# Mathematical modeling of the problem of compression of a rock sample with friction at the end

• Erkinbek Kyiyanovich Abdyldayev Al-Farabi Kazakh National University
• Makpal Orazbayevna Nogaibayeva al-Farabi Kazakh National University

### Abstract

Analytical solutions of the problem of the stress-strain state of the environment around the workings with non-uniform compression in the elastic-plastic formulation, with account for transboundary deformation are few. Some solutions of the problem under the conditions of Tresk and Coulomb plasticity are obtained.
In these solutions, there are simplifying assumptions that the area of inelastic deformations cover the entire contour of the mine, the angle of internal friction is zero, etc. The features of the post-limit deformation of rock masses near underground mines consist on the formation of destruction zones around the mine workings, zones of plastic and elastic deformation, covering the part of contour or the entire contour depending on the boundary conditions and contour profiles, and a given law of the state of the environment. The mathematical description of the process of formation of inelastic deformations areas near the workings and obtaining a solution by the analytical method is rather difficult. Due to the lack of knowledge of this problem to date, it is advisable to use numerical methods of mathematics and mechanics using modern information technology and technology. The article presents mathematical models and results of solving a geomechanical problem based on information technology and the finite element method. The developed procedures and programs allow solving with the help of modern computers a wide class of mining tasks in which it is required to determine the stress-strain state of the rock mass weakened by mine workings in different mining and geological conditions.

