Аннотації

Дослідження другої основної задачі теорії пружності для шару з N циліндричними включеннями

Автор(и):
Мірошніков В.Ю. , Денисова Т.В.
Автор(и) (англ)
Miroshnikov V.Yu., Denysova T.V.
Дата публікації:

28.05.2021

Анотація (укр):

При проектуванні конструкцій у вигляді армованого шару доводиться стикатися з ситуацією, коли стержні армування розташовані близько один до одного. У цьому випадку зростає вплив їх один на одного. Для отримання напружено-деформованого стану в зоні контакту шару і включення необхідно мати метод, який би дозволяв отримати результат з високою точністю. У цій роботі запропоновано аналітико-чисельний підхід до вирішення просторової задачі теорії пружності для шару з заданою кількістю поздовжніх циліндричних включень і заданих на межах шару переміщеннях. Розв’язок задачі отримано на основі узагальненого методу Фур'є щодо системи рівнянь Ламе в локальних циліндричних координатах, пов'язаних з включеннями і декартових координатах, пов'язаних з межами шару. Нескінченні системи лінійних алгебраїчних рівнянь, які отримані в результаті задоволення граничних умов і умов сполучення шару з включеннями, розв’язано методом редукції. В результаті отримані переміщення і напруження в різних точках розглянутого середовища. При порядку системи рівнянь 6, точність виконання граничних умов склала 10-2 для значень від 0 до 1. Чисельні дослідження алгебраїчної системи рівнянь дають підстави стверджувати, що її рішення може бути з будь-яким ступенем точності знайдено методом редукції, що підтверджується високою точністю виконання граничних умов. У чисельному аналізі порівнювалися варіанти шару з одним і з трьома включеннями. Результат показав, що близьке розташування стержнів армування збільшує напруження на поверхні цих включень. Також були отримані значення напружень на поверхнях контактів шару з включеннями. Запропонований алгоритм розв’язання можна використовувати при проектуванні конструкцій, розрахунковою схемою яких є шар з поздовжніми циліндричними включеннями і заданими на межах шару переміщеннях.

Анотація (рус):

Анотація (англ):

When designing structures in the form of a reinforced layer, one has to deal with the situation when the reinforcement bars are located close to each other. In this case, their influence on each other increases. In order to obtain the stress-strain state in the contact zone of the layer and the inclusion, it is necessary to have a method that would allow obtaining a result with high accuracy. In this work, an analytical-numerical approach to solving the spatial problem of the theory of elasticity for a layer with a given number of longitudinal cylindrical inclusions and displacements given at the boundaries of the layer has been proposed. The solution of the problem has been obtained by the generalized Fourier method with respect to the system of Lame's equation in local cylindrical coordinates associated with inclusions and Cartesian coordinates associated with layer boundaries. Infinite systems of linear algebraic equations obtained by satisfying the boundary conditions and conjugation conditions of a layer with inclusions have been solved by the reduction method. As a result, displacements and stresses have been obtained at different points of the considered medium. When the order of the system of equations is 6, the accuracy of fulfilling the boundary conditions was 10-2 for values from 0 to 1. Numerical studies of the algebraic system of equations give grounds to assert that its solution can be found with any degree of accuracy by the reduction method, which is confirmed by the high accuracy of fulfilling the boundary conditions. In the numerical analysis, variants of the layer with 1 and 3 inclusions have been compared. The result has shown that close placement of reinforcement bars increases stresses on the surface of these inclusions. The values of stresses on the layer contact surfaces with inclusions have also been obtained. The proposed solution algorithm can be used in the design of structures, the computational scheme of which is the layer with longitudinal cylindrical inclusions and displacements specified at the layer boundaries.

Література:

