Dynamics of non-equilibrium phase boundaries in a heat conducting non-linearly elastic medium

The phase transformation of the first kind in a non-linearly elastic heat conducting medium is simulated by the relationships on a strong discontinuity. A generalization of the Stefan formulation is given. An existence condition for stationary flow, analogous to the Gibbs phase equilibrium condition, is obtained for non-equilibrium phase boundaries. A pure dilatational phase transition in a compressible fluid and pure shear transformation of the twinning type in non-linearly elastic crystals are considered as model examples. The problem of the structure is solved for closure of the system of relationships on the shock.

A phase transformation ordinarily turns out to be localized in a narrow domain of space and it can be simulated in terms of the conditions on a strong discontinuity /1/. Formulation of the problem of the static equilibrium of liquid phases as well as of liquid and (non-linearly elastic) solid phases was given by Gibbs, who proposed a phase equilibrium criterion and formulated appropriate conditions on the shock; the extension of the Gibbs conditions to the case of the equilibrium of two solid phases is known in both the linear /2/ and non-linear /3/ theories of elasticity. The dynamic problem of the propagation of the equilibrium phase boundary is considered in the Stefan formulation as a rule, including the assumption about the continuity of the density (the strain tensor component) on the shock; the thermal problem is here separated from the mechanical one. Simulating the interphasal surface on the shock the temperature fields are merged by using the well-known Stefan conditions as well as the phase equilibrium condition that reduces to giving the temperature on the front.

The purpose of this paper is to extend the Stefan-Gibbs formulation to the case of the motion of a coherent isothermal phase boundary in a non-linearly elastic heat conducting medium and to derive the dynamic analogue of the phase equilibrium condition (and the Stefan conditions) with possible dissipation at the transformation front. Two dissipative mechanisms are examined, viscous and kinetic. The case of equilibrium phase boundaries was investigated in /4–6/.     

A theorem of instability by linear approximation for a one-dimensional non-linearly elastic body

On the basis of and in a development of the ideas and results of A.A. Movchan (Sr.), that extend to continuous bodies the definitions and main fundamental theorems of Lyapunov on stability and instability, a criterion for instability of the equilibrium position of a one-dimensional non-linearly elastic body subject to potential external forces is established. For the specified simplest type of continuous elastic system (which possesses, however, a number of fundamental properties of continuous elastic systems including unboundedness of the operator of linear approximation and discreteness of its spectrum) a theorem of instability by linear approximation is stated and proved. The method of proof is a version of Persidskii's sector method.     

The forms of the relation between the stress and strain tensors in a non-linearly elastic material

New relations for the stress and strain tensors, which comprise energy pairs, are obtained for a non-linearly elastic material using a similar method to that employed by Novozhilov, based on a trigonometric representation of the tensors. Shear strain and stress tensors, not used previously, are introduced in a natural way. It is established that the unit tensor, the deviator and the shear tensor form an orthogonal tensor basis. The stress tensor can be expanded in a strain-tensor basis and vice versa. By using this expansion, the non-linear law of elasticity can be written in a compact and physically clear form. It is shown that in the frame of the principal axes the stresses are expressed in terms of the strains and vice versa using linear relations, while the non-linearity is contained in the coefficients, which are functions of mixed invariants of the tensors, introduced by Novozhilov, the generalized moduli of bulk compression and shear and the phase of similitude of the deviators. Relations for different energy pairs of tensors are considered, including for tensors of the true stresses and strains, where the generalized moduli of elasticity have a physical meaning for large strains.     

A two-dimensional model of the dynamics of sharp bending of a non-linearly elastic plate

To describe the dynamics of the bending of a thin non-linearly elastic plate, a version of perturbation theory is proposed which correctly takes into account the non-linearity of the medium, the non-uniformity of the deformations along the plate thickness and the boundary conditions on its surface. An effective (2 + 1)-dimensional model is constructed which generalizes the static non-linearly geometrical Föppl-Karman equations. Two-dimensional solitons of the longitudinal deformation are obtained. The conditions for their existence and stability are investigated.     

Propagation of axially symmetric elastic vibrations along a waveguide in an elastic medium

The scatter equation is derived for axially symmetric (longitudinal and torsional) normal vibrations of an infinitely long elastic cylindrical waveguide embedded in an elastic medium. The low-frequency vibrations are analyzed thoroughly. It is shown that low-frequency longitudinal vibrations follow either of two branches of the scatter curve. Recommendations are made concerning the use of waveguide methods for studying the physicomechanical properties of polymers.     

Neutral stability of compression solitons in the bending of a non-linear elastic rod

The spectral stability of compression solitons in non-linear elastic rods with respect to perturbations of the flexural mode of the oscillations of the rod is investigated. The system of equations of the isotropic theory of elasticity, taking account of the non-linear corrections corresponding to the interaction being studied, is used to describe the interaction of longitudinal and flexural waves in the rod. This system of equations describes long longitudinal-flexural waves of small but finite amplitude. It is shown that trapped flexural modes exist, which propagate together with a compression soliton. It is established that these modes, which are the least stable, do not increase with time.     

Scattering of elastic waves through a heterogeneous medium

In this paper we study the elastic wave diffraction in R ³ through a heterogeneous medium, with periodic structure, which occupies a bounded domain. We show that, as the period tends to zero, the solution tends, in some sense, to the solution corresponding to the diffraction by an obstacle made of the classical “homogenized medium”. An analogous result is also proved for the scattering frequencies and the associated scattering functions.     

