Tangential tractions

Assume that the normal pressure and displacements are not affected by the existence of the tangential traction and resulting displacements. Hence, the normal pressure and contact area can be determined by the Hertzian theory. For the sliding contact of spheres, substituting Eq. (2.69) into Eq. (2.77) gives rise to the tangential traction as... 

With the tangential traction parallel to the x-axis, the general displacements on the contact surface due to the tangential traction can be expressed by... 

It should be noted that the displacement in the z-direction due to tangential traction is usually not zero at the interface. Therefore, the actual interface with tangential force mismatches the flat surface caused by the normal loading [Johnson, 1985], In this text, it is assumed that this mismatch of normal displacements can be neglected this turns out to be an acceptable approximation for most cases. 

This distribution leads to a uniform tangential displacement of the contact surface. q0 is the maximum tangential traction at the center of the contact surface. By integrating Eq. (2.94) over the contact surface, q0 is found to be... 

Several methods were presented by Lubkin (1951) for the determination of the radius of the no-slip core and the distribution of tangential traction in this region. Only the final results of Lubkin s work are introduced here. 

Figure 2.14. Twist angle and tangential traction by torsion (from Lubkin, 1951) (a) Twist moment Mz versus total twist angle fi (b) Typical distribution of tangential traction.

The foregoing formulation for crazing focuses on the description along the normal direction. The formulation of Tijssens et al. [20,21] also accounts for a tangential component in the cohesive properties with a tangential traction at and a tangential displacement At being related by... 

At first glance dead-weight tangential traction devices of the type depicted in Fig. 7-1 la appear to be well suited for the determination of the coefficient of static friction, the assumption being that the slider assembly is put in motion when the force exerted on it just exceeds the force of static friction. But experiments by Burwell and Rabinowicz [10]... 

Figure 8-3. Relation of the tangential traction coefficient to the area ratio A /A. 0 Clean platinum, t Lubricated platinum. Data computed by Tabor [5] from results by Courtney-Pratt and Eisner [10].

Figure 8-12 shows some typical results for platinum against platinum and for steel against steel. The tangential traction coefficient ((i = T/N, where T is the tangential tractive force and N is the normal loading force. The displacements at low values of T are almost entirely irreversible when T is relaxed and hence are due to the plastic shear of asperity junctions. The tangential traction coefficient approaches the coefficient of friction (as measured in the usual way) while the dis-... 

Figure 8-12. Initiation of sliding by tangential traction. (a) Polished platinum, (b) Steel. 1 Unlubricated. 2 Laurie acid in cetane. Normal...

Dayson [17] applied the concept of junction growth by tangential traction to determine the effective value of a and adjusted the value of e as follows ... 

When two particles are in contact, a collision analysis can be conducted to obtain the velocities of the particles after the collision. To simplify the analysis, it can be assumed that the tangential traction and the resulting displacements have no effect on the normal collision. For the collision between particles a and b, the normal components after collision can be obtained by solving the equations for the restitution coefficient and the conservation of momentum. 

The Young s modulus of the film can be obtained from the tangential traction for a sphere-plane contact given by the Hertz theory and Amontons law of friction (49) ... 

Bt = admissible set of tangential tractions Bn = admissible set of normal contact tractions... 

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