Vol. 37 No. 3 (2026): July - September
2026-09-30
Open Access

Stress and strain state of a square bar under complex loading

Javlonbek Turdibekov

Received: 2026-10-13

Published: 2026-09-30

PDF (Russian)

Abstract

Background: Elastoplastic tension and torsion of a homogeneous square bar are examined for sequential and proportional loading paths.
Methods: A displacement formulation is used: plastic shear strains contribute to the source in the warping Poisson equation and enter the
Neumann boundary conditions. The Prandtl–Reuss model is compared with an average curvature model that accounts for deformation-
path memory. A five-point finite-difference scheme and successive over-relaxation are employed. Results: Three elastic benchmark meshes
exhibit second-order convergence; the torque error on the 181 × 181 mesh is 0.000941%. In the reported comparison at the same final
kinematic state, the average curvature model predicts a torque 3.15–9.44% above the Prandtl–Reuss result. Conclusions: Local trajectories
reveal sensitivity to the loading sequence. The conclusions are restricted to the adopted reduced cross-sectional description and the elastic
verification of the algorithm.

Keywords

List of references

  1. Timoshenko SP, Goodier JN. Theory of Elasticity. McGraw-Hill; 1970. 3rd ed.; McGraw-Hill.

  2. Reddy JN. An Introduction to Continuum Mechanics. Cambridge University Press; 2013. 2nd ed.; Cambridge University Press; DOI 10.1017/CBO9781139178952.

  3. Borja RI. Plasticity: Modeling and Computation. Springer; 2013. Springer; DOI 10.1007/978-3-642-38547-6.

  4. de Souza Neto EA, Peri´c D, Owen DRJ. Computational Methods for Plasticity: Theory and Applications. Wiley; 2008. Wiley; DOI 10.1002/9780470694626.

  5. Sidebottom OM. Evaluation of finite-plasticity theories for nonproportionate loading of torsion-tension members. Experimental Mechanics. 1972. Experimental Mechanics, 12, 18–24; DOI 10.1007/BF02320785.

  6. Yoshida K, Tsuchimoto T. Plastic flow of thin-walled tubes under nonlinear tension-torsion loading paths and an improved pseudo-corner model. International Journal of Plasticity. 2018. International Journal of Plasticity, 104, 214–229; DOI 10.1016/j.ijplas.2018.02.013.

  7. Ilyushin AA. Plastichnost. Osnovy obshchei matematicheskoi teorii [Plasticity. Fundamentals of the general mathematical theory]. Izdatelstvo Akademii nauk SSSR; 1963. Izdatelstvo Akademii nauk SSSR; in Russian.

  8. Zubchaninov VG. The general mathematical theory of plasticity and the Ilyushin postulates of macroscopic definability and isotropy. Moscow University Mechanics Bulletin. 2018. Moscow University Mechanics Bulletin, 73, 101–116; DOI 10.3103/S0027133018050011.

  9. Babamuratov KS, Abirov RA. On physical reliability in the theory of plasticity. Strength of Materials. 2001. Strength of Materials, 33, 1–7; DOI 10.1023/A:1010478208161.

  10. Abirov RA. On Plastic Behaviour of Materials under Complex Loading. Journal of the Mechanical Behavior of Materials. 2007. Journal of the Mechanical Behavior of Materials, 18(5–6), 341–348; DOI 10.1515/JMBM.2007.18.5-6.341.

  11. Abirov RA. On physical reliability and taking complex loading into account in plasticity. Materials Science. 2008. Materials Science, 44, 512–516; DOI 10.1007/s11003-009-9114-6.

  12. Xu B, Zhang R, Yang K, Yu G, Chen Y. Application of generalized finite difference method for elastoplastic torsion analysis of prismatic bars. Engineering Analysis with Boundary Elements. 2023. Engineering Analysis with Boundary Elements, 146, 939–950; DOI 10.1016/j.enganabound.2022.11.028.

  13. Moayyedian F, Grabski JK. Elastic-plastic torsion problem with non-linear hardenings using the method of fundamental solution. Archives of Civil and Mechanical Engineering. 2021. Archives of Civil and Mechanical Engineering, 21, 155; DOI 10.1007/s43452-021-00307-2.

  14. Pires PFM, Buriol TM, dos Santos T. An open-source finite element framework for Saint-Venant torsion of heterogeneous beams with arbitrary cross-sections. Journal of Engineering Mathematics. 2025. Journal of Engineering Mathematics, 155, 9; DOI 10.1007/s10665-025-10493-0.

  15. Chen H, Gomez J, Pindera MJ. Saint Venant torsion of homogeneous and composite bars by the finite volume method. Composite Structures. 2020. Composite Structures, 242, 112128; DOI 10.1016/j.compstruct.2020.112128.

  16. Abirov R, Turdibekov J. Axial-Torsional Path Dependence in an Elastoplastic Rod with a Multiply Connected Cross-Section. Applied Mechanics. 2026. Applied Mechanics, 7(3), 71; DOI 10.3390/applmech7030071.

  17. LeVeque RJ. Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems. SIAM; 2007. SIAM; DOI 10.1137/1.9780898717839.

  18. Strniˇsa F, Janˇciˇc M, Kosec G. Numerical analysis of small-strain elasto-plastic deformation using local Radial Basis Function approximation with Picard iteration. Applied Mathematical Modelling. 2025. Applied Mathematical Modelling, 137, 115714; DOI 10.1016/j.apm.2024.115714.

About the Authors

Javlonbek Turdibekov
javlonbekturdibekov9@gmail.com
Institute of Mechanics and Seismic Stability of Structures named after M.T. Urazbaev, Academy of Sciences of the Republic of Uzbekistan

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How to Cite

Stress and strain state of a square bar under complex loading. (2026). Uzbek Journal “Problems of Mechanics”, 37(3). https://pmjournal.uz/jour/article/view/24

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