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


Calculation method used for the reference solution
--------------------------------------------------------

This problem requires an analytical solution. We determine the relationship which associates the level of homogeneous damage :math:`a` with the imposed deformation :math:`\varepsilon` (reciprocal problem where more generally non-local no longer admits a simple analytical solution). Since the problem is homogeneous, the damage (under load) and the deformation are linked by the coherence relationship (threshold function):

:math:`{A}^{\text{'}}(a)\Gamma (\varepsilon )+{\omega }^{\text{'}}(a)=0`

with

:math:`\omega (a)=\mathrm{ka},A(a)=\frac{{(1-a)}^{2}}{{(1-a)}^{2}+\mathrm{ma}(1+\mathrm{pa})}\text{et}\Gamma (\varepsilon )=\left[{c}_{T}\mathrm{tr}\varepsilon +\sqrt{{c}_{H}{\mathrm{tr}}^{2}\varepsilon +{c}_{S}{\varepsilon }_{\mathrm{eq}}^{2}}\right]`

where parameters :math:`{c}_{T},{c}_{H}\text{et}{c}_{S}` are derived from the problem data by:

:math:`{c}_{S}=\frac{E}{2}\left[(1-2\nu ){c}_{\mathrm{comp}}+(1+\nu )\sqrt{\frac{1-2\nu }{2(1+\nu )}{c}_{\mathrm{volu}}+1}\right];{c}_{T}={c}_{\mathrm{comp}}\sqrt{{c}_{S}};{c}_{H}=\frac{1+\nu }{2(1-2\nu )}{c}_{\mathrm{volu}}{c}_{S}`

A uniaxial deformation of the form is adopted:

:math:`\varepsilon \mathrm{=}\varepsilon n\mathrm{\otimes }n\text{où}∥n∥\mathrm{=}1\text{et}\varepsilon >0`

In this case the deformation threshold function is simply written: :math:`\Gamma (\varepsilon )=\left[{c}_{T}+\sqrt{{c}_{H}+{c}_{S}}\right]\varepsilon`

To achieve the given damage :math:`a`, by stressing the deformation in the :math:`n` direction, it is therefore necessary to impose a deformation intensity:

:math:`\varepsilon =\frac{-k}{{A}^{\text{'}}(a)\left[{c}_{T}+\sqrt{{c}_{H}+{c}_{S}}\right]}`

For the reference solution, we therefore adopt the following strategy: we set the damage level and we verify by EF calculations that for an estimated theoretical deformation we reach this same damage level.


Benchmark results
----------------------

In plane deformations, a direction of stress :math:`n=(1/\sqrt{\mathrm{5,}}2/\sqrt{5})` is adopted. In :math:`\mathrm{3D}`, it is worth :math:`n=(1/\sqrt{\mathrm{14,}}2/\sqrt{\mathrm{14,}}3/\sqrt{14})`. Damage :math:`a=0.6` is the target; this corresponds to an intensity of stress :math:`\varepsilon =9.574237\times {10}^{-4}` according to the reference solution above.

The load is applied using the control technique PRED_ELAS in which the maximum limit is fixed so as to reach the deformation level :math:`\varepsilon` above. It will be verified that the corresponding damage actually reaches :math:`0.6`.


Uncertainty about the solution
----------------------------

Néant


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

Not applicable