摘要
功能梯度板中的弹性波的传播在工程实际中有着广泛应用，其分析方法有常见的传统方法的推广和新的求解技术的应用。为了精确的计算波速和弹性板变形，我们分别采用了分层法、Frobenius法和同伦分析法，对结果进行了互相验证，并对这些方法进行了比较。我们也分析了几种常见的材料性能变化模式如指数和幂级数等，对结果进行了讨论，发现表面波的两种变化模式在功能梯度材料中的效应是不同的，在应用中一定要针对具体问题和模态选择材料。我们也发现，分层法虽然概念上简单，但收敛速度比较慢；Frobenius法可以求出精确解，但求解过程较繁，而且对数值计算过程的精度要求也比较高；同伦法可以求得包括Frobenius解的一般解，而收敛速度和精度可以根据选择适当的参数来调整。我们从计算方法和材料性能变化两个方面研究了功能梯度材料板中表面波的传播，求得了相应的速度和变形，为实际应用提供了重要的理论结果。
在第一章，我们简单回顾了功能梯度材料的基本概念和发展过程，并对用于功能梯度结构的方法作了介绍和分析。针对功能梯度板中的表面波问题，我们提出了基于分层法、Frobenius精确解和同伦法的求解思路。
在第二章和第三章，利用均匀层状模型和Frobenius级数方法分析了功能梯度板中表面波(Rayleigh波)的传播。通过对已有解析解的情形进行分析，说明了均匀层状模型和Frobenius方法在处理功能梯度板中表面波传播问题的可行性。我们还对弹性模量和质量密度按指数变化和多项式形式变化的情况进行了数值模拟。
在第四章，采用同伦分析方法对功能梯度板中表面波的传播进行了研究。首先采用幂函数作为基函数分析了该问题，并且得到了与前面两种方法相同的结果。考虑到幂函数的收敛速度和收敛半径的限制，我们又利用了均匀板中表面波的解为四个指数函数的组合，选取幂函数和指数函数的乘积函数作为基函数再次分析上面的问题，得到了很好的结果。
在第五章，对上面几种方法的有效性进行了比较。我们发现Frobenius级数方法仅是同伦分析方法的一个特例。
在第六章，从对功能梯度板中表面波的分析结果的总结，得出了一些对表面声波谐振器和结构抗震有指导意义的结论。
The propagation of elastic waves in functionally graded plates has wide applications in engineering. The analytical methods for the consideration of finite solids with structural complications have been subjecting to extensive research efforts. To obtain the velocity and deformation accurately, layered model, Forbenius method, and homotopy analysis method are employed to study the propagation of surface acoustic waves in functionally graded plates. Comparisons with results from different methods are made for validation and ranking. Cases with polynomial and exponential material property variation schemes are calculated with these methods, which show that the influences to the velocity and displacement model are different. In applications, the grading scheme must be chosen according to the displacement model. Also, we found that the layered model is simple in concept and implementation, but its convergence is slow. Although we can get the analytical solutions from Frobenius method, the derivation is complicate and its requirement for precise numerical evaluation is difficult to meet. Homotopy analysis method, on the other hand, can obtain the general solutions which include the Frobenius solution, and the rate of convergence can be improved by properly selecting parameters. We study the surface acoustic waves in functionally graded plate with the solution methods and property variation schemes, and obtained the velocity and deformation. This study provides a theoretical foundation for applications in engineering.
In the first chapter, we briefly reviewed the concept and development of functionally graded material, and introduced the methods for the analysis of functionally graded structures. For the surface acoustic waves in functionally graded plates, we provide the layered model, Frobenius method, and the homotopy analysis method.
In the second and third chapters, we analyzed the propagation of surface acoustic waves (Rayleigh waves) in functionally graded plates with the layered model and Frobenius method, respectively. Solutions from plates with exponential property grading schemes are obtained from the layered model and Frobenius method and comparisons with known analytical results are made to validate and evaluate the methods. We also analyzed cases with polynomial material property variation schemes.
In the fourth chapter, we employed the homotopy analysis method (HAM) to analyze the propagation of surface acoustic wave in functionally graded plate. First, power series are used as the primary basis function, and obtained the same results as from the layered model and Frobenius method. To improve the rate and radius of convergence of power series, we made use of the analytical solutions of homogeneous plates and analyzed the problem again with the combination of power function and exponential function as primary basis function.
In the fifth chapter, through comparison of the three methods, we found that the solution of Frobenius method is included in that of the HAM.
In the sixth chapter, by analyzing the propagation of surface acoustic waves in functionally graded plates, we obtained some conclusions significant for resonator and other engineering applications.
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