Chapter 6
Construction of the Solutions
of Non-stationary Dynamic Problems
for Linear Viscoelastic Bodies
with a Constant Poisson’s Ratio
Leonid Igumnov, Ekaterina A. Korovaytseva, and Sergey G. Pshenichnov
Abstract The problems of propagation of non-stationary waves in linear viscoelastic
bodies on condition that the Poisson’s ratio of the material does not change through
the time are considered. The issues of finding of the solutions of such problems
by the method of Laplace transform in time are discussed. Some properties of the
solution in Laplace transforms, which relate to their singular points and simplify
finding the originals, have also been considered. The case when a hereditary kernel
is an exponential two-parametrical one is considered. We have demonstrated that in
such case the singular points of the Laplace transforms are connected by a simple
relation with the singular points of the Laplace transforms for the corresponding
elastic body. The conditions under which the poles of the solution in transforms have
the first order have been established. As an example, the solution of the problem of
one-dimensional non-stationary wave propagation in a linear viscoelastic cylinder is
presented.
Keywords Viscoelastic bodies · Non-stationary waves · Laplace transform
6.1 Introduction
In studies of transitional wave processes in viscoelastic bodies, an important role
is played by the analytical methods of building the solutions of non-stationary
dynamic problems of linear viscoelasticity. One of such methods developed by
Ilyasov involves a special type of convolution of solution of a dynamic problem
of the elasticity theory with the solution of a one-dimensional problem where hereditary kernels of a viscoelastic body take part (Ilyasov 2011). There is a numerical
L. Igumnov
Research Institute for Mechanics, National Research Lobachevsky State University of Niznhy
Novgorod, Niznhy Novgorod, Russia
E. A. Korovaytseva (B) · S. G. Pshenichnov
Institute of Mechanics, Lomonosov Moscow State University, Moscow, Russia
e-mail: katrell@mail.ru
© 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_6
89
Construction of the Solutions
of Non-stationary Dynamic Problems
for Linear Viscoelastic Bodies
with a Constant Poisson’s Ratio
Leonid Igumnov, Ekaterina A. Korovaytseva, and Sergey G. Pshenichnov
Abstract The problems of propagation of non-stationary waves in linear viscoelastic
bodies on condition that the Poisson’s ratio of the material does not change through
the time are considered. The issues of finding of the solutions of such problems
by the method of Laplace transform in time are discussed. Some properties of the
solution in Laplace transforms, which relate to their singular points and simplify
finding the originals, have also been considered. The case when a hereditary kernel
is an exponential two-parametrical one is considered. We have demonstrated that in
such case the singular points of the Laplace transforms are connected by a simple
relation with the singular points of the Laplace transforms for the corresponding
elastic body. The conditions under which the poles of the solution in transforms have
the first order have been established. As an example, the solution of the problem of
one-dimensional non-stationary wave propagation in a linear viscoelastic cylinder is
presented.
Keywords Viscoelastic bodies · Non-stationary waves · Laplace transform
6.1 Introduction
In studies of transitional wave processes in viscoelastic bodies, an important role
is played by the analytical methods of building the solutions of non-stationary
dynamic problems of linear viscoelasticity. One of such methods developed by
Ilyasov involves a special type of convolution of solution of a dynamic problem
of the elasticity theory with the solution of a one-dimensional problem where hereditary kernels of a viscoelastic body take part (Ilyasov 2011). There is a numerical
L. Igumnov
Research Institute for Mechanics, National Research Lobachevsky State University of Niznhy
Novgorod, Niznhy Novgorod, Russia
E. A. Korovaytseva (B) · S. G. Pshenichnov
Institute of Mechanics, Lomonosov Moscow State University, Moscow, Russia
e-mail: katrell@mail.ru
© 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_6
89