Cylindrical Coordinate System Research Articles

RESEARCH ON ANALYTICAL METHOD OF CIRCULAR CYLINDRICAL SCATTERED WAVE PRESSURE SUBJECTED TO UNDERWATER EXPLOSION

In order to improve the anti-blast and protection design of offshore structures such as bridges and wharves, this paper proposes an efficient analytical method for calculating the circular cylindrical scattered wave pressure subjected to underwater explosion, aiming at the common cylindrical structures with constant cross-sections in offshore structures. Based on the theory of diffracted waves, this paper derives the time-domain solution of the scattered wave pressure of the cylindrical structure by the method of separation of variables in the cylindrical coordinate system according to the water body state equation and boundary conditions; The calculation accuracy of the scattered wave pressure time-domain solution proposed in this paper is verified by numerical simulation comparison. The numerical example shows that the calculation errors of the two methods are within the engineering allowable range. Based on the time-domain substructure analysis method that considers the compressibility of water, by comparing the analytical methods proposed in this paper without considering the compressibility of water, the influence of water compressibility on scattered wave pressure is analyzed.

Accurate coupled vibration analysis of a piezoelectric ceramic cylinder by the superposition method

The radial and thickness extensional vibration modes in piezoelectric cylinders are always inevitably coupled due to the finite dimension, Poisson’s ratio, and piezoelectric effect. In this paper, an analytical model based on the superposition method is developed to obtain the coupled dynamic response of a piezoelectric cylinder under an applied voltage. The problem can be described mathematically by three partial differential equations with mixed boundary conditions in the cylindrical coordinates system. To solve this, the problem is decomposed first into two building block – vibrations in radial and thickness directions. In each building block, the expressions of displacements and electric potential are assumed and then the induced dynamic responses, such as in-plane stress and electric displacements, are calculated. Finally, the vibration responses of the two building blocks are superimposed to satisfy the mixed boundary conditions using Fourier and Fourier-Bessel series expansions. Electrical impedance of a typical piezoelectric disk and frequency spectrum of piezoelectric cylinders of different diameter-to-thickness ratios are calculated by the present analytical method as well as by finite element method. Comparison shows an excellent agreement. This analytical model can be applied to material characterization and the design and the optimization of the active elements of piezoelectric devices.

About one method of calculation in the arbitrary curvilinear basis of the Laplace operator and curl from the vector function

Abstract A simple algorithm for calculating Christoffel symbols, a covariant projection of the result of the Laplace operator's action on the vector, vector curl and other similar operations in an arbitrary oblique base are proposed. For an arbitrary base with ortho ei is found the expressions of vector projections (ΔA) i and (rot A) i , where A is a counter variant vector. Examples of orthonormal bases are considered and general expressions for (ΔA) i and (rot A) i for the bases are also given. As a demonstration of the working capacity of the common formulas obtained, detailed calculations of (ΔA) i and (rot A) i as an example are made in cases of spherical and cylindrical coordinate systems.

Solving a one mixed problem in elasticity theory for half-space with a cylindrical cavity by the generalized Fourier method

Designing and constructing underground structures for various purposes, such as tunnels, mines, mine workings, necessitate the development of procedures for calculating their strength and reliability. The physical model of such objects worth considering is a homogeneous isotropic half-space that contains an infinitely long hollow cylinder, located parallel to its border. One can explore problems related to the mechanics of deformable solids for such a multiply connected body. This paper reports the proofs of addition theorems for the basic solutions to the Lamé equation regarding the half-space and cylinder recorded, respectively, in the Cartesian and cylindrical coordinate systems. This result is important from a theoretical point of view in order to substantiate a numerical-analytical method ‒ the generalized Fourier method. This method makes it possible to solve spatial boundary problems from the theory of elasticity and thermo-elasticity for isotropic and transversal-isotropic multiply connected bodies. Similar to the classical Fourier method, the general solutions to equilibrium equations have been used here but in several coordinate systems rather than one. From a practical point of view, this method has made it possible to investigate the combined problem of elasticity theory regarding the multiply-connected body described above. The analysis of the stressed-strained state of this elastic body has made it possible to draw conclusions on determining those regions that are most vulnerable to destruction. The highest values are accepted by normal stresses in the region between the boundaries of the half-space and the cylinder. Changing the σy component along the Ox axis corresponds to the displacements assigned on the half-space. The τxy component contributes less to the distribution of stresses than σx and σy. The applied aspect of using the reported results is the possibility to apply them when designing underground structures

