Construction of mathematical models of the statics of grain media considering the Reynolds effect

Authors

DOI:

https://doi.org/10.15587/1729-4061.2019.184592

Keywords:

granular materials, equilibrium thermodynamics, Reynolds effect, horizontal grain layer, boundary-value problem, boundary conditions

Abstract

This study addresses the construction of a mathematical model, the statement of boundary-value problems on the statics of a grainy material related to the technological processes of agricultural production. A working apparatus employed to construct the model of a grainy material is the methods of equilibrium thermodynamics. We have stated the main thermodynamic equality, which makes it possible to derive a rheological ratio that establishes the connection between stresses and deformations of the granular material. The chosen grainy material is a granular medium that manifests a Reynolds effect. This effect occurs in the case of small deformations and indicates the presence of a dependence of dilation on the stress tensor deviator. In contrast to the classical methods that consider a model of continuous medium with the non-deformed and smooth grain’s particles, the present work takes into consideration both a Reynolds effect and the presence of elastic deformations. The resulting rheological ratio produces the dependence for a stress tensor on the deformation tensor corresponding to ratios from the linear theory of elasticity.

For the case of an isothermal process of deformation, a boundary-value problem on the grain material’s statics in the field of gravity has been stated. This paper shows the statement and solution to two particular tasks on the balance of a granular layer along the horizontal plane: in the absence of surface forces and under the action of tangent surface forces on a free surface.

The boundary problems on the equilibrium of a granular material are nonlinear in character, and the resulting solution represents a complex mathematical apparatus involving numerical methods.

The obtained models for the statics of a continuous environment precede the consideration of dynamic problems, in particular, the study of equilibrium stability

Author Biographies

Alexander Nanka, Kharkiv Petro Vasylenko National Technical University of Agriculture Alchevskykh str., 44, Kharkiv, Ukraine, 62001

PhD, Professor, Rector

Department of Technical Systems and Animal Husbandry Technologies

Ivan Iyevlev, Kharkiv Petro Vasylenko National Technical University of Agriculture Alchevskykh str., 44, Kharkiv, Ukraine, 62001

PhD, Associate Professor

Department of Technical Systems and Animal Husbandry Technologies

Vitaliy Sementsov, Kharkiv Petro Vasylenko National Technical University of Agriculture Alchevskykh str., 44, Kharkiv, Ukraine, 62001

PhD, Associate Professor

Department of Technical Systems and Animal Husbandry Technologies

Denis Boiko, Kharkiv Petro Vasylenko National Technical University of Agriculture Alchevskykh str., 44, Kharkiv, Ukraine, 62001

PhD, Senior Lecturer

Department of Technical Systems and Animal Husbandry Technologies

Viktor Duhanets, State Agrarian and Engineering University in Podilya Shevchenka str., 13, Kamianets-Podilskyi, Ukraine, 32300

Doctor of Pedagogical Sciences, Associate Professor

Department of Tractors, Cars and Power Tools

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Published

2019-11-22

How to Cite

Nanka, A., Iyevlev, I., Sementsov, V., Boiko, D., & Duhanets, V. (2019). Construction of mathematical models of the statics of grain media considering the Reynolds effect. Eastern-European Journal of Enterprise Technologies, 6(7 (102), 53–62. https://doi.org/10.15587/1729-4061.2019.184592

Issue

Section

Applied mechanics