The paper by Rodney J. Hill presents a self-consistent mechanics for composite materials, focusing on the estimation of macroscopic elastic moduli of two-phase composites. The method, similar to the Hershey-Kröner theory for crystalline aggregates, accounts for the inhomogeneity of stress and strain. The phases can be arbitrarily acelotropic and in any concentrations, but are required to have the character of a matrix and effectively ellipsoidal inclusions. Detailed results are provided for an isotropic dispersion of spheres. Hill introduces the concept of an 'overall constraint' tensor for each phase, which is derived from the solution of an auxiliary elastic problem involving a uniformly loaded infinite mass with an ellipsoidal inhomogeneity. The method is applied to statistically homogeneous dispersions, where the inclusions can be treated as variously-sized spheres or similar ellipsoids. The equations for the overall stiffness and compliance tensors are derived, and the phase concentration factors are defined. For an isotropic dispersion of spheres, Hill provides explicit equations for the bulk and shear moduli, showing that the theoretical rigidity lies between the Hashin-Shtrikman bounds at any concentration. The theory is found to be reliable under moderate conditions, especially when the dispersed phase is sufficiently dilute. The paper concludes with a discussion on the mechanical significance of the overall constraint tensor and its implications for composite materials.The paper by Rodney J. Hill presents a self-consistent mechanics for composite materials, focusing on the estimation of macroscopic elastic moduli of two-phase composites. The method, similar to the Hershey-Kröner theory for crystalline aggregates, accounts for the inhomogeneity of stress and strain. The phases can be arbitrarily acelotropic and in any concentrations, but are required to have the character of a matrix and effectively ellipsoidal inclusions. Detailed results are provided for an isotropic dispersion of spheres. Hill introduces the concept of an 'overall constraint' tensor for each phase, which is derived from the solution of an auxiliary elastic problem involving a uniformly loaded infinite mass with an ellipsoidal inhomogeneity. The method is applied to statistically homogeneous dispersions, where the inclusions can be treated as variously-sized spheres or similar ellipsoids. The equations for the overall stiffness and compliance tensors are derived, and the phase concentration factors are defined. For an isotropic dispersion of spheres, Hill provides explicit equations for the bulk and shear moduli, showing that the theoretical rigidity lies between the Hashin-Shtrikman bounds at any concentration. The theory is found to be reliable under moderate conditions, especially when the dispersed phase is sufficiently dilute. The paper concludes with a discussion on the mechanical significance of the overall constraint tensor and its implications for composite materials.