Benchmark solution
=====================


.. _RefHeading__2736_1308541645:


Calculation method
-----------------

The calculation of stress intensity factors is obtained in [:ref:`1 <1>`] by a semi-analytical procedure which consists in superimposing displacement profiles verifying free stress conditions on the crack lips.


Reference quantities and results
-----------------------------------

The values of :math:`{K}_{1}` and :math:`{K}_{2}` given in [:ref:`1 <1>`] are compared to crack bottoms.


+----------+---------------+---------------++-----------------+---------------+---------------+
|**Points**|:math:`{K}_{1}`|:math:`{K}_{2}`||**Points**       |:math:`{K}_{1}`|:math:`{K}_{2}`|
+----------+---------------+---------------++-----------------+---------------+---------------+
|:math:`A` |1,7943         |2.8522         ||:math:`A\text{'}`|3,7215         |2.3379         |
+----------+---------------+---------------++-----------------+---------------+---------------+
|:math:`B` |1,9932         |2.4042         ||:math:`B\text{'}`|2.6700         |1.0248         |
+----------+---------------+---------------++-----------------+---------------+---------------+
|:math:`C` |-1.6920        |-0.1337        ||:math:`C\text{'}`|5.3966         |-0.1143        |
+----------+---------------+---------------++-----------------+---------------+---------------+
|:math:`F` |0.0510         |0.2894         ||:math:`F\text{'}`|4.3255         |-0.1661        |
+----------+---------------+---------------++-----------------+---------------+---------------+
|:math:`G` |-0.5317        |0.1885         ||:math:`G\text{'}`|3.6812         |0.9279         |
+----------+---------------+---------------++-----------------+---------------+---------------+
|:math:`H` |-0.0517        |-0.1979        ||:math:`H\text{'}`|0.4157         |-0.3947        |
+----------+---------------+---------------++-----------------+---------------+---------------+
|:math:`I` |-0.1933        |0.0213         ||:math:`I\text{'}`|1.0043         |0.0648         |
+----------+---------------+---------------++-----------------+---------------+---------------+

Table 2: stress intensity factors at the bottom of cracks obtained in [:ref:`1 <1>`].


Uncertainty about the solution
----------------------------

The number of significant figures given in Table 2 reflects the quality of the semi-analytical solution. In fact, convergence towards these figures can be verified in the case of an increasingly fine mesh.


Bibliographical references
---------------------------


1. A. K. YAVUZ, S. L. PHOENIX, "Multiple Crack Analysis in Finite Plates", AIAA Journal, Vol. 44, No. 11, November 2006.


2. M. SIAVELIS, M. GUITON, P. MASSIN, N. MOËS "Large sliding contact along branched discontinuities with X- FEM", under review, 2012.