Аннотації
29.12.2019
Розв’язано просторову задачу теорії пружності для шару з повздовжньою циліндричною порожниною, коли на межах шару та на межі порожнини задані напруження. Розв’язок задачі отримано узагальненим методом Фур’є стосовно системи рівнянь Ламе в циліндричних координатах, пов’язаних із циліндром, та декартових координатах, пов’язаних із межами шару. Нескінченні системи лінійних алгебраїчних рівнянь, які отримані в результаті задоволення граничних умов, вирішено методом зрізання. В результаті були отримані напруження в різних точках пружного тіла. Проведено аналіз напружено – деформованого стану шару від дії зрівноваженого навантаження на його межах.
Решена пространственная задача теории упругости для слоя с продольной цилиндрической полостью, когда на границах слоя и на границе полости заданы напряжения. Решение задачи получено обобщенным методом Фурье относительно системы уравнений Ламе в цилиндрических координатах, связанных с цилиндром, и декартовых координатах, связанных с границами слоя. Бесконечные системы линейных алгебраических уравнений, полученные в результате удовлетворения граничных условий, решено методом редукции. В результате были получены напряжения в различных точках упругого тела. Проведен анализ напряженно - деформированного состояния слоя от воздействия уравновешенной нагрузки на его границах.
An analytic - numerical algorithm for solving a substantially spatial problem of the theory of elasticity for a layer with a longitudinal cylindrical cavity and the conditions of the first principal problem, given on the boundary surfaces, is developed. The solution is based on the generalized Fourier method applied to the Lame equation system. The cavity is considered in the cylindrical coordinate system, the layer - in Cartesian. By satisfying the boundary conditions and using special formulas for the transition between coordinate systems for basic solutions, we create infinite systems of linear algebraic equations, which are solved by the method of cutting. The numerical study of the determinant gives reason to claim that this system of equations has a single solution. As a result, stresses were obtained at different points in the elastic body. Cutting parameters were chosen so that the accuracy of the boundary conditions reaches 10-2. As the cutoff parameter increases, the accuracy of boundary conditions increases, but the duration of the calculation increases. The numerical solution of the problem is performed for a layer with a cylindrical cavity and a normal balanced load at the boundary of the layer. The analysis of the stress state gives grounds to state: 1. The solution of the algebraic system of equations can be found with any degree of accuracy by the method of reduction, which is confirmed by the high accuracy of boundary conditions. 2. The presence of a cavity gives rise to a redistribution of stresses, in which maximum values occur on the surface of the cavity, and tensile stresses occur at the top of the layer (which is compressed without a cavity). This method can be used in the calculation of structures, the calculation scheme of which is a layer with a cylindrical cavity, under given boundary conditions in the form of balanced stresses. Stress analysis enables the selection of geometric characteristics at the initial design stage.
1. Гузь А.Н., Космодамианский А.С., Шевченко В.П. и др. Механика композитов. Том 7. Концентрация напряжений. Київ: Наук. думка. – 1998. – С. 114 – 137.2. Vaysfel’d N., Popov G., Reut V. The axisymmetric contact interaction of an infinite elastic plate with an absolutely rigid inclusion. Acta Mech. 2015. vol. 226. P. 797–810. doi: https://doi.org/10.1007/s00707-014-1229-73. Попов Г.Я., Вайсфельд Н.Д. Осесимметричная задача теории упругости для бесконечной плиты с цилиндрическим включением при учете ее удельного веса. Прикладная механика. 2014. Т. 50, № 6. С. 27–38.4. Bobyleva T. Approximate Method of Calculating Stresses in Layered Array / Procedia Engineering. – 2016. – Vol.153. – P.103 – 106. https://doi.org/10.1016/j.proeng.2016.08.0875. Гринченко В.