## 2 Materials and Methods
### Specim Mechanical Design
#### 2.2.2 Vhcf Specim Design: FE Model Development and Analyses  
This section revises the key passages that are adopted for preparing and analyzing a suitable Vhcf specimen, in accordance with [43], and exposes the current numerical outcomes that have been extracted to approach the experimental ultrasonic/low-frequency test campaign. The Vhcf specimen was designed to work in resonance conditions within the UFTM operative range. The development of FE models exploited the Ansys APDL capabilities. Axisymmetric four-node bi-dimensional elements were adopted for preparing the models. Two classes of FE models were prepared: (i) a global model for studying the overall structural behavior, keeping the number of elements, nodes, and degrees of freedom low, and (ii) a local model to study the effects introduced by the removal of the stress singularity in detail. Indeed, for the sake of clarity, the FE global model works with the presence of a singular joint, whereas the removal of this feature takes place separately, within the local description. This approach, called the sub-modelling technique or the global-local technique [57], is particularly helpful for investigating local structural details such as fillets or unions of elements that require a specific modelling strategy for their analysis. The global model, presented in Figure 4, is constituted by 2500 elements, and aims to simulate the overall testing configuration (i.e., the horn and specimen) under resonance conditions. Moreover, the global description is adopted for design purposes, and it is necessary to define the local boundary conditions to apply to local models. The main steps to prepare the global model are as follows:  
1. The numerical model of the titanium horn is first analyzed through a modal analysis under free-free conditions to collect the closest natural frequency and mode shape to the UFTM operative range. An operative frequency of \(f_{n}=20,195\) Hz is thus selected and adopted for design purposes. 2. The 1D elastic wave equations [58] reported in Equation (4) permit the definition of the specimen total length \((L_{tot})\), where \(f_{n}\) is the frequency of interest and \(E,\rho\) are the mechanical properties of the aluminium. This calculation results in \(L_{tot}=125.5\) mm: \[L_{tot}=\frac{1}{2f_{n}}\sqrt{E/\rho}\] (4)  
Figure 4: Vhcf specimen—Global model [40]. A frequency response analysis (FRA) is run at \(f_{n}=20,195\) Hz, where the current numerical model is composed using the \(L_{tot}\) of the titanium horn and the aluminum bar, with a displacement excitation at the horn base in the range of 2.2-18 \(\upmu\)m, which corresponds to the capability of the UFTM. Longitudinal stress distributions are collected along the aluminium specimen axis and are depicted in Figure 5. This assessment was useful and necessary to explore the stress magnitudes applied to the whole specimen. 4. A 0.3 mm thick adhesive layer is inserted into the mechanical system, thus dividing the aluminium bar in two separated components, now adhesively joined. The adhesive is placed at a suitable distance in order to impose the desired stress magnitude on the adhesive layer. In this work, the stress range from 5.5-46 MPa was selected. In this way, the \(L_{tot}\) is divided into \(L_{1}\) = 114.15 mm and \(L_{2}\) = 11.35 mm, as shown in Figure 5.
5. A final numerical re-assessment is performed through an FRA. Once the VHEF global model was prepared and the operative adhesive stress interval was defined, the local description was prepared by including the suitable geometrical modification for removing the stress singularity. This modification is based on the procedure elaborated in Section 2.2.1. Indeed, as presented in Table 2 and Figure 3, many configurations can avoid singularities and, in general, all of them are good candidates for testing specimens. As mentioned in Section 2.2.1, the selected adherent-adhesive configuration is 43.5\({}^{\circ}\) and 90\({}^{\circ}\).