Elastic contact deformation with Hertz

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

Hertz [27] solved the problem of the contact between two elastic elliptical bodies by modeling each body as an infinite half plane which is loaded over a contact area that is small in comparison to the body itself. The requirement of small areas of contact further allowed Hertz to use a parabola to represent the shape of the profile of the ellipses. In essence. Hertz modeled the interaction of elliptical asperities in contact. Fundamental in his solution is the assumption that, when two elliptical objects are compressed against one another, the shape of the deformed mating surface lies between the shape of the two undeformed surfaces but more closely resembles the shape of the surface with the higher elastic modulus. This means the deformed shape after two spheres are pressed against one another is a spherical shape. 

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] ... 

Modelling of the true contaet area between surfaees requires consideration of the deformation that occurs at the peaks of asperities as they come into contact with mating surfaces. Purely elastic contact between two solids was first described by H Hertz [7], The Hertzian contact area (yl between a sphere of radius r and a flat surface... 

Only small particles, less than 1 pm in diameter, would show this effect. Krupp explained this in terms of the equations for London-van der Waals attractive forces between rigid spheres, together with the Hertz equations of contacL Because the attraction is proportional to particle diameter, the force at the particle contact decreases with D. However, the elastic area of the contact spot decreases faster, from the Hertz Equation (9.1), with Thus, as the particle gets smaller, the contact pressure must rise to the point at which plastic deformation occurs. 

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... 

E. Winkler, F. Grashof, H. Hertz,8 etc., have studied the stresses which are set up when two elastic isotropic bodies are in contact over a portion of their surface, when the surfaces of contact are perfectly smooth, and when the press, exerted between the surfaces is normal to the plane of contact. H. Hertz showed that there is a definite point in such a surface representing the hardness defined as the strength of a body relative to the kind of deformation which corresponds to contact with a circular surface of press. and that the hardness of a body may be measured by the normal press, per unit area which must act at the centre of a circular surface of press, in order that in some point of the body the stress may first reach the limit consistent with perfect elasticity. If H be the hardness of a body in contact with another body of a greater hardness than H, then for a circular surface of pressure of diameter d press. p radius of curvature of the line p and the modulus of penetration E,... 

AFM can also be used to probe local mechanical properties of thin films of food biopolymers, which are difficult to measure using traditional rheological methods. Several mechanical models have been developed to analyze the Young s modulus of food systems. One of the simplest models, the Hertz model, assumes that only the elastic deformation exists in a surface with spherical contacts, and the adhesion force can be neglected (Hugel and Seitz 2001). Equation (8.2) describes the relationship between the loading force, F and the penetration depth, d, where a is the radius of contact area, R the curvature of the tip radius, Vi and the Poisson s ratios of the two contact materials that have Young s modulus, Ei and E2. ... 

The resolution of the contact mode depends on a contact area at the tip apex. The contact diameter (a) can be estimated with the Hertz model in which the contact area increases with applied force (F) on the tip due to elastic deformation between tip and sample. 

When two surfaces with very small surface asperites are brought together, there may be no plastic deformation whatsoever, only elastic deformation, and Ar will be greater. That means Ar > Up. This is true of a highly polished surface such as that of a bearing ball that is pressed into a flat polished surface. It will produce an area of contact as given by Hertz s equation for elastic deformation, namely ... 

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.

For spheres of the same isotropic elastic material with plane strain modulus E and the same undeformed radius R, the deformed contact area is flat and its boundary is circular due to the symmetry of the configuration. According to the Hertz point of view, the normal stress distribution predicted for the contact area z = 0,0[Pg.645]

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... 

During uniaxial loading of a comparatively soft spherical elastic pellet with a smooth stiff wall (flat surface), the contact area of the particle deforms as a circle. The model of Hertz (Hertz, 1882) describes the force and internal pressure distribution as well as the contact radius depending on the particle radius and moduh of elasticity behavior of the two contacting materials. 

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