John Douglas Eshelby's paper "The determination of the elastic field of an ellipsoidal inclusion, and related problems" presents a method for determining the elastic field inside an ellipsoidal inclusion embedded in an isotropic elastic medium. The paper introduces a technique involving a sequence of imaginary operations—cutting, straining, and welding—to find the elastic field. It shows that if the inclusion is an ellipsoid, the strain inside is uniform and can be expressed using elliptic integrals. The paper also addresses the problem of how an ellipsoidal inhomogeneity affects an applied stress field. It demonstrates that the elastic field inside the ellipsoid can be used to solve various problems related to stress and strain, including the determination of the elastic field far from the inclusion, the stress and strain components at a point outside the inclusion, the total strain energy, and the interaction energy with another elastic field. The paper also discusses the effective elastic constants of a material containing a dilute dispersion of ellipsoidal inhomogeneities. The results are derived using a combination of analytical methods and the properties of elliptic integrals. The paper concludes with a discussion of the implications of these results for the understanding of elastic fields in materials with inhomogeneities.John Douglas Eshelby's paper "The determination of the elastic field of an ellipsoidal inclusion, and related problems" presents a method for determining the elastic field inside an ellipsoidal inclusion embedded in an isotropic elastic medium. The paper introduces a technique involving a sequence of imaginary operations—cutting, straining, and welding—to find the elastic field. It shows that if the inclusion is an ellipsoid, the strain inside is uniform and can be expressed using elliptic integrals. The paper also addresses the problem of how an ellipsoidal inhomogeneity affects an applied stress field. It demonstrates that the elastic field inside the ellipsoid can be used to solve various problems related to stress and strain, including the determination of the elastic field far from the inclusion, the stress and strain components at a point outside the inclusion, the total strain energy, and the interaction energy with another elastic field. The paper also discusses the effective elastic constants of a material containing a dilute dispersion of ellipsoidal inhomogeneities. The results are derived using a combination of analytical methods and the properties of elliptic integrals. The paper concludes with a discussion of the implications of these results for the understanding of elastic fields in materials with inhomogeneities.