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


Theoretical framework
---------------

The unknowns of the problem are the degrees of freedom of movement and nodal damage. It is then a question of minimizing an energy of the form:

:math:`\phi (u,\alpha )\mathrm{=}\frac{1}{2}A(d)E{\epsilon }^{2}+\psi (d)+\frac{c}{2}\mathrm{\nabla }\alpha \mathrm{.}\mathrm{\nabla }\alpha`

Where :math:`E` is the Young's modulus of the material, :math:`A(d)` the stiffness function, the stiffness function, :math:`\psi (d)` the dissipation, and :math:`c` the non-local coefficient.

In the case of law ENDO_CARRE:

:math:`A(d)={(1-d)}^{2}` and :math:`\psi (d)=\frac{{\sigma }_{y}^{2}}{E}d`

The criterion corresponding to law ENDO_CARRE, for a homogeneous solution (:math:`\nabla \alpha =0`), is therefore written:

:math:`d=1-(\frac{{W}_{y}}{{W}_{\mathrm{el}}})`

Where :math:`{W}_{\mathrm{el}}` is the elastic deformation energy and:

:math:`{W}_{y}=\frac{{\sigma }_{y}^{2}}{2E}`


Geometry
---------

Consider a square with side :math:`L=1\text{m}`.

Dx

**Figure 1**: Representation of the problem


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


Law of damage: material ENDO_CARRE
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Elastic characteristics:

:math:`E=1\text{Pa}`

:math:`\nu =0.`

Characteristics related to the law of damage:

Elasticity limit:

:math:`\mathrm{SY}=0.01\text{Pa}`

Non-local characteristics:

:math:`c=1.0\text{N}`


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

**Embed**: Imposed movements zero DY = 0 :math:`m` on the bottom and top horizontal edges (:math:`y=0.` and :math:`y=1.`) and DX = 0 :math:`m` on the left edge (:math:`x=0.`). See Figure 1.

**Loading 1**: Imposed displacement :math:`{U}_{1}` on the right vertical edge:

At moment :math:`{t}_{1}`: DX = :math:`0.01\text{m}`

At moment :math:`{t}_{2}`: DX = :math:`0.0125\text{m}`

At moment :math:`{t}_{3}`: DX = :math:`0.02\text{m}`