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

## Authors

• Alexander Nanka Kharkiv Petro Vasylenko National Technical University of Agriculture Alchevskykh str., 44, Kharkiv, Ukraine, 62001, Ukraine
• Ivan Iyevlev Kharkiv Petro Vasylenko National Technical University of Agriculture Alchevskykh str., 44, Kharkiv, Ukraine, 62001, Ukraine
• Vitaliy Sementsov Kharkiv Petro Vasylenko National Technical University of Agriculture Alchevskykh str., 44, Kharkiv, Ukraine, 62001, Ukraine
• Denis Boiko Kharkiv Petro Vasylenko National Technical University of Agriculture Alchevskykh str., 44, Kharkiv, Ukraine, 62001, Ukraine
• Viktor Duhanets State Agrarian and Engineering University in Podilya Shevchenka str., 13, Kamianets-Podilskyi, Ukraine, 32300, Ukraine

## 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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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

## Section

Applied mechanics