Finite Element AnalYsis of cracking Localization: The Smeared crack Approach with a Mixed Formulation

In the analysis of cracking localization, consideration of stability and bifurcation of equilibrium states is necessary. Therefore, it is more suitable to have an energy expression written in terms of discrete irreversible variables, which will allow the variations of the energy with respect to the irreversible variables to be considered easily. This implies that, for this kind of analysis ' the discrete crack approach should be appropriate. Nevertheless, the discrete crack approach may not be the best choice fbr problems with many cracks, which are unavoidable for the analysis of the cracking localization. In this aspect, the smeared crack approach may be more appropriate ' To avoid this dilemma, a mixed finitl element formulation based on the smeared crack finite element approach is p.opor"j in this study. In the formulation, both displacement and crack strain fields are discretized. bonsequently, ttre staUitity of crack patterns with respect o the discrete irreversible variables, which are the nodal local crack strain variables, can be easily investigated. This will allow the cracking localization to be discussed even when the smeared crack approach is used in the analysis' 1. Introduction cracking localization either by allowing many The failure of quasi-brittle materials such cracks to open or grow without consideration o