Polymers toughened with rubber microspheres: an analytical solution for stresses and strains in the rubber particles at equilibrium and rupture

Large strains in rubber toughened polymers cause void formation and growth in the rubber particles and yielding in the matrix. Void formation usually precedes plasticity in the matrix around the particle and previous papers have proposed models for the relationship between rubber surface energy, volume strain energy and void growth. In this paper, it is shown that another volume criterion must also be satisfied arising from the fact that in all these models, no decohesion is allowed at the particle-matrix interface. A fracture mechanics approach, where linear and nonlinear elasticity are assumed for the matrix and the rubber particle, respectively, is used to define a void formation criterion depending on the rubber fracture surface energy. After formation, the stability of the void is examined, taking into account the volume conservation between matrix and particle and the stress due to surface tension when the void size is very small. A size effect is observed, indicating that voids cannot grow in small particles. The required value of fracture energy in a particle on a microscopic scale is discussed.