A model for describing near-resonance oscillations in an elastic layer

One-dimensional transverse oscillations in a layer of a non-linear elastic medium are considered, when one of the boundaries is subjected to external actions, causing periodic changes in both tangential components of the velocity. In a mode close to resonance, the non-linear properties of the medium may lead to a slow change in the form of the oscillations as the number of the reflections from the layer boundaries increases. Differential equations describing this process were previously derived. The equations obtained are hyperbolic and the change in the solution may both keep the functions continuous and lead to the formation of jumps. In this paper a model of the evolution of the wave patterns is constructed as integral equations having the form of conservation laws, which determine the change in the functions describing the oscillations of the layer as “slow” time increases. The system of hyperbolic differential equations previously obtained follows from these conservation laws for continuous motions, in which one of the variables is slow time, for which one period of the actual time serves as an infinitesimal quantity, while the second variable is the real time. For the discontinuous solutions of the same integral equations, conditions on the discontinuity are obtained. An analogy is established between the solutions of the equations obtained and non-linear waves propagating in an unbounded uniform elastic medium with a certain chosen elastic potential. This analogy enable discontinuities which may be physically realised to be distinguished. The problem of steady oscillations of an elastic layer is discussed.     

Non-linear radial oscillations of a transversely isotropic hyperelastic incompressible tube

The constitutive equation for a transversely isotropic incompressible hyperelastic material is written in a covariant form for arbitrary orientation of the anisotropic director. Three non-linear differential equations are derived for radial oscillations in radial, tangential and longitudinal transversely isotropic thin-walled cylindrical tubes of generalised Mooney-Rivlin material. A Lie point symmetry analysis is performed. The conditions on the strain-energy function and on the net applied surface pressure for Lie point symmetries to exist are determined. For radial and tangential transversely isotropic tubes the differential equations are reduced to Abel equations of the second kind. Radial oscillations in a longitudinal transversely isotropic tube and in an isotropic tube are described by the Ermakov-Pinney equation.     

Near-resonace transverse oscillations in an elastic layer. Steady-state solutions

The solutions of the equations of the non-linear evolution of transverse oscillations in a layer of an incompressible elastic medium under conditions close to resonance conditions are investigated qualitatively and using analytical methods. The oscillations are created by a small periodic motion of one of the boundaries in its plane, with a period that is close to the period of the natural oscillations of the layer. It is assumed that the medium can possess slight anisotropy and that the amplitude of the oscillations which arise is small. Previously obtained differential equations are used, which describe the slow evolution of the wave pattern of non-linear transverse waves. Two possible formulations of problems for these equations are considered. In the first formulation, it is determined what the external action must be in order that the non-linear evolution of oscillations or periodic oscillations occurs according to a (previously specified) desired law. In the second formulation it is assumed that the periodic motion of one of the boundaries is given. It is shown that a steady-state solution, that does not vary from period to period, can be represented by a continuous solution and also by a solution which contains discontinuities in the strain and velocity components. The mechanism of the overturn of a non-linear wave during its evolution and the formation of a discontinuity are qualitatively described.     

Asymptotic solution of the axisymmetric contact problem for an elastic layer of incompressible material

The solution of the axisymmetric contact problem for an elastic layer made of incompressible material and clamped along the base is constructed by regular and singular asymptotic methods.     

Axisymmetric contact problems for a prestressed incompressible elastic layer

Two axisymmetric problems of the indentation without friction of an elastic punch into the upper face of a layer when there is a uniform field of initial stresses in the layer are considered. The model of an isotropic incompressible non-linearly elastic material, specified by a Mooney potential, is used. Two cases are investigated: when the lower face of the layer is rigidly clamped after it is prestressed, and when the lower face of the layer is supported on a rigid base without friction after it is prestressed. It is assumed that the additional stresses due to the action of the punch on the layer are small compared with the initial stresses; this enables the problem of determining the additional stresses to be linearized. The problem is reduced to solving integral equations of the first kind with symmetrical irregular kernels relative to the pressure in the contact area. Approximate solutions of the integral equations are constructed by the method of orthogonal polynomials for large values of the parameter characterizing the relative layer thickness. The case of a punch with a plane base is considered as an example.     

