Elastic contact deformation with Hertz theory

Description of Elastic Contact Deformation with the Hertz Theory... 

When the particle deformation is small compared to the size of the undeformed spheres, the contacts obey Hertzian contact mechanics. According to Hertz s theory, the elastic energy associated with a single contact is [118] ... 

Fig. 3.4. The thickness of the adsorbed trilayer of the liquid crystal 5CB on DMOAP covered BK7 glass substrate. The full line is a fit to the Hertz theory with E = E/(l — v ) = 1.2 X 10 (1 0.3)Nm. R om point B to the point C, the surface adsorbed molecular trilayer is elastically deformed. At the instability point C, the layer ruptures and the AFM tip is in hard contact with the surface at D. The inset shows the linear relation between the thickness of the adsorbed layer and the length of fully extended liquid crystal molecule.

When two solids are in contact, deformation takes place in the local contact zone resulting in a contact force. This suggests that the contact force is directly related to the amount of local deformation or indentation of the two solids. The best-known force model for the contact between two spheres of isotropic material was developed by Hertz based on the theory of elasticity [9]. With radii Rj and R of the two spheres i and j, and masses mj and mj, the contact force f follows the relation... 

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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