Dynamics of a rod undergoing a longitudinal impact by a body

A longitudinal elastic impact caused by a body on a thin rod is considered. The results of theoretical, finite element, and experimental approaches to solving the problem are compared. The theoretical approach takes into account both the propagation of longitudinal waves in the rod and the local deformations described in the Hertz model. This approach leads to a differential equation with a delayed argument. The three-dimensional dynamic problem is considered in terms of the finite element approach in which the wave propagation and local deformation are automatically taken into account. A benchmark test of these two approaches showed a complete qualitative and satisfactory quantitative agreement of the results concerning the contact force and the impact time. In the experiments, only the impact time was determined. The comparison of the measured impact time with the theoretical and finite element method’s results was satisfactory. Owing to the fact that the tested rod was relatively short, the approximate model with two degrees of freedom was also developed to calculate the force and the impact time. The problem of excitation of transverse oscillation after the rebound of the impactor off the rod is solved. For the parametric resonance, the motion has a character of beats at which the energy of longitudinal oscillation is transferred into the energy of transverse oscillation and vice versa. The estimate for the maximum possible amplitude of transverse oscillation is obtained.     

Waves induced by the longitudinal impact of a rod against a stepped rod in contact with a rigid barrier

The longitudinal elastic central impact of a rod system, consisting of a uniform rod having a pre-impact velocity against a stepped rod which is in a state of rest and interacts by means of unilateral constraints with a rigid barrier, is modelled.     

Formation of cracks on compressing an unbounded brittle body with a circular opening

A simplified model of a brittle body /1/ is used as a basis for investigating the appearance of cracks originating at the boundary of a circular cavity in a body in a state of plane deformation caused by uniaxial compression at infinity. The singular integral equation of the problem is reduced to an integral Fredholm equation with a degenerate kernel. The solution is obtained in the form of a Fourier series in terms of Legendre polynomials.     

The equilibrium of a rigid body on a plane with anisotropic dry friction

The problem of the conditions for the static equilibrium of a body, resting on a rough plane at one, two or three points, is considered. It is assumed that an arbitrary system of active forces is applied to the body, while the friction on the rough supporting plane is anisotropic. This model generalizes the well-known isotropic model of Coulomb dry friction. Explicit analytic formulae, which express the necessary and sufficient conditions for static equilibrium, are obtained. The investigation procedure uses the idea of an anisotropic force of static friction, which enables analytical results for the equilibrium conditions to be obtained more easily.     

Investigation of longitudinal wave propagation in a polyethylene rod

Longitudinal wave propagation in a polyethylene rod has been investigated. A computer has been used to obtain a numerical solution of the problem of wave propagation in a rod, whose material is described by the Boltzmann-Volterra equation. The results of the computations are compared with the experimental data. The method of characteristics is employed to construct the dynamic – diagrams for the case of wave propagation in a prestressed rod on the basis of the model of a viscoelastic standard solid.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 59–67, January–February, 1970.     

The interaction of a magnetoelastic shear wave with longitudinal cavities in a conducting layer

We study the influence of a strong magnetic field on the interaction of a shear wave with longitudinal cylindrical cavities in an elastic ideally conducting layer. The resulting singular integral equation of the boundary-value problem under consideration is implemented numerically for the case of a single cavity. We present the results of computation of the stresses on the edge of a circular cavity and an elliptical cavity. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 74–78.     

Bending of an anisotropic rod with longitudinal cavities

We study the stress state of hollow anisotropic rods bent by a force at the end. The complex potential is represented by a series in a specially constructed system of fundamental solutions of a suitable differential operator. As an example we consider the bending of a rod of square cross section with a centrally located square hole. Two figures. One table. Bibliography: 3 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 44–48.     

The Signorini problem with Coulomb friction for a hyperelastic body under finite deformations

The quasi-static three-dimensional problem of elasticity theory for a hyperelastic body under finite deformations, loading by bulk and surface forces, partial fastening and unilateral contact with a rigid punch and in the presence of time-dependent anisotropic Coulomb friction is considered. The equivalent variational formulation contains a quasi-variational inequality. After time discretization and application of the iteration method, the problem arising with “specified” friction is reduced to a non-convex miniumum functional problem, which is studied by Ball's scheme. The operator in contact stress space is determined. It is shown that a threshold level of the coefficient of friction corresponds to each level of loading, below which there is at least one fixed point of the operator. If the solution at a certain instant of time is known, the iteration process converges to the solution of the problem at the next, fairly close instant of time.     

Permanent rotations of a body with a viscous filler suspended on a rod

Problems of the existence, stability, and branching of the permanent rotations of a heavy, dynamically symmetrical rigid body suspended on a rod and which has an axisymmetric ellipsoidal cavity filled with a fluid are discussed. The phenomenological model of the friction of the fluid against the cavity wall proposed by Samsonov is used. All the trivial permanent rotations of the system and the non-trivial rotations that branch off from the trivial ones are found. Their stability and branching are investigated using a modified Routh's theory. The results obtained are presented in the form of an atlas of bifurcation diagrams.     

Parametric resonances in the problem of longitudinal impact on a thin rod

The longitudinal impact on a thin elastic rod, which generates a periodic system of longitudinal waves in it, is considered. At definite values of the parameters of the problem in the linear approximation, these waves induce parametric resonances, which are accompanied by an unlimited increase in the amplitude of the transverse vibrations. To obtain finite values of the amplitudes, a quasilinear system is considered in which the effect of the transverse vibrations on the longitudinal vibrations is taken into account. This system was previously solved using the Bubnov–Galerkin method and beats accompanied by energy transfer between the transverse and longitudinal vibrations were obtained. In this work, an approximate analytical solution of the system has been derived that is based on double-scale expansions. A qualitative analysis of this solution has been carried out. An estimate of the maximum transverse bending has been obtained for various methods of loading. Both shortand long-term pulses have been considered. It has been shown that, in the case of a spontaneously applied long-term pulse that is lower than the Euler critical load, intensive transverse vibrations can occur.     

