of a rigid body in a resisting medium under the action of a follower force that causes the center of mass to move rectilinearly and uniformly. The body, which is axisymmetric and homogeneous, interacts with the medium by its frontal area that has the form of a flat circular disk. Since there is no exact analytic description of the forces and torques exerted by the medium on the disk, the problem is "immersed" in a wider class of problems. Partial solutions and phase portraits in the three-dimensional space of quasivelocities are obtained for the dynamic systems under consideration. The transcendental first integrals of the dynamic part of the equations of motion are listed

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Non-stationary stress state of an elastic layer at the mixed boundary conditions

Kubenko¹

2016

Z. Angew. Math. Mech.

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An exact analytical solution has been constructed for the plane problem on action of a non-stationary load on the surface of an elastic layer for conditions of a 'mixed' boundary problem when normal stress and tangent displacement (the fourth boundary problem) are specified on one boundary and normal displacement and shear stress (the second boundary problem) for another boundary. Laplace and Fourier integral transforms are used. Their inversions were obtained with the help of tabular relationships and the convolution theorem for a wide range of acting non-stationary loads. Expressions for stresses (displacements) were obtained in explicit form. The obtained expressions allow determining the wave process characteristics in any point of the layer at an arbitrary moment of time. Some variants of non-stationary loads acting on an area with fixed boundaries or an area with boundaries changing by a known function are considered. Distinctive features of wave processes are analyzed.The boundary conditions in problems of elasticity theory belong to one of four types; see e.g. [1]. The principal types are the first and second, wherein the stress and displacement vectors, respectively, are specified. The third and fourth types are called mixed boundary conditions because each employs both displacement and stress, i.e. type three specifies the normal component of the displacement vector and the tangential components of the stress vector while type four species the normal stress and tangential displacement components. The mixed boundary conditions are physically less realistic and sometimes considered approximations of types one and two. The choice of which boundary condition to use could significantly inform the likelihood of obtaining an analytic solution for non-stationary problems.The effectively analytical and numerically analytical methods, grounded mainly in the application of the Laplace transform in the time domain and Fourier (Hankel) transform for the linear coordinate, are developed for a solution of relevant boundary value problems. The problem thus is the build-up of originals which primarily depends on the character of the space-time distribution of the load on the boundary and the type thereof. Interested readers are directed to refs.[2-5] for studies of non-stationary wave processes at an operation of the concentrated or distributed surface actions in an isotropic elastic half-space (half-plane).Refs.[6] and [7] give the solutions of many non-stationary problems. The displacements of points on the surface of the semi-space as well as on the axis of symmetry were studied in refs.[8] and [9]. Thereat, for joint inversion of transforms, the authors used the Cagniard technique with account of transform uniformity with respect to transform parameters [10]. The saddle-point was used in ref. [9] for an approximate search for displacements of points remote from the axis of symmetry. Similarly, with the inverse Laplace transform, using the theory of residues, refs. [11][12][13] solved for the displacements and stress...

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

Cited by 14 publications

References 12 publications

Impact indentation of a rigid body into an elastic layer. Axisymmetric problem

Impact indentation of a rigid body into an elastic layer. Axisymmetric problem

Spatial motion of a rigid body in a resisting medium

Non-stationary stress state of an elastic layer at the mixed boundary conditions

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