Optimal design of 1-3 composite piezoelectrics

1 I n t r o d u c t i o n Piezoelectric transducers have been employed as sensors and transmitters of acoustic signals in ultrasound medical imaging, nondestructive testing and underwater acoustics (hydrophones). In this paper we consider the optimal design of a hydrophone composite consisting of parallel piezoceramic rods that are embedded in a porous polymer matrix. The hydrophone is assumed to operate in the low-frequency range and hence its behaviour can be described in the quasistatic limit. One may ask why is pure piezoceramic not used since it is the only material with piezoelectric properties? The basic problem is that under hydrostatic load, the anisotropic piezoelectric response of pure piezoelectric is such that it has poor hydrophone performance characteristics such as hydrostatic piezoelectric coefficient d h , voltage coeJ:ficien~ gh = dh/~33 (where g33 is a dielectric constant in the x3-direction), the hydrophone figure of meri t dhg h , and the electromechanical coupling factor k h = ~ (where s h is a dilatational compliance). It was shown in a number of papers (see e.g. Klieker et al. 1981; Newnham 1986; Newnham and Ruschau 1991; Ting et al. 1990) that composites with high hydrophone sensitivity can be achieved by making a composite consisting of piezoceramic rods in a soft polymer matrix. Figure 1 schematically depicts such a "1-3 piezocomposite" when exposed to a hydrostatic pressure field. An appropriately designed piezocomposite is capable of converting an applied hydrostatic field into a predominantly tensile stress on the rods, thus enhacing all of the hydrophone characteristics. Using simple models in which the elastic and electric fields were taken to be uniform in the different phases, Haun and Newnham (1986), Chan and Unsworth (1989), and Smith (1991, 1993) qualitatively explained the enhancement due to the Poisson's ratio effect. Smith (1991) proposed that even greater enhancement in hydrophone characteristics can be achieved by using matrix materials with negative Poisson's ratio. A more sophisticated analysis was recently given by Avellaneda and Swart (1994) using the so-called differential-effective-medium approximation. It was found that the performance of the composite depends significantly on the properties and the volume fraction of the rods, and on the mechanical properties of the polymer matrix. For example, the use of a matr ix with negative Poisson's ratio or a porous matrix increases the sensitivity of the hydrophone by an order of magnitude. This paper extends the analyses of Avellaneda and Swart (1994). Our main contribution is that we depart from the assumption of isotropy of the matrix, and require only transverse isotropy of this material. We treat the matr ix material itself as a composite; it is assumed to be prepared from a polymer with given properties, weakened by an optimal arrangement of pores. The mierostructure of the matr ix material is an additional control in the problem that we study. As we will see, the optimal matr ix is highly anisotropic, with a large ratio of the minimal and maximal eigenvalues of the stiffness tensor. Here we only give a summary of the results and a brief description of the method. The detailed derivation of our results will be published elsewhere (Gibiansky and Torquato 1997). The paper is organized as follows. In Section 2, we give a brief summary of the formulae that describe performance characteristics of hydrophones. In Section 3, we discuss the design parameters of the problem. Section 4 presents the results of numerical optimization. Section 5 summarizes the results of the paper. 2 H y d r o p h o n e p e r f o r m a n c e c h a r a c t e r i s t i c s In this section we give a brief summary of the formulae that describe piezoelectric hydrophones (see e.g. Smith 1991, 1993; Avellaneda and Swart 1994). The object under study is a composite of PZT-ceramic rods in a porous polymer. If the wavelength of the applied field is much larger than the spacing between rods, the behaviour of a composite can be characterized by the averaged equations of piezoelectricity, i.e.