Hertz theory of impact

Hertz Theory of Impact is described in the Mathematical Theory of Elasticity by A.E.H Love, Dover Publications, NY (1944), p 198. This theory was applied by M.P. Murgai, JChemPhys 22, 1687-9 (1954) CA 49,... 

Deresiewicz, H. (1968) A note on Hertz theory of impact. Acta Mech. 6, 110-112 Doi, M., Edwards, S.F. (1986) The Theory of Polymer Dynamics (Clarendon Press, Oxford) Dugdale, D.S. (1960) Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8, 100-104 Dundurs, J. (1975) Properties of Elastic Bodies in Contact , in The Mechanics of the Contact between Deformable Bodies, ed. by A. D. de Pater, J.J. Kalker (Delft University Press) pp. 54-66 Dundurs, J., Stippes, M. (1970) Role of elastic constants in certain contact problems. J. Appl. Mech. 37, 965-970... 

Pao, YH. (1955) Extension of the Hertz theory of impact to the viscoelastic case. J. Appl. Phys. 26, 1083-1088... 

In this chapter, two simple cases of stereomechanical collision of spheres are analyzed. The fundamentals of contact mechanics of solids are introduced to illustrate the interrelationship between the collisional forces and deformations of solids. Specifically, the general theories of stresses and strains inside a solid medium under the application of an external force are described. The intrinsic relations between the contact force and the corresponding elastic deformations of contacting bodies are discussed. In this connection, it is assumed that the deformations are processed at an infinitely small impact velocity and for an infinitely long period of contact. The normal impact of elastic bodies is modeled by the Hertzian theory [Hertz, 1881], and the oblique impact is delineated by Mindlin s theory [Mindlin, 1949]. In order to link the contact theories to collisional mechanics, it is assumed that the process of a dynamic impact of two solids can be regarded as quasi-static. This quasi-static approach is valid when the impact velocity is small compared to the speed of the elastic... 

Contact problems have their origins in the works of Hertz (1881) and Boussinesq (1885) on elastic materials. Indentation problems are an important subset of contact problems (17,18). The assessment of mechanical properties of materials by means of indentation experiments is an important issue in polymer physics. One of the simplest pieces of equipment used in the experiments is the scleroscope, in which a rigid metallic ball indents the surface of the material. To gain some insight into this problem, we consider the simple case of a flat circular cylindrical indentor, which presents a relatively simple solution. This problem is also interesting from the point of view of soil mechanics, particularly in the theory of the safety of foundations. In fact, the impacting cylinder can be considered to represent a circular pillar and the viscoelastic medium the solid upon which it rests. 

The elastic contribution of the impact energy, absorbed during the compression, is released during the restitution phase of the impact and leads to the force that separates the contact partners. The elastic part of the contact force Fei h described by the approach based on the Hertz theory (Hertz, 1882). The elastic force is proportional to the equivalent modulus of elasticity E and the equivalent shear modulus G for normal and shear direction respectively, which are given as ... 

In the case of a collision between two particles, the contact forces are calculated according to a contact model based on the theory developed by Hertz (1882) for the normal impact and a non-slip approximation of the model by Mindlin and Dere-siewicz (1953) for the tangential part of the contact force, as proposed by Tsuji et al. (1992). The normal contact force is... 

Based on the presented theory, it is intended to construct the contact force model between the two spheres. The initial indentation velocity between the two spheres is = 0.3 m/s. The speed of deformation waves is 2.6xlO m/s, which provides a limiting value of 0.(126 m/s for the impact to be considered elastic. Hence, the Hertz contact force model with permanent indentation is a valid one. The generalized parameter K is calculated from equation (2), with v = 0.33, to be equal to 5.50x10 N/m -. The equivalent mass of the two spheres is obtained from equation (7) as m = 0.046 kg. From equations (12), (13), and (14), the unknown parameters in the contact force model are evaluated as... 

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