Wave Dynamics of Generalized Continua

Багдоев, А. Г.; Erofeyev, Vladimir I.; Shekoyan, Ashot V.

doi:10.1007/978-3-642-37267-4

Cited by 25 publications

(5 citation statements)

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“…The work continues the studies of wave processes begun in [22][23][24][25], generalizing them with allowance for the micropolarity of the medium, and also continues the studies begun in [26], generalizing them with allowance for the electrical conductivity of a nonlinear micropolar continuum.…”

Section: Introductionmentioning

confidence: 84%

“…The physical nonlinearity which is important for solid mechanics given for example in [21], for magnetoelasticity was taken into account in [22][23][24][25], in which the main attention was paid to the formation of spatially localized waves. The problems of estimating the influence of a magnetic field on the localization of a strain wave were previously considered by the authors in one-dimensional (rod) [22], two-dimensional (plate) [23], and three-dimensional [24,25] formulations.…”

Section: Introductionmentioning

confidence: 99%

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Spatially localized nonlinear magnetoelastic waves in an electrically conductive micropolar medium

Ерофеев

Malkhanov

2023

Z Angew Math Mech

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A nonlinear model of an electrically conductive micropolar medium interacting with an external magnetic field is proposed. The deformable state of such a medium is described by two asymmetric tensors: tensor of deformations and bending‐torsion tensor. Both tensors consider both linear and nonlinear terms in rotation and displacement gradients (geometric nonlinearity). The components of the bending‐torsion tensor which have the same indices, describe torsional deformations, while the rest describe bending deformations. The stress state of the medium is described by two asymmetric tensors: stresses tensor and moment stresses tensor. It is assumed, as is customary in magnetoelasticity, that the action of an electromagnetic field on a strain field occurs through Lorentz forces. From the system of Maxwell's equations follow the equations for electric and magnetic induction, which together with electromagnetic equations of state, should be added to the group of equations which describe the dynamics of a micropolar medium. In the frames of the proposed model, a one‐dimensional nonlinear magnetoelastic shear‐rotation wave is considered. In the equations of dynamics, a nonlinear term is considered. This term makes the most significant contribution to wave processes. It is shown that two factors will influence the wave propagation: dispersion and nonlinearity. Nonlinearity leads to the emergence of new harmonics in the wave, which contributes to the appearance of sharp drops in the moving wave profile. Dispersion, on the contrary, smoothed out differences due to differences in phase velocities of harmonic component waves. The combined action of these factors can lead to the formation of stationary waves that propagate at a constant speed without changing shape. Only those cases where a constant component is absent in the deformation wave are physically realizable. Stationary waves can be either periodic or aperiodic. The latter are spatially localized waves—solitons. It is shown that the behavior of “subsonic” and “supersonic” solitons will be qualitatively different.

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Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

Spatially localized nonlinear magnetoelastic waves in an electrically conductive micropolar medium

Ерофеев

Malkhanov

2023

Z Angew Math Mech

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“…Çäåñü E − íàïðÿaeåííîñòü ýëåêòðè÷åñêîãî ïîëÿ, B − âåêòîð ìàãíèòíîé èíäóêöèè, j − âåêòîð ïëîòíîñòè òîêîâ, ρ e − îáúåìíàÿ ïëîòíîñòü ýëåêòðè÷åñêèõ çàðÿäîâ. Âåêòîðû F nonlinear , G nonlinear âêëþ÷àþò â ñåáÿ ñëàãàåìûå, îáóñëîâëåííûå ó÷åòîì óïðóãîé ãåî-ìåòðè÷åñêîé íåëèíåéíîñòè (7).…”

Section: óðàâíåíèÿ äèíàìèêè ìàãíèòîóïðóãîé ìèêðîïîëÿðíîé ñðåäûmentioning

confidence: 99%

Spatially-Localized Nonlinear Magnetoelastic Waves in a Micropolar Electrical Conducting Medium

