An Energy Criterion for the Onset of Shear Localization in Thermal Viscoplastic Materials, Part I: Necessary and Sufficient Initiation Conditions

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Abstract
Necessary and/or sufficient conditions for the onset of shear flow localization in thermal viscoplastic materials are derived. Initiation of shear band formation is examined through the analysis of a one-dimensional model of dynamic simple shear with velocity-controlled boundaries. An exact energy estimate is derived for the fully nonlinear initial boundary value problem which illustrates the weak effect of elasticity 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 dynamic simple shear problem with velocity controlled and thermally insulated boundaries. The former solution is also shown to be the unique static solution of the simple shear problem, with no elastic or heat conduction 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 discussion regarding the notions of instability and localization motivates the introduction of a novel localization criterion, based on an appropriately selected energy norm, that takes full account of the time variation of the homogeneous solution. Taking advantage of the admissible spatial form of the perturbations together with the foregoing criterion allows for the association of the positive rate of change of the total kinetic energy of the absolute perturbations with the onset of shear strain localization.

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