Theory of Filler Reinforcement in Natural and Synthetic Rubber. The Stresses in and about the Particles

An attempt is made to develop a general theory of filler reinforcement by determining the stresses occurring in and about a spherical particle imbedded in a rubberlike medium subjected to an applied tension. For a system containing a single particle rigidly attached to the adjacent medium, an application of the theory of elasticity shows that, for infinitesimal deformations, the stress components within the particle are independent of the radial distance from the origin, taken at the center of the particle. The stress components at a given point in the surrounding medium depend on the elastic constants both of the particle and of the medium, on the radius of the sphere, on the distance from the origin, and on the angle between the direction vector and the applied tension. Expressions are given for the average stresses in media containing many (independent) particles. Theoretical values of the bulk moduli of the synthetic rubbers considered in the treatment are derived from sound velocity data. Curves showing the spatial distribution of radial and shear stresses are presented for a range of values of elastic constants to be expected for different kinds of filler particles and rubberlike materials.