Erofeev

Shekoyan

Belubekyan

2019

PSP

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A nonlinear model of an electrically conducting micropolar medium interacting with an external magnetic field is proposed. The deformable state of such a medium is described by two asymmetric tensors: tensor of deformations and bending-torsion tensor. In both tensors, linear and nonlinear terms are taken into account in rotation gradients and displacement gradients (geometric nonlinearity). The components of the bending-torsion tensor, which have identical indices, describe torsional deformations, and the rest - bending deformations. The stress state of the medium is described by two asymmetric tensors: stress tensor and moment stress tensor. It is assumed, as it is usual in magnetoelasticity, that the action of the electromagnetic field on the deformation field occurs through the Lorentz forces. From the system of Maxwell equations follow the equations for electrical and magnetic inductions, which, together with the electromagnetic equations of state, must be added to the equations of the dynamics of a micropolar medium. Within the framework of the proposed model, a one-dimensional nonlinear shear-rotation magnetoelastic wave is considered. The nonlinear term is selected and taken into account in the equations of dynamics, making the most significant contribution to wave processes. It is shown that two factors will influence the wave propagation: dispersion and nonlinearity. Nonlinearity leads to the emergence of new harmonics in the wave, which contributes to the appearance of a sharp drop in the moving profile. The dispersion, on the contrary, smoothes the differences due to the difference in the phase velocities of the harmonic components of the waves. The combined effect of these factors can lead to the formation of stationary waves that propagate at a constant speed without changing the shape. Only those cases are physically feasible when there is no constant component in the deformation wave. Stationary waves can be both periodic and aperiodic. The latter are spatially localized waves - solitons. It is shown that the behavior of "subsonic" and “supersonic” solitons will be qualitatively different.

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“…Влияние свойств поверхностного слоя образца на скорость и затухание рэлеевских волн позволяет использовать последнее для определения остаточных напряжений поверхностного слоя металла, термических и механических свойств поверхностного слоя образца [3,4]. Наряду с моделью классического континуума в механике деформируемого твердого тела широко применяются модели обобщенных континуумов [5][6][7][8][9][10][11]. К числу обобщенных континуумов принадлежит, в частности, градиентно-упругая среда.…”

Section: Acknowledgementsunclassified

Excitation of the Waves by a Focused Source,moving Along the Border of Gradient-Elastic Half-Space

Antonov¹,

Ерофеев²

2018

BSTD

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Аннотация. В рамках математической модели градиентно-упругого континуума, т.е. среды, напряженно-деформированное состояние которой описывается тензором деформаций, вторыми градиентами вектора перемещений, несимметричным тензором напряжений и тензором моментных напряжений, рассматривается задача о генерации возмущений движущимся источником. Предполагается, что источник движется с постоянной скоростью вдоль границы полупространства. Задача рассматривается в двумерной постановке, когда все процессы однородны вдоль горизонтальной поперечной координатной оси. Вектор перемещений содержит две компоненты: продольную и вертикальную поперечную. В результате аналитических исследований показано, что движущийся источник будет генерировать волны, распространяющиеся вдоль границы полупространства и экспоненциально убывающие в его глубину. Такая волна, в отличие от классической поверхностной волны Рэлея, обладает дисперсией, поскольку ее фазовая скорость не является постоянной величиной, а зависит от частоты. Амплитуды перемещений изменяются в зависимости от величины нагрузки движущегося источника и его скорости. Ключевые слова: градиентно-упругое полупространство, движущийся источник, поверхностная волна.

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Wave Dynamics of Generalized Continua

Cited by 25 publications

References 0 publications

Spatially localized nonlinear magnetoelastic waves in an electrically conductive micropolar medium

Spatially localized nonlinear magnetoelastic waves in an electrically conductive micropolar medium

Spatially-Localized Nonlinear Magnetoelastic Waves in a Micropolar Electrical Conducting Medium

Excitation of the Waves by a Focused Source,moving Along the Border of Gradient-Elastic Half-Space

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