 | | 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 |
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| In English (Mech. Solids): | | 4810 |
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| A.A.Markin, M.Yu. Sokolova, and D.V. Khristich, "A. A. Il'yushin's Postulate for Anisotropic Materials and a Version of Constitutive Relations," Mech. Solids. 46 (1), 30-35 (2011) |
| Year |
2011 |
Volume |
46 |
Number |
1 |
Pages |
30-35 |
| DOI |
10.3103/S0025654411010055 |
| Title |
A. A. Il'yushin's Postulate for Anisotropic Materials and a Version of Constitutive Relations |
| Author(s) |
A.A.Markin (Tula State University, pr-t Lenina 92, Tula, 300600 Russia, markin@tsu.tula.ru)
M.Yu. Sokolova (Tula State University, pr-t Lenina 92, Tula, 300600 Russia, sokolova@tula.net)
D.V. Khristich (Tula State University, pr-t Lenina 92, Tula, 300600 Russia, dmitro@tula.net) |
| Abstract |
The particular isotropy postulate proposed by A. A. Il'yushin permits stating and implementing programs for construction and experimental justification of relations determining the thermomechanical properties of initially isotropic materials for trajectories of various degree of complexity.
In the present paper, we propose an extended statement of the isotropy postulate, which takes into account the initial elastic anisotropy as well as possible dilatation effects for initially isotropic materials. The construction is based on the consideration (following the approach proposed by Ya. K. Rykhlevskii) of eigensubspaces of the linear elasticity operator. The strain and stress tensors in the eigensubspaces are similar, and their vector images in the six-dimensional space are coaxial and do not vary under orthogonal transformations within the eigensubspace. The form of the particular postulate generalized to nonlinear operators takes into account the fact that the vector images of stresses can leave the eigensubspace in which the strain trajectories are located. In this case, the process image must be invariant under rotations and reflections of the strain trajectory within the eigensubspace.
In the framework of the extended isotropy postulate, we consider a version of the constitutive relations of physically nonlinear thermoelasticity in which the effects of multimodulus behavior of materials, as well as dilatation and thermodynamic effects, are taken into account. |
| Keywords |
particular postulate, anisotropy, thermoelasticity, nonlinearity, strain trajectory, vector images of processes |
| References |
| 1. | A. A. Il'yushin,
"Problems of the General Theory of Plasticity,"
Prikl. Mat. Mekh.
24 (3), 399-411 (1960)
[J. Appl. Math. Mech. (Engl. Transl.)
24 (3), 587-603 (1960)]. |
| 2. | A. A. Il'yushin,
Plasticity. Foundations of Mathematical Theory
(Izd-vo AN SSSR, Moscow, 1963)
[in Russian]. |
| 3. | V. S. Lenskii,
"Experimental Verification of Laws of Isotropy and Retardation under Complex Loading,"
Izv. Akad. Nauk SSSR. OTN. Mekh. Mashinostr.,
No. 11, 15-24 (1958). |
| 4. | R. A. Vasin,
"On Experimental Verification of a Law Describing the Vector Properties of Plastic Materials,"
in Strength, Plasticity, and Viscoelasticity of Materials and Constructions
(UNTs AN SSSR, Sverdlovsk, 1986),
pp. 41-45
[in Russian]. |
| 5. | V. G. Zubchaninov,
Mathematical Theory of Plasticity
(TGTU, Tver, 2002)
[in Russian]. |
| 6. | A. A. Markin and M. Yu. Sokolova,
"Eigenstates of Anisotropic Materials and Particular Postulate of Isotropy,"
in Problems of Mechanics of Deformable Solids and Rocks,
Collected Papers Dedicated to the 75th Birthday of E. I. Shemyakin
(Fizmatlit, Moscow, 2006),
pp. 423-433
[in Russian]. |
| 7. | A. A. Markin and M. Yu. Sokolova,
"Generalization of A. A. Il'yushin's Particular Postulate to the Case of Anisotropic Materials,"
in Modern Problems of Strength, Plasticity, and Stability,
Collection of Papers Dedicated to the 75th Birthday of the V. G. Zubchaninov
(TGTU, Tver, 2007),
pp. 201-209
[in Russian]. |
| 8. | J. Rychlewski,
"On Hooke's Law,"
Prikl. Mat. Mekh.
48 (3), 420-435 (1984)
[J. Appl. Math. Mech. (Engl. Transl.)
48 (3), 303-314 (1984)]. |
| 9. | B. D. Annin and N. I. Ostrosablin,
"Anisotropy of Elastic Properties of Materials,"
Zh. Prikl. Mekh. Tekhn. Fiz.
