Metaharmonic potentials in mechanics of micropolar media

Рассматриваются дифференциальные уравнения для потенциалов перемещений и микровращений, замещающие связанные векторные дифференциальные уравнения линейной теории микрополярной упругости. Исследуются только гармонические зависимости от времени. Опираясь на представление векторов перемещений и микровращений с помощью четырех винтовых векторов, обеспечивающее выполнимость связанных векторных дифференциальных уравнений линейной теории микрополярной упругости, получены новые представления в терминах двух метагармонических векторных полей. Проблема нахождения вихревых составляющих перемещений и микровращений приводится к решению двух несвязанных между собой векторных метагармонических уравнений. Часто указанные уравнения могут быть решены разделением пространственных переменных. Поэтому полученные представления могут находить применение в прикладных задачах механики деформируемого твердого тела, связанных с распространением гармонических волн перемещений и микровращений, характеризующихся заданным азимутальным числом, вдоль длинных цилиндрических волноводов. The coupled vector differential equations of the linear theory of micropolar elasticity formulated in terms of displacements and micro-rotations are studied. A harmonic dependence of the physical fields on time is assumed. By employing the displacements and micro-rotations representation formula in the terms of four screw vectors a new representation based on two metaharmonic vectors are obtained. Thus the problem of determination of the vortex parts of the displacement and micro-rotation fields is reduced to solution of two uncoupled vector metaharmonic equations. The latter can be oftenly solved by the separation of variables technique. For this reason obtained results can be applied to various problems of the micropolar elasticity related to harmonic wave propagation in waveguides. In particular this is true for waves of a given azimuthal number in a long cylindrical waveguide.

Mathematical modeling on the propagation of guided elastic waves

The main cause of train derailment is related to transverse defects that arise in the railhead. These consist typically of opened or internal flaws that develop generally in a plane that is orthogonal to the rail direction. Most of the actual inspection techniques of rails relay on eddy currents, electromagnetic induction, and ultrasounds. Ultrasounds based testing is performed according to the excitation-echo procedure [1]. It is conducted conventionally by using a contact excitation probe that rolls on the railhead or by a contact-less system using a laser as excitation and air-coupled acoustic sensors for wave reception. The ratio of false predictions either positive or negative is yet too high due to the low accuracy of the actual devices. The inspection rate is also late; new numerical method has been developed in this context: The semi-analytical finite element method SAFE. This method has been applied in the case of anisotropic media [2], composite plates [3] and media in contact with fluids [4]. This method has been used successfully for several structures and especially in the case of beams of any cross-section such as rails that are the subject of this work and we were interested in wave propagation in waveguides of any arbitrary cross-section in the case of beams or rails.

Marching schemes for Cauchy wave propagation problems in laterally varying waveguides

This paper intends to develop practical marching schemes for Cauchy problems of the Helmholtz equation in laterally varying waveguides. We arrive at a stable representation of the marching solutions in waveguides. Based on the representation, a second-order marching scheme is then constructed to eliminate the ill-conditioning and compute the wave propagation in waveguides with laterally variable mediums. In the end, extensive experiments are implemented to verify the efficiency and accuracy of the marching scheme in various waveguides, and we also point out the application scope of the scheme.

Lamb wave propagation in elastic waveguides with variable thickness

The problem of Lamb wave propagation in waveguides with varying height is treated by a multimodal approach. The technique is based on a rearrangement of the equations of elasticity that provides a new system of coupled mode equations preserving energy conservation. These coupled mode equations avoid the usual problem at the cut-offs with zero wavenumber. Thereafter, we define an impedance matrix that is governed by a Riccati equation yielding a stable numerical computation of the solution. Incidentally, the versatility of the multimodal method is exemplified by treating analytically the case of slowly varying guide and by showing how to get easily the Green tensor in any geometry. The method is applied for a waveguide whose height is described by a Gaussian function and the energy conservation in verified numerically. We determine the Green tensor in this geometry.