Displacement Functions Research Articles

Modeling and experiments on the vibro-acoustic analysis of ring stiffened cylindrical shells with internal bulkheads: A comparative study

The vibro-acoustic response of ring stiffened cylindrical shells with internal bulkheads under forced excitation is presented. The numerical analysis model is established using the Jacobi Ritz-Boundary element method. The first-order shear deformation theory, multi-segment technique and artificial spring technology are applied to establish the theoretical model, and the Jacobi orthogonal polynomials are introduced to represent the displacement functions. The Newmark-β integration method is used to obtain the vibration response of the structure, and the time-domain Kirchhoff boundary integral formulation is applied to describe the exterior acoustic field. The vibration and sound radiation of the ring stiffened cylindrical shell under impact and narrow band random loading were measured. The comparative study reveals that the results obtained from the proposed method agree well with the experimental results. The stiffeners and bulkheads have a significant effect on the vibration characteristics of the structure. Additionally, some physical insights into the resonant peaks and sound pressure directivity of the cylindrical shells are provided.

The modified in-plane rigid-elastic-coupled tire modal model: dynamic response to short wavelength road profiles

The 400-degree-of-freedom deformed in-plane rigid-elastic-coupled tire model was enhanced by incorporating the vertical movement of the wheel as a rigid body. This refined model was then utilised to build the tire modal model. The tire modal model was developed to derive the transfer function (TF) covering a range of 0–300 Hz, which connects road irregularities as input to the contact patch and the wheel's vertical motion as output. The TF was operated under a quasi-static condition to obtain the tire's enveloping characteristics rolling over short wavelength obstacles as a direct function of vertical wheel displacement under varying contact patch length constraints. The tire modal model was combined with the quarter car model to analyse the vehicle's response when passing through short wavelength obstacles and random road profiles. This analysis was performed under varying load conditions represented by the contact patch length. The model and methodology demonstrated their effectiveness in handling rolling tires over short-wavelength obstacles and random road profiles.

Why do hybrid nanoparticles perform better in improving heat transfer performance of lubricating oil during lubrication process: An experiment and molecular dynamics simulation

Lubricants are indispensable ingredients for alleviating friction and wear in mechanical components especially in automobile. However, due to its poor heat transfer capabilities, traditional lubricants are denatured under extreme working conditions, leading to lubrication failure during the friction process. Therefore, this study elaborately put forward to a novel MoS2/h-BN hybrid nanoparticle as efficient nano-additive for engine oil. The results of experiments and molecular dynamics simulation jointly show that compared with mono nanoparticles, hybrid nano-lubricants (NL) thermal conductivity was significantly improved than that of base oil owing to the synergistic effect. The corresponding mechanism for the improving heat transfer performance was suitably proposed based on the Mean Square Displacement (MSD) and Radial Distribution Function (RDF) results at the microscopic atomic level.

Nonlocal nonlinear higher-order finite element model for static bending of curved nanobeams including graphene platelets reinforcement

The aim of this work is concerned with the nonlocal nonlinear bending characteristics of non-classical isotropic and composite curved beams based on graphene platelets reinforcement, viz., nano size/micro beams. The size-dependant effect prominent in micro/nanobeams is incorporated in the formulation based on the nonlocal theory using size-dependent continuum approach. The proposed structural model includes both moderately large displacement and rotations, respectively. The system governing equations are established in terms of displacement functions using the principle of virtual work statements. A three-noded shear deformable curved beam finite element using a higher-order sinusoidal function based beam theory is employed; the solutions are obtained by adopting direct iterative procedure. The efficacy of the model is tested against the analytical studies available for the nano-size beams. A detailed analysis is made to highlight the influence of various design variables like thickness ratio, curvature of beam, nonlocal parameter, and the support conditions on the nonlinear bending behavior of curved nano/micro-size beams.

