原子力显微镜 (AFM：Atomic Force Microscope) 自被发明以来，一直在不断被完善。从最初的静态 AFM 发展到如今各式各样的动态 AFM，已经成为纳米科学和纳米材料领域研究的有力工具。尽管 AFM 在成像能力方面表现的已足够出色，并得到了广泛的应用，但其在定量表征方面仍然存在众多挑战。而同一种材料在纳米尺度下的性能，与在宏观尺度相比，存在很多差异。材料的组成是影响其力学性能的重要因素，微结构的细节对其预期寿命也会产生很大影响。随着纳米技术的飞速发展，以及各类新型材料的不断应用，在纳米尺度下对材料的各类性能参量特别是是力学性能进行表征显得尤为重要，包括对材料摩擦、表面黏弹性、黏附等性能参量的表征。此外，新材料的不断出现，也对材料内部结构、缺陷、杂质等的无损检测提出了要求。

    在此背景下，本论文针对 AFM 分辨率的提高和材料性能的振动力学反演两大关键科学问题展开，围绕 AFM 悬臂梁结构设计和利用 AFM 对材料性能相关力学参量定量表征两个方面，通过理论分析和数值求解，采用反问题的方法开展研究工作。首先基于共振放大效应，增强了 AFM 的高阶谐波信号；然后建立了 AFM 振动信号与样品弹性模量、基体内部填充结构弹性模量等一些力学性能之间的定量关系，为利用 AFM 表征材料相关力学参量提供了理论基础。主要完成的工作如下：

     (1) 增强了 AFM 的高阶谐波信号。在 AFM 中，样品性能信息更多地包含在高阶谐波信号中。然而均匀矩形悬臂梁不存在恰好为基频整数倍的高阶频率，因而其高阶谐波信号被抑制，从而导致样品的性能难以被探测。本文以增强高阶谐波信号为目标，建立并求解了悬臂梁的运动方程。设计了阶梯型悬臂梁结构和 T 型悬臂梁结构，调谐其固有频率，使其第二阶或第三阶自然频率为基频的整数倍，基于共振放大效应，同时对前两阶或前三阶模态进行激发，实现了高阶谐波信号的增强。同时给出了可供参考的悬臂梁尺寸图和表。

   (2) 提出并求解了反问题：如何依据压痕深度–载荷曲线，同时获取薄膜等效杨氏模量和厚度。在压痕实验中，薄膜/衬底双层异质结构的等效杨氏模量会随压痕深度的变化而变化。在以往的研究中，只存在一个未知量，即薄膜等效杨氏模量。然而在实际情况中，薄膜厚度的测量有时是非常困难的，甚至无法被测量。因此在反问题中将薄膜的厚度和等效杨氏模量均视为未知量并对其进行了求解。

     (3) 提出了一种定量测定样品杨氏模量和泊松比的有效方法。建立了 T 型悬臂梁在接触模式下，弯扭耦合非线性振动理论模型。随着接触刚度的增加，接触共振频率在高阶模态下出现分岔现象。利用共振频率分岔点解耦了等效杨氏模量表达式中的泊松比和杨氏模量。

     (4) 分析了超声波遇到圆柱形以及椭圆柱形障碍物时发生散射所引起的相位变化，建立了 AFM 悬臂梁探针与样品表面相互作用的非线性振动理论模型，得到了共振差频 AFM 的相位信号的解。研究了材料内部填充结构的尺寸、埋深、弹性模量以及超声激励频率等对相位对比度的影响。

    本文深入地研究了 AFM 的动力学问题，通过反问题方法，建立了 AFM 信号与相关材料力学性能间的定量关系，为利用 AFM 对材料的力学性能进行定量表征提供了理论基础与参考。

    Atomic force microscope (AFM) has been continuously improved since its invention. From the initial static AFM to the various currently used dynamic AFMs, AFM is a powerful tool in the field of the nanoscience and the nanomaterials. Although AFM is with the capable of excellent imaging and is extensively used, there are still many limits on its performance of the quantitative characterization of a sample. Some properties of a material can be very different when its size varies from the microscopic to the macroscopic scale. The mechanical properties of a material are largely determined by their composition. The details of the microstructure have a great influence on their life expectancy. With the rapid development of the nanotechnology and the applications of new materials, the characterization of the various properties of the materials at the nanometer scale, especially the mechanical properties, is urgently required. These properties include the characterization of the friction, surface viscoelasticity, adhesion and other performance parameters. Besides, the development of the new materials requires nondestructive testing of the structure, defects and impurities inside the material.

    This dissertation aims to solve two key scientific problems, i.e., the improvement of the AFM resolution and the inversion of the mechanical properties of a material based on vibration. It focuses on the structural design of the AFM cantilever and the quantitative characterization of the mechanical parameters related with the material properties via AFM. Through theoretical analysis and numerical solution, this work is carried out by using the inverse problem method. Firstly, the higher harmonic signals are enhanced based on the resonance amplification effect. Then the quantitative relationships between the AFM signals and some mechanical properties of a sample are established, such as the elastic modulus of the sample and the elastic modulus of the inclusion of the matrix. This dissertation provides a theoretical basis for the use of the AFM to characterize the mechanical properties of the materials. The main work done in this dissertation is as follows:

    (1) The higher harmonic signals of the AFM are enhanced. In AFM, the rich information of the sample is contained in the higher harmonics. However, for a uniform rectangular cantilever, the natural frequencies of the higher modes are not integral multiples of the fundamental frequency. Therefore, its higher harmonic signals are suppressed. As a result, the properties of the samples are difficult to be detected. In order to solve this problem, the equation of motion of the cantilever is developed and solved, and a step-like/T–shaped cantilever structure is designed to change the natural frequencies of the cantilever. The natural frequency of a mode can be tuned to coincide with a specific harmonic. The first two or three modes of the flexural vibration are excited based on the resonance amplification effect, and the higher harmonic signals of the AFM are enhanced. Besides, a comprehensive map and a table of the size of the cantilever are given for reference.

    (2) The inverse problem is proposed and solved: How to extract the effective Young's modulus and the thickness of the film from the indentation depth-load curve. In the indentation experiment, the effective Young's modulus of the thin film/substrate heterostructure varies as the indentation depth varies. In the previous studies, only the effective Young's modulus of the film is unknown. However, in practice, sometimes the measurement of the film thickness can be extremely difficult if not impossible. Therefore, the thickness and effective Young's modulus of the film are treated as the two unknowns in the inverse problem and solved.

    (3) A method of quantitatively determining the Young's modulus and Poisson’s ratio of the material is proposed. The nonlinear model of the coupled flexural and torsional motions of the T-shaped cantilever in the contact mode is developed. With the increase of the contact stiffness, the bifurcation of the contact resonance frequencies of the coupled structure occurs in a higher mode. The Poisson’s ratio and the Young's modulus are decoupled by using the resonance frequency bifurcation point.

    (4) The phase change caused by scattering when an incident ultrasonic wave encounters the circular cylindrical and the elliptic cylindrical obstacles is analyzed. The nonlinear vibration model of the AFM cantilever tip–sample interaction is developed, and the solution of the phase of the resonance difference AFM is obtained. The effects of the internal structure size, embedding depth, elastic modulus, and the ultrasonic excitation frequency on phase contrast are studied.

    In this dissertation, the in-depth study on the AFM dynamics is presented. Through the inverse problem, the quantitative relationships between the AFM signals and some mechanical properties of a material are established, which provides the theoretical basis for the quantitative characterization of the mechanical properties of a material with the AFM.