The article deals with the issues of qualitative behavior and methods for calculating the spectra of natural vibrations of a layered composite consisting of elastic and viscoelastic phases with dissipation. As a viscoelastic phase, it is proposed to consider a viscoelastic material with exponential aftereffect kernels, with aftereffect kernels in the form of Rabotnov functions, a material with Kelvin-Voigt friction and a material with fractional Kelvin-Voigt friction. A general scheme is given for studying the qualitative properties of the spectra of one-dimensional oscillations and a method for calculating one-dimensional oscillations, which consists in reducing the spectral problem to solving algebraic equations. The question of the convergence of natural frequencies of oscillations of composite samples to the natural frequencies of averaged boundary value problems, which are non-self-adjoint, is discussed.

The paper considers the principles of constructing a mathematical model of water treatment based on the use of a biologically active layer, the bacteria of which absorb harmful impurities contained in water. A system of equations is presented on the basis of which a model of water purification is constructed in the simplest element, which is a rod covered with a biofilm. The system of equations is a system that includes a parabolic equation in a three-dimensional domain and a hyperbolic equation on a part of the surface of this domain, connected to each other through a boundary condition and a potential in an equation of hyperbolic type. Next, an asymptotic analysis of this system is carried out, which allows us to reduce the model of an individual element to the solution of a simple ordinary differential equation. On this basis, a model of the entire water treatment device is proposed.

The paper considers the problem of damping vibrations of a membrane and a plate with the help of forces distributed over their entire area. The proposed method allows us to consider restrictions not only on the absolute value of the control, but also on the absolute value of the derivatives of the functions that specify the control. Sufficient conditions are given for the initial conditions under which the problem of bringing the system to rest in a finite time is solvable, and the time of bringing to rest is estimated.

The article deals with the problems of constructing effective characteristics of a layered composite material, the layers of which are viscoelastic material. In this case, the hereditary property of viscosity is modeled using the Rabotnov functions, which have singularities at zero. It is proved that the average properties of a material can be analytically described in the form of a homogeneous viscoelastic material with memory, which is also specified using kernels, which are also specified analytically by the Rabotnov functions. It is of interest to study the natural vibrations of such a material. The easiest way is to obtain analytical formulas for the natural frequencies of oscillations transverse with respect to the layers. Such analytical expressions for natural frequencies are obtained in the present work. The spectra are compared for the case of composites made of elastic materials, materials in which nonlocal terms are specified with the help of exponential functions and with the help of Rabotnov functions.

The article considers the problem of viscoelastic vibrations of a layered composite material of two pairwise alternating layers. Three types of layer materials with different properties are considered. The first case is two elastic materials with internal dissipation of mechanical energy, which is described by an integral term of the convolution type with an exponential kernel. The second case is a layered composite of elastic materials with Kelvin-Voigt friction, and finally, in the third case, the internal dissipation in the layers of the material is described by the fractional Kelvin–Voigt friction. We consider the transverse vibrations of this layered composite and give a qualitative picture of the behavior of the spectra. The purpose of this work is to reveal the influence of internal friction for its various models on the vibration spectrum. This technique can be used to create building materials with predetermined properties, for example, for sound insulation.

The article investigates the problems of control of dynamical systems with nonlocal terms of the convolution type using a force distributed over the area in which the motion is considered. The conditions are established under which it is possible to bring the system to complete rest with the help of a distributed force limited in absolute value. The case of one spatial variable is considered, but the proposed spectral method of research can be generalized to dimensions two and three. A connection is established between the behavior of the Fourier coefficients of the initial condition and the solvability of the control problem. The method can be used for mechanical systems with hereditary properties that arise in structural mechanics.

This article deals with control problems for dynamical systems with non-local convolution type terms. A method is proposed to get conditions under which the moving system will go into complete rest. The force acting on the system is distributed over the entire moving domain. Domains of one, two and three dimensions are considered. For these three cases of dimension and two types of fluids (Oldroyd fluid and Kelvin-Voigt fluid), the initial conditions are formulated for the problem posed, with the help of which the proposed method can bring these systems to complete rest in a finite time. Sufficient conditions are given that must be satisfied by the initial oscillations of the systems, under which the spectral method we use can bring these initial oscillations to complete rest. This is a condition on the smoothness of the initial functions and some additional boundary conditions for them. The article presents a new technique for damping unwanted vibrations in visco-elastic building materials.

Multilayer composite materials are often used in building structures. The direct calculation of layered structures requires large expenditures of computer time. Therefore, the homogenization method is used. This method reduces the problem of a layered material with isotropic layers to the problem of a homogeneous transversely isotropic medium. The material considered in the article is also elastic-creeping. In the equations of state of such a material, terms of the convolution type with difference creep (relaxation) kernels are added to the terms of the usual theory of elasticity. The creep (relaxation) kernels are represented by decreasing exponential functions depending on two parameters. This problem becomes a problem of the theory of elasticity with a parameter after applying the Laplace transform in time to it. The inverse Laplace transform can be done in a computer algebra package, for example, Wolfram Mathematica, Wolfram-alpha. The obtained characteristics of the material are used to solve the problem of a layered elastic-creeping beam with hinge support. Formulas are given for determining displacements in the case of layers parallel to the beam axis.

The paper considers a mechanical system defined by a linear system of integro-differential equations with nonlocal convolution type terms. The problem of controllability is studied for various types of control (boundary and volume-distributed control) and various types of kernels. These kernels simulate the aftereffect of the system. It is proved that in some cases there is no possibility of damping the oscillations of the system for arbitrary initial conditions. In this work, those cases are distinguished when there is the possibility of damping oscillations over a finite period of time for any initial conditions.

