微梁力学性能尺寸效应的研究
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摘要
随着微机电系统(Micro-electro-mechanical-systems,简写为MEMS)技术的兴起和发展,功能各异的MEMS器件和产品已经被成功研制并得到了广泛应用。根据这些MEMS器件中主要结构的特征尺寸和受力特点,可以将其简化为一些简单的结构形式,微梁就是一种在微机电系统中得到了广泛应用的典型结构形式,由于其几何尺寸的微小化以及制备工艺的特殊化,微梁的力学性能与宏观尺寸下有着很大的不同。目前,微尺度实验证实微结构的力学性能存在尺寸效应,传统力学理论的本构关系中不包含任何与尺度相关的参数,不能够描述和解释力学性能的尺寸效应现象。因此,发展和完善能够解释和描述微结构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,通过在其本构关系中引入具有长度量纲的参数来考虑应变梯度的影响,能够描述和解释微构件力学性能的尺寸效应现象。
本文以微梁为研究对象,基于应变梯度理论(包括偶应力理论和全应变梯度弹性理论)对其力学性能的尺寸效应进行了研究,建立了相应的理论模型,并对其力学性能的变化规律进行了深入分析。研究工作的主要内容如下:
考虑微结构力学性能的尺寸效应对微梁动态特性的影响,基于偶应力理论并利用哈密尔顿原理,建立了微梁动态性能的理论模型。该模型中除了2个传统的材料参数之外,只包含1个材料内禀特征尺寸参数。结合悬臂梁和简支梁的特征尺寸分析了微梁固有频率的变化规律,结果表明,微梁的固有频率存在尺寸效应现象,所得结论可以为微梁的结构设计和实验测试提供理论基础。
针对细长压杆受到轴向压缩载荷时的屈曲问题,基于偶应力理论并利用能量变分法建立了细长压杆屈曲特性的理论模型。分析细长压杆屈曲特性的变化规律并给出了几种典型边界条件下细长压杆的屈曲载荷和屈曲模态的解析表达式,结果表明,细长压杆的屈曲载荷存在尺寸效应现象,而其屈曲模态与特征尺寸无关,所得结论可以为屈曲型微构件的设计和优化以及相关的实验研究提供理论依据。
考虑微结构尺寸效应对力电耦合环境下微梁Pull-in特性的影响,基于偶应力理论建立了静电驱动微梁Pull-in特性的理论模型。考虑到静电载荷固有的非线性特性和静电驱动微梁的力电耦合特性,采用瑞恩里茨法来近似处理静电载荷,利用能量变分法给出了微梁Pull-in特性的近似解析表达式,分析了静电驱动微梁Pull-in电压和Pull-in位移的变化规律。结果表明:当微梁的厚度h等于材料内禀特征尺寸参数l时,其Pull-in电压增大2.7倍,呈现出明显的尺寸效应现象;而当微梁的厚度h远大于材料内禀特征尺寸参数l时,其Pull-in电压与传统理论值基本相等。另外,静电驱动微梁的无量纲Pull-in位移与材料内禀特征尺寸参数l无关。
基于全应变梯度弹性理论,建立了微梁静动态弯曲特性的理论模型,给出其控制方程和边界条件。该模型中除了2个传统的材料常数之外,还包括3个分别对应于拉伸梯度张量偏斜分量、膨胀梯度张量和旋转梯度张量的材料内禀特征尺寸参数;而边界条件是由传统边界条件和高阶边界条件所组成。结合微梁的特征尺寸和边界条件,分析了微梁弯曲刚度和固有频率的变化规律。结果表明,微梁的无量纲弯曲刚度和无量纲固有频率存在尺寸效应。所建模型不仅包含了应变张量和旋转梯度张量的影响,而且还包含了拉伸梯度张量偏斜分量和膨胀梯度张量的影响,能全面地反映出微梁静动态弯曲特性的尺寸效应现象。
本文所建立的微梁力学模型能够反映出其弯曲、动态、屈曲和在力电耦合环境下的Pull-in特性等力学性能的尺寸效应,研究结果可以为MEMS产品的结构设计和实验研究提供理论依据。
With the advances of Micro-electro-mechanical systems (MEMS), a wide range of MEMS devices and products have been fabricated. According to geometry and loaded forms of those devices and products, main structures can be simplified to be some typical structural forms. Micro-beam is one of most used structural forms in MEMS devices and products. Due to small size and special processing technology, mechanical properties of micro-beams are very different from that in the macro-scale range. Therefore, thoroughly understanding and accurate characterization of mechanical properties of micro-beams is critical for optimal design and reliability of MEMS devices and products. Nowadays, many micro-scale experiments have verified that mechanical properties of some materials at the micron scale are size dependent. No material length scale parameter is introduced into the constitutive relation of conventional theory, so it cannot characterize and interpret size effect on the mechanical properties of materials and structures at the micron scale. Strain gradient theories, which include effects of strain gradient tensor and possess material length scale parameter, can characterize and interpret size effect on the mechanical properties of micro-structures at the micron scale.
Based on strain gradient theories (including couple stress theory and strain gradient elasticity theory), size effect on the mechanical properties of micro-beams are studied in this work. Theoretical models are established and influence of size effect on mechanical properties of micro-beams is analyzed thoroughly. Conclusions can render basic guidelines for optimal design and experimental tests of MEMS devices and products. Main contents of this work are as follows:
In order to analyze influence of size effect of materials on the dynamic property of micro-beams, a theoretical model of dynamic property of micro-beams is established on the basis of a modified couple stress theory. It can be seen that in the new model, only one additional material length scale parameter is included besides two classical material constants. In the light of feature sizes of simply supported micro-beams and cantilever micro-beams, the governing equations of micro-beams are solved and size effect on natural frequencies of micro-beams are assessed. It is found that natural frequencies predicted by the newly established model are 2.6 and 3 times as that calculated from classical beam model when feature sizes of micro-beams (for rectangular and circular micro-beams, feature sizes are thickness and diameter, respectively) equal to material internal length scale. However, Natural frequencies predicted by the newly established model are approximate to that calculated from classical beam model when feature sizes of micro-beams are by far larger than material internal length scale parameter. Therefore, the new model can capture size effect on natural frequencies of micro-beams and can provide the theoretical basis for optimal design and experimental test of dynamic properties of MEMS products and devices.
For buckling problems of slender columns under the action of axial compressive loads, a theoretical model for buckling characteristics of slender columns is established by a combination of the basic equations of the modified couple stress theory and the energy variational method. It can be seen that in the new model, only one additional material length scale parameter is included besides two classical material constants. Solutions of corresponding boundary value problems for buckling are presented and size effect on normalized buckling loads of slender columns with retangular and circular cross sectional shapes are analyzed. It is shown that the buckling load of slender columns predicted by the newly established model are 7 and 9 times as that calculated from classical beam model when feature sizes (for rectangular and circular micro-beams, feature sizes are thickness and diameter, respectively) of slender columns equal to material internal length scale parameter. However, the buckling load of slender columns predicted by the newly established model conforms to that calculated from classical beam model when feature sizes of micