Prediction of the Stress Field and Effective Shear Modulus of Composites Containing Periodic Inclusions Incorporating Interface Effects in Anti-plane Shear

We consider the anti-plane shear of a composite containing a periodic array of circular inclusions which incorporate separate interface effects in the presence of uniform remote loading. Using complex variable methods, the corresponding stress distributions and effective shear modulus of the composite are obtained by analyzing a representative unit cell subjected to periodic boundary conditions imposed on its edge. We present several examples to illustrate the interfacial stress field and effective shear modulus relative to the interface parameter and volume fraction of the inclusions. We show that when the volume fraction of the inclusions falls below approximately 9 %, the interfacial stress recovers effectively to that of a single inclusion with the same interface parameter in an infinite plane. We find also that when the shear modulus of the inclusions exceeds twice the shear modulus of the matrix, we can essentially treat each inclusion-matrix interface as being perfectly bonded without inducing significant errors in the effective shear modulus of the composite. Finally, we show that the use of effective medium theories may induce significant errors in the determination of the effective shear modulus of the composite when the inclusions are much softer than the matrix and simultaneously the volume fraction of the inclusions exceeds 50 %.