УПРУГОПЛАСТИЧЕСКОЕ КРУЧЕНИЕ С КОНЕЧНЫМИ ДЕФОРМАЦИЯМИ: СРАВНЕНИЕ АНАЛИТИЧЕСКОГО И МКЭ-МОДЕЛИРОВАНИЯ ДЛЯ НЕМОНОТОННО УПРОЧНЯЮЩИХСЯ ПОЛИМЕРОВ

Севастьянов, Г.М.; Бормотин, К.С.

doi:10.15593/perm.mech/2023.3.11

УПРУГОПЛАСТИЧЕСКОЕ КРУЧЕНИЕ С КОНЕЧНЫМИ ДЕФОРМАЦИЯМИ: СРАВНЕНИЕ АНАЛИТИЧЕСКОГО И МКЭ-МОДЕЛИРОВАНИЯ ДЛЯ НЕМОНОТОННО УПРОЧНЯЮЩИХСЯ ПОЛИМЕРОВ

Г.М. Севастьянов, К.С. Бормотин

Сведения об авторах

Г.М. Севастьянов
Комсомольский-на-Амуре государственный университет, Комсомольск-на-Амуре, Россия
К.С. Бормотин
Комсомольский-на-Амуре государственный университет, Комсомольск-на-Амуре, Россия

https://doi.org/10.15593/perm.mech/2023.3.11

Полимерные материалы в зависимости от структуры и химического состава демонстрируют различные типы изотропного деформационного упрочнения. В том числе в диапазоне пластического деформирования на истинной кривой «деформация – напряжение» может присутствовать нисходящий участок разупрочнения, вызванный ослаблением межмолекулярных связей, сменяющийся далее степенным упрочнением. Законы деформирования материалов могут быть установлены из простых опытов, одним из которых часто выступает кручение. Для кручения тонкостенных цилиндрических образцов напряженно-деформированное состояние практически однородно, поэтому такие опыты просто интерпретировать. Однако при больших деформациях тонкостенных образцов возникают проблемы устойчивости. Для полнотелых цилиндрических образцов напряженное состояние неоднородно, интерпретация таких опытов возможна на базе МКЭ-моделирования или с использованием точных или приближенных аналитических решений соответствующих начально-краевых задач механики. В настоящем исследовании представлено точное аналитическое решение упругопластической задачи кручения цилиндрического образца, справедливое для произвольного закона изотропного упрочнения. В качестве кинематики упругопластического деформирования используется мультипликативная формулировка. Нелинейно-упругие свойства материала описываются моделью Муни – Ривлина. В условии пластического течения используется эквивалентное напряжение Треска, что позволяет получить замкнутое решение. Рассчитаны интегральные характеристики процесса – крутящий момент и осевая сила (эффект второго порядка). Аналитические результаты сопоставлены с результатами численного моделирования в MSC.Marc, а также с доступными экспериментальными данными. Аналитическое решение для крутящего момента очень близко соответствует численному решению методом конечных элементов. Также удовлетворительно совпадают кривые осевой силы. При умеренных деформациях аналитическое решение достаточно точно описывает экспериментальные результаты.

Ключевые слова

кручениеконечные деформацииупругопластическая задачаизотропное упрочнениеполимерыконечно-элементный анализ.

Finite-strain elastic-plastic torsion: Analytical and FEM modeling for nonmonotonically hardening polymers

Polymeric materials, depending on the structure and chemical composition, exhibit various types of isotropic strain hardening. In particular, in the range of plastic deformation on the "true strain – true stress" curve, there may be a descending section of softening caused by the weakening of intermolecular bonds. This softening region is further replaced by a power-law hardening. The laws of deformation of materials can be established from simple experiments, one of which is often torsion. For torsion of thin-walled cylindrical specimens, the stress-strain state is practically uniform; therefore, such experiments are easy to interpret. However, stability problems arise at large deformations of thin-walled specimens. For solid cylindrical specimens, the stress state is inhomogeneous; interpretation of such experiments is possible on the basis of FEM modeling or using exact or approximate analytical solutions of the corresponding initial-boundary value problems of mechanics. In the present study, an exact analytical solution of the elastic- plastic problem of torsion of a cylindrical sample is presented, which is valid for an arbitrary law of isotropic hardening. The multiplicative decomposition is utilized as the kinematics of elastic- plastic deformation. The non-linear elastic properties of the material are described by the Mooney – Rivlin model. In the plasticity condition, the Tresca equivalent stress is used, which makes it possible to obtain a closed solution. The integral characteristics of the process (torque and axial force that represents a second order effect) are calculated. The analytical results are compared with the results of numerical simulations in MSC.Marc, as well as with the available experimental data. The analytical solution for torque corresponds very closely to the numerical solution obtained by the finite element method. Also, the curves of axial force coincide satisfactorily. At moderate strains, the analytical solution accurately describes the experimental results.