Wave attenuation in an elastic medium

This article examines the propagation of viscoelastic (elastic) waves in a medium consisting of two layers of finite thickness. It is found that there is a mechanical effect manifest in the monotonic dependence of the damping factor on the parameters of the system. These dependences have distinct maxima and minima, thus making it possible to optimize the damping properties of systems by varying their geometric parameters.Translated from Dinamicheskie, Sistemy, No. 4, pp. 57–62, 1985.     

Wave propagation in thermo-chiral elastic medium

The propagation of time harmonic waves through an infinite thermo-chiral elastic material has been investigated. The elastic field of thermo-chiral medium has been described by extending the governing equations and constitutive relations of hemitropic micropolar material to include temperature field. Seven basic waves consisting of three coupled dilatational waves and four coupled shear waves traveling with distinct speeds may exist in the medium. All the waves are found to be dispersive, however the coupled dilatational waves are attenuating and temperature dependent, while the coupled shear waves are independent of temperature field. The phase speeds and corresponding attenuation quality factors of all the coupled dilatational waves have been computed numerically for a specific model. The effect of chirality and temperature field have been shown graphically.     

Recovery of residual stress in a vertically heterogeneous elastic medium

We study the problem of identifying residual stress within athin subsurface layer in an elastic medium occupying a region = {(x₁, x₂, x₃) , R³: 0 < x₃ < L, where L } in space,where all parameters depend only on the depth x₃. Under thetheoretical framework of linear elasticity with initial stress,the incremental elasticity tensor of each material point iswritten as a sum of two terms, namely the elasticity tensorand the acoustoelastic tensor, both of which are taken hereas isotropic functions of their arguments. By imposing impulsiveloads and measuring the displacements at the boundary x₃ = 0,we recover the residual stress and its gradient there. If theresidual stress has a diagonal form, we can recover the residualstress inside the subsurface layer.     

A circular rotational dislocation loop in a nonlocally elastic medium

In the framework of the nonlocal theory of elasticity we determine the stresses caused by a circular rotational dislocation loop in an unbounded body. In contrast to the solution of the corresponding problem in the classical theory of elasticity, the stress field is regular at all points of the medium, including the line of dislocation. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

An asymmetric wave field in a transversely isotropic elastic medium

A transversely isotropic homogeneous elastic medium excited by a point force perpendicular to the anisotropic axis is considered. The wave field in this medium is constructed and investigated. The front sets of the SV and SH waves are in contact with one another at a point. The front sets in the vicinity of this point are investigated additionally. If we consider the SH wave (or the SV wave) separately, then a false plane front set arises in this region. In considering the SH and SV waves in combination, this false front set disappears. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 163–174.     

The propagation of elastic waves in a viscous-fluid-saturated porous medium

Elastic shock waves in a viscous-fluid-saturated porous medium are investigated. The porosity is only taken into account with respect to pores communicating with one another, and isolated pores are considered as elements of the elastic part of the porous skeleton. It is shown, using the theory of discontinuity, that in such a medium there are two types of vortex-free waves and one equivoluminal wave. Differential equations and their solution for determining the change in the wave-front intensity are obtained. The effect of the fluid viscosity and porosity on the propagation of spherical waves is demonstrated using an example.     

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.     

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.     

Averaged model of vibration of a damped elastic medium

We consider an initial boundary-value problem used to describe the nonstationary vibration of an elastic medium with large number of small cavities filled with a viscous incompressible fluid. We study the asymptotic behavior of the solution in the case where the diameters of the cavities tend to zero, their number tends to infinity, and the cavities occupy a three-dimensional region. We construct an averaged equation to describe the leading term of the asymptotics. This equation serves as a model of propagation of waves in various media, such as damped soil, rocks, and some biological tissues.     

Successive scatterings of stress waves by a train of parallel elastic cylinders in an elastic medium

Zusammenfassung Die Streuung von Spannungswellen durch eine Reihe paralleler elastischer Kreiszylinder, eingebettet in einem elastischen Kontinuum, wurde untersucht. Alle Zylinder sind unendlich lang und haben einheitlichen Querschnitt und identische Materialeigenschaften, verschieden von denjenigen des Kontinuums. Die Achsen der Zylinder liegen in einer Parallelebene zum Propagationsvector einer ebenen Spannungswelle. Durch Anwendung der Kirchhoffschen Methode des retardierten Potentials [1–4] wurden in dem beschriebenen System Wellenfrontspannungen und Wellenfrontverschiebungen gefunden, welche von der Streuung der ersten und zweiten Generation verursacht wurden.Die Ergebnisse zeigen die Intensität der Spannungen; Verschiebungen klingen sehr schnell ab und verschwinden beinahe mit dem fünften Zylinder. Die Resultate wurden mit dem Abklingen von Wellen in einem geschichteten System verglichen, das aus zwei Materialien hergestellt ist. Die Analyse der Wellenzerstreuung kann auch auf Systeme ausgedehnt werden, welche mehrere Materialphasen von allgemeiner Form enthalten.

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This work was performed under the sponsorship of Southwest Research Institute Internal Research Program under SwRI Project 02-9009.     

Following screened perturbations in an elastic medium with a thin layer

The result of the calculation of the field of a longitudinal wave which has passed through an elastic layer is studied and the character of the propagation of different phases of this wave is determined.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im V. A. Steklova AN SSSR, Vol. 165, pp. 159–165, 1987.