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IssuesArchive of Issues2010-3pp.493-496

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D.V. Georgievskii, "On Potential Isotropic Tensor Functions of Two Tensor Arguments in Mechanics of Solids," Mech. Solids. 45 (3), 493-496 (2010)
Year 2010 Volume 45 Number 3 Pages 493-496
DOI 10.3103/S0025654410030192
Title On Potential Isotropic Tensor Functions of Two Tensor Arguments in Mechanics of Solids
Author(s) D.V. Georgievskii (Lomonosov Moscow State University, GSP-2, Leninskie Gory, Moscow, 119992 Russia, georgiev@mech.math.msu.su)
Abstract In solid mechanics, the phenomenological description of processes that occur at micro- and nanolevel means that new material parameters modeling some characteristics of the object structure are introduced in the mathematical model and, first of all, in the constitutive relations. These parameters can be either of scalar or tensor nature.

In what follows, we discuss several properties of isotropic tensor functions in 3 and possibly in 2, encountered in solid mechanics, which depend on two tensor arguments and have a potential with respect to one of them. It is admissible that the second tensor argument can be the above-mentioned parameter characterizing the structure.
Keywords tensor function, potentiality, isotropy, invariant, quasilinearity
References
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2.  V. V. Lokhin and L. I. Sedov, "Nonlinear Tensor Functions of Several Tensor Arguments," Prikl. Mat. Mekh. 27 (3), 393-417 (1963) [J. Appl. Math. Mech. 27 (3), 597-629 (1963)].
3.  A. J. M. Spenser, Continuum Physics, Vol. 1, Pt. III: Theory of Invariants (New York-London, 1971; Mir, Moscow, 1974).
4.  B. D. Annin, "Lagrange-Silvester Formula for a Tensor Function Depending on Two Tensors," Dokl. Akad. Nauk. SSSR 133 (4), 743-744 (1960)
5.  R. V. Goldstein and V. M. Entov, Qualitative Methods in Continuum Mechanics (Nauka, Moscow, 1989; Wiley, New York, 1994).
6.  B. E. Pobedrya and D. V. Georgievskii, Foundations of Continuum Mechanics, Lecture Course (Fizmatlit, Moscow, 2006) [in Russian].
7.  B. E. Pobedrya, Lectures on Tensor Analysis (Izd-vo MGU, Moscow, 1986) [in Russian].
8.  D. V. Georgievskii, "Tensor Nonlinear Effects under Isothermal Strain of Continua," Uspekhi Mekh. 1 (2), 150-176 (2002).
9.  Yu. I. Dimitrienko, Nonlinear Continuum Mechanics (Fizmatlit, Moscow, 2009) [in Russian].
10.  M. U. Nikabadze, Several Problems of Tensor Calculus (TsPI MGU, Moscow, 2007) [in Russian].
11.  M. Hanin and M. Reiner, "On Isotropic Tensor-Functions and the Measure of Deformation," ZAMP 7 (5), 377-393 (1956).
Received 11 January 2010
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