Chapter 2
Excitation of the Waves with a Focused
Source, Moving Along the Border
of Gradient-Elastic Half-Space
Artem M. Antonov, Vladimir I. Erofeev, Aleksey O. Malkhanov,
and Nadezhda A. Novoseltseva
Abstract In the frames of the mathematical model of the gradient-elastic continuum,
i.e., the medium with the stress–strain state described by the strain tensor, the second
gradients of the displacement vector, the asymmetric stress tensor, and the moment
stress tensor, we consider the problem of generating disturbances by a moving source.
It is assumed that the source moves at a constant speed along the border of the
half-space. The problem is considered in a two-dimensional formulation, when all
processes are homogeneous along the horizontal transverse coordinate axis. The
displacement vector contains two components: longitudinal and vertical transverse.
As a result of analytical studies, it has been shown that a moving source will generate
waves propagating along the border of a half-space and decreasing exponentially
into its depth. Such a wave, in contrast to the classical surface Rayleigh wave, has
a dispersion, since its phase velocity is not constant, but depends on the frequency.
The displacement amplitudes vary depending on the magnitude of the load of the
moving source and its speed.
Keywords Gradient-elastic half-space · Moving source · Surface wave
2.1 Introduction
In 1885, the English scientist Lord Rayleigh theoretically showed that along the flat
border of a solid elastic half-space with a vacuum or a sufficiently diluted medium
(for example, with air) waves can propagate, whose amplitude rapidly decreases with
depth (Lord Rayleigh 1885). These waves, later called Rayleigh surface waves, are
the main type of waves observed during earthquakes. Therefore, they are studied in
detail in seismology (Aki and Richards 2009).
A. M. Antonov · V. I. Erofeev (B) · A. O. Malkhanov
Mechanical Engineering Research Institute of RAS, Nizhny Novgorod, Russia
e-mail: erof.vi@yandex.ru
V. I. Erofeev · N. A. Novoseltseva
Research Institute for Mechanics, National Research Lobachevsky State University of Nizhny
Novgorod, 23, Gagarin Av., Nizhny Novgorod 603950, Russia
© Springer Nature Switzerland AG 2021
F. dell’Isola and L. Igumnov (eds.), Dynamics, Strength of Materials and Durability
in Multiscale Mechanics, Advanced Structured Materials 137,
https://doi.org/10.1007/978-3-030-53755-5_2
17
Excitation of the Waves with a Focused
Source, Moving Along the Border
of Gradient-Elastic Half-Space
Artem M. Antonov, Vladimir I. Erofeev, Aleksey O. Malkhanov,
and Nadezhda A. Novoseltseva
Abstract In the frames of the mathematical model of the gradient-elastic continuum,
i.e., the medium with the stress–strain state described by the strain tensor, the second
gradients of the displacement vector, the asymmetric stress tensor, and the moment
stress tensor, we consider the problem of generating disturbances by a moving source.
It is assumed that the source moves at a constant speed along the border of the
half-space. The problem is considered in a two-dimensional formulation, when all
processes are homogeneous along the horizontal transverse coordinate axis. The
displacement vector contains two components: longitudinal and vertical transverse.
As a result of analytical studies, it has been shown that a moving source will generate
waves propagating along the border of a half-space and decreasing exponentially
into its depth. Such a wave, in contrast to the classical surface Rayleigh wave, has
a dispersion, since its phase velocity is not constant, but depends on the frequency.
The displacement amplitudes vary depending on the magnitude of the load of the
moving source and its speed.
Keywords Gradient-elastic half-space · Moving source · Surface wave
2.1 Introduction
In 1885, the English scientist Lord Rayleigh theoretically showed that along the flat
border of a solid elastic half-space with a vacuum or a sufficiently diluted medium
(for example, with air) waves can propagate, whose amplitude rapidly decreases with
depth (Lord Rayleigh 1885). These waves, later called Rayleigh surface waves, are
the main type of waves observed during earthquakes. Therefore, they are studied in
detail in seismology (Aki and Richards 2009).
A. M. Antonov · V. I. Erofeev (B) · A. O. Malkhanov
Mechanical Engineering Research Institute of RAS, Nizhny Novgorod, Russia
e-mail: erof.vi@yandex.ru
V. I. Erofeev · N. A. Novoseltseva
Research Institute for Mechanics, National Research Lobachevsky State University of Nizhny
Novgorod, 23, Gagarin Av., Nizhny Novgorod 603950, Russia
© Springer Nature Switzerland AG 2021
F. dell’Isola and L. Igumnov (eds.), Dynamics, Strength of Materials and Durability
in Multiscale Mechanics, Advanced Structured Materials 137,
https://doi.org/10.1007/978-3-030-53755-5_2
17