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A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
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| I. Yu. Kadashevich and Yu. I. Kadashevich, "Theory of Plasticity with Cross Couplings," Mech. Solids. 38 (5), 79-84 (2003) |
| Year |
2003 |
Volume |
38 |
Number |
5 |
Pages |
79-84 |
| Title |
Theory of Plasticity with Cross Couplings |
| Author(s) |
I. Yu. Kadashevich (St. Petersburg)
Yu. I. Kadashevich (St. Petersburg) |
| Abstract |
Being a result of development of the multi-surface plasticity theory, the theory with cross couplings proposed here is a fairly simple phenomenological theory of plasticity which describes a wide range of complex loading phenomena.
In 1968, Novozhilov and Kadashevich [1] proposed a version of the statistical theory of plasticity which took into account micro-stresses. That theory had some effect on the development of new ideas in the theory of plasticity and creep. The elastic-plastic deformation of structural materials at variable temperature is studied in [2] on the basis of multi-element models. An original structural model with subelements subject to linear hardening is examined in detail in [3] in connection with complex loading. An extension of the flow theory is described in [4, 5]. A generalization of the structural model of a material subjected to non-isothermal loading is given in [6]. A new approach to the incorporation of the third invariant is described in [7, 8]. We should also mention the review [9], which gives a detailed exposition of new trends in the theory of mathematical models of inelastic behavior and traces the links between the views outlined by Novozhilov and his students and the theory of elastic-plastic processes in the form proposed by Il'yushin [10], as well as the endochronic approaches to plasticity related to the ideas by Valanis [11]; a link is also traced to theories of sliding [12]. A detailed study of the relationship between micro and macro properties in elastic-plastic materials can be found in the comprehensive review by Vakulenko [13].
This paper describes the results of an investigation of a version of the theory, which the authors have called the theory with cross couplings. Originally, that theory found no approval among the readers, because it had no clear rheological interpretation. Moreover, after the corresponding rheological model and thermodynamical constraints on the constitutive relations had been proposed in [14], the possibilities of the theory were deemed too narrow, so that its further development was stopped and the theory was almost forgotten.
In the present paper, we analyze the constitutive relations, show that the theory successfully combines mathematical simplicity with the ability to describe a wide range of phenomena, and study the issues of thermodynamical consistency and the uniqueness of solutions. We formulate exact rules under which the constitutive relations are functional. Due to the unique cross coupling mechanism, which involves an automatic choice of the yield surface and the alteration of loading modes, it is possible to use the phenomenological theory for the description of delicate effects of complex deformation of materials. |
| References |
| 1. | Yu. I. Kadashevich and V. V. Novozhilov,
"On taking into account micro-stresses in the theory of plasticity,"
Inzh. Zh. MTT, No. 3, pp. 82-91, 1968. |
| 2. | V. S. Zarubin and M. A. Kuz'min,
"Elastic-plastic deformation of structural materials at variable temperature,"
Izv. Vuzov. Mashinostroenie, No. 12, pp. 57-60, 1969. |
| 3. | Yu. N. Shevchenko and V. Yu. Marina,
"A structural model of a medium subjected to non-isothermal loading,"
Prikl. Mekhanika, Vol. 12, No. 12, pp. 18-27, 1976. |
| 4. | A. N. Suprun, "On the basic experiment in rheonomic plasticity," in
Applied Problems of Strength and Plasticity. Methods of Solution [in Russian], pp. 74-84,
Izd-vo Gorkovsk. Un-ta, Gorky, 1987. |
| 5. | Yu. G. Korotkikh,
"On the basic experiment for the model of thermoviscoplasticity," in
Problems of Strength and Plasticity [in Russian], No. 6, pp. 3-20, Izd-vo Gorkovsk. Un-ta,
Gorky, 1977. |
| 6. | D. A. Gokhfel'd and O. S. Sadakov,
Plasticity and Creep of Structural Elements under Repeated Loading [in Russian],
Mashinostroenie, Moscow, 1984. |
| 7. | V. S. Bondar' and A. N. Frolov,
"Mathematical modeling of inelastic behavior and damage accumulation in materials subject to complex loading," Izv. AN SSSR. MTT [Mechanics of Solids],
No. 6, pp. 99-107, 1990. |
| 8. | Yu. I. Kadashevich and S. P. Pomytkin,
"On taking into account the deviator similarity phase in the theory of creep,"
Izv. AN. MTT [Mechanics of Solids], No. 5, pp. 129-133, 1992. |
| 9. | R. A. Vasin,
"Constitutive relations in the theory of plasticity," in
Advances in Science and Technology. Mechanics of Solids [in Russian],
Vol. 21, pp. 3-75, VINITI, Moscow, 1990. |
| 10. | A. A. Il'yushin, Plasticity: Fundamentals of the General Mathematical Theory [in Russian], Izd-vo AN SSSR, Moscow, 1963. |
| 11. | K. C. Valanis,
"A theory of plasticity without a yield surface,"
Arch. Mech., Vol. 23, No. 4, pp. 517-533, 1971. |
| 12. | V. V. Novozhilov, Yu. I. Kadashevich, and Yu. A. Chernyakov,
"The theory of plasticity with micro-strains taken into account,"
Doklady AN SSSR, Vol. 284, No. 4, pp. 821-823, 1985. |
| 13. | A. A. Vakulenko,
"On the relation between the micro and macro properties in elastic-plastic media,"
in Advances in Science and Technology. Mechanics of Solids [in Russian], Vol. 22,
pp. 3-54, VINITI, Moscow, 1991. |
| 14. | V. A. Pal'mov,
"On a version of the theory of plastic flow," Izv. AN SSSR. MTT
[Mechanics of Solids], No. 6, pp. 179-182, 1977. |
| 15. | Yu. G. Korotkikh and G. A. Makovkin,
"On modeling processes of non-proportional elastic-plastic deformation on the basis of plasticity equations with combined hardening," in
Applied Problems of Strength and Plasticity [in Russian], pp. 5-10, Tovarishch.
Nauch. Izdanii KMK, Moscow, 1997. |
| 16. | Z. Xia and F. Ellyin,
"A constitutive model with capability to simulate complex multiaxial ratchetting behavior of materials," Intern. J. Plasticity, Vol. 13, No. 1/2, pp. 127-142, 1997. |
| 17. | D. L. McDowell,
"Description of nonproportional cyclic ratchetting behavior,"
Eur. J. Mech. A., Vol. 13, No. 5, pp. 593-604, 1994. |
| 18. | W. T. Koiter,
"Stress-strain relations, uniqueness, and variational theorems for elastic-plastic material with a singular yield surface,"
Quart. Appl. Math., Vol. 11, pp. 350-354, 1953. |
|
| Received |
03 May 2000 |
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