Mathematical Sciences: Dynamical Problems in Piezocomposites for Transducer Applications

9622927研究人员将研究传感器用聚合物压电复合材料的研究和设计中出现的各种数学问题。他将尝试使用均匀化理论的方法结合Bloch-Floque展开式来理解由嵌入在聚合物基质中的压电陶瓷棒阵列组成的复合材料中波传播的各种类型的共振。他将研究这类问题的两大类。第一个是声波波长与复合微结构的尺寸相当的地方(长波近似)。在这里，他将试图找出机电功率转换的质量与压电陶瓷相的频率和体积分数的关系，并从理论上解释现有的厚度模式共振的数值数据。这包括“背衬效应”，即共振频率对背衬介质的硬度的依赖。研究人员将研究的第二类主要问题是当声波波长与微结构尺度相当时的高频区域。这里的主要工作是将均化思想扩展到复合材料在压电陶瓷棒之间表现出显著的“串扰”的频率范围。他还将研究这两个频率制度之间的过渡行为。这项拟议的工作位于数学和复合材料之间的前沿，研究人员计划通过在宾夕法尼亚州立大学材料研究实验室获得的实验和数值数据来支持理论结果。%复合材料是两种或两种以上单相材料(成分)的混合物，其性能远远好于每一种成分。这就是复合材料被广泛应用于现代技术和制造的许多领域的原因。特别是，压电复合材料是现代声学TiC换能器的主要元件。高频声学换能器用于超声、医学成像和无损检测受损材料。低频换能器用于水下声学(所谓的水听器)以及发现鱼、跟踪船只和深海地震学。在过去的30年里，数学科学的快速发展使人们有可能理解将这种复合材料的整体性质与其组成成分的性质联系起来的一般原理和关系。这反过来又使得设计更高效、更低成本的换能器成为可能。然而，昂贵和耗时的实验以及现象的巨大复杂性阻碍了进展。复合材料性能测量能力的巨大进步导致了对其性能的更好表征。然而，这种测量精度并没有得到相应的数学理论的改进，这些改进可以指导换能器的实验研究和工业设计。该项目将涉及设计和改进用于传感器应用的压电陶瓷复合材料的研究和设计的数学技术和方法。研究人员将尝试使用现代数学技术和工具来了解聚合物压电陶瓷复合材料中波传播的各种类型的共振。这项研究将为提高传感器的灵敏度和分辨率范围提供实用的建议。***

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. ***