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1

Differential equations expressing the thermal balance of metals and alloys

Mustafayev Amir G., Aghalarova Ismat U.

Nafta-Gaz

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2025

|

R. 81, nr 3

199--209

EN

One of the main directions of development in the oil and gas industry is the creation of new and efficient technological equipment. From 2020 to 2025, the increase in oil and gas production could be achieved through the correct selection of repair equipment and adherence to existing regulations for restoring wells that have been taken out of operation and returning them to service. Since the working parts of the equipment used in repair and restoration works in wells are constantly in contact with degradable objects, temperatures and thermal stresses increase due to friction at their contact points. High temperatures (1000–1200°C) create a stress-deformed state in the contact areas between tools and equipment. The stress-deformed conditions in the cutting zones lead to the formation of microcracks in the working part of the tool. Over time, these cracks grow, increasing temperature and thermal stresses, leading to tool wear, premature failure, and, in some cases, jamming. To maintain equipment and tools in working condition and ensure their periodic maintenance, the correct selection of milling parameters is necessary. The high productivity achieved during well drilling, repair, and restoration works depends on economic efficiency, durability, the choice of materials for cutting and milling tools that meet modern requirements, the compactness of construction, and the size and condition of the repaired well. One of the main factors ensuring the safe operation of equipment and tools in a well during repair work is maintaining the thermal-physical conditions of the moving parts of the tool at the required level. The thermal regime of the cutting tool largely depends on the physical and mechanical properties of the objects being destroyed: rock, metal, hardened cement, etc. When interacting with degradable objects, thermomechanical stresses affect the contacting surfaces of the tool, and a significant amount of heat is released at the working surface. As a result of extensive scientific and experimental research on the management of thermal-physical processes to enhance the stability of oil production, operational, and repair equipment, differential equations have been formulated to describe thermal processes. These equations incorporate coefficients of thermal conductivity, heat transfer, and cooling, which are fundamental to solving thermal problems.

PL

Jednym z głównych kierunków rozwoju przemysłu naftowego i gazowego jest projektowanie nowoczesnego i wydajnego sprzętu technologicznego. W latach 2020–2025 wzrost wydobycia ropy naftowej i gazu ziemnego można było osiągnąć dzięki prawidłowemu doborowi sprzętu naprawczego i przestrzeganiu obowiązujących przepisów dotyczących przywracania do eksploatacji odwiertów wyłączonych z użytkowania. Ponieważ elementy robocze urządzeń wykorzystywanych do naprawy i renowacji odwiertów są w ciągłym kontakcie z obiektami ulegającymi degradacji, następuje wzrost temperatury i naprężeń termicznych w wyniku tarcia w punktach ich styku. Wysokie temperatury (1000–1200°C) prowadzą do odkształceń naprężeniowych w miejscach styku narzędzi i sprzętu. Z kolei odkształcenia naprężeniowe w strefach skrawania prowadzą do powstawania mikropęknięć w części roboczej narzędzia. Z czasem pęknięcia te powiększają się, zwiększając temperaturę i naprężenia termiczne, co prowadzi do zużycia narzędzia, jego przedwczesnej awarii, a w niektórych przypadkach także zakleszczenia. Aby utrzymać sprzęt i narzędzia w stanie gotowości do pracy oraz zapewnić możliwość ich okresowej konserwacji, konieczny jest właściwy dobór parametrów frezowania. Wysoka wydajność osiągana podczas wiercenia, napraw i renowacji odwiertów zależy od efektywności ekonomicznej, trwałości, doboru materiałów na narzędzia skrawające i frezujące spełniające współczesne wymagania, a także od kompaktowości konstrukcji oraz wielkości i stanu naprawianego odwiertu. Jednym z kluczowych czynników zapewniających bezpieczną eksploatację sprzętu i narzędzi w odwiercie podczas prac naprawczych jest utrzymanie warunków termofizycznych ruchomych części narzędzia na wymaganym poziomie. Parametry termiczne narzędzia skrawającego zależą w dużej mierze od właściwości fizycznych i mechanicznych obrabianych obiektów: skał, metalu, stwardniałego cementu itp. Podczas pracy z obiektami ulegającymi degradacji, na powierzchniach styku narzędzia występują naprężenia termomechaniczne, a na powierzchni roboczej uwalniana jest znaczna ilość ciepła. W wyniku szeroko zakrojonych badań naukowych i eksperymentalnych nad regulacją procesów termofizycznych w celu zwiększenia stabilności produkcji ropy naftowej, sprzętu operacyjnego i naprawczego, opracowano równania różniczkowe opisujące procesy termiczne. Równania te wykorzystują współczynniki przewodnictwa cieplnego, wymiany ciepła i chłodzenia, które stanowią podstawę do rozwiązywania problemów termicznych.

