Reference problem
=====================


Geometry
---------

The geometry, dimensions and materials are taken to be identical to those of Bittencourt et al. [:ref:`1 <1>`] and Ventura et al. [:ref:`2 <2>`].

Structure :math:`\mathrm{2D}` is a rectangular plate (:math:`20\mathrm{mm}\times 8\mathrm{mm}`) with 3 holes, including a through crack (). The length of the initial crack is :math:`a=\mathrm{1,5}\mathrm{mm}`.

The nodes marked :math:`\mathrm{P1}`, :math:`\mathrm{P2}`, and :math:`\mathrm{P3}` on the serve to impose boundary conditions, which are explained in paragraph [§ :ref:`1.3 <Ref193800026>`].


.. image:: images/100000000000037B000001D1FEB09A073CEAB243.jpg
    :width: 4.5646in
    :height: 2.5165in


.. _RefImage_100000000000037B000001D1FEB09A073CEAB243.jpg:

Figure 1.1-1: geometry of the cracked plate


Material properties
----------------------

Young's module: :math:`E=205000\mathrm{MPa}`

Poisson's ratio: :math:`\nu \mathrm{=}\mathrm{0,3}`


.. _Ref193800026:


Boundary conditions and loads
-------------------------------------

In order to block rigid modes, we block the movements of nodes :math:`\mathrm{P1}` and :math:`\mathrm{P2}` as follows:


*
    *
        * :math:`{\mathrm{DY}}^{\mathrm{P1}}={\mathrm{DY}}^{\mathrm{P2}}=0`;


        * :math:`{\mathrm{DX}}^{\mathrm{P1}}=0.`

In order to simulate fatigue propagation, a unit nodal force is applied in :math:`\mathrm{P3}`: :math:`\mathit{FY}\mathrm{=}\mathrm{-}1`. A load cycle will correspond to: zero load → max load → zero load. 35 propagation steps are simulated. At each propagation step, the crack advances by an imposed length equal to 0.1 m.


Benchmark solution
---------------------

Given the lack of precision of the diagrams in article :ref:`[1] <RefNumPara__20520_1002211838>`, it is not possible to derive/to establish precise numerical values. All we have to do is check that the crack paths have the same appearance (see § :ref:`2.3 <RefNumPara__20300_1002211838>`).

For the test, the values of the stress intensity factors :math:`{K}_{I}` and :math:`{K}_{\mathrm{II}}` calculated by modeling A at the end of propagation are used as a reference:

:math:`\begin{array}{c}{K}_{I}^{\mathit{ref}}\mathrm{=}\mathrm{1,142045}{\mathit{MPa.mm}}^{\mathrm{0,5}}\\ {K}_{\mathit{II}}^{\mathit{ref}}\mathrm{=}\mathrm{-}\mathrm{0,057097}{\mathit{MPa.mm}}^{\mathrm{0,5}}\end{array}`


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



.. _Ref112833703:


.. _RefNumPara__20520_1002211838:

1. T.N. Bittencourt, P.A. Wawrzynek, A.R. Ingraffea, A.R. Ingraffea, J.L. Sousa, Quasi-automatic simulation of crack propagation for 2D LEFM problems, Engineering Fracture Mechanics, vol. 55, vol. 55, pp. 321—334, 1996


2. G. Ventura, J.X. Xu, T. Belytschko, T. Belytschko, A vector level set method and new discontinuity approximations for crack growth by EFG, *International Journal for Numerical Methods in Engineering*, vol. 54, vol. 54, pp. 923—944, 2002