Keywords

torsionfinite strainelastoplasticityisotropic hardeningpolymersFEA.

Литература

1.Haward R.N., Thackray G. The use of a mathematical model to describe isothermal stress-strain curves in glassy thermoplastics // Proceedings of the Royal Society of London A. – 1967. – Vol. 302, no. 1471. – P. 453–72. DOI: 10.1098/rspa.1968.0029 Scholar
2.Wu P.D., Van Der Giessen E. On improved network models for rubber elasticity and their applications to orientation hardening in glassy polymers // Journal of the Mechanics and Physics of Solids. – 1993. – Vol. 41. – P. 427–456. DOI: 10.1016/0022- 5096(93)90043-F Scholar
3.Wu P.D., Van der Giessen E. On large-strain inelastic torsion of glassy polymers // International Journal of Mechanical Sciences. – 1993. – Vol. 35, no. 11. – P. 935–951. DOI: 10.1016/0020-7403(93)90031-O Scholar
4.Meijer H.E.H., Govaert L.E. Mechanical performance of polymer systems: The relation between structure and properties // Progress in Polymer Science. – 2005. – Vol. 30. – P. 915–938. DOI: 10.1016/j.progpolymci.2005.06.009 Scholar
5.Cheng L., Guo T.F. Void interaction and coalescence in polymeric materials // International Journal of Solids and Structures. – 2007. – Vol. 44. – P. 1787–1808. DOI: 10.1016/j.ijsolstr.2006.08.007 Scholar
6.Wu P.D., Neale K.W., Van der Giessen E. Large strain torsion of axially-constrained solid rubber bars // Acta Mechanica Sinica. –1994. – Vol. 10. – P. 136–149. DOI: 10.1007/BF02486584 Scholar
7.Material parameters identification: Gradient-based, genetic and hybrid optimization algorithms / B.M. Chaparro, S. Thuillier, L.F. Menezes, P.Y.Manach, J.V. Fernandes // Computational Materials Science. – 2008. – Vol. 44, no. 2. – P. 339–346. DOI: 10.1016/j.commatsci.2008.03.028 Scholar
8.Hosseinzadeh A.R., Mahmoudi A.H. Determination of mechanical properties using sharp macro-indentation method and genetic algorithm // Mechanics of Materials. – 2017. – Vol. 114. – P. 57–68. DOI: 10.1016/j.mechmat.2017.07.004 Scholar
9.Sevenois R.D.B., Van Papegem W. Fatigue Damage Modeling Techniques for Textile Composites: Review and Comparison with Unidirectional Composite Modeling Techniques // Applied Mechanics Reviews. – 2015. – Vol. 67, iss. 2. – P. 020802. Scholar
10.Optimization of Chaboche kinematic hardening parameters for 20MnMoNi55 reactor pressure vessel steel by sequenced genetic algorithms maintaining the hierarchy of dependence / S. Mal, S. Bhattacharjee, M. Jana, P. Das, S.K. Acharyya // Engineering Optimization. – 2021. – Vol. 53, no. 2. – P. 335–347. DOI: 10.1080/0305215X.2020.1726340 Scholar
11.Sener B. Description of anomalous behavior of aluminum alloys with Hill48 yield criterion by using different experimental inputs and weight coefficients // Journal of Applied and Computational Mechanics. – 2021. – Vol. 7, no. 3. – P. 1606– 1619. DOI: 10.22055/jacm.2021.36297.2821 Scholar
12.Grabski J.K., Mrozek A. Identification of elastoplastic properties of rods from torsion test using meshless methods and a metaheuristic // Computers & Mathematics with Applications. – 2021. – Vol. 92. – P. 149–158. DOI: 10.1016/j.camwa.2021.03.024 Scholar
13.Liu I.-S. A note on the Mooney – Rivlin material model // Continuum Mechanics and Thermodynamics. – 2012. – Vol. 24. – P. 583–590. DOI: 10.1007/s00161-011-0197-6 Scholar
14.Levitas V.I. Large deformation of materials with complex rheological properties at normal and high pressure. – New York: Nova Science Publishers, 1996. Scholar
15.Phase-field simulation of stress-induced martensitic phase transformations at large strains / V.A. Levin, V.I. Levitas, K.M. Zingerman, E.I. Freiman // International Journal of Solids and Structures. – 2013. – Vol. 50, no. 19. – P. 2914–2928. DOI: 10.1016/j.jsolistr.2013.05.003 Scholar
16.Feng B., Levitas V.I., Li W. FEM modeling of plastic flow and strain-induced phase transformation in BN under high pressure and large shear in a rotational diamond anvil cell // International Journal of Plasticity. – 2019. – Vol. 113. – P. 236–254. DOI: 10.1016/j.ijplas.2018.10.004 Scholar
17.Sevastyanov G.M. Adiabatic heating effect in elastic- plastic contraction / expansion of spherical cavity in isotropic incompressible material // European Journal of Mechanics – A/Solids. – 2021. – Vol. 87. – Article 104223. DOI: 10.1016/j.euromechsol.2021.104223 Scholar
18.Poroboví A.A. Termodinamika упруго-неупругого процесса при конечных деформациях // Прикладная механика и техническая физика. – 2007. – Т. 48, № 4. – С. 144–153. Scholar