2

An overview of the effect of various interlayer materials and boundary conditions on delaminated glass plates

Dural Ebru

Advances in Science and Technology. Research Journal

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2025

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Vol. 19, no. 6

70--83

EN

This study presents a large-deflection mathematical analysis of a laminated glass plate that has suffered initial delamination. Three distinct interlayer types are taken into consideration in order to examine how the type of interlayer affects the behavior of delaminated glass plates. The plate is subjected to uniform pressure. The analysis is based on solving five nonlinear partial differential equations relating the lateral deflections and stresses to the applied load. The established solution approach is presented in a simple form suitable for analyzing various loads, geometries, material properties and boundary conditions. Two boundary conditions—fixed and simply supported edges—are taken into consideration for plates. It is established that interlayer type has a major effect in determining the delamination strength of laminated glass. Design engineers can use the current research findings to build laminated glass for structural applications.

3

Research of dynamic processes in a layer during collision with an impactor

Pyr'yev Yuriy, Pawlikowski Marek, Drobnicki Rafał, Penkul Andrzej

Acta Mechanica et Automatica

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2024

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Vol. 18, no. 4

585--592

EN

The article concerns the modeling of the transverse impact of an impactor (test sample) on the surface of an infinite elastic layer. The Laplace transform with respect to time and the Hankel transform with respect to the radius for the axisymmetric case were applied. The propagation of elastic waves in the layer and local deformations in the contact zone are taken into account. Impact force, impact time and the coefficient of restitution were examined. The results are compared with the elastic half-space. The calculations carried out showed that for layer thicknesses of more than five impactor diameters, the layer can be considered as a half-space.

4

Talerz z barwnego szklanego alabastru o koronkowym obrzeżu

Żelasko Stefania, Lichota Lutosław

Szkło i Ceramika

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2022

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R. 73, nr 4

54--57

PL

Ten alabastrowy majstersztyk pochodzi z schaffgotschowskiej fabryki szkła Josephinenhütte Glasfabrik.

5

Ultimate compressive strength assessment of uncleaned and cleaned corroded plates with locked crack

Woloszyk Krzysztof, Garbatov Yordan

Polish Maritime Research

|

2021

|

nr 3

117--127

EN

The work presented here investigates the structural response of cleaned corroded plates, subjected to compressive load in the presence of a locked crack, where the change of mechanical properties as a result of corrosion development and the cleaning process is also accounted for. A Finite Element model for assessing the compressive strength, considering geometric and material nonlinearities, is developed, and the analysed plates are compared with the available experimental data. An experimental design plan is generated using the Design of Experiments techniques, which quantifies the influence of the governing variables and their interactions with respect to the plate’s ultimate compressive strength. With a limited number of observations, the most significant effects are identified. The corrosion degradation is revealed to be the most crucial effect leading to an effective strength reduction. It was found that, in the case of a corroded plate with a locked crack subjected to a compressive load, the most severe case is when the crack is transversely oriented. The strength reduction is slightly lower than when the corrosion degradation and the presence of a crack are considered to be a simple summation of these two effects but acting separately. The outcome of the analysis is the development of several empirical formulations that allow a fast estimation of the ultimate strength of a corroded plate, subjected to compressive load in the presence of a locked crack, accounting for different cleaning.

