Abstract
The idea based on the transition between governing differential equations is used in this paper to study localization problems. First, the implication of moving jump forms of conservation laws is explored on the initiation and evolution of localization. It is shown that the jump conditions would result in the necessary condition for the loss of ellipticity if there is a jump in mass density. As a result, the initiation of localization might be identified via monitoring the transition between different types of governing differential equations, and the evolution of localization might be traced by using a moving material boundary without invoking higher order terms in space and/or time. One- and two-dimensional sample problems are then considered to demonstrate the proposed numerical procedure, with the use of local models, to simulate the evolution of localization under quasi-static and dynamic loading conditions. Finally, conclusions and future work are given based on current research results. Key words: softening, localization, transition, discontinuity, ellipticity, hyperbolicity.
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