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IssuesArchive of Issues2016-5pp.619-622

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D.V. Georgievskii, "Potentiality of Isotropic Nonlinear Tensor Functions Relating Two Deviators," Mech. Solids. 51 (5), 619-622 (2016)
Year 2016 Volume 51 Number 5 Pages 619-622
DOI 10.3103/S0025654416050162
Title Potentiality of Isotropic Nonlinear Tensor Functions Relating Two Deviators
Author(s) D.V. Georgievskii (Lomonosov Moscow State University, Moscow, 119992 Russia; Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr. Vernadskogo 101, str. 1, Moscow, 119526 Russia, georgiev@mech.math.msu.ru)
Abstract In the theory of constitutive relations, isotropic quadratic nonlinear tensor functions modeling media with second-order effects, in particular, with misalignment of the force and kinematic tensors, are considered. It is very interesting to consider tensor functions with a scalar potential relating two symmetric deviators of rank two. In this case, the potentiality conditions are integrated, and it is shown that the first integral contains two arbitrary functions of the quadratic invariant of the tensor argument and one arbitrary function of the cubic invariant. A tensorially nonlinear generalization of the rigid-viscoplastic model (a two-contact Bingham solid) is carried out.
Keywords tensor function, scalar potential, constitutive relations, material function, invariant, rigid-viscoplastic model
References
1.  R. S. Rivlin and J. L. Ericksen, "Stress-Deformation Relations for Isotropic Materials," J. Rat. Mech. Anal. 4, 323-425 (1955).
2.  M. A. Brutyan and P. L. Krapivskii, "Hydrodynamics of Newtonian Fluids," in Ser. Complex and Special Fields of Mechanics, Vol. 4 (VINITI, Moscow, 1991), pp. 3-98 [in Russian].
3.  B. E. Pobedrya, Numerical Methods in Elasticity and Plasticity Theory (Izd-vo MGU, Moscow, 1995) [in Russian].
4.  D. V. Georgievskii, "Tensor Nonlinear Effects in Isothermal Deformation of Continuous Media," Uspekhi Mekh. 1 (2), 150-176 (2002).
5.  D. V. Georgievskii, "Tensor-Nonlinear Shear Flows: Material Functions and Diffusion-Vortex Solutions," Nelin. Din. 7 (3), 451-463 (2011).
6.  D. V. Georgievskii, "On "Orthogonal Effects" of Stress-Strain State in Continuum Mechanics," Vestnik Kiev Nats. Un-tu, Ser. Fiz.-Mat. Nauki, No. 3, 114-116 (2013).
7.  D. V. Georgievskii, "The Angle between the Stress Deviator and the Strain-Rate Deviator in a Tensor Nonlinear Isotropic Medium," Vestnik Moskov. Univ. Ser. I Mat. Mekh., No. 6, 63-66 (2013) [Moscow Univ. Mech. Bull. (Engl. Transl.) 68 (6), 149-151 (2013)].
8.  D. V. Georgievskii, W. H. Müller, and B. E. Abali, "Establishing Experiments to Find Material Functions in Tensor Nonlinear Constitutive Relations," Izv. Ross. Akad. Nauk. Ser. Fiz. 76 (12), 1534-1537 (2012) [Bull. Russ. Acad. Sci. Phys. (Engl. Transl.) 76 (12), 1374-1377 (2012)].
9.  D. V. Georgievskii, "Establishing Experiments in Tensor Nonlinear Theories of Continuum Mechanics," Vestnik Moskov. Univ. Ser. I Mat. Mekh., No. 2, 66-68 (2016) [Moscow Univ. Mech. Bull. (Engl. Transl.) 71 (2), 49-50 (2016)].
10.  R. V. Goldstein, V. A. Gorodtsov, and D. S. Lisovenko, "Pointing Effect for Cylindrically Anisotropic Nano/Microtubes," Fiz. Mezomekhanika 19 (1), 5-14 (2016).
Received 03 April 2016
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