A unified and rigorous solution for quasi-static cylindrical cavity expansion in plasticity constitutive models

This paper presents a unified and rigorous solution for the quasi-static cylindrical cavity expansion problem in all existing plasticity constitutive models. Rigorous three-dimensional definitions of p’ and q are used as proposed by Chen & Abousleiman. Then, the p-q stress space is transformed to the three stress components in a cylindrical coordinate system. The governing partial differential equations (PDEs) for cylindrical cavity expansion, including the stress equilibrium equation, the constitutive equation, the consistency equation, the continuity equation, and the drainage conditions, are written with respect to the three stress components in a cylindrical coordinate system. The PDEs are subsequently reduced to ordinary differential equations (ODEs), which can be summarized in unified matrix form by means of a similarity transformation. A rigorous semi-analytical solution can immediately be obtained by numerically solving the ODEs using commercial ODE solvers, such as MATLAB. The proposed solution procedure is general and can be applied to both the drained and undrained conditions in all plasticity constitutive models. Finally, the partially drained effect was discussed through numerical analysis, and empirical equations considering the partially drained effect were proposed for calculating the limit effective radial stress and excess pore pressure at the cavity wall.

Efficient coupling of down-converted photon pairs into single mode fiber

We present the analysis of the down-conversion emission profile from a critically phase-matched type-I nonlinear crystal. The emission profile of the down-conversion modes shows asymmetry in cylindrical coordinate system due to the birefringent effects of the nonlinear crystal, i.e. pump walk-off. This effect leads to reduced collection of the pairs from the source to the single mode fiber, which limits the brightness as a function of the crystal length. We propose a method to improve the symmetry of the down-conversion modes by inserting a lens off-centered with respect to the optical axis. Our theoretical and experimental results show that the asymmetry of the system increases with the crystal length by L0.5, and the brightness is proportional to crystal length by L0.72 and L0.91 for uncorrected and corrected down conversion mode profiles, respectively. The proposed method allows more efficient use of photon pair sources in terms of brightness which is to be used in quantum optics applications.

Large deformation analysis of spontaneous twist and contraction in nematic elastomer fibers with helical director

A cylindrical rubber fiber subject to a twist will also elongate: a manifestation of Poynting’s effect in large strain elasticity. Here, we construct an analogous treatment for an active rubber fiber actuated via an axisymmetric pattern of spontaneous distortion. We start by constructing an exact large-deformation solution to the equations of elasticity for such fiber subject to imposed twist and stretch, which reveals spontaneous warping and twisting of the fiber cross section absent in passive rubbers. We then compute the corresponding non-linear elastic energy, which encompasses the Poynting effect but is minimized by a finite spontaneous twist and stretch. In the second half of the paper, we apply these results to understand the twist-contraction actuation of nematic elastomer fibers fabricated with director fields that encode helical patterns of contraction on heating. We first consider patterns making a constant angle with respect to the local cylindrical coordinate system (conical spiral director curves) and verify the predicted spontaneous twist, contraction, and cross-section deformation via finite elements. Second, we consider realistic director distributions for the experimentally reported fibers fabricated by cross-linking while simultaneously applying stretch and twist. Counterintuitively, we find that the maximum actuation twist is produced by applying a finite optimal twist during fabrication. Finally, we illustrate that spontaneously twisting fibers will coil into spring-like shapes on actuation if the ends are prevented from twisting relative to each other. Such a twist–torsion coupling would allow us to make a tendril-like “soft-spring” actuator with low force and high linear stroke compared to the intrinsic contraction of the elastomer itself.

