Stress softening in rubber vulcanizates. Part I. Use of a strain amplification factor to describe the elastic behavior of filler‐reinforced vulcanized rubber

Measurements of Young's modulus of vulcanized rubbers containing thermal carbon black show the predicted dependence on the volume concentration given by relationships derived for a suspension of spherical particles, for example, that due to Guth, Simha, and Gold which gives E = E0 (1 + 2.5c + 14.1c2). A simple interpretation of the results is that the strain in the rubber is increased by the presence of filler so that the ratio of the average strain to the measured overall strain is given by the factor X = 1 + 2.5c + 14.1c2. This factor was used to analyze simple extension stress–strain data obtained at larger extensions. For this purpose the Mooney-Rivlin relation was used to describe the behavior of the rubber phase. Values of C1 independent of the volume concentration and in close accord with measurements of the equilibrium volume swelling of the rubbers were obtained. Values of λ* were also consistent with those of C1. Analysis of stress–strain data obtained on rubbers containing smaller particle-sized carbon blacks is more complex. For these materials the relation due to Guth, viz., E = E0 (1 + 0.67fc + 1.62f2c2), was chosen. By the choice of suitable values of f, good agreement with the Mooney-Rivlin stress–strain relation was achieved at volume concentrations less than about 0.15.