On the reducibility of solutions for the generalized yield criterion to solutions for Tresca’s yield criterion under axial symmetry

E.A. Lyamina, N.V. Kalenova, A.R. Pirumov

Author details

E.A. Lyamina
Ishlinsky Institute for Problems in Mechanics RAS, Moscow, Russian Federation; Samara National Research University, Samara, Russian Federation
N.V. Kalenova
Moscow Aviation Institute, Moscow, Russian Federation
A.R. Pirumov
Russian Technological University – MIREA, Moscow, Russian Federation

https://doi.org/10.15593/perm.mech/2023.2.04

Many continuum mechanics models are reduced to simpler models at certain parameter values. However, solutions for the general model may not converge to the corresponding solu-tions for a simpler model. In the mathematical theory of plasticity, the yield criterion completely determines the material's behavior if the associated plastic flow rule is accepted. In this paper, the reducibility of axisymmetric solutions for the generalized yield criterion to the corresponding solutions for Tresca’s criterion is investigated when the generalized yield condition tends to Tresca’s criterion. It is shown that there is no convergence if the maximum friction law is one of the boundary conditions. In this case, the solutions for both yield criteria are singular. In particu-lar, the quadratic invariant of the strain rate tensor tends to infinity near the friction surface. The strain rate intensity factor controls the magnitude of this invariant in the vicinity of the friction surface. The strain rate intensity factor is involved in some constitutive equations for predicting the evolution of material properties near frictional interfaces in metal forming processes. In this paper, using the solution of a specific boundary value problem, the behavior of this factor is in-vestigated when the generalized yield criterion tends to Tresca’s criterion. It is shown that the strain rate intensity factor continuously changes when the generalized yield criterion deviates from Tresca’s yield criterion. This behavior of the strain rate intensity factor justifies its use in the constitutive equations for the evolution of material properties near friction surfaces.

Keywords

generalized yield criterionmaximum frictionsingularityreducibility.

О сходимости решений по обобщенному условию текучести к решениям по условию Треска при осевой симметрии

Многие модели механики сплошной среды сводятся к более простым моделям при определенных значениях параметров. Однако решения по общей модели могут не сходиться к соответствующим решениям по более простой модели. В математической теории пластичности условиетекучести полностью определяет поведение материала, если принимается ассоциированный закон пластического течения. В публикуемой работе исследуется сходимость осесимметричных идеально жесткопластических решений по обобщенному условию текучести к соответствующим решениям по условию Треска при стремлении обобщенного условия текучести к условию Треска.Показано, что сходимость отсутствует, если закон максимального трения является одним из краевых условий задачи. В этом случае решения по обоим условиям текучести являются сингу-лярными. В частности, квадратичный инвариант тензора скорости деформации стремится к бес-конечности при приближении к поверхности трения. Коэффициент интенсивности скорости де-формации контролирует величину этого инварианта в окрестности поверхности трения. Коэф-фициент интенсивности скорости деформации входит в некоторые определяющие уравнениядля предсказания эволюции свойств материала вблизи поверхностей трения в процессах обра-ботки давлением. В настоящей работе на примере конкретной краевой задачи исследуется по-ведение этого коэффициента при стремлении обобщенного условия текучести к условию Треска. Показано, что коэффициент интенсивности скорости деформации непрерывно изменяется при отклонении обобщенного условия текучести от условия текучести Треска. Такое поведение ко-эффициента интенсивности скорости деформации оправдывает его использование в определяющих уравнениях для эволюции свойств материала вблизи поверхностей трения.

Ключевые слова

обобщенное условие текучестимаксимальное трениесингулярностьсходимость.

