About the impact of a viscoelastic on a fixed semi-space

Abstract

The impact interaction of a viscoelastic body bounded by a quadratic paraboloid in the contact zone with an absolutely rigid fixed half-space having a flat boundary surface is considered. We have introduced the assumption that the force of resistance to the deformation of a body that hits depends not only on its elastic characteristics, but also on the square of the velocity of its center of mass. Additionally, the well-known dependences of G. Hertz on the distribution of contact deformations and dynamic pressure were used. Under these assumptions, the first integral of a non-linear differential equation of motion of the second order is expressed in elementary functions. In the same functions, the expression for the maxima was obtained: dynamic compression, impact force, dimensions of the elliptical contact area, pressure at the center of this area. An extremum study has established that with a viscoelastic impact, the maxima of the impact force and contact pressure can be reached not at the end of the body compression phase, but somewhat earlier during this process. A condition is derived when the maxima of the forces of viscoelastic and elastic impacts are the same. It is related to the coefficient of viscosity and the square of the velocity of the collision of bodies. It is shown that at the end of the compression stage, the impact force and the contact pressure during viscoelastic impact can be significantly less than the maximum, as well as those that the theory of impact of perfectly elastic bodies leads to. A compact formula for calculating the coefficient of speed recovery with a direct central impact is derived. It is shown that it depends on the coefficients of viscosity for compression and decompression and on the square of the velocity of collision of bodies. With the increase of this value, the speed recovery coefficient decreases. The duration in time of the stages of compression and expansion are represented by improper convergent integrals of the second kind, which are not expressed analytically in terms of known functions. The integrands in them have an algebraic singularity in the series 1/2. Therefore, it is recommended to single out the singular part in quadratures and integrate it analytically, and calculate the regular part on a computer. Examples of calculations are given and a comparative analysis of numerical results is carried out.