BOOSTING SEGMENTATION ACCURACY OF THE DEEP LEARNING MODELS BASED ON THE SYNTHETIC DATA GENERATION

Abstract. In the era of data-driven machine learning algorithms, data represents a new oil. The application of machine learning algorithms shows they need large heterogeneous datasets that crucially are correctly labeled. However, data collection and its labeling are time-consuming and labor-intensive processes. A particular task we solve using machine learning is related to the segmentation of medical devices in echocardiographic images during minimally invasive surgery. However, the lack of data motivated us to develop an algorithm generating synthetic samples based on real datasets. The concept of this algorithm is to place a medical device (catheter) in an empty cavity of an anatomical structure, for example, in a heart chamber, and then transform it. To create random transformations of the catheter, the algorithm uses a coordinate system that uniquely identifies each point regardless of the bend and the shape of the object. It is proposed to take a cylindrical coordinate system as a basis, modifying it by replacing the Z-axis with a spline along which the h-coordinate is measured. Having used the proposed algorithm, we generated new images with the catheter inserted into different heart cavities while varying its location and shape. Afterward, we compared the results of deep neural networks trained on the datasets comprised of real and synthetic data. The network trained on both real and synthetic datasets performed more accurate segmentation than the model trained only on real data. For instance, modified U-net trained on combined datasets performed segmentation with the Dice similarity coefficient of 92.6±2.2%, while the same model trained only on real samples achieved the level of 86.5±3.6%. Using a synthetic dataset allowed decreasing the accuracy spread and improving the generalization of the model. It is worth noting that the proposed algorithm allows reducing subjectivity, minimizing the labeling routine, increasing the number of samples, and improving the heterogeneity.

The study for maritime radar coverage based on cylindrical coordinate parabolic equationmethod

In order to predict the omnidirectional propagation characteristics of radar waves over the sea surface, a radio wave propagation model based on cylindrical coordinate parabolic equation is proposed and used to study the coverage range of marine radar. In this paper, the forward parabolic equation in cylindrical coordinate system is derived from Helmholtz Equation. By using the orthogonality of trigonometric functions in the general solution of the parabolic equation, the excitation coefficients of each wave propagation modes are solved, and the stepwise iterative algorithm of the cylindrical coordinate parabolic equation is realized by combining the Split Step Fourier method, to predict the propagation characteristics of the radio waves in space. Numerical examples show that the method proposed in this paper can overcome the azimuth limitation of the traditional three-dimensional cartesian coordinate parabolic equation method, which propagates about 30° along the axis, and realize the omnidirectional calculation. Compared with quasi-three dimensional parabolic equation, it has higher calculation accuracy. Based on the proposed method, the influences of atmospheric waveguide and rough sea surface on radar detection performance are studied, the simulation results have important significance for constructing digital battlefield at sea.

The surface effects on solid and hollow nanowires under diametral loading

The size effects of energetic surfaces on the elastic behavior of solid and hollow nanowires are investigated under diametral loading for the first time. In order to model the diametral compression test on nanowires, the structural version of the locally-exact theory is developed for solid and hollow nanowires (LETNW) by considering energetic surfaces attached at the boundaries of bulk materials and modeled using the Gurtin-Murdoch model. The developed analytical solutions employ Fourier series representations of displacement fields in the cylindrical coordinate system that exactly satisfy the governing differential equations, facilitating rapid convergence of full-field stress distributions. The solution's accuracy is validated against the in-house finite-element (FE) program which is currently the computational standard, where excellent agreement is obtained. The surface effects on the local stresses are then illustrated by varying the surface parameters, loading angle, dimension of nanowires, as well as residual surface stress. The efficiency of the LETNW is attributed to its analytical nature and thus consumes much less execution time relative to the FE implementation. What's more, instabilities associated with the solution of stress fields are observed with surface properties in certain crystallographic orientations for both LETNW and FE approaches. Contrary to the LETNW technique, the bifurcation phenomenon is shown to depend on the finite-element mesh refinement. The locally-exact theory provides an alternative and independent way of identifying surface bifurcations in solid and hollow nanowires and hence is a good tool in assessing the finite-element method as well as other approaches’ predictive capability for this class of materials.

An approach based on the porous media model for numerical simulation of 3D finned-tubes heat exchanger

