In English. Summaries in Estonian

Proceedings of the Estonian Academy of Sciences.

Physics * Mathematics


Volume 52 No. 1 March 2003


Special issue on nonlinear waves in microstructured solids

Preface; 3–4

Nonlinear wave mechanics of complex material systems; 5–11

Gérard A. Maugin

Abstract. Inspired by the original ideas of L. de Broglie on wave mechanics, which were at the basis of the wave-like interpretation of quantum mechanics, we tentatively develop possible fruitful analogies between the conservation equations of nonlinear continuum mechanics in the canonical framework as expressed on the material manifold and the dispersive kinematic wave theory of Whitham in order to construct a nonlinear wave mechanics of structured solid continua. The final purpose of this approach obviously is not quantization but the relationship between dynamical localized concentrations of continuous fields, such as solitary waves of the envelope type, and the notion of quasi-particles.

Key words: waves, action, hyperelasticity, wave mechanics, kinematics, complex materials, nonlinear waves, solitons.

Waves in microstructured solids with nonlinearities in microscale; 12–20

Jüri Engelbrecht and Franco Pastrone

Abstract. Consistent modelling of wave motion in microstructured solids is discussed. Based on the Mindlin model, the simplest model equation of motion is derived. The fundamental properties of such a model are (i) hierarchical structure distinguishing the macro- and microbalance, (ii) changes in the wave speed, and (iii) definite influence of dispersion. The nonlinearity in the microlevel, although presented by a simple energy density function, leads to a complicated nonlinear term in the equation of motion as well as in the corresponding evolution equation. Such consistent modelling opens up direct ways for determining material constants characterizing the microstructure.

Key words: microstructured solids, wave hierarchy, nonlinearity, evolution equation.

Waves in solids with vectorial microstructure; 21–29

Franco Pastrone

Abstract. A general model of solids with vectorial microstructures is introduced. Field equations are obtained via a variational principle, with natural boundary conditions. It is proved that scalar unidimensional bodies and Cosserat solids are included in this model. Waves and stability problems are briefly discussed.

Key words: microstructured solids, nonlinear waves, stability.

Numerical simulation of waves and fronts in structured materials: a thermodynamic approach; 30–42

Arkadi Berezovski, Jüri Engelbrecht, and Gérard A. Maugin

Abstract. A thermodynamically consistent form for the finite-volume numerical method for thermoelastic wave and front propagation is proposed. Such reformulation provides applicability of the Godunov-type numerical schemes based on averages of field variables to the description of nonequilibrium situations. The nonequilibrium description uses contact quantities instead of numerical fluxes. These quantities satisfy the thermodynamic consistency conditions which generalize the classical equilibrium conditions.

Key words: thermoelastic waves, phase transition fronts, finite-volume methods, thermodynamics of discrete systems.

Nonlinear waves guided at a liquid–solid interface; 43–62

Andreas P. Mayer and Alexander S. Kovalev

Abstract. Nonlinear acoustic waves propagating at the interface between a solid and a fluid with a compressibility much higher than that of the solid are considered. It is shown that their waveform evolution in the fluid is governed by the two-dimensional Zabolotskaya–Khokhlov (ZK) equation, with a linear boundary condition determined by the acoustic mismatch between fluid and solid. Two evolution equations used for the interpretation of recent experiments are derived as two different limiting cases of the ZK equation, with the corresponding boundary condition at the interface. The possibility of the formation of solitary waves is discussed for the case of Scholte waves becoming dispersive due to inhomogeneity of the solid.

Key words: Scholte waves, interface waves, solitons.

Lattice modelling of nonlinear waves in a bi-layer with delamination; 63–75

Karima R. Khusnutdinova and Vadim V. Silberschmidt

Abstract. A lattice model consisting of two one-dimensional periodic chains with linear links between elements and nonlinear interaction between the chains is suggested to study nonlinear dynamics of a bi-layer. The properties of the model are discussed, and the influence of a delamination zone on the propagation of solitary waves is studied numerically.

Key words: bi-layer, delamination, Frenkel–Kontorova model, coupled Klein–Gordon equations, nonlinear waves.

Exact discrete breather solutions and conservation laws of lattice equations; 76–84

Mikhail M. Bogdan and Gérard A. Maugin

Abstract. Exactly-solvable differential-difference equations describing the nonlinear dynamics of one-dimensional lattices and electrical transmission lines are investigated. The equations considered are discrete modified Korteweg–de Vries equation, nonlinear self-dual network equations, and the Hirota lattice equation. Explicit expressions for principal integrals of motion of the equations are presented and discussed. All the above-mentioned differential-difference equations have exact discrete breather solutions. Conservation laws and values of the basic integrals, energy, energy flow, and total momentum for main types of solitons, as well as the adiabatic invariant for the discrete breather, are found. Quasi-classical quantization of the discrete breather oscillation is performed and the breather energy spectrum found.

