微尺度下结构的静动力学行为研究
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摘要
随着微机械加工技术水平的提高,微机电系统这一新兴科学技术领域得以发展,并在信息通信、航空航天、生物医学和军事等领域都有着十分广阔的应用前景。由于微器件和微机械的结构尺寸一般在微米甚至纳米量级,在研究微结构力学行为时,微尺度效应影响不容忽视。因此,对微机电系统的研究需要建立考虑微尺度效应的理论模型来精确预测微结构的各种力学行为。
本文运用修正偶应力理论或应变梯度理论构建完成一系列用于分析微结构静动力学行为(静态变形、振动、屈曲等)的理论模型。为了描述微尺度效应,修正偶应力和应变梯度理论模型分别引入一个和三个与微结构有关的材料内禀长度。通过三种微结构(微梁、微板和微输液管)的静动力学行为研究,探讨了微尺度效应的影响和尺度效应强弱的影响因素。主要研究工作如下：
1、基于修正偶应力理论,分别建立用于研究微尺度下悬臂输液管的动力特性和稳定性的Bernoulli-Euler梁及Timoshenko梁模型。算例分析表明：悬臂微输液管的振动频率和阻尼特性均具有尺度效应,非经典梁模型预测的临界流速值偏高,输液管系统偏于稳定。
2、采用应变梯度理论建立一种可预测微尺度效应的微输液管动力学模型。与修正偶应力理论模型结果比较发现,在包含旋转梯度张量,同时考虑膨胀梯度张量和拉伸梯度张量的偏斜分量后,微输液管具有更高的固有频率和临界流速,尺度效应更显著。
3、以修正偶应力理论为基础,建立分析微板振动特性的理论模型。以两类不同边值的矩形板问题为例,研究了微尺度效应对微板固有频率的影响规律。
4、利用修正偶应力理论建立了非线性的微梁模型并研究了微梁的静态弯曲、屈曲和自由振动。研究表明：微梁在横向载荷下的静态变形、轴向载荷下的临界屈曲载荷和屈曲形态以及初始侧向位移下的非线性固有频率皆具有尺度效应。
5、应用修正偶应力理论进一步建立力-电耦合非经典梁模型,研究了静电驱动微梁的pull-in特性。结果表明：静电驱动微梁的微尺度效应表现在更小的静态挠度和更高的pull-in电压值,且该模型预测的微尺度效应强弱与微梁的长宽比无关。
上述研究均表明结构特征尺寸是反映尺度效应强弱的一个重要参数。微尺度效应随结构特征尺寸的增大而减弱。当结构特征尺寸(管道外径、板厚或梁厚)与材料内禀长度相当时,微结构表现出强烈的微尺度效应；当结构特征尺寸远大于材料内禀长度时,微尺度效应消失。
本文所建立的理论模型可用于微机电系统的刚度、强度及稳定性等性能的定量评估,研究结果可为微机电系统的设计提供理论支持。
With the advancement of the micromachining technology, a new scientific and technological undertaking, micro-electro-mechanical systems (MEMS), has been developing and has a very broad application prospect in various fields, such as the information communication, aerospace, biomedical, and military. In various micro devices and micro machinery, the characteristic sizes of their structures are typically on the order of microns or even nanometers. In this scale, the size effect can not be ignored in the study on mechanical behavior of micro-structures. Therefore, the research on MEMS raises the demand of establishing theoretical models considering size effect to predict accurately the various mechanical behaviors of microscale structures.
By using the modified couple stress theory or strain gradient theory, a set of theoretical models are established to analyze the static and dynamic behaviors (static deformation, vibration, buckling and so on) in this paper. In order to capture the size effect, the models based on the modified couple stress and strain gradient theories introduce one and three material length scale parameters dependent on the micro-structure respectively. By the analysis on the static and dynamic behavior of three microscale structures (microbeams, microplates and microscale pipes conveying fluid), this paper investigates the size effect on microscale structures and the parameters influencing the intensity of size effect. The main results are shown as follows:
1. Based on the modified couple stress theory, the non-classical Bernoulli-Euler and Timoshenko beam models are established to analyze the dynamic characteristics and stability of microscale cantilevered pipes conveying fluid. The numerical examples show that the natural frequencies and damping properties are size-dependent. The results also show that the critical flow velocities predicted by the non-classical beam models are generally higher than those predicted by the classical beam models, indicating that the stability of microscale pipes is enhanced.
