zldmЗаводская лаборатория. Диагностика материаловIndustrial laboratory. Diagnostics of materials1028-68612588-0187ООО «Издательство «ТЕСТ-ЗЛ»10.26896/1028-6861-2023-89-3-80-86zldm-1890Research ArticleМАТЕМАТИЧЕСКИЕ МЕТОДЫ ИССЛЕДОВАНИЯMATHEMATICAL METHODS OF INVESTIGATIONThe stress-strain curve model in the form of an extremal of a non-integrable linear variation formThe stress-strain curve model in the form of an extremal of a non-integrable linear variation formGolovinaN. Y.GolovinaN. Y.

Natalia Ya. Golovina

625000, Tyumen, Volodarskogo ul., 38

Natalia Ya. Golovina

625000, Tyumen, Volodarskogo ul., 38

golovinanj@tyuiu.ruTyumen Industrial UniversityРоссияTyumen Industrial UniversityRussian Federation2023230320238938088Copyright © Golovina N.Y., 20232023Golovina N.Y.Golovina N.Y.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.zldm.ru/jour/article/view/1890

The article develops an idea that the stress-strain curve for an arbitrary material is the extremum of some functional. However, for irreversible processes, the using of the principle of stationarity of some functional is incorrect, because due to the dissipation of the deformation process, the possible work of internal forces is non-integrable. Therefore, it is proposed to use the generalized variational principle of L. I. Sedov for modeling the stress-strain curve of elastoplastic materials. A concept of sequential inclusion of certain deformation mechanisms on different segment of the stress-strain curve is proposed. According to this concept, each section of the stress-strain curve must correspond either to the stationary value of the corresponding functional, or to the stationary value of the non-integrated form of variations of the corresponding stress derivatives. The combination of naturally obtained spectra of boundary conditions at the ends of each segment leads to a variation-consistent formulation of the system of boundary and contact conditions of solutions of different differential equations on each segment of stress-strain curve. As a result, it is possible to construct a differentiable stress-strain curve over the entire area of the stress-strain curve definition. The resulting solution, in contrast to the Ramberg - Osgood empirical law, has a strictly liner segment. The obtained mathematical model was tested on experimental data of materials for various industrial purposes. The achieved accuracy of the mathematical model is sufficient for engineering applications.

The article develops an idea that the stress-strain curve for an arbitrary material is the extremum of some functional. However, for irreversible processes, the using of the principle of stationarity of some functional is incorrect, because due to the dissipation of the deformation process, the possible work of internal forces is non-integrable. Therefore, it is proposed to use the generalized variational principle of L. I. Sedov for modeling the stress-strain curve of elastoplastic materials. A concept of sequential inclusion of certain deformation mechanisms on different segment of the stress-strain curve is proposed. According to this concept, each section of the stress-strain curve must correspond either to the stationary value of the corresponding functional, or to the stationary value of the non-integrated form of variations of the corresponding stress derivatives. The combination of naturally obtained spectra of boundary conditions at the ends of each segment leads to a variation-consistent formulation of the system of boundary and contact conditions of solutions of different differential equations on each segment of stress-strain curve. As a result, it is possible to construct a differentiable stress-strain curve over the entire area of the stress-strain curve definition. The resulting solution, in contrast to the Ramberg - Osgood empirical law, has a strictly liner segment. The obtained mathematical model was tested on experimental data of materials for various industrial purposes. The achieved accuracy of the mathematical model is sufficient for engineering applications.