19.Sevastyanov G.M. Analytical solution for high-pressure torsion in the framework of geometrically nonlinear non- associative plasticity // International Journal of Solids and Structures. – 2020. – Vol. 206. – P. 383–395. DOI: 10.1016/j.ijsolstr.2020.09.028 Scholar
20.Leonov A.I. Nonequilibrium thermodynamics and rheology of viscoelastic polymer media // Rheologica Acta. – 1976. – Vol. 15. – P. 85–98. DOI: 10.1007/BF01517499 Scholar
21.Лурье А.И. Нелинейная теория упругости. – М: Наука, 1980. Scholar
22.Feng B., Levitas V.I., Hemley R.J. Large elastoplasticity under static megabar pressures: Formulation and application to compression of samples in diamond anvil cells // International Journal of Plasticity. – 2016. – Vol. 84. – P. 33–57. DOI: 10.1016/j.ijplas.2016.04.017 Scholar
23.Bowden P.B., Jukes J.A. The plastic flow of isotropic polymers // Journal of Materials Science. – 1972. –Vol. 7. – P. 52–63. DOI: 10.1007/BF00549550 Scholar
24.Yield criteria for amorphous glassy polymers / R. Quinson, J. Perez, M. Rink, A. Pavan // Journal of Materials Science. – 1997. – Vol. 32. – P. 1371–1379. DOI: 10.1023/A:1018525127466 Scholar
25.Rottler J., Robbins M.O. Yield conditions for deformation of amorphous polymer glasses // Physical Reviews E. – 2001. – Vol. 64. – Article 051801. DOI: 10.1103/PhysRevE.64.051801 Scholar
26.Anisotropic mechanical behavior of semi-crystalline polymers: Characterization and modeling of non-monotonic loading including damage / R.B. Arieby, K. Mrabet, O.A. Terfas, C. Laurent, R. Rahouadj // Journal of Applied Polymer Science. – 2017. – Vol. 134, no. 7. – Article 44468. DOI: 10.1002/app.44468 Scholar
27.Rivlin R.S. Large elastic deformations of isotropic materials. VI. Further results in the theory of torsion, shear and flexure // Philosophical Transactions of the Royal Society A. – 1949. – Vol. 242. – P. 173–195. DOI: 10.1098/rsta.1949.0009 Scholar
28.Арутюнян Н.Х., Радаев Ю.Н. Упругопластическое кручение цилиндрического стержня при конечных деформациях // Прикладная математика и механика. – 1989. – Т. 53, № 6. – С. 1014–1022. Scholar
29.Севастьянов Г.М., Буренин А.А. О больших деформациях при кручении несжимаемого упругопластического цилиндра // Доклады Академии наук. – 2018. – Т. 482, № 3. – С. 285–287. Scholar
30.Севастьянов Г.М., Буренин А.А. Адиабатический нагрев материала при упругопластическом кручении с конечными деформациями // Прикладная механика и техническая физика. – 2019. – Т. 60, № 6. – С. 149–161. Scholar
31.Marc. Volume B: Element Library. MSC.Software Corporation, 2021 [Электронный ресурс]. – URL: http://www.mscsoft ware.com/product/marc (дата обращения: 04.09.2022). Scholar
32.Гаришин О.К., Корляков А.С., Шадрин В.В. Моделирование упруго-вязко-пластических свойств термопластических полимеров. Комплексный экспериментально-теоретический подход // Вычислительная механика сплошных сред. – 2014. – Т. 7, № 2. – С. 208–218. DOI: 10.7242/1999-6691/2014.7.2.21 Scholar
33.Bathe K.-J. Finite element procedures. – New Jersey: Prentice Hall, 1982. Scholar
34.A thermo-mechanically coupled theory for large deformations of amorphous polymers. Part II: Applications / N.M. Ames, V. Srivastava, S.A. Chester, L. Anand // International Journal of Plasticity. – 2009. – Vol. 25, no. 8. – P. 1495–1539. DOI: 10.1016/j.ijplas.2008.11.005 Scholar
35.Карпов Е.В., Ларичкин А.Ю. Влияние осевого сжатия и крутящего момента на локализацию деформаций и разрушение при сложном циклическом нагружении стержней из оргстекла // Прикладная механика и техническая физика. – 2014. – Т. 55, № 1. – С. 115–126. Scholar
36.Mulliken A.D., Boyce M.C. Mechanics of the rate-dependent elastic-plastic deformation of glassy polymers from low to high strain rates // International Journal of Solids and Structures. – 2006. – Vol. 43, no. 5. – P. 1331–1356. DOI: 10.1016/j.ijsolstr.2005.04.016 Scholar
37.Influence of temperature and strain rate on the mechanical behavior of three amorphous polymers: Characterization and modeling of the compressive yield stress / J. Richeton, S. Ahzi, K.S. Vecchio, F.C. Jiang, Adharapurapu R.R. // International Journal of Solids and Structures. – 2006. – Vol. 43. – P. 2318–2335. DOI: 10.1016/j.ijsolstr.2005.06.040 Scholar
38.G'Sell C., Boni S., Shrivastava S. Application of the plane simple shear test for determination of the plastic behaviour of solid polymers at large strains // Journal of Materials Science. – 1983. – Vol. 18. – P. 903–918. DOI: 10.1007/BF00745590 Scholar

