Triply periodical particulate matrix compositesinvarying external stress fields

We consider a linear elastic composite medium\ which consists of a homogeneous matrix containing aligned ellipsoidal uncoated or coated inclusions arranged in a periodic array and subjected to inhomo! geneous boundary conditions[ The hypothesis of e}ective _eld homogeneity near the inclusions is used[ The general integral equation obtained reduces the analysis of in_nite number of inclusion problems to the analysis of a _nite number of inclusions in some representative volume element "RVE#[ The integral equation is solved by the Fourier transform method as well as by the iteration method of the Neumann series " _rst! order approximation#[ The nonlocal macroscopic constitutive equation relating the unit cell averages of stress and strain is derived in explicit closed forms either of a di}erential equation of a second!order or of an integral equation[ The employed of explicit relations for numerical estimations of tensors describing the local and nonlocal e}ective elastic properties as well as average stresses in the composites containing simple cubic lattices of rigid inclusions and voids are considered