6

Wykrywanie uszkodzeń w konstrukcjach inżynierskich przy zastosowaniu metody dyskretnej transformacji falkowej

Knitter-Piątkowska Anna, Guminiak Michał

Przegląd Budowlany

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2020

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R. 91, nr 11

61--64

PL

Badania związane z wykrywaniem uszkodzeń i osłabień elementów konstrukcyjnych stanową bardzo ważny element kompleksowej analizy budowli inżynierskich. W analizie identyfikacji uszkodzeń wiodącą rolę odgrywają tzw. metody nieniszczące, które pozwalają dostatecznie precyzyjnie zlokalizować powstałe uszkodzenia. Prezentowana praca poświęcona jest zastosowaniu dyskretnej transformacji falkowej w procesie lokalizacji uszkodzeń konstrukcji. Dowolne uszkodzenie, np. w postaci lokalnego osłabienia sztywności konstrukcji (pęknięcia), jest przyczyną zaburzenia w rejestrowanym sygnale odpowiedzi - ugięciu, deformacji przekroju lub np. przyspieszeniu wybranego punktu konstrukcji. Zaburzenie sygnału jest na tyle małe, że dopiero jego przetworzenie za pomocą analizy falkowej pozwala zlokalizować miejsce uszkodzenia. Zaletą przedstawionej procedury jest wykorzystanie wyłącznie sygnału odpowiedzi rzeczywistej konstrukcji uszkodzonej. Przedstawiono krótki przegląd dotychczasowych analiz konstrukcji płytowych (płyt cienkich).

EN

Research related to the detection of damage and weakening of structural elements is a very important element of a comprehensive analysis of engineering structures. In the analysis of damage identification, the leading role is played by the so-called non-destructive methods that allow for sufficiently precise localization of the damage. The presented work is devoted to the application of the discrete wavelet transformation (DWT) to the process of identification and localization damages in structures. Any damage, e.g. in the form of a local weakening of the structure stiffness (cracks), causes disturbances in the recorded response signal - deflection, deformation of the cross-section or e.g. acceleration of a selected point of the structure. However, the signal disturbance is so small that only its processing by means of wavelet analysis allows to locate the damage site. The advantage of the presented procedure is the use of the response signal only of the real - damaged structure. The presented work is an overview of the results obtained so far. The slabs were analyzed as the basic surface structural systems that form the building structure.

7

Analysis of ultimate compressive strength of cracked plates with the use of doe techniques

Woloszyk Krzysztof, Garbatov Yordan

Polish Maritime Research

|

2020

|

nr 3

109--120

EN

The objective of this work is to investigate the structural compressive response of plates with locked cracks accounting for all relevant factors and correlation between them. The nonlinear FE model considering both geometric and material nonlinearities is employed herein, and the FE model of the structural response of intact plates is validated with the available experimental data. In the common studies, based on One Factor at a Time analysis, some of the parameters and interactions between them are excluded. In the present study, the numerical investigations are conducted with the use of the Design of Experiments techniques, where all essential parameters and their interactions are adequately considered. With a total of 32 numerical analyses, the most influential factors and their interactions are identified. As a study outcome, empirical formulations, which allow for a fast estimation of the ultimate compressive strength of intact plates, plates with locked cracks, and repaired cracked plates, are derived. The developed formulations represent a fast and practical tool for estimating the ultimate compressive strength of intact, cracked, and repaired plates, which can be easily employed in the reliability analysis.results followed by their discussion.

8

Nonlinear homogenization of heterogeneous periodic plates of Reissner-Mindlin type

Pruchnicki Erick

Journal of Theoretical and Applied Mechanics

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2020

|

Vol. 58 nr 2

317--323

EN

In this paper, we propose a multiscale finite-strain plate theory for a composite nonlinear plate described by a repetitive periodic heterogeneity. We consider two scales, the macroscopic scale is linked to the entire plate and the microscopic scale is linked to the size of the heterogeneity. At the macroscopic scale, we approximate the displacement field by the Reissner-Mindlin model. By considering the equivalence between variations of the macroscopic elastic energy at each point of the mid surface and the microscopic one, we deduce that the macroscopic stress resultants can be expressed in terms of the microscopic stress.

9

Analiza przyczyn zarysowania posadzki fibrobetonowej w magazynie wysokiego składowania

Kamiński Krzysztof

Inżynieria i Budownictwo

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2020

|

R. 76, nr 9

424--429

EN

In recent years, fiber concrete industrial floors have practically replaced traditional cross-reinforced concrete slabs. The analytical calculation methods used generate too low values of internal forces compared to the more accurate methods based on FEM. Designing the plate for equivalent strength can lead to excessive scratching and damage to the top surface preventing the use of forklift trucks of the high-bay warehouse.

PL

Fibrobetonowe posadzki przemysłowe w ostatnich latach praktycznie wyparły tradycyjne krzyżowo zbrojone płyty żelbetowe. Na podstawie analitycznych metod obliczeń uzyskuje się wartości sił wewnętrznych mniejsze, niż wykorzystując metodę elementów skończonych. Projektowanie płyty na wytrzymałość równoważną może być przyczyną powstania nadmiernych zarysowań i uszkodzeń górnej powierzchni posadzki, a wskutek tego uniemożliwić stosowanie wózków widłowych do transportu.

10

Effect of punch profile on deformation behaviour of AA5052 sheet in stretch flanging process

Kumar Surendra, Ahmed M., Panthi S. K.

Archives of Civil and Mechanical Engineering

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2020

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Vol. 20, no. 1

247--263

EN

Stretch flanging is a type of bending process widely used in automobile and aerospace industries. Forming of the stretch flange is mainly affected by three important parameters: materials of the sheet, the geometry of tools and different process parameters. This work focuses on the effect of punch profile on deformation behavior of AA5052 alloy sheet to form the stretch flange. Six punches of different geometry i.e. cylindrical, two stepped, three stepped, six stepped, conical and hemispherical are used. Results are presented in the form of edge crack in the sheet at edge corner and its propagation towards center, forming load comparison for different punch profile and distribution of radial and circumferential strain in the sheet. It is observed that the punch profile has a considerable effect on the deformation behavior of the sheet. Circumferential strain, radial strain and load requirement to form the flange are found to be minimum in hemispherical punch profile as compared to other punch profiles. Experiments are performed to validate the FE simulation results and results are found in very good agreement in terms of edge crack length. Fractography study shows uniform and large number of small size dimples at the fractured surface for hemispherical and conical punch profile.

11

A circular inclusion and two radial coaxial cracks with contacting faces in a piecewise homogeneous isotropic plate under bending

Sulym Heorhiy, Opanasovych Viktor, Zvizlo Ivan, Seliverstov Roman, Bilash Oksana

Acta Mechanica et Automatica

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2020

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Vol. 14, no. 1

16--21

EN

The bending problem of an infinite, piecewise homogeneous, isotropic plate with circular interfacial zone and two coaxial radial cracks is solved on the assumption of crack closure along a line on the plate surface. Using the theory of functions of a complex variable, complex potentials and a superposition of plane problem of the elasticity theory and plate bending problem, the solution is obtained in the form of a system of singular integral equations, which is numerically solved after reducing to a system of linear algebraic equations by the mechanical quadrature method. Numerical results are presented for the forces and moments intensity factors, contact forces between crack faces and critical load for various geometrical and mechanical task parameters.

12

Experimental damage assessment of support condition for plate structures using wavelet transform

Abdulkareem Muyideen, Bakhary Norhisham, Vafaei Mohammadreza, Md Noor Norhazilan, Abd Samat Roslida

Journal of Theoretical and Applied Mechanics

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2019

|

Vol. 57 nr 2

501--518

EN

Wavelet transforms (WTs) have gained popularity due to their ability to identify singularities by decomposing mode shapes of structures. In VBDD, the support condition of a structure influences structural responses and modal properties. In fact, the structural responses and modal properties are a lot more sensitive to changing boundary conditions than to crack and fatigue damage, resulting in inaccurate damage detection results. Therefore, in this study, sensitivity tests to estimate a suitable distance range which allows damage detection by imposing single support damage are carried out. The estimated appropriate distance is then applied to detect damage at multiple supports. This involved the applicability of response acceleration of plate structures to support assessment by applying continuous wavelet transform (CWT) and discrete wavelet transform (DWT). The damage cases have been introduced by releasing bolts at specified fixed supports of the plate to simulate the damage. The response accelerations of the rectangular plate at points close to the supports were measured and decomposed using CWT and DWT to assess the structural integrity of each support. The results showed that an appropriate distance range was necessary for accurate damage detection, and both, CWT and DWT could provide reliable outputs. However, the first- and fourth-level detail coefficients of DWT failed to indicate damage in some cases. A more detailed investigation of the effect of different wavelet scale ranges on damage detection using CWT demonstrated that the accuracy of damage detection increased as the scale decreased.

13

Numerical investigations of stress concentration in reinforcement steel structure by composite overlays

Szybiński Bogdan, Wygoda Mateusz

Journal of KONES

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2019

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Vol. 26, No. 4

249--256

EN

The stress concentration observed in the vicinity of cut-outs and holes in structural elements significantly influences the fatigue endurance of machines subjected to cyclic loads. Numerous studies have been made so far to improve this situation and increase the structure lifetime. Several design recommendations have also been worked out to avoid the problem of premature failure. The proposed article illustrates the influence of the composite overlays applied around the cut-outs made in flat steel constructional elements subjected to axial tension. The detailed study concerns the reinforcement made from the FRP (fibre reinforcement polymer) composite applied around the notches. Two types of composite materials were used, namely: TVR 380 M12/R-glass (glass fibres embedded in epoxy resin matrix) and AS4D/9310 (carbon fibres embedded in epoxy resin matrix). In the first step, the detailed numerical studies (finite element analysis) were performed for the steel samples (with no overlays added) with cut-outs made in the form of circle, square and triangle hole (the last two with rounded corners). The results of these studies were compared with the existing analytical solutions with respect to the stress concentration factors (SCF) estimation. The relatively good conformity was observed when using dense meshes of finite elements placed around the void vicinity. In the next step, the composite overlays were applied around cut-outs and their influence on the stress concentration was investigated. The influence of the fibre orientation, numbers of layers, sizes of the composite overlay used were considered. It was proved that the application of composite overlays evidently decreases the stress concentration around the notches.

14

Numerical analysis of frequencies and forms of own collars of different forms with free zone

Borysenko Maksym, Zavhorodnii Andrii, Skupskyi Ruslan

Journal of Applied Mathematics and Computational Mechanics

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2019

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Vol. 18, nr 1

5-13

EN

Thin plates of three different forms with different physical-mechanical characteristics and free edges are examined in this work. Modeling of geometry and numerical calculation of frequencies and forms of free oscillation of plates is accomplished by the finite element method, which is realized using the licensed computer program FEMAP with the NASTRAN solver. A comparative analysis of the calculated eigenfrequencies is carried out. The dependence of the corresponding frequencies on the physical and mechanical characteristics of the material in the form of coefficients is established.

15

Analysis the state of stress in a stainless steel plate with stress concentrator in the form of circular hole

Konieczny Mateusz, Achtelik Henryk, Gasik Grzegorz

Inżynieria Materiałowa

|

2019

|

Vol. 40, nr 6

146--153

EN

The paper presents an analytical, numerical and experimental analysis of the state of stress in a plate with stress concentrator in the form of circular hole subjected to uniaxial tension. This type of the plate have found wide spread applications in the field of aerospace, marine and automobile engineering. The maximum difference in the values of reduced stresses calculated analytically in relation to the results obtained experimentally was δσa red = –19.74%. However, the maximum difference in the values of reduced stress calculated numerically using the finite element method (FEM) with the ANSYS program in relation to the results obtained experimentally δσn red = –31.86%, The study showed that the analytically calculated and numerically determined stresses are lower than the stresses obtained experimentally, which results in less operational safety of the structural element. However, the numerical method in the form of the finite element method becomes, in the case of a very complicated element shape, the only method for determining the magnitude and distribution of stresses near the notch in the designed element.

PL

Celem pracy jest analityczna, numeryczna oraz eksperymentalna analiza stanu naprężenia w tarczy z koncentratorem naprężeń w postaci otworu kołowego o średnicy d poddanej jednoosiowemu rozciąganiu siłą Pi i na tej podstawie zlokalizowanie stref spiętrzenia naprężeń.

16

Analiza stosowanych systemów transportu popiołu z odpylaczy spalin w kotłach energetycznych i ciepłowniczych z paleniskami rusztowymi

Janik Kazimierz, Janik Ireneusz, Janik Adrianna, Rataj Zbigniew Leon

Instal

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2019

|

nr 12

15--21

PL

W artykule przeprowadzono analizę stosowanych przenośników w systemach odbioru i transportu pyłu z kotłów z paleniskiem rusztowym w energetyce przemysłowej i ciepłowniczej. Zwrócono uwagę na konstrukcję najczęściej stosowanych rodzajów przenośników, ich parametry konstrukcyjne i eksploatacyjne. Dokonano oceny porównawczej zużycia energii elektrycznej do napędu urządzeń. Przeprowadzona analiza może być pomocna przy podejmowaniu decyzji techniczno-finansowych w ww. zakresie.

EN

This paper analyzes the conveyors used in dust collection and transport systems from grate boilers in industrial and heating energy. Attention was paid to the construction of the most commonly used types of dust conveyors, their construction and operational parameters. A comparative assessment was made of electricity consumption for the drive of the equipment. The analysis helps in making technical and financial decisions.

17

Effect of the thickness of the piezoelectric patches on the active control of a thin plate

Latrache M., Amrane M. N.

Mechanics and Mechanical Engineering

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2018

|

Vol. 22, nr 4

1345--1354

EN

This paper presents a numerical study pertaining to on the active vibration control (AVC) of the 3-D rectangle simply supported plate bonded of the piezoelectric sensor/actuator pairs. A LQR controller is designed based on the independent mode space control techniques to stifle the vibration of the system. The change in the thickness of the patches was a clear impact on the control results, and also in the values of the voltage in actuator. The results were established by simulating in ANSYS and MATLAB.

18

Bending analysis of plates by superposing cylindrical deflections

Jemielita G. E.

Journal of Theoretical and Applied Mechanics

|

2018

|

Vol. 56 nr 1

95--105

EN

In this paper, the present exact solutions in the plate theory using the cylindrical deflection method and represented by the Fourier series corresponding to the oblique or Cartesian coordinates are given for a hingely supported triangular plate subject to a distributed load in form of a hexagonal pyramid and for a rectangular hingely supported plate subject to loading in form of a truncated octagonal pyramid. In the case of the rectangular hingely supported plate under the truncated octagonal pyramid load, a series of parametric solutions has been obtained. The solutions depend on the parameter ε. For various values of the parameter ε, various load cases are found.

19

Analysis of the impact of slot geometry on the stresses distribution in a plate of S235JR steel subjected at biaxial load

Konieczny M., Gasiak G.

Inżynieria Materiałowa

|

2018

|

Vol. 39, nr 6

228--235

EN

The numerical calculations the state of reduced stress in the plate with the slot at the biaxial load made it possible to determine areas of stress concentration around the top of the slot. It was shown that the highest of reduced stress concentration occurred in the plate with a flat slot with dimensions of l = 20 mm, r = 0.25 mm, c = 0.5 mm when the slope of the main axis of the slot forms an angle α = 30° and α = 60° for biaxial load, i.e. Pz = 100 kN and compression Py = –100 kN (pure shear case). The stress concentration factor k depending on the method of plate load, slot geometry and angle of inclination the main axis of the slot takes values from k = 3.07 to k = 18.69 for a flat slot when the inclination of the main axis of the slot formed an angle α = 0° and angle α = 90°; from k = 2.81 to k = 18.42 in the case of a flat slot when the slope of the main axis of the slot formed an angle α = 45° and from k = 2.83 to k = 18.48 for a flat slot when the inclination of the main axis of the slot formed an angle α = 30° and angle α = 60°. Significant values of reduced stresses concentration in the top area of the slot are usually the cause of microcracks, which under the influence of variable loads may initiate the propagation of fatigue cracks and lead to destruction of the machine element or construction element.

