Keywords: cohesive crack, crack, enrichment scheme, extended-generalised finite element method (XFEM), stress recovery.

In the modelling of crack problems using the extended-generalised finite element method (XFEM) [1,2,3,4], the crack faces behind the crack tip are modelled by enrichment of discontinuous Heaviside functions and the crack tip region can be modelled by enrichment of proper branch functions such as the true crack tip asymptotic displacement fields [1,2,5,6], or discontinuous Heaviside functions with proper partition of the element including the crack tip [7]. This partition divides the partially cracked element into fully cracked and uncracked parts, and makes the enriched discontinuity conform to the crack.

This study investigates the advantages and disadvantages of various crack tip enrichment schemes, and the accuracy of the resulting crack tip displacement and stress fields. Three methods for evaluating the stresses are compared: direct differentiation of the displacements, simple interpolation of the averaged nodal stress values evaluated from adjacent elements by bi-linear extrapolation from the Gauss points using shape functions (AVG), and the statically admissible stress recovery (SAR) [8]. Both cracks with traction-free faces and cohesive cracks are considered. For cohesive cracks, the accuracy of load-deformation curves, evolution of the fracture process zone (FPZ), crack opening profile, and distribution of the traction in the FPZ are also studied.

If the crack tip region is enriched with branch functions such as the true crack tip asymptotic displacement field, partially cracked elements can be handled directly without further partition. However, sophisticated quadrature rules are required to handle the possible singularity at the tip and/or the nature of the angular oscillations of the enrichment functions [8]. On the other hand, if only the jump function is used in the crack tip region, the partially cracked elements need to be partitioned into a fully cracked part and an uncracked part. However, this enrichment requires only the standard Gauss-Legendre quadrature.

If the true crack tip asymptotic displacement field is used as the enrichment function at the crack tip, but the coefficients appearing in it are assumed to be independent at each enriched node, the accuracy of the method is no different from that obtained by the use of the jump function only for enrichment. However, when the enriched fields at different nodes are enforced to be the same, the enrichment approximation reduces to the real crack tip asymptotic field. The accuracy is then improved and the SIFs can be obtained directly for linear problems [5,6].

Although the above discussion is mainly based on homogeneous crack problems, it is obviously not limited to these problems.

1

T. Belytschko, T. Black, "Elastic crack growth in finite elements with minimal remeshing", Int J Numer Meth Engng, 45, 601-620, 1999. doi:10.1002/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S

2

N. Moës, J. Dolbow, T. Belytschko, "A finite element method for crack growth without remeshing", Int J Numer Meth Engng, 46, 131-150, 1999. doi:10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J

3

T. Strouboulis, K. Copps, I. Babuska, "The generalized finite element method", Comput Meth Appl Mech Eng, 190, 4081-4193, 2001. doi:10.1016/S0045-7825(01)00188-8

4

B.L. Karihaloo, Q.Z. Xiao, "Modelling of stationary and growing cracks in FE framework without remeshing: a state-of-the-art review", Comput Struct, 81, 119-129, 2003. doi:10.1016/S0045-7949(02)00431-5

5

Q.Z. Xiao, B.L. Karihaloo, "Direct evaluation of accurate coefficients of the linear elastic crack tip asymptotic field", Fatig Fract Engng Mater Struct, 26, 719-730, 2003. doi:10.1046/j.1460-2695.2003.00648.x

6

X.Y. Liu, Q.Z. Xiao, B.L. Karihaloo, "XFEM for direct evaluation of mixed mode SIFs in homogeneous and bi-materials", Int J Numer Meth Engng, 59, 1103-1118, 2004. doi:10.1002/nme.906

7

G. Zi, T. Belytschko, "New crack-tip elements for XFEM and applications to cohesive cracks", Int J Numer Meth Engng, 57, 2221-2240, 2003. doi:10.1002/nme.849

8

Q.Z. Xiao, B.L. Karihaloo, "Improving the accuracy of XFEM crack tip fields using higher order quadrature and statically admissible stress recovery", Int J Numer Meth Engng, 66, 1378-1410, 2006. doi:10.1002/nme.1601

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