A MATHEMATICAL TREATMENT OF A THEORY OF RUBBER STRUCTURE

Abstract

The theory that the elasticity of rubber is due to the heat vibrations of very long chain molecules, bound to one another at occasional points along their length, but able to move freely relatively to one another at all other points, is susceptible of mathematical treatment. In the present treatment it is assumed that the rubber molecule has a restricted rotation about the axis formed by joining two adjacent junction points.A stress-strain curve has been developed mathematically on this assumption, and this curve, which is a reasonably close approximation to the curve obtained experimentally, serves as a standard with which to compare the rubber stress-strain curve and as a starting point for further mathematical work on the structure of rubber. The discrepancy between the mathematical and the experimental curve is explained on the very probable assumption that there is a wave motion or other vibration along the length of the rotating chain, as well as a rotation of the chain as a whole.An explanation of the peculiar S-shape of the beginning of the experimental stress-strain curve develops automatically from the mathematically deduced relation between stress and strain, and it is also shown why the S-shape appears to vanish when the calculation of the stress is based on the actual cross section of the stretched rubber.In addition, the following values, calculated from the above assumption and X-ray data, of certain constants, were obtained:(i) The average distance between junction points.(ii) The average length of molecular chain between junction points and, incidentally, the ratio between the number of freely swinging carbon atoms and those bound at junction points. This gives the number of freely swinging carbon atoms on the molecular chain between junction points and an idea of the length of the rubber molecule.(iii) The quantity of kinetic energy per cubic centimetre causing the elastic effect in rubber.(iv) The percentage of sulphur necessary to form the junction points in vulcanized rubber and, consequently, the minimum quantity of sulphur needed for vulcanization, both for hard and soft rubber. This minimum agrees closely with practical experiment.