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.

The Solution with Singularity
1. Introduction

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  and this scheme easily generalizes for systems of the linear equations. In work  such transition is described for unidimensional inhomogeneous system of the linear theory of elasticity. The system of equations with mass forces is not described by this case whereas the task about action of single force, i.e. creation of Green’s function, gets to this case. The purpose of article is to liquidate this lacune. Such work partly repeats calculations of works   , but it needs to be made in an explicit form.

2. General Scheme: The System of Integral Equations

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 are modules of elasticity and on the repeating indexes summation is made.

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, are Kronecker’s symbols, is delta function; displacements of infinity vanish.

Let’s present elastic modules in the form

where α is numerical parameter.

If system

has solution (Green’s function), then from (2.5) we receive

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,

and.

3. General Scheme: The Series Expansion of Solution

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.

4. Special Case: The Isotropic Medium with Constant Coefficient of Poisson

In these conditions of the balance equations in displacements has the kind

where, and

If to rewrite system (4.1) without, then we will receive

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

5. Some Data on Fourier’s Transformation

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. We will determine direct and inversion Fourier’s transformations by formulas

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, and is the Bessel’s function.

6. Unidimensional Inhomogeneous Medium

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, and Δ is the Laplacian operator.

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 and.

7. Conclusion

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  . It is possible to consider two-dimensional heterogeneity. In Section 6 it is possible to apply Fourier’s transformation on the final interval to finite bodies.

Cite this paper

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

ReferencesDobrovolsky, I.P. (2014) The Integral Equation, Corresponding to the Ordinary Differential Equation. Open Access Library Journal, 1, e1058. http://dx.doi.org/10.4236/oalib.1101058Dobrovolsky, I.P. (2015) Unidimensional Inhomogeneous Isotropic Elastic Half-Space. Open Access Library Journal, 2, e1670. http://dx.doi.org/10.4236/oalib.1101670Dobrovolskiy I.P. ,et al. (2013)Inhomogeneity and Inclusion in the Linear Elasticity Theory. Nauka i Studia NR 35, 49-58.