Linearization of tensor nonlinear constitutive relations in the problems on stability of flows

D. V. Georgievskii

Čebyševskij sbornik, Tome 18 (2017) no. 3, pp. 202-209.

Résumé

The apparatus of tensor nonlinear functions occupies an important place in the nonlinear mechanics of a continuous medium, both in hydrodynamic applications and in problems of mechanics of a deformed solid, strength and fracture [1]. Tensor nonlinear defining correlations simulate the so-called orthogonal effects of the stress-strain state (see in [2] a review on the issue), characterized by noncollinearity of voltage deviators and the corresponding kinematic tensor. Such a noncollinearity can explain the Poynting effect and ratchet [3–9]. The scientific works pays much attention both to the definition of the main flow parameters and to the stability of such a flow with respect to small perturbations belonging to a particular class. The statement of the boundary value problem in perturbations assumes the linearization of all the system equations near the main process, including the defining correlations. Along with the general form of the tensor-nonlinear determining relations, the paper considers tensor-linear isotropic media, tensor linear potential media, the Bingham body (a two-constant viscoplastic model), the Saint-Venant flow (ideally rigid-plastic model), and the Newtonian fluid.

Détail

Keywords: constitutive relations, linearization, tensor nonlinear functions, stress, strain rate, potential media, Bingham solid, Newtonian viscous fluid.

Bibliographie

[1] Levin V.A., Nonlinear Computational Mechanics of Strength, v. 1, Formation and Development of Defects, Fizmatlit, M., 2015, 456 pp.

[2] Georgievskii D.V., “On the Orthogonal Effects of Stress-Strain State in Mechanics of Continuum”, Vestnik Kievskogo Nats. Univ. Ser. Fiziko-Matematich. Nauki, 2013, no. 3, 114–116 | Zbl

[3] Poynting J.H., “On the changes in the dimensions of a steel wire when twisted, and on the pressure of distorsional waves in steel”, Proc. Roy. Soc. London. Ser. A, 86 (1912), 534–561 | DOI | Zbl

[4] Green A.E., “A note on second-order effect in the torsion of incompressible cylinders”, Proc. Cambridge Philos. Soc., 50:3 (1954), 488–490 | DOI | MR | Zbl

[5] Chen M., Chen Z., “Second-order effect of an elastic circular shaft during torsion”, Appl. Math. Mech., 12:6 (1991), 769–776

[6] Chaboche J.L., “Modeling of ratcheting: evaluation of various approaches”, Europ. J. Mech. Ser. A. Solids, 13:4 (1994), 501–518

[7] Delobelle P., Robinet P., Bocher L., “Experimental study and phenomenological modelization of ratchet under uniaxial and biaxial loading on an austenitic stainless steel”, Internat. J. Plasticity, 11:4 (1995), 295–330 | DOI

[8] Batra R.C., dell'Isola F., Ruta G.C., “Generalized Poynting effects in prismatic bars”, J. Elasticity, 50:2 (1998), 181–196 | DOI | MR | Zbl

[9] Akinola A., “An energy function for transversely-isotropic elastic material and the Poynting effect”, Korean J. Comput. Appl. Math., 6:3 (1999), 639–649 | MR | Zbl

[10] Rivlin R.S., Ericksen J.L., “Stress-deformation relations for isotropic materials”, J. Rational Mech. Anal., 4:2 (1955), 323–425 | MR | Zbl

[11] Pobedria B.E., Lectures on Tensor Analysis, Moscow State University Ed., M., 1986, 263 pp.

[12] Bell J. F., Mechanics of Solids, v. I, The Experimental Foundations of Solid Mechanics, Springer, Berlin, 1973 | MR

[13] Georgievskii D.V., Müller W.H., Abali B.E, “Establishing experiments for obtaining of the material functions in tensor nonlinear constitutive relations”, Izvestiya RAN. Ser. fizich., 76:12 (2012), 1534–1537

[14] Georgievskii D.V., “Establishing experiments in tensor nonlinear theories of continuum mechanics”, Vestnik Moskovskogo Univ. Ser. 1. Matematika, mekhanika, 2016, no. 2, 66–68

[15] Georgievskii D.V., Stability of Viscoplastic Solids Deformation Processes, URSS, M., 1998