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Nonlinear dynamics and complex systems

by Marko Vendelin last modified 2011-11-02 13:45

Complexity of wave motion in solids, coherent wave fields, solitonics and surface waves, phase-transformation fronts, acoustodiagnostics of material properties, microstructured materials, impact, nonlinear integrated photoelasticity. Turbulent diffusion, statistical topography and flooding, heart rate variability, econophysics.

 Based on the theory of continua with scale analysis, several mathematical models are formulated for describing the dynamical behaviour of new materials (functionally graded materials, metal-ceramic composites, etc). Complicated dispersive and nonlinear effects and possible phase-transition are taken into account.

For martensitic-austenitic materials the mechanism of phase transition fronts is described with specific thermodynamical consistency conditions and the types of solitary waves (solitons) including the plaited solitons are found.  The emergence of such waves is shown to be dependent on a certain threshold. The existence of hierarchical waves  is shown to be characteristic to cases when the scale factor exist (wave lenght vrs structural scale). The concepts of internal variables are introduced into the models of continua. A special formalism of dual internal variables is proposed.

The emergence of solitons in garnular materials is analysed. A composite wave-propagation algorithm is proposed based on the finite volume method for the calculation of phase-transition boundaries and wave fields in layered composites and functionally graded materials as well for the propagation of Mode I cracks in brittle materials.

It is shown that in case of the classical third-order dispersion and quadratic nonlinearity (the Korteweg-de Vries model), the trajectories of emerging solitons form a certain pattern. It is also shown that the external force field may result in soliton ensembles where the mechanism is based on hidden solitons and resonance. In case of higher order dispersion (the Mindlin model ) the emergence of counter-propagating soliton trains is analysed.

 The theoretical basis of the nonlinear acoustodiagnostics is elaborated and methods for Nondestructive Testing (NDT) are derived based on the informative properties of nonlinear wave fields for determining the prestress of the material properties. A resonance phenomenon is found based on two-wave method that enhanges considerably the accuracy of NDT. Inverse propblems are solved for determining the properties of microstructured materials. The novel ideas proposed are based on using of asymmetry of solitary waves and different phase velocities of harmonic waves.

A mathematical model for describing the behaviour  of the piano hammer is derived, according with the hysteretic properties of the material (felt).  An experimental device is constructed for determining the dynamical parameters of hammers.

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