Key words: lattice equations, exact soliton solutions, discrete breathers, integrals of motion.

Selection of localized nonlinear seismic waves; 85–93

Alexey V. Porubov, Vitaly V. Gurski, and Gérard A. Maugin

Abstract. The asymptotic solution is obtained for the nonlinear evolution equation governing seismic wave propagation in the Earth’s crust. The conditions are found under which the amplitude and velocity of an initial solitary wave tend to the finite values prescribed by the equation coefficients. Numerical simulations demonstrate validity of these predictions in case of an arbitrary localized pulse evolution, and in the presence of the solitary wave interactions.

Key words: solitary wave, nonlinearity, dispersion, dissipation.

Nonlinear excitations of incommensurate surface structures; 94–102

Alexander S. Kovalev, Igor V. Gerasimchuk, and Gérard A. Maugin

Abstract. Nonlinear dynamics of the incommensurate surface layer with a spatially periodical structure is investigated analytically. In the framework of the Frenkel–Kontorova model the nonlinear excitations of the periodic soliton lattice, such as moving additional kinks and gap solitons, are discussed.

Key words: incommensurate structure, kink, gap soliton.

Wave propagation in dissipative microstructured materials; 103–114

Tarvo Sillat and Jüri Engelbrecht

Abstract. One-dimensional deformation waves in microstructured materials are described by an hierarchical evolution equation that clearly distinguishes macro- and microstructural behaviour. The pseudospectral method is used for numerical simulation supported by the analytical solution for the linear case. It is shown how dissipative effects on various scales affect the harmonic wave. The shock wave formation on macroscale is strongly influenced by the microstructure. The results of this study can be used for material processing.

Key words: wave hierarchy, dissipation, microstructure.

On the existence of bulk solitary waves in plexiglas; 115–124

Alexander M. Samsonov, Galina V. Dreiden, and Irina V. Semenova

Abstract. The study of the nonlinear waves in plexiglas was aimed to prove the nonexistence of positive strain (compression) solitary waves in this polymer. The estimation and calculation were based on the only, relatively old data available on plexiglas’ elasticity. We succeeded to generate and observe for the first time a compression solitary wave in plexiglas and prove that this polymer is a transparent material suitable for observation of the compression soliton in an elastic solid wave guide, and may be of interest for applications in fracture or nondestructive testing.

Key words: soliton, nonlinear elasticity, solids, plexiglas, holographic interferometry.

Solitons in hierarchical Korteweg–de Vries type systems; 125–134

Lauri Ilison and Andrus Salupere

Abstract. Wave propagation in dilatant granular materials is studied by using a hierarchical Korteweg–de Vries type evolution equation. The model equation is solved numerically under harmonic initial conditions. The behaviour of the solution is described and analysed over a wide range of material parameters (two dispersion parameters and one microstructure parameter). Two main solution types with different subtypes are introduced. The character of the both solution types is found to be solitonic.

Key words: dilatant granular materials, solitons, wave hierarchies, Korteweg–de Vries type evolution equations.

On the formation of solitons in media with higher-order dispersive effects; 135–144

Olari Ilison and Andrus Salupere

Abstract. Wave propagation in microstructured materials is strongly influenced by dispersive effects. In the present paper two Korteweg–de Vries type model equations, with the third- and fifth-order dispersion, are studied. Both model equations are solved numerically, under harmonic initial and periodic boundary conditions, by making use of the pseudospectral method. The character of the solution is found to be solitonic in both cases. The number of visible and hidden solitons in the emerging train is detected. Phenomena of recurrence and super-recurrence are examined.

Key words: microstructure, dispersion, nonlinearity, Korteweg–de Vries type evolution equations, solitons, pseudospectral method.

Periodically forced solitonic structures in dispersive media; 145–156

Andrus Salupere and Martti Kukk

Abstract. The influence of the amplitude-dependent periodic driven field on the formation and propagation of solitary waves in nonlinear dispersive media is studied. The model equation – the forced Korteweg–de Vries equation – is integrated numerically under harmonic initial and periodic boundary conditions by using the pseudospectral method. Main attention is paid to solitonic solutions. The driven field is classified as weak, moderate, strong or dominating, according to the character of the solution. The solution is found to be solitonic in the case of weak, moderate, and strong fields.

Key words: periodic driven field, forced Korteweg–de Vries equation, solitary waves, solitons, pseudospectral method.

Nonlinear interaction of waves with material inhomogeneity; 157–168

Andres Braunbrück and Arvi Ravasoo

Abstract. A relatively simple method for nondestructive evaluation of weak and smooth variation of the physical properties of the material from their constant values is proposed. The method is based on the analysis of nonlinear effects of simultaneous propagation, reflection, and interaction of two ultrasonic waves in the material. The results of the analysis enable one to solve several problems of material parameter evaluation provided some preliminary information about the material is available.

Key words: longitudinal waves, nonlinear interaction, inhomogeneity, material characterization.