Non-linear analysis of the consolidation of an elastic saturated soil with incompressible fluid and variable permeability by FEM

Research interest in the mechanical behaviour of soils is growing as a result of an increasing number of geomechanical problems involving consolidation effects. The main aim of this paper is to validate and to solve a model for consolidation of an elastic saturated soil with incompressible fluid and variable permeability. Firstly, we prove the existence and uniqueness of the solution of the variational problem corresponding to an initial and boundary value problem (IBVP): a special case of the Biot’s ‘consolidation of clay’ model (where the applied forces depend on time). Secondly, we prove the convergence of the method using a technique based on the proof of solution’s existence. Finally, we then solved this constitutive model by the finite element method (FEM) employing repeated fixed point techniques in order to obtain the results for displacement and pore water pressure. The pore fluid is considered incompressible. The results of the numerical experiments are compared with analytical solutions and, in cases where such solutions do not exist, with experimental data. Therefore, the model can be used for quantitative predictions of consolidation behaviour of soils with permeability dependent on the settlement.     

Torsional oscillations of a circular stamp on an elastic layer with a cylindrical cavity

The problem of torsional oscillations of a stamp that is linked with an elastic stratum which contains a cylindrical cavity is considered. The problem is formulated in the form of conjugate integral equations that are related to the integral Weber transforms. The conjugate equations are reduced to an equivalent Fredholm equation of the second kind.Translated from Dinamicheskie Sistemy, No. 9, pp. 54–59, 1990.     

Non-linear self-excited acoustic oscillations

Summary The response of a gas column contained in a tube to a destabilizing boundary condition is investigated. For the special case where the particle velocity at an end wall of the tube is proportional to the pressure disturbance, a non-linear analysis is given for closed and open tubes (with and without mean flow). The inviscid solutions for closed tubes are sawtooth waves, for open tubes without mean flow they turn out to be essentially square waves and for open tubes with mean flow the tops and bottoms of the square waves become ramp-like.The analysis given may be regarded as a guideline for the investigation of certain types of combustion instabilities.

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Zusammenfassung Betrachtet werden nichtlineare Schwingungen in einem mit Gas gefüllten Rohr, die durch eine destabilisierende Randbedingung angetrieben werden. Für den Spezialfall der Proportionalität der Teilchengeschwindigkeit an einer Endwand des Rohres zur Druckstörung wird eine nichtlineare Analysis für geschlossene und offene Rohre (mit und ohne Durchströmung) angegeben. Die reibungsfreien Lösungen für geschlossene Rohre sind sägezahnförmige Wellen, für undurchströmte offene Rohre findet man im wesentlichen rechteckförmige Wellen und für durchströmte offene Rohre gehen die Rechteckwellen in trapezförmige Wellen über. Die gezeigte Analysis soll als Leitlinie zur Untersuchung bestimmter Typen von Verbrennungsinstabilitäten dienen.

     

Harmonic oscillations of a rigid punch on a porous elastic layer

The plane dynamic contact problem of the harmonic oscillations of a rigid punch on the free surface of an elastic layer of porous isotropic material with linear properties is considered. The Fourier transformation of the problem is reduced to a Fredholm integral equation of the first kind in the contact pressure. The properties of the kernel of the fundamental integral equation are investigated and a numerical method of solving it is constructed. Numerical results are compared with existing results in classical limiting cases.     

Non-linear oscillations of DNA base pairs

Non-linear oscillations of the nitrogenous bases of DNA, which form pairs: adenine – thymine or guanine – cytosine, are considered. An approach is used in which this biophysical problem can be reduced to a mechanical problem of the non-linear oscillations of two coupled dissimilar pendulums, which oscillate in a plane orthogonal to the principal axis of the molecule. The dynamics of such a model system are analysed. Singular points are obtained in phase space. The eigenvalues of the Jacobi matrix are calculated. The behaviour of the model system close to the singular points is described. The results are illustrated in the form of graphs of the solutions of the corresponding model equations and graphs of the trajectories of motion in configuration space.     

