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 )=\frac{1}{2}A(d)E{\epsilon }^{2}+\psi (d)+\frac{c}{2}\nabla \alpha \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 cube with side :math:`L=1\text{m}`.

Dy

Dx

Dz

**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**: Null imposed moves DY = 0 :math:`m`. on the bottom side (:math:`y=0.`), DX = 0 :math:`m`. on the left side (:math:`x=0.`) and DZ = 0 :math:`m`. on the back side (:math:`z=0.`). See figure 1.

**Loading 1**: Imposed linear displacement :math:`{U}_{1}` on the right side (:math:`x=1.`):

:math:`{U}_{1}=0.0\text{m}` for INST =0, :math:`{U}_{1}=0.02\text{m}` for INST =1.0.

**Loading 1**: Imposed linear displacement :math:`{U}_{2}` on the right side (:math:`y=1.`):

:math:`{U}_{2}=0.0\text{m}` for INST =0, :math:`{U}_{2}=0.02\text{m}` for INST =1.0.

**Loading 1**: Imposed linear displacement :math:`{U}_{3}` on the right side (:math:`z=1.`):

:math:`{U}_{3}=0.0\text{m}` for INST =0, :math:`{U}_{3}=0.02\text{m}` for INST =1.0.