Respecting common reasoning, we consider stress-strain state of a half-space with a rectangular parallelepiped shape cavity with coordinates of geometric center *e*_{i} and faces *a*_{1, 2}, *b*_{1, 2}, *c*_{1, 2} (Fig. 1), here cavity sides are parallel to the coordinate planes. In this case, six faces *S*_{1}. . *S*_{6} form the *S* surface of the cavity. First, we consider the surface surrounding the cavity space without the cavity itself. The forces acting on *S*_{1}. . *S*_{6} faces without the cavity are determined by stress tensor flow:

$$ P=\underset{S}{\iint}\sigma \cdot ndS $$

(8)

Components of this vector are equal (convolution is made with second indices of stress tensor):

$$ {P}_i=\underset{S}{\iint }{\sigma}_{ik}{n}_k dS=\underset{S}{\iint}\left({\sigma}_{i1}{n}_1+{\sigma}_{i2}{n}_2+{\sigma}_{i3}{n}_3\right) dS $$

(9)

When calculating stress tensor components, flow surface integrals of quite long functions have to be taken; therefore, they can be presented in the form of low-degree polynomials around the cavity center *e*_{i}. Then taking into account the differential operator *D*:

$$ D=\left({x}_1-{e}_1\right)\frac{\partial }{\partial {x}_1}+\left({x}_2-{e}_2\right)\frac{\partial }{\partial {x}_2}+\left({x}_3-{e}_3\right)\frac{\partial }{\partial {x}_3} $$

(10)

we get

$$ {\sigma}_{ij}^{\ast}\left({x}_1,{x}_2,{x}_3\right)=\sum \limits_{k=0}^{m=2}\frac{D^k{\sigma}_{ij}\left({e}_1,{e}_2,{e}_3\right)}{k!}+{R}_m\left({x}_1,{x}_2,{x}_3\right) $$

(11)

For stress tensor first column, the equation can be written as follows:

$$ {\displaystyle \begin{array}{l}{\sigma}_{n1}^{\ast}\left({x}_1,{x}_2,{x}_3\right)=\frac{3}{4}{K}_1^n{x}_1^2+\frac{1}{4}\left(6{K}_2^n{x}_2+6{K}_3^n{x}_3+8{K}_4^n\right){x}_1+\\ {}+\frac{3}{4}{K}_5^n{x}_2^2+\frac{1}{4}\left(6{K}_6^n{x}_3+8{K}_7^n\right){x}_2+\frac{3}{4}{K}_8^n{x}_3^2+2{K}_9^n{x}_3+3{K}_{10}^n,\\ {}n=\mathrm{1..3}\end{array}} $$

(12)

where \( {K}_m^n\left(n=\mathrm{1..3},m=\mathrm{1..10}\right) \) are constants determined by the medium elastic characteristics, force acting, and cavity location.

Components of vector of force *P*_{nk} (*k* is the face number) acting on the *S*_{1} face of the cavity:

$$ {\displaystyle \begin{array}{l}{P}_{n1}=\int \int {\sigma}_{n1}^{\ast } dS=\underset{c_1}{\overset{c_2}{\int }}\underset{b_1}{\overset{b_2}{\int }}{\sigma}_{n1}^{\ast }{dx}_2{dx}_3=\frac{1}{4}{K}_8^n\left({b}_2-{b}_1\right)\left(-{c}_1^3+{c}_2^3\right)+\\ {}+\frac{1}{2}\left(\frac{3}{4}{K}_6^n\left(-{b}_1^2+{b}_2^2\right)+\frac{3}{2}{K}_3^n{a}_1\left({b}_2-{b}_1\right)\right)\left(-{c}_1^2+{c}_2^2\right)+\\ {}+\frac{1}{4}{K}_5^n\left(-{b}_1^3+{b}_2^3\right)\left({c}_2-{c}_1\right)+\frac{1}{2}\left(\frac{3}{2}{K}_2^n{a}_1+2{K}_7^n\right)\left(-{b}_1^2+{b}_2^2\right)\left({c}_2-{c}_1\right)+\\ {}+\frac{3}{4}{K}_1^n{a}_1^2\left({b}_2-{b}_1\right)\left({c}_2-{c}_1\right)+2{K}_4^n{a}_1\left({b}_2-{b}_1\right)\left({c}_2-{c}_1\right)+\\ {}+3{K}_{10}^n\left({b}_2-{b}_1\right)\left({c}_2-{c}_1\right),\\ {}n=\mathrm{1..3}\end{array}} $$

