 | | 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
| Total articles in the database: | | 11223 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8011
|
| In English (Mech. Solids): | | 3212 |
|
| << Previous article | Volume 50, Issue 6 / 2015 | Next article >> |
| V.A. Peleshko, "Applied and Engineering Versions of the Theory of Elastoplastic Processes of Active Complex Loading. Part 1: Conditions of Mathematical Well-Posedness and Methods for Solving Boundary Value Problems," Mech. Solids. 50 (6), 650-656 (2015) |
| Year |
2015 |
Volume |
50 |
Number |
6 |
Pages |
650-656 |
| DOI |
10.3103/S0025654415060060 |
| Title |
Applied and Engineering Versions of the Theory of Elastoplastic Processes of Active Complex Loading. Part 1: Conditions of Mathematical Well-Posedness and Methods for Solving Boundary Value Problems |
| Author(s) |
V.A. Peleshko (Central Scientific Research Institute for Engineering (TsNIIMash), ul. Pionerskaya 4, Korolev, Moscow Oblast, 141070 Russia, peleshkobva@inbox.ru) |
| Abstract |
In the proposed theory of plasticity, the deviator constitutive relation has a trinomial form (the vectors of stresses, stress rates, and strain rates, which are formed form the deviators, are coplanar) and contains two material functions; one of these functions depends on the modulus of the stress vector, and the other, on the angle between the stress vector and the strain rate, the length of the deformation trajectory arc, and the moduli of the stress and strain vectors. The spherical parts of the stress and strain tensors satisfy the relations of elastic variation in the volume.
We obtain conditions on the material functions of the model which ensure the mathematical well-posedness of the statement of the initial-boundary value problem (i.e., the existence and uniqueness of the generalized solution, and its continuous dependence on the external loads). We also describe the scheme for solving the initial-boundary value problem step by step using the model and present the expression for the Jacobian of the boundary value problem at the time step. These results are formalized as a subprogram for prescribing the mechanical properties of the user material in the finite-element complex ABAQUS, which allows one to calculate the structure deformations on the basis of the proposed theory. |
| Keywords |
plasticity, active complex loading process, initial-boundary value problem, finite element calculations |
| References |
| 1. | A. A. Il'yushin,
Plasticity
(Izd-vo AN SSSR, Moscow, 1963)
[in Russian]. |
| 2. | A. A. Il'yushin,
"On Foundations of General Mathematical Theory of Plasticity,"
in Problems of the Theory of Plasticity
(Izd-vo AN SSSR, Moscow, 1961),
pp. 3-29
[in Russian]. |
| 3. | V. S. Lenskii,
"Hypothesis of Local Determinacy in Plasticity,"
Izv. Akad. Nauk SSSR. OTN. Mekh. Mashinostr.,
No. 5, 154-158 (1962). |
| 4. | A. A. Il'yushin,
"SN-EVM Method in Theory of Plasticity,"
in Problems of Applied Mathematics and Mechanics
(Nauka, Moscow, 1971),
pp. 166-178
[in Russian]. |
| 5. | R. A. Vasin and A. A. Il'yushin,
"On One Representation of Elasticity and Plasticity Laws in Plane Problems,"
Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela,
No. 4, 114-118 (1983)
[Mech. Solids (Engl. Transl.)]. |
| 6. | V. S. Lenskii and E. V. Lenskii,
"Three-Term Relation of General Plasticity,"
Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela,
No. 4, 111-115 (1985)
[Mech. Solids (Engl. Transl.)]. |
| 7. | V. G. Zubchaninov,
"Determining Relations of the Theory of Elastoplastic Processes,"
Prikl. Mekh.
25 (5), 3-12 (1989)
[Int. Appl. Mech. (Engl. Transl.)
25 (5), 427-435 (1989)]. |
| 8. | I. N. Molodtsov,
"A Version of Thermal Mechanics of Elastoplastic Processes under Complex Loading,"
in Elasticity and Inelasticity
(MGU, Moscow, 2011),
pp. 202-209
[in Russian]. |
| 9. | 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)]. |
| 10. | R. A. Vasin,
"On the Relationship between Stresses and Strains for Two-Link Polygonal Deformation Trajectories,"
Prikl. Mekh.
