The problem solution of deformable power surfaces from anisotropic material interaction in elements of structure constructions

The paper presents an analytical method of examining deformable power surfaces from anisotropic material and their interaction. Materials with anisotropic properties are often used in the elements of transport structures as well as in building construction [1-7]. Most frequently the following materials are used in construction: rolled metal products, products obtained by drawing, wires of various kinds, reinforced concrete structures, bridge bearings, supporting walls and other structures, composite materials, plastics and other [8-13]. For forecasting strength properties of structures strength joints and their behaviour under various external influences, the authors developed a method of calculation of their stress-strain behaviour [9]. This method can be applied to strength joints of complex form made of anisotropic elastic elements under specific conditions of deformation of the contact surfaces.


Materials and methods
The problem solution is shown by the analytical method.In the presented problem, the following interaction conditions of the contact surfaces are considered: 1. Movements are specified on the entire surface of the deformable solid body.The area in which they are different from zero is only a certain time-varying part of the surface of the body.

Equations and mathematics
The deformable body is assumed to have finite dimensions, its material being homogeneous.Its state is described by the following problem in the theory of elasticity: Let's represent the Green's function Boundary-value problem solution (1) can be presents as: Problem solution (3) is defined as the volume potential The volume of integration V1 contains volume V occupied by the deformed body, and the tensor Fpn (y, t) is mass forces tensor distributed in the volume V1.
The integration in (6) in fact was carried out from the difference in the volumes V and V1.The function Wnm (x, t) represented by the formula (6) satisfies the equation of the boundary value problem (1) by the properties of the fundamental solution, and the selection of Fpn (y, t) is carried out taking into account the boundary conditions of problem (3).
Carried out a number of transformations, using the equalities to define the tensor components N(xs)e -ikxs , integrating the intermediate expression on the surface S with the MATEC Web of Conferences 196, 02013 (2018) https://doi.org/10.1051/matecconf/201819602013XXVII R-S-P Seminar 2018, Theoretical Foundation of Civil Engineering subsequent transition to integration on the surface V and introducing a number of notations with the expansion of the divergence operator, taking into account the convolution theorem, made it possible to determine the Fourier images of the coordinates of the mass force: In formulae (8) integrating volumesя V 2 are defined by V 1 , V and the correlation y x z   .In the congruence (7) equation is a matrix of the third order, the elements of which are functions of the parameter k , and the determinant is different from zero.From the construction of the matrix k inverse to the matrix of the system, we obtain the formula Allowing to write down x -the solution of the problem (3) in the form: From the proof of the theorem it follows that the tensor function ) , ( t W mn x ,defined by ( 10) is a solution of the boundary value problem (3) and relation (11) determines the Green's function of the boundary value problem (1).
External forces are prescribed on the entire surface of the deformable element, and the area in which these forces are different from zero also constitute just a part of the body surface, and vary some time later.
In this case, the stress-strain state of the body is described by the following boundaryvalue problem of the theory of elasticity: Problem solution ( 14) is in the form: The choice ) , ( t F pn y is carried out from the condition for the satisfaction of boundary conditions (12).
The Fourier transform of the boundary problem (14), integration by parts, the application of the Ostrogradskii-Gauss formula and the convolution theorem for the Fourier images of the stress tensor coordinates made it possible to obtain: The system of linear algebraic equations (20), equivalent to the original boundary-value problem, has also a unique solution because of the uniqueness of the solution, and, consequently, its determinant is different from zero. ) , has a reverse matrix ) , ( t K k  , using which, the solution of the system of equations (20) From the proofs of the theorems presented in the paper: -function ) , ( t W mn x , defined by relation ( 22), satisfies the system of equations of the boundary-value problem (14); -function ) , ( t W mn x , defined by relation (22), satisfies boundary problem conditions ( 14) and ( 22) boundary problem solution (14); According to the statement of the problemexternal forces are specified on the part surface of the deformable element, and movements on the parts.The areas in which they are given make up just a part of the body surface and they change some time later.This loading type is described by the following problem in the theory of elasticity: on S U (t) part of the deformed body surface U * io (x s t) interchanges are given, on the other part ) (t S  -there are surface power P * i (x s ,t).It is assumed that the material of the deformed body is anisotropic.Then, as before, the Kelvin-Somigliana tensor for the anisotropic Lamé operator makes it possible to reduce the inhomogeneous boundary value problem (23) to a homogeneous one: To solve the problem (24) the solutions of the first and second boundary value problems presented above are used.
The boundary problem solution (24) with the Stokes formula for the Lame operator can be written in the form: Relation (28) is a system of three equations (i=1,2,3) for defining three unknown functions F m (k,t) and is integral type of initial boundary problem.Matrix introduction R mi (x,t),and inverse system matrix allowed to put down the solution of the system as follows: The relations ( 27) and ( 29) allow to write down the solution of the original problem in the form

Conclusions
From the evidence presented in the paper and the method proposed by the authors, it follows: 1.The quadrature (30) satisfies the system of differential equations of the boundary value problem (24).
3. The quadrature ( 30) is a solution of the boundary-value problem (24).
MATEC Web of Conferences 196, 02013 (2018) https://doi.org/10.1051/matecconf/201819602013XXVII R-S-P Seminar 2018, Theoretical Foundation of Civil Engineering Thus, the problem of creating an effective method for calculating the deformed state of homogeneous anisotropic bodies of complex shape under static loading with external influences varying in magnitude, direction, and area of application is proposed and considered in this paper. ,