Differential equationsof balansed continuum for planar deformation in cylindrical coordinates at bilinear approximation of clossing equations

S. V. Bakushev

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 69 (2021), pp. 69-85

Résumé

Problems of the formulation of differential equations of equilibrium in terms of displacements for a plane strain of continuous media at bilinear approximation of closing equations are considered leaving out of account geometric nonlinearity in the cylindrical coordinate system. Based on the assumption that the curves of volumetric and shear strain are independent from each other, six main cases of physical dependencies are considered, which are the functions of the relative position of break points on the bilinear curves of the volumetric and shear strain. Obtaining of bilinear physical dependencies is based on the calculation of secant moduli of the volumetric and shear strain. On the first line of the curves, secant moduli are constant for both volumetric and shear strain, while on the second line, the secant modulus of the volumetric strain is a function of the volumetric strain, and the secant modulus of the shear strain is a function of the shear strain intensity. Putting the corresponding bilinear physical equations into differential equations of continuum equilibrium, which disregard geometrical nonlinearity, the resulting differential equations of equilibrium are obtained in terms of displacements for a one-dimensional plane strain of continuum in the cylindrical coordinate system. These equations can be used when determining stress-strain state of continuous media under one-dimensional plane strains with no regard for geometrical nonlinearity, and whose physical relations are approximated by bilinear functions.

Détail

Keywords: solid body, plane strain, cylindrical coordinates, differential equations of equilibrium, closing bilinear equations, geometrically linear model.

Bibliographie

[1] Novozhilov V. V., Teoriya uprugosti, Sudpromgiz, L., 1958, 370 pp.

[2] Bakushev S. V., Geometricheski i fizicheski nelineinaya mekhanika sploshnoi sredy: Ploskaya zadacha, Knizhnyi dom «Librokom», M., 2013, 312 pp.

[3] Rudykh O. L., “Prakticheskie voprosy approksimatsii eksperimentalnykh krivykh stepennymi i drobno-lineinymi funktsiyami”, Vestnik Tomskogo gosudarstvennogo arkhitekturno-stroitelnogo universiteta, 2010, no. 1 (26), 110–122

[4] Gonik E. G., Kibets A. I., Petrov M. V., Fedorova T. G., Frolova I. A., “Vliyanie approksimatsii diagrammy deformirovaniya na kriticheskie nagruzki pri poperechnom izgibe tsilindricheskoi obolochki”, Problemy prochnosti i plastichnosti, 79:2 (2017), 169–181

[5] Korol E. Z., Opredelenie diagramm nelineinogo deformirovaniya raznosoprotivlyayuschikhsya materialov pri neodnorodnom napryazhenno-deformirovannom sostoyanii, Deponirovannaya rukopis No 86-V2005, 2005

[6] Zubchaninov V. G., “Obschaya matematicheskaya teoriya plastichnosti i postulaty makroskopicheskoi opredelimosti i izotropii A. A. Ilyushina”, Vestnik Moskovskogo universiteta. Seriya 1: Matematika. Mekhanika, 2018, no. 5, 29–46 | Zbl

[7] Sadovskii V. M., “Reologicheskie modeli raznomodulnykh i sypuchikh sred”, Dalnevostochnyi matematicheskii zhurnal, 4:2 (2003), 252–263

[8] Morschinina A. A., “Nelineinaya osesimmetrichnaya zadacha teorii uprugosti dlya poloi sfery”, Vestnik Sankt-Peterburgskogo universiteta. Matematika. Mekhanika. Astronomiya, 2009, no. 4, 84–88

[9] Fedotova I. A., “Matematicheskaya model raboty nelineinogo sterzhnevogo elementa konstruktsii”, Izvestiya Peterburgskogo universiteta putei soobscheniya, 2004, no. 1, 7–12

[10] Mitskevich S. A., Krysko A. V., Zhigalov M. V., Krysko V. A., “Dinamicheskaya ustoichivost pologikh obolochek na pryamougolnom plane s uchetom geometricheskoi i fizicheskoi nelineinosti”, Problemy prochnosti i plastichnosti, 79:3 (2017), 249–258

[11] Makhmetova N. M., “Ob odnom algoritme nelineinykh zadach statiki podzemnykh sooruzhenii”, Vestnik Kazakhskoi akademii transporta i kommunikatsii im. M. Tynyshpaeva, 2014, no. 5 (90), 80–87

[12] Krysko V. A., Papkova I. V., Saltykova O. A., Babenkova T. V., Kashubina A. A., “Slozhnye kolebaniya balok Eilera–Bernulli s uchetom geometricheskoi i fizicheskoi nelineinostei”, Mezhdunarodnyi nauchno-issledovatelskii zhurnal, 2014, no. 3-1 (22), 14–16

[13] Petrov V. V., “Raschet neodnorodnykh po tolschine obolochek s uchetom fizicheskoi i geometricheskoi nelineinostei”, Academia. Arkhitektura i stroitelstvo, 2016, no. 1, 112–117

[14] Egorov A. V., “Deformirovanie tsentralno-szhatogo gibkogo sterzhnya”, Inzhenernyi zhurnal: nauka i innovatsii, 2018, no. 4 (76), 1 pp. | DOI

[15] Lure A. I., Nelineinaya teoriya uprugosti, Nauka, M., 1980, 512 pp.

[16] Demidov S. P., Teoriya uprugosti, uchebnik dlya vuzov, Vyssh. shkola, M., 1979, 432 pp.

[17] Papkovich P. F., Teoriya uprugosti, Oborongiz, M.–L., 1939, 643 pp.

