微构件的尺寸效应及数值方法研究
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摘要
微机电系统(Micro-Electro-Mechanical Systems,简写为MEMS)器件具有高谐振频率、高的品质因数、超低功率等特性,故MEMS微构件在谐振器、混频器生物传感器等方面具有广阔的应用前景,是目前国际研究的热点课题之一。这些微型器件中的微机械构件根据形状尺寸与受力特点,可以简化为微梁、微板、微膜和微杆等力学模型。当微构件的几何尺寸在微米、亚微米或纳米等范畴时,微构件的力学性能与宏观尺寸下构件的力学性能有很大的不同。
目前,在微小尺度实验中已经证实,当微构件的几何尺寸在微米、亚微米或纳米量级上时,微构件的力学性能具有明显的尺寸效应,而这种尺寸效应现象不能用经典的连续介质力学来描述。因此,发展和完善能够解释和描述微构件的尺寸效应现象的理论和模型就显得至关重要。应变梯度理论通过在本构关系中引入材料的内禀特征长度来考虑应变梯度的影响,为解释和描述微构件的尺寸效应提供了重要的力学理论基础。同时,如果不考虑应变梯度的影响,则应变梯度理论退化为经典的弹性理论。因此,应变梯度理论是经典弹性理论的推广和扩展。
本论文以微机电系统中的微构件为研究对象,基于应变梯度理论建立微构件的尺寸效应模型以及微构件的数值方法,对微构件力学性能的尺寸效应进行深入的研究,研究工作的主要内容如下：
基于应变梯度理论和哈密顿变分原理,建立非线性欧拉-伯努利微梁的尺寸效应模型。在该模型中,引入了材料的三个内禀特征长度参数,可以预测微梁力学性能的尺寸效应；同时,考虑中面伸长和残余应力的影响,可以预测中面伸长以及残余应力对微梁尺寸效应的影响。如果忽略膨胀梯度张量和拉伸梯度张量偏张量的影响,则新模型退化为偶应力欧拉-伯努利非线性微梁尺寸效应模型；如果忽略材料所有的内禀特征长度参数,则新模型退化为经典的欧拉-伯努利非线性微梁模型。结合简支梁和固支梁的特征尺寸,研究了非线性微梁的弯曲变形、屈曲载荷和频率的尺寸效应。研究结果表明：当微梁的特征尺寸与材料的内禀特征长度尺寸参数接近时,微梁的非线性力学特性表现出明显的尺寸效应,然而,随着微梁特征尺寸的增加,这种尺寸效应现象逐渐降低直至最终消失。微梁中的残余应力和中面伸长对微梁的力学性能尺寸效应影响显著,考虑中面伸长,微梁的尺寸效应现象降低；随着残余应力的增加,微梁的尺寸效应现象也显著降低。因此,该模型能够反映出非线性微梁弯曲变形、屈曲载荷和频率的尺寸效应,可以为微梁的结构设计和实验测试提供理论基础。
基于应变梯度理论和哈密顿变分原理,推导并建立了曲线边界克希霍夫微板的尺寸效应模型。在该模型中,引入了材料的三个内禀特征长度参数,可以预测微板力学性能的尺寸效应,同时得到微板在曲线边界上的边界条件以及角点处应满足的边界条件。由于模型中不仅包含了应变张量和旋转梯度张量的影响,同时还包含了拉伸梯度偏斜张量以及膨胀梯度张量的影响,能全面地反映微板的弯曲变形和固有频率的尺寸效应。以固支圆板和简支圆板为研究对象,研究了圆板的弯曲变形和固有频率的尺寸效应,并将结果与偶应力微板模型和经典的微板模型进行比较。结果表明,圆形微板的无量纲弯曲变形和无量纲固有频率存在明显的尺寸效应,同时,不同约束形式的圆板尺寸效应也不同。
基于应变梯度理论和哈密顿变分原理,建立了静电驱动非线性微梁吸合电压的尺寸效应模型。在该模型中,不但考虑了静电载荷固有的非线性特性,同时考虑微梁中面伸长的几何非线性特性。采用瑞利-利兹法近似处理静电载荷和中面伸长,利用变分法给出了确定微梁吸合电压的近似表达式,分析静电驱动非线性微梁的吸合电压的变化规律。结果表明：当微梁的特征尺寸与材料的内禀特征长度参数接近时,吸合电压呈现出明显的尺寸效应现象；当微梁内的残余应力增大时,其吸合电压的尺寸效应现象明显减弱,同时,中面伸长对微梁的吸合电压也有明显的影响。该模型可以为微机电系统中微梁结构的设计和实验验证提供重要的理论基础。
基于应变梯度理论和哈密顿变分原理,建立了平面杆件系统中微梁的尺寸效应模型以及数值分析模型。该尺寸效应模型以及数值分析模型中包含了材料的内禀特征长度参数,可以分析平面杆件系统中微梁的尺寸效应。同时,基于该数值分析模型可以分析非线性微梁弯曲变形的尺寸效应以及静电吸合电压的尺寸效应。由于在微机电系统中存在大量的微型弹簧,因此,采用该数值模型对微弹簧的刚度进行分析,发现微弹簧的刚度具有明显的尺寸效应,为复杂微型弹簧的计算和设计提供理论和分析基础。
本文所建立的微构件的尺寸效应模型能够反映微构件的弯曲变形、固有频率、屈曲载荷等力学性能的尺寸效应以及力电耦合环境下的吸合特性的尺寸效应,研究结果可以为微机电系统中微构件的设计和实验研究提供理论依据。
Attributed to the small mass and size, MEMS (Micro Electro Mechanical Systems, abbreviated as MEMS) come with high fundamental resonance frequencies, high quality factors and low power dissipation. MEMS have been widely used in the field of resonators, mixers, sensors and actuators. According to geometry and loaded forms of those devices and products, main structures can be simplified to be some typical structural forms, such as micro-beams, micro-plates, micro-membranes and micro-bars etc. In the micro scale range, the sub-micro scale range and the nano scale range, the mechanical properties of micro-component are very different from that in the macro scale range.
