Numerical Modeling of a Boundary-Value Problem of Thermoelastoplasticity in Deformations
Article Sidebar
Main Article Content
Abstract
In this paper, using the deformation compatibility condition, similar to the well-known displacement equation, differential deformation equations are written. These, in combination with the equilibrium equation and boundary and additional boundary conditions, form a boundary-value problem of thermoelastoplasticity in deformations. The mathematical model is formulated directly in deformations, which ensures a more robust numerical implementation when analyzing the nonlinear behavior of materials taking into account temperature effects. Ilyushin's deformation theory of plasticity is used as the theoretical basis, allowing for the description of the evolution of plastic deformations through strain tensor invariants without explicitly introducing a yield surface in the classical stress formulation. The nonlinearity of the problem is due to both plastic effects and the temperature dependence of mechanical properties. The finite difference method is applied to numerically solve the boundary-value problem. An iterative solution algorithm is developed taking into account the nonlinear components of the model. An assessment of the convergence and stability of the numerical scheme for various mesh parameters and physical and mechanical properties of the material is carried out. The numerical results obtained in the elastic domain are compared with the solutions obtained in displacement and stress domains. The proposed approach can be used in the numerical modeling of structural elements operating under combined mechanical and thermal loading. The developed model and numerical solution algorithm are designed for further integration into engineering analysis software packages and can serve as the basis for constructing more complex multidimensional thermoelastoplastic models.
Article Details
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
Ильюшин А.А. Пластичность // Часть 1. Упруго-пластические деформации. – М.: Логос, – 2004, – 388 с.
Chen C.,·Wang T.·A strain-rate-dependent thermo-elasto-plastic constitutive model for crystal-line metallic materials // Scientific· Reports. – 2021, – Vol. 11, – DOI: 10.1038/s41598-021-88333-1.
Dafalias Y.F., Popov E.P. A model of thermo-plasticity for cyclic loading // Journal of Engi-neering Mechanics. –2016, –Vol. 142(3), – P. 1-12.
Abirov R.A., Khusanov B.E., Sagdullaeva D.A. Numerical modeling of the problem of indenta-tion of elastic and elastic-plastic massive bodies // IOP Conference Series: Materials Science and Engineering. – 2020. – Vol. 971, P. 1-9. – DOI:10.1088/1757-899X/971/3/032017.
Халджигитов А.А. Об уравнениях термопластичности // Проблемы механики. – 1998, – №1.
Abouelregal A. E., Marin M., Foul A., Askar S. Spatiotemporal Nonlocal Thermoelastic Model with Caputo-Tempered Fractional Derivatives for Infinite Thermoelastic Porous Half-Space with Voids. // Journal of Computational Applied Mechanics, – 2025, – Vol. 56(2), – P. 276-295. – DOI: 10.22059/jcamech.2025.390564.1372.
Chaboche J.L. Constitutive equations for cyclic plasticity and thermo-plasticity // International Journal of Plasticity. –2017, –Vol. 98, –P. 1-24.
Lubliner J. Plasticity theory with thermal coupling // Dover Publications. –2015, – P. 97-164.
Ortiz M., Stainier L. The variational formulation of thermo-plasticity // Computer·Methods in Applied Mechanics and Engineering. – 2016, –Vol. 193, – P. 249-303.
Simo J.C, Hughes T.J.R. Computational inelasticity with applications to thermo-plasticity // Springer. – 2015. – 392 p.
Zienkiewicz O C., Taylor L R. The Finite Element Method // 5th edn. Butterworth-Heinemann, Oxford. – 2000. – Vol. 1. – 736 p.
Zienkiewicz O.C., Taylor R.L., Zhu J.Z. The finite element method: nonlinear problems // Else-vier. –2018, –P. 221-305.
Belytschko T., Liu W. K., Moran B. Nonlinear finite elements for thermo-elasto-plastic prob-lems // Wiley. –2016, –P. 421-487.
Houlsby G.T., Puzrin A.M. Principles of hyperplasticity with thermal effects // Springer. –2018, –P. 211-260.
Reddy J.N. Energy principles and variational methods in thermo-elastic-plastic analysis // Wiley. –2017, –P. 333-379.
Sluys L. J., de Borst R. Wave propagation in thermo-elasto-plastic continua // Computer Meth-ods in Applied Mechanics and Engineering. –2015, –Vol. 289, –P. 336-361.
Voyiadjis G. Z., Woelke P. Thermo-viscoplasticity at high strain rates // Acta Mechanica. – 2019, –Vol. 230, – P. 289-315.
Khaldjigitov A. A., Djumayozov U. Z., Sagdullayeva D. A. Numerical Solution of Coupled Thermo-Elastic-Plastic Dynamic Problems // Mathematical Modelling of Engineering Prob-lems. – 2023, – Vol. 8(4), – P. 510-518. – DOI: 10.18280/mmep.080403.
Khaldjigitov A. A., Djumayozov U. Z., Ibodulloev S. R. Effective finite-difference method for elastoplastic boundary value problems // AIP Conference Proceedings. – 2020, – Vol. 2365(1), – 020008. – DOI: 10.1063/5.0057047.
Khaldjigitov A., Djumayozov U., Khasanova Z., Rakhmonova R. Coupled Problems of Thermoe-lasticity in Strains // E3S Web of Conferences. – 2024, – Vol. 497, – 02008. – DOI: 10.1051/e3sconf/202449702008.
Khaldjigitov A., Djumayozov U., Khasanova Z., Rakhmonova R. Numerical Solution of the Plane Problem of Thermo-Elasticity in Strains // E3S Web of Conferences. – 2024, – Vol. 563, – 02019.
Khaldjigitov A A., Djumayozov U Z., Sagdullayeva D A. Numerical Simulation of Elastoplastic Problems in Strains and Displacements // E3S Web of Conferences. – 2024, – Vol. 563, – 02030.