Mathematical Sciences: Dynamical Problems in Piezocomposites for Transducer Applications

9622927 Berlyand The investigator will study a variety of mathematical problems which arise in the study and design of polymer piezoceramic composites for transducer applications. He will attempt to use methods of homogenization theory in combination with the Bloch-Floquet expansion to understand various types of resonances for wave propagation in composite materials consisting of an array of piezoceramic rods embedded in a polymer matrix. He will study two main classes of such problems. The first is where the acoustic wavelength is comparable with dimensions of the composite microstructure (long wave approximation). Here he will attempt to find the dependence of the quality of electromechanical power conversion on the frequency and the volume fraction of the piezoceramic phase and explain theoretically existing numerical data for the thickness-mode resonance. This includes the 'backing effect, i.e., the dependence of the resonance frequencies on the stiffness of the backing medium. The second main class of problems that the investigator will study is the high-frequency regime when the acoustic wavelength is comparable to the microstructure scale. The main effort here is to extend homogenization ideas to the range of frequencies where the composite material exhibits significant 'cross-talk' between piezoceramic rods. He will also study the transitional behavior between these two frequency regimes. The proposed work lies at the frontier between mathematics and composites, and the investigator is planning to support theoretical results by experimental and numerical data obtained at the Penn State Materials Research Lab. %%% A composite material is a mixture of two or more single phase materials (constituents) whose properties are far better than that of each constituent. That is why composite materials are widely used in many areas of modern technology and manufacturing. In particular, piezocomposites are the principal elements of modern acous tic transducers. High-frequency acoustic transducers are used for ultrasound medical imaging and non-destructive testing of damaged materials. Low- frequency transducers are used for underwater acoustics (so-called hydrophones) as well as for finding fish, tracking vessels and deep-sea seismology. Rapid advances in mathematical sciences within the last 30 years open the possibility of understanding the general principles and relationships linking the overall properties of such composite materials to the properties of their constituents. This in turn makes it possible to design more efficient, lower cost transducers. However, progress has been hampered by expensive and time consuming experiments and by the enormous complexity of the phenomena. Huge improvements in the ability to measure the properties of composite materials have led to better characterization of their properties. However, this precision of measurement has not been matched by corresponding improvements in the mathematical theory, which could guide the experimental study and industrial design of the transducers. This project will involve devising and improving mathematical techniques and methods for the study and design of piezoceramic composites for transducer applications. The investigator will attempt to use modern mathematical techniques and tools to understand various types of resonances for wave propagation in polymer piezoceramic composites. This study will provide practical recommendations for increasing the sensitivity and resolution range of the transducers. ***