Non-linear reflection of a plane wave in an elastic medium

Exact formulae are derived for the reflected and refracted waves which occur for the inclined incidence of a plane horizontally polarized transverse wave of arbitrary profile on a horizontal interface between two elastic half-spaces experiencing non-linear friction when they move with respect to one another. A smooth function of general form is adopted as the friction function, which depends on the difference between the horizontal velocities of the elements of the boundaries of the half-spaces considered. It is shown that if the friction function depends non-monotonically on the relative velocity of displacement of the sides of a slit, then even when the profile of the incident wave is smooth, the reflected and refracted waves may contain strong discontinuities.     

Free and nonsteady-state oscillations of elastic fluid-filled systems situated on an elastic base

We develop a method of computing the nonsteady-state and free oscillations of a framed elastic structure situated on an elastic base and containing an ideal compressible fluid. The solution uses the method of integral transforms in conjunction with the method of orthogonal polynomials. In the transform space the problem reduces to systems of linear algebraic equations. The Fourier transform is applied to return to the original space. Examples of the computation are given.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 69–72.     

A two-dimensional model for extensional motion of a pre-stressed incompressible elastic layer near cut-off frequencies

A two-dimensional model for extensional motion of a pre-stressedincompressible elastic layer near its cut-off frequencies isderived. Leading-order solutions for displacement and pressureare obtained in terms of the long wave amplitude by direct asymptoticintegration. A governing equation, together with correctionsfor displacement and pressure, is derived from the second-orderproblem. A novel feature of this (two-dimensional) hyperbolicgoverning equation is that, for certain pre-stressed states,time and one of the two (in-plane) spatial variables can changeroles. Although whenever this phenomenon occurs the equationstill remains hyperbolic, it is clearly not wave-like. The second-ordersolution is completed by deriving a refined governing equationfrom the third-order problem. Asymptotic consistency, in thesense that the dispersion relation associated with the two-dimensionalmodel concurs with the appropriate order expansion of the three-dimensionalrelation at each order, is verified. The model has particularapplication to stationary thickness vibration of, or transientresponse to high frequency shock loading in, thin walled bodies.     

The oscillations of a rod in an inhomogeneous elastic medium

The local length-dependence of the natural frequencies and forms of plane transverse oscillations of a thin inhomogeneous rod in an elastic medium with a variable stiffness and arbitrary elastic-fastening boundary conditions is investigated. It is established that the presence of an external elastic medium, described by the Winkler model, can lead to an anomalous effect – an increase in the natural frequencies of lower oscillation modes as the length of the rod increases continuously. The extremely fine properties of this change as a function of the length, the mode number and the method of fastening are revealed. The oscillations in the case of standard methods of fastening are investigated separately. Simple examples, which illustrate the anomalous dependence of the natural oscillation frequencies of the rod in an extremely inhomogeneous elastic medium with different boundary conditions are calculated.     

Indentation of an incompressible inhomogeneous layer by a rigid circular indenter

This work deals with an incompressible inhomogeneous layer bondedto a rigid substrate and indented without friction by a rigidcircular indenter. The corresponding mixed boundary-value problemof elasticity is reduced to equivalent dual integral equations.It is shown that the pliability function in these equationsmay be found from a system of nonlinear differential equationsand that its behaviour is peculiar when the elastic medium isincompressible. A novel technique taking into account this peculiarityis developed in order to reduce the dual integral equationsto Fredholm integral equations of the second kind with symmetricstrictly coercive operators. For a homogeneous layer and a flatindenter, the structure of the Fredholm integral equations permitsan approximate analytical solution which is very accurate forany layer thickness. For an indenter of three-dimensional profile,leading asymptotic terms of the solution are derived in thecase of a thin inhomogeneous layer.     

Non-linear elastic behavior of filler aggregates

A filled rubber network consists of polymer chains which are suspended between filler aggregates. In this contribution, the nonlinear elastic behavior of the aggregated filler particles inside the rubber matrix is investigated. Previously, by using scaling theory, the influence of initial length and fractal dimension of aggregates on the elastic response of aggregated structures was studied. Here we additionally take into account a deformation induced evolution of the aggregate structure. To this end, the directional topology of the aggregate structure is represented by its backbone chain. Thus, the analytical approach proposed describes not only the geometrically but also the physically non-linear behavior of aggregates. Our solution can further be generalized for colloidal structures as for example granular materials or suspended solid structures. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)