2. Benchmark solution#
2.1. Calculation method#
The study of this case is based entirely on the article by Mariani and Perego. Three initial crack configurations are chosen: \(\chi \mathrm{=}\mathrm{0,}25\) and \(50\). In this test case, we only chose \(\chi \mathrm{=}50\). We therefore compare the propagation path with the experimental path in the article [bib1].
The reference expressions for stress intensity factors \({K}_{I}\) and \({K}_{\mathit{II}}\) are those for the mesh method. We will therefore compare the values of the simplex, upwind and geometric methods to the values given by the mesh method.
For the propagation of the crack, we use the Paris law:
\(\begin{array}{c}\frac{\mathit{da}}{\mathit{dN}}\mathrm{=}C\Delta {K}^{m}\end{array}\) where a is the crack length, \(C\) and \(m\) are material constants, \(\begin{array}{}\Delta K\end{array}\) is the difference between two consecutive \(\mathrm{FICs}\)”s, and \(N\) is the number of cycles.
The bifurcation criterion used is the maximum hoop stress criterion:
\(\begin{array}{c}\beta \mathrm{=}2\mathrm{arctan}\mathrm{[}\frac{1}{4}(\frac{{K}_{I}}{{K}_{\mathit{II}}}\mathrm{-}\mathit{sign}({K}_{\mathit{II}})\sqrt{(\frac{{K}_{I}}{{K}_{\mathit{II}}})\mathrm{²}+8})\mathrm{]}\end{array}\)
With the numerical values of the test:
No spread: \(\mathrm{0,3}m\)
\({x}_{0}:65\mathrm{mm}\)
\({y}_{0}:19\mathrm{mm}\)
Number of propagation steps: 13
\(\mathrm{RI}:3\mathrm{mm}\)
\(\mathrm{RS}:12\mathrm{mm}\)
\(\mathrm{RP}:12\mathrm{mm}\)
2.2. Reference quantities and results#
Reference (mesh method) |
|||
\(x(\mathrm{mm})\) |
\(y(\mathrm{mm})\) |
\({K}_{I}({\mathrm{MPa.m}}^{\mathrm{0,5}})\) |
\({K}_{\mathrm{II}}({\mathrm{MPa.m}}^{\mathrm{0,5}})\) |
65 |
19 |
2.43961 10-1 |
4,27722 10-2 |
66,129 |
22,313 |
2.90147 10-1 |
1.21013 10-4 |
67,261 |
25,625 |
3,30840 10-1 |
7,10255 10-3 |
68,533 |
28,885 |
3.75984 10-1 |
1.94683 10-3 |
69,839 |
32,132 |
4.33606 10-1 |
1.20266 10-3 |
71,164 |
35,372 |
4.96975 10-1 |
8,82542 10-4 |
72.5 |
38,607 |
5,73785 10-1 |
-1.23199 10-3 |
73,821 |
41,848 |
6,70222 10-1 |
-3,54655 10-3 |
75,109 |
45,103 |
7,89716 10-1 |
-4.54122 10-3 |
76,359 |
48,372 |
9,39463 10-1 |
-8,18030 10-3 |
77,552 |
51,662 |
1.15201 |
-1.55772 10-2 |
78,655 |
54,984 |
1.45163 |
-2.31849 10-2 |
79,652 |
58,339 |
1.91885 |
-3,52229 10-2 |
Table 2.2-1: reference values for \({K}_{I}\) and \({K}_{\mathrm{II}}\)
2.3. Bibliographical references#
Mariani S, Perego U — Extended Finite Element Method for Quasi-Brittle Fracture, International Journal for Numerical Methods in Engineering, 58:103-126 (2003)