2. By utilizing the strain gradient theory, another dynamic model of micropipes conveying fluid is presented to capture the size effect. From the comparison between results of the present model and the model based on the modified couple stress theory, it is found that micropipes conveying fluid take larger natural frequencies and higher critical flow velocities, and thus the size effect is stronger. This is because the former model has introduced the additional dilatation gradient tensor and the deviatoric stretch gradient tensor in addition to the rotation gradient tensor.
3. On the bases of the modified couple stress theory, a theoretical model is developed for analyzing the vibration characteristic of microscale plates. Taking two different boundary value problems of rectangular micro-plates for example, the size effect on natural frequencies of microscale plates is investigated.
4. A nonlinear model of microbeams is established using the modified couple stress theory and the static bending, postbuckling and free vibration of microbeams is analyzed. The results show that the static deflections of a bending beam subjected to transverse force, the critical buckling loads and buckled configurations of an axially loaded beam, and the nonlinear frequencies of a beam with initial lateral displacement are size-dependent.
5. Applying the modified couple stress theory, a non-classical beam model accounting for the electromechanical coupling is further presented and the pull-in characteristic of electrostatically actuated microbeams is investigated. It is found that the size effect on electrostatically actuated microbeams exhibits smaller deflection and larger pull-in voltage. Furthermore, the intensity of size effect predicted by this model is independent on the length-width ratio.
The all above researches explore that the characteristic size is an important parameter reflecting the intensity of size effect. The size effect becomes weaker with the increase of the characteristic size. When the characteristic size of a microscale structure (outside diameter of a pipe, plate thickness or beam thickness) becomes comparable to the material length scale parameter, the microstructure displays strong size effect. While the characteristic size is far greater than the material length scale parameter, the size effect is almost diminishing.
The theoretical models established in this paper can be utilized to evaluate quantitatively the mechanical properties of MEMS structures, such as the strength, stiffness and stability. Moreover, the results can provide theoretical support for the design of MEMS devices.
本文运用修正偶应力理论或应变梯度理论构建完成一系列用于分析微结构静动力学行为(静态变形、振动、屈曲等)的理论模型。为了描述微尺度效应,修正偶应力和应变梯度理论模型分别引入一个和三个与微结构有关的材料内禀长度。通过三种微结构(微梁、微板和微输液管)的静动力学行为研究,探讨了微尺度效应的影响和尺度效应强弱的影响因素。主要研究工作如下：
1、基于修正偶应力理论,分别建立用于研究微尺度下悬臂输液管的动力特性和稳定性的Bernoulli-Euler梁及Timoshenko梁模型。算例分析表明：悬臂微输液管的振动频率和阻尼特性均具有尺度效应,非经典梁模型预测的临界流速值偏高,输液管系统偏于稳定。
2、采用应变梯度理论建立一种可预测微尺度效应的微输液管动力学模型。与修正偶应力理论模型结果比较发现,在包含旋转梯度张量,同时考虑膨胀梯度张量和拉伸梯度张量的偏斜分量后,微输液管具有更高的固有频率和临界流速,尺度效应更显著。
3、以修正偶应力理论为基础,建立分析微板振动特性的理论模型。以两类不同边值的矩形板问题为例,研究了微尺度效应对微板固有频率的影响规律。
4、利用修正偶应力理论建立了非线性的微梁模型并研究了微梁的静态弯曲、屈曲和自由振动。研究表明：微梁在横向载荷下的静态变形、轴向载荷下的临界屈曲载荷和屈曲形态以及初始侧向位移下的非线性固有频率皆具有尺度效应。
5、应用修正偶应力理论进一步建立力-电耦合非经典梁模型,研究了静电驱动微梁的pull-in特性。结果表明：静电驱动微梁的微尺度效应表现在更小的静态挠度和更高的pull-in电压值,且该模型预测的微尺度效应强弱与微梁的长宽比无关。
上述研究均表明结构特征尺寸是反映尺度效应强弱的一个重要参数。微尺度效应随结构特征尺寸的增大而减弱。当结构特征尺寸(管道外径、板厚或梁厚)与材料内禀长度相当时,微结构表现出强烈的微尺度效应；当结构特征尺寸远大于材料内禀长度时,微尺度效应消失。
本文所建立的理论模型可用于微机电系统的刚度、强度及稳定性等性能的定量评估,研究结果可为微机电系统的设计提供理论支持。
With the advancement of the micromachining technology, a new scientific and technological undertaking, micro-electro-mechanical systems (MEMS), has been developing and has a very broad application prospect in various fields, such as the information communication, aerospace, biomedical, and military. In various micro devices and micro machinery, the characteristic sizes of their structures are typically on the order of microns or even nanometers. In this scale, the size effect can not be ignored in the study on mechanical behavior of micro-structures. Therefore, the research on MEMS raises the demand of establishing theoretical models considering size effect to predict accurately the various mechanical behaviors of microscale structures.