1 (11), 89-94 (1965). |
| 11. | Dao Zui Bik,
"On the Local Determinacy Hypothesis in Plasticity,"
Vestnik Moskov. Univ. Ser. I Mat. Mekh., No. 2, 67-75 (1965). |
| 12. | V. P. Degtyarev,
Plasticity and Creep of Engineering Structures
(Mashinostroenie, Moscow, 1967)
[in Russian]. |
| 13. | I. M. Korovin,
"Several Problems of Material Plasticity under Loading along a Trajectory with a Kink Point,"
Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela,
No. 3, 152-158 (1969)
[Mech. Solids (Engl. Transl.)]. |
| 14. | L. V. Abramova and I. V. Kryukova,
"Elastoplastic Metal Strain Theory for Two-Link Broken Paths,"
Probl. Prochn.,
No. 1, 8-12 (1981)
[Strength of Materials (Engl. Transl.)
13 (1), 5-10 (1981)]. |
| 15. | V. G. Zubchaninov,
"On Laws of Theory of Elastoplastic Processes under Complex Loading in Plane Problems,"
in Strength of Materials and Structure Elements in Complex Stress State
(Naukova Dumka, Kiev, 1986),
pp. 110-117
[in Russian]. |
| 16. | V. I. Malyi,
"Study of Several Functionals of the Theory of Elastoplastic Processes,"
in Elasticity and Inelasticity, No. 5
(MGU, Moscow, 1978),
pp. 202-209
[in Russian]. |
| 17. | Dao Zui Bik,
"Modification of Relations for Elastoplastic Mean Curvature Processes,"
Vestnik Moskov. Univ. Ser. I Mat. Mekh.,
No. 5, 103-106 (1981). |
| 18. | S. V. Ermakov,
"Study of the Statement of the Boundary-Value Problem
of the Theory of Elastoplastic Mean Curvature Processes,"
Vestnik Moskov. Univ. Ser. I Mat. Mekh.,
No. 2, 88-92 (1982). |
| 19. | A. V. Muravlev,
"Several General Properties of the Stress-Strain Relationship in Plasticity,"
Izv. Akad. Nauk SSSR.
Mekh. Tverd. Tela,
No. 6, 178-179 (1984)
[Mech. Solids (Engl. Transl.)]. |
| 20. | R. A. Vasin and R. I. Shirov,
"On the Study of the Vector and Scalar Properties of Metals in Complex Loading Tests,"
in Strength of Materials and Structure Elements in Complex Stress State
(Naukova Dumka, Kiev, 1986),
pp. 57-61
[in Russian]. |
| 21. | V. A. Peleshko,
"Experimental Investigation of a Variant of the Theory of Elastoplastic Deformation
of Metals under Complex Loading,"
Probl. Prochn.,
No. 12, 48-53 (1990)
[Strength of Materials (Engl. Transl.)
22 (12), 1773-1779 (1990)]. |
| 22. | V. A. Peleshko,
"Conditions of Mathematical Well-Posedness of a Version
of Differential-Nonlinear Plasticity Relations
and Methods for Solving Boundary-Value Problems,"
Vestnik Moskov. Univ. Ser. I Mat. Mekh.,
No. 6, 37-43 (1990). |
| 23. | I. I. Vorovich and Yu. P. Krasovskii,
"On the Method of Elastic Solutions,"
Dokl. Akad. Nauk SSSR
126 (4), 740-743 (1959)
[Soviet Math. Dokl. (Engl. Transl.)]. |
| 24. | D. L. Bykov,
"On Some Methods for Solving Problems of the Theory of Plasticity,"
in Elasticity and Nonelasticity, No. 4
(Izd-vo MGU, Moscow, 1975),
pp. 119-139
[in Russian]. |
| 25. | B. E. Pobedrya,
Numerical Methods in Elasticity and Plasticity Theory
(Izd-vo MGU, Moscow, 1995)
[in Russian]. |
| 26. | G. L. Brovko,
"On a Method of Successive Approximations in the Class of Problems of General Plasticity,"
Vestnik Moskov. Univ. Ser. I Mat. Mekh.,
No. 6, 76-83 (1982). |
| 27. | S. V. Ermakov,
"Investigation of the Formulation of the Boundary-Value Problem of
the Local Theory of Elastoplastic Processes,"
Prikl. Mat. Mekh.
56 (2), 321-330 (1992)
[J. Appl. Math. Mech. (Engl. Transl.)
56 (2), 274-282 (1992)]. |
| 28. | H. Gajewski, K. Gröger, and K. Zacharias,
Nonlinear Operator Equations and Operator Differential Equations
(Akademic-Verlag, Berlin, 1974; Mir, Moscow, 1978). |
|
| Received |
23 September 2013 |
| Link to Fulltext |
|
| << Previous article | Volume 50, Issue 6 / 2015 | Next article >> |
|
 If you find a misprint on a webpage, please help us correct it promptly - just highlight and press Ctrl+Enter
|
|
Russian 
English