 | | 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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| V.N. Paimushin, "A Study of Elasticity and Plasticity Equations under Arbitrary Displacements and Strains," Mech. Solids. 46 (2), 213-224 (2011) |
| Year |
2011 |
Volume |
46 |
Number |
2 |
Pages |
213-224 |
| DOI |
10.3103/S0025654411020099 |
| Title |
A Study of Elasticity and Plasticity Equations under Arbitrary Displacements and Strains |
| Author(s) |
V.N. Paimushin (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) |
| Abstract |
Equations of geometrically nonlinear theory of elasticity
with finite displacements and strains are analyzed. The equations are composed using three versions
of physical relations and applied to solve the problem of tension-compression
of a straight bar.
It is shown that the use of the classical relations
between the components of the stress tensor and the Cauchy-Green strain tensor
in the problem of compression of the bar
results in the appearance of "spurious" static loss of stability
such that the bar axis remains straight
if the stresses are referred to unit areas
before the deformation (conditional stresses).
However, in the problem of tension, the classical relations do not permit one to describe the phenomenon of static instability (neck formation as the plastic instability occurs). These drawbacks disappear if one uses the third version of the physical equations, composed as relations between the true stresses referred to unit areas of the deformed faces on which they act and the true elongations and shears. The relations of the third version are most correct; they permit one to pass to self-consistent equations of elasticity and plasticity under small strains and finite displacements, and they should be recommended for practical use. As an example, such relations are composed for the flow theory. |
| Keywords |
geometrically nonlinear theory of elasticity, bar tension, static instability, theory of flow |
| References |
| 1. | V. V. Novozhilov,
Foundations of Nonlinear Theory of Elasticity
(Gostekhizdat, Leningrad-Moscow, 1948)
[in Russian]. |
| 2. | 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)]. |
| 3. | K. F. Chernykh,
Nonlinear Elasticity in Engineering
(Mashinostroenie, Leningrad, 1986)
[in Russian]. |
| 4. | A. I. Golovanov and L. U. Sultanov,
Theoretical Foundations of Computational Nonlinear Mechanics of Deformable Bodies, Lecture Course
(Izd-vo Kazansk. Univ., Kazan, 2008)
[in Russian]. |
| 5. | 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)]. |
| 6. | 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)]. |
| 7. | V. N. Paimushin,
"Problems of Geometric Non-Linearity and Stability in the Mechanics
of Thin Shells and Rectilinear Columns,"
Prikl. Mat. Mekh.
71 (5), 855-893 (2007)
[J. Appl. Math. Mech. (Engl. Transl.)
71 (5), 772-805 (2007)]. |
| 8. | V. N. Paimushin and N. V. Polyakova,
"The Consistent Equations of the Theory of Plane Curvilinear Rods
for Finite Displacements and Linearized Problems of Stability,"
Prikl. Mat. Mekh.
73 (2), 303-324 (2009)
[J. Appl. Math. Mech. (Engl. Transl.)
73 (2), 220-236 (2009)]. |
| 9. | Ya. G. Panovko and I. I. Gubanova,
Stability and Oscillations of Elastic Systems
(Consultant Bureau, New York 1965; Nauka, Moscow, 1979). |
| 10. | D. V. Berezhnoi and V. N. Paimushin,
"Two Statements of Elastoplastic Problems and Theoretical Determination
of the Neck Formation Location in Samples under Tension,"
Prikl. Mat. Mekh.
75, (2011) (in press)
[J. Appl. Math. Mech. (Engl. Transl.)
75, (2011) (in press)]. |
| 11. | K. Z. Galimov, V. N. Paimushin, and I. G. Teregulov,
Foundations of the Nonlinear Theory of Shells
(Fen, Kazan, 1996)
[in Russian]. |
| 12. | A. I. Golovanov and D. V. Berezhnoi,
Finite-Element Method in Mechanics of Deformable Solids
(DAS, Kazan, 2001)
[in Russian]. |
|
| Received |
06 December 2010 |
| Link to Fulltext |
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