Crack opening conditions at 'corner nodes' in FE analysis with cracking along mesh lines

On the basis of a recent work proposed by the authors on a double-minimization method for evaluating inter-element forces and stresses transmitted across mesh lines, the crack opening conditions at a corner node of the FE mesh, from where several lines (potential cracks) emanate, is examined in this paper. The study is developed locally as a post-processing step of a standard displacement-based FE calculation, in terms of an always-increasing external (macroscopic) load factor l. The cracking laws for each potential crack line are assumed rigid-plastic with hyperbolic failure criterion in terms of normal and shear components of the stress traction at that point. It is observed that, as l increases, in general such point may undergo up to four phases of evolution until a crack can effectively open through it. First, while stress tractions across mesh lines at the point are all below cracking criterion, forces may be evaluated with the double minimization method recently proposed. Second, cracking criterion is reached for one of the lines only. Stress evaluation requires a modified minimization method with one (hyperbolic) constraint; however, crack still does not open at the node because of the lack of kinematic continuity. Third, cracking criterion is satisfied for a second of the lines converging at the nodal point. Stress tractions may then be calculated with a system of equations involving the two hyperbolic constraints alone and no minimization is needed. But in general the through crack cannot open yet at this stage because of non-coincident flow rules, until either (i) a third line reaches the cracking criterion, or (ii) these get reoriented to exhibit parallel directions in the global reference system. Two simple examples of application are provided which illustrates the development of the various cracking stages and shows different situations that may take place.