The Vibro-Acoustic Characteristics Analysis of the Coupled System between Composite Laminated Rotationally Stiffened Plate and Acoustic Cavities

In order to study vibro-acoustic characteristics between composite laminated rotationally stiffened plate and acoustic cavities in the coupled system, first-order shear deformation theory (FSDT) and modified Fourier series are used to construct a unified analysis model. The involved coupled systems primarily encompass three types: the coupled system between composite laminated rotationally stiffened plate and cylindrical-cylindrical cavities, spherical-cylindrical cavities, and conical-cylindrical cavities. First, the first-order shear deformation theory and the modified Fourier series are applied to construct the allowable displacement function of the composite laminated rotationally stiffened plate and the allowable sound pressure function of the acoustic cavities. Second, the energy functionals for the structural domain and the acoustic field domain are established, respectively. According to the continuity condition of the particle vibration velocity at the coupling boundary between the composite, laminated cylindrical shell and the enclosed cavity, the coupling potential energy between the stiffened plate and two acoustic cavities is introduced to obtain the energy functional of the coupled system. Third, the Rayleigh-Ritz method is utilized to solve the energy functional and, when combined with artificial virtual spring technology, the suggested theory may be used to study the vibro-acoustic characteristics of a coupled system with arbitrary elastic boundary conditions. Finally, based on validating the fast convergence and correctness of the model, this paper will analyze the impact of crucial parameters on vibro-acoustic characteristics. Furthermore, by incorporating internal point forces and point-sound source stimulation, a steady-state response analysis of the coupled system will be conducted. This research can give a theoretical foundation for the vibration and noise reduction of a vibro-acoustic coupling system.

Layering Ni to mediate the Kirkendall effect (KE) in cu/ta nonequilibrium diffusion for enhanced abrasion resistance

This study delves into the KE occurring at the interface of the Cu/Ta system during thermal diffusion, along with the mediating impact of integrating a Ni interlayer on the nonequilibrium diffusion of Cu/Ta. In the Cu/Ta system, nonequilibrium thermal diffusion results in the generation and aggregation of Cu side Kirkendall voids (KVs). In the Cu/Ni/Ta system, the Ni interlayer ensures the equilibrium of system atom migration through the movement and annihilation of vacancies, thereby enhancing the bonding between the layers. Calculations of interdiffusion coefficients using the Boltzmann-Matano (B-M) method reveal that interdiffusion coefficients at the coating-substrate interface are 4–5 orders of magnitude higher for the Cu/Ta system compared to the Cu/Ni/Ta system. The analysis of mean square displacement (MSD) and radial distribution function (RDF) provides insights into the role of the Ni interlayer as a potential barrier mediating the atomic-level diffusion behavior of the system. Moreover, the Cu/Ni coating exhibits superior abrasion resistance and bond strength compared to the Cu coating, attributed to the inhibition of KVs generation at the interface. This exploration introduces a novel approach to inhibit defects stemming from the KE and advances comprehension of interfacial phenomena in composites.

Investigations on the dynamic snap-through of MFC bonded self-resetting bistable laminates

Unsymmetric composite laminates having two stable equilibrium configurations have been studied extensively in the recent past due to their potential applications in morphing structures. Surface bonded Macro Fiber Composites (MFC) actuators have been considered as a viable solution to trigger the snap-through transition in bistable laminates. Although MFC bonded bistable laminates are widely used in morphing applications, they might require considerably high voltage inputs to achieve the required levels of actuation control during the shape transition. As a solution, other possible energy sources can be combined with active MFC patches to reduce the snap-through energy required from the single source of MFC actuation. In this work, we examine the dynamic behavior of bistable composite plates actuated using MFC actuators, where external vibration energy has been used to assist with the MFC-controlled actuation between stable states. A refined semi-analytical model based on the Rayleigh–Ritz formulation has been proposed, where the membrane energy and the bending energy are separately evaluated. Bending components are directly evaluated using the approximated transverse displacement functions, whereas the membrane components are evaluated separately by combining compatibility conditions and equilibrium equations. Results from the proposed semi-analytical framework are compared with a full geometrically nonlinear finite element framework and necessary experimental observations. The results show a significant reduction in the snap-through energy demand on MFC layers where external dynamic excitation assists the snap-through process. Additionally, a parametric study is performed using variable stiffness (VS) fiber orientation parameters, achieving bistable laminate-MFC configurations that lower snap-through requirements through the proposed morphing strategy. Thus, the study offers to aid a multi-efficient snap-through strategy for the morphing of multistable composite structures.

Statistical analysis of coupled oscillations with a single fixed endpoint and its application to carbon nanomaterials

Squared deviations from the equilibrium positions of one-dimensional coupled harmonic oscillators with fixed and free endpoints are calculated, and the time averages are expressed as a function of the initial displacements and velocities. Furthermore, we consider the averages of squared deviations over an ensemble of initial displacements and velocities, which distribute based on a product of the same distribution functions with variances of the initial coordinates σx2 and velocities σv2 , respectively. We demonstrate that the mean squared deviation linearly increases as the oscillator separates further from the fixed endpoint because of the asymmetrical boundary conditions, and that the increase rate depends only on σv2 and not on σx2 . This simple statistical property of harmonic oscillation is similarly observed in the oscillations of a graphene sheet and carbon nanotube in molecular dynamics simulations, in which the interacting forces are nonlinear.

Elastodynamic Green's functions for sandwich panels with aluminum foam core and transversely isotropic face sheets using potential functions method

Green's functions of stress and displacement due to wave propagation in a sandwich panel consisting of transversely isotropic face-sheets and aluminum foam core are developed in the present paper. Dynamic partial differential equations of equilibrium for the considered face-sheets and core are expressed in cylindrical coordinates (r, θ, z), and converted into two separate equations using two unknown potential functions. Then, by writing the potential functions as Fourier series in the circumferential direction and using Hankel transform in the radial direction, an analytical solution to the potential functions in Hankel transformed space is developed. Using the boundary conditions and continuity of the problem as well as the place of application of harmonic forces, Green's functions for displacements and stresses in the frequency domain are derived using Hankel's inverse integral transform. These functions are expressed as complex integrals and calculated numerically. To verify the methodology and results, a comparison study is performed for particular cases showing a high level of the accuracy. Green's functions obtained from this paper will give information about the loading models or materials causing the maximum normal stress, shear stress, vertical and radial displacements. In addition, based on the results, with an increase in the core thickness compared to that of face-sheets in a Sandwich panel, an increment in Green's functions of stress and displacement is observed in both real and imaginary parts.

Approximate contact force model for viscoelastic materials based on generalized Maxwell model

This paper proposes a novel approximate model for estimating the normal contact force between a rigid sphere and a viscoelastic half-space based on the generalized Maxwell model. The relaxation function derived from the generalized Maxwell model is applied to the solution to the elastic contact problem based on the elastic–viscoelastic correspondence to obtain an integral expression of pressure for viscoelastic contact. Using the approximate method on the basis of Taylor series, the integral expression is translated into an explicit function of displacement, velocity, time and material properties. A normal contact force model is established by integrating the pressure over contact area. A relaxation test of Nafion® XL is conducted to capture the viscoelastic parameters, and these parameters are employed to simulate the contact behaviors using the finite element method (FEM) to determine the scale factor in the contact force model. In order to verify the validity of the approximate model, the outcomes obtained from the model are compared with those generated by the FEM. A parametric study is conducted to analyze the effects of variations in parameters on normal contact behaviors. The results indicate that the established model is able to predict the contact responses of the viscoelastic materials with complex mechanical characteristics.

Investigating the Sommerfeld effect in the rotating shaft with internal and external damping using the Timoshenko beam theory

The Sommerfeld effect is a destructive phenomenon in rotating systems with a non-ideal electrical source, which causes instability in the system by applying a dynamic jump around critical speeds. In this article, the Sommerfeld effect has been investigated for the first time using the Timoshenko beam theory for an eccentric continuous shaft with internal and external damping. After deriving the governing equations and finding displacement functions using the semi-analytical method, the Sommerfeld effect near the critical speeds is detected using the instantaneous power balance method. As confirmation of the correctness of the derived relations, it has been shown that for thin shafts, there is a good consistency between the results obtained from the Euler–Bernoulli and Timoshenko theories in the early modes. However, it was observed that at higher critical speeds, the jump amplitude decreases, and the unstable speed range increases significantly, so the probability of entering the vicinity of the instability range in the next mode is not unexpected. Since no effect has been ignored in this study, the dynamic analysis of the Sommerfeld jump in thick shafts is also possible. Despite the common belief that Timoshenko beam theory is only considered suitable for studying thick shafts, it has been shown that the effects of shear deformation are significant in high-speed systems, even for non-thick shafts, and regardless of them in higher modes, it causes a calculation error in determining the point of occurrence of the Sommerfeld phenomenon.

