Hertzian Theory for Frictionless Spheres in Contact

When two elastic and frictionless spheres are brought into contact under compressional forces or pressures, deformation occurs. The maximum displacement and contact area depend not only on the compressional force but also on the elastic material properties and radii of the spheres. The contact between two elastic and frictionless spherical bodies under compression was first investigated by Hertz (1881) and is known as the Hertzian contact. 

Consider two frictionless spheres in contact as shown in Fig. 2.9. M is a point on the surface of sphere 1 with a distance r from the Z -axis. N is the opposite point on the surface of sphere 2 with the same distance r from the Z2-axis. O is the contact point and also serves as the origin of both Zi-axis and Z2-axis. N, M, and O are in the same plane. The distances from M or N to the tangential plane which is normal to the Z i -axis and Z2-axis are denoted by i and Z2, respectively. In Fig. 2.9, the triangle O2AO is similar to the triangle NBO. Hence, the theorem of similarity of triangles gives 

During contact, location M tends to be moved in the direction of Z by a vertical displacement lz0. Similarly, N is going to be moved by a distance /z02 in the direction of Z2. When the center of sphere 1 approaches the center of sphere 2 by a distance a (note that a here is an approaching distance rather than the distance between the two centers of the spheres), the change of distance between M and N is then given by a — (lzoi + Im2). Consider the case of deformation in which M and N are on the edge of the surface of contact as a result of local compression. We have 

It is necessary to find an appropriate form for the pressure distribution p so that Eq. (2.65) can be satisfied for any r provided that r is small compared to the radii of spheres. One such pressure distribution is the Hertz pressure distribution in which the pressure at any location within the contact area is represented, as illustrated in Fig. 2.10, by 

It is noted that the total loading force Fz is equal to the pressure integrated over the contact area so that the constant C can be expressed by 

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