Ramberg - Osgood lawempirical stress-strain curvesstress-strain curve as a solution of the ordinary differential equation of the fourth orderstress-strain curves as an extreme of functionalprocessing of experimental dataRamberg - Osgood lawempirical stress-strain curvesstress-strain curve as a solution of the ordinary differential equation of the fourth orderstress-strain curves as an extreme of functionalprocessing of experimental dataReferencesRamberg W., Osgood W. R. Description of stress-strain curves by three parameters. National Advisory Committee For Aeronautics / Technical Note N 902. Washington, DC, 1943.Ramberg W., Osgood W. R. Description of stress-strain curves by three parameters. National Advisory Committee For Aeronautics / Technical Note N 902. Washington, DC, 1943.Mendelson A. Plasticity: Theory, and Application. — Malabar: Krieger, 1968. — 183 p. [in Russian].Mendelson A. Plasticity: Theory, and Application. — Malabar: Krieger, 1968. — 183 p. [in Russian].Papirno R. Goodness-of-Fit of the Ramberg - Osgood Analytic Stress-Strain Curve to Tensile Test Data / J. Testing Eval. 1982. Vol. 6. N 10. E 263 - 268. DOI:10.1520/JTE10264JPapirno R. Goodness-of-Fit of the Ramberg - Osgood Analytic Stress-Strain Curve to Tensile Test Data / J. Testing Eval. 1982. Vol. 6. N 10. E 263 - 268. DOI:10.1520/JTE10264JHollomon J. H. Tensile deformation / Trans. AIME. 1945. Vol. 162. E 268 - 290.Hollomon J. H. Tensile deformation / Trans. AIME. 1945. Vol. 162. E 268 - 290.Ludwigson D. C. Modified stress — strain relation for FCC metals and alloys / Metall Trans. 1971. Vol. 2. N 10. E 2825 - 2828.Ludwigson D. C. Modified stress — strain relation for FCC metals and alloys / Metall Trans. 1971. Vol. 2. N 10. E 2825 - 2828.Ludwik P. Elemente der technologischen Mechanik. — Berlin: Springer, 1909. — 57 p. [in Russian].Ludwik P. Elemente der technologischen Mechanik. — Berlin: Springer, 1909. — 57 p. [in Russian].Swift Pi. W. Plastic instability under plane stress / J. Mech. Phys. Solids. 1952. Vol. 1. N 1. E 1 - 18.Swift Pi. W. Plastic instability under plane stress / J. Mech. Phys. Solids. 1952. Vol. 1. N 1. E 1 - 18.Voce E. The relationship between stress and strain for homogeneous deformation / J. Inst. Metals. 1948. N 74. E 537 - 562.Voce E. The relationship between stress and strain for homogeneous deformation / J. Inst. Metals. 1948. N 74. E 537 - 562.Gao Pi. S. Modeling Stress Strain Curves for Nonlinear Analysis / Materials Science Forum 2009. E 575 - 578, 539 - 544. DOI:10.4028/www.scientific.net/msf.575-578.539Gao Pi. S. Modeling Stress Strain Curves for Nonlinear Analysis / Materials Science Forum 2009. E 575 - 578, 539 - 544. DOI:10.4028/www.scientific.net/msf.575-578.539Rasmussen K. Full-range stress-strain curves for stainless steel alloys / J. Constr. Steel Res. 2003. Vol. 59. N 1. E 47 - 61. DOI:10.1016/S0143-974X(02)00018-4Rasmussen K. Full-range stress-strain curves for stainless steel alloys / J. Constr. Steel Res. 2003. Vol. 59. N 1. E 47 - 61. DOI:10.1016/S0143-974X(02)00018-4Belov P. A., Golovina N. Ya. Generalization of the Ramberg - Osgood Model for Elastoplastic Materials / J. Mater. Eng. Perform. 2019. Vol. 28. N 12. E 7342 - 7346. DOI:10.1007/sll665-019-04422-3Belov P. A., Golovina N. Ya. Generalization of the Ramberg - Osgood Model for Elastoplastic Materials / J. Mater. Eng. Perform. 2019. Vol. 28. N 12. E 7342 - 7346. DOI:10.1007/sll665-019-04422-3Belov P. A., Golovina N. Ya. Stress-strain curve as an extremal of some functional / Science and Business: Development Ways. 2019. Vol. 10. N 100. E 44 - 52.Belov P. A., Golovina N. Ya. Stress-strain curve as an extremal of some functional / Science and Business: Development Ways. 2019. Vol. 10. N 100. E 44 - 52.Abdella K. Inversion of a full-range stress — strain relation for stainless steel alloys / Int. J. Non-Lin. Mech. 2006. N 41. E 456 - 463.Abdella K. Inversion of a full-range stress — strain relation for stainless steel alloys / Int. J. Non-Lin. Mech. 2006. N 41. E 456 - 463.Golovina N. Ya. Comparative analysis of fatique models of plastic materials / XII All-Russian Congress on Fundamental Problems of Theoretical and Applied Mechanics Collected Works. 2019. Vol. 4. E 611 - 613.Golovina N. Ya. Comparative analysis of fatique models of plastic materials / XII All-Russian Congress on Fundamental Problems of Theoretical and Applied Mechanics Collected Works. 2019. Vol. 4. E 611 - 613.Gardner L., Yun X., Fieber A., Macorini L. Steel Design by Advanced Analysis: Material Modeling and Strain Limits / Engineering. 2019. N 5. E 243 - 249.Gardner L., Yun X., Fieber A., Macorini L. Steel Design by Advanced Analysis: Material Modeling and Strain Limits / Engineering. 2019. N 5. E 243 - 249.Golovina N. Ya., Krivosheeva S. Y. Research in area of longevity of sylphon scraies / IOP Conference Series: Earth and Environmental Science. Current Problems and Solutions. 