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


Geometry
---------

We represent a square with side :math:`l\mathrm{=}\mathrm{1m}`.


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    :width: 1.961in
    :height: 2.4709in


.. _RefImage_10000200000000BC000000EDED7B21EA295799F1.png:


Illustration 1: Geometry of the test case
-------------------------------------

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

**Material:**

Elasticity parameters:

Young's modulus :math:`{E}_{1}={20.10}^{9}\mathrm{MPa}`, Poisson's ratio :math:`{\nu }_{1}=\mathrm{0,25}`

Parameters of law ENDO_HETEROGENE:

Elastic limit :math:`{\sigma }_{y}\mathrm{=}5\mathit{MPa}`

Weibull module :math:`m=6`

Tenacity :math:`{K}_{c}=1{\mathrm{MPa.m}}^{1/2}`

Thickness of sample :math:`\mathit{ep}\mathrm{=}\mathrm{1m}`

Seed :math:`\mathrm{GR}=121`

Parameter of the non-local model:

Characteristic length :math:`{l}_{c}=\mathrm{0,2}m`


Boundary conditions and loading
------------------------------------

The lower edge is blocked in movement in the vertical direction, the lateral edges are subject to a field of movement that varies linearly in height. Either:

At the bottom:

:math:`{u}_{y}(x,y=0)=0`

To the right

:math:`{u}_{y}(x=l,y)={\mathrm{c.u}}_{d}(1-y/l)`

with :math:`{u}_{d}=\mathrm{0,0001}`

To the right

:math:`{u}_{y}(x=\mathrm{0,}y)=-{u}_{y}(x=l,y)`

:math:`c` corresponds to a loading ramp varying between 0 and 1 over the simulation time (1 s).


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    :width: 2.7339in
    :height: 1.8366in


.. _RefImage_1000020000000147000000C397BCB8B820E6BC3D.png:

Figure 2: Boundary condition diagrams

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

There is no reference solution here and the test is of a non-regression type. It uses the coupling law ENDO_HETEROGENE, which is itself based on regularized constraints.

The result is therefore purely qualitative. The aim here is to observe the onset of a crack following lateral loading.