Keywords: extended isogeometric analysis, curved crack, enrichment functions, linear elastic fracture mechanics, mapping, stress intensity factor.

The fracture analysis of an isotropic two-dimensional body with a curved crack using the extended isogeometric analysis (XIGA) is discussed in this paper. The extended finite element method (XFEM) belongs to the class of partition of unity methods (PUM). The principal feature of this method is the ability to add, locally, a priori knowledge of the solution to the approximation space. This enrichment allows for capturing particular features such as discontinuities and singularities which are present in the solution exactly. The XFEM, in particular, has been used successfully to solve crack initiation and propagation problems with no or minimal remeshing of the moving boundaries. On the other hand, isogeometric analysis (IGA) is another new robust computational approach, which offers the possibility of integrating the finite element analysis (FEA) into conventional NURBS-based CAD design tools. This allows models to be designed, tested and adjusted in one go, using a common data set.

The isogeometric formulation using NURBS basis functions has been recently enriched by the general XFEM methodology to solve linear fracture mechanics problems with incompatible meshes while obtaining solutions with higher order convergence rates and high levels of accuracy. This approach, entitled the extended isogeometric analysis (XIGA), benefits from the advantages of its origins: XFEM and IGA; while it is capable of analysing crack growth problems without any remeshing requirement, complex geometries can be modelled with few elements and higher order inter-element continuities are satisfied.

XIGA has been successfully employed for analysis of bodies with straight cracks in the parametric space and parent element. For improving the accuracy of integration using the Gauss quadrature rule, the "sub-triangles approach" and the "almost polar technique" have been utilized for split and crack tip elements, respectively. The principal difficulty with curved cracks is the special treatment required for the integration of sub-triangles which is that one of its sides is not straight in the parametric space.

In this paper, specific mappings based on the blending function method are used for integration over the elements cut by a non-straight part of the crack, and is extended to the analysis of structures having curved cracks in the parametric space. Consequently, no limitation remains for considering arbitrary non-smooth and smooth strong discontinuities in the XIGA.

In order to impose Dirichlet boundary conditions, the Lagrange multiplier method is used. Mixed-mode stress intensity factors (SIFs) are evaluated by means of the interaction integral to determine the fracture properties of the domain.

Finally, a plate with an arc-shaped central crack with different arc angles is analysed using the proposed method. Comparison of the numerical results with analytical solutions available in the literature shows the efficiency and validity of the approach.

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