2018. Vol. 12. N 3. E 012043.Golovina N. Ya., Krivosheeva S. Y. Research in area of longevity of sylphon scraies / IOP Conference Series: Earth and Environmental Science. Current Problems and Solutions. 2018. Vol. 12. N 3. E 012043.Golovina N. Ya. The nonlinear stress-strain curve model as a solution of the fourth order differential equation / Int. J. Press. Vess. Piping. 2021. N 189. E 104258. DOI:10.1016/j.ijpvp.2020.104258Golovina N. Ya. The nonlinear stress-strain curve model as a solution of the fourth order differential equation / Int. J. Press. Vess. Piping. 2021. N 189. E 104258. DOI:10.1016/j.ijpvp.2020.104258Golovina N. Ya. Modeling the Stress-Strain Curve of Elastic-Plastic Materials / Solid State Phenomena. 2021. N 316. E 936 - 941.Golovina N. Ya. Modeling the Stress-Strain Curve of Elastic-Plastic Materials / Solid State Phenomena. 2021. N 316. E 936 - 941.Quach W. M. Three-Stage Full-Range Stress-Strain Model for Stainless Steels / J. Struct. Eng. 2008. N 134. E 1518 - 1527.Quach W. M. Three-Stage Full-Range Stress-Strain Model for Stainless Steels / J. Struct. Eng. 2008. N 134. E 1518 - 1527.Plertele S., De Waele W., Denys R. A generic stress — strain model for metallic materials with two-stage strain hardening behavior / Int. J. Non-Lin. Mech. 2011. Vol.46. N3 . E 519-531.Plertele S., De Waele W., Denys R. A generic stress — strain model for metallic materials with two-stage strain hardening behavior / Int. J. Non-Lin. Mech. 2011. Vol.46. N3 . E 519-531.Li Т., Zheng J., Chen Z. Description of full-range strain hardening behavior of steels / Springer Plus. 2016. N 5. E 1316. DOI:10.1186/s40064-016-2998-3Li Т., Zheng J., Chen Z. Description of full-range strain hardening behavior of steels / Springer Plus. 2016. N 5. E 1316. DOI:10.1186/s40064-016-2998-3Belov P. A., Gorshkov A. G, Lurie S. A. Variational model of nonholonomic 4D media / Rigid Body Mech. 2006. N 6. E 29 - 46.Belov P. A., Gorshkov A. G, Lurie S. A. Variational model of nonholonomic 4D media / Rigid Body Mech. 2006. N 6. E 29 - 46.Belov P. A., Lurie S. A. Variation model of non-holonomic media / Mech. Composite Mater. Designs. 2001. Vol. 7. N 2. P 266 - 276.Belov P. A., Lurie S. A. Variation model of non-holonomic media / Mech. Composite Mater. Designs. 2001. Vol. 7. N 2. P 266 - 276.Golovina N. Ya. PhD thesis, Tyumen State Oil and Gas University, 2002. https://perma.cc/53VQ-2L2H (accessed 2021-03-22).Golovina N. Ya. PhD thesis, Tyumen State Oil and Gas University, 2002. https://perma.cc/53VQ-2L2H (accessed 2021-03-22).Walport E, Gardner L., Real E., et al. Effects of material nonlinearity on the global analysis and stability of stainless steel frames / J. Constr. Steel Res. 2019. N 152. E 173 - 182. DOI:10.1016/j.jcsr.2018.04.019Walport E, Gardner L., Real E., et al. Effects of material nonlinearity on the global analysis and stability of stainless steel frames / J. Constr. Steel Res. 2019. N 152. E 173 - 182. DOI:10.1016/j.jcsr.2018.04.019Arrayago I., Real E., Gardner L. Description of stress-strain curves for stainless steel alloys / Materials and Design. 2015. N 87. E 540 - 552. DOI:10.1016/j.matdes.2015.08.001Arrayago I., Real E., Gardner L. Description of stress-strain curves for stainless steel alloys / Materials and Design. 2015. N 87. E 540 - 552. DOI:10.1016/j.matdes.2015.08.001Mirambell E., Real E. On the calculation of deflections in structural stainless steel beams: an experimental and numerical investigation / J. Constr. Steel Res. 2000. N 54. E 109 - 133.Mirambell E., Real E. On the calculation of deflections in structural stainless steel beams: an experimental and numerical investigation / J. Constr. Steel Res. 2000. N 54. E 109 - 133.Yun X., Gardner L. The continuous strength method for the design of cold-formed steel non-slender tubular cross-sections / Engineering Structures. 2018. N 175. E 549 - 564. DOI:10.1016/j.engstruct.2018.08.070Yun X., Gardner L. The continuous strength method for the design of cold-formed steel non-slender tubular cross-sections / Engineering Structures. 2018. N 175. E 549 - 564. DOI:10.1016/j.engstruct.2018.08.070Golovina N. Ya. Stress-Strain Curve as Extremal of Some Functional / J. Mater. Eng. Perform. 2021. Vol. 30. N 6. E 4641-4650. DOI:10.1007/sll665-021-05768-3Golovina N. Ya. Stress-Strain Curve as Extremal of Some Functional / J. Mater. Eng. Perform. 2021. Vol. 30. N 6. E 4641-4650. DOI:10.1007/sll665-021-05768-3Golovina N. Ya., Belov P. A. Analysis of empirical models of deformation curves of elastoplastic materials (review). Part 1 / Math. Model. Comput. Meth. 2022. N 1. E 63 - 96.Golovina N. Ya., Belov P. A. Analysis of empirical models of deformation curves of elastoplastic materials (review). Part 1 / Math. Model. Comput. Meth. 2022. N 1. E 63 - 96.

The authors declare that there are no conflicts of interest present.