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


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

The reference solution comes from THAU and LU [:ref:`bib1 <bib1>`] and L.B. book FREUND [:ref:`bib2 <bib2>`]. The following figure represents the infinite medium and not the geometry of the test.


.. image:: images/Object_3.png
    :width: 3.861in
    :height: 2.5555in


.. _RefSchema_Object_3.png:

:math:`{K}_{I}^{D}(t)\mathrm{=}\mathrm{2P}H(t)\frac{\sqrt{1\mathrm{-}2\nu }}{(1\mathrm{-}\nu )}\sqrt{\frac{{c}_{d}t}{\pi }},0<t<\mathrm{2a}\mathrm{/}{c}_{d}`

The expression for the energy recovery rate is as follows:

:math:`G(t)\mathrm{=}\frac{{K}_{I}^{{D}^{2}}(t)}{E}(1\mathrm{-}{\nu }^{2})\mathrm{=}{\mathrm{4P}}^{2}H(t)\frac{(1\mathrm{-}2\nu )(1+\nu )}{(1\mathrm{-}\nu )E}\frac{{c}_{d}t}{\pi },0<t<\mathrm{2a}\mathrm{/}{c}_{d}`

:math:`G\mathrm{=}\frac{(1\mathrm{-}{\nu }^{2})}{E}{K}_{1}^{2}` with :math:`{K}_{1}\mathrm{=}\frac{\alpha E}{P(1\mathrm{-}\nu )}{T}_{0}\sqrt{\mathit{Pa}}`

That is: :math:`G\mathrm{=}\frac{(1\mathrm{-}{\nu }^{2})}{P{(1\mathrm{-}\nu )}^{2}}{\alpha }^{2}{\mathit{ET}}_{0}^{2}a`


Benchmark result
---------------------

So the reference result is: :math:`G\mathrm{=}5.9115{10}^{\mathrm{-}7}N\mathrm{/}\mathit{mm}`


.. csv-table::

    ":math:`t` "," :math:`{G}_{\mathit{freund}}`"
    "2e-3", "5.677e5"
    "4e-3", "1.135e6"
    "6e-3", "1.703e6"



Bibliographical reference
-------------------------


1. Transient stress intensity factors for a finite crack in an elastic solid caused by a dilatational wave, International Journal of Solids and Structures 7, THAU and LU (1971)


2. Dynamic Fracture Mechanics L.B FREUND.