Numerical modeling approaches

In the past decades, numerical modeling approaches have been extensively investigated to simulate piezoelectric structures. The numerical modeling path

MODELING OF MFC TRANSDUCERS INTEGRATED INTO A THIN HOST STRUCTURE 33

provided numerous possibilities: Early FEM formulation derived from the classical laminated plate theory is reported in [16] for active vibration control of composite plates with distributed piezoelectric sensors and actuators. The mesh-free model of piezoelectric composite structures has been reported in [19].

The method demonstrated a higher convergence rate than standard First-Order-Shear-Deformation (FOSD) FEM method. A shallow shell is presented in [17], which deals with the piezoelectric induced bending and twisting deformation of laminated composite structures. A rigorous multi-scale approach, based on the asymptotic expansion homogenization method, has been developed in [141] to analyze the behavior of laminated structures with integrated MFC sensors and actuators.

FEM methods are certainly the most widely used approach in different research tracks and application of piezoelectric transducers [11, 18, 21, 23, 157–160].

The piezoelectric effect is simulated through strain field so that the mechanical and electric domains are strongly coupled together in FEM formulations. It has been reported in [22] that the two-dimensional FEM models of MFC transducers, based on the Kirchhoff or the Mindlin-Reissner assumptions, agree well with both experimental results and 3D FEM models. Regarding to the anisotropic transducers such as MFC transducers, a material-structural coordinates transformation regarding MFC transducers is required in the mentioned FEM methods, as shown in Figure 2.20. That is because the properties of the transducers are defined in the material coordinates (1, 2), which is different from the structural coordinates (x,y). Recently, this transformation is taken into consideration in a FOSD-based FEM in [23].

2 1

x y

Host plate MFC

Figure 2.20: Material-structural coordinates transformation

In FEM methods, the discretizations of the electric field and the displacement field are independent with each other [161,162]. An accurate electric assumption for the piezoelectric elements is essential for ensuring that both the mechanical displacement and the electric potential field converge to the exact solutions.

Hybrid finite element methods have been developed to improve the discretization of the electrical field [20,162–165], but the modeling complexity is increased.

The conventional assumption of the electric field distribution on the transducer might be too strong to simulate the MFC transducers. For example, the effective distribution of the electric field on an MFC-d33 transducer is shown in Figure 2.7.

A uniform electric field leads to an over-estimation of the performance of the MFC-d33 transducer. A detailed description of the transducer in modeling would be too complicated because of the multilayer constitution. An alternative is to correct the uniform electric field assumption in modeling based on the effective electric field distribution shape of MFC transducers [44,166,167].

The performance of a piezoelectric transducer on a host structure also highly depends upon the quality of the adhesive bond layer, which transfers the stress/strain between the transducer and the host plate, as shown in Figure 2.21.

Assuming that the stain in the thickness-wise of the transducer is linear, the adhesive layer undergoes shear deformation when the transducer functions as an actuator. At last, the deformation of the host plate is smaller than the transducer. A similar situation can also be found when the transducer is used as a sensor. This effect is called shear-lag effect and cannot be negligible in experimental studies [150,168–171]. However, the adhesive layer between the host structure and transducer is commonly assumed to be perfect in numerical simulations because of the lack of material information. The corrections of both effects must be considered in the modeling in order to achieve an experimental validation.

Host plate Contraction MFC

Expansion Bond layer

z

x

Figure 2.21: Strain transfer mechanism through adhesive bond layer

Furthermore, an obvious drawback of FEM methods concerning dynamic applications is that they generate large-scale models because a sufficiently fine mesh is required to discrete both the host structure and the integrated transducers. For plate-type structures, it is possible that the host structure and the integrated transducers share the same mesh to avoid the mesh coupling between them. As a result, the material properties of the transducers should be homogenized together with the host structure at the location of the transducers, and the adhesive layer is neglected in most of cases. Thus, the size of the model is mainly determined by the mesh of the host structure. A model order

CONCLUDING REMARKS 35

reduction needs to be performed on the FEM models for reducing computational time in dynamic application.

To sum up, MFC transducers must be simultaneously modeled together with host structures in FEM methods so that the bending coupling between them can be formed. The piezoelectric coupling matrix depends on both the placement and the properties of these transducers, and the structural model is also influenced by them too. For example, the piezoelectric fibrous orientation leads to changes in both of them. Thus, it is challenging to efficiently evaluate the influences of these parameters directly in dynamic application, where a reduced-order model is always required. The conventional model order reduction techniques, often used for designing control solutions such as Krylov subspace projection [24] and balanced truncation [25], transform the original model into an equivalent vector space, so that these useful physical parameters are not accessible anymore after the reduction.

A semi-analytical modeling approach based on the substructuring concept would efficiently evaluate the performance of MFC transducers in dynamic application.

We can divide the system into subcomponents including the MFC transducers, and the size of each subcomponent can be efficiently reduced. The dynamics of the overall system is obtained by assembling the reduced-order models of all the subcomponents. The closed-form analytical solutions can be used to express the direct and inverse piezoelectric couplings. This concept retains the structure of full order system models so that each subcomponent can be easily updated including the placement and the properties of MFC transducers. Moreover, it is also essential to ensure the stability of the reduced-order model, for designing active control algorithms and performing real-time simulations.