The dynamics of a rigid body with a sharp edge in contact with an inclined surface in the presence of dry friction

In this paper we consider the dynamics of a rigid body with a sharp edge in contact with a rough plane. The body can move so that its contact point is fixed or slips or loses contact with the support. In this paper, the dynamics of the system is considered within three mechanical models which describe different regimes of motion. The boundaries of the domain of definition of each model are given, the possibility of transitions from one regime to another and their consistency with different coefficients of friction on the horizontal and inclined surfaces are discussed.     

The opening of a nonthrough longitudinal crack in a cylindrical shell taking account of elastoplastic deformation

In the context of an analog of the _k —model the problem of the stress-strain state of a shallow cylindrical shell with a nonthrough longitudinal crack is reduced to a system of singular integral equations that is solved by the small-parameter method. For a shell under an internal pressure we study the dependence of the opening of the crack on the geometric and physico-mechanical parameters of the shell.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 31, 1990, pp. 66–70.     

Studying the longitudinal flow of polymers with periodic deformation

Conclusions 1. At low rates of elongation the state of a polymer melt depends neither on the strain nor on the strain rate. The frequency dependence of the components of the complex longitudinal modulus, measured by superposition of a periodic deformation on a steady longitudinal flow, resemble, at low elongation rates, the frequency characteristics of the components of the complex shear modulus.2. Relaxation processes due to steady longitudinal flow affect the components of the complex longitudinal modulus measured by periodic deformation.3. A melt of a polydisperse polymer under conditions of longitudinal flow begins, at some definite strain rate, to respond to periodic deformation as a rubber-like body and continues to do so as the amount of elongation increases, which is manifested by the trend of the frequency dependence of the modulus component E', this trend becoming similar to that for a solid body.4. As the strain increases during elongation, the upper boundary of the flow range, which is determined by the periodic deformation, shifts toward low frequencies.5. The viscoelasticity characteristics of a material subject to elongation can be accurately enough described by nonlinear equations with the aid of the linear shear-relaxation spectrum.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 507–513, May–June, 1978.     

Antiplane deformation of an orthotropic body with a crack and elliptical cavities

A study was made of the stress distribution in an anisotropic body with a tunnel crack and one or two elliptical cavities. Examined as special cases were problems for a body with cavities and edge cracks with one or two sides, a body with cavities connected by a narrow slit, and a body with two or three longitudinal-transverse cracks. It is shown that anisotropy of the material has a substantial effect on the SIF and stresses around the cavities.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 61–66, 1988.     

Viscoelastic body colliding against a rigid wall with and without dry friction effects

The impact of a body against a rigid wall is studied. The body consists of a rigid block and a viscoelastic rod described by the distributed-order fractional model and in particular its solid-like and fluid-like special cases. Translatory motion of a body is studied in two cases: without and with the influence of dry friction. When present, dry friction is modeled by the Coulomb friction law. The problem is treated analytically by the use of the Laplace transform method and solutions are obtained in a convolution form.     

The interaction of surface topography and friction dynamics in brake systems described by a Cellular Automaton

The friction coefficient μ , which is the quotient of the friction force R and the normal force N is in principal not a stationary material parameter, but also dependent on for instance the relative velocity, the normal load, the temperature, the climate conditions, the location and the event itself. The dynamics in the boundary layer between a brake disc and a brake pad is closely linked with the surface topography dynamics. Growing and destroying processes of hard, thin patches, carrying the friction power, determine the time-dependence of the friction coefficient. This interaction between friction and wear has already been simulated with a set of differential equations [2-4], which give an idea about the equilibrium of flow in the contact zone and which are able to describe the fading effect, for example. Based on this assumption we discretised the boundary layer with a Cellular Automaton [5], which makes it possible to have a more detailed look at the processes in the contact area. This paper will show new conclusions concerning the interdependencies of the friction behaviour and the surface topography. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Antiplane strain of a multiconnected anisotropic body with longitudinal cavities and plane cracks

Applying the theory of generalized complex potentials and the method of least squares, we solve the problem of the stressed state of a multiconnected anisotropic body under an antiplane strain. The problem is reduced to a system of linear algebraic equations in the unknown constants that occur in the required functions. By numerical studies we exhibit the influence of the elastic and geometric characteristics on the stress distribution and the variation of the stress intensity factors in a cylinder with one or two cracks and in an infinite body with circular cavities and cracks. One figure. Six tables. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25. 1995, pp. 56–62.     

Contact interaction of a slotted cylindrical shell and a deformable filler with regard for dry friction

The statement of the mixed problem of the friction interaction of a deformable filler with a slotted cylindrical shell is formulated. Using one-dimensional models of a shell and a filler, we obtain an integral equation for calculating contact stresses. On the basis of a numerical solution, the influence of geometric sizes, the number of slots in the shell, and the physical properties of the interacting bodies on the rigidity and strength of the system is investigated.     

The eigenmode expansion method for oscillations of an elastic body with internal and external friction

A method is proposed for solving dynamical problems for a viscoelastic body (the Kelvin-Voigt model) in a massless viscous medium. Interaction with the external medium produces on the boundary of the body stresses proportional to the rate of displacement. The model of external friction is that used for modelling dynamical processes in elastic media filling an infinite domain [1, 2]. The implementation of numerical methods of solution requires an equivalent restatement of the problem in a finite domain, using external viscous friction to allow for the radiation of energy at infinity.