At present, the size effects of the mechanical properties for the micro-component have been discovered in the mechanical properties experiments. The size effects of the micro-compoment cannot be described using classical continuum mechanics. Therefore, the creation of new theory which is able to explain and describe the effect of the micro-components is crucial. In order to describe and explain the size effect of the micro-components, the strain gradient is introduced into the strain gradient theory when the material intrinsic characteristic scale parameters are introduced into the constitutive relations in the new theory. At the same time, the strain gradient theory simplified as the classic elasticity theory when the material intrinsic characteristic scale parameters equal to zero. Therefore, the strain gradient theory is the expansion of the classical elasticity theory.
In our paper, the micro-components in MEMS are mainly analyzed and the size effect models are established based on the strain gradient theory. The size effects of mechanical property are widely studied for the components in the micro scale range. Main content of our work are as follows:
A nonlinear size effect model of micro-beam is established based on the strain gradient theory and the Hamiltonian variational principle. In the new model, the middle plane elongation and the residual stress are included while the three intrinsic characteristic scale paramenters are introduced. Therefore, the new model can predict not only the size effect of the micro-beams but also the effect of the middle plane elongation and the residual stress on the size effects. When the two material intrinsic characteristic scale parameters regarding as the dilatation gradient tensor and deviatoric stretch gradient tensor are zero, the new model can be simplified as the nonlinear model based on the couple stress theory. When the three material intrinsic characteristic scale parameters are zero, the new model can be simplified as the nonlinear model based on the classical theory. For the simply supported micro-beam and the clamped-clamped micro-beam, the size effects of the nonlinear static deformation, the nonlinear natural frequency and the nonlinear buckling load are analyzed. The size effect on the static deformation, the natural frequency and the buckling load of micro-beams are assessed when the feature size of the micro-beams approaches to the material intrinsic characteristic scale parameter. However, the size effect gradually reduced when the feature sizes of micro-beams are by far larger than the material intrinsic characteristic scale parameter. The effects of the middle plane elongation and the residual stress on the effect of micro-beams are significant. At the same time, the size effect phenomenon significantly reduced when the residual stress increases. The bending deformation, the buckling load and the nonlinear natural frequency based on the strain gradient theory are equal to the results based the classical theory when the feature size of the micro-beams are much larger than the material intrinsic characteristic length scales. Therefore, the new model can capture size effect on the nonlinear bending deformation, the nonlinear buckling load and the nonlinear natural frequency of micro-beams and can provide a theoretical basis for the structural design and experimental testing of the micro-beams.
Based on the strain gradient theory and Hamiltonian variational principle, the size effect model for arbitrary boundary shape micro-plates is derived. The new model can captured the size effect of the mechanical properties of the micro-plates based on the introduction of three material intrinsic characteristic scale parameters. At the same time, the boundary conditions are derived at the arbitrary shape boundary and the coner of the plate. The new model not only includes the effect of the strain tensor and rotation gradient tentor, but also contains the stretch gradient skew tensor and dilatation strain gradient tensor, can fully descrie the size effects of the static deformation and the natural frequency of the micro-plate. For the simply supported micro-plate and the clamped micro-plate, the size effects of the static and the natural frequency are analyzed. The numerical results are compared with that of the couple stress micro-plates and the classical micro-plates. The results show that the normalized static deformation and the normalized nature frequency of circular micro-plates have significant size effects. The size effect is different when the constraint of the circular plate is different.
The size effect model of the electrostatically acturated nonlinear micro-beams is established based on the strain gradient theory and Hamiltonian variational principle. Taking into account size effect on mechanical propreties of materials and the inherent nonlinear property of electrostatic force and the middle plane elongation, approximate expression of the pull-in voltage of electrostatically actuated micro-beams are obtained by using the Rayleigh-Ritz method. The results show that the dimensionless pull-in voltage increases significantly with the dimensionless micro-beam thickness decreasing, showing a significant size effect. When the material intrinsic characteristic length increases, the size effect of the dimensionless pull-in voltage is more significantly, indicating that the effect of the strain gradient on the micro-beam pull-in voltage is remarkable. When the dimensionless micro-beam thickness decreases, the effect of the residual stress on the dimensionless pull-in voltage is significant. The size effect of the normalized pull-in voltage is weakened when the residual stress increases, indicating the residual stress can decrease the effect of the strain gradient on the pull-in voltage. The effect of the middle plane elongation on the pull-in voltage is reduced markedly when the strain gradient is considered. The results can prove the reference in the design of micro structures in MEMS.
Based on the strain gradient theory and the Hamiltonian variational principle, the size effect model and the numerical simulation model of the micro-beam in the plane bar system are derived. The size effect model and the numerical simulation model can analyze the size effect of micro-beam in the plane bar system due to the introducation of the material intrinsic characteristic scale parameters. At the same time, the size effects of the static deformation and the electrostatic pull-in voltage for the nonlinear micro-beam can be analyzed based on the numerical model. Due to a large number of micro-springs are used in MEMS, the size effects of the stiffness of the plane micro-spring are analyzed. The results show that the stiffnesses of micro-spring have significant size effect and provide a theoretical basis for the analysis and design of complex plane micro-spring.
The above mechanical models can capture size effect on the static deformation, the natural frequency, the buckling load and the pull-in voltage of the micro components. The research results can provide a theoretical guideline for the optimal design and experimental tests of the micro-components in MEMS devices.
目前,在微小尺度实验中已经证实,当微构件的几何尺寸在微米、亚微米或纳米量级上时,微构件的力学性能具有明显的尺寸效应,而这种尺寸效应现象不能用经典的连续介质力学来描述。因此,发展和完善能够解释和描述微构件的尺寸效应现象的理论和模型就显得至关重要。应变梯度理论通过在本构关系中引入材料的内禀特征长度来考虑应变梯度的影响,为解释和描述微构件的尺寸效应提供了重要的力学理论基础。同时,如果不考虑应变梯度的影响,则应变梯度理论退化为经典的弹性理论。因此,应变梯度理论是经典弹性理论的推广和扩展。
本论文以微机电系统中的微构件为研究对象,基于应变梯度理论建立微构件的尺寸效应模型以及微构件的数值方法,对微构件力学性能的尺寸效应进行深入的研究,研究工作的主要内容如下：
基于应变梯度理论和哈密顿变分原理,建立非线性欧拉-伯努利微梁的尺寸效应模型。在该模型中,引入了材料的三个内禀特征长度参数,可以预测微梁力学性能的尺寸效应；同时,考虑中面伸长和残余应力的影响,可以预测中面伸长以及残余应力对微梁尺寸效应的影响。如果忽略膨胀梯度张量和拉伸梯度张量偏张量的影响,则新模型退化为偶应力欧拉-伯努利非线性微梁尺寸效应模型；如果忽略材料所有的内禀特征长度参数,则新模型退化为经典的欧拉-伯努利非线性微梁模型。结合简支梁和固支梁的特征尺寸,研究了非线性微梁的弯曲变形、屈曲载荷和频率的尺寸效应。研究结果表明：当微梁的特征尺寸与材料的内禀特征长度尺寸参数接近时,微梁的非线性力学特性表现出明显的尺寸效应,然而,随着微梁特征尺寸的增加,这种尺寸效应现象逐渐降低直至最终消失。微梁中的残余应力和中面伸长对微梁的力学性能尺寸效应影响显著,考虑中面伸长,微梁的尺寸效应现象降低；随着残余应力的增加,微梁的尺寸效应现象也显著降低。因此,该模型能够反映出非线性微梁弯曲变形、屈曲载荷和频率的尺寸效应,可以为微梁的结构设计和实验测试提供理论基础。
基于应变梯度理论和哈密顿变分原理,推导并建立了曲线边界克希霍夫微板的尺寸效应模型。在该模型中,引入了材料的三个内禀特征长度参数,可以预测微板力学性能的尺寸效应,同时得到微板在曲线边界上的边界条件以及角点处应满足的边界条件。由于模型中不仅包含了应变张量和旋转梯度张量的影响,同时还包含了拉伸梯度偏斜张量以及膨胀梯度张量的影响,能全面地反映微板的弯曲变形和固有频率的尺寸效应。以固支圆板和简支圆板为研究对象,研究了圆板的弯曲变形和固有频率的尺寸效应,并将结果与偶应力微板模型和经典的微板模型进行比较。结果表明,圆形微板的无量纲弯曲变形和无量纲固有频率存在明显的尺寸效应,同时,不同约束形式的圆板尺寸效应也不同。
基于应变梯度理论和哈密顿变分原理,建立了静电驱动非线性微梁吸合电压的尺寸效应模型。在该模型中,不但考虑了静电载荷固有的非线性特性,同时考虑微梁中面伸长的几何非线性特性。采用瑞利-利兹法近似处理静电载荷和中面伸长,利用变分法给出了确定微梁吸合电压的近似表达式,分析静电驱动非线性微梁的吸合电压的变化规律。结果表明：当微梁的特征尺寸与材料的内禀特征长度参数接近时,吸合电压呈现出明显的尺寸效应现象；当微梁内的残余应力增大时,其吸合电压的尺寸效应现象明显减弱,同时,中面伸长对微梁的吸合电压也有明显的影响。该模型可以为微机电系统中微梁结构的设计和实验验证提供重要的理论基础。
基于应变梯度理论和哈密顿变分原理,建立了平面杆件系统中微梁的尺寸效应模型以及数值分析模型。该尺寸效应模型以及数值分析模型中包含了材料的内禀特征长度参数,可以分析平面杆件系统中微梁的尺寸效应。同时,基于该数值分析模型可以分析非线性微梁弯曲变形的尺寸效应以及静电吸合电压的尺寸效应。由于在微机电系统中存在大量的微型弹簧,因此,采用该数值模型对微弹簧的刚度进行分析,发现微弹簧的刚度具有明显的尺寸效应,为复杂微型弹簧的计算和设计提供理论和分析基础。
本文所建立的微构件的尺寸效应模型能够反映微构件的弯曲变形、固有频率、屈曲载荷等力学性能的尺寸效应以及力电耦合环境下的吸合特性的尺寸效应,研究结果可以为微机电系统中微构件的设计和实验研究提供理论依据。
Attributed to the small mass and size, MEMS (Micro Electro Mechanical Systems, abbreviated as MEMS) come with high fundamental resonance frequencies, high quality factors and low power dissipation. MEMS have been widely used in the field of resonators, mixers, sensors and actuators. According to geometry and loaded forms of those devices and products, main structures can be simplified to be some typical structural forms, such as micro-beams, micro-plates, micro-membranes and micro-bars etc. In the micro scale range, the sub-micro scale range and the nano scale range, the mechanical properties of micro-component are very different from that in the macro scale range.
At present, the size effects of the mechanical properties for the micro-component have been discovered in the mechanical properties experiments. The size effects of the micro-compoment cannot be described using classical continuum mechanics. Therefore, the creation of new theory which is able to explain and describe the effect of the micro-components is crucial. In order to describe and explain the size effect of the micro-components, the strain gradient is introduced into the strain gradient theory when the material intrinsic characteristic scale parameters are introduced into the constitutive relations in the new theory. At the same time, the strain gradient theory simplified as the classic elasticity theory when the material intrinsic characteristic scale parameters equal to zero. Therefore, the strain gradient theory is the expansion of the classical elasticity theory.