[18] Zaripov R. M., Masalimov R. B., Lisin Yu. V., “Modelirovanie napryazhenno-deformirovannogo sostoyaniya pryamolineinykh i krivolineinykh uchastkov truboprovoda”, Neftegazovoe delo, 13:3 (2015), 103–109

[19] Strelnikov V. N., Sukov G. S., Voloshin A. I., Chibisov Yu. V., Lesnyak G. A., “Uravneniya uprugosti v bipolyarnykh koordinatakh”, Progresivni tekhnologiï i sistemi mashinobuduvannya, 2010, no. 2 (40), 248–253

[20] Buntov A. E., Gotsev D. V., “Neodnorodnoe napryazhenno-deformirovannoe sostoyanie uprugogo tsilindricheskogo tela s uchetom vnutrennei struktury materiala”, Vestnik Dagestanskogo gosudarstvennogo tekhnicheskogo universiteta. Tekhnicheskie nauki, 45:1 (2018), 8–21

[21] Nemirovskii Yu. V., Matveev K. A., Mokhovnev D. V., “Ratsionalnoe proektirovanie armirovannykh koltsevykh plastin”, Problemy optimalnogo proektirovaniya sooruzhenii, sbornik dokladov 3-i Vserossiiskoi konferentsii, Novosibirskii GASU (Sibstrin), Sibirskoe otdelenie Rossiiskoi akademii arkhitektury i stroitelnykh nauk, Sibirskoe otdelenie mezhdunarodnoi akademii nauk vysshei shkoly, 2014, 272–282

[22] Radchenko V. P., Popov N. N., “Ispolzovanie metoda malogo parametra dlya resheniya stokhasticheskikh nelineinykh zadach teorii ustanovivsheisya polzuchesti”, Vestnik Chuvashskogo gosudarstvennogo pedagogicheskogo universiteta im. I.Ya. Yakovleva. Seriya: Mekhanika predelnogo sostoyaniya, 2013, no. 1 (15), 185–194

[23] Verveiko N. D., Frolova O. A., “Predelnoe osesimmetrichnoe napryazhennoe sostoyanie szhimaemogo sypuchego materiala s tsilindricheskoi polostyu”, Vestnik Chuvashskogo gosudarstvennogo pedagogicheskogo universiteta im. I.Ya. Yakovleva. Seriya: Mekhanika predelnogo sostoyaniya, 2015, no. 3 (25), 83–92

[24] Khristich D. V., Astapov Yu. V., “Uchet vzaimnogo vliyaniya polei napryazhenii, deformatsii i temperatur pri reshenii zadachi Lame”, Izvestiya Tulskogo gosudarstvennogo universiteta. Estestvennye nauki, 2015, no. 1, 67–73

[25] Kovalev A. V., Sviridov I. E., Scheglova Yu. D., “Primenenie metoda vozmuschenii pri opredelenii napryazhenno-deformirovannogo sostoyaniya dvukhsloinogo slaboanizotropnogo sterzhnya nekrugovogo poperechnogo secheniya pri uprugoplasticheskom kruchenii”, Vestnik Samarskogo gosudarstvennogo tekhnicheskogo universiteta. Seriya: Fiziko-matematicheskie nauki, 21:2 (2017), 292–307 | DOI | Zbl

[26] Chan L. T., Tarlakovskii D. V., “Uprugoe momentnoe poluprostranstvo pod deistviem osesimmetrichnykh nestatsionarnykh poverkhnostnykh kinematicheskikh vozmuschenii”, Problemy prochnosti i plastichnosti, 81:1 (2019), 40–52 | DOI

[27] Nilson Tadeu Mascia, Francisco Antonio Rocco Lahr, “Remarks on orthotropic elastic models applied to wood”, Materials Research, 9:3 (2006), 301–310 | DOI

[28] Xiaowen Lei, Toshiaki Natsuki, Jinxing Shi, Qing-Qing Ni, “Analysis of carbon nanotubes on the mechanical properties at atomic scale”, Journal of Nanomaterials, 2011 (2011) | DOI

[29] Bakushev S. V., “Differentsialnye uravneniya ravnovesiya sploshnoi sredy dlya ploskoi deformatsii v dekartovykh koordinatakh pri bilineinoi approksimatsii zamykayuschikh uravnenii (geometricheski lineinaya model)”, Regionalnaya arkhitektura i stroitelstvo, 2019, no. 1(38), 76–85

[30] Bakushev S. V., “Differentsialnye uravneniya ravnovesiya sploshnoi sredy dlya ploskoi deformatsii v dekartovykh koordinatakh pri bilineinoi approksimatsii zamykayuschikh uravnenii (geometricheski nelineinaya model)”, Regionalnaya arkhitektura i stroitelstvo, 2019, no. 2(39), 86–100

[31] Bakushev S. V., “Differentsialnye uravneniya ravnovesiya osesimmetrichnoi deformatsii pri bilineinoi approksimatsii zamykayuschikh uravnenii”, Stroitelnaya mekhanika i raschet sooruzhenii, 2019, no. 1, 8–17

[32] Bakushev S. V., “Differentsialnye uravneniya ravnovesiya tsentralno-simmetrichnoi deformatsii pri bilineinoi approksimatsii zamykayuschikh uravnenii”, Izvestiya vuzov. Stroitelstvo, 2018, no. 11(719), 5–19

[33] Bakushev S. V., “Approksimatsiya diagramm deformirovaniya bilineinymi funktsiyami”, Stroitelnaya mekhanika i raschet sooruzhenii, 2019, no. 2 (283), 2–11