P. Pallav, PhD

This page deals with the distribution of the wear rate over the surface of each molar.

We continue with the same theoretical set up as in page Chewing Abstractly. As explained there, the lower molar moving up creates both a velocity gradient (v'), and pressure (p) in the food fluid.
Because neither is distributed uniformly over the surfaces, the wear rate (proportional to p x v', Erosive Activity) is not the same in all places on the surfaces either.

Velocity Gradient

At any moment during the closing motion the velocity gradient (at the surfaces) increases linearly with r from zero at the center to a higher value at the circumference.

As the height (h) decreases, the velocity gradient increases too. That is, it stays zero at the center but it increases to an increasingly higher value at the circumference.

The figure shows the velocity gradient at three different distances (h). The values of h have no units, these merely indicate the ratio (3:2:1, in that order) of the heights at three positions during the closing motion, but you may think of millimeters.

Pressure

The pressure is not the same everywhere between the molars either. The highest pressure occurs at the center. From there it decays to zero at the circumference.

In fluid mechanics, zero means equal to the static pressure of the environment. With this pressure, the food is not pushed against the surfaces anymore.

Similar to the velocity gradient, as the height (h) decreases when the lower molar moves up, the pressure increases in proportion. That is, as the height decreases the pressure at the center increases, but it will still decay to zero at the circumference.

Erosive Activity

Equation 4 was used for the EA plot in the figure to the left.

Zero times anything equals zero. Therefore the Erosive Activity is zero at the center where the velocity gradient is zero, and at the circumference because of zero pressure. The maximum Erosive Activity occurs in between these places.

The absolute magnitude of the plots, that is the values at the y-scales, depend on many parameters and are as such not relevant in this discussion.

Equations

The velocity gradient (v') at either surface can be found with Equation 2 in Chewing Abstractly and Equations 1 and 2 in Velocity in Fluids. b is the exponent in equation 3 in Shearing. Because we assume pseudo plasticity, b = ½.

Where V_v = vertical closing velocity

The highest pressure occurs at the center:

From there it decays to zero at the circumference:

The average pressure (P_av) can be found with Equation 3 in Chewing Abstractly.

The Erosive Actvity is the product (=multiplication) of the pressure (p_r equation3) and the velocity gradient (v'_r equation 1) at each point (r) on the surface. With the Equations above and Equation 3 in Chewing Abstractly the Erosive Activity (EA) can be expressed as

Equation 4 can be integrated over, and divided by the occlusal area. This results in the average Erosive Activity (EA_av)

Integration of Equation 5 over the closing trajectory (h = ∞ to h_min) results in the average Erosive Activity in a chewing cycle. h refers to the average height (vertical clearance) between two molars. h_min refers to the average height at centric occlusion.

A great value of R, meaning large molars and / or a small value of h_min, meaning well fitting occlusal surfaces with little space between the contacts, create a high wear rate. 10 % more radius or diameter will yield 25% more, 10% less average height will give 30% more erosive activity. (b=0.5)

Equations 2 to 6 are limited to plane (flat) and parallel occlusal surfaces.

Slanted Closing Motion

The shearing, which is caused by the vertical closing motion (V_v) is much greater than the shearing, introduced by a horizontal component of the motion (V_h). That is why we are not taking it into account.

With a vertical motion only, the point of zero velocity gradient will be at the center of the occlusal surfaces. When a horizontal component is added to the closing motion, this point will move away from the center. Where this point will be (r_(v'=0)) depends on the height (h), both velocities (V_v and V_h) and the type of fluid (b).

With a horizontal velocity (V_h) twice as great as the vertical velocity (V_v) the angle of motion changes by more than 60 degrees. With pseudo plastic flow and a height of 1 millimeter, the point of zero velocity gradient will move just 1 millimeter away from the center. The direction of this shift will be opposite on both surfaces.

It doesn't seem likely that the variation of the angle at which the molars come together during chewing will exceed 60 degrees in any direction from an average angle, so the result will be that the erosive activity at the center will not be completely zero.