Rodney J Hill's paper presents a self-consistent method for estimating the macroscopic elastic moduli of two-phase composites. The method accounts for the inhomogeneity of stress and strain in a manner similar to the Hershey–Kröner theory of crystalline aggregates. The phases are assumed to be arbitrarily anisotropic and in any concentrations, but are required to have the character of a matrix and effectively ellipsoidal inclusions. Detailed results are given for an isotropic dispersion of spheres. The method draws on the solution to an auxiliary elastic problem, namely a uniformly loaded infinite mass containing an ellipsoidal inhomogeneity. The properties and orientation of a typical crystal are assigned to the inclusion, and the macroscopic properties of the polycrystal to the matrix. For self-consistency, the orientation average of the inclusion stress or strain is set equal to the overall stress or strain. The result is an implicit tensor formula for the macroscopic moduli. The analysis for the composite proceeds in a similar spirit but differs in an important respect: only the particulate phase can reasonably be treated on this footing. However, as is well known, a knowledge of average stress or strain in this one phase suffices to determine the overall properties when the matrix is homogeneous. The entire analysis remains structurally close to that for a crystal aggregate. The paper introduces symbolic notation for tensors and matrices, and discusses the auxiliary problem of a single ellipsoidal inclusion embedded in a homogeneous mass. The solution has the character of a uniform field locally perturbed in the neighbourhood of the inclusion. The inclusion is strained uniformly, though not necessarily coaxially. The paper then presents the self-consistent theory for statistically homogeneous dispersions, where the inclusions can be treated, on average, either as variously-sized spheres or as similar ellipsoids with corresponding axes aligned. Each phase may be arbitrarily anisotropic but is assumed homogeneous in situ. The paper discusses the isotropic dispersion of spheres and derives equations for the bulk and shear moduli. The results are compared with known exact solutions and the theory is shown to be reliable under moderate conditions. The paper concludes with an acknowledgment of the support received for the research.Rodney J Hill's paper presents a self-consistent method for estimating the macroscopic elastic moduli of two-phase composites. The method accounts for the inhomogeneity of stress and strain in a manner similar to the Hershey–Kröner theory of crystalline aggregates. The phases are assumed to be arbitrarily anisotropic and in any concentrations, but are required to have the character of a matrix and effectively ellipsoidal inclusions. Detailed results are given for an isotropic dispersion of spheres. The method draws on the solution to an auxiliary elastic problem, namely a uniformly loaded infinite mass containing an ellipsoidal inhomogeneity. The properties and orientation of a typical crystal are assigned to the inclusion, and the macroscopic properties of the polycrystal to the matrix. For self-consistency, the orientation average of the inclusion stress or strain is set equal to the overall stress or strain. The result is an implicit tensor formula for the macroscopic moduli. The analysis for the composite proceeds in a similar spirit but differs in an important respect: only the particulate phase can reasonably be treated on this footing. However, as is well known, a knowledge of average stress or strain in this one phase suffices to determine the overall properties when the matrix is homogeneous. The entire analysis remains structurally close to that for a crystal aggregate. The paper introduces symbolic notation for tensors and matrices, and discusses the auxiliary problem of a single ellipsoidal inclusion embedded in a homogeneous mass. The solution has the character of a uniform field locally perturbed in the neighbourhood of the inclusion. The inclusion is strained uniformly, though not necessarily coaxially. The paper then presents the self-consistent theory for statistically homogeneous dispersions, where the inclusions can be treated, on average, either as variously-sized spheres or as similar ellipsoids with corresponding axes aligned. Each phase may be arbitrarily anisotropic but is assumed homogeneous in situ. The paper discusses the isotropic dispersion of spheres and derives equations for the bulk and shear moduli. The results are compared with known exact solutions and the theory is shown to be reliable under moderate conditions. The paper concludes with an acknowledgment of the support received for the research.