Force relaxation response in the poroelastic axisymmetric Boussinesq problem for an indenter of arbitrary profiles I: Permeable and impermeable surface drainage conditions

In this study, we have derived a theoretical solution for the poroelastic axisymmetric Boussinesq problem when an indenter with arbitrary profiles is subjected to step displacement loading. The analysis is conducted within the framework of Biot’s theory, employing the McNamee–Gibson displacement function method. We explore two distinct scenarios for surface drainage boundary conditions: one where the surface allows full permeability and another where it is fully impermeable. The formulation of the mechanical boundary condition at the surface relies on an equation that establishes the relationship between the depth of indentation and the contact radius. Our solution encompasses two aspects: force asymptotes at the undrained and drained limits, and a normalized force relaxation that accounts for transient responses. Specific findings are presented for three indenter shapes: paraboloidal, conical, and cylindrical. Of particular significance, we demonstrate that for these specific indenter shapes, the normalized force relaxation at ω=0 can be expressed in closed form in the Laplace domain. In addition to this analytical breakthrough, we conducted numerical simulations employing actual indenters. Remarkably, the normalized force relaxation curves from actual indenters show negligible dependence on material properties and closely align with our closed-form solution at ω=0. This finding suggests that the closed-form solution could act as a universal master curve capable of characterizing the normalized force relaxation response in instances of a real indenter, irrespective of material properties.

Экспериментальное определение нелинейной функции демпфирования механических систем

In this article, the coefficients of the nonlinear damping function of a mechanical system with one translational degree of freedom are determined from an experimentally obtained oscillogram of free vibrations. The function is modeled using three types of damping: coulomb damping, linear viscous, and nonlinear viscous damping. Numerical values of the damping coefficients are identified. The characteristic of the dissipative force as a function of displacement is obtained, and is used to find the amount of energy dissipated over a time period. An equivalent relative damping ratio is approximated using the energy balance method and then used to perform numerical integration of the equation of motion. A satisfactory match of the envelope curve and the phase of the vibrational process is demonstrated by comparing the calculated oscillogram to the experimental one. The damping function parameters can be further refined by approximating the experimental amplitudes. The obtained value of the relative damping coefficient can be used to solve nonlinear problems in the area of dynamics of weakly damped systems.

A third-order shear deformation plate bending formulation for thick plates: first principles derivation and applications

A third-order shear deformation plate bending formulation is presented in this study from the first principles. The derivation assumed a displacement field constructed using third-order polynomial function of the transverse (z) coordinate; and made to apriori satisfy the linear three-dimensional (3D) kinematics relations as well as the transverse shear stress free boundary conditions at the top and bottom plate surfaces. The formulation thus has no need for shear stress correction factors of the first-order shear deformation plate theories. The domain equations of equilibrium are obtained as a set of three coupled differential equations in terms of three unknown displacements. The system of coupled equations is solved for simply supported rectangular and square plates subjected to four cases of loading distributions: sinusoidal loading, uniformly distributed loading, linearly distributed loading and point load at the plate center. Navier’s double trigonometric series method is used to construct trial solutions for the three displacement functions such that the boundary conditions are satisfied identically. The integration problem is thus reduced to an algebraic problem and is solved for each considered loading. It is found that the present formulation gives exact results for the normal stresses σxx for sinusoidal and uniformly distributed loads. The study further showed that the results for deflection and stresses agreed with Krishna Murty’s higher order shear deformation plate theory results. The present formulation gave accurate results because of the inclusion of transverse normal strain effects in the formulation. The formulation gives a quadratic variation of the transverse shear stresses across the thickness in consonance with the theory of elasticity method.

Penyusunan Program Komputasi Pengidentifikasi Parameter Dinamik Balok Ditumpu Jepit-Bebas

