## 2 Modeling of Strain Actuation on Curved Beams
### FE Modeling  
The finite element modeling approach was implemented to investigate the effect of the curvature on the transmission performance of a piezoelectric patch bonded onto a curved structure. The complex geometry of the architecture and the presence of curved interfaces led to us considering the finite element simulation approach, which is also appropriate for more complex and realistic applications, such as the protection of aircraft leading edges, inlets, and nacelles from ice [21]. The model hereafter illustrated was conceived to meet the following requirements:  
1. To determine a relation between the strain and displacement transmitted by an expanding piezoelectric element and by pin forces;
2. To describe the strain displacement field along the radial and tangential coordinates;
3. To investigate the effects of specific parameters such as the piezo device/structure thickness ratio, the angular position of the piezo device, and its tangential extension;
4. To estimate the stress-strain field within the bonding layer;
5. Finally, to ensure adequate computational lightness. The structural system under investigation is shown in Figure 2.  
In line with these requirements and considering the plane nature of the problem, 2D finite elements (CQUAD) were considered for the piezo device, the structure, and the boundary layer for the parameterization process. The system is constituted by a 90 degree PVC arc with a piezo patch bonded on the outer side. A radius of 75 mm was assumed for the mean line of the arc; the out-of-plane dimension was set at one-quarter of a millimeter, at least 1 order thinner than the other dimensions, to make negligible any output variation along this direction. Four elements were placed along the thickness of the piezo patch and the beam structure to represent the displacement by a higher-order polynomial. A detailed view of the mesh close to the zone of connection between the structure and the piezo device is shown in Figure 3. The physical connection among the piezo patch, the boundary layer, and the beam was obtained through rigid links (RBE2) among coincident nodes. The perfect bonding condition was simulated by removing the adhesive layer and connecting the nodes of the piezo patch directly to the structure. Figure 3: Mesh details for the piezo patch and the structure below. Figure 2: Structural system of a cylindrical beam with a piezoelectric patch. The static behavior of the system was investigated by considering two different types of actions:  
* The thermal load: The expansion (contraction) caused by the application of a voltage between the top and bottom faces of the piezo device was simulated by assigning to the piezo material a fictious expansion coefficient, \(\alpha\_\)PZT: \[\alpha_{PZT}=\frac{d_{31}}{t_{p}}\] (10) This approach approximates the behavior of the system well when electromechanical cross-related effects are negligible; that is to say, far from resonance peaks. * The pin forces: Three forces were placed at the interface between the piezo device and the structure--on the first and last nodes, oriented in opposite ways along the tangential direction, and on the middle node, oriented along the radial direction. This last radial force was also considered to allow the force balance along the radial direction, which is already intrinsically satisfied by the thermal loading. The scheme in Figure 4 illustrates the relation between the two opposite tangential forces, \(F_{t}\), and the radial one, \(F_{r}\). In practice, the equilibrium requires that the radial force is 2 times the sine of the halved tangential extension, \(\theta\), of the piezo device. The scope of the work is to find a relation between the results achieved by the above load conditions.