Bibliography

1.Hill R. The Mathematical Theory of plasticity. Oxford, Clarendon Press, 1950, 356 p. Scholar
2.Alexandrov S., Miszuris W. The transition of qualitative behaviour between rigid perfectly plastic and viscoplastic solutions, 2016, J. Eng. Math., 97, pp. 67-81. doi:10.1007/s10665-015-9797-7 Scholar
3.Dealy J.M., Wissbrun K.F. Melt rheology and its role in plastic processing: theory and applications. New York, Van Nostrand Reinhold, 1990, 665 p. Scholar
4.Sińczak J., Kusiak J., Łapkowski W., Okoń R. The influence of deformation conditions on the flow of strain rate sensitive materials, 1992, J. Mater. Process. Technol., 34, pp. 219-224. doi:10.1016/0924-0136(92)90110-E Scholar
5.Mitsoulis E., Hatzikiriakos S.G. Capillary extrusion flow of a fluoropolymer melt, 2013, Int. J Mater. Form., 6, pp. 29-40. doi:10.1007/s12289-011-1062-7 Scholar
6.Alexandrov S., Harris D. Comparison of solution behaviour for three models of pressure-dependent plasticity: A simple analytical example, 2006, Int. J. Mech. Sci., 48, pp. 750-762. doi:10.1016/j.ijmecsci.2006.01.009 Scholar
7.Harris D., Grekova E. A hyperbolic well-posed model for the flow of granular materials. In: Hill J.M., Selvadurai A. (eds) Mathematics and Mechanics of Granular Materials. Springer, Dordrecht, 2005, pp.107-135. doi:10.1007/1-4020-4183-7_7 Scholar
8.Spencer A.J.M. A theory of the kinematics of ideal soils under plane strain conditions, 1964, J. Mech. Phys. Solids, 12, pp. 337-351. doi:10.1016/0022-5096(64)90029-8 Scholar
9.Ishlinsky A.Yu. O ploskom dviginii peska [To plane flow of sand]. Ukrainskiy mathematicheskiy journal, 1954, Vol. 6, No. 4, pp. 430-441. Scholar
10.Ostrowska-Maciejewska J., Harris D. Three-dimensional constitutive equations for rigid/perfectly plastic granular materials,1990, Math. Proc. Camb. Philos. Soc., 108, pp. 153-169. doi:10.1017/S0305004100069024 Scholar
11.Druyanov B. Technological mechanics of porous bodies. New-York, Clarendon Press, 1993, 198 p. Scholar
12.Barlat F., Kuwabara T., Korkolis Y.P. Anisotropic plasticity and application to plane stress. In: Altenbach H., Öchsner A. (eds) Encyclopedia of Continuum Mechanics. Springer, Berlin, Heidelberg, 2018, pp. 1-22. doi:10.1007/978-3-662-53605-6_225-1 Scholar
13.Giraldo-Londoño O., Paulino Glaucio H. A unified approach for topology optimization with local stress constraints considering various failure criteria: von Mises, Drucker–Prager, Tresca, Mohr–Coulomb, Bresler– Pister and Willam–Warnke, Proc. R. Soc. A., 476, 2020, p. 20190861. doi:10.1098/rspa.2019.0861 Scholar
14.Hosford W. F. A generalized isotropic yield criterion, 1972, ASME. J. Appl. Mech., 39(2), pp. 607–609. doi:10.1115/1.3422732 Scholar
15.Billington, E.W. Generalized isotropic yield criterion for incompressible materials, 1988, Acta Mechanica, 72, pp. 1–20. doi:10.1007/BF01176540 Scholar
16.Dodd B., Naruse K. Limitations on isotropic yield criteria, 1989, Int. J. Mech. Sci.,. 31(7), pp. 511-519. doi: 10.1016/0020-7403(89)90100-8 Scholar
17.Cazacu O. New expressions and calibration strategies for Karafillis and Boyce (1993) yield criterion, 2020, Int. J. Solids Struct., 185–186, pp. 410-422. doi:10.1016/j.ijsolstr.2019.09.004. Scholar
18.Alexandrov S., Richmond O. Singular plastic flow fields near surfaces of maximum friction stress, 2001, Int. J. Non-Linear Mech., 36(1), pp. 1-11. doi:10.1016/S0020-7462(99)00075-X Scholar
19.Alexandrov S., Richmond O. Asymptotic behavior of the velocity field in an axisymmetric flow of a material satisfying the Tresca condition, Dokl. Ross. Akad. Nauk, 1998, Vol. 360, No. 4, pp. 480–482. Scholar
20.Goldstein R.V., Alexandrov S.E. An approach to prediction of microstructure formation near friction surfaces at large plastic strains, 2015, Phys. Mesomech., 18, pp.223–227. doi:10.1134/S1029959915030066. Scholar
21.Kim Y.-T., Ikeda K. Flow behavior of the billet surface layer in porthole die extrusion of aluminum, 2000, Metallurg. Mater. Trans., 31A, pp. 1635-1643. doi:10.1007/s11661-000-0173-4 Scholar
22.Trunina T.A., Kokovikhin E.A. Formation of a finely dispersed structure in steel surface layers under combined processing using hydraulic pressing, 2008, J. Mach. Manuf. Reliab., 37, pp.160–162. doi:10.3103/S1052618808020118 Scholar
23.Sasaki T.T., Morris R.A., Thompson G.B., Syarif Y., Fox D. Formation of ultra-fine copper grains in copper-clad aluminum wire, 2010, Scripta Mater., 63, pp. 488-491. doi:10.1016/j.scriptamat.2010.05.010 Scholar
24.Li X., Zu G., Ding M., Mu Y., Wang P. Interfacial microstructure and mechanical properties of Cu/Al clad sheet fabricated by asymmetrical roll bonding and annealing, 2011, Mater. Sci. Technol., 529A, pp. 485-491. doi:10.1016/j.msea.2011.09.087 Scholar
25.Alexandrov S., Jeng Y.-R., Kuo C.-Y., Chen C.-Y. Towards the theoretical/experimental description of the evolution of material properties at frictional interfaces in metal forming processes, 2022, Trib. Int., 171, Paper 107518. doi:10.1016/j.triboint.2022.107518 Scholar
26.Spencer A.J.M. A theory of the failure of ductile materials reinforced by elastic fibres, 1965, Int. J. Mech. Sci., 7(3), pp. 197-209. doi:10.1016/0020-7403(65)90018-4 Scholar
27.Alexandrov S., Erisov Y., Grechnikov F. Effect of the yield criterion of matrix on the brittle fracture of fibres in uniaxial tension of composites, 2016, Adv. Mater. Sci. Eng., Article 3746161. doi:10.1155/2016/3746161 Scholar
28.Alexandrov S., Mustafa Y. The strain rate intensity factor in the plane strain compression of thin anisotropic metal strip, 2014, Meccanica, 49, pp. 2901–2906. doi:10.1007/s11012-014-0039-2 Scholar
29.Lyamina E. Prediction of a material property gradient near the friction surface in axisymmetric extrusion and drawing, 2022, Metals, 12(8), Article 1310. doi:10.3390/met12081310 Scholar
30.Alexandrov S. The strain rate intensity factor and its applications: a review, 2009, Materials Science Forum, 623, pp. 1–20. doi:10.4028/www.scientific.net/MSF.623.1 Scholar

1.Hill R. The Mathematical Theory of plasticity. – Oxford: Clarendon Press, 1950. – 356 p. Scholar
2.Alexandrov S., Miszuris W. The transition of qualitative behaviour between rigid perfectly plastic and viscoplastic solutions // J. Eng. Math. 2016. – Vol. 97. – P. 67–81. DOI: 10.1007/s10665-015-9797-7 Scholar
3.Dealy J.M., Wissbrun K.F. Melt rheology and its role in plastic processing: theory and applications. – New York: Van Nostrand Reinhold, 1990. – 665 p. Scholar
4.The influence of deformation conditions on the flow of strain rate sensitive materials / J. Sińczak, J. Kusiak, W. Łapkowski, R. Okoń // J. Mater. Process. Technol. – 1992. – Vol. 34. – P. 219–224. DOI: 10.1016/0924-0136(92)90110-E Scholar
5.Mitsoulis E., Hatzikiriakos S.G. Capillary extrusion flow of a fluoropolymer melt // Int. J Mater. Form. – 2013. – Vol. 6. – P. 29–40. DOI: 10.1007/s12289-011-1062-7 Scholar
6.Alexandrov S., Harris D. Comparison of solution behaviour for three models of pressure-dependent plasticity: A simple analytical example // Int. J. Mech. Sci. – 2006. – Vol. 48. – P. 750–762. DOI: 10.1016/j.ijmecsci.2006.01.009 Scholar
7.Harris D., Grekova E. A hyperbolic well-posed model for the flow of granular materials // Hill J.M., Selvadurai A. (eds) Mathematics and Mechanics of Granular Materials. – Springer, Dordrecht, 2005. – P. 107–135. DOI: 10.1007/1-4020-4183-7_7 Scholar
8.Spencer A.J.M. A theory of the kinematics of ideal soils under plane strain conditions // J. Mech. Phys. Solids. – 1964. – Vol. 12. – P. 337–351. DOI: 10.1016/0022-5096(64)90029-8 Scholar
9.Ишлинский А.Ю. О плоском движении песка // Украинский математический журнал. – 1954. – Т. 6, No 4. – С. 430–441. Scholar
10.Ostrowska-Maciejewska J., Harris D. Three-dimensional constitutive equations for rigid/perfectly plastic granular materials // Math. Proc. Camb. Philos. Soc. – 1990. – Vol. 108. – P. 153–169. DOI: 10.1017/S0305004100069024 Scholar
11.Druyanov B. Technological mechanics of porous bodies. – New-York: Clarendon Press, 1993. – 198 p. Scholar
12.Barlat F., Kuwabara T., Korkolis Y.P. Anisotropic plasticity and application to plane stress // Altenbach H., Öchsner A. (eds) Encyclopedia of Continuum Mechanics. – Springer, Berlin, Heidelberg, 2018. – P. 1–22. DOI: 10.1007/978-3-662-53605-6_225-1 Scholar
13.Giraldo-Londoño O., Paulino Glaucio H. A unified approach for topology optimization with local stress constraints considering various failure criteria: von Mises, Drucker – Prager, Tresca, Mohr – Coulomb, Bresler – Pister and Willam – Warnke // Proc. R. Soc. A. – 2020. – Vol. 476. – P. 20190861. DOI: 10.1098/rspa.2019.0861 Scholar
14.Hosford W.F. A generalized isotropic yield criterion // ASME. J. Appl. Mech. – 1972. – Vol. 39, no. 2. – P. 607–609. DOI: 10.1115/1.3422732 Scholar
15.Billington E.W. Generalized isotropic yield criterion for incompressible materials// Acta Mechanica. – 1988. – Vol. 72. – P. 1–20. DOI: 10.1007/BF01176540 Scholar
16.Dodd B., Naruse K. Limitations on isotropic yield criteria // Int. J. Mech. Sci. – 1989. Vol. 31, no. 7. – P. 511–519. DOI: 10.1016/0020-7403(89)90100-8 Scholar
