Macroscopic elastic moduli of spherically-symmetric-inclusion composites and the microscopic stress-strain fields

Explicit algebraic expressions of the estimates for the microscopic elastic stress-strain fields and the macroscopic elastic moduli of the (n-component) multi-coated sphere assemblage (called n-spheres) under the imposed macroscopic strains or stresses are obtained from the minimum energy principles, and the near-interaction approximations (NIAs) for certain geometric correlation parameters (in the case of macroscopically-deviatoric stress-strain states). Limiting procedures are developed to derive the respective results for the inclusion composites with surface-stress (stiff) or spring-layer (compliant) imperfect interfaces, and those with the spherically-symmetric anisotropic coating positioned between the isotropic inner inclusion and the outer matrix phases. When the volume proportion of the outermost spherical shell (the matrix) increases to be the predominant one, one obtains the respective results for the most important specific case: the dilute solutions for the complex inhomogeneities suspended in the major matrix phase, which are needed also for other effective medium approximations and microscopic fields’ analyses. The differential substitution scheme is developed to construct the initial-valued ordinary differential equations estimating the effective moduli of the inclusion composites with radially-variable-moduli coating, and those with radially-variable anisotropic-moduli coating. The equations are integrated to give explicit algebraic estimates of the moduli in a number of cases, including that with constant-anisotropic-moduli-coating.