## 2 Materials and Methods
### Specim Mechanical Design
#### 2.2.1 Removal of the Stress Singularity
In contrast, the \"inverse\" problem consists of manipulating Equation (1) to understand the system conditions that prevent the stress singularities under prescribed acting loads. It was first explored by [PERSON] [39] and its solution requires knowledge of the mechanical or geometrical parameters to insert within the mathematical framework, as well as a guess value of the singularity exponent. [PERSON] [39] illustrated that, to obtain a valid solution, the value of the real part of the solution must satisfy the condition \(p_{R}\geq 1\) in any circumstances to achieve a non-singular geometry. Moreover, the convergency is guaranteed using \(p_{R}=1+10^{-4}\) in a non-linear strategy framework. For this work, mechanical data were considered totally known and applicable. The unknown of the problem was the substrate angle at the material interface, which ensured the absence of a singular field at a fixed adhesive angle. In this way, by restricting the problem to a 1D variable, one can easily explore multiple adhesive-adherent conditions and choose the final configuration that fits better within the whole mechanical project. The work logic described above was applied in the angular adhesive range of 10-120\({}^{\circ}\) to explore a broad set of solutions. A Newton-Raphson strategy manages the non-linear problem defined by the \(A\)-\(F\) functions within the determinant definition. In Table 2, the obtained substrate angles are reported for both plane stress and plane strain conditions, whereas in Figure 3, these are graphically subdivided. The trends illustrated in Figure 3 should be interpreted as the upper conditions to avoid stress singularities and can be adopted as a design tool for similar bi-material joints. Thus, with a fixed adhesive angle and plane strain/stress conditions, substrate values lower than those lying on the curve present singularity-free circumstances. Three separated regions can exist and are highlighted: (i) a safe region in which the local adhesive-adherent geometry always prevents stress singularities for any kind of geometrical combination; (ii) a critical region in which the presence of the singular feature strongly depends on the modelling assumption. If plane strain conditions are assumed, the singularity is expected, whereas, in the case of plane stress, this characteristic is removed; and (iii) an unsafe region where a singularity is surely present for any modelling case. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Adhesive Angle \(\mathbf{\theta_{1}}[^{\circ}]\)**} & \multicolumn{2}{c}{**Substrate Angle \(\mathbf{\theta_{2}}[^{\circ}]\)**} \\ \cline{2-3}  & **Plane Stress** & **Plane Strain** \\ \hline
10 & 179.6 & 179.6 \\ \hline
20 & 179.3 & 179.2 \\ \hline
30 & 178.8 & 178.7 \\ \hline
40 & 177.9 & 177.5 \\ \hline
50 & 175.5 & 171.8 \\ \hline
60 & 121.2 & 70.28 \\ \hline
70 & 61.35 & 51.09 \\ \hline
80 & 51.68 & 45.7 \\ \hline
90 & 48.17 & 43.5 \\ \hline
100 & 46.97 & 42.64 \\ \hline
110 & 47.78 & 42.65 \\ \hline
120 & 60.60 & 44.83 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Obtained solutions from the inverse problem. Figure 3: Adhesive – adherent solution avoiding the stress singularity. As mentioned above, in general, all the valid substrate-adhesive configurations are suitable for removing the singularity from the mechanical system. However, in this work a pair of angles 43.5\({}^{\circ}\) and 90\({}^{\circ}\) is selected, as this configuration is easy to achieve and a 90\({}^{\circ}\) adhesive angle can be reached using simple manufacturing procedures. Even though the inverse mathematical model provides information concerning singularity-free circumstances, nothing is stated for the stress experienced by the current joint under prescribed loads. This was made accessible through numerical analyses as described in the following section.