Т., Улитко А.Ф. Пространственные задачи теории упругости и пластичности. Равновесие упругих тел канонической формы. – Київ: Наук.думка. – 1985. – 280 с.6. Гузь А.Н., Кубенко В. Д., Черевко М.А. Дифракция упругих волн. –Київ: Наук. Думка. – 1978. – 307 с.7. Гринченко В.Т., Мелешко В. В. Гармонические колебания и волны в упругих телах. – Київ: Наук. Думка. – 1981. – 284 с.8. Волчков В.В., Вуколов Д.С., Сторожев В.И. Дифракция волн сдвига на внутренних туннельных цилиндрических неоднородностях в виде полости и включения в упругом слое со свободными гранями/ Механика твердого тела. – 2016. – Вып. 46. – С. 119 – 133.9. Николаев А.Г., Проценко В.С. Обобщенный метод Фурье в пространственных задачах теории упругости. – Харьков: Нац. аэрокосм. университет им. Н.Е. Жуковского «ХАИ», 2011. – 344 с.10. Проценко В.С., Николаев А.Г. Пространственная задача Кирша / Математические методы анализа динамических систем. – 1982. – Вып. 6. – С. 3 – 11.11. Проценко В.С., Украинец Н.А. Применение обобщенного метода Фурье к решению первой основной задачи теории упругости в полупространстве с цилиндрической полостью / Вісник Запорізького національного університету. 2015. Вып. 2. С. 193–202.12. Николаев А.Г., Орлов Е.М. Решение первой осесимметричной термоупругой краевой задачи для трансверсально-изотропного полупространства со сфероидальной полостью / Проблеми обчислювальної механіки і міцності конструкцій. – 2012. – Вип.20. – С. 253-259.13. Miroshnikov V.Yu. First basic elasticity theory problem in a half-space with several parallel round cylindrical cavities / Journal of Mechanical Engineering. – 2018. – Vol. 21, № 2. – P. 12 – 18.14. Protsenko V., Miroshnikov V. Investigating a problem from the theory of elasticity for a half-space with cylindrical cavities for which boundary conditions of contact type are assigned / Eastern-European Journal of Enterprise Technologies. – 2018. – Vol 4, № 7 (94). – P. 43 – 50.15. Николаев А.Г., Танчик Е.А. Упругая механика многокомпонентных тел. – Харьков: Нац.аэрокосм.ун-т им. Н.Е. Жуковского «ХАИ». – 2014. – 272 с.16. Miroshnikov V.Yu. The study of the second main problem of the theory of elasticity for a layer with a cylindrical cavity / Strength of Materials and Theory of Structures. – 2019. – №102. – P. 77–90. https://doi.org/10.32347/2410-2547.2019.102.77-9017. Miroshnikov V.Yu., Medvedeva A.V., Oleshkevich S. V. Determination of the Stress State of the Layer with a Cylindrical Elastic Inclusion / Materials Science Forum. – 2019. – Vol. 968. – pp. 413-420. https://doi.org/10.4028/www.scientific.net/MSF.968.41318. Miroshnikov V. Investigation of the Stress Strain State of the Layer with a Longitudinal Cylindrical Thick-Walled Tube and the Displacements Given at the Boundaries of the Layer / Journal of Mechanical Engineering. – 2019. – Vol. 22, N 2. – P. 44-52. https://doi.org/10.15407/pmach2019.02.044.
1. Guz’ A.N., Kosmodamianskiy A.S., Shevchenko V.P. and others. Mekhanika kompozitov (Mechanics of composites). Vol 7. Kontsentratsiya napryazheniy (Concentration of stresses). Kiev: Nauk. Dumka. – 1998. – P. 114 – 137. (In Russian).2. Vaysfel’d N., Popov G., Reut V. The axisymmetric contact interaction of an infinite elastic plate with an absolutely rigid inclusion / Acta Mech. – 2015. – vol. 226. – P. 797–810. doi: https://doi.org/10.1007/s00707-014-1229-73. Popov G., Vaysfel’d N. Osesimmetrichnaya zadacha teorii uprugosti dlya beskonechnoy plity s tsilindricheskim vklyucheniyem pri uchete yeye udel'nogo vesa (The axisymmetric problem of the theory of elasticity for an infinite plate with a cylindrical inclusion, taking into account its specific gravity) / Prikladnaya mekhanika (Applied mechanics). – 2014. – Vol. 50, № 6. – P. 27–38.4. Bobyleva T. Approximate Method of Calculating Stresses in Layered Array / Procedia Engineering. – 2016. – Vol.153. – P.103 – 106. https://doi.org/10.1016/j.proeng.2016.08.0875. Grinchenko V.T., Ulitko A.F. Prostranstvennyye zadachi teorii uprugosti i plastichnosti. Ravnovesiye uprugikh tel kanonicheskoy formy (Spatial Problems of the Theory of Elasticity and Plasticity. Equilibrium of elastic bodies of canonical form). – Kiyev: Naukova Dumka. – 1985.