PL

Jedną z przyczyn powstawania i rozwoju pęknięć zmęczeniowych jest występowanie spiętrzenia naprężeń miejscowych, powstających wokół otworów, szczelin, podcięć, wgłębień kątowych w materiale oraz kołnierzy. Pęknięcia zmęczeniowe rozwijają się od tzw. ognisk, które powstają w miejscach spiętrzenia naprężeń. Znanych jest szereg metod analitycznych, eksperymentalnych i numerycznych podejmujących problem analizy stanu naprężenia w płytach lub tarczach zawierających otwory i szczeliny. Celem pracy jest numeryczna analiza stanu naprężenia w tarczy ze szczeliną, wykonanej ze stali S235JR, poddanej dwuosiowemu obciążeniu i na tej podstawie wskazanie miejsc spiętrzenia naprężeń wokół wierzchołka szczeliny.

20

Bending of composite plate under magnetic field

Kędzia P., Kosma Z.

Modelowanie Inżynierskie

|

2017

|

T. 34, nr 65

57--63

EN

The main objective of presented study is composite rectangular plate subjected to load generated by magnetic field. The field changes around the area limited by magnetic elements. The plate is made of polyethylene and consists of three layers: two faces and porous core. The core is filled with ferrofluid. Porous structure prevents fluid from flowing out between cells in the middle plane. The load influences perpendicular to the plate and along middle plane.The load is generated by the system build of Helmholtz and Golay coils which are frequently used in modern MRI tomographs (Magnetic Resonance Imaging). Bending of the plate is affected by the magnetic field which influences on ferrofluid in porous cells. The function of bending of the plate is approximated by bicubic spline function presented by normalized B-spline functions. Required approximations of partial derivatives appearing in equation of bending of the plate and in boundary conditions are obtained with the property, that bicubic spline function is polynomial spline function respect to each of independent variable separately. Efficiency of these algorithms is proved by comparing test results with numerical simulations, with FEM in Autodesk Simulation v.14 program. Prepared numerical algorithm determines bending of the plate under applied load which is induced by spatial changes of magnetic field.

PL

Przedmiotem badań jest kompozytowa płyta prostokątna poddana obciążeniu wywoływanemu przez zmienne w przestrzeni pole magnetyczne. Płyta jest wykonana z polietylenu i składa się z trzech warstw: dwóch okładzin zewnętrznych oraz porowatego rdzenia, wypełnionego płynem ferromagnetycznym. Porowata struktura płyty zapobiega przelewaniu się płynu ferromagnetycznego wzdłuż płaszczyzny środkowej. Obciążenie działające jednocześnie prostopadle do powierzchni płyty oraz wzdłuż płaszczyzny środkowej wywołane jest przez cewki Helmholtza oraz cewki Golaya stanowiące podstawowy trzon układu magnesów współczesnych tomografów do obrazowania MRI (Magnetic Resonance Imaging). Wygenerowane pole magnetyczne tomografu poprzez oddziaływanie na płyn ferromagnetyczny wywołuje ugięcie płyty. Funkcję ugięcia płyty aproksymowano bikubiczną funkcją sklejaną przedstawioną przez znormalizowane B-funkcje sklejane. Niezbędne aproksymacje pochodnych cząstkowych występujących w równaniu ugięcia płyty i w warunkach brzegowych uzyskano, korzystając z własności, że bikubiczna funkcja sklejana jest wielomianową funkcją sklejaną względem każdej zmiennej niezależnej z osobna. Skuteczność algorytmów została w pełni potwierdzona poprzez porównanie wyników obliczeń testowych z symulacjami numerycznymi wykonanymi metodą elementów skończonych za pomocą programu Autodesk Simulation v.14. Przy wykorzystaniu opracowanej metody numerycznej wyznaczone zostały ugięcia płyty z uwzględnieniem przestrzennych zmian natężenia pola magnetycznego podczas jej zginania.