The Splitting Method and the GFEMin the Two-Dimensional Analysis of Linear Elastic Domains with Multiple Cracks

Abstract

an essential role in terms of modeling this kind of phenomenon. In fact, it can be said nowadays that failure analysis governed by unstable propagation from a single crack is common practice.However, failure analysis may become a more serious issue when many cracks which interact with each other must be considered. Furthermore, as well having many cracks, it is often necessary to predict and compute a large number of possible scenarios of multiple cracks. These requirements can make the analysis very complex or almost impracticable. Choosing an efficient numerical approach plays a crucial role in enabling computational analysis, and the Finite Element Method is the most commonly adopted option. Moreover, there is still a lack of studies with regards to establishing a computational framework that combines a representative mechanical model with an efficient numerical approach to solve such problems. This issue is addressed in this work.Domain decomposition methods such as the overlapping and non-overlapping Schwarz alternating methods, Le Tallec (1994), Wang and Atluri (1996) were most successful in overcoming the above mentioned mechanical modeling difficulties. This class of methods has proven to be very efficient and accurate in terms of analyzing multiple crack problems, despite requiring an iterative procedure to impose the restriction of stress vector nullity at the crack faces.The Splitting Method, proposed by Andersson and Babuska (1996), (2005), also belongs to the class of decomposition methods, and is hereby adopted for several reasons. Firstly, a very important characteristic of this method is that it was designed to efficiently search and identify the worst scenario among a set of possible multisite crack scenarios. This means that each pattern of cracked domains can be analyzed without substantially increasing the computational effort. For each pattern, the results can be obtained from the resolution of a single linear system. Moreover, it is important to stress that this method does not require an iterative procedure to impose the nullity restriction of stress vectors at the crack faces.On the other hand, regarding numerical approaches, the methods derived from the Partition of Unity Finite Element Method, introduced by Babuska and Melenk (1995), (1997), such as the Generalized Finite Element Method or the Extended Finite Element Method (GFEM/ XFEM) have been successfully used in linear fracture mechanics problems, Duarte et al.(2000), Sukumar et al. (2000), Belytschko et al. (2009), Kim et al.(2012 and Barros et al.(2004). Using the GFEM, for example, the approximation to the solution can be improved locally by exploring a priori known accurate solutions, and therefore avoiding costly mesh refinements. For instance, concerning crack problems, the Westergaard Complex Stress Function scan be explored in order to reproduce stress fields more accurately in the proximity of the crack tip. In fact, in Alves (2010), the referred enrichment strategies were primarily used to improve accuracy to analyze frac...