Rigorous solution of the problem of the state of a linearly elastic isotropic body under the action of polynomial bulk forces

Cover Page

Cite item


When solving boundary value problems about the construction of the stress-strain state of an linearly elastic, isotropic body, an important step is finding the internal state generated by the forces, distributed over the area occupied by the body. In the classical version, there is a numerical method for estimating the state at any point of the body based on the singular-integral representation of Cesaro. In the variant of conservative bulk forces, it is possible to construct solutions in an analytical form. With arbitrary regular effects of mechanical and other physical nature the force is not potential and the approaches of Papkovich–Neiber and Arzhanykh–Slobodyansky are powerless. In addition, the solution of nonlinear problems of elastostatics by means of the perturbation method, as well as the use of the Schwarz algorithm in solving problems for the study of multi-cavity solids, lead to the need to solve a sequence of linear problems. At the same time, fictitious bulk forces are necessarily generated, which as a rule have a polynomial nature.

The method proposed by the authors earlier for estimating the stress-strain state of a solid caused by the action of polynomial bulk forces represented in Cartesian coordinates has been improved. The internal state is restored in strict accordance with the forces statically acting on a simply connected bounded linear-elastic body. An effective method for constructing a solution and an algorithm for its computer implementation are proposed and described. Test calculations are demonstrated. The analysis of the state of the ball under the action of a superposition of bulk forces of different nature at different ratios of parameters that emphasize the level of influence of these factors is performed. The results are presented graphically. Conclusions are drawn:

a) the procedure for writing out the stress-strain state on the volume forces represented by polynomials from Cartesian coordinates is justified;
b) the algorithm is implemented in the Mathematica computing system and tested on high-order polynomials;
c) the analysis of the quasi-static state of a linear-elastic isotropic ball exposed to the forces of gravity and inertia at various combinations of parameters corresponding to the variants of slow, fast, compensatory (inertial forces are proportional to the gravitational) rotations is carried out.

The prospects for the development of a new approach to the class of bounded and unbounded bodies containing an arbitrary number of cavities are noted.

About the authors

Viktor B. Penkov

Lipetsk State Technical University

Email: vbpenkov@mail.ru
ORCID iD: 0000-0002-6059-1856
SPIN-code: 3720-0060
Scopus Author ID: 56490841400

Dr. Phys. & Math. Sci.; Professor; Dept. of General Mechanics

30, Moskovskaya st., 398055, Russian Federation

Lyubov V. Levina

Lipetsk State Technical University

Email: satalkina_lyubov@mail.ru
ORCID iD: 0000-0002-7441-835X
SPIN-code: 6294-4940
Scopus Author ID: 57201669457
ResearcherId: ABF-3858-2020

Cand. Phys. & Math. Sci.; Associate Professor; Dept. of Applied Mathematics

30, Moskovskaya st., 398055, Russian Federation

Evgeny A. Novikov

Lipetsk State Technical University

Author for correspondence.
Email: 89513027802@mail.ru
ORCID iD: 0000-0001-5606-5207
Scopus Author ID: 57208702566