17.Cazacu O. New expressions and calibration strategies for Karafillis and Boyce (1993) yield criterion // Int. J. Solids Struct. – 2020. – Vol. 185–186. – P. 410–422. DOI: 10.1016/j.ijsolstr.2019.09.004 Scholar
18.Alexandrov S., Richmond O. Singular plastic flow fields near surfaces of maximum friction stress // Int. J. Non-Linear Mech. – 2001. – Vol. 36, no. 1. – P. 1–11. DOI: 10.1016/S0020-7462(99)00075-X Scholar
19.Александров С.Е., Ричмонд О. Асимптотическое поведение поля скорости при осесимметричном течении материала, подчиняющегося условию Треска // Докл. РАН. – 1998. – Т. 360, No 4. – С. 480–482. Scholar
20.Гольдштейн Р.В., Александров С.Е. Подход к предсказанию формирования микроструктуры материала вблизи поверхностей трения при развитых пластических деформациях // Физ. Мезомех. – 2014. – Т. 17, No 5. – С. 15–20. Scholar
21.Kim Y.-T., Ikeda K. Flow behavior of the billet surface layer in porthole die extrusion of aluminum // Metallurg. Mater. Trans. – 2000. – Vol. 31A. – P. 1635–1643. DOI: 10.1007/s11661-000-0173-4 Scholar
22.Трунина Т.А., Коковихин Е.А. Формирование мелкодисперсной структуры в поверхностных слоях стали при комбинированной обработке с применением гидропрессования // Пробл. Машиностр. Надежн. Машин. – 2008. – No 2. – С. 71–74. Scholar
23.Formation of ultra-fine copper grains in copper-clad aluminum wire / T.T. Sasaki, R.A. Morris, G.B. Thompson, Y. Syarif, D. Fox // Scripta Mater. – 2010. – Vol. 63. – P. 488–491. DOI: 10.1016/j.scriptamat.2010.05.010 Scholar
24.Interfacial microstructure and mechanical properties of Cu/Al clad sheet fabricated by asymmetrical roll bonding and annealing / X. Li, G. Zu, M. Ding, Y. Mu, P. Wang // Mater. Sci. Technol. – 2011. – Vol. 529A. – P. 485–491. DOI: 10.1016/j.msea.2011.09.087 Scholar
25.Towards the theoretical/experimental description of the evolution of material properties at frictional interfaces in metal forming processes / S. Alexandrov, Y.-R. Jeng, C.-Y. Kuo, C.-Y. Chen // Trib. Int. – 2022. – Vol. 171. – P. 107518. DOI: 10.1016/j.triboint.2022.107518 Scholar
26.Spencer A.J.M. A theory of the failure of ductile materials reinforced by elastic fibres // Int. J. Mech. Sci. – 1965. – Vol. 7, no. 3. – P. 197–209. DOI: 10.1016/0020-7403(65)90018-4 Scholar
27.Alexandrov S., Erisov Y., Grechnikov F. Effect of the yield criterion of matrix on the brittle fracture of fibres in uniaxial tension of composites // Adv. Mater. Sci. Eng. – 2016. – Article 3746161. DOI: 10.1155/2016/3746161 Scholar
28.Alexandrov S., Mustafa Y. The strain rate intensity factor in the plane strain compression of thin anisotropic metal strip // Meccanica. – 2014. – Vol. 49. – P. 2901–2906. DOI: 10.1007/s11012-014-0039-2 Scholar
29.Lyamina E. Prediction of a material property gradient near the friction surface in axisymmetric extrusion and drawing // Metals. – 2022. – Vol. 12, no. 8. – Article 1310. DOI: 10.3390/met12081310 Scholar
30.Alexandrov S. The strain rate intensity factor and its applications: a review // Materials Science Forum. – 2009. – Vol. 623. – P. 1–20. DOI: 10.4028/www.scientific.net/MSF.623.1 Scholar

Article information

Published: 23 June 2023
Issue date: 23 June 2023
Issue: No. 2 (2023)