  Necking instabilities, in which tensile (extensional) deformation localizes
into a small spatial region, are generic failure modes in elasto-viscoplastic
materials. Materials in this very broad class --- including amorphous,
crystalline, polycrystalline and other materials --- feature a predominantly
elastic response at small stresses, plasticity onset at a rather well-defined
yield stress and rate-dependence. Necking instabilities involve a unique
coupling between the system's geometry and its constitutive behavior. We
consider generic elasto-viscoplastic constitutive relations involving an
internal-state field, which represents the structural evolution of the material
during plastic deformation, and study necking in the long-wavelength
approximation (sometimes termed the lubrication or slender-bar approximation).
We derive a general expression for the largest time-dependent eigenvalue in an
approximate WKB-like linear stability analysis, highlighting various basic
physical effects involved in necking. This expression is then used to propose
criteria for the onset of necking and more importantly for the emergence of
strong localization. Applications to strain-softening amorphous plasticity, in
the framework of the Shear-Transformation-Zone model, and to strain-hardening
crystalline/polycrystalline plasticity, in the framework of the Kocks-Mecking
model, are presented. These quantitative analyses of widely different material
models support the theoretical predictions, most notably for the strong
localization during necking.
