摘要
压电材料作为典型的多场耦合材料已经广泛应用于各种智能结构以及传感器和驱动器等器件中。随着微机电和纳机电系统的发展,薄膜或层状介质形式的压电材料越来越得到重视。准晶材料由于其独特的准周期结构以及声子场和相位子场的耦合特性也引起了广泛的研究兴趣。大量的研究开始采用压痕实验技术进行多场耦合材料的性能测试。准确的压痕响应理论分析将十分有助于设计压痕实验,也是定量分析材料常数的必要依据。本文工作基于压痕技术目前应用于多场耦合材料性能测试的实际需求,进行了相应接触问题的理论建模和求解。
基于线弹性力学中混合边值问题的求解体系,本文研究了横观各向同性压电材料以及一维六方和二维六方准晶材料的轴对称接触问题。首先基于Hankel积分变换或势理论方法推导出表面作用各种点荷载的Green函数解,接着将其代入相应的接触边界条件推导出控制接触问题的Fredholm积分方程,最后求解积分方程得出压痕响应的解。
本文研究了压电薄膜-弹性基底系统和压电层状介质的无摩擦接触问题,压电材料半空间的完全粘附接触问题以及一维和二维准晶材料半空间的无摩擦接触问题。本文考虑了平底圆柱、球形和锥形三种典型形状压头与材料的接触,在分层模型中还考虑了各种层间界面模型。对于半空间材料的接触问题给出了精确解析解,对于分层介质接触问题进行了数值求解。数值算例分析了各种条件对压痕响应的影响,对比了层状介质接触问题的解与半空间材料的解以及完全粘附接触问题与无摩擦接触问题的解,绘出了准晶体在压头作用下的应力云图。
本文得到的薄膜-基底系统接触问题解和层状介质接触问题的解可以直接用来指导压痕实验；而半空间材料的接触解还可以作为标准解以校核其它模型(如有限厚度层模型、层状介质模型)接触问题数值求解或近似求解的结果。本文给出的准晶体与刚性压头无摩擦接触时位移和应力各分量的精确解析表达式将对进一步分析和理解声子场-相位子场的耦合特性提供一定的帮助。
The piezoelectric material, which is one of the typical multi-field coupling materials, has been extensively used in intelligent structures or devices such as sensors and actuators. With the development of the MEMS and NEMS, piezoelectric materials in the form of thin film or layered medium have become more and more important. Quasicrystal material is also attracting increasing research interests due its quasi-periodic structure and the phonon-phason coupling property. Recently, indentation technique has been widely applied in testing piezoelectric and quasicrystal materials. Theoretical studies on indentation responses of these materials would be significant to the experimental study of indentation and quantitative analysis of the material constants. In view of the application of indentation technique in characterization of material property for multi-field coupling materials, this dissertation presents the modeling and solving procedure of relevant contact problems of piezoelectric and quasicrystal materials.
By the appropriate methods for solving mixed boundary value problems in linear elasticity, this dissertation studies the axisymmetric contact problems of transversely isotropic piezoelectric material, one-dimensional hexagonal and two-dimensional hexagonal quasicrystal materials. First of all, the Green's functions of the half-space subjected to point sources on the surface are derived by applying Hankel integral transform or using potential theory method. By substituting the Green's function into the contact boundary conditions, the Fredholm integral equation, which serves as the final governing equation of the contact problem, is then derived. Finally, the solutions of the indentation responses are obtained by solving the Fredholm integral equation.
This dissertation studies not only the frictionless contact problems of the piezoelectric film-elastic substrate system, piezoelectric layered half-space, one-dimensional and two-dimensional quasicrystal half-space, but also the adhesive (no-slip) contact problem of piezoelectric half-space. The rigid flat-ended cylindrical, conical and spherical indenters are considered in contact with the materials, and several interface models are used to characterize the interface between the film and the substrate or layers. For half-space model, the analytic solution of the contact problem is presented. However, the contact problem of layered medium is solved numerically. In the numerical examples, we analyze the influences of various parameters on the indentation responses, compare not only the solution of film-substrate system or layered medium with the solution of the half-space material, but also compare the solution of adhesive contact problem with the solution of frictionless contact problem, and plot the stress contour of the quasicrystals under indentation.
The contact solutions of the film-substrate system and layered medium obtained in this dissertation could directly guide the indentation test. The solutions of half-space material can not only serve as benchmarks for numerical or approximate solutions of thin-film model or layered model, but also have important value for experimental studies. The exact and complete analytic expressions for the displacements and stresses of quasicrystals under indentation presented in terms of elementary functions in this dissertation would be very helpful to the studies and understanding of the coupling property of phonon field and phason field.
引文
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