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


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

The analytic expressions of the stress intensity factors :math:`{K}_{I}` and :math:`{K}_{\mathrm{II}}` are functions of the distributed force :math:`p`, of the length of the crack a, of the width of the plate :math:`\mathrm{Lx}`:

:math:`\begin{array}{}{K}_{I}=p\sqrt{\pi a}f(\frac{a}{\mathrm{Lx}})\\ {K}_{\mathrm{II}}=0\end{array}`

where function :math:`f` can be determined in several different ways. We choose the one obtained by [:ref:`1 <1>`], and that is true for :math:`\begin{array}{}\frac{a}{\mathrm{Lx}}<\mathrm{0,6}\end{array}`:

:math:`\begin{array}{}f(\frac{a}{\mathrm{Lx}})=\mathrm{1,12}-\mathrm{0,231}(\frac{a}{\mathrm{Lx}})+\mathrm{10,55}{(\frac{a}{\mathrm{Lx}})}^{2}-\mathrm{21,72}{(\frac{a}{\mathrm{Lx}})}^{3}+\mathrm{30,39}{(\frac{a}{\mathrm{Lx}})}^{4}\end{array}`

We are moving the crack forward thanks to the Paris law:

:math:`\begin{array}{}\frac{\mathrm{da}}{\mathrm{dN}}=C\Delta {K}^{m}\end{array}` where a is the crack length, :math:`C` and :math:`m` are material constants, :math:`\begin{array}{}\Delta K\end{array}` is the difference between two consecutive FICs's, and :math:`N` is the number of cycles.

With the numerical values of the test:

 No spread: :math:`\mathrm{0,25}m`

 :math:`\mathrm{Lx}`: :math:`10m`


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


+------------+-----------------------------------------------+---------------------------------------------------------+
|Reference                                                                                                             |
+------------+-----------------------------------------------+---------------------------------------------------------+
|:math:`a(m)`|:math:`{K}_{I}({\mathrm{Pa.m}}^{\mathrm{0,5}})`|:math:`{K}_{\mathrm{II}}({\mathrm{Pa.m}}^{\mathrm{0,5}})`|
+------------+-----------------------------------------------+---------------------------------------------------------+
|2.5         |4,205998 106                                   |0                                                        |
+------------+-----------------------------------------------+---------------------------------------------------------+
|2.75        |4,63286 106                                    |0                                                        |
+------------+-----------------------------------------------+---------------------------------------------------------+
|3           |5,09492 106                                    |0                                                        |
+------------+-----------------------------------------------+---------------------------------------------------------+
|3.25        |5,59908 106                                    |0                                                        |
+------------+-----------------------------------------------+---------------------------------------------------------+
|3.5         |6,15349 106                                    |0                                                        |
+------------+-----------------------------------------------+---------------------------------------------------------+
|3.75        |6,76776 106                                    |0                                                        |
+------------+-----------------------------------------------+---------------------------------------------------------+
|4           |7,4531 106                                     |0                                                        |
+------------+-----------------------------------------------+---------------------------------------------------------+
|4.25        |8,2224 106                                     |0                                                        |
+------------+-----------------------------------------------+---------------------------------------------------------+
|4.5         |9,0905 106                                     |0                                                        |
+------------+-----------------------------------------------+---------------------------------------------------------+
|4.75        |1,0074 107                                     |0                                                        |
+------------+-----------------------------------------------+---------------------------------------------------------+
|5           |1,1192 107                                     |0                                                        |
+------------+-----------------------------------------------+---------------------------------------------------------+
|5.25        |1,2465 107                                     |0                                                        |
+------------+-----------------------------------------------+---------------------------------------------------------+
|5.5         |1,3916 107                                     |0                                                        |
+------------+-----------------------------------------------+---------------------------------------------------------+
|5.75        |1,55716 107                                    |0                                                        |
+------------+-----------------------------------------------+---------------------------------------------------------+
|6           |1,74586 107                                    |0                                                        |
+------------+-----------------------------------------------+---------------------------------------------------------+

**Table** 2.2-1 **: reference values for** :math:`{K}_{I}` **and** :math:`{K}_{\mathrm{II}}`


Uncertainties about the solution
----------------------------

None, analytical solution.


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


1. TADA H., PARIS P., IRWIN G.:The stress analysis of cracks, Handbook. Del Research Corporation, Hellertown, PA, 1973.