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logo SMC logo SMC  DYNA logo SMC  DYNA  SHM Static periodic homogenization of piezocomposite transducers


Arnaud Deraemaeker

Keywords Piezoelectricity, piezocomposite transducers, static periodic homogenization, macro fiber composite


Prof A. Benjeddou (Supméca Paris)

Dr H. Nasser (CRPHT Luxembourg)



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 d31 and the d33 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 d33 MFCs and P2 for d31 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 ( 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 post-processed. Figure 3 (bottom) shows the evolution of the equivalent d31 and d33 coefficients for P2 and P1-type 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 P2-type MFCs and evolution of the d31 and d32 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 multi-layer 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 P1-type transducer. Computation of the static response using multi-layer 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