1.Haward R.N., Thackray G. The use of a mathematical model to describe isothermal stress-strain curves in glassy thermoplastics. Proceedings of the Royal Society of London A, 1967, 302(1471), pp. 453–72. doi: 10.1098/rspa.1968.0029 Scholar
2.Wu P.D., Van Der Giessen E. On improved network models for rubber elasticity and their applications to orientation hardening in glassy polymers. Journal of the Mechanics and Physics of Solids, 1993, 41, pp. 427–456. doi: 10.1016/0022-5096(93)90043-F Scholar
3.Wu P.D., Van der Giessen E. On large-strain inelastic torsion of glassy polymers. International Journal of Mechanical Sciences, 1993, 35(11), pp. 935–951. doi: 10.1016/0020- 7403(93)90031-O Scholar
4.Meijer H.E.H., Govaert L.E. Mechanical performance of polymer systems: The relation between structure and properties. Progress in Polymer Science, 2005, 30, pp. 915–938. doi: 10.1016/j.progpolymci.2005.06.009 Scholar
5.Cheng L., Guo T.F. Void interaction and coalescence in polymeric materials. International Journal of Solids and Structures, 2007, 44, pp. 1787–1808. doi: 10.1016/j.ijsolstr.2006.08.007 Scholar
6.Wu P.D., Neale K.W., Van der Giessen E. Large strain torsion of axially-constrained solid rubber bars. Acta Mechanica Sinica, 1994, 10, pp. 136–149. doi: 10.1007/BF02486584 Scholar
7.Chaparro B.M., Thuillier S., Menezes L.F., Manach P.Y., Fernandes J.V. Material parameters identification: Gradient-based, genetic and hybrid optimization algorithms. Computational Materials Science, 2008, 44(2), pp. 339–346. doi: 10.1016/j.commatsci.2008.03.028 Scholar
8.Hosseinzadeh A.R., Mahmoudi A.H. Determination of mechanical properties using sharp macro-indentation method and genetic algorithm. Mechanics of Materials, 2017, 114, pp. 57–68. doi: 10.1016/j.mechmat.2017.07.004 Scholar
9.Sevenois R.D.B., Van Papegem W. Fatigue Damage Modeling Techniques for Textile Composites: Review and Comparison with Unidirectional Composite Modeling Techniques // Applied Mechanics Reviews. – 2015. – Vol. 67, iss. 2. – P. 020802. Scholar
10.Mal S., Bhattacharjee S., Jana M., Das P., Acharyya S.K. Optimization of Chaboche kinematic hardening parameters for 20MnMoNi55 reactor pressure vessel steel by sequenced genetic algorithms maintaining the hierarchy of dependence. Engineering Optimization, 2021, 53(2), pp. 335–347. doi: 10.1080/0305215X.2020.1726340 Scholar
11.Sener B. Description of anomalous behavior of aluminum alloys with Hill48 yield criterion by using different experimental inputs and weight coefficients. Journal of Applied and Computational Mechanics, 2021, 7(3), pp. 1606–1619. doi: 10.22055/jacm.2021.36297.2821 Scholar
12.Grabski J.K., Mrozek A. Identification of elastoplastic properties of rods from torsion test using meshless methods and a metaheuristic. Computers & Mathematics with Applications, 2021, 92, pp. 149–158. doi: 10.1016/j.camwa.2021.03.024 Scholar
13.Liu L-S. A note on the Mooney – Rivlin material model. Continuum Mechanics and Thermodynamics, 2012, 24, pp. 583– 590. doi: 10.1007/s00161-011-0197-6 Scholar