Developing efficient and compact heat exchangers is of important significance to improve the energy utilization efficiency of entire society. 3D finned-tubes, as a kind of passive enhanced heat exchange technology, are widely used for improving the thermal-hydraulic performance of the heat exchanger. In the numerical study of 3D finned-tube heat exchanger, on account of the limit by computing capacity, it is very hard to carry out the full-size numerical simulation, so the method based on porous media model becomes a common approach for numerical simulation of the heat exchanger. In the present study with the discrete 3D finned-tube heat exchanger as the study object, a kind of approach based on the porous media model under the cylindrical coordinate system is proposed. In this approach, only the fin region on the surface of base tube is simplified to the annular porous media zone while the region beyond the fin tips is still regarded as the pure gas flow region. The resistance coefficients of annular porous media, which are used to represent the retardation effects of the fin on the fluid motion, are given under the cylindrical coordinate system. Based on serialized numerical simulations of the fin configuration model seen by the air when it flows in the circumferential, axial and radial direction, the viscous resistance coefficients and inertial resistance coefficients of the annular porous media zone in each direction are obtained through fitting. Through comparison of simulated values between the porous media model and 3D finned-tube practical physical model, it is found that the pressure drop calculated by the method based on porous media model proposed in this study agree well with that based on the real physical model. The pressure drop predicted by the porous media model also agree well with the experimental results. In the approach based on porous media model, the grid number and calculation cost are smaller than those in practical physical model. Especially in case of tube bundles, number of grids applied in the simulation based on porous media model is far less than the grid quantity in the practical physical model simulation, but the pressure drop obtained by the porous media model is almost equal to the result obtained from the practical physical model. Besides, in the present method, only the fin zone is deemed as the porous media, so the flow field in the region beyond the fin tips will still be solved as same as that in the full-size simulation. Thus more flow details within the heat exchanger can be captured. The approach based on porous media model under the cylindrical coordinate system proposed can be used in the simulation study of the internal flow field of 3D finned-tube heat exchangers in actual scale. On this basis, this approach can contribute to optimization design of the 3D finned-tube heat exchanger.

Variable seismic motions of P-wave scattering by a layered V-shaped canyon of the second stratification type

Seismic waves scattering especially waves SH by different topographies has been widely studied in the field of earthquake engineering. However, most of these studies mainly focus on the topographies with the ideal assumption (elastic, isotropic and homogeneous) of uniform medium or the first stratification type of canyon (canyon that is not crossing by layer-interfaces). Compared with SH wave scattering, there exists a challenge to derive the theoretical solution for the scattering of P or SV wave especially by relative complicated canyon with non-uniform medium, which is defined as the second stratification type or type II here. The paper focuses on the study on the two-dimensional scattering of plane P waves by a symmetrical V-shaped canyon embedded in a two-layered elastic half-plane of type II. The challenge is to deal with three issues simultaneously, namely satisfaction of boundary conditions in the vertical direction, more complex reflections and more coordinate system transformations. Here more cylindrical coordinate systems and the skew coordinate transformation formulas need to be defined and used so as to satisfy the boundary conditions along the canyon surface, in which coordinate transformation and its solution process become more complex. The derived solution of symmetrical V-shaped canyon is rigorous theoretically and reliable, and further demonstrate that it is affected directly by the second layered type of symmetrical V-shaped canyon. Moreover, the effects of the layer medium, the incident frequencies and incident angles on the ground motion also are investigated by the double-layer considering nonhomogeneous cases. This study shows that inhomogeneous-material, frequencies of incident waves and the existence of the canyon result in conspicuous even non-ignorable effect on variability of surface ground motions.

High-resolution numerical simulation of detonation diffraction

<p indent=0mm>High-resolution scheme presents an important application value in discontinuous problems, such as in explosive detonation with high pressure. In this study, a high-resolution Eulerian reaction hydrodynamic code is written based on a two-dimensional axisymmetry in a cylindrical coordinate system. The Level-set method is used to track the motion interface between the explosive/reaction product and inert medium. The entropy-based CREST model and pressure-based ignition and growth model are both used to calculate the detonation diffraction at the corner turning in the TATB-based explosive. A multi-criterion reaction-rate model, which combines the two aforementioned reaction models, is also proposed. The results show that the multi-criterion reaction model can directly calculate the dead zone without extra desensitization modification, and the height of the dead zone is basically consistent with that in the experiment.

СРАВНЕНИЕ РЕШЕНИЙ ЗАДАЧ ИЗГИБА ТРЕХСЛОЙНЫХ ПЛАСТИН НА ОСНОВАНИЯХ ВИНКЛЕРА И ПАСТЕРНАКА

Solutions of problems on axisymmetric bending of an elastic three-layer circular plate on the Winkler and Pasternak foundations are given. The bearing layers are taken as isotropic, for which Kirchhoff’s hypotheses are fulfilled. In a sufficiently thick lightweight, incompressible in thickness aggregate, the Timoshenko model is valid. The cylindrical coordinate system, in which the statements and solutions of boundary value problems are carried out, is connected with the median plane of the filler. On the plate contour, it is assumed that there is a rigid diaphragm that prevents the relative shear of the layers. The system of differential equations of equilibrium is obtained by the variational method. Three types of boundary conditions are formulated. One- and two-parameter Winkler and Pasternak models are used to describe the reaction of an elastic foundation. The solution to the boundary value problem is reduced to finding three desired functions, plate deflection, shear, and radial displacement in the filler. The general analytical solution to the boundary value problem is written out in the case of the Pasternak model in Bessel functions. At the Winkler foundation, the known solution is given in Kelvin functions. A numerical comparison of the displacements and stresses obtained by both models with a uniformly distributed load and rigid sealing of the plate contour is carried out.

