 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
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| In English (Mech. Solids): | | 4810 |
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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 ℝ3 and possibly in ℝ2, 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 |
| 1. | R. S. Rivlin,
"Further Remarks on the Stress - Deformation Relations
for Isotropic Materials,"
J. Rational Mech. Anal.
4 (5), 681-702 (1955). |
| 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 |
| Link to Fulltext |
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