3. Precision of the proposed methods#
The method for extrapolating movement jumps has been validated on tests whose analytical solutions are known. Some results, in 2D and 3D, for a meshed or non-meshed crack are presented below. The results are also compared to the theoretically more accurate method based on the calculation of the energy restoration rate and on singular functions (operator CALC_G: theta method).
3.1. Test SSLP313: 2D C_ PLAN (mesh crack)#
It is an inclined crack in an infinite medium subject to a uniform stress field in one direction (analytical reference solution in plane stresses, exact in an infinite medium). The crack opens in mixed mode (\(\mathit{K1}\) and \(\mathit{K2}\)) [V3.02.313].
For the test, the crack is meshed into a fairly large plate. The quadratic mesh is very fine. The results are as follows:
Reference solution (analytical solution)
\(\mathit{K1}\) |
|
|
3.58E+06 |
2.69E+06 |
1.00E+02 |
Calculation using the theta method (CALC_G)
\(\mathit{K1}\) |
|
|
||
CALC_G without quarter knots |
3.60E+06 |
2.70E+06 |
1.01E+02 |
|
Deviation/ref. |
0.8 % |
0.2 % |
1.1 % |
1.1 % |
CALC_G with quarter knots |
3.60E+06 |
2.70E+06 |
1.01E+02 |
|
Deviation/ref. |
0.8 % |
0.2 % |
1.2 % |
1.2 % |
POST_K1_K2_K3: knotless mesh of quarter edges
method |
\({\mathit{K1}}_{\mathit{max}}\) |
\({\mathit{K1}}_{\mathit{min}}\) |
\({\mathit{K2}}_{\mathit{max}}\) |
\({\mathit{K2}}_{\mathit{min}}\) |
\({G}_{\mathit{max}}\) |
\({G}_{\mathit{min}}\) |
Gap \({G}_{\mathit{max}}\) / ref. |
Gap \({G}_{\mathit{min}}\) / ref. |
1 |
3.54E+06 |
3.19E+06 |
2.63E+06 |
1.92E+06 |
9.73E+01 |
6.94E+01 |
— 3.33% |
— 30.70% |
2 |
3.51E+06 |
3.33E+06 |
2.61E+06 |
2.25E+06 |
9.57E+01 |
8.08E+01 |
— 4.50% |
— 19.32% |
3 |
3.50E+06 |
2.59E+06 |
9.47E+01 |
-5.47% |
||||
POST_K1_K2_K3: mesh with quarter edge knots
method |
\({\mathit{K1}}_{\mathit{max}}\) |
\({\mathit{K1}}_{\mathit{min}}\) |
\({\mathit{K2}}_{\mathit{max}}\) |
\({\mathit{K2}}_{\mathit{min}}\) |
\({G}_{\mathit{max}}\) |
\({G}_{\mathit{min}}\) |
Gap \({G}_{\mathit{max}}\) / ref. |
Gap \({G}_{\mathit{min}}\) / ref. |
1 |
3.61E+06 |
3.60E+06 |
2.70E+06 |
2.69E+06 |
1.01E+02 |
1.01E+02 |
1.29% |
1.07% |
2 |
3.60E+06 |
3.53E+06 |
2.69E+06 |
2.65E+06 |
1.01E+02 |
9.75E+01 |
1.02% |
— 2.67% |
3 |
3.56E+06 |
2.66E+06 |
9.88E+01 |
-1.42% |
||||
On this test, we see that the « Barsoum » type mesh is essential if accurate results are desired. With « Barsoum » method 1 is more stable. It provides values of \(G\) (from \(\mathit{K1}\) and \(\mathit{K2}\)) to approximately 1% of the analytical solution. Methods 2 and 3 lead to errors of 1 to 2.5%. It should be noted that in this case, the method by extrapolation of movements is as accurate as the theta method.
On the other hand, with a normal mesh, the results of the extrapolation method vary a lot (between — 3% and -30% of the solution). It is the same with linear elements. In the case of a mesh without « Barsoum » elements, method 3 is the most accurate.
3.2. Test SSLV134: 3D (mesh crack)#
It is a flat disk-shaped crack in an infinite 3D medium subject to a uniform stress field in one direction (analytical reference solution known as a « penny shape crack »). The crack opens in pure mode 1, and \(\mathit{K1}\) is constant along the crack bottom [V3.04.134].
For this test, the crack is meshed in a parallelepiped block. The mesh is relatively coarse.
