纳米结构材料具有较大的比表面积，其力学性能呈现出明显的尺寸相关特性，即纳米结构材料的表面效应。经典连续介质力学由于不包含任何内禀尺度参数，无法预测纳米材料特有的尺寸效应。针对这一问题，研究者们广泛使用表面弹性理论（Gurtin-Murdoch理论）来研究纳米材料的力学行为，通过建立表面能密度和表面应力的线弹性本构方程，引入表面弹性常数作为刻画表面效应的关键材料参数。该参量仅能通过分子动力学方法计算确定，需要耗费大量计算资源，且数值结果有时为负值，给理论应用带来一定困难。

本文将采用一种基于表面能密度的新弹性理论对纳米尺度表面效应进行表征。该理论不再引入表面弹性常数，仅采用块体材料表面能密度和表面晶格弛豫参数刻画纳米材料表面性质，该参量能够通过简单的实验或数值计算得到，较之表面弹性常数更易确定，且物理意义清晰。应用新的弹性理论，本文针对若干典型纳米结构材料的力学行为开展研究，包括Timoshenko纳米梁静态弯曲、自由振动，平面应变、轴对称纳米Hertz接触，轴对称纳米黏附接触，锂离子电池中纳米电极的应力和位移分布等问题。进一步揭示了表面效应对纳米材料力学性能的影响机理。主要研究内容及成果如下：

首先，建立了两端固支及Timoshenko悬臂纳米梁静态弯曲及自由振动模型，同时考虑表面效应和剪切变形效应对纳米梁等效弹性模量和固有频率的耦合影响。研究发现：当纳米梁长度保持不变，随纳米梁横截面尺寸的增加，梁等效弹性模量和固有频率均经历了表面效应占主导、表面效应和剪切变形效应共同影响、再到剪切变形效应占主导的变化过程；表面效应使两端固支纳米梁发生硬化，悬臂纳米梁则发生软化；而剪切变形效应使上述两种纳米梁均发生软化。上述研究成果与已有实验数据符合较好，能够为纳米机电系统设计提供理论指导。

系统研究了纳米尺度平面应变和轴对称接触问题。建立考虑表面效应的Boussinesq接触模型（任意对称法向分布载荷），得到其一般解，并系统分析了三种平面应变纳米接触问题，包括均布法向载荷、刚性平压头压入和刚性圆柱压头压入以及三种轴对称纳米接触问题，包括轴对称法向均布载荷、刚性圆柱平压头压入和刚性球压头压入。发现当接触宽度（或接触半径）与纳米量级内禀尺度参数（基底的体材料表面能密度和剪切模量的比值）相当时，表面效应对接触应力和变形产生显著影响。与经典Hertz解相比，表面效应使接触区域的正应力分布更均匀和光滑，法向位移减小且分布更均匀，同时剪切应力非零；表面效应随着基底体材料表面能密度的增加或剪切模量、接触宽度（接触半径）的减小而增强。同时预测了刚性球压头压入问题中纳米压痕硬度的尺寸效应，发现压头半径越小或外载越小，纳米压痕硬度越大。

采用新的弹性理论和基于L-J势的黏附接触理论，分析了轴对称纳米黏附接触力学行为。建立刚性纳米球压头压入半无限大弹性基底的黏附接触模型，基底体材料表面能密度作为刻画接触区域表面性质的唯一参量。研究表明，基底剪切模量越小或压头尺寸越小，表面效应对纳米黏附接触行为的影响越大；与经典黏附接触理论的预测结果相比，压入深度给定时，最大撕脱力更小，基底变形更加平缓；当基底剪切模量较小或压头尺寸较大时，外加载荷—压入深度曲线更易发生回滞行为，回滞拐点表示压头发生突然黏附或突然撕脱的失稳行为，所对应的压入深度要比经典解更大；当压头和基底发生黏附自平衡时，压入深度较之经典解减小，表明表面效应使基底发生硬化，与已有实验数据定性一致。

最后，应用新的弹性理论和扩散理论，研究了锂离子电池纳米电极在充放电过程中的力学行为，预测了纳米球颗粒电极和纳米线电极内部化学场扩散诱导的应力和位移分布。研究发现：表面效应能够有效抑制纳米电极在充放电过程中的体积膨胀，使纳米电极中正应力及平均应力降低；此外，纳米线电极中von Mises应力降低，但纳米球颗粒电极中von Mises应力保持不变。上述研究结果能够为锂离子电池的电极优化设计提供理论指导，通过合理选择纳米电极的特征尺寸和几何形状，能够避免电极内部应力集中或膨胀变形引发的失效现象，从而有效延长锂离子电池的使用寿命。

