Application of the method of the specific forces in the mechanics of a viscoelastic collision between a solid body and a semi-space

Abstract

The Mechanics of a viscoelastic collision between a solid body and a semi-space at impact at an arbitrary angle of attack have been examined in this paper. The radius of the contact area has been determined, using a geometry of mutual approach between two spherical bodies. It was proposed that the forces of viscosity and the forces of elasticity can be found by integration of the specific forces acting inside an elementary volume of the contact zone, and on this basis the principally new method of defining the viscoelastic forces has been developed. In this method, the volume of deformation is considered as a system comprising an infinitely large number of infinitesimal discrete elements connected to each other in a definite way. It is shown here that this method allows finding the viscoelastic forces for any theoretical or experimental dependencies between the distance of mutual approach of two solid bodies and the diameter of the contact area. Also the differential equations of the displacement (the movement) of the centre of mass of the body have been obtained. Equations for the calculation of work and energy in the compression and in the restitution phases, and also in the rolling shear phase have been derived. Approximate solutions for the differential equations of movement (displacement) by using the method of equivalent work have been derived. Equations for the normal contact stresses have been obtained. Also, equations for kinematic and dynamic parameters of the viscoelastic collision have been derived in this article. Examples of the comparison of theoretical results and conclusions have been given in the paper.

Solutions of the Problems of a Viscoelastic Dynamic Contact between Smooth Curvilinear Surfaces of two Solid Bodies by the Application of the “Method of the Specific Forces”

“…In the past many old papers and others published recently (Mindlin, 1949;Simon, 1967;Johnson, 1985 ;Goloshchapov, , 2015Cundall & Strack,1979;Hertzsch, Spahn , Schafer, Dippel, & Wolf, 1996;Ramírez, Poeschel, Brilliantov, & Schwager, 1999;Stronge, 2000;Roylance, 2001;Brilliantov & Poeschel, 2004;Makse, Gland, Jnohnson, & Schwartz, 2004;Schwager & Poschel, Van Zeebroeck, 2005;Dintwa, 2006;Schwager & Poschel, 2007Cheng, Subic, & Monir Takla, 2008;Becker, Schwagerand, & Pöschel, 2008;Schwager & Poschel, 2008;Thornton, 2009; have been used traditional theoretical rheological models, such as the "Linear Spring Dashpot Model"-(LS+D), the "Hertz Mindlin Spring Dashpot Model"-(HM+D), and the "Discrete Elements Method" -(DEM) and others . In all of these methods and models, for the definition of the effective parameter of elasticity (Some authors name it like a stiffness, or spring parameter), the Hertz's theory of elastic contact between two surfaces Lifshitz, 1944, 1965) has been used.…”

“…For example, the dynamic elasticity modules are very yieldable to a changing of a velocity and temperature of the matter in the area of deformations (Ferry, 1948(Ferry, , 1963Lee, 1962;Van Krevelen, 1972;Moor 1978;Nilsen, 1978Nilsen, , 1994Lakes, 1998;Meyers, 1994;Menard, 1999;Roylance, 2001;Goloshchapov, 2001Goloshchapov, , 2015Hosford, 2005;Popov, 2010, Popov & Hess, 2015. Also, it is necessary to mention that, the researches in the field of the collision of viscoelastic particles (granules) with identical mechanical properties have been made by Brilliantov, Spahn, Hertzsch and Pöschel (1996).…”

“…Also it is very important to understand indeed that, these specific forces Fxi, Fyi and Fzi are the elementary axial loads distributed on the infinitesimal sizes, da, db and dx parallel to axes X, Y and Z (Figure2). The name "specific force" already was applied by author of article (Goloshchapov, 2015), and this name is the short name, which usually is used to denote the elementary distributed axial loads.…”

“…The new conception is proposed here, how to find the elastic and the viscous forces by an integration of the specific forces in the boundaries of the contact area, which can be found by considering of the geometry of the contact. This method has been already used by author for case of the collision between a spherical solid body and a semi-space (Goloshchapov, , 2015.…”

“…The general tangential force can be found as sum (2) As it is known (Brilliantov et al, 1996;Goloshchapov, , 2015Jonson, 1985;Mindlin, 1949;Ramirez et al, 1999;Schafer et al, 1996;Schwager & Poschel, 2007;Stronge, 2000;Thornton, 2009 we can write all these viscoelastic forces, simply as: …”