The article discusses the problem of diffraction of an acoustic wave by a multitude of obstacles (or cavities) enclosed in a finite subdomain of a homogeneous material domain. It is assumed that the distance between obstacles ( cavities) is much greater than the size of each obstacle (cavity). Conditions are indicated under which the total influence of obstacles on the transmitted wave is expressed as an additional term of the potential type in the wave equation. Conditions at the boundary of the cavities are accepted as boundary conditions of the third kind. The results of the work can be used in the development of non-destructive testing procedures.

The work is devoted to the construction of analytical solutions for the stress-strain state of a cylindrical hollow elastic rod with a layered structure along the radius. Earlier, the problem of finding the stress-strain state of a rod of composite material fixed at one end with the applied forces and moments of forces at the other end was considered. An approximate representation of the solutions was given, which included auxiliary problems on one fragment of the cylinder, consisting of periodically repeating similar fragments. Such auxiliary problems in the general case do not have an analytical solution. In this paper it is shown that in the presence of radial symmetry of the rod section, it is possible to construct a stress-strain state in an analytical form. In addition, tensile and bending stiffness can be presented in an analytical form. The latter circumstance allows us to set a problem of optimizing the stiffness characteristics of a rod with its fixed weight. Optimization is carried out by varying the thickness of the layers of the same materials.

The paper considers the question of constructing the effective characteristics of a layered composite material when periodically repeating layers consist of an isotropic elastically creeping material. For the elastic composites, such characteristics are well known and can be obtained explicitly. In this paper, the effective characteristics of the layered elastic-creeping composite are obtained explicitly when the creep kernel are the power-law functions. The elements of the compliance matrix are presented as the sum of the terms corresponding to the instantaneous elastic compliance, unlimited and limited as the loading time increases due to the creep of the composite individual phases.

Composite materials consisting of several phases are widely used in modern construction. Numerous experiments have shown that the properties of structurally heterogeneous materials can differ significantly from those of the individual components making up the composition. Besides, rapidly changing coefficients of differential equations describing such composite materials greatly complicate the solution of boundary value problems even with the help of computer calculation methods. Therefore, the homogenization method is used. In this paper the two approaches propose to obtain in explicit analytical form the effective model of the problem of loading a heterogeneous pipe made of layered material, provided that the elastic properties of the material depend only on the distance from the center of the section of the pipe. We point to a method that obviously leads to an analytical result. It follows from the article that it is possible to choose the function that determines the structure of the “winding” in such a way as to obtain the stiffness characteristics of the pipe as close as possible to the desired with fixed mass fractions of the materials used. A similar approach can be applied to the study of creep properties of pipes made of composite materials.

Composite materials consisting of several phases are widely used in modern construction. The mechanical characteristics of elastic-creeping layered composite materials are considered in the article. Each of the constituent phases has the properties of elasticity, viscosity or creep. Numerous experiments have shown that the properties of structurally heterogeneous materials can differ significantly from those of the individual components making up the composition. Besides, rapidly changing coefficients of differential equations describing such composite materials greatly complicate the solution of boundary value problems even with the help of computer calculation methods. Therefore, the homogenization method is used to solve such problems. Creep kernels are given by the sum of a finite number of decreasing exponential functions. The use of creep kernels of this type is experimentally justified. In this paper, it is shown that an effective (averaged) model for a composite material with the indicated properties is a homogenized medium that is described by a creep kernel, also represented by a sum of exponential functions. An algorithm for the rapid and accurate calculation of averaged creep kernels of a homogenized material is proposed.

The paper deals with a cylindrical hollow rod with a layered structure along the radius. Layers consist of isotropic elastic creeping materials. An analytical solution is given for the problem of the stress-strain state of such rod. Tensile and bending stiffness can be presented in analytical form. An algorithm is proposed for constructing an analytical solution to the problem of various loading of a layered rod consisting of elastic-creeping layers in the presence of radial symmetry.

Composite materials consisting of several phases are widely used in modern construction. Numerous experiments have shown that the properties of structurally heterogeneous materials can differ significantly from those of the individual components making up the composition. Besides, rapidly changing coefficients of differential equations describing such composite materials greatly complicate the solution of boundary value problems even with the help of computer calculation methods. Therefore, the homogenization method is used. In this paper the approach propose to obtain in explicit analytical form the effective model of the problem of loading a heterogeneous pipe made of layered material, provided that the elastic properties of the material only depend on the distance from the center of pipe cross section. We point to a method that obviously leads to an analytical result. It follows from the article that it is possible to choose the function that determines the structure of the “winding” in such a way as to obtain the stiffness characteristics of the pipe as close as possible to the desired with fixed mass fractions of the materials used.

The article deals with the problem of a layered composite pipe. The layers alternate in pairs and consist of elastic-creeping materials. The work is devoted to the study of the mechanical properties of the composite material with the help of its microstructure. Processes are studied with a typical pipe radius which is much more than the typical winding width. Heterogeneous layered material behaves as a certain “effective” material without layers in such processes. The method of calculation of effective moduli based on mathematical homogenization theory is described. Creep kernels are given by the sum of a finite number of decreasing exponential functions.

A pipe made of composite material is considered. Two approaches are proposed. The first is based on the averaging method. In this case, an effective model is a model of a hollow circular cylinder. In the second approach, a one-dimensional beam is an effective model. We also represent the problem of minimizing the weight of the pipe at fixed stiffness of the layers. The paper shows that this problem is equivalent to a well-studied nonlinear programming problem.

The article suggests a method for modelling the stress-strain state of layered elastic-creeping tube. This method is based on a combination of the theory of averaging and creep theory. This theory takes into account the previous stress state and its effect on the subsequent deformation period. Creep kernels for the averaged problem are found in the explicit analytical form. These formulas can be used to estimate displacement and stress fields during long-term loading of a composite tube.