 | | 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 |
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| D. V. Georgievskii, "Perturbations of flows of incompressible nonlinearly viscous and viscoplastic fluids caused by variations in material functions," Mech. Solids. 42 (3), 375-381 (2007) |
| Year |
2007 |
Volume |
42 |
Number |
3 |
Pages |
375-381 |
| Title |
Perturbations of flows of incompressible nonlinearly viscous and viscoplastic fluids caused by variations in material functions |
| Author(s) |
D. V. Georgievskii (Lomonosov Moscow State University, GSP-2, Leninskie Gory, Moscow, 119992, Russia, georgiev@mech.math.msu.su) |
| Abstract |
Material functions are necessary element of the constitutive relations determining any model of continuum. These functions can be defined as a collection of objects from which the operator of constitutive relations can be reconstructed completely. The material functions are found in test experiments and show the differences between a given medium and other media in the framework of the same model [1]. The "test experiment theory" is an important part of modern experimental mechanics.
Just as in any experiment, from determining the viscosity coefficient by using the rotational viscosimeters to constructing the yield surface by using machines combined loading, the material functions are determined with an unavoidable error. For example, experimenters know that, in experiments with arbitrary accuracy, the moduli of elasticity can only be measured with an unimprovable tolerance of about 7%. Starting already from [2], the investigators’ attention has been repeatedly drawn to the fact that it is necessary to take into account this tolerance in determining the material constants, functions, and functionals in problems of mechanics and especially in analyzing the stability of deformation processes. Mathematically, this means that problems of stability under perturbations of the initial data, external constantly acting forces, domain boundaries, etc. should be supplemented with the assumption that the material functions have unknown perturbations of a certain class [3].
The variations of material functions in the framework of the linearized stability theory were considered in [2, 4, 5]. In what follows, we study isotropic tensor functions in the most general case of scalar and tensor nonlinearity. These functions are assigned the meaning of constitutive relations between the stress and strain rate tensors in continuum. These constitutive relations contain scalar material functions of invariants on which, as follows from the above, some variations proportional to a small physical parameter α can be imposed. These variations imply perturbations of the tensor function itself. The components of such perturbations linear and quadratic in α are determined. In each of the approximations, we write out a closed system of equations consisting of the equations of motion (linear in the variables of the respective approximation) and the incompressibility condition.
We analyze tensor-linear functions with arbitrary scalar rheology inmore detail. Materials with such constitutive relations include non-Newtonian viscous fluids and viscoplastic materials. Viscoplastic materials are characterized by the existence of rigidity zones, where the stress intensity is less than the yield strength. We derive equations for the boundaries of the rigidity zones in the perturbed motion, in particular, for the case in which the unperturbed medium is a viscous Newtonian fluid. Throughout the paper, index-free notation is used. |
| References |
| 1. | B. E. Pobedrya,
Mechanics of Composite Materials
(Izd-vo MGU, Moscow, 1984)
[in Russian]. |
| 2. | A. A. Il'yushin,
"Stability of Plates and Shells Beyond the Yield Strength,"
Prikl. Mat. Mekh.
8 (5), 337-360 (1944)
[J. Appl. Math. Mech. (Engl. Transl.)]. |
| 3. | D. V. Georgievskii,
"Perturbation of Material Functions in Constitutive Relations of
Ideally- and Viscoplastic Media,"
in Elasticity and Inelasticity
(Izd-vo MGU, Moscow, 2006),
pp. 35-43
[in Russian]. |
| 4. | D. V. Georgievskii,
"Problem of Stability of Quasilinear Flows with Respect to Perturbations
of the Strengthening Function,"
Prikl. Mat. Mekh.
63 (5), 826-832 (1999)
[J. Appl. Math. Mech. (Engl. Transl.)]. |
| 5. | D. V. Georgievskii, "On the Linearization of the Constitutive Relations
for Tensorially Nonlinear Isotropic Media," Izv. Akad. Nauk. Mekh.
Tverd. Tela, No. 5, 21-25 (2001) [Mech. Solids (Engl. Transl.)]. |
| 6. | B. E. Pobedrya, Lectures on Tensor Analysis (Izd-vo MGU,
Moscow, 1986) [in Russian]. |
| 7. | D. V. Georgievskii, "Small Perturbations of Undeformed State in
Media with Yield Strength," Dokl. Ross. Akad. Nauk 392 (5),
634-637 (2003) [Russian Acad. Sci. Dokl. Math.]. |
| 8. | B. E. Pobedrya,
"On Quasilinear Tensor Operators,"
Vestnik Moskov. Univ. Ser. I. Mat. Mekh.,
No. 5, 97-100 (1971)
[Moscow Univ. Math. Bull.]. |
| 9. | A. A. Il'yushin,
"Deformation of Viscoplastic Bodies,"
Uchen. Zap. MGU
39, 3-81 (1940). |
| 10. | D. M. Klimov, A. G. Petrov, and D. V. Georgievskii,
Viscoplastic Flows: Dynamical Chaos, Stability, Mixing
(Nauka, Moscow, 2005)
[in Russian]. |
|
| Received |
12 May 2006 |
| Link to Fulltext |
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