含应变梯度效应的弹性理论及其应用研究

发布时间：2018-08-09 15:36

【摘要】：微构件是微机电系统中的基本构件,准确掌握微构件的力学行为是实现微机电系统精确控制的基础。对特征尺寸在微米及亚微米量级的构件而言,不论其力学性能还是多场耦合性能均表现出明显尺寸效应。传统理论不能描述微构件力学行为的尺寸效应,已有研究初步证明考虑高阶变形量影响的应变梯度理论可以解释这种尺寸效应。然而,现有应变梯度理论中,有的尺度参量不独立,有的引入了不恰当的平衡条件,有的是近似理论。这些理论描述的微构件力学行为的尺寸效应均不能真实、合理地反映实际情况。因此,无论是从应变梯度理论的发展来说,还是准确描述微构件的力学行为,建立正确合理的有效理论,对微构件的力学性能包括其力电耦合性能进行深入研究是当前的重要研究内容。本文的研究内容主要包括：发展了应变梯度张量的对称／反对称和静水／偏量分解方法,将应变梯度张量分解为两种形式的独立分量。用应变梯度独立分量重构了应变梯度弹性理论的本构方程,构造了各向同性弹性尺度张量。根据高阶张量理论及弹性尺度张量间的约束关系,从理论上证明了独立的尺度参量只有3个,解决了应变梯度弹性理论本构关系的基本理论问题。进而,提出了尺度参量独立的应变梯度弹性理论,发展了独立分量形式的应变梯度理论变分原理,推导了控制方程和边界条件,并给出了正交曲线坐标系下应变梯度理论变形量的可用公式。应用新发展的应变梯度弹性理论,采用平面应变假设,发展了梁弯曲理论。定义了梁弯曲中的轴力、剪力、弯矩、高阶轴力和高阶弯矩,并给出了用这些应力合力量表达的平衡方程和边界条件。具体求解了两种情况下的悬臂梁弯曲问题：一种考虑为平面应变问题,另一种考虑为伯努利-欧拉梁问题。两种情况下的理论分析结果分别与环氧树脂和硅悬臂梁的弯曲实验结果相吻合,验证了新理论的有效性。应用新理论与Aifantis单参数理论分别分析了杆扭转、固定层剪切、薄梁纯弯曲和球体膨胀四个典型问题中的尺寸效应。通过对比两种理论下的分析结果,说明了新理论可统一有效的描述各种变形问题中的尺寸效应,而单参数理论具有局限性,揭示了应变梯度弹性理论包含多尺度参数的必要性。针对微机电系统广泛采用的层复合结构,分别提出了适用于层合微梁和微板的位移模式。应用新发展的位移模式和最小势能原理,建立了双层微梁和微板的尺寸效应模型,推导了相应的平衡方程和边界条件。具体分析了受均布载荷作用的简支双层微梁和四边简支双层微板的弯曲问题。分析结果发现,双层微梁和微板的弯曲挠度及轴向应力均表现出明显的尺寸效应。且当一层微梁或微板厚度远远大于另一层微梁或微板厚度时,双层微梁或微板的挠度变形接近于单层微梁或微板的挠度变形。将尺度参量独立的应变梯度弹性理论拓展到中心对称介电材料,发展了只包含3个独立尺度参量和2个挠曲电耦合参量的挠曲电理论。给出了内能密度函数的具体表达形式以及独立分量形式的本构关系。发展了独立分量形式的挠曲电理论变分原理,推导了控制方程和边界条件,给出了广义静电力的表达形式。新发展的挠曲电理论同时考虑了尺寸效应、极化梯度效应和正逆挠曲电效应的影响。通过理论推证发现,并非所有的应变梯度量都能诱导极化,而是只有膨胀梯度和偏转动梯度的反对称部分能够诱导极化,拉伸梯度和偏转动梯度的对称部分不能诱导极化。应用新发展的挠曲电理论,建立了伯努利-欧拉梁和基尔霍夫圆板的挠曲电效应模型,给出了平衡方程和边界条件。分别针对自由端受集中力和上下表面间受电压作用的悬臂梁以及受均布载荷和上下表面间受电压作用的简支轴对称圆板,研究了悬臂梁和简支轴对称圆板的正逆挠曲电效应特性。研究发现,悬臂梁和简支轴对称圆板的正逆挠曲电效应均表现出明显尺寸依赖性,且随着挠曲电耦合系数的减小而减弱,特别的当挠曲电耦合系数为零时,正逆挠曲电效应消失。本文提出的应变梯度弹性理论及挠曲电理论能够有效预测微构件力学性能的尺寸效应及力电耦合性能的挠曲电效应。研究成果对MEMS产品的设计分析、性能预测和实验研究都具有重要的指导意义。
[Abstract]:Micro components are the basic components of microelectromechanical systems. The accurate grasp of the mechanical behavior of the microstructures is the basis for precise control of the microelectromechanical systems. For the components of the size of micrometers and submicrons, the mechanical properties and the multi field coupling performance show obvious size effects. The traditional theory can not describe the force of the micro component. The size effect of learning behavior has been studied. It has been proved that the strain gradient theory considering the influence of high order deformation can explain this size effect. However, in the existing strain gradient theory, some scale parameters are not independent, some have introduced improper equilibrium conditions and some are similar to theory. The size effect can not be true and reflects the actual situation reasonably. Therefore, it is an important research content to study the mechanical behavior of the micro component, whether it is from the development of the strain gradient theory, or to accurately describe the mechanical behavior of the micro component, and to study the mechanical properties of the microstructures, including the mechanical and electrical coupling properties. The main contents include: developing the symmetric / antisymmetric and hydrostatic / partial decomposition method of strain gradient tensor, decomposing the strain gradient tensor into two forms of independent component. The constitutive equation of strain gradient elastic theory is reconstructed with the strain gradient independent component, and the isotropic elastic tensor is constructed. The constraint relation between the theory and the elastic scale tensor proves that only 3 independent scale parameters have been theoretically proved, and the basic theoretical problem of the constitutive relation of strain gradient elasticity is solved. Then, the strain gradient elastic theory with independent scale parameter is proposed, and the variational principle of strain gradient theory is developed, and the theory of strain gradient theory is developed. The equation and boundary condition are controlled and the formula of strain gradient deformation in the orthogonal curvilinear coordinate system is given. Using the new developed strain gradient elasticity theory and the assumption of plane strain, the beam bending theory is developed. The axial force, shear, bending moment, high order axial force and high order bending moment are defined in the bending of the beam, and the use of these stresses is given. The equilibrium equations and boundary conditions expressed by force force are solved in two cases: one