Postgraduate Student; Dept. of General Mechanics

Russian Federation, 30, Moskovskaya st., 398055, Russian Federation


  1. Truesdell C. A First Course in Rational Continuum Mechanics. Vol. 1: General Concepts, Pure and Applied Mathematics, vol. 71. New York, San Francisco, London, Academic Press, 1977, xxiii+280 pp.
  2. Rabotnov Yu. N. Mekhanika deformiruemogo tverdogo tela [Mechanics of a Deformable Rigid Body]. Moscow, Nauka, 1988, 712 pp. (In Russian)
  3. Lurie A. I. Theory of Elasticity, Foundations of Engineering Mechanics. Berlin, Springer-Verlag, 2005, iv+1050 pp. https://doi.org/10.1007/978-3-540-26455-2.
  4. Muskhelishvili N. I. Nekotorye osnovnye zadachi matematicheskoi teorii uprugosti [Some Basic Problems of the Mathematical Theory of Elasticity]. Moscow, Nauka, 1966, 707 pp. (In Russian)
  5. Green A. E., Zerna W. Theoretical Elasticity. New York, Dover Publications, 1992, xvi+457 pp.
  6. Arfken G. B., Weber H. J. Mathematical Methods for Physicists. Amsterdam, Elsiver/Academic Press, 2005, xii+1182 pp.
  7. Neuber H. Ein neuer Ansatz zur Lösung räumlicher Probleme der Elastizitätstheorie. Der Hohlkegel unter Einzellast als Beispiel, ZAMM, 1934, vol. 14, no. 4, pp. 203–212. https://doi.org/10.1002/zamm.19340140404.
  8. Matveenko V. P., Schevelev N. A. Analytical study of the stress-strain state of rotation bodies under the action of mass forces, In: Stress-Strain State of Structures Made of Elastic and Viscoelastic Materials. Sverdlovsk, 1977, pp. 54–60 (In Russian).
  9. Vestyak V. A., Tarlakovsky D. V. Unsteady axisymmetric deformation of an elastic space with a spherical cavity under the action of bulk forces, Moscow Univ. Mech. Bull., 2016, vol. 71, no. 4, pp. 87–92. https://doi.org/10.3103/S0027133016040038.
  10. Sharafutdinov G. Z. Functions of a complex variable in problems in the theory of elasticity with mass forces, J. Appl. Math. Mech., 2009, vol. 73, no. 1, pp. 69–87. https://doi.org/10.1016/j.jappmathmech.2009.03.008.
  11. Zaytsev A. V, Fukalov A. A. Exact analytical solutions of equilibrium problems for elastic anisotropic bodies with central and axial symmetry, which are in the field of gravitational forces, and their applications to the problems of geomechanics, Matemat. Model. Estestv. Nauk., 2015, vol. 1, pp. 141–144 (In Russian).
  12. Pikul V. V. To anomalous deformation of solids, Physical Mesomechanics, 2013, no. 2, pp. 93–100 (In Russian).
  13. Kozlov V. V. The Lorentz force and its generalizations, Nelin. Dinam., 2011, vol. 7, no. 3, pp. 627–634 (In Russian).
  14. Satalkina L. V. The method of boundary states in problems of the theory of elasticity of inhomogeneous bodies and thermoelasticity, Thesis of Dissertation (Cand. Phys. & Math. Sci.). Lipetsk, 2010, 108 pp. (In Russian)
  15. Penkov V. B., Novikov E. A., Novikova O. S., Levina L. V. Combining the method of boundary states and the Lindstedt–Poincaré method in geometrically nonlinear elastostatics, J. Phys.: Conf. Ser., 2020, vol. 1479, 012135. https://doi.org/10.1088/1742-6596/1479/1/012135.
  16. Ivanychev D. A., Novikov E. A. The solution of physically nonlinear problems for isotropic bodies by the method of boundary states, In: Problems of Strength, Plasticity, and Stability in Solid Mechanics. Tver, Tver State Techn. Univ., 2021, pp. 43–47 (In Russian).
  17. Penkov V. B., Ivanychev D. A., Levina L. V., Novikov E. A. Using the method of boundary states with perturbations to solve physically nonlinear problems of the theory of elasticity, J. Phys.: Conf. Ser., 2020, vol. 1479, 012134. https://doi.org/10.1088/1742-6596/1479/1/012134.
  18. Kuzmenko V. I., Kuzmenko N. V., Levina L. V., Penkov V. B. A method for solving problems of the isotropic elasticity theory with bulk forces in polynomial representation, Mech. Solids, 2019, vol. 54, no. 5, pp. 741–749. https://doi.org/10.3103/S0025654419050108.
  19. Penkov V. B., Levina L. V., Novikova O. S. Analytical solution of elastostatic problems of a simply connected body loaded with nonconservative volume forces: theoretical and algorithmic support, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2020, vol. 24, no. 1, pp. 56–73 (In Russian). https://doi.org/10.14498/vsgtu1711.
  20. Ivanychev D. A. A boundary state method for solving a mixed problem of the theory of anisotropic elasticity with mass forces, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2021, no. 71, pp. 63–77 (In Russian). https://doi.org/10.17223/19988621/71/6.
  21. Penkov V.B., Rybakova M. R., Satalkina L. V. Application of the Schwarz algorithm to spatial problems of elasticity theory, Vesti Vuzov Chernozemya, 2015, no. 2 (40), pp. 23–31 (In Russian).

Supplementary files

There are no supplementary files to display.

Copyright (c) 2021 Authors; Samara State Technical University (Compilation, Design, and Layout)

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies