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

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