4. Conclusion

The results obtained with the displacement extrapolation method are generally satisfactory, with less than 5% error of \(G\), especially if the elements of the crack background are of the Barsoum type (case of mesh crack) or if several layers of elements are enriched around the bottom (case of non-meshed crack, method X-FEM). In both cases, it is a question of capturing the asymptotic behavior of the displacement as best as possible.

It should in fact be noted that the asymptotic expression for displacements is only valid for \(r\) tending towards 0. This is why care must be taken not to choose an extrapolation domain that is too large (distance \(\mathit{dmax}\) from the POST_K1_K2_K3 operator of the order of 4 to 5 elements).

On the tests presented for a mesh crack, method 1 gives the most accurate and stable results, whether in 2D or 3D, if there are Barsoum elements. If the mesh does not include Barsoum elements, it is then recommended to use the results of method 3. For a non-meshed crack, method 1 also seems to be the most accurate.

On a study for which no reference solution is known, it is possible to estimate the quality of the calculation a posteriori. In fact, POST_K1_K2_K3 systematically provides for the first two methods the maximum values and the minimum values (over all the calculated points) of the stress intensity factors, as well as the G value recalculated by the Irwin formula. In contrast, method 3 provides only one value for each stress intensity factor. This method is a weighted average of the stress intensity factors extrapolated at each node.

A result can be considered satisfactory if the 5 values thus supplied (min and max of methods 1 and 2, and method 3) are similar. It is also recommended to compare the results obtained with this method with those obtained from the calculation of the energy restoration rate and the singular functions (operator CALC_G).