Operational Modal Analysis (OMA) is a new procedure to identify a structural vibrating system. In this paper, the operational modal analysis based on Ibrahim Time Domain Method. This method requires some displacement data from free vibration response of structure model. Here, a beam with clamped-free condition is used as a test model. Based on Finite Element Method, this beam is divided into 3 elements that consist of 6 degree of freedoms for each element (3 translations and 3 rotations). In this case, a fixed lumped mass is added to each translation node. The beam is excited in its clamped support by sinusoidal displacement during 0.1 s. After 0.1 s, the excitation is stopped and then the displacement data is recorded. Therefore, this condition could be assumed that recorded data represents free vibration response. For this purpose, some computation program was developed to predict the response of free vibration system. First, we make a program SINUS.FOR to produce a sinus displacement function. This function as a support motion is used as an input for program BALKEN.FOR that consist of various program, like Guyan Condensation, Cholesky Decomposition, Jacobi Method, Matrix Iteration Method, Phase Plane Method, Finite Difference Method, Houbolt Method, etc. This entire program is compiled by FORTRAN programming language. Next, the output of BALKEN.FOR that called matrix [A] become an input of program EIGENWERT-ITD.MW which is written in MAPLE programming languages. In this research, it is assumed that only the displacement of x1(t), x2(t) and x3(t) is known. It is known that to identify a system with 6 degrees of freedom, The Ibrahim Time Domain Method requires displacement responses. To fulfill this condition, we need 9 pseudo measurement stations. It means that each station with known displacement response has to be manipulated in such way so that it can produce 3 pseudo displacement response groups. The Result shows that the common error is below 5 %. Except, the damping ratio of mode number 5 is more than 10 %. While the other result can be identified better, especially for the natural frequency is identified very well.

A nano-positioning system based on an optimal two-stage linear displacement amplifier

Nano-positioning systems with large motion ranges often employ displacement amplifiers with poor linearity and a large footprint. Here, we propose a compact nano-positioning system comprising an optimally designed two-stage displacement amplifier with linear input-output relationship over a large motion range. The first stage is a truncated bridge amplifier whose size is 25% of a conventional bridge amplifier, while the second stage is a modified lever-type displacement amplifier whose amplified motion direction can be chosen as desired. It is proposed to exploit the compliance of the structure to improve its linearity by operating the truncated bridge amplifier at a specific optimal angle. A simple reduced-order model is proposed and is shown to agree with finite element analysis to within 1%. Further, the variation in the amplification ratio at various payloads are analysed as function of input displacement. Subsequently, a two-stage amplifier with amplification 28 demonstrating a linear range of 940 µm is fabricated using acrylic. This amplifier’s nonlinearity is less than 5.2%, representing about 10-fold improvement compared to a conventional bridge amplifier. Lastly, a nano-positioner employing an optimal amplifier is fabricated using spring steel and demonstrated to possess nonlinearity less than 1.5% over a range of 360 µm and a natural frequency of 210 Hz with a footprint of 21 cm2, which represents a substantial improvement in range-to-footprint ratio compared to other amplifiers of similar natural frequencies.

Molecular Dynamics Simulation of Silane Inserted CSH Nanostructure.

Herein, the toughening mechanism and effects of 3-(aminopropyl)triethoxysilane (3-APTES) intercalation in calcium-silicate-hydrate (CSH) structures were investigated through molecular dynamics simulations. CSH established a model using 11 Å-tobermorite to simulate the tensile properties, toughness, adsorption energy, average orientation displacement and radial distribution function of 3-APTES intercalation at different Ca/Si ratios under conditions of a CVFF force field, an NVT system, and 298 K temperature. Simulation results demonstrate that 3-APTES alters the fracture process of CSH and effectively enhances its tensile properties and toughness. The presence of 3-APTES molecules increases the energy required to destroy CSH, thereby increasing the adsorption energy of CSH crystals. Furthermore, 3-APTES molecules effectively increase the atom density within the CSH structure. As the Ca/Si ratio increases, Ca-O bond formation is enhanced, with noticeable aggregation occurring because of modification by 3-APTES within the CSH structure. This study found that 3-APTES organic compounds can effectively improve the tensile, toughness, adsorption and other properties of the CSH structure, and further improve the microstructure of CSH.

Application of the Double Approximation Method for Constructing Stiffness Matrices of Volumetric Finite Elements