– 280 p. (in Russian).6. Guz' A.N., Kubenko V.D. , Cherevko M.A. Difraktsiya uprugikh voln (Diffraction of elastic waves). – Kiev: Nauk. Dumka. – 1978. – 307 p. (In Russian).7. Grinchenko V.T., Meleshko V.V. Garmonicheskiye kolebaniya i volny v uprugikh telakh (Harmonic vibrations and waves in elastic bodies). – Kiev: Nauk. Dumka. – 1981. – 284 p. (In Russian).8. Volchkov V.V., Vukolov D.S., Storogev V.I. Difraktsiya voln sdviga na vnutrennikh tunnel'nykh tsilindricheskikh neodnorodnostyakh v vide polosti i vklyucheniya v uprugom sloye so svobodnymi granyami (Diffraction of shear waves by internal tunneling cylindrical non-homogeneities in the form of a cavity and inclusion in an elastic layer with free faces) / Mekhanika tverdogo tela (Solid mechanics). – 2016. – Vol. 46. – P. 119 – 133. (In Russian).9. Nikolaev A.G., Protsenko V.S. Obobshchennyy metod Fur'ye v prostranstvennykh zadachakh teorii uprugosti (Generalized Fourier method in spatial problems of the theory of elasticity). –Kharkov: Nats. aerokosm. universitet im. N.Ye. Zhukovskogo «KHAI» (National Aerospace University "KhAI"), 2011. – 344 с. (In Russian).10. Protsenko V.S., Nikolaev A.G. Prostranstvennaya zadacha Kirsha (Kirsch spatial problem) / Matematicheskiye metody analiza dinamicheskikh sistem (Mathematical methods for analyzing dynamic systems). – 1982. – Vol. 6. – P. 3 – 11. (In Russian).11. Protsenko V.S., Ukrainec N.A. Primeneniye obobshchennogo metoda Fur'ye k resheniyu pervoy osnovnoy zadachi teorii uprugosti v poluprostranstve s tsilindricheskoy polost'yu (Application of the generalized Fourier method to the solution of the first main problem of the theory of elasticity in a half-space with a cylindrical cavity) / Visnyk Zaporizʹkoho natsionalʹnoho universytetu (Bulletin of Zaporizhzhya National University). 2015. Vol. 2. P. 193–202. (In Russian).12. Nikolaev A.G., Orlov E.M. Resheniye pervoy osesimmetrichnoy termouprugoy krayevoy zadachi dlya transversal'no-izotropnogo poluprostranstva so sferoidal'noy polost'yu (Solution of the first axisymmetric thermoelastic boundary value problem for a transversely isotropic half-space with a spheroidal cavity) / Problemy obchyslyuvalʹnoyi mekhaniky i mitsnosti konstruktsiy (Problems of Computational Mechanics and Strength of Structures). – 2012. – Vol.20. – P. 253-259. (In Russian).13. Miroshnikov V.Yu. First basic elasticity theory problem in a half-space with several parallel round cylindrical cavities / Journal of Mechanical Engineering. – 2018. – Vol. 21, № 2. – P. 12 – 18.14. Protsenko V., Miroshnikov V. Investigating a problem from the theory of elasticity for a half-space with cylindrical cavities for which boundary conditions of contact type are assigned / Eastern-European Journal of Enterprise Technologies. – 2018. – Vol 4, № 7 (94). – P. 43 – 50.15. Nikolaev A.G., Tanchyk E.A. Uprugaya mekhanika mnogokomponentnykh tel (Elastic mechanics of multicomponent bodies). – Kharkov: Nats. aerokosm. universitet im. N.Ye. Zhukovskogo «KHAI» (National Aerospace University "KhAI"). – 2014. – 272 p. (In Russian).16. Miroshnikov V. Yu. The study of the second main problem of the theory of elasticity for a layer with a cylindrical cavity / Strength of Materials and Theory of Structures. – 2019. – №102. – P. 77–90. https://doi.org/10.32347/2410-2547.2019.102.77-9017. Miroshnikov V.Yu., Medvedeva A.V., Oleshkevich S.V. Determination of the Stress State of the Layer with a Cylindrical Elastic Inclusion / Materials Science Forum. – 2019. – Vol. 968. – pp. 413-420. https://doi.org/10.4028/www.scientific.net/MSF.968.41318. Miroshnikov V. Investigation of the Stress Strain State of the Layer with a Longitudinal Cylindrical Thick-Walled Tube and the Displacements Given at the Boundaries of the Layer / Journal of Mechanical Engineering. – 2019. – Vol. 22, N 2. – P. 44 – 52. https://doi.org/10.15407/pmach2019.02.044