Damage detection in a 3D wind turbine tower by using extensive multilevel 2D wavelet decomposition and heat map, including soil-structure interaction

The wind-turbine tower is the structure’s supporting pole and accounts for 15% of the total cost. Damage to it may cause the structure’s catastrophic failure. Advances have been made to conditional monitoring systems for the blades and mechanical components inside the nacelle, while less attention has been given to the tower’s health-monitoring efforts. Moreover, the interaction between soil and structure may be significant. This article describes and numerically demonstrates the multi-level 2D wavelet decomposition’s extensive analysis to identify a damaged location in a wind-turbine tower with and without soil-structure interaction. The finite-element model based on the NREL 5-MW reference onshore wind turbine is established and verified. This method uses the 3D mode shape to detect the tower’s damaged location. A novel algorithm was introduced to accelerate and facilitate the damage finite element modeling process of wind turbine towers. An application code was provided to determine the correct 3D directions of the turbine tower meshes. In order to perform extensive calculations, a new algorithm named “Automatic Wavelet Analysis Aid” used to the extraction and automation of the whole wavelets families and their wavelets. Additionally, the conversion of the cylindrical coordinate system to the Cartesian coordinate system was used, and the relationship between the detection of damage accuracy for this proposed transformation is presented. Finally, the heat map illustrations of data and envelope curves are used as a roadmap for damage detection.

Optimizing temperature-dependent molar volume fraction of biodiesel fuel in a pseudo-binary mixture through Bragg fiber waveguide sensor having defect layer

In the present communication, the temperature-dependent molar volume fraction of various kinds of renewable biodiesel fuel resources in a pseudo-binary mixture is theoretically optimized through the Bragg fiber waveguide (BFW) sensor with a geometrical defect in periodic cylindrical Bragg reflectors. BFW structure is modelled using a transfer matrix method (TMM) and Henkel formalism in a cylindrical coordinate system. The temperature dependence molar volume fraction is linked to the change in the refractive index of the pseudo-binary mixture, which is further predicted using the capacity of various models; Lorentz-Lorenz, Dale-Gladstone, Eykman, Newton, and Kay. In the presence of a geometric defect layer, a sharp transmission peak of 0.1nmfull width at half maxima (FWHM) is obtained in the considered photonic band gap (PBG), which is sensitive to the change in core refractive index. The temperature-dependent (due to different weather conditions) maximum sensitivity of the proposed sensor is found to be 1280 nm/RIU, which is further compared with the phoxonic crystal-based static temperature sensor having sensitivity 142 nm/RIU only. Along with improved sensitivity in the proposed sensor, other sensing performance parameters detection accuracy and quality parameter (which are inversely proportional to the FWHM) are also improved due to least FWHM of resonant transmission peak about 0.1 nm comparatively.

ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ РАВНОВЕСИЯ ИДЕАЛЬНО УПРУГОПЛАСТИЧЕСКОЙ СПЛОШНОЙ СРЕДЫ ДЛЯ ПЛОСКОЙ ДЕФОРМАЦИИ В ЦИЛИНДРИЧЕСКИХ КООРДИНАТАХ ПРИ БИЛИНЕЙНОЙ АППРОКСИМАЦИИ ЗАМЫКАЮЩИХ УРАВНЕНИЙ

