An Adiabatic Shear Band (ASB) is a relatively narrow band presenting large deformation and high temperature, occuring in various ductile materials. It is well established that ASBs can cause mesh dependence in the numerical simulation of this localization phenomenon. In this respect, several discontinuous models have been proposed and widely applied for overcoming this difficulty. Yet some crucial conditions are substantially required to build and improve these models, such as the accurate description of physical profiles, additional constitutive relations in multi-physical approaches and the prevision of bandwidth evolution. Without a mesh to discretize the physical domain, we propose a new energy-based variational model for adiabatic shear banding structure, including elasticity, work hardening, heat conduction and thermal boundary condition. Balance and constitutive equations are transformed into a mathematical optimization problem with respect to a limited set of scalars. Consequently by means of canonical expressions of displacement and temperature profiles, the bandwidth and central temperature can be accurately tracked as internal variables of the total incremental potential in steady and transient state. As an application of our 1D variational modelling for shear localization, we extend it and propose a variational two-scale model resorting to a strain localization element. Compared to existing work, the advantage of our approach is that an inhomogeneous plastic deformation and temperature distribution in the localized region are introduced by canonical analytical expressions. Moreover bandwidth evolution can be accurately calculated by the optimization of an incremental potential. The variational derivation theoretically validates the feasibility of our two-scale modelling. Furthermore finite element implementation is derived and gives a good base for future implementation.
