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


Calculation method used for the reference solution
--------------------------------------------------------



Modeling A
~~~~~~~~~~~~~~~

The reference solution comes from WILSON and YU [:ref:`bib1 <bib1>`]:


.. image:: images/Object_1.svg
    :width: 221
    :height: 108


.. _RefImage_Object_1.svg:

In plane deformations, the IRWIN formula gives:

.. image:: images/Object_2.svg
    :width: 221
    :height: 108


.. _RefImage_Object_2.svg:

or numerically:

.. image:: images/Object_3.svg
    :width: 221
    :height: 108


.. _RefImage_Object_3.svg:


B modeling
~~~~~~~~~~~~~~~

The reference solution for the stress intensity factor for mode I :math:`{K}_{I}` is expressed as follows [:ref:`bib3 <bib3>`]:

:math:`{K}_{I}=\mathrm{\sigma }\sqrt{\mathrm{\pi }a}F\left(\frac{a}{W}\right)`

with :math:`F\left(\frac{a}{W}\right)=\mathrm{1,122}-\mathrm{0,231}\left(\frac{a}{W}\right)+\mathrm{10,550}{\left(\frac{a}{W}\right)}^{2}-\mathrm{21,710}{\left(\frac{a}{W}\right)}^{3}+\mathrm{30,382}{\left(\frac{a}{W}\right)}^{4}`

The :math:`G` energy return rate is obtained thanks to Irwin's formula: :math:`G=\frac{\left(1-{\mathrm{\nu }}^{2}\right)}{E}{K}_{I}^{2}`.


Benchmark results
----------------------


Modeling A
~~~~~~~~~~~~~~~

The reference results are those from the reference solution from WILSON and YU [:ref:`bib1 <bib1>`]:


.. image:: images/Object_4.svg
    :width: 221
    :height: 108


.. _RefImage_Object_4.svg:


B modeling
~~~~~~~~~~~~~~~

The reference results are those from the reference solution from [:ref:`bib3 <bib3>`]: :math:`G=\mathrm{572,05}J\mathrm{.}{m}^{-2}`


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


1. The Use of J-Integrals in thermal stress crack problems - International Journal of Fracture (1979) WILSON and YU.


2. Complementary qualification of codes INCA/MAYA in linear thermo-elasticity. Technical note DRE/STRE/LMA 84/598


3. H. Tada, P. Paris, G. Irwin, G. Irwin, The Stress Analysis of Cracks Handbook, 3rd edition, 2000