EVALUATION OF THE STRESS-STRAIN STATE OF HALF-SPACE WITH CYLINDRICAL CAVITIES

A three-dimensional problem was solved in the theory of elasticity for an elastic uniform halfspace with cylindrical cavities parallel to each another and the half-space boundary. Stresses rapidly decaying to zero at big distances from the origin of coordinates are specified on the boundaries of cylindrical cavities and on the half-space boundary. The problem of solving such problems is topical. It is encountered in practice and solved using approximate methods. The approach used herein yields solutions of the stated problem with an a priori specified accuracy depending on the system order. In contrast to publications referred to in the paper, the focus, apart from another approach, was on analysing the stress state of half-space to study the mutual influence of cylindrical cavities and of the cavities with half-space boundary. The problem was solved using the generalised Fourier method for Lame equations in cylindrical coordinates linked to cylinders and Cartesian coordinates linked to half-space. For transition between basic solutions of Lame’s equations, special formulas were used for transition between local cylindrical systems of coordinates and between the Cartesian and cylindrical systems of coordinates. The truncation method was used to solve infinite systems of linear algebraic equations to which the problem was reduced. This yielded displacements and stresses in an elastic body. The numerical results were derived for the case of half-space and two cylinders under a load applied to the half-space boundary. Separate computations were performed for a load applied to the surface of the cylindrical cavity. In both cases the analysis of the stress-strain state indicates that space weakening due to cylindrical cavities or the half-space boundary gives rise to extremal stresses in these sites. The method can also be used for other boundary conditions.

Introduction.Design of different kinds of constructions and underground utility systems requires the knowledge of the stress-strain state of half-space with cavities and loads specified on the boundaries of cavities and half-space.This information is obtained by solving the relevant elasticity theory problem.
The approach to solving the problem stated in the title is based on the generalised Fourier method [2].This method is used to solve the first basic elasticity theory problem for space with round cylindrical cavities [1], the mixed problem for space with cylindrical cavities when displacements are specified on the boundaries of some parallel cylindrical cavities; on others, stresses; and on other ones, tangential forces and normal displacements [9].Studies [3][4][5][6]8] also use the generalised Fourier method for transverse-isotropic bodies bounded by coordinate surfaces in cylindrical and paraboloid coordinates with a lumped force applied to the half-space boundary with a lumped force.Paper [7] considers the first basic elasticity theory problem for half-space with one cylindrical cavity and extremely partial conditions on the half-space boundary Objective of the paper.Analysis of the stress-strain state for half-space with several cylindrical cavities parallel to each another and the half-space boundary, and studying their mutual influence.The generalised Fourier method is used to achieve this goal.The problem is being considered for the first time.
Problem statement.Let us consider an elastic uniform half-space with N round cylindrical parallel cavities, nonintersecting with each another and the half-space boundary.The cavities will be considered in the local cylindrical coordinate system ( z , , p p   ), where p is cylinder number, and the half-space is in the Cartesian system of coordinates (x,y,z) oriented identically and linked to the system of coordinates of the cylinder with number p=1.The half-space boundary is located at the distance y=h.The boundary equation of half-space Sd is y=h, and that of the surfaces of cylinders Sp is . Stresses are specified on the half-space boundary and on the boundaries of cavities.They will be considered rapidly decaying to zero at big distances from the origin of coordinates.It is required to find the solution of the Lame's equation ( )


will be assumed rapidly decaying to zero over coordinates z and x at big distances from the origin of coordinates.
Solution method.In the given systems of coordinates we take basic solutions of the Lame's equation: are space points, respectively, in the Cartesian system of coordinates identified with the system q x , q y (fig.1): , and in the cylindrical system of coordinates linked to the p-th cylinder; ( ) ( ) ( ) respectively, are the internal and external solutions of the Lame's equation for the cylinder; ( ) ( ) are solutions of the Lame's equation for half-space.
The problem solution is given in the form are basic solutions specified by ( 1) and ( 2), with sought for functions being found from boundary conditions.
For transition between systems of coordinates (fig. 1) we use the next formulas.

Fig. 1. Half-space with cylindrical cavities
Formulas using for transition from coordinates of cylinder with number p to the half-space coordinate ( ) Formulas using for transition from half-space coordinates to coordinates of cylinder p where are coordinates of cylinder p relative to the first cylinder; Formulas using for transition from the coordinates of cylinder with number p to coordinates of cylinder q where To satisfy the boundary conditions on the boundary of half-space y=h we rewrite the right-hand side of (3) with transition formulas (4) to the Cartesian system of coordinates through basic solutions ( )  k u  .For the vector derived, we find the stress and set it at y=h to specified ( ) , which first will be represented with a double Fourier integral, and designate this expression as Using formulas for transition from the Cartesian system to the cylindrical one ( 5), and from one cylinder to another one (6), we write (3) in the coordinates of the cylinder with number p through basic solutions . If now we find stresses caused by U  on the surface of cylindrical cavities and account for boundary conditions, we will find equality From ( 7) we find and substitute . This yields a set of 3N infinite systems of linear algebraic equations for finding unknowns For the systems obtained, their unique solvability can be proved.Moreover, these systems can be solved using the truncation method, with convergence of approximate solutions to an exact one.Functions ( )  .On the boundaries of the cylinders stresses are specified as The obtained infinite system of equations was reduced to a finite one for parameter msystem order (parameter m is investigated in [7]), and the integration limits for the given function for  and  were set to -10 to 10.The integrals were computed using Filon and Simpson quadrature formulas.The accuracy of satisfying the boundary conditions where they are continuous, due to these parameters, was brought to 10 -4 .
Fig. 2 shows the stress state of half-space for z=0 at different distances from the halfspace boundary y=h.As compared to the stress state in space (fig.5), x  and z  on the half-space boundary (y = 40) take non-zero values due to the influence of the half-space boundary.More over, stresses x  and z  on the cylinder surface (y = 10) are maximum.In approaching to half-space, the stresses decrease, and close to the half-space boundary, they increase.
Conclusions.The generalised Fourier method is used for solving the spatial elasticity theory problem when stresses are specified on the half-space boundary and on the boundaries of several parallel cylindrical round cavities.The problem is reduced to a set of infinite systems of linear algebraic equations.The systems can be solved using the reduction method.
Numerical analysis of the algebraic system for two cylinders in half-space demonstrates that this method enables finding its solution with any degree of accuracy.This is confirmed by the high accuracy of satisfying the boundary conditions.
The graphs show the stress patterns in half-space in the most interesting zones and the mutual influence of half-space boundaries and cavities.

Fig. 4 .Fig. 5 .
Fig. 4. Stresses in half-space with cylindrical cavities in plane z=0 when the half-space boundary is unloaded and the cylindrical cavity is loaded: ax  ; b -y  ; c -z  ; d -xy 
3, c where stresses decrease gradually with a bigger distance from the halfspace boundary.With weakened space, tangential forces xy  (fig.2, d) have extremal values near the surfaces of cylinders, in contrast to half-space with no cylindrical cavities where, in moving away from the boundary of half-space, xy  first rise sharply in sites where loads terminate, and then they decrease gradually.If the half-space boundary is unloaded and the cylindrical cavity is loaded, for the next boundary conditions