## 2 Materials and Methods
### Specim Mechanical Design
#### 2.2.1 Removal of the Stress Singularity  
The detection and quantification of model features that can endanger the structural integrity of a component has always been an important topic in structural engineering. Among these features, stress singularities are point regions where stress theoretically trends toward infinite values. In homogenous materials, stress singularities originate in presence of sharp angular notches [26] and crack fronts, whereas for multi-material unions such features arise at the material interface under prescribed loads [28]. Stress singularities are particularly dangerous as their presence has been demonstrated to critically impact safety and to be a root source of failures [25; 56]. To properly understand the physical conditions for removing singularities, the basic mathematical framework should first be introduced and clarified. In this study, the problem is restricted to the special case of a bi-material union, which is the simplest and the most straightforward problem to address, as shown in Figure 2, where x1 and x2 are Cartesian coordinates and \(r\) and \(\theta\) are polar coordinates that are introduced for the treatment of the mathematical problem. Figure 2: Local material wedge. The wedge prototype presented in Figure 2 is characterized by a specific set of material properties such as the elasticity modulus \((E_{j})\), the Poisson's ratio \(\left(\
u_{j}\right)\), and the shear modulus \(\left(\mu_{j}\right)\), where the \(j\) index represents the material number. [PERSON] [29] approached the problem by manipulating the equilibrium equations in the frame of the Mellin transformation for passing from a physical to an auxiliary plane (i.e., the p-plane), thus solving the system in terms of the singularity exponent. Under this hypothesis, the so-called Bogy determinant reported in Equation (1) became one of the fundamental theoretical tools for studying stress singularities in the case of bi-material unions. \[\mathbf{D}(\theta_{1},\theta_{2},\alpha,\beta;p)=A\alpha^{2}+B\alpha\beta+C\beta^{ 2}+D\alpha+E\beta+F, \tag{1}\]  
Here, \(\theta_{1},\ \theta_{2}\) are the opening angles of material wedges; \(p\) is a complex root defined within the interval \(]0,\ 1[\) that identifies the singularity exponent; and A-F are functions that depend on trigonometric expressions and on the singularity exponent. The \(\alpha\) and \(\beta\) parameters are the Dundurs' parameters shown in Equation (2). \[\alpha=\frac{\overline{E_{1}}-\overline{E_{2}}}{\overline{E_{1}}+\overline{E_ {2}}},\ \beta=\frac{\mu_{1}(1-2k_{2})-\mu_{2}(1-2k_{1})}{2[\mu_{1}(1-k_{2})+\mu_{2}(1- k_{1})]} \tag{2}\]  
Here, \(\overline{E_{j}}\) is the elasticity modulus under plane stress or plane strain and \(k_{j}\) is the Kolosoff parameter, which depends on the Poisson ratio. In general, solving Equation (1) leads to a complex solution \(\left(p=p_{R}+ip_{I},\ i=\sqrt{-1}\right)\) that is associated with one of the following possible stress forms: (i) hyperbolic solutions; (ii) oscillatory unbounded solutions; (iii) logarithmic solutions; and (iv) constant solutions, in proximity to the singular corner \((\tau\to 0)\). The latter are reported in Equation (3), where \(\mathbf{\sigma}\) is the stress vector. \[\mathbf{\sigma}= \begin{cases}O\big{(}r^{p-1}\big{)}&if\ p=p_{R}\\ O\big{[}r^{p_{R}-1}\text{cos}\big{(}p_{I}\text{log}r\big{)}\ or\ r^{p_{R}-1}\text{sin}(p_{I}\text{log}r\big{)}\ \big{]}&if\ p=p_{R}+ip_{I}\\ O(\text{log}r)&if\ D\
eq 0\ f\ or\ p_{R}\in]0-1\big{[},\frac{dp}{dp}=0\ at\ p=1\\ O(1)&if\ D\
eq 0\ f\ or\ p_{R}\in]0-1\big{[},\frac{dp}{dp}\
eq 0\ at\ p=1\\ \end{cases}  \tag{3}\]  
Mathematically, Equations (1)-(3) represent the \"direct\" Bogy problem as it is classically developed, where the principal unknown is the magnitude of the stress singularity. However, from a design perspective, this framework uniquely states whether the bi-material joint is singular or not and, in the case of a singular field, it is informed about the severity of the singularity (eigenvalue) and the associated stress form (eigenvector).