### References

1.Fadeev A.B. Metod konechnyih elementov v geomehanike [The finite element method in geomechanics]. M.: Nedra, 1987. – p. 221.
2. Zenkevich O. Metodyi konechnyih elementov v tehnike [Finite-element methods in engineering]. - M.: Mir, 1975. – p. 542.
3. Erzhanov Z.S., Karimbaev T. Metod konechnyih elementov v zadachah mehaniki gornyih porod [The finite element method in problems of rock mechanics]. - Alma-Ata: Science, 1975. – p. 238.
4. Bieniawski Z.T., Denkhaus H.G., Volger V.W. Failure of fractured rock. Int. J. Rock Mech. Min. Sci., 1969, vol.6, №3, p.323-341.
5. Bredy B.T. Nonlinear mechanical behavior of the brittle rock. Int.J. Rock Mech. Min. Sci., 1969, vol.6, №3, p. 211-225.
6. Ghang G.Y., Duncan J.M. Analysis of soil movement around a deep excavation. Proc. ASCE, 1970, vol. 96, №5, p.1655-1681.
7. Goodman R.E., Toylor R.L., Brekke T.L. A Model for the mechanics of jointed rock. Proc ASCE, 1968, vol. 94, №SM3, p. 32. 637-659.
8. Tatsuoka F. Fundamental Research for the Deformability of Sand by the Triaxle Apparatus. Ph.D. dissertation presented to Tokyo Univesity in Japanese, 1972.
9. Turner M.J., Clough R.W., Martun H.C., Topp L.J. Stiffness and deflection analysis of complex structures. J. Aero. Sci., 1965, vol. 23, p. 805-823.
10. Zienkiewicz J.C., Valliopan S., King I.P. Stress analysis of rock as a no tension material. Gtotechnique, 1968, vol. 18, p. 56-66.
11. Abdyldaev E.K. Metod konechnyih elementov pri reshenii prikladnyih zadach [The finite element method for solving applied problems]. - Almaty: Polygraphy-service, 2011. - p.111.
12. Kozhakhmetov K.Kh., Nurmambetova RD, Ramatov K.S. Pryamoy metod granichnyih integralov dlya resheniya zadach gornoy geomehaniki [The direct method of boundary integrals for solving problems of mining geomechanics] // Bulletin of the Academy of Sciences of Kazakhstan SSR, 1987, №9.
13. Telles D. Primenenie metoda granichnyih elementov dlya resheniya neuprugih zadach [Application of the boundary element method for solving inelastic problems]. - M.: Stroiizdat, 1987.
14. Krauch S., Starfield A. Metody granichnyih elementov v mehanike tverdogo tela [Methods of boundary elements in solid mechanics]. - M.: Mir, 1987.
15. Kartashov, Yu. M., Matveev B.V., Mikheev G.V., Fadeev A.B. Prochnost i deformiruemost gornyih porod [Strength and deformability of rocks] - M.: Nedra 1979.-p.269.
16. Kazikaev D.M., Kozyrev A.A., Kasparyan E.V., Iosiv M.A. Upravlenie geomehanicheskimi protsessami pri razrabotke mestorozhdeniy poleznyih iskopaemyih. Uchebnoe posobie [Management of geomechanical processes in the development of mineral deposits]. Tutorial - M. Publishing House "Mining Book" 2016. –p.490.
17. Rakishev B.R., Mashanov A.A., Abdyldaev E.K. Struktura massiva i deformiruemost gornyih porod [The massif structure and deformability of rocks], Almaty, 2011, p.281.
18. Abdyldaev E. K., Mashanov A. A.,Metodyi otsenki ustoychivosti sklonov: Monografiya [Methods for assessing the stability of slopes: Monograph]. – Almaty, 2013. – p.203.
19. Moldoshev R.A., Abdyldaev Ch.E., Abdyldaev E.K. Deformatsionnyie modeli, uchityivayuschie dilatansii i razuprochneniya gornyih porod [Deformation models that take into account dilatancy and rock softening], Bulletin of the Kazakh National Technical University named after KI Satpayev №1 (95), -Almaty, -2013. -p. 21-24
20. Abdyldaev E.K, Abdim Farabi. Metodika organizatsi i mobilnogo obucheniya v usloviyah vnedrenie sistemyie-Learning [Methods of organizing mobile learning in the context of e-Learning system implementation]. News KSTU. I.Razzakova No. 1 (41) part 1, 2017, Tokmok-ITS "Technik", 2017, - p.205-209
21. Abdyldaev E.K. Uchet vliyaniya strukturyi massiva pri opredelenii ustoychivosti otkosa karera [Accounting for the influence of the structure of the array in determining the stability of the slope of the pit]. Bulletin of the Kyrgyz-Russian Slavic University. A series of natural and technical sciences, Volume 17, Number 8, 2017, Bishkek 2017, - p.65-67
22. Abdyldaev E.K. Napryazhenno-deformirovannoe sostoyanie massiva gornyih porod vblizi vyirabotok [The stress-strain state of the rock mass near the workings]. - Frunze: Ilim, 1990.-p.164.
23. Abdyldaev, E.K., Zhumazhanov, B.ZH, Salimova, G.E.Kriteriy otsenki ustoychivosti porodnogo massiva vblizi vyirabotok [Criterion for assessing the sustainability of the rock mass near the workings]. Bulletin of the Kazakh National University. AL-FARABI, Almaty, 3 (54), 2007.
24. Abdyldaev E.K. Modeli otsenki napryazhennogo sostoyaniya massiva vblizi vyrabotok pri kombinirovannoy razrabotke [Models for estimating the stress state of the massif near the workings during combined mining]. Weeds abstracts of the International scientific - practical conference "Actual problems of computer science, mechanics and robotics. Digital technologies in mechanical engineering" Almaty - 2018, p. 21-22
25. Abdyldaev, E.K., Orazayeva L., Uypalakova, D.M. Informatsionnaya tehnologiya pri chislennom reshenii prikladnyih zadach [Information technology in the numerical solution of applied problems]. The weed of materials from the International Scientific and Practical Conference "Modern Information and Communication Technologies in Education, Science and Practice" in the framework of the third modernization of Kazakhstan. Kazakh State Women's Pedagogical University, Almaty-2018, p.3.
Published
2020-04-06
How to Cite
ABDYLDAYEV, Erkinbek Kyiyanovich; NOGAIBAYEVA, Makpal Orazbayevna. Mathematical modeling of the problem of compression of a rock sample with friction at the end. Journal of Mathematics, Mechanics and Computer Science, [S.l.], v. 105, n. 1, p. 99-108, apr. 2020. ISSN 2617-4871. Available at: <https://bm.kaznu.kz/index.php/kaznu/article/view/693>. Date accessed: 07 june 2020.
Citation Formats
Section
Mechanics
Keywords geomechanical tasks, field structures, mathematical models, array heterogeneity, rock properties, rock samples