Static periodic homogenization of piezocomposite transducers 
Contact 

Keywords  Piezoelectricity, piezocomposite transducers, static periodic homogenization, macro fiber composite 
Collaborations 
Prof A. Benjeddou (Supméca Paris) Dr H. Nasser (CRPHT Luxembourg) 
Motivation 
Piezoelectric actuators and sensors have been widely used in active vibration control applications and structural health monitoring. PZT ceramics are commonly used due to their good actuation capability and very wide bandwidth. The major drawbacks of these ceramics are their brittle nature, and the fact that they cannot be easily attached to curved structures. In order to overcome these drawbacks, two techniques have been developed: (i) thick film deposition of PZT which requires that the part be heated at 900C for sintering, and (ii) using packaged PZT composites which can be glued on curved structures. This work focuses on the second alternative. A typical piezocomposite transducer is made of an active composite layer sandwiched between two soft thin encapsulating layers (Figure 1). The packaging plays two different roles: (i) applying prestress to the active layer in order to avoid cracks, and (ii) bringing the electric field to the active layer through the use of a specific surface electrode pattern. Figure 1 : Layout of macro fiber composites (MFC) using the d_{31} and the d_{33} piezoelectric modes and associated representative volume element (RVE) for periodic homogenization 
Static periodic homogenization of MFCs 
The active laye of Macro Fiber Composites (MFCs) are made of rectangular fibers surrounded by an epoxy matrix. There exists two main types of MFCs based on the piezoelectric mode used (P1 for d_{33} MFCs and P2 for d_{31} MFCs). For each type of MFC, we have developed a methodology to find equivalent homogeneous properties of the composite active layer. This requires to define a representative volume element (RVE, Figure 1). Six local problems, defined in Figure 2 are then defined on the RVE in order to compute the equivalent mechanical, piezoelectric and dielectric properties of the active layer. Figure 2 : six local problems on the RVE to compute the equivalent homogeneous mechanical, piezoelectric and dielectric properties of P1 and P2 MFCs 
Finite element modeling 
The six local problems are solved using the piezoelectric module in the Structural Dynamics Toolbox (www.sdtools.com). Figure 3 shows the solution of the first local problem (applied voltage and faces with normal in the plane blocked) for both P1 and P2 type MFCs. From the solutions of the six local problems, the equivalent properties can be postprocessed. Figure 3 (bottom) shows the evolution of the equivalent d_{31} and d_{33} coefficients for P2 and P1type MFCs as a function of the volume fraction of fibers (note that MFCs have a volume fraction of rho=0.86) Figure 3 : Finite element solution of the first local problem for P1 and P2type MFCs and evolution of the d_{31} and d_{32} piezoelectric coefficients as a function of the volume fraction of fibers. 
Modeling of plates equipped with MFCs 
Now that the equivalent properties are known, it is possible to use the multilayer plate elements of the Structural Dynamics Toolbox to model plate structures equipped with MFC transducers. An example is shown in Figure 4. The active layer properties are derived from the 3D finite element computations performed on the RVE. Figure 4 : Example of a plate equipped with a MFC P1type transducer. Computation of the static response using multilayer plate elements in SDT 
Selected publications 
[1] A. Deraemaeker, H. Nasser, A. Benjeddou, and A. Preumont. Mixing rules for the piezoelectric properties of Macro Fiber Composites. Journal of Intelligent Material Systems and Structures, 20(12):1391–1518, 2009 [2] A. Deraemaeker and H. Nasser. Numerical evaluation of the equivalent properties of Macro Fiber Composite (MFC) transducers using periodic homogenization. International Journal of Solids and Structures, 47:3272–3285, 2010 