 | | 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: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4760 |
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| D. V. Berezhnoi, V. N. Paimushin, and V. I. Shalashilin, "Studies of Quality of Geometrically Nonlinear Elasticity Theory for Small Strains and Arbitrary Displacements," Mech. Solids. 44 (6), 837-851 (2009) |
| Year |
2009 |
Volume |
44 |
Number |
6 |
Pages |
837-851 |
| DOI |
10.3103/S0025654409060041 |
| Title |
Studies of Quality of Geometrically Nonlinear Elasticity Theory for Small Strains and Arbitrary Displacements |
| Author(s) |
D. V. Berezhnoi (Scientific and Technical Center for Problems in Dynamics and Strength, Tupolev Kazan State Technical University, K. Marksa 10, Kazan, 420111 Russia, dsm@dsm.kstu-kai.ru)
V. N. Paimushin (Scientific and Technical Center for Problems in Dynamics and Strength, Tupolev Kazan State Technical University, K. Marksa 10, Kazan, 420111 Russia)
V. I. Shalashilin (Scientific and Technical Center for Problems in Dynamics and Strength, Tupolev Kazan State Technical University, K. Marksa 10, Kazan, 420111 Russia) |
| Abstract |
Earlier it was shown in [1, 2] that the equations of classical nonlinear elasticity constructed for the case of small strains and arbitrary displacements are ill posed, because their use in specific problems may result in the appearance of "spurious" bifurcation points. A detailed analysis of these equations and the construction, in their stead, of consistent equations of geometrically nonlinear theory of elasticity can be found in [3]. Certain steps in this direction were also made in [4, 5]. In [3], it was also stated that the methods and applied program packages (APPs) based on the use of the classical relations of nonlinear elasticity require some revision and correction. In the present paper, this conclusion is justified and confirmed by numerical finite-element solutions of several three-dimensional geometrically nonlinear deformation problems and linearized problems on the stability of equilibrium of a rectilinear beam. These solutions were obtained by using two APPs developed by the authors and the well-known APP "ANSYS." It is shown that the classical equations of the geometrically nonlinear theory of elasticity, which underly the first of the developed APP and the well-known APP "ANSYS," often lead to overestimated buckling loads for structural members as compared with the consistent equations proposed in [1-3]. |
| Keywords |
geometric nonlinearity, small strains, large displacements |
| References |
| 1. | V. N. Paimushin and V. I. Shalashilin,
"Consistent Variant of Continuum Deformation Theory
in the Quadratic Approximation,"
Dokl. Ross. Akad. Nauk
396 (4), 492-495 (2004)
[Dokl. Phys. (Engl. Transl.)
49 (6), 374-377 (2004)] |
| 2. | V. N. Paimushin and V. I. Shalashilin,
"The Relations of Deformation Theory in the Quadratic Approximation and the Problems of Constructing Improved Versions of the Geometrically Non-Linear Theory of Laminated Structures,"
Prikl. Mat. Mekh.
69 (5), 861-881 (2005)
[J. Appl. Math. Mech. (Engl. Transl.)
69 (5), 773-791 (2005)]. |
| 3. | V. N. Paimushin,
"The Equations of the Geometrically Non-Linear Theory of Elasticity and Momentless Shells for Arbitrary Displacements,"
Prikl. Mat. Mekh.
72 (5), 822-841 (2008)
[J. Appl. Math. Mech. (Engl. Transl.)
72 (5), 597-610 (2008)]. |
| 4. | L. H. Donnell,
Beams, Plates, and Shells
(McGraw-Hill, New York, 1976; Nauka, Moscow, 1982). |
| 5. | F. N. Shklyarchuk,
Analysis of the Strain State and Stability
of Geometrically Nonlinear Elastic Systems,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 1, 140-146 (1998)
[Mech. Solids (Engl. Transl.)
33 (1), 114-119 (1998)]. |
| 6. | K. Washizu,
Variational Variational Methods in Elasticity and Plasticity
(Pergamon Press, New York, 1972; Mir, Moscow, 1987). |
| 7. | A. I. Golovanov and D. V. Berezhnoi,
Finite-Element Method in Mechanics of Deformable Solids
(DAS, Kazan, 2001)
[in Russian]. |
| 8. | V. V. Novozhilov,
Foundations of the Nonlinear Theory of Elasticity
(Gostekhizdat, Leningrad-Moscow, 1948; Graylock Press, Rochester, New York, 1953). |
| 9. | V. V. Petrov,
"Finite Deflection Analysis of Shallow Shells,"
Nauch. Dokl. Vyssh. Shkoly. Stroit.,
No. 1, 27-35 (1959). |
| 10. | E. I. Grigolyuk and V. I. Shalashilin,
Problems of Nonlinear Deformation: The Continuation Method Applied to Nonlinear Problems in Solid Mechanics
(Nauka, Moscow, 1988; Kluwer, Dordrecht, 1991; Springer, New York, 1991). |
|
| Received |
27 July 2007 |
| Link to Fulltext |
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