This paper by R. J. Asaro and J. R. Rice, published in 1977, explores the conditions under which strain localization occurs in ductile single crystals undergoing single slip. The authors model plastic flow as rate-insensitive and identify localization as a bifurcation from homogeneous deformation to a concentrated "shear band." They find that localization is possible only when the plastic hardening modulus for the slip system reaches a critical value, \( h_{cr} \), which is sensitive to the constitutive law governing incremental shear. The paper develops a general form of the constitutive law, incorporating deviations from the Schmid rule, and shows that \( h_{cr} \) can be positive when there are such deviations. The authors suggest that micromechanical processes like "cross-slip" in crystals provide specific cases where stresses other than the Schmid stress influence plastic response, and there is an experimental association between localization and large amounts of cross-slip. They derive the specific form of \( h_{cr} \) for a constitutive model that accounts for non-Schmid effects in cross-slip and develop a dislocation model to estimate the parameters involved. The work supports the idea that localization can occur with positive strain hardening, \( h_{cr} > 0 \), and that the attainment of an ideally plastic or strain-softening state may not be necessary for localization.This paper by R. J. Asaro and J. R. Rice, published in 1977, explores the conditions under which strain localization occurs in ductile single crystals undergoing single slip. The authors model plastic flow as rate-insensitive and identify localization as a bifurcation from homogeneous deformation to a concentrated "shear band." They find that localization is possible only when the plastic hardening modulus for the slip system reaches a critical value, \( h_{cr} \), which is sensitive to the constitutive law governing incremental shear. The paper develops a general form of the constitutive law, incorporating deviations from the Schmid rule, and shows that \( h_{cr} \) can be positive when there are such deviations. The authors suggest that micromechanical processes like "cross-slip" in crystals provide specific cases where stresses other than the Schmid stress influence plastic response, and there is an experimental association between localization and large amounts of cross-slip. They derive the specific form of \( h_{cr} \) for a constitutive model that accounts for non-Schmid effects in cross-slip and develop a dislocation model to estimate the parameters involved. The work supports the idea that localization can occur with positive strain hardening, \( h_{cr} > 0 \), and that the attainment of an ideally plastic or strain-softening state may not be necessary for localization.