Introduction. When numerically solving problems of elasticity theory in a three-dimensional formulation by the finite element method, finite elements (FE) in the form of parallelepipeds, prisms and tetrahedra are used. Regularly, the construction of stiffness matrices of volumetric FE is based on the principle of isoparametricity, which involves the Lagrange polynomials to approximate the geometry and displacements. In computational practice, the most widespread FE are the so-called multilinear isoparametric FE with a linear law of approximation of displacements. The main disadvantage of these elements lies in the “locking” effect when modulating bending deformations. Moreover, the error of the numerical solution increases drastically in the case when the structure, in comparison to conventional deformations, undergoes significant displacements as a rigid whole. Long-term experience in solving problems of deformable solid mechanics by the finite element method has shown that existing volumetric FE have slow convergence, specifically, when modeling bending deformations of plates and shells. This study aims at constructing stiffness matrices of multilinear volumetric FE of increased accuracy allowing for rigid displacements based on the double approximation method.Materials and Methods. The mathematical apparatus of the double approximation method based on the principle of a separate representation of the distribution functions of displacements and deformations inside the element, was used to construct the stiffness matrices of volumetric FE. The storage and processing of the resulting system of equations was implemented in algorithmic terms of sparse matrices. Software development and computational experiments were carried out using the Microsoft Visual Studio 2013 64-bit computing platform and the Intel ® Parallel Studio XE 2019 compiler with the integrated Intel ® Visual Fortran Composer XE 2019 text editor. Visualization of the calculation results was performed using the descriptor graphics of the MATLAB computer mathematics package. A large eight-node SOLID185 CE of the ANSYS Mechanical software complex was used as a test sample.Results. Mathematical tool and software were developed to study the stress-strain state of massive structures under various types of external actions. The authorized application software package was verified on test examples with known analytical solutions. It has been shown that the constructed FE accurately satisfy the basic requirements for finite element modeling of spatial problems of elasticity theory.Discussion and Conclusion. The performed testing of the developed mathematical and program toolkit has shown that the finite elements constructed on the basis of the double approximation method can successfully compete with similar SOLID185 volumetric elements of the ANSYS Mechanical software complex. The proposed elements can be integrated into domestic import-substituting software systems that implement the finite element method in the form of the displacement method.

НЕЛИНЕЙНЫЙ АНАЛИЗ МАССИВНЫХ ЖЕЛЕЗОБЕТОННЫХ КОНСТРУКЦИЙ С УЧЕТОМ ТРЕЩИНООБРАЗОВАНИЯ

A finite element method, as well as the algorithm and the program for solid reinforced concrete structures analysis have been developed, taking into account plastic deformations of concrete. A modified Willam & Warnke failure criterion was used, supplemented by a flow criterion. Two models of volumetric deformation of concrete have been developed: an elastic model under brittle fracture and an ideal elastic-plastic model. An eight-node solid finite element with linear approximation of displacement functions, which implements the deformation models above mentioned, is constructed. This finite element is adapted to the PRINS computational software, and as part of this program it can be used for physically nonlinear analysis of building structures containing three-dimensional reinforced concrete elements. Modern building codes prescribe to carry out calculations of concrete and reinforced concrete structures in a nonlinear formulation, taking into account the real properties of concrete and reinforcement. To verify the developed finite element, a series of test calculations of a beam in the condition of pure bending was carried out. Comparison of the calculation results with experimental data confirmed the high accuracy and reliability of the results obtained.

On the features of inhomogeneous residual stress identification using the digital speckle interferometry and the hole-drilling method

A phenomenological approach to the actual problem of determining the inhomogeneous residual stress-strain state in the components of high-tech engineering systems at the stages of their design and operation is presented. The approach is based on physical and mechanical methods of measuring displacements. Current physical models describe the physical regularities of the residual states attributed to changes in the structure by the interaction of defects and dislocations in the field of micro- and meso-stresses. At the same time, there are the problems of the transition to the macrolevel, the construction of multilevel models, and the conversion of these models in engineering practice. In the framework of phenomenological approaches, in the general case, the solution of this problem requires the solution of three-dimensional inverse problems of thermoelasticity. A well-known mechanical method for determining a uniform field of residual elastic stresses recommended by ASTM E837 is described. The method proposed earlier by one of the authors for determining an inhomogeneous (in the plane) field of residual elastic stresses is discussed. A method of the three-dimensional inhomogeneous residual elastic stress-strain state determination based on the experimental determination of the displacement vector components by the method of step-by-step point hole-drilling and data of digital speckle interferometry and digital image correlation is developed. The constitutive relations for the components of the displacement vector are written in the form of Volterra integral operators. The basic operator functions are the functions of four variables, i.e., the coordinates of the cylindrical system (r, θ, z) associated with the hole, and the hole depth h. A method for verification of the basic functions is presented. The problem is reduced to the determination of three displacement functions of three variables: hole radius r, h, and z. Numerical simulation of basic functions is carried out. The obtained results are consistent with the known experimental data and calculated values of the deformation on the surface depending on the depth of the hole according to the ASTM E837 Standard.