AN ENERGY CRITERION FOR THE ONSET OF SHEAR LOCALIZATION IN THERMAL VISCOPLASTIC MATERIALS .1. NECESSARY AND SUFFICIENT INITIATION CONDITIONS

Necessary and/or sufficient conditions for the onset of shear flow loc alization in thermal viscoplastic materials are derived. Initiation of shear band formation is examined through the analysis of a one-dimens ional model of dynamic simple shear with velocity-controlled boundarie s. An exact energy estimate is derived for the fully nonlinear initial boundary value problem which illustrates the weak effect of elasticit y as regards the onset of localization. The existence, uniqueness and well-posedness properties of a spatially homogeneous solution are also examined. A unique homogeneous solution is shown to exist for the dyn amic simple shear problem with velocity controlled and thermally insul ated boundaries. The former solution is also shown to be the unique st atic solution of the simple shear problem, with no elastic or hear con duction effects, consistent with the foregoing boundary conditions. A linear system of equations is derived for the perturbations superposed on the reference, time-dependent homogeneous (static) solution. A dis cussion regarding the notions of instability and localization motivate s the introduction of a novel localization criterion, based on an appr opriately selected energy norm, that takes full account of the time va riation of the homogeneous solution. Taking advantage of the admissibl e spatial form of the perturbations together with the foregoing criter ion allows for the association of the positive rate of change of the t otal kinetic energy of the absolute perturbations with the onset of sh ear strain localization.