is the problem of plane strain and the other is the Bernoulli Euler beam problem. The theoretical analysis results in two cases are in agreement with the experimental results of the bending of epoxy resin and silicon cantilever beam respectively. The new theory and Aifantis single parameter theory are applied to analyze the size effects of four typical problems: rod torsion, fixed layer shear, pure bending of thin beams and spheroid expansion. By comparing the analysis results under the two theories, it shows that the new theory can describe the size effect of various deformation problems in a unified and effective way, and the single parameter is single parameter. The theory of number is limited, which reveals the necessity of multi scale parameters in the strain gradient elasticity theory. According to the laminated structure widely used in microelectromechanical systems, the displacement modes suitable for laminated microbeams and microplates are put forward respectively. The size effect of double microbeams and microplates is established by applying the newly developed displacement mode and the principle of minimum potential energy. The corresponding equilibrium equations and boundary conditions are derived, and the bending problems of simple supported double layer microbeams and four simply supported double-layer microplates are analyzed. The results show that the bending deflection and axial stress of the double-layer microbeam and the micro plate show a clear scale effect, and the thickness of a layer of micro beam or the microplate is far from the same. The deflection deformation of the double layer microbeam or microplate is close to the deflection and deformation of the single layer microbeam or the microplate when the thickness of the micro beam or microplate is larger than that of the other layer. The strain gradient elastic theory which is independent of the scale parameter is extended to the central symmetric dielectric material, and the flexure theory, which contains only 3 independent scale parameters and 2 flexural coupling parameters, is developed. The specific expression of the internal energy density function and the constitutive relation of the independent component form. The variational principle of the flexural theory of the independent component is developed, the control equation and the boundary condition are derived, and the expression of the generalized static electricity is given. The new developed flexure theory takes into account the size effect, the polarization gradient effect and the positive inverse. It is found by theoretical evidence that not all strain gradient quantities can induce polarization, but only the anti symmetric part of the expansion gradient and partial rotational gradient can induce polarization, and the symmetric part of the tensile gradient and partial rotational gradient can not induce polarization. The new developed flexo theory has been used to establish Bernoulli Europe. The balance equation and boundary condition are given for the flexural effect model of the tension beam and the Kirchhoff circular plate. The positive inverse of the cantilever beam and the simply supported axisymmetric circular plate are studied for the cantilever beam of the free end subjected to the concentrated force and the voltage acting on the upper and lower surfaces, as well as the simple supported axisymmetric circular plates subjected to the voltage action between the uniform load and the upper and lower surfaces. It is found that the positive and inverse flexure effect of the cantilever beam and the simply supported axisymmetric circular plate shows a significant dimension dependence, and decreases with the decrease of the flexure coupling coefficient, especially when the flexure coupling coefficient is zero. The strain gradient elasticity theory and the flexure electricity proposed in this paper are proposed. The theory can effectively predict the size effect of the mechanical properties of microstructures and the flexural effect of the mechanical and electrical coupling properties. The research results have important guiding significance for the design analysis, performance prediction and experimental research of MEMS products.
【学位授予单位】：山东大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：TH-39

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