微纳米结构的力学行为及其有限单元方法研究

发布时间：2018-01-19 20:18

本文关键词： MEMS 尺寸效应应变梯度理论铁木辛柯梁单元　出处：《山东大学》2017年硕士论文　论文类型：学位论文

【摘要】：随着科技的快速发展以及制造工艺水平的不断提升,微机电系统(英文缩写为MEMS)因其具有众多优点而被广泛地应用在电子产品、医疗器械、汽车工业以及航空航天等众多高精尖领域。因此,需要对MEMS器件进行细致地分析和研究。在MEMS的结构设计以及分析计算过程中,可以将一些结构简化为微板或者微梁,这也正是MEMS器件中典型的结构。所以,分析这类微尺寸结构的力学性能就变得非常重要。然而微观结构的性能和宏观尺寸结构的性能有着明显的不同,因此传统理论不适用于分析微观结构的力学性能。许多研究学者做了相关的微米结构的力学试验,试验表明复合材料、多晶硅、聚合物以及金属材料都具有尺寸效应,即随着微结构尺寸的不断缩小,其材料的力学性能在不断的增强,而这一现象无法用传统理论进行解释。于是需要发展和完善一种能够适用于研究微纳米结构力学性能的理论与模型。虽然许多学者发展了多种用于解释尺寸效应现象的理论,例如非局部理论、偶应力理论、表面能理论以及应变梯度理论等,但是最成功的理论之一当属应变梯度理论。非局部理论更加适用于结构软化的研究。表面能理论只考虑了表面效应的影响而忽略了结构体内部的影响,因此该理论更加适用于表面与体积之比更大的纳米量级的结构。梯度弹性理论虽然计算简单但精度较低。偶应力理论只考虑了一种应变梯度项,而应变梯度理论则考虑了三种应变梯度项,集合了多种理论的优点,避免了一些缺点,因此该理论得到了广泛应用。应用应变梯度理论构造了一种适用于复杂问题的非传统微纳米尺度铁木辛柯梁单元。新单元中包含三个能够预测尺寸效应的材料长度尺寸参数,同时这个单元可以通过设置材料长度尺寸参数退化为修正偶应力理论单元或者传统单元。对于传统的单元,满足C_0型连续,每个单元包含两个节点,每个节点又有两个自由度。但对于新构造的单元,则满足C_1型连续,每个单元包含两个节点,每个节点又有四个自由度。新单元采用挠度和转角独立差值,有限元方程、刚度矩阵和质量矩阵通过积分弱形式可推导得出。为了检验新单元的精确性和可靠性,研究了新单元的收敛情况以及剪切锁死问题。接着研究了铁木辛柯梁不同边界条件下的静态问题和动态问题。为了精确地研究微纳米结构的尺寸效应,本文还同时考虑了表面影响和体影响。从物理角度来说,尺寸效应不仅是由体产生的,还与表面有关。表面能理论和应变梯度理论分别用于说明表面影响和体影响。本文构造了基于应变梯度理论以及表面能理论的伯努利-欧拉梁模型和铁木辛柯梁模型。控制方程、边界条件和初始条件可以应用哈密顿原理得出。两个新模型中都包含三个材料长度尺寸参数和三个表面弹性常数。新构造的非传统模型不仅能够退化为只考虑体影响或者只考虑表面影响的模型,还可以退化为修正偶应力理论模型和传统模型。除此之外,当不考虑剪切变形影响时,新构造的模型可以退化为伯努利-欧拉梁模型。为了说明新构造的模型,研究了微纳米尺度的伯努利-欧拉梁以及铁木辛柯梁的静态问题和动态问题。结果显示当梁的尺寸很小时,不同模型之间的差别很大,即尺寸效应很明显。随着纳米梁尺寸的不断变大,模型之间的差别不断减小。本文还基于应变梯度理论研究了具有弹性支座的Kirchhoff微板模型。此模型引入了三个额外的材料参数,使得新模型能够预测弹性支撑微板结构的尺寸效应。通过设置一些材料参数等于零,能够将此模型退化为传统理论模型或修正偶应力理论模型。结果讨论中给出了四边简支的方形微板静态弯曲问题、稳定性问题和自由振动问题等的研究,同时比较了当前模型和其他退化模型之间的差别,并研究了弹性支撑对板的影响。此研究可以有效地指导具有弹性支撑方形微板的分析与设计,因此具有很好的应用前景。
[Abstract]:With the rapid development of technology and manufacturing level rising, microelectromechanical systems (abbreviated as MEMS English) because it has many advantages and is widely used in electronic products, medical devices, automotive industry and aerospace and other high-tech fields. Therefore, the need for detailed analysis and Research on the MEMS device. The MEMS structure design and analysis of the calculation process, can be simplified as micro plate or micro beam, which is the typical structure of the MEMS device. Therefore, the analysis of mechanical properties of micro size structures have become very important. However, the performance of microstructure and macro size structure has obvious is different, so the traditional theory is not applicable to the analysis of mechanical properties of micro structures. Many scholars have done research on mechanical test of micron structure of the relevant test results show that the composite, polycrystalline silicon, polymer And metal materials have size effect, with the size of the micro structure is shrinking, the mechanical properties of the materials in the unceasing enhancement, and this phenomenon can not use the traditional theory to explain. So we need to develop and perfect the theory and a model can be applied to the research on micro nano mechanical properties. Although many scholars have developed many used to explain the size effect of the theory, such as the non local theory, couple stress theory, surface energy theory and the strain gradient theory, but the theory is undoubtedly one of the most successful. The strain gradient theory of nonlocal theory is more suitable to study the structure of softening. The theory of surface energy only considering the influence of the surface effect and ignore influence of structure internal structure, so the theory is more suitable for the surface to volume ratio the greater the nanometer level. Gradient elasticity theory is simpler but more precision Low. The couple stress