The paper by Rodney J. Hill, "Continuum Micro-Mechanics of Elastoplastic Polycrystals," published in the Journal of the Mechanics and Physics of Solids in 1965, addresses the theoretical evaluation of stress and strain in an arbitrarily deformed aggregate of elastoplastic crystals. Hill assumes a tensor constitutive law for individual crystals and estimates the mechanical properties of the aggregate as a whole using a self-consistent model similar to those used by Hershey (1954), Kröner (1958, 1961), and Budiansky and Wu (1962). The primary focus is on determining the average strain concentration factor in grains with given lattice orientations, considering the inhomogeneity of distortion within a polycrystal. Hill introduces a symbolic notation for curvilinear tensors and defines concentration-factor tensors \(A_c\) and \(B_c\) to relate the local and overall strain and stress rates. He then discusses the auxiliary problem, where a single crystal is embedded in a finite homogeneous mass, and the stress and strain rates are determined for both phases. This problem is used to derive the macroscopic constitutive relations for the aggregate. The paper also explores the extension of the analysis to elastoplastic behavior, where the crystal is assumed to be elastic/plastic or rigid/plastic. The plastic part of the crystal strain-rate is derived, and the macroscopic constitutive law for an elastoplastic polycrystal is approximated piecewise linearly. Hill compares this approach with Kröner's method, which assigns the outer phase an isotropic constraint that the aggregate would exert if its incremental deformation were purely elastic. Finally, Hill proposes a self-consistent model with plastic flow, combining the hypothesis of the previous section with the self-consistent method. The goal is to determine the macroscopic constitutive law itself, and the analysis is further developed in specific cases, such as when the crystals are elastically isotropic and only one mode is activated. The paper concludes with a discussion of the inhomogeneity of internal fields and the conditions for uniquely determining the stress-rate field.The paper by Rodney J. Hill, "Continuum Micro-Mechanics of Elastoplastic Polycrystals," published in the Journal of the Mechanics and Physics of Solids in 1965, addresses the theoretical evaluation of stress and strain in an arbitrarily deformed aggregate of elastoplastic crystals. Hill assumes a tensor constitutive law for individual crystals and estimates the mechanical properties of the aggregate as a whole using a self-consistent model similar to those used by Hershey (1954), Kröner (1958, 1961), and Budiansky and Wu (1962). The primary focus is on determining the average strain concentration factor in grains with given lattice orientations, considering the inhomogeneity of distortion within a polycrystal. Hill introduces a symbolic notation for curvilinear tensors and defines concentration-factor tensors \(A_c\) and \(B_c\) to relate the local and overall strain and stress rates. He then discusses the auxiliary problem, where a single crystal is embedded in a finite homogeneous mass, and the stress and strain rates are determined for both phases. This problem is used to derive the macroscopic constitutive relations for the aggregate. The paper also explores the extension of the analysis to elastoplastic behavior, where the crystal is assumed to be elastic/plastic or rigid/plastic. The plastic part of the crystal strain-rate is derived, and the macroscopic constitutive law for an elastoplastic polycrystal is approximated piecewise linearly. Hill compares this approach with Kröner's method, which assigns the outer phase an isotropic constraint that the aggregate would exert if its incremental deformation were purely elastic. Finally, Hill proposes a self-consistent model with plastic flow, combining the hypothesis of the previous section with the self-consistent method. The goal is to determine the macroscopic constitutive law itself, and the analysis is further developed in specific cases, such as when the crystals are elastically isotropic and only one mode is activated. The paper concludes with a discussion of the inhomogeneity of internal fields and the conditions for uniquely determining the stress-rate field.