Abstract
The paper presents the multimodulus approach to the continuum mechanics problem. In the introduction, a number of multimodulus models are presented that were developed by V.P. Myasnikov for the isotropic elasticity case. The numerical finite-element approximation of the multimodulus equilibrium state equations forming the sophisticated Lame system is also considered. In the final part, the results of two-dimensional calculations of the stress-strain state for various multimodulus media such as steel, concrete, ceramics, and soil are presented. The effect is discussed of the influence of the different moduli in tension and compression on the strain intensity
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. A. Ambartsumyan, Multimodulus Elasticity Theory (Nauka, Moscow, 1982) [in Russian].
R. M. Jones, AIAA J. 15, 16 (1977).
A. A. Lebedev, B. I. Koval’chuk, F. F. Giginyak, and V. P. Lamashevskii, Mechanical Properties of the Construction Materials under Combined Stressed State. Handbook (Naukova Dumka, Kiev, 1983) [in Russian].
M. Ya. Leonov, V. A. Panyaev, and K. N. Rusinko, “The Strain-Stress Dependency for the Semi-Fragile Bodies,” Inzh. Zh. MTT (Engineering J. MTT), No. 6, 26–32 (1967) [in Russian].
V. V. Rzhevskii and G. Ya. Novik, Foundations of the Rock Physics. 4th Edition (Nedra, Moscow, 1984) [in Russian].
R. N. Schock, “The Response of Rocks to Large Stresses,” in Impact and Explosion Cratering (Pergamon Press, N.Y., 1977), 657–688.
V. N. Nikolaevskii, “The Mohorovicic Discontinuity as the Limiting Depth of the Fragile-Dilatansion Rock State,” Doklady Acad. Nauk SSSR (Sov. Phys. Dokl.), 249, 817–821 (1979) [in Russian].
E. V. Lomakin, “The Constitutive Relationships of the Strain Theory for the Dilating Media,” Izvestia Acad. Nauk SSSR. MTT (Proc. Acad. Sci. SSSR. MTT), No. 6, 66–75, (1991) [in Russian].
V. Mori, “Destruction Mechanisms in the Walls of Boreholes, Underground Constructions, and excavations,” in Rock Mechanics Applied to the Problems of Oil Prospecting and Production (Mir, Moscow, 1994), 361–411 (Translation of: Mecanique des roches appliquee aux problemes d’exploration de production petrolieres, Boussens, 1993) [in Russian].
F. G. Santarelli and I. T. Braun, “Behavior of the Deep Well Borehole Drilled in a Rock with the Elasticity Module Depending on the Rock Pressure,” in Rock Mechanics Applied to the Problems of Oil Prospecting and Production (Mir, Moscow, 1994), 88–96 (Translation of: Mecanique des roches appliquee aux problemes d’exploration de production petrolieres, Boussens, 1993) [in Russian].
A. Geno, “Stresses and Destructions in the Oil Well Walls,” in Rock Mechanics Applied to the Problems of Oil Prospecting and Production (Mir, Moscow, 1994), 73–87 (Translation of: Mecanique des roches appliquee aux problemes d’exploration de production petrolieres, Boussens, 1993) [in Russian].
A. Geno, The Problem of the Well Borehole Instability at Large Depths, in Rock Mechanics Applied to the Problems of Oil Prospecting and Production (Mir, Moscow, 1994), 97–107 (Translation of: Mecanique des roches appliquee aux problemes d’exploration de production petrolieres, Boussens, 1993) [in Russian].
V. A. Lyakhovskii and V. P. Myasnikov, “On the Behavior of the Media with Micro-disruptions,” Izv. Acad. Nauk SSSR. Fizika Zemli (Proc Acad. Sci. USSR. Phys. Earth), No. 10, 71–75 (1984) [in Russian].
V. P. Myasnikov and A. I. Oleinikov, The Main General Relatinships of the Model of the Isotropic Elastic Differently Resisting Media, Doklady Acad. Nauk SSSR (Reports Acad. Sci. USSR), 1992. vol. 322, no. 1, pp. 57–60 [in Russian].
V. A. Gasilov, M. V. Golovin, V. P. Myasnikov, et al., “Mathematic Modeling of the Stress-Strain Rock State on the Base of the Multimodulus Model of the Continuous Media,” Preprint of the Keldysh Institute of Applied Mechanics, Moscow (1997) No. 112 [in Russian].
V. A. Gasilov, M. V. Golovin, V. P. Myasnikov, et al., “Application of the Multimodulus Model of the Continuous Media to the Analysis of the Rock Behavior under Large Stresses,” Izv. Ross. Akad. Nauk, MTT (Proc Russian Acad. Sci. MTT), No. 2, 86–92 (2000) [in Russian].
O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method. 5th Ed. (McGrow Hill, London, 2000).
J. M. Ortega and W. C. Reinbolt, Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970).
V. I. Maslyankin, M. I. Mukhina, and A. A. Reznik, “To the Solution of the Linear Equation System with the Rarified Matriz by Iteration Methods,” Matematicheskoe modelirovanie (Mathematical Modeling), 4, 100–108 (1992) [in Russian].
Author information
Authors and Affiliations
Additional information
Original Russian Text © V.A. Gasilov, M.V. Golovin, 2008, published in Matematicheskoe Modelirovanie, 2008, Vol. 20, No. 4, pp. 117–127.
Rights and permissions
About this article
Cite this article
Gasilov, V.A., Golovin, M.V. On the numerical investigations of the stress-strain behavior of materials based on the isotropic elasticity model of the multimodulus media. Math Models Comput Simul 1, 220–227 (2009). https://doi.org/10.1134/S2070048209020069
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S2070048209020069