The homogeneous system of the equations of the linear theory of elasticity for the isotropic environment with one-dimensional continuous heterogeneity is considered. Bidimensional transformation Fourier is applied and the problem for images is led to the ordinary differential equations. Generally, the differential equations are transformed in integro-differential and the algorithm of such transformation is resulted. Solutions of specific problems are resulted.
The elastic half-space is considered inhomogeneous along depth. Such problems were investigated in many works (for example, [
Research is made with application of bidimensional transformation Fourier which leads to the ordinary differential equations. The method of transition to integral equations [
In cartesian coordinates (x, y, z) it is considered isotropic inhomogeneous on z an elastic half-space z ≥ 0 with the shear modulus μ(z) and coefficient of Poisson ν(z). In this case the homogeneous system of the equations of the theory of elasticity in displacements (u, v, w) looks like
where
In [
where η = d(lnμ)/dz. β = 1 − ν, Δxy is Laplace operator on two variables.
Displacements are expressed by formulas
and stresses are
. (2.4)
Formulas (2.1)-(2.4) are received in the monograph [
To a half-space (or a lay) we shall apply bidimensional transformation Fourier in the form of
where i is imaginary unit.
Then the basic operators will be transformed by formulas
For functions
Transformations of stresses on a plane z = const are
The problem is reduced to a solution of the ordinary linear differential equations for functions
The first equation from (2.7) we will write down in a kind
One of ways of search of the approached the general solution of the Equation (3.1) for any smooth function μ(z) is transition to the integro-differential equation on algorithm [
For a finite segment we use the general solution of the inhomogeneous equation
It allows to construct the equation for a semi-infinite segment.
As a result we come to two integro-differential equations of the II kind:
for layer
and for a half-space
By integration by parts (3.3) and (3.4) will be transformed to integral equations
If for the equation
Let’s note the second equation of system (2.7) in the form
where
If to apply to the Equation (4.1) operator, inverse to
For a layer
And for a half-space
At μ = 1/(az+b) function Q(z) = 0 and Formulas (4.2) and (4.3) give the exact general solution of the equation (4.1).
We shall consider a problem about unit force on a surface of a half-space z ≥ 0 in the origin of coordinates. Apparently from system (2.8), in this case (and, in general, for problems with axial symmetry) it is possible to put
Let’s put
Then the Equation (4.1) has the solution limited on infinity
Here C1 and C2 are arbitrary constants, ζ = ρ(1 + z), Ψ(α, β; x) is degenerate hypergeometric function of 2-nd sort or Kummer’s function of 2-nd kind. In computer program Maple this function is designated as Kummer U (α, β, z) and it has integral representation
where Γ(a) is the gamma-function.
It is simple to receive the solution of the problem (5.1)-(5.3) in Fourier transformations, but to carry out the inverse Fourier transform through known functions is not receive. However it is possible to approximate special functions by combinations of elementary functions. Approximation can be made with a demanded exactitude and simplifies the further calculations. In particular for (5.3) we have
The error of this approximation does not exceed 0.5 %.
Level lines of stress σzz are shown on
(the Boussinesq’s problem) such zone does not arise.
It is interesting to compare the received solution to the approximate solution. As the approximate solution we take the zero approximation for the Equation (4.3). Such solution turns out at Q = 0 and it has the form
Using boundary conditions (5.1) we receive function
According to (2.8) transformation of stress σzz gets the form
To receive stress σzz it is necessary to calculate integral
where
To calculate (5.9) we will allocate the whole part in the integrand.
The integral (5.9) is calculated in elementary functions for first three items from a right part of (5.10). The fourth item leads to the integral
which is not expressed in known functions.
However by means of approximation
integral (5.11) also can be calculated in elementary functions. This operation is easily supervised because the integral (5.11) can be found in separate points numerically by means of known mathematical programs (for example, Maple or Mathematica).
Level lines of stress σzz calculated by the described algorithm are shown on
basically are similar. In the approximate solution the area of small tensile stresses remains but it becomes more extensive and moves further from an origin of coordinates.
Transition to integro-differential (or integral) equations is an effective method of a solution of problems for a half-space (or a layer) with arbitrary heterogeneity. It is very important that the solution of an integral equation gives the approximate general solution of the ordinary differential equation. Procedure of transition to integro- differential (or integral) equations allows constructing such equations for the inhomogeneous differential equations.
Igor Petrovich Dobrovolsky, (2015) Unidimensional Inhomogeneous Isotropic Elastic Half-Space. Open Access Library Journal,02,1-6. doi: 10.4236/oalib.1101670