2. Benchmark solution

2.1. Calculation method used for the reference solution

This problem requires an analytical solution. During the traction phase, the relationship between the imposed deformation \(\varepsilon\) and the homogeneous damage level \(a\) is determined. Here, a uniaxial deformation of the form is adopted:

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

It is therefore a problem in confined uniaxial traction. Since the problem is homogeneous, the damage (under load) and the deformation are linked by the coherence relationship:

\(A\text{'}(a)\Gamma (\epsilon )+\omega \text{'}(a)=0\) with \(\omega (a)=\mathit{ka},A(a)=\frac{{(1-a)}^{2}}{{(1-a)}^{2}+\mathit{ma}(1+\mathit{pa})}\)

The G function is based on the form of the criterion that defines the elasticity domain, cf. [R5.03.28] or, in more detail, [Lorentz, 2016]. In the present case of confined uniaxial tension/compression, it is simply written:

\(\Gamma (\epsilon )=\frac{1}{2}{E}_{c}{\epsilon }^{2}\) where \({E}_{c}=\lambda +2\mu\)

So we simply deduce the relationship between \(\epsilon\) and \(a\):

\(\epsilon =\sqrt{\frac{-2k}{{E}_{c}A\text{'}(a)}}\)

For the test case, the deformation will increase to a level such that the damage level is \(a=0.2\) and it will be tested that the damage reaches its target value.

In a second step, the deformation is released and then compression is imposed up to a level \({\epsilon }_{\mathit{comp}}=-2{\sigma }^{c}/{E}_{c}\). The damage no longer evolves. The stress-deformation relationship is written as:

\(\sigma =A(a){E}_{c}\epsilon +\frac{1}{2}(1-A(a)){E}_{c}S\text{'}(\epsilon )\) where \(S(\epsilon )={\langle -\epsilon \rangle }^{2}\mathrm{exp}(\frac{1}{\gamma \epsilon })\)

The discharge is therefore carried out linearly with a secant stiffness module \(A(a){E}_{c}\). Then, in the compression phase, the model gradually restores the stiffness according to the above relationship. In the test, it is verified that the level of constraint is that corresponding to the analytical relationship. In practice, rather than projecting the stress tensor onto the direction of stress, we ensure that the deformation energy obtained via the POST_ELEM command (keyword factor TRAV_EXT, component TRAV_ELAS) is indeed equal to the expected value, i.e.:

\(\frac{\mathit{vol}(\Omega )}{2}\sigma \mathrm{:}ϵ=\frac{\mathit{vol}(\Omega )}{2}\sigma ϵ\) where \(\mathit{vol}(\Omega )\) refers to the volume of the element (4 mm2 or 8 mm3)

2.2. Benchmark results

In plane deformations, a direction of stress \(n=(1/\sqrt{\mathrm{5,}}2/\sqrt{5})\) is adopted. In 3D, it’s worth \(n=(1/\sqrt{\mathrm{14,}}2/\sqrt{\mathrm{14,}}3/\sqrt{14})\). Damage \(a=0.2\) is the target; this corresponds to an intensity of stress \(\epsilon =2.86\times {10}^{-4}\) 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 0.2.

In compression, the imposed deformation level is \({\epsilon }_{\mathit{comp}}=-1.8\times {10}^{-4}\) for a stress of \({\sigma }_{\mathit{comp}}=-4.537\text{MPa}\).

2.3. Uncertainty about the solution

Nil.

2.4. Bibliographical references

Lorentz E. (2016) A nonlocal damage model for plain concrete consistent with cohesive fracture. Submitted to J. Mech. Phys. Solids.