The paper by James R. Rice discusses the localization of plastic deformation into a shear band as an instability of plastic flow and a precursor to rupture. It reviews experimental observations, presents a general theoretical framework, and calculates critical conditions for various material models. The interplay between inelastic constitutive descriptions, such as deviations from normality and vertex-like yielding, and the onset of localization is emphasized. The paper begins with an introduction, highlighting that ductile solids often transition from smooth deformation to localized deformation in the form of a shear band. Examples include Lüders bands in metals, ductile fracture in single crystals, and localization in metal polycrystals and structural alloys. The paper also discusses geological materials, such as clays and rocks, where localization is common. The paper then explores mechanisms of localization, considering both material and geometric instabilities. It discusses the role of initial imperfections and the response of materials with different constitutive relations. The paper also addresses the limiting nature of the localization instability, noting that other deformation modes can precede localization, but the latter is often the limiting mode. The paper presents a general theory of localization, considering rate-independent, thermally decoupled constitutive models. It discusses the conditions for bifurcation from homogeneous deformation into a localized shear band. The paper also explores the dynamic growth of disturbances and the role of initial imperfections in localization. The paper then presents results for various material models, including rigid-plastic materials, elastic-plastic materials, and plastically dilatant materials with pressure-sensitive yielding. It discusses the critical conditions for localization in these models and the implications for the behavior of materials under deformation. The paper concludes with the importance of understanding the mechanisms of localization for predicting the behavior of materials under various loading conditions.The paper by James R. Rice discusses the localization of plastic deformation into a shear band as an instability of plastic flow and a precursor to rupture. It reviews experimental observations, presents a general theoretical framework, and calculates critical conditions for various material models. The interplay between inelastic constitutive descriptions, such as deviations from normality and vertex-like yielding, and the onset of localization is emphasized. The paper begins with an introduction, highlighting that ductile solids often transition from smooth deformation to localized deformation in the form of a shear band. Examples include Lüders bands in metals, ductile fracture in single crystals, and localization in metal polycrystals and structural alloys. The paper also discusses geological materials, such as clays and rocks, where localization is common. The paper then explores mechanisms of localization, considering both material and geometric instabilities. It discusses the role of initial imperfections and the response of materials with different constitutive relations. The paper also addresses the limiting nature of the localization instability, noting that other deformation modes can precede localization, but the latter is often the limiting mode. The paper presents a general theory of localization, considering rate-independent, thermally decoupled constitutive models. It discusses the conditions for bifurcation from homogeneous deformation into a localized shear band. The paper also explores the dynamic growth of disturbances and the role of initial imperfections in localization. The paper then presents results for various material models, including rigid-plastic materials, elastic-plastic materials, and plastically dilatant materials with pressure-sensitive yielding. It discusses the critical conditions for localization in these models and the implications for the behavior of materials under deformation. The paper concludes with the importance of understanding the mechanisms of localization for predicting the behavior of materials under various loading conditions.