Modeling of deformation of a thin-walled cylinder as an isotropic body in the form of a rubber

Abstract

The article deals with the problem of deformation a liner of a milking cup, as an isotropic body in the form of a thin-walled cylinder. Based on the developed analytical model of a liner, taking into account its pre-tension, structural and technological parameters and physical and mechanical characteristics of the rubber material for modeling deformations in the radial, longitudinal and circular planes.
The main parameter of the closure of the liner is its radial deformation on the entire active part, depending on the design parameters and the physical and mechanical characteristics of the rubber. The parameters for deformation modeling are R - radius of the liner, E - modulus of elasticity, h - thickness of the liner, pH - vacuum pressure, l - length of the active part of the liner, ν - Poisson coefficient for rubber, Fn - force of the actual liner. Depending on the central angle, the deformation shape of the liner along its entire working length is modeled in the radial cross-section plane. Under pre-tensioning of the liner, there is a deformation of up to 1 mm in the form of a slightly expressed ellipsoid immediately after the pinched ends. At a distance of up to 10 mm from both stranded ends at a vacuum of 48 KPa, the liner of the circular cross section is deformed in the form of a “dumbbell”, however, with a pre-tension of 60 N, the maximum deformation at the place of greatest deflection reaches 7 mm from the radius of the liner 11 mm towards the flattening, and without pre-tensioning the maximum deflection at the point of maximum deflection reaches 5 mm.
The analysis of the simulation results shows that the pre-tension of the rubber does not affect the nature of its closure under vacuum pressure. The tension partially changes the amount of deformation. However, pretensioning of the rubber provides initial deformation in the cross section immediately after pinching, reducing the diameter, and allowing the milking cup to be held securely by the cow.