Abstract
The physical phenomena now widely known as the “Mullins effect” was apparently first studied in detail by Holt in about 1930. He showed that if a vulcanized rubber which contains carbon black is stretched to a relative elongation αm=L/L0 and released, it will not follow the same stress-strain curve when it is stretched once again to this same elongation. Instead, the rubber appears much softer on the second stretch for elongations below αm. Holt examined this behavior in some detail and showed that additional pre-stretches to αm softened the rubber further, but to a lesser degree than was observed on the first prestretch. In addition, Holt showed that the rubber regains a portion of its stiffness if allowed to rest in the relaxed state. Although this recovery was very slow at room temperature, up to about 50% recovery was noted after about an hour at 100° C. Similar softening effects were noted in gum stocks at exceedingly high elongations, but the effects were much less marked than in the filled stocks. This same effect was examined in more detail by Mullins after 1940. His results confirmed and greatly extended the earlier results of Holt. In addition, Mullins speculated about the mechanisms involved but came to no definite conclusion in that regard. Later, however, he and Tobin presented a phenomenological theory for the effect wherein they considered the rubber to be composed of hard and soft regions. They showed that their data could be described by assuming that some fraction of the hard regions became soft after a prestretch. No definite molecular basis for this process was proposed by Mullins and Tobin, although they speculated that either the breaking up of filler particle aggregates or the breaking loose of rubber to filler bonds might be involved. Later work by Blanchard and Parkinson confirmed and extended the data of Holt and Mullins. Further, these authors concluded that the softening was due to the breaking of rubber to filler bonds. They incorporated this idea into a semiempirical theory which agreed with their experiments. In addition, they obtained what they believed to be a distribution for the strength of the rubber-filler bonds. It will be seen in that which is to follow that their distribution probably does not represent what they thought it did, even though their basic ideas were correct.
Optimum Collective Submanifold in Resonant Cases by the Self-Consistent Collective-Coordinate Method for Large-Amplitude Collective Motion