Abstract. The article considers the construction of differential equations of equilibrium in displacements for plane deformation of elastic perfectly plastic regarding shear deformations continuous medium and nonlinearly elastic continuous medium with respect to volumetric deformations with bilinear approximation of the closing equations, both regarding and regardless geometrical nonlinearity in a cylindrical coordinate system. Nonlinear diagrams of volumetric and shear deformation are approximated by bilinear functions. Proceeding from the assumption of independence, generally speaking, of volume and shear deformation from each other, five main cases of physical dependencies are considered, depending on the relative position of the break points of bilinear diagrams of volume and shear deformation. The construction of bilinear physical dependencies is based on the calculation of the secant moduli of volumetric and shear deformation. In this case, in the first section of the diagrams, the secant modulus of both volumetric and shear deformation is constant, while in the second section of the diagrams, the secant modulus of volumetric deformation is a function of volumetric deformation, and the secant shear modulus is a function of the intensity of shear deformations. Substituting the corresponding bilinear physical relations into the differential equations of equilibrium of a continuous medium, written both regardless and regarding geometrical nonlinearity, the resolving differential equations of equilibrium in displacements for plane deformation in a cylindrical coordinate system are received. The received differential equations of equilibrium in displacements in cylindrical coordinates can be applied in determining the stress-strain state of elastic perfectly plastic with respect to shear deformations continuous medium and nonlinearly elastic with respect to volumetric deformations continuous medium under conditions of plane deformation, both regarding and regardless geometrical nonlinearity, physical relations for which are approximated by bilinear functions.

Determining track-induced lateral thermal expansion forces on a curved railway track

This research studies the development of lateral thermal expansion forces on a curved railway track. The geometric alignment of a railway right of way often requires railway tracks to be curved. This curvature which is usually defined by the radius of curvature or degree of curvature represents a higher level of complexity in the track’s analysis and design process. Particularly, presence of curvature on the track introduces multiple sources of force in the lateral (radial) direction, including, but not limited to, lateral thermal expansion, lateral wheel/rail forces due to centrifugal action, lateral components of vertical loads, bogie hunting and nosing effects of locomotives, and vehicle curving dynamics. Some of these forces are well understood such as centrifugal forces while some are not as well understood, such as lateral thermal expansion forces. To bridge this gap, this research studies the development of track-induced lateral thermal expansion forces on a curved railway track. In this research, the curved track is assumed to be an arbitrary arc section of a circular track and is modeled as an equivalent idealized circular ring for analysis. Owing to its importance, three analytical methods are used to include: 1) Timoshenko thermoelastic stress analysis in cylindrical coordinate system, 2) mechanics of thin wall cylinders and 3) adaptation of a variational calculus formulation method from a previous comparable study. A fourth analysis approach is also introduced using a commercially available finite element analysis package. The results of these analyses are compared through a wide range of parametric studies and are then validated by the finite element analysis. The results of this study showed that the several methods presented in this paper, could be used to approximate thermally induced expansion behavior (pre-buckling) on a curved railway track. While all three techniques are effective, the Timoshenko stress analysis method appears to be the most suitable as it is a direct method that examines the stress build up from the element level and takes into account additional material properties, such as the Poisson effect. The research resulted in a methodology for determining load transfer from thermally induced forces in curved railroad track to the fastener and supporting structure.

FREE TORSIONAL VIBRATIONS OF A ROUND ELASTIC CONE-SHAPED ROD

t. The article deals with the problem of torsional vibrations of circular elastic conical rods in the system of cylindrical coordinates. It is believed that vibrations of the rod are caused by external dynamic loads applied to its surface. The cross-sectional radius is taken as a linear function of the longitudinal coordinates. On the basis of the equation of motion of the theory of elasticity for torsional vibrations, the equation of torsional vibrations of a conical rod with a circular cross-section is derived using the Fouret and Laplace integral substitutions. On the basis of the obtained equation, the problem of harmonic torsional vibrations of a rod is solved. The frequency equation and the relationship between frequency and number of waves are given. Based on the obtained numerical results, appropriate conclusions were drawn

STRESS-STREIN STATE OF THE ROCK MASS AROUND THE VERTICAL MINE

The article deals with the problem of the stress-strain state of a rock mass around a vertical working of a circular cross-section. An exact formulation of the three-dimensional problem of the deformation of a half-space weakened by a deep cylindrical cavity is used. The stress-strain state of a half-space, as a three-dimensional body, strictly obeys the basic requirements of the three-dimensional linear theory of elasticity and is described by its corresponding equations and relations in a cylindrical coordinate system. The specific problem of rock mechanics has been solved, i.e. the considered rock mass works only in compression. The deformation process and stress state around vertical shafts of circular cross-section are expressed in terms of stress functions. Calculation formulas are derived for all nonzero components of the strain and stress tensors, taking into account the axisymmetry of the problem under consideration, represented in terms of stress functions.