14.Levitas V.I. Large deformation of materials with complex rheological properties at normal and high pressure. New York, Nova Science Publishers, 1996. Scholar
15.Levin V.A., Levitas V.I., Zingerman K.M., Freiman E.I. Phase-field simulation of stress-induced martensitic phase transformations at large strains. International Journal of Solids and Structures, 2013, 50(19), pp. 2914–2928. doi: 10.1016/j.ijsolstr.2013.05.003 Scholar
16.Feng B., Levitas V.I., Li W. FEM modeling of plastic flow and strain-induced phase transformation in BN under high pressure and large shear in a rotational diamond anvil cell. International Journal of Plasticity, 2019, 113, pp. 236–254. doi: 10.1016/j.ijplas.2018.10.004 Scholar
17.Sevastyanov G.M. Adiabatic heating effect in elastic- plastic contraction / expansion of spherical cavity in isotropic incompressible material. European Journal of Mechanics - A/Solids, 2021, 87, 104223. doi: 10.1016/j.euromechsol.2021.104223 Scholar
18.Rogovoi A.A. Thermodynamics of finite strain elastic- inelastic deformation. Jornal of Applied Mechanics and Technical Physics, 2007, 48(4), 591-598. doi: 10.1007/s10808-007-0074-z Scholar
19.Sevastyanov G.M. Analytical solution for high- pressure torsion in the framework of geometrically nonlinear non-associative plasticity. International Journal of Solids and Structures, 2020, 206, pp. 383-395. doi: 10.1016/j.ijsolstr.2020.09.028 Scholar
20.Leonov A.I. Nonequilibrium thermodynamics and rheology of viscoelastic polymer media. Rheologica Acta, 1976, 15, pp. 85-98. doi: 10.1007/BF01517499 Scholar
21.Lurie A.I. Nonlinear theory of elasticity. New York, North-Holland, 1990. Scholar
22.Feng B., Levitas V.I., Hemley R.J. Large elastoplasticity under static megabar pressures: Formulation and application to compression of samples in diamond anvil cells. International Journal of Plasticity, 2016, 84, pp. 33-57. doi: 10.1016/j.ijplas.2016.04.017 Scholar
23.Bowden P.B., Jukes J.A. The plastic flow of isotropic polymers. Journal of Materials Science, 1972, 7, pp. 52-63. doi: 10.1007/BF00549550 Scholar
24.Quinson R., Perez J., Rink M., Pavan A. Yield criteria for amorphous glassy polymers. Journal of Materials Science, 1997, 32, pp. 1371-1379. doi: 10.1023/A:1018525127466 Scholar
25.Rottler J., Robbins M.O. Yield conditions for deformation of amorphous polymer glasses. Physical Reviews E, 2001, 64, 051801. doi: 10.1103/PhysRevE.64.051801 Scholar
26.Arieby R.B., Mrabet K., Terfas O.A., Laurent C., Rahouadj R. Anisotropic mechanical behavior of semi-crystalline polymers: Characterization and modeling of non-monotonic loading including damage. Journal of Applied Polymer Science, 2017, 134(7), 44468. doi: 10.1002/app.44468 Scholar
27.Sevastyanov G.M., Burenin A.A. Finite strain upon elastic- plastic torsion of an incompressible circular cylinder. Doklady Physics, 2018, 63, pp. 393-395. doi: 10.1134/S1028335818090094 Scholar
28.Rivlin R.S. Large elastic deformations of isotropic materials. VI. Further results in the theory of torsion, shear and flexure. Philosophical Transactions of the Royal Society A, 1949, 242, pp. 173-195. doi: 10.1098/rsta.1949.0009 Scholar
