Effects of proximity on mis-matched composites

Composite materials typically have some degree of disorder in the distribution of particles of the second phase, leading to instances of close proximity of such particles. It is well known that under stress the two phases deform incompatibly giving rise to induced internal stresses. We show that proximity can lead to additional stresses in two different ways. First, and lesser, the incompatibility at the interface of a particle with the matrix is affected by the presence of a neighboring particle. Second, and much greater, the finite nature of the displacements associated with the incompatibility means that at close approach these displacements become significant with respect to the interparticle spacing with correspondingly large strains and stresses.