2. Benchmark solution

2.1. Calculation method used for the reference solution

This problem requires an analytical solution. We determine the relationship which associates the level of homogeneous damage \(a\) with the imposed deformation \(\varepsilon\) (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):

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

with

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

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

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

A uniaxial deformation of the form is adopted:

\(\varepsilon \mathrm{=}\varepsilon n\mathrm{\otimes }n\text{où}∥n∥\mathrm{=}1\text{et}\varepsilon >0\)

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

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

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

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.

2.2. Benchmark results

In plane deformations, a direction of stress \(n=(1/\sqrt{\mathrm{5,}}2/\sqrt{5})\) is adopted. In \(\mathrm{3D}\), it is worth \(n=(1/\sqrt{\mathrm{14,}}2/\sqrt{\mathrm{14,}}3/\sqrt{14})\). Damage \(a=0.6\) is the target; this corresponds to an intensity of stress \(\varepsilon =9.574237\times {10}^{-4}\) 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 \(\varepsilon\) above. It will be verified that the corresponding damage actually reaches \(0.6\).

2.3. Uncertainty about the solution

Néant

2.4. Bibliographical references

Not applicable