The scheme of creation of systems of the integro-differential equations for evaluation of Green’s function in non-uniform elastic boundless medium is described. The summand with singularity is allocated. The isotropic medium wit h constant coefficient of Poisson and unidimensional inhomogeneous isotropic medium are considered.
Transition from differential equations to integral equations (or to integro-differential equations) is an alternative possibility of the solution of the differential equations. It is sometimes simpler to receive the solution of the integral equation, than differential equation.
The general scheme of such transition for one linear differential equation is described in work [
Let’s consider nonuniform non-isotropic linearly elastic medium. We enter Cartesian coordinate system of xi and we will designate displacements as ui. Small deformations are defined by Cauchy’s formulas
where the comma in the inferior index means the derivative on the corresponding coordinate.
Stresses are described by Hooke’s law
where
Then balance equations in displacements with single forces in the point ξ receive the kind
where s is number of force, i is number of the equation,
Let’s present elastic modules in the form
Then (2.3) receives the kind
where α is numerical parameter.
If system
has solution
where integration is made on all space.
From (2.7) we have system of integro-differential equations (at s = 1, 2, 3)
For isotropic medium
where λ and μ are Lame’s coefficients.
If to accept the condition
then Green’s function (Kelvin’s tensor) in an isotropic homogeneous medium has kind
where ν is Poisson’s coefficient,
We look for the solution of system (2.5) in shape
Then (2.5) takes the form
or
From (3.3) at identical degrees α we come to the following set of systems of equations at zero boundary conditions
By means of (2.7) and (2.8) of (3.4) we have
In essence, (3.1) and (3.5) is the solution of the equation (2.8) by method of successive iterations.
In these conditions of the balance equations in displacements has the kind
where
If to rewrite system (4.1) without
The system (4.3) at ν = const has Green’s function
It is easy to be convinced of it. If to divide the equations of system (4.3) on μ, then we receive
and then the solution (4.4) becomes obvious.
Remark. Expressions (2.11) and (4.4) formally match up, but between them there is the important difference. The formula (2.11) is the consequence of the assumption (2.10), and the formula (4.4) is the solution of system of equations.
Now for (4.1) by analogy with (2.5)-(2.8) it is possible to write the system of the integro-differential equations
Let’s note some properties of Fourier’s transformation which will be used further. In this section and further we will designate Cartesian axials (x, y, z) and Fourier’s transformation by the sign “~” or by the arrow
Transformation of the derivative on x leads to multiplication of the transform on
We have some useful formulas. The Fourier’s transformation of product of functions is
Differentiating the first formula (5.1) on p, we receive
We have also formula
We will determine Fourier’s double transformation by formulas
In the axisymmetric case from (5.6) we receive Hankel’s transformation
where
Let’s consider unidimensional inhomogeneous on the axis z medium. Such problems matter in sciences of the Earth. Let’s designate displacements on axes (x, y, z) as (u, v, w). In these conditions of the balance equations in displacements has the kind
where
It is possible to apply the general methods stated in Sections 2 and 3 to the solution of system (6.1). However, it is better to make double Fourier’s transformation on (x, y) according to Section 5. Then (6.1) takes the form
(6.2)
Application of the general methods to (6.2) is more reasonable as in this case we receive system of the one-dimensional integro-differential equations. For the system
with constant elastic moduli Green’s function is Calvin’s tensor (2.11) transformed by Fourier’s transformation.
As in Section 4, it is possible to investigate the case of constant coefficient of Poisson. In this case the system (6.2) can receive other form if to divide the Equations (6.2) on
(6.4)
where
Formulas (2.11) and (4.4) can be considered as zero-order approximation of Green’s function for the inhomogeneous medium. These formulas allocate part of the formula of Green with singularity. For the half-space it is necessary to apply Mindlin’s tensor [
Dobrovolsky, I.P. (2016) About the Integral Equations for Calculation of Green’s Tensor in Elastic Inhomogeneous Medium. Open Access Library Journal, 3: e3077. http://dx.doi.org/10.4236/oalib.1103077