This PhD internship proposal aims to contribute with the proper development of the ongoing research project under the number 2016/23649-0 granted by the São Paulo State Foundation for Research (FAPESP). The main objective of this project is the proposition of numerical models based on the elastostatic formulation of the Boundary Element Method (BEM) to simulate the fracture and fatigue phenomena in multi-cracked bodies. The BEM, especially its dual formulation, is an efficient numerical technique for handling crack growth problems due to the non-requirement of a domain discretization. This aspect enables the accurate description of the elastic fields surrounding the crack tip and simplifies the remeshing procedures during crack propagation. In the conventional isoparametric BEM approach, both the geometry and the mechanical fields along the boundary elements are approximated by polynomial shape functions. For complex geometries, these functions may not be able to efficiently describe the structural boundary and, therefore, a very dense mesh is required to obtain satisfactory results. As an alternative, the non-uniform rational B-splines (NURBS) used by computer-aided design (CAD) software may be applied for an accurate boundary geometry description. The unknown fields may also be approximated by the NURBS, giving rise to the isogeometric BEM (IGABEM). However, both isoparametric and isogeometric BEM approaches suffer from the drawback that the mechanical fields at the boundary element may not follow the behaviour of the shape functions. To overcome this deficiency, the enrichment strategy may be used by augmenting the numerical approximation with special functions that are able to improve the mechanical response. These functions are determined from previous knowledge of the solution for the investigated problem. In this context, this internship project intends to develop an enriched formulation for the IGABEM to obtain a numerical model able to accurately describe the geometry and the mechanical fields in fracture problems. To capture the classical square root behaviour near the crack tips predicted by the linear elastic fracture mechanics, the displacement approximation of the elements near crack tip will be enriched with functions associated with Williams's asymptotic expansions. This strategy also allows the direct evaluation of the stress intensity factors (SIFs). Additionally, the discontinuous Heaviside sign step function will be used for enrichment of elements intercepted by cracks to avoid remeshing. The extended approach will also be investigated to the imposition of the Dirichlet and Newmann boundary conditions. The collaboration with Trevelyan's research group at Durham University is of great value for this study since they are at the forefront of the use of both enriched and isogeometric formulations within the BEM framework.
News published in Agência FAPESP Newsletter about the scholarship: