Abstract
An analytical algorithm for studying the stability of the equilibrium of compressible hyperelastic sphere with Lagrange variables is proposed. The problem is solved in a spherical coordinate system in a general three-dimensional formulation using linearized stability theory and the method of separation of variables with respect to the radial displacement, the displacement due to the rotation, and the resulting strain in the principal directions. Results of numerical and graphical analysis of the stress–strain state for a three-layer-sphere are used to analyze the gravity stress–strain state of the lithosphere of the Kuril island arc system.
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References
L. S. Leibenzon, “Deformation of an Elastic Sphere in Relation to the Problem of the Structure of the Earth,” in Collected Papers of the USSR Academy of Sciences (Izd. Akad. Nauk SSSR, Moscow, 1955), Vol. 4, pp. 186–266 [in Russian].
L. V. Baev and V. N. Solodovnikov, “Solving the Problem of Gravitational Compression of a Layered Sphere (by the Example of the Earth),” Prikl. Mekh. Tekh. Fiz. 45 (6), 103–115 (2004) [J. Appl. Mech. Tech. Phys. 45 (6), 860–870 (2004)].
Yu. L. Rebetskii and A. V. Mikhailova, “Role of Gravity in the Formation of the Deep Structure of Shear Zones,” Geodin. Tektonofiz. 2 (1), 45–67 (2011).
V. E. Panin, “Fundamentals of Physical Mesomechanics,” Fiz. Mezomekh., No. 1, 5–22 (1998).
G. I. Gurevich, “On the Initial Assumptions of the Approach to Modeling in Tectonics,” in Some Problems of the Mechanics of Deformable Media (collected papers) (Izd. Akad. Nauk SSSR, Moscow, 1959), pp. 75–144 [in Russian].
K. F. Chernykh, Introduction to Anisotropic Elasticity (Nauka, Moscow, 1988) [in Russian].
A. M. Dziewonski, A. L. Hales, E. R. Lapwood, “Parametrically Simple Earth Models Consistent with Geophysical Data,” Phys. Earth Planet. Interiors 10, 12–48 (1975).
F. Birch, “Elasticity and Constitution of the Earth’s Interior,” J. Geophys. Res. 57, 227–286 (1952).
F. D. Murnaghan, Finite Deformation of an Elastic Solid, (John Wiley and Sons, New York, 1951).
M. A. Biot, “Non-Linear Theory of Elasticity and the Linearized Case for a Body under Initial Stress,” Philos. Mag. 27, 89–115 (1939).
A. N. Guz’, Fundamentals of the Theory of Elastic Stability of Deformable Bodies (Vishcha Shkola, Kiev, 1986) [in Russian].
V. V. Bolotin, Nonconservative Problems of the Theory of Elastic Stability (Fizmatgiz, Moscow, 1961) [in Russian].
A. I. Lur’e, Theory of Elasticity (Nauka, Moscow, 1970) [in Russian].
F. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York,1953), Part 1.
E. B. Osipova, “Finite Deformations and Stability of Equilibrium of a Compressible Elastic Hollow Sphere at Follow-Up Internal Pressure,” Fiz. Mezomekh. 12 (6), 79–86 (2009).
ETOPO2 Digital Topographic Model (National Geophysical Data Center, 1988); http://topexucsdedu/cgibin/ get datacgi.
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Original Russian Text © E.B. Osipova.
Translated from PrikladnayaMekhanika i Tekhnicheskaya Fizika, Vol. 56, No. 4, pp. 160–169, July–August, 2015.
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Osipova, E.B. Stability of equilibrium of a compressible hyperelastic hollow sphere. J Appl Mech Tech Phy 56, 679–687 (2015). https://doi.org/10.1134/S002189441504015X
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DOI: https://doi.org/10.1134/S002189441504015X