## 3 Numerical studies
### Interaction between a macrocrack and diffuse microcracks  
To study the effect of the presence of a macrocrack in the microcracked domain we analyze the specimen of Figure 9, where a horizontal crack in the middle of the right side induces a geometrical singularity. The same example was originally investigated in [10], that used a variation of the well-known X-FEM methods to deal with the description of the macro-discontinuity embedded in the microcracked medium. The mechanical parameters, as declared in [10], are reported in Table 3.2. for completeness sake. Boundary conditions refer in this case to the only macro displacements field, as straightforwardly derived from a classical description of the Cauchy type. As detailed in the original contribution [10], macro displacements are not affected by any localization phenomenon. A remarkable concentration conversely characterizes the micro vectors, see Figures 10 and 11. The geometrical singularity, i.e. the crack tip, activates a \(45\)-degree displacement localization that has the shape of a regular four-bra \begin{table}
\begin{tabular}{||c|c|c|c|c|c|c||} \hline \hline \(\ell_{m}\;[mm]\) & \(\ell_{M}\;[mm]\) & \(E\;[N/mm^{2}]\) & \(A\;[mm^{2}]\) & \(\widehat{A}\;[mm^{2}]\) & \(\ell_{C}\;mm\) & \(\chi\) \\ \hline
75 & 5 & \(10^{3}\) & 1 & 0.0314 & 1 & 50 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Physical parameters. Figure 9: The cracked plane membrane. Figure 12, 13 and 14 show plots of both the macro and micro stress tensors on the whole domain. The macro stress diagrams show results that are well-known within the field of fracture mechanics for classical Cauchy media. The performed numerical analysis capture the macro stress concentration that typically arises at the tip of a cohesive fracture. It is quite interesting to notice that the micro displacement localizations have a counterpart in the micro stress field. All the components of the micro stress tensors are affected by a remarkable localization in the vicinity of the crack tip. In this case the magnitude of the micro quantities is lower than the macro ones and one may conclude that the macro singularities due to the crack tip overcome the localization phenomena peculiar to the micro level. The example is also used to introduce some preliminary investigations on the convergence properties of the adopted mixed discretization with respect to the macro and micro variables that have been directly discretized in the model. To this purpose the point A, located in the middle of the vertical right side of the specimen, has been firstly used to draw the convergence curves plotted in Figures 15 and 16. The point A is also represented in Figure 9.  
At a first glance the diagrams point out that all the numerical unknowns of the methods, i.e. two macro  
Figure 11: Macro–displacements \(u_{y}\) - Micro–displacements \(d_{y}\). Figure 10: Macro–displacements \(u_{x}\) - Micro–displacements \(d_{x}\). Figure 14: Macro–stress \(T_{xy}\) - Micro–stress \(S_{xy}\). Figure 12: Macro–stress \(T_{xx}\) - Micro–stress \(S_{xx}\). Figure 13: Macro–stress \(T_{yy}\) - Micro–stress \(S_{yy}\). displacement vectorial components \(u_{x}\) and \(u_{y}\), two micro displacement vectorial components \(d_{x}\) and \(d_{y}\), three macro stress tensorial components \(T_{xx}\), \(T_{yy}\) and \(T_{xy}\), three micro stress tensorial components \(S_{xx}\), \(S_{yy}\) and \(S_{xy}\), approximately enjoy a similar convergence rate. This is mainly due to the adopted mixed polynomial discretization, see i.e. Section 2.1.. Improvements in the accuracy of the results, especially for what concerns the stress interpolation, may be straightforwardly achieved by the adoption of finer polynomial interpolations. One may also resort to the dual formulation of Section 2.2., which principally takes advantage of the role of the stresses that are discretized as main variables of the problem.