When the disk wheel model is heated or cooled, the annular polymer strips covering its surface attempt to expand or contract.
This strain-energy decomposition is a key to understanding the warpage mechanism; it will be applied to the disk wheel model where the rate of increase in bending energy with warpage remains constant, but the rate of reduction of in-plane extension-compression energy with possible warpage increases with "temperature load." Thus a critical temperature is reached where if the disk warps, lower strain-energy results.
This will be demonstrated analytically for the disk wheel model where less tangential restraint on polymer strip movement is shown to substantially increase saddle warpage growth.
provides an expression of the rate of increase of warpage with temperature for the more rigid disk wheel model.
Then, although the minimum temperature to initiate warpage is the same as the unmodified disk wheel model, subsequent warpage growth is greater for the more relaxed wheel with
To summarize, the saddle warpage equation for the disk wheel relates the square of the out-of-plane disk movement, [W.sup.2], with disk temperature change, [Delta]T, measured from the thermally unstressed disk temperature, as follows:
The disk constant, C, governing the growth of warpage with temperature, is equal to 3/4 for the disk wheel with the concentric annular polymer strips rigidly bonded to the radial metal spokes; and equal to 9/4 for the more relaxed disk wheel model.
Note, to simplify the derivation of warpage formulae while focusing on its key features, the disk's polymer stiffness (Young's modulus), is assumed independent of temperature, However, a similar argument leads to the same (saddle and cup) warpage formulae without this stiffness assumption; and, similar warpage formulae also apply if a layered structure, symmetric about a central plane, with each layer having the same general form as the disk wheel model, is considered.
If cooled, the disk wheel model will warp into a cup shape [ILLUSTRATION FOR FIGURE 1 OMITTED], due to positive hoop stress development in the polymer strips.
where [Delta]T ([less than] 0) is the temperature change measured from the thermally unstressed disk wheel temperature.
The mechanism causing warpage in the continuous disk model is very similar to that discussed previously for the disk wheel model.
The result of a series of numerical calculations at different disk temperature, property and geometry values can be summarized in a similar form to the disk wheel warpage result as follows.