29.Arutyunyan N.Kh., Radayev Yu.N. Elastoplastic torsion of a cylindrical rod for finite deformations. Journal of Applied Mathematics and Mechanics, 1989, 53(6), pp. 804-811. doi: 10.1016/0021-8928(89)90090-7 Scholar
30.Sevast’yanov G.M., Burenin A.A. Local adiabatic heating effect in finite-strain elastic-plastic torsion. Journal of Applied Mechanics and Technical Physics, 2019, 60(6), pp. 1104-1114. doi: 10.1134/S0021894419060166 Scholar
31.Marc. Volume B: Element Library. MSC.Software Corporation, 2021 (http://www.mscsoftware.com/product/marc) Scholar
32.Garishin O.K., Korylakov A.S., Shadrin V.V. Modelirovanie uprugo-vyazko-plasticheskih svojstv termoplasticheskih polimerov. Kompleksnyj eksperimental'no- teoreticheskij podhod [Modeling of elastic-viscoplastic properties of thermoplastic polymers. Integrated experimental-theoretical approach]. Vychislitel'naya mekhanika sploshnyh sred [Computational Continuum Mechanics], 2014, 7(2), c. 208-218. doi: 10.7242/1999-6691/2014.7.2.21 Scholar
33.Bathe K.-J., 1982. Finite element procedures. New Jersey, Prentice Hall, 1982. Scholar
34.Ames N.M., Srivastava V., Chester S.A., Anand L. A thermo-mechanically coupled theory for large deformations of amorphous polymers. Part II: Applications. International Journal of Plasticity, 2009, 25(8), pp. 1495-1539. doi: 10.1016/j.ijplas.2008.11.005 Scholar
35.Karpov E.V., Larichkin A.Y. Impact of axial compression and torque on strain localization and fracture under complex cyclic loading of Plexiglas rods. Journal of Applied Mechanics and Technical Physics, 2014, 55(1), pp. 95-104. doi: 10.1134/S0021894414010131 Scholar
36.Mulliken A.D., Boyce M.C. Mechanics of the rate- dependent elastic-plastic deformation of glassy polymers from low to high strain rates. International Journal of Solids and Structures, 2006, 43(5), pp. 1331-1356. doi: 10.1016/j.ijsolstr.2005.04.016 Scholar
37.Richeton J., Ahzi S., Vecchio K.S., Jiang F.C., Adharapurapu R.R. Influence of temperature and strain rate on the mechanical behavior of three amorphous polymers: Characterization and modeling of the compressive yield stress. International Journal of Solids and Structures, 2006, 43, pp. 2318-2335. doi: 10.1016/j.ijsolstr.2005.06.040 Scholar
38.G'Sell C., Boni S., Shrivastava S. Application of the plane simple shear test for determination of the plastic behaviour of solid polymers at large strains. Journal of Materials Science, 1983, 18, pp. 903-918. doi: 10.1007/BF00745590 Scholar

Другие статьи выпуска

Весь выпуск (№ 3, 2023)

Скачать PDF

Информация о статье

Опубликована: 03.07.2023
В выпуске: 03.07.2023
Выпуск: № 3 (2023)

УПРУГОПЛАСТИЧЕСКОЕ КРУЧЕНИЕ С КОНЕЧНЫМИ ДЕФОРМАЦИЯМИ: СРАВНЕНИЕ АНАЛИТИЧЕСКОГО И МКЭ-МОДЕЛИРОВАНИЯ ДЛЯ НЕМОНОТОННО УПРОЧНЯЮЩИХСЯ ПОЛИМЕРОВ

Аннотация

Ключевые слова

Finite-strain elastic-plastic torsion: Analytical and FEM modeling for nonmonotonically hardening polymers

Keywords

Литература

Другие статьи выпуска

Информация о статье