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IssuesArchive of Issues2013-5pp.546-552

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Yu.N. Radaev, "Asymptotic Axes of Stress Tensors and Strain Increment Tensors in Mechanics of Compressible Continua," Mech. Solids. 48 (5), 546-552 (2013)
Year 2013 Volume 48 Number 5 Pages 546-552
DOI 10.3103/S0025654413050105
Title Asymptotic Axes of Stress Tensors and Strain Increment Tensors in Mechanics of Compressible Continua
Author(s) Yu.N. Radaev (Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr-tVernadskogo101, str.1, Moscow, 119526 Russia, radayev@ipmnet.ru, y.radayev@gmail.com)
Abstract New tensor representations of the stress state and the kinematics of compressible flows are obtained in the paper with the use of the notion of asymptotic directions of the symmetric stress tensor and the strain increment tensor. The exposition is based on terminology and notation typical of the mathematical theory of plasticity, but all main results remain valid for stresses and strains in compressible continua. The simplest and most efficient forms of the stress tensor for "completely plastic," "semiplastic," and "nonplastic" spatial stress states are found, where the asymptotic stress axes serve as the most natural reference frame ensuring new symmetric tensor representations of stresses different from the spectral ones. Similar representations can be extended to the stress increment tensor. Two-dimensional curvilinear grids such that the strain rates of their elements are always zero are chosen on the surfaces orthogonal to the directions of the "intermediate" principal strain increment. Incremental relations for the sliding rates along the grid lines are obtained, and these relations generalize the Geiringer equations along the characteristic lines, which are well known in the theory of plane deformation of perfectly plastic bodies. The generalization readily applies to spatial flows, and the possible flow compressibility is taken into account as well.
Keywords compressibility, stress, strain increment, asymptotic direction, plane deformation, Geiringer equations
References
1.  D. D. Ivlev, Theory of Perfect Plasticity (Nauka, Moscow, 1966) [in Russian].
2.  Yu. N. Radaev, Spatial Problem of Mathematical Theory of Plasticity, 2nd ed. (Izd-vo Samara Univ., Samara, 2006) [in Russian].
3.  L. M. Kachanov, Foundations of the Theory of Plasticity (Nauka, Moscow, 1969) [in Russian].
4.  V. V. Sokolovskii, The Theory of Plasticity (Vysshaya Shkola, Moscow, 1969) [in Russian].
5.  A. A. Il'yushin, Plasticity, Part 1 (Gostekhizdat, Moscow-Leningrad, 1948) [in Russian].
6.  A. Haar and T. Karman, "To the Theory of Stress States in Plastic and Loose Media," in Theory of Plasticity. Collection of Papers, Ed. by Yu. N. Rabotnov (Izdat. Inostr. Liter., Moscow, 1948), pp. 41-56 [in Russian].
7.  A. Yu. Ishlinskii, "On Equations of Body Deformation beyond the Limit of Elasticity," Uch. Zapiski Moskov. Univ. Ser. Mekh., No. 117, 90-108 (1946). (The paper is also published in the book: A. Yu. Ishlinskii, Applied Problems of Mechanics, Vol. 1 (Nauka, Moscow, 1986), pp. 62-83 [in Russian].)
8.  W. Lode, "Influence of the Average Principal Stress on the Yield of Metal," in Theory of Plasticity. Collection of Papers Ed. by Yu. N. Rabotnov (Izdat. Inostr. Liter., Moscow, 1948), pp. 168-205 [in Russian].
9.  A. Nadai, Theory of Flow and Fracture of Solids, Vol. 2 (Mir, Moscow, 1969) [in Russian].
10.  H. Polaczek-Geiringer, "Beitrag zum vollständigen ebenen Plastizitätsproblem," in Verhandlungen d. 3 Intern. Kongress f. techn. Mechank, Stockholm, 1930, pp. 185-190.
Received 27 May 2013
Link to Fulltext http://link.springer.com/article/10.3103/S0025654413050105
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