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IssuesArchive of Issues2011-2pp.213-224

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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
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