  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/s00707-014-1229-7
  3. 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.2016.08.087
  5. Guz' A.N., Kubenko V. D. , Cherevko M. A. Difraktsiya uprugikh voln (Diffraction of elastic waves). – Kiev: Nauk. Dumka. – 1978. – 307 p. (In Russian).
  6. 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).
  7. 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).
  8. 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 p. (In Russian).
  9. 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).
  10. 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).
  11. 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).
  12. Nikolaev A. G., Tanchik 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.
  13. Nikolaev A. G., Tanchik E. A.  The first boundary-value problem of the elasticity theory for a cylinder with N cylindrical cavities / Numerical Analysis and Applications. – 2015. – Vol. 8. – P. 148–158.
  14. Nikolaev A. G., Tanchik E. A. Stresses in an Infinite Circular Cylinder with Four Cylindrical Cavities / Journal of Mathematical Sciences. –2016. – Vol. 217, Iss. 3. – P. 299–311.
  15. Nikolaev A. G., Tanchik E. A.  Model of the Stress State of a Unidirectional Composite with Cylindrical Fibers Forming a Tetragonal Structure / Mechanics of Composite Materials. –2016. –Vol. 52.  – P. 177–188.
  16. Nikolaev A. G., Tanchik E. A. Stresses in an elastic cylinder with cylindrical cavities forming a hexagonal structure / Journal of Applied Mechanics and Technical Physics. – 2016. –Vol. 57. – P. 1141–1149.
  17. 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/10.32347/2410-2547.2019.102.77-90
  18. Miroshnikov V. Y. Stress State of an Elastic Layer with a Cylindrical Cavity on a Rigid Foundation / International Applied Mechanics. – 2020. –№56(3). – P. 372–381.
  19. Miroshnikov V. Determination of the stress state of a layer with a cylindrical cavity, located on an elastic base and given boundary conditions in the form of displacements / European Journal of Technical and Natural Sciences. Section 3. Machinery construction. – 2019. – №5-6. – P.21–25 https://doi.org/10.29013/EJTNS-19-5.6-21-26
  20. Miroshnikov V. Yu., Protsenko V.S.  Determining the stress state of a layer on a rigid base weakened by several longitudinal cylindrical cavities / Journal of Advanced Research in Technical Science. – 2019. – Iss. 17. – P.11–21  https://doi.org/10.26160/2474-5901-2019-17-11-21
  21. Miroshnikov V., Denysova T., Protsenko V. The study of the first main problem of the theory of elasticity for a layer with a cylindrical cavity / Strength of Materials and Theory of Structures. – 2019. – №103. – P. 208–218. DOI: https://doi.org/10.32347/2410-2547.2019.103.208-218
  22. 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.968.413
  

References:

  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/s00707-014-1229-7
  3. 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.2016.08.087
  5. Guz' A.N., Kubenko V. D. , Cherevko M. A. Difraktsiya uprugikh voln (Diffraction of elastic waves). – Kiev: Nauk. Dumka. – 1978. – 307 p. (In Russian).
  6. 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).
  7. 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).
  8. 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 p. (In Russian).
  9. 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).
  10. 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).
  11. 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).
  12. Nikolaev A. G., Tanchik 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.
  13. Nikolaev A. G., Tanchik E. A.  The first boundary-value problem of the elasticity theory for a cylinder with N cylindrical cavities / Numerical Analysis and Applications. – 2015. – Vol. 8. – P. 148–158.
  14. Nikolaev A. G., Tanchik E. A. Stresses in an Infinite Circular Cylinder with Four Cylindrical Cavities / Journal of Mathematical Sciences. –2016. – Vol. 217, Iss. 3. – P. 299–311.
  15. Nikolaev A. G., Tanchik E. A.  Model of the Stress State of a Unidirectional Composite with Cylindrical Fibers Forming a Tetragonal Structure / Mechanics of Composite Materials. –2016. –Vol. 52.  – P. 177–188.
  16. Nikolaev A. G., Tanchik E. A. Stresses in an elastic cylinder with cylindrical cavities forming a hexagonal structure / Journal of Applied Mechanics and Technical Physics. – 2016. –Vol. 57. – P. 1141–1149.
  17. 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/10.32347/2410-2547.2019.102.77-90
  18. Miroshnikov V. Y. Stress State of an Elastic Layer with a Cylindrical Cavity on a Rigid Foundation / International Applied Mechanics. – 2020. –№56(3). – P. 372–381.
  19. Miroshnikov V. Determination of the stress state of a layer with a cylindrical cavity, located on an elastic base and given boundary conditions in the form of displacements / European Journal of Technical and Natural Sciences. Section 3. Machinery construction. – 2019. – №5-6. – P.21–25 https://doi.org/10.29013/EJTNS-19-5.6-21-26
  20. Miroshnikov V. Yu., Protsenko V.S.  Determining the stress state of a layer on a rigid base weakened by several longitudinal cylindrical cavities / Journal of Advanced Research in Technical Science. – 2019. – Iss. 17. – P.11–21  https://doi.org/10.26160/2474-5901-2019-17-11-21
  21. Miroshnikov V., Denysova T., Protsenko V. The study of the first main problem of the theory of elasticity for a layer with a cylindrical cavity / Strength of Materials and Theory of Structures. – 2019. – №103. – P. 208–218. DOI: https://doi.org/10.32347/2410-2547.2019.103.208-218
  22. 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.968.413