Stiffness Reduction Analysis of Composite due to Reinforcement Break in Reinforcement Clusters

2nd Report, Stress-Strain Relation and Equivalent Expression for a Multi-phase Triple Inhomogeniety

Abstract

Generally, reinforcement clusters are formed in a composite. For the case that the rapture strain of the reinforcement in such a composite is smaller than that of the matrix, some of the reinforcements may break in the cluster when the composite is subjected to the external force. Thus the broken reinforcements and the unbroken reinforcements coexist within the cluster. Such a cluster can be regarded as a multi-phase inhomogeneity because the cluster contains different inhomogeneities correspond to the broken and the unbroken reinforcements. From another viewpoint, the cluster can be regarded as a triple inhomogeneity because a crack exists in the reinforcement and the reinforcements are gathered to the cluster. Therefore, the reinforcement cluster in the composite can be modeled as a multi-phase triple inhomogeneity. In the present study, the average interaction stress is derived for the material containing many multi-phase triple inclusions by adopting the Mori-Tanaka theorem. The stress-strain relations to every regions in the multi-phase triple inclusion are formulated by using the resultant average interaction stress. Moreover, the stress-strain relations to the multi-phase triple inhomogeneity are derived and they are compared with those to the multi-phase triple inclusion, and the equivalent expressions for the multi-phase triple inhomogeneity are derived to the every regions in it.