# Smooth Generalized/eXtended FEM approximations in the computation of configurational forces in linear elastic fracture mechanics

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## Abstract

The computation of crack severity parameters in the linear elastic fracture mechanics (LEFM) modeling is strongly dependent on the local quality of the approximated stress fields right at the crack tip vicinity. This work investigates the behavior of extrinsically enriched smooth mesh-based approximations, obtained via \(C^{k}\)-GFEM framework (Duarte et al. in Comput Methods Appl Mech Eng 196:33–56, 2006), in the computation of \(\mathcal {J}\)-integral in both pure mode I and mixed-mode loadings for two-dimensional problems of the LEFM. The method of configurational forces is used for this purpose as shown in Steinmann et al. (Int J Solids Struct 38:5509–5526, 2001), for instance, by performing some adaptations according to Häusler et al. (Int J Numer Methods Eng 85:1522–1542, 2011). As such method provides vector quantities, it is also possible to compute the angle \(\theta _{{\mathrm{ADV}}}\) of probable crack advance. The \(C^{k}\)-GFEM is quite versatile and shares similar features with the standard FEM regarding the domain partition and numerical integration (Mendonça et al. in Finite Elem Anal Des 47:698–717, 2011). The tests were conducted using three-noded triangular element meshes and numerical integrations were performed using only global coordinates. The evaluations combined different schemes of polynomial and discontinuous/singular (Moës et al. in Int J Numer Methods Eng 46:131–150, 1999) enrichments. The use of a smooth partition of unity (PoU) can influence the accuracy of computed crack severity parameters. The configurational forces computation is favored by the smoothness, reducing the dependence on the way the crack severity parameters are evaluated.

## Keywords

Smooth generalized/extended FEM Crack modeling Configurational forces Eshelbian mechanics J-integral Angle of crack advance Convergence analysis Linear elastic fracture mechanics Mixed mode Crack severity parameters## Notes

### Acknowledgements

Diego A. F. Torres, Clovis S. de Barcellos and Paulo T. R. Mendonça gratefully acknowledge the financial support provided by the Brazilian government agency National Council for Scientific and Technological Development–CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) for this research, under research Grants 163.461/2012-0, 304.698/2013-0 and 304.702/2013-7, respectively. The authors also gratefully acknowledge the Grant provided by DIRPPG/UTFPR–Câmpus Londrina, through the DIRPPG 02/2017 Official Notice, for the final preparation of the manuscript.

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