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

Numerical Solution of the Plane Dynamic Problem of Elasticity in Terms of Stresses

Akmal Bobonazarov
Abduvali Khaldjigitov
Otajon Tilovov

Received: 2026-10-13

Published: 2026-09-30

PDF (Russian)

Abstract

This study addresses the formulation and numerical solution of a plane dynamic elasticity problem in which the stress tensor components are the primary unknowns. The objective is to develop a numerical algorithm for directly determining the dynamic stress field in a homogeneous isotropic elastic medium. The equations of motion, strain--displacement relations, and Hooke's law are used to derive a system of three differential equations in terms of stresses, supplemented by initial and boundary conditions. Spatial discretization is performed using the Galerkin finite element method. Time integration is carried out using the Newmark method in a C++ program and a finite difference scheme in a FreeFEM++ implementation. A rectangular domain is subjected to a time-dependent normal load on its upper boundary, while the remaining boundaries are traction-free. Spatial stress distributions and the time history of the normal stress at the centre of the domain are obtained. Comparison of the two implementations shows qualitative agreement in the stress distributions and their oscillatory behaviour. The results demonstrate the applicability of the approach to the numerical investigation of plane dynamic elasticity problems formulated in terms of stresses.

Keywords

List of references

  1. Newmark NM. A Method of Computation for Structural Dynamics. Journal of the Engineering Mechanics Division, ASCE. 1959;85(EM3):67–94.

  2. Benthien GW, Gurtin ME, Ralston TD. On the semi-discrete Galerkin method for hyperbolic problems and its application to problems in elastodynamics. Archive for Rational Mechanics and Analysis. 1972;48:51–63.

  3. Yu Z, Xie X. Semi-Discrete and Fully Discrete Hybrid Stress Finite Element Methods for Elastodynamic Problems. Numerical Mathematics: Theory, Methods and Applications. 2015;8(4):582–604.

  4. Shemyakin EI. Dinamicheskie zadachi teorii uprugosti i plastichnosti: kurs lekcij [Dynamic problems of elasticity and plasticity: lecture course]. Novosibirsk: NGU; 1968. 337 p. (In Russian).

  5. Novackij V. Dinamicheskie zadachi termouprugosti [Dynamic problems of thermoelasticity]. Moscow: Mir; 1970. 256 p. (In Russian).

  6. Shemyakin EI. Vvedenie v teoriyu uprugosti [Introduction to elasticity theory]. Moscow: Moscow State University Publishing House; 1993. 95 p. (In Russian).

  7. Konovalov AN. Dinamicheskaya zadacha teorii uprugosti v postanovke skorosti–napryazheniya [Dynamic elasticity problem in velocity–stress formulation]. Differencial’nye uravneniya. 1999;35(2):238–248. (In Russian).

  8. Konovalov AN. Reshenie zadach teorii uprugosti v napryazheniyakh [Solving problems of elasticity theory in stresses]. Novosibirsk: NGU; 1979. (In Russian).

  9. Zienkiewicz OC, Taylor RL, Zhu JZ. The Finite Element Method: Its Basis and Fundamentals. 7th ed. Oxford: Elsevier; 2013.

  10. Zienkiewicz OC, Taylor RL. The Finite Element Method. Vol. 2: Solid Mechanics. 5th ed. Oxford: Butterworth-Heinemann; 2000.

  11. Hughes TJR. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Mineola, New York: Dover Publications; 2000.

  12. Kalandarov A, Khaldjigitov A, Atabayev K. Coupled plane problem of dynamic thermal elasticity in stresses. In: AIP Conference Proceedings, Vol. 3265, No. 1; 2025:050024. https://doi.org/10.1063/5.0267727.

  13. Khaldjigitov AA, Kalandarov AK, Yusupov YS. Svyazannye zadachi termouprugosti i termoplastichnosti [Coupled problems of thermoelasticity and thermoplasticity]. Tashkent: Fan va texnologiya; 2019. 193 p. (In Russian).

  14. Khaldjigitov AA, Tilovov OU. Chislennoe reshenie chastichno-svyazannoj zadachi termouprugosti v napryazheniyakh [Numerical solution of a partially coupled thermoelasticity problem in stresses]. Problemy mekhaniki. 2023;(3). (In Russian).

About the Authors

Akmal Bobonazarov
akmalbobo1991@gmail.com
National University of Uzbekistan
Abduvali Khaldjigitov
akhald@mail.ru
National University of Uzbekistan
Otajon Tilovov
otajontilovov95@gmail.com
National University of Uzbekistan

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How to Cite

Numerical Solution of the Plane Dynamic Problem of Elasticity in Terms of Stresses. (2026). Uzbek Journal “Problems of Mechanics”, 37(3). https://pmjournal.uz/jour/article/view/20

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