Steady-state spinning of a disk in contact with a foundation

When the stress in the material of a spinning body is influenced by the rate of strain, such as in a viscoelastic material, the deformation depends on the angular velocity of the body. During the spinning motion, material entering the contact area (leading edge) is compressed by the sudden increase in contact pressure, while material leaving the foundation relaxes. For a perfectly elastic material the deformation is reversible, so the contact area (and stress state) is symmetrical about a plane normal to the foundation and containing the axle. A viscoelastic material, on the other hand, responds instantaneously to the sudden increase in contact pressure but requires a finite time to relax after leaving the contact area. During such a loading/unloading stress-strain cycle some strain energy is dissipated. In other words, in contrast to a perfectly elastic material, the deformation is not reversible, and the loading and unloading stress-strain paths do not coincide. Consequently, the point at which material leaves the foundation is closer to the center plane than the point at which material enters the contact zone. Furthermore, since the contact pressure is asymmetrical, rolling is resisted by a moment around the axle.

The nature of viscoelastic material effects in this problem is illustrated in Figure 2 through Figure 4. Figure 2 shows the reaction force normal to the foundation; Figure 3 shows the moment around the axle as a function of the angular spinning velocity. The bullet points in the two figures represent the reference transient Lagrangian solution. Figure 4 shows the contact pressure at different angular velocities. These figures indicate that at low angular velocities, when the time that a material point is in contact with the foundation is long compared to the relaxation time of the material, the behavior of the disk corresponds to the fully relaxed long-term elastic solution. The vertical reaction force (Figure 2) is at a minimum, and the stress state is symmetrical about the midplane (Figure 4), so the moment around the axle is zero (Figure 3). At high angular velocities the solution corresponds to the instantaneous (or dynamic) elastic solution with the vertical reaction force reaching a limiting value. The stress state is still symmetrical about the midplane, so the moment around the axle is zero. The viscoelastic effects become important when the time that a material point is in contact with the foundation is of the same order of magnitude as the relaxation time of the material. Under these conditions energy is dissipated in each loading/unloading cycle, so the contact area becomes asymmetrical (Figure 4) and rolling is resisted by a moment around the axle (Figure 3).

Figure 5 through Figure 7 compare the radial, circumferential, and shear stress between the two analysis methods for the case where the viscoelastic effects are a maximum ($\omega =$ 2.5 rad/s). The solid lines represent the steady-state transport solution; the broken lines represent the reference transient Lagrangian solution. The figures plot the stress near the outer surface along a streamline—the angle is measured about the $z$-axis along the direction of material flow, with the $x$-axis defining $\theta =$0°. The reference solution is obtained by monitoring the stress at one integration point on the streamline during the analysis history. Since the solution is steady, the time variation of stress can be converted to a variation along the streamline. The figures show very good agreement between the two solution methods.

The steady-state transport solution obtained with cylindrical elements also agrees closely with the reference solution. The results of this simulation are not reported here.