Analytical reference solution:
\(\mathit{K1}\) |
|
1.59 106 |
11.59 |
Calculation using the theta method (CALC_G)
\(G\) |
|
CALC_G with quarter knots |
11.75 |
Deviation/ref. |
1.3% |
POST_K1_K2_K3: mesh without quarter edge knots
method |
\({\mathit{K1}}_{\mathit{max}}\) |
\({\mathit{K1}}_{\mathit{min}}\) |
\({G}_{\mathit{max}}\) |
\({G}_{\mathit{min}}\) |
Gap \({G}_{\mathit{max}}\) / ref. |
Gap \({G}_{\mathit{min}}\) / ref. |
1 |
1.56E+06 |
1.45E+06 |
1.11E+01 |
9.63E+00 |
— 4.32% |
— 16.91% |
2 |
1.53E+06 |
1.49E+06 |
1.06E+01 |
1.01E+01 |
— 8.35% |
— 13.08% |
3 |
1.52E+06 |
1.05E+01 |
— 9.51% |
|||
POST_K1_K2_K3: mesh with quarter edge nodes
method |
\({\mathit{K1}}_{\mathit{max}}\) |
\({\mathit{K1}}_{\mathit{min}}\) |
\({G}_{\mathit{max}}\) |
\({G}_{\mathit{min}}\) |
Gap \({G}_{\mathit{max}}\) / ref. |
Gap \({G}_{\mathit{min}}\) / ref. |
1 |
1.61E+06 |
1.59E+06 |
1.18E+01 |
1.16E+01 |
1.32% |
— 0.06% |
2 |
1.59E+06 |
1.53E+06 |
1.15E+01 |
1.07E+01 |
— 0.42% |
— 7.87% |
3 |
1.55E+06 |
1.10E+01 |
— 5.16% |
|||
On this test, we can still see that the « Barsoum » type mesh is essential if accurate results are desired. With « Barsoum » method 1 is the most stable, with a deviation from the reference solution of less than 1.5% for \(G\). The mesh is relatively coarse, which is why the theta method is more accurate.
3.3. Test SSLV134: 3D (unmeshed crack)#
The case under consideration is the same as in the previous paragraph, but this time the crack is not meshed. It is set directly in the command file, using the X- FEM [R7.02.12] method. Since the mesh is not regular with respect to the crack, the values of \(K\) and \(G\) calculated vary along the crack bottom. For the comparison below, the value corresponding to a particular point chosen arbitrarily is retained (middle of the crack bottom shown).
The mesh is linear and relatively coarse. In the X- FEM method, the user can choose the zone over which the elements around the crack bottom are enriched with the asymptotic displacements (key words RAYON_ENRI and NB_COUCHES of DEFI_FISS_XFEM). This enrichment aims to improve the accuracy of the calculation. Here we compare the results obtained with an enrichment limited to only the elements containing the crack bottom and with an enrichment over four layers of elements around the crack bottom.
Calculation with the theta method (CALC_G * — smoothing by default type LEGENDREde degree 5)
\(G\) |
|
CALC_G with enrichment on one layer |
11.42 |
Variance/ref. |
-1.4% |
CALC_G with 4-layer enrichment |
11.61 |
Variance/ref. |
0.2% |
POST_K1_K2_K3: enrichment on a single layer
method |
\({\mathit{K1}}_{\mathit{max}}\) |
\({\mathit{K1}}_{\mathit{min}}\) |
\({G}_{\mathit{max}}\) |
\({G}_{\mathit{min}}\) |
Gap \({G}_{\mathit{max}}\) / ref. |
Gap \({G}_{\mathit{min}}\) / ref. |
1 |
1.65E+06 |
1.43E+06 |
12.4 |
9.34 |
6.99% |
— 19.41% |
2 |
1.52E+06 |
1.44E+06 |
10.5 |
9.45 |
— 9.41% |
— 18.46% |
3 |
1.47E+06 |
9.81 |
— 15.35% |
|||
POST_K1_K2_K3: enrichment on four layers
method |
\({\mathit{K1}}_{\mathit{max}}\) |
\({\mathit{K1}}_{\mathit{min}}\) |
\({G}_{\mathit{max}}\) |
\({G}_{\mathit{min}}\) |
Gap \({G}_{\mathit{max}}\) / ref. |
Gap \({G}_{\mathit{min}}\) / ref. |
1 |
1.58E+06 |
1.58E+06 |
11.3 |
11.3 |
— 2.51% |
— 2.51% |
2 |
1.55E+06 |
1.47E+06 |
10.9 |
9.88 |
— 5.95% |
— 14.65% |
3 |
1.51E+06 |
10.4 |
— 10.26% |
|||
On this test, we note that it is essential to enrich several layers of elements around the bottom of the crack in order to obtain satisfactory results. Note that the mesh used here is linear and relatively coarse: with a finer mesh, the results are significantly improved. A convergence study on a similar case is presented in [bib5].
With an enrichment over four layers, method 1 is the one that leads to the most accurate results. In both cases, the maximum curvilinear abscissa corresponds to the distance of approximately four elements. For its part, the theta method is here less sensitive to the enrichment parameter.