(13)

Or in a compact form relating to *a*_{1} coordinate of the *S*_{1} face:

$$ {P}_{n1}=\frac{3}{4}{C}_1^{n1}{a}_1^2+\frac{1}{4}{C}_2^{n1}{a}_1+\frac{1}{8}{C}_3^{n1} $$

(14)

Here \( {C}_{\mathrm{1..3}}^{nk} \) are determined by shape and dimensions of cavity projection on a plane that is perpendicular to axis with *a*_{1} coordinate. Following in the same way for the rest faces of the cavity, we obtain:

$$ {\displaystyle \begin{array}{l}{P}_{nk}=\frac{3}{4}{C}_1^{nk}{a}_1^2+\frac{1}{4}{C}_2^{nk}{a}_1+\frac{1}{8}{C}_3^{nk},n=\mathrm{1..3},k=\mathrm{1..6}\\ {}\end{array}} $$

(15)

Let us select a certain volume in the body and consider the total force acting on it. It can be represented as ∫**F***dV* where **F** is the force acting per unit volume. This force can be considered as the sum of the forces that act on the given volume from the parts surrounding it. The action of these forces is carried out through the surface surrounding this volume, then the resultant force can be written as an integral over this surface:

$$ \int {\mathbf{F}}_i dV=\int \frac{\partial {\sigma}_{ik}}{\partial {x}_k} dV=\oint {\sigma}_{ik}{df}_k. $$

In this expression, the integral over the surface is the force acting on the volume bounded by this surface from the side of the surrounding parts of the body. Conversely, the force with which this volume acts on the surrounding surface itself has the opposite sign

$$ -\oint {\sigma}_{ik}{df}_k $$

This is true for a continuous medium. In the presence of a cavity, the forces *P*_{nk} acting on the surface *S* will not be compensated by this volume of the medium, then applying the force − ∮ *σ*_{ik}*df*_{k} on this surface, one can approximately describe the picture of the stress-strain state in the vicinity of the cavity. It is convenient to imagine this as a case of the action of a distributed force on an elastic half-space. Of course, we must remember that such an assumption would make an error in the results and of course the expected faster growth of tension as the distance from the boundary of the cavity. It should also be taken into account that the solution used for the half-space gives acceptable results at points located in subareas well approximated by this half-space, that is, near the base of these cavities, but far from the edge of the base. Nevertheless, our task is to show the principle possibility of using this method.

Generally, distribution of forces on cavity surface is not always uniform; particularly, it takes place with high-strain gradient. In this case, deformation under distributed force action is defined by the integral:

$$ {u}_i=\iint {G}_{ik}\left({x}_1-{x_1}^{\prime },{x}_2-{x_2}^{\prime },{x}_3\right){\sigma}_{km}\left({x_1}^{\prime },{x_2}^{\prime}\right){dx_1}^{\prime }{dx_2}^{\prime } $$

(16)

To avoid heavy calculations while integrating components of Green’s tensor and stresses, we can represent action of distributed forces as a system of concentrated loads.

Let us consider *S*_{2} surface of cavity and put the Cartesian coordinate system *y*_{i} in its center so that its unit axis are aligned with coordinate axis of system *x*_{i}. Deformation \( {u}_i^y \) from concentrated force *P*_{n2} in this coordinate system:

$$ {\displaystyle \begin{array}{l}{u}_1^y=\frac{1+\nu }{2\pi E}\left(\left[\frac{2\left(1-\nu \right)}{r}+\frac{y_1^2}{r^3}\right]{P}_{12}+\left[\frac{1-2\nu }{r\left(r+{y}_1\right)}+\frac{y_1}{r^3}\right]\left({y}_2{P}_{22}+{y}_3{P}_{32}\right)\right),\\ {}{u}_2^y=\frac{1+\nu }{2\pi E}\left(\left[\frac{y_1{y}_2}{r^3}-\frac{\left(1-2\nu \right){y}_2}{r\left(r+{y}_1\right)}\right]{P}_{12}+\frac{2\left(1-\nu \right)r+{y}_1}{r\left(r+{y}_1\right)}{P}_{22}+\frac{\left[2r\left(\nu r+{y}_1\right)+{y}_1^2\right]{y}_1}{r^3{\left(r+{y}_1\right)}^2}\left({y}_2{P}_{22}+{y}_3{P}_{32}\right)\right),\\ {}{u}_3^y=\frac{1+\nu }{2\pi E}\left(\left[\frac{y_1{y}_3}{r^3}-\frac{\left(1-2\nu \right){y}_3}{r\left(r+{y}_1\right)}\right]{P}_{12}+\frac{2\left(1-\nu \right)r+{y}_1}{r\left(r+{y}_1\right)}{P}_{32}+\frac{\left[2r\left(\nu r+{y}_1\right)+{y}_1^2\right]{y}_3}{r^3{\left(r+{y}_1\right)}^2}\left({y}_2{P}_{22}+{y}_3{P}_{32}\right)\right),\\ {}r=\sqrt{y_1^2+{y}_2^2+{y}_3^2}.\end{array}} $$

(17)

Using (6), we can easily get expressions for strain \( {\varepsilon}_{ij}^y \) and stress \( {\sigma}_{ij}^y \) tensor components, for example:

$$ {\displaystyle \begin{array}{l}{\varepsilon}_{11}^y=\frac{\left({r}^2v-\frac{3}{2}{y}_1^2\right)\left({P}_{12}{y}_1+{P}_{22}{y}_2+{P}_{32}{y}_3\right)\left(1+v\right)}{r^5 E\pi}\\ {}{\sigma}_{11}^y=-\frac{6\left(\left({y}_1^2+\frac{1}{4}{y}_2^2+\frac{1}{4}{y}_3^2\right)r+{y}_1\left({y}_1^2+\frac{3}{4}{y}_2^2+\frac{3}{4}{y}_3^2\right)\right)\left({P}_{12}{y}_1+{P}_{22}{y}_2+{P}_{32}{y}_3\right){y}_1^2}{r^5{\left(r+{y}_1\right)}^3\pi },\\ {}{\varepsilon}_{12}^y=-\frac{3{y}_1{y}_2\left({P}_{12}{y}_1+{P}_{22}{y}_2+{P}_{32}{y}_3\right)\left({ry}_1+{y}_1^2+\frac{1}{2}{y}_2^2+\frac{1}{2}{y}_3^2\right)\left(1+v\right)}{r^5{\left(r+{y}_1\right)}^2 E\pi},\\ {}{\sigma}_{12}^y=-\frac{3}{2}\frac{y_1{y}_2\left({P}_{12}{y}_1+{P}_{22}{y}_2+{P}_{32}{y}_3\right)\left(2{ry}_1+2{y}_1^2+{y}_2^2+{y}_3^2\right)}{r^5{\left(r+{y}_1\right)}^2\pi }.\end{array}} $$

(18)

Let us break *S*_{2} surface of cavity into *n* equal parts \( {S}_2^{n\left(k,l\right)} \)^{2.}^{Footnote 1} We take that *P*_{n2} forces are distributed uniformly, then forces *P*_{n2}/*n* act on each \( {S}_2^n \) part. In the center of each \( {S}_2^n \) part, there is a coordinate system *y*_{i}(*n*) and deformation under action of *P*_{n2}/*n* forces in the system has the form of the expression (17). We can make similar expressions for the rest of the cavity surface.

Then at some point, *A* around cavity stresses can be represented as a sum of stresses from concentrated force **F** and forces acting on cavity surface:

$$ {\sigma}_{ij}^A={\sigma}_{ij}+\sum \limits_1^6\sum \limits_{i=1}^n{\sigma}_{ij}^{y(n)} $$

(19)

However, depending on reference point position regarding cavity, only some members are to be considered in the double sum of expression. It is obvious that some cavity faces and corresponding forces are separated from the reference point by cavity space and their action can be neglected. Following in a similar way, we can set additive series for any point around cavity.