theory considers only a strain gradient, and the strain gradient theory is considered the three strain gradient, combines the advantages of a variety of theory, to avoid some disadvantages, so the theory has been widely applied. Non traditional application of strain gradient theory is constructed for complex problem of micro nanometer scale Timoshenko beam element. The new unit contains three material length dimensions can predict the size effect parameters, and the unit can be set through the material length parameter degradation stress theory unit or the traditional unit. The modified couple for the traditional unit, meet the C_0 continuous, each containing two nodes each. Nodes have two degrees of freedom. But for the new structure of the unit, will meet C_1 continuous, each unit contains two nodes, each node has four degrees of freedom. The new unit adopts deflection and rotation Independent difference, finite element equation, the stiffness matrix and mass matrix can be derived through integral weak form. In order to test the accuracy and reliability of the new unit, a new research unit and convergence of shear locking problem. Then study the static problem of iron Xin Keliang under different boundary conditions and dynamic problems. In order to accurately study the size effect of micro nano structure, this paper also considers the influence of the surface effect and volume. From the physical point of view, the size effect is not only by the body, but also with the surface. The surface energy theory and strain gradient theory respectively used to illustrate the influence of surface effect and volume. This paper constructed the strain gradient theory and the theory of surface energy the Bernoulli Euler beam model and Timoshenko beam model. Based on the control equations, initial and boundary conditions can be obtained using the Hamilton principle. All two new models Three material length parameters and three surface elastic constants. Non traditional model of the new structure can not only reduced to only consider the impact or only consider the surface of the model, also can be reduced to the modified couple stress theory model and traditional model. In addition, when not considering the effect of shear deformation, new structure the model can be reduced to the Bernoulli Euler beam model. In order to illustrate the new structure model of micro nano scale Bernoulli Euler beam and Timoshenko beam static problems and dynamic problems. The results show that when the beam size is very small, the difference between the different models, namely the size effect is obvious. With the development of nano the beam size becomes larger and larger, the difference between the model decreases. The strain gradient theory is studied based on the Kirchhoff micro plate with elastic support model. This model introduces three additional material The material parameters, the new model can predict the size effect of micro elastic support plate structure. By setting some material parameters is equal to zero, this model can degenerate stress theoretical model for the traditional theory model. The results are given in the discussion I simply supported square plate micro static bending, free vibration and stability studies so, at the same time were compared between the current model and other degradation model differences, and study the influence of the elastic support plate. The analysis and design of this study can effectively guide the square plates with elastic support, so it has good application prospects.

【学位授予单位】：山东大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O342

【相似文献】