Mathematical modeling of multilayer road surfaces

The paper is devoted to the mathematical study of elastic states and deformations of multilayer road surfaces under the force of water. The filtration coefficient in the multilayer strip is piecewise constant. Young’s modulus and Poisson’s ratio are given in each band. The problem is reduced to a planar boundary value problem for a system of partial differential equations. A method of constructing solutions to this problem in the form of almost-periodic Bohr functions is investigated. A computational experiment was carried out. Graphs of the required mechanical parameters are constructed, their convergence to the boundary values is shown. The Maple software package was used to build the solutions. The constructed algorithm can be used in solving problems in continuum mechanics.


Introduction
When solving plane problems for heterogeneous media in the theory of elasticity, filtration theory, diffusion theory, thermal conductivity, electrodynamics and magneto dynamics in the case when the region is an l-layer band, boundary value problems for systems of differential equations with boundary conditions given both on the band boundary and on the gluing line are solved [1,2]. It is especially important to solve such problems in the study of deformations and elastic states of multilayer road surfaces [3]. Due to the porosity of the road surface, liquid can be filtered through it [4]. Under the influence of water, the strength properties of the material can change, as a result of which it can collapse. This is typical of any porous materials. For example, Boskovich D., Clark P. and others showed in [5] that in the development of coal seams for methane production, injection under water pressure with subsequent rapid pumping is used When water is injected into the well, the strength properties of the coal can change, as a result, the coal can collapse and clog the cleavages, making it difficult to extract gas from the coal seams.
In many textbooks on mathematical physics, analytical solutions to boundary value problems for systems of partial differential equations are written using Fourier integrals, which entails great computational difficulties. The representation of solutions in the form of absolutely convergent Fourier series greatly simplifies the task and makes the further process of computer simulation quite simple. It is worth noting that if a periodic load is applied to a region having a periodic structure, the stress-strain state will also be determined by periodic functions. However, in practice, it is quite difficult to create a periodic load on the body. Then the load will be expressed in the general case by almost-periodic functions (Bohr-Fourier series), the use of which for the study of the stress state of a multilayer elastic band will be devoted to this article.
The apparatus of almost-periodic functions and generalized discrete Fourier transform previously proposed in this paper was used by the author to solve other problems [1,6], as well as by other authors [7,8]

General statement of the boundary value problem for a system of differential equations
The General mathematical formulation of the problem has the following form. It is required to find functions ( , ) km U x y such that they satisfy a system of differential equations: in each of the m-th band ( figure 1): The boundary conditions must be set at the strip boundary, and the gluing conditions must be set at the media interface lines. For example, for a two-layer strip , x     0, ay    0, ya  these conditions might look like this: We will construct almost-periodic in the sense of Bohr solutions on each line m ya  , using the generalized discrete Fourier transform, considered in [9].
We assume that the functions , . A more detailed description of almost-periodic functions and their properties is given in [10,11]. The unknowns in formulas (3)   Using the properties of this operator we move from a system of partial differential equations to a system of ordinary differential equations for each n with respect to a function ( , ) km km u u y  , where n  is a parameter: The order of the system is not higher 2n . The solution to this system will be where   qkm n p  is constant at fixed , , , q k m  . They are found from the boundary conditions and gluing conditions. To determine these constants, a finite system of linear algebraic equations is obtained.

Construction of a mathematical algorithm for studying the elastic-deformed state of a multilayer strip under the force of water
Consider the following problem. Similar problems arise in the study of elastic states and deformations of road surfaces under the force of water.
where the functions  1, . ml  The problem is reduced to solving a system of differential equations where ( ) ( ) 0 , mm  -volume weights of a unit of liquid and a unit of a porous body with water enclosed in it. The system of equations (4) for this case will take the form for each n:    for each value of the variable 0  are derived from a system of linear algebraic equations, which is a consequence of the boundary conditions and coupling conditions. The determinant of this system is nonzero at any 0  . In General, the system is quite cumbersome, so let's consider a special case.
The software module is developed in the Maple environment. Graphs of tangent and normal stresses, as well as functions of filtration rate potentials are presented in figure 2(a, b), 3 (a, b), 4 (a, b).     (1)  .