The aim of this study is to develop analytical dependencies for the uniaxial stiffness of a spatial composition of elastic balls of same diameter, considering its volumetric structure. A review of the literature was conducted regarding types of ball packings that have practical applications for describing the structure of crystals, composite materials, and ball mill loadings of various types. For calculating the stiffness of a three-dimensional composition of balls, the study is based on G. Hertz's theory of elastic ball contact. According to this theory, the relationship between compressive force and the center-to-center displacement of balls is nonlinear with an exponent of 1.5. By spatially combining individual ball contacts, the nonlinear stiffness for simple cubic and face-centered cubic packings of balls under uniaxial compression was determined. These packing types were chosen as boundary cases of regular ball packings: the former as the least dense possible packing and the latter as the densest.
Initially, the stiffness of a single layer of ball packing in a plane perpendicular to the compression force was determined by summing the parallel-connected stiffnesses of all balls. Next, the total stiffness of the spatial composition of balls compressed between two massive plates was calculated through sequential combination of the stiffnesses of all single layers along the height of the composition. Differences in the stiffness of elemental ball contacts, both between themselves and with the bounding plate layer, were taken into consideration.
As a result, formulas were derived for determining the uniaxial stiffness of the spatial ball composition for the two boundary packing types, depending on the elastic properties of the ball material and massive boundaries, the ball diameter, and the dimensions of the deformed ball composition.
The comparison of packing stiffnesses did not account for the friction coefficient due to its minor influence and its significant reduction under conditions of vibration or the presence of liquid at ball contacts. It was concluded that, firstly, the stiffness of a ball composition in a face-centered cubic packing slightly exceeds that of a simple cubic packing, within the permissible error margins of engineering calculations. Secondly, the formulas for face-centered cubic ball packing are more suitable for practical calculations. Thirdly, the results of the study can be used for modeling the stress-strain state of technological ball loadings in vibratory, planetary, and other types of mills; for modeling the behavior of layers made of solid bulk materials with approximately isometric particle shapes; and for determining the elasticity of frames in composite material fillers with significant differences in the elastic properties of their components.
