Strain, Speed, and Microslip | Mechanics and Machines

Microslip in rolling motion is often very complicated, but the net effect can sometimes be estimated pretty easily based on strain and the resulting changes in velocity.

Consider a belt or a sheet of material drawn between rollers as sketched above. The roller at $A$ applies a traction on the sheet to pull it to the right, and the roller at $B$ resists the motion, so that the sheet is in tension between $A$ and $B$ . To the left of roller $B$ the speed of the sheet is $V_0$ . But as the sheet is pulled over the roller, the tension within it increases, causing it to stretch. This stretch causes its speed to change as described below.

Suppose the tension in the sheet causes a strain $\epsilon_x(x)$ , which is related to a displacement field $u(x)$ according to the usual definition of strain in a stationary material, so that

$\epsilon_x = \frac{d\, u(x)}{dx}$

We can visualize the relationship between $u(x)$ and the total displacement of a particle as in the sketch below.
In the absence of any strain, a particle on the moving sheet would move from position $P$ to position $P_1$ during the time $\Delta t$ . Its speed is $\Delta x / \Delta t$ which we define as the reference speed $V_0$ . But due to the strain in the material, each point is displaced relative to its zero-strain location by a distance $u(x)$ . Therefore, the particle starts the time interval at $Q$ rather than $P$ and ends at $Q_1$ rather than $P_1$ .

Then the total distance it travels during the interval $\Delta t$ is $\Delta x + u(x+\Delta x)-u(x)$ . Taking the limit as $\Delta t$ goes to zero, we can write the speed as

$V(x) = \frac{\Delta x}{\Delta t} \left( 1 + \frac{d \, u(x)}{d x} \right)$

or simply

$V(x) = V_0 \left( 1+\epsilon_x \right)$

So we see that the speed of a body moving under strain is modified by the factor $(1+\epsilon)$ .

Rolling creep and energy losses can often be determined directly from the velocity differences in contact regions. Returning to the example of the sheet drawn between rollers, the sheet will stick to the roller as it comes onto the roller and slip as it leaves, so the surface speed of roller $B$ will be $V_0$ . Likewise, the surface speed of roller $A$ will be $V_0 (1+\epsilon_x)$ . The difference between these two speeds represents lost motion or creep in the system.

The classic book Contact Mechanics by K. L. Johnson lays out the details of web and roller stick and slip in some detail at the beginning of Chapter 8, and follows with an excellent exposition on rolling contact in general. The relationship between velocity and strain derived above is stated but not derived in the text.

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