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


Calculation method used for the 2D reference solution
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We use the elastoplastic fracture energetic method based on the parameter :math:`\mathrm{Gp}` [:ref:`1 <1>`], [:ref:`2 <2>`].

The notch bottom is formed by a semicircle with radius :math:`R`. The zone :math:`\mathrm{Ze}` of length :math:`\Delta l` corresponds to the virtual propagation of the notch and is cut into "chips".


.. image:: images/10001A1C000069BB0000140D95B54754556914A7.svg
    :width: 480
    :height: 90


.. _RefImage_10001A1C000069BB0000140D95B54754556914A7.svg:

It determines at each moment the evolution of quantity :math:`\mathrm{Gp}(\Delta l)` defined by:

:math:`\mathit{Gp}(\Delta l)=2[{W}_{\mathit{elas}}^{\mathit{traction}}(\Delta l)]/\Delta l`

where :math:`{W}_{\mathit{elas}}^{\mathit{traction}}(\Delta l)` is the elastic tensile energy calculated on zone :math:`\mathrm{Ze}`. We must then calculate the maximum of this quantity in relation to :math:`\Delta l`, which we call ":math:`\mathrm{Gp}`".

:math:`\mathrm{Gp}=\underset{\Delta l}{\mathrm{Max}}\{\mathrm{Gp}(\Delta l)\}`

The critical moment when the defect will begin to spread is then the moment when tenacity :math:`\mathrm{Kj}={\mathrm{Kj}}_{\mathrm{crit}}`. It is then said that :math:`\mathrm{Gp}` reaches the critical value ":math:`\mathrm{Gpc}`".


Bibliographical references
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1. WADIER Y.: "Brief presentation of the energetic approach to elastoplastic rupture applied to breakage by cleavage", Note EDF R&D HT-64/03/001/A, January 2003.


2. WADIER Y., LORENTZ E.: "Fracture mechanics in the presence of plasticity: modeling of the crack by a notch". C.R.A.S. t. 332, series IIb, 2004.