By using the modified couple stress theory or strain gradient theory, a set of theoretical models are established to analyze the static and dynamic behaviors (static deformation, vibration, buckling and so on) in this paper. In order to capture the size effect, the models based on the modified couple stress and strain gradient theories introduce one and three material length scale parameters dependent on the micro-structure respectively. By the analysis on the static and dynamic behavior of three microscale structures (microbeams, microplates and microscale pipes conveying fluid), this paper investigates the size effect on microscale structures and the parameters influencing the intensity of size effect. The main results are shown as follows:
1. Based on the modified couple stress theory, the non-classical Bernoulli-Euler and Timoshenko beam models are established to analyze the dynamic characteristics and stability of microscale cantilevered pipes conveying fluid. The numerical examples show that the natural frequencies and damping properties are size-dependent. The results also show that the critical flow velocities predicted by the non-classical beam models are generally higher than those predicted by the classical beam models, indicating that the stability of microscale pipes is enhanced.
2. By utilizing the strain gradient theory, another dynamic model of micropipes conveying fluid is presented to capture the size effect. From the comparison between results of the present model and the model based on the modified couple stress theory, it is found that micropipes conveying fluid take larger natural frequencies and higher critical flow velocities, and thus the size effect is stronger. This is because the former model has introduced the additional dilatation gradient tensor and the deviatoric stretch gradient tensor in addition to the rotation gradient tensor.
3. On the bases of the modified couple stress theory, a theoretical model is developed for analyzing the vibration characteristic of microscale plates. Taking two different boundary value problems of rectangular micro-plates for example, the size effect on natural frequencies of microscale plates is investigated.
4. A nonlinear model of microbeams is established using the modified couple stress theory and the static bending, postbuckling and free vibration of microbeams is analyzed. The results show that the static deflections of a bending beam subjected to transverse force, the critical buckling loads and buckled configurations of an axially loaded beam, and the nonlinear frequencies of a beam with initial lateral displacement are size-dependent.
5. Applying the modified couple stress theory, a non-classical beam model accounting for the electromechanical coupling is further presented and the pull-in characteristic of electrostatically actuated microbeams is investigated. It is found that the size effect on electrostatically actuated microbeams exhibits smaller deflection and larger pull-in voltage. Furthermore, the intensity of size effect predicted by this model is independent on the length-width ratio.
The all above researches explore that the characteristic size is an important parameter reflecting the intensity of size effect. The size effect becomes weaker with the increase of the characteristic size. When the characteristic size of a microscale structure (outside diameter of a pipe, plate thickness or beam thickness) becomes comparable to the material length scale parameter, the microstructure displays strong size effect. While the characteristic size is far greater than the material length scale parameter, the size effect is almost diminishing.
The theoretical models established in this paper can be utilized to evaluate quantitatively the mechanical properties of MEMS structures, such as the strength, stiffness and stability. Moreover, the results can provide theoretical support for the design of MEMS devices.
引文
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