49 (6), 131-151 (2008)
[J. Appl. Mech. Tech. Phys. (Engl. Transl.)
49 (6), 998-1014 (2008)]. |
| 10. | Yu. N. Shevchenko and R. G. Terekhov,
"Test of a Consequence of the Postulate of Isotropy,"
Probl. Prochn.,
No. 2, 42-45 (1972)
[Strength of Materials (Engl. Transl.)
4 (2), 168-171 (1972)]. |
| 11. | A. A. MarkinMarkin and M. Yu. Sokolova,
"Non-Linear Relations of Anisotropic Elasticity and a Particular Postulate of Isotropy,"
Prikl. Mat. Mekh.
71 (4), 587-594 (2007)
[J. Appl. Math. Mech. (Engl. Transl.)
71 (4), 536-542 (2007)]. |
| 12. | B. I. Koval'chuk, V. V. Kosarchuk, and A. A. Lebedev,
"Plastic Strains of Initially Anisotropic Materials under Simple and Complex Loading,"
in Strength, Plasticity, and Viscoelasticity of Materials and Constructions
(UNTs AN SSSR, Sverdlovsk, 1986),
pp. 74-81
[in Russian]. |
| 13. | V. V. Kosarchuk and B. I. Koval'chuk,
"Formulation of the Law of Delay of the Vector Properties of Initially Anisotropic Materials,"
Probl. Prochn.,
No. 11, 3-6 (1986)
[Strength of Materials (Engl. Transl.)
4 (11), 1435-1439 (1986)]. |
| 14. | A. A. Markin and M. Yu. Sokolova,
"Constitutive Relations of Nonlinear Thermoelasticity of Anisotropic Bodies,"
Zh. Prikl. Mekh. Tekhn. Fiz.
44 (1), 170-175 (2003)
[J. Appl. Mech. Tech. Phys. (Engl. Transl.)
44 (1), 141-145 (2003)]. |
| 15. | A. A. Markin and M. Yu. Sokolova,
Thermo-Mechanical Models of Reversible Finite Deformation
(Izd-vo TulGU, Tula, 2010)
[in Russian]. |
| 16. | G. L. Brovko,
"The Concepts of Process Transform and Five-Dimensional Isotropy of the Properties
of Materials for Finite Deformations,"
Dokl. Akad. Nauk SSSR
308 (3), 565-570 (1989)
[Sov. Phys. Dokl. (Engl. Transl.)
34, 856 (1989)]. |
| 17. | Yu. N. Shevchenko and N. N. Tormakhov,
"Isotropy Postulate for Finite Deformations,"
Prikl. Mekh.
35 (1), 17-27 (1999)
[Int. Appl. Mech. (Engl. Transl.)
35 (1), 13-23 (1999)]. |
| 18. | A. A. Pozdeev, P. V. Trusov, and Yu. I. Nyashin,
Large Elastoplastic Deformations: Theory, Algorithms, Applications
(Nauka, Moscow, 1986)
[in Russian]. |
| 19. | A. A. Markin and A. A. Tolokonnikov,
"Measures of Finite Deformation Processes,"
Izv. Sev.-Kavk. Nauchn. Tsentra Vyssh. Shk. Estestv. Nauki,
No. 2, 49-53 (1987). |
| 20. | R. A. Vasin, A. A. Il'yushin, and P. A. Mossakovskii,
"Investigation of the Constitutive Relations and Fracture Criteria
for Solid and Thin-Walled Tubular Cylindrical Samples,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 2, 177-184 (1994)
[Mech. Solids (Engl. Transl.)]. |
| 21. | L. S. Andreev,
"Verifying the Isotropy Postulate ,"
Prikl. Mekh.
5 (7), 122-125 (1969)
[Int. Appl. Mech. (Engl. Transl.)
5 (7), 762-765 (1969)]. |
| 22. | O. L. Tolokonnikov,
"Machine for Testing Tubular Samples of Materials in a High Pressure Medium,"
Izv. Akad. Nauk. SSSR. Mekh. Tverd. Tela,
No. 3, 185-188 (1985)
[Mech. Solids (Engl. Transl.)]. |
| 23. | N. N. Tormakhov,
Laws of Elastoplastic Strain of a Body Element
along Trajectories of Small Curvature under Large Deformations,
Author's Abstract of Candidate's Dissertation in Technical Sciences
(Kiev, 1988)
[in Russian]. |
|
| Received |
02 June 2010 |
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