The mechanical properties of nanostructured materials exhibit size effect due to their large ratio of surface area to volume which is also called surface effect. Such phenomenon cannot be predicted within the framework of the classical continuum mechanics since no internal length scale is involved. As for this issue, the surface elastic theory (Gurtin-Murdoch theory) is widely used to study the mechanical properties in nanomaterials. The linear surface constitutive relation related to surface energy density or surface stress is established, in which surface elastic constants are involved to characterize the surface effect. Such parameters can only be determined by molecular dynamics and require a significant amount of computing resources. Furthermore, a negative value of the surface elastic constant is often found which brings out some difficulties for the theoretical applications. In this paper, a novel elastic theory based on the surface energy density is adopted to characterize the surface effect. The surface elastic constants are no longer involved in the theory. Only the bulk surface energy density and surface relaxation are introduced to characterize the surface properties of nanomaterials. Both the two parameters are easy to determine with clearly physical meanings. Based on the novel elastic theory, the mechanical properties in several typical nanostructured materials are investigated, including the static bending and vibration of Timoshenko nanobeam, plain-strain and axisymmetric nanocontact problems, adhesive nanocontact problem and the stress and displacement distributions of nanoelectrodes in lithium-ion battery. The influential mechanisms of surface effect on the mechanical properties of nanostructured materials are further revealed. The main contents and contributions of this paper are as follows:

Firstly, a fixed-fixed and cantilevered Timoshenko nanobeam model are established to predict the size-dependent effective elastic modulus and resonant frequency of nanobeam, in which both the surface effect and the shear deformation are considered. It can be found that when the length of a nanobeam is fixed, with the increase of the cross-section size, both the effective elastic modulus and resonant frequency experience a transition from the dominant of the surface effect to the coupling influences of the surface effect and shear deformantion, and further to the dominant of the shear deformation. The surface effect stiffens a fixed–fixed nanobeam while softens a cantilevered one. The shear deformation always makes a nanobeam soft. The above results agree well with the existing experiment data and could give a theoretical guidance of the design of nano-electro-mechanical systems.

The plain-strain and axisymmetric nanocontact problems are systematically investigated. Boussinesq contact models with considering surface effect are established and general solutions are obtained. Three two-dimensional nanocontact models (an elastic half-space under an uniform pressure, an elastic half-space punched by a flat-ended indenter or a cylindrical one) and three axisymmetric ones (an elastic half-space under an axisymmetrically uniform pressure, an elastic half-space punched by a rigid flat-ended cylindrical indenter or a spherical one) are analyzed respectively. It can be found that surface effect on the nanocontact behaviors cannot be neglected when the contact width or contact radius is on the same magnitude with the intrinsic length scale, i.e., the ratio of the bulk surface energy density to the bulk shear modulus of the indented material. Compared with the classical contact solutions, surface effect could not only make the normal stresses smoother and more uniform, but also lead to smaller and more uniform displacements as well as a non-zero shear stress at the contact surface. Furthermore, the size effect of nanoindentation hardness is predicted for the case of rigid spherical indenter. It can be found that nanoindentation hardness increases with the reduction of the indenter radius or the external load.

The adhesive contact behaviors at nanoscale are investigated by combining the new elastic theory and classical adhesive contact model based on Lennard-Jones force law. An axisymmetric adhesive contact model between a rigid spherical indenter and an elastic half-space with surface effect is established. The surface energy density of the indented bulk substrate, as only one additional parameter, serves as an important factor to characterize the surface property. It is found that the surface effect can be enhanced by decreasing the shear modulus of the indented material or the radius of the spherical indenter. As compared with the classical adhesive contact solutions based on Lennard-Jones force law, surface effect could not only lead to a smaller maximum pull-off force and flatter deformation of the substrate, but also increase the corresponding approaches at turning points in hysteretic phenomenon which occurs more easily for the case of soft substrate or large indenter. When neglecting the external load, the corresponding approaches become smaller than classical predictions, which demonstrates the substrate becomes hardened when surface effect is considered. The results agree qualitatively with the existing experimental data.

Lastly, combining the new elastic theory and diffusion theory, the diffusion-induced stresses and displacements of nanostructured electrodes in lithium-ion batteries are analyzed, including spherical nanoparticle and nanowire electrode. It is found that when the characteristic size of the electrode is at nanoscale, the surface effect could not only restrain the volume expansion of the nanoelectrodes, but also reduce the normal stresses and average one. The von Mises stress in nanowire electrodes decreases as compared with the classical solutions while the von Mises stress in spherical nanoparticle could not be influenced by surface effect. The results provide a theoretical guidance on the design of the electrodes. The tendency for mechanical degradation of electrodes caused by stress concentration and huge volume expansion could be avoided by choosing nanoeletrodes with reasonable size and geometry, which could prolong the service life of lithium-ion batteries.