Successive iteration method applied to composites containing sliding inclusions: effective modulus and anelasticity

The effective shear modulus and Poisson's ratio of a body containing spherical sliding inclusions are calculated. First, the amount of sliding on the surface of a single spherical sliding inclusion is determined so that the tangential traction of the stresses due to the sliding, the Somigliana dislocations, cancels that of the external stresses on the surface of the inclusion. Next, the influence of other inclusions is accounted for by using a successive iteration method based on the average field theory. The successive iteration converges into closed forms, leading to analytical forms of the effective elastic constants. It is shown that the sliding occurs in a first order kinetics, the relaxation time of which is proportional to the radius of the inclusions with a constant depending on the volume fraction of the inclusions. The two-dimensional problem of a body containing aligned cylindrical fibers is also solved.