In our paper, the micro-components in MEMS are mainly analyzed and the size effect models are established based on the strain gradient theory. The size effects of mechanical property are widely studied for the components in the micro scale range. Main content of our work are as follows:
A nonlinear size effect model of micro-beam is established based on the strain gradient theory and the Hamiltonian variational principle. In the new model, the middle plane elongation and the residual stress are included while the three intrinsic characteristic scale paramenters are introduced. Therefore, the new model can predict not only the size effect of the micro-beams but also the effect of the middle plane elongation and the residual stress on the size effects. When the two material intrinsic characteristic scale parameters regarding as the dilatation gradient tensor and deviatoric stretch gradient tensor are zero, the new model can be simplified as the nonlinear model based on the couple stress theory. When the three material intrinsic characteristic scale parameters are zero, the new model can be simplified as the nonlinear model based on the classical theory. For the simply supported micro-beam and the clamped-clamped micro-beam, the size effects of the nonlinear static deformation, the nonlinear natural frequency and the nonlinear buckling load are analyzed. The size effect on the static deformation, the natural frequency and the buckling load of micro-beams are assessed when the feature size of the micro-beams approaches to the material intrinsic characteristic scale parameter. However, the size effect gradually reduced when the feature sizes of micro-beams are by far larger than the material intrinsic characteristic scale parameter. The effects of the middle plane elongation and the residual stress on the effect of micro-beams are significant. At the same time, the size effect phenomenon significantly reduced when the residual stress increases. The bending deformation, the buckling load and the nonlinear natural frequency based on the strain gradient theory are equal to the results based the classical theory when the feature size of the micro-beams are much larger than the material intrinsic characteristic length scales. Therefore, the new model can capture size effect on the nonlinear bending deformation, the nonlinear buckling load and the nonlinear natural frequency of micro-beams and can provide a theoretical basis for the structural design and experimental testing of the micro-beams.
Based on the strain gradient theory and Hamiltonian variational principle, the size effect model for arbitrary boundary shape micro-plates is derived. The new model can captured the size effect of the mechanical properties of the micro-plates based on the introduction of three material intrinsic characteristic scale parameters. At the same time, the boundary conditions are derived at the arbitrary shape boundary and the coner of the plate. The new model not only includes the effect of the strain tensor and rotation gradient tentor, but also contains the stretch gradient skew tensor and dilatation strain gradient tensor, can fully descrie the size effects of the static deformation and the natural frequency of the micro-plate. For the simply supported micro-plate and the clamped micro-plate, the size effects of the static and the natural frequency are analyzed. The numerical results are compared with that of the couple stress micro-plates and the classical micro-plates. The results show that the normalized static deformation and the normalized nature frequency of circular micro-plates have significant size effects. The size effect is different when the constraint of the circular plate is different.
The size effect model of the electrostatically acturated nonlinear micro-beams is established based on the strain gradient theory and Hamiltonian variational principle. Taking into account size effect on mechanical propreties of materials and the inherent nonlinear property of electrostatic force and the middle plane elongation, approximate expression of the pull-in voltage of electrostatically actuated micro-beams are obtained by using the Rayleigh-Ritz method. The results show that the dimensionless pull-in voltage increases significantly with the dimensionless micro-beam thickness decreasing, showing a significant size effect. When the material intrinsic characteristic length increases, the size effect of the dimensionless pull-in voltage is more significantly, indicating that the effect of the strain gradient on the micro-beam pull-in voltage is remarkable. When the dimensionless micro-beam thickness decreases, the effect of the residual stress on the dimensionless pull-in voltage is significant. The size effect of the normalized pull-in voltage is weakened when the residual stress increases, indicating the residual stress can decrease the effect of the strain gradient on the pull-in voltage. The effect of the middle plane elongation on the pull-in voltage is reduced markedly when the strain gradient is considered. The results can prove the reference in the design of micro structures in MEMS.
Based on the strain gradient theory and the Hamiltonian variational principle, the size effect model and the numerical simulation model of the micro-beam in the plane bar system are derived. The size effect model and the numerical simulation model can analyze the size effect of micro-beam in the plane bar system due to the introducation of the material intrinsic characteristic scale parameters. At the same time, the size effects of the static deformation and the electrostatic pull-in voltage for the nonlinear micro-beam can be analyzed based on the numerical model. Due to a large number of micro-springs are used in MEMS, the size effects of the stiffness of the plane micro-spring are analyzed. The results show that the stiffnesses of micro-spring have significant size effect and provide a theoretical basis for the analysis and design of complex plane micro-spring.
The above mechanical models can capture size effect on the static deformation, the natural frequency, the buckling load and the pull-in voltage of the micro components. The research results can provide a theoretical guideline for the optimal design and experimental tests of the micro-components in MEMS devices.
引文
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