## 2 Modeling of Strain Actuation on Curved Beams  
### Theoretical Models  
The first models concerning the strain actuation from a piezoelectric ceramic transducer to a straight beam were proposed by [PERSON] and [PERSON] in 1987 [23]. For bending excitations, they considered the classical butterfly diagram for the strain within the beam thickness and constant deformation along the actuator thickness. This latter assumption is reasonable as the piezo device is significantly thinner than the driven structure. The model took into account the presence of the adhesive layer between the two elements and reached important conclusions, among which were an expression for the quality of the bonding and an expression for the transmitted strain along the contact zone. The formulas are synthetically reported herein:  
\[\text{Actuated Strain Function}\quad\frac{\varepsilon_{t}}{\Lambda}=\frac{6}{ \psi+6}-\frac{6}{\psi+6}\frac{\text{ch}\Gamma x^{\prime}}{\text{ch}\Gamma} \tag{1}\]  
\[\text{Adhesive Layer Quality}\quad\Gamma^{2}=\left(\frac{L}{2}\right)^{2}\frac{G_{a }/E_{p}}{t_{a}t_{p}}\bigg{(}\frac{\psi+6}{\psi}\bigg{)} \tag{2}\]  
The main conclusions of the theory may be summarized as follows, directly linked to Formulas (1) and (2) above, respectively:  
1. The strain transmitted to the structure is made of a constant and an exponential term, which vanishes for values of the \(\Gamma\) parameter higher than 10, and the adhesion is called \"perfect\";  
Figure 1: Structural system of a cylindrical beam with a piezoelectric patch. 2. For usual values of the physical and geometrical characteristics, the adhesion quality constant is always higher than 10. Indeed, the first point should be better clarified by adding that at the extremities of the contact region, the structural deformation is always null (boundary condition); for larger and larger values of \(\Gamma\), in the region where the strain is different from a constant value (left term), the expression results are smaller and smaller. This formulation loses its validity as the thickness of the piezoelectric element approaches the structural one; in such cases, the hypothesis of considering the strain along the vertical direction of the actuator to be constant cannot hold anymore. Different theoretical approaches have tried to overcome this limitation, although assuming that the adhesive layer could be always considered perfect, or in other words that the bonding quality always attains high values. This further hypothesis is reasonable if standard physical and geometrical values are considered, as reported before. Among the different proposals, the one introduced by [PERSON] and [PERSON] in 1995 is herein reported. The transmitted strain, which is now constant along the contact region extension and is now a scalar, assumes a slightly different form, as follows: \[\text{Actuated Strain}\quad\frac{\varepsilon_{t}}{\Lambda}=\frac{6(\vec{t}^{2 }+\vec{t})}{\phi\vec{t}^{3}+6\vec{t}^{2}+12\vec{t}+8}\] (3) It should be noted that by moving from the favorable assumption of a perfect bonding layer, these models allowed deal with the problem from a 2D standpoint, so that the Poisson module does appear. Finally, it can be seen how the novel resembles the former expression as the piezoelectric thickness vanishes, according to the classical rules of polynomial fractions, and the Poisson modules are equal to each other. All of these expressions are retrieved for planar systems, without any curvature. This eventuality may cause the models to express results that are very far from reality for many applications, such as for the use of piezoelectric actuators as dynamic exciters of curved beams or plates. This may be the case for wing sections and for the implementation of compact mechanical de-icing systems. The reported models may be used as a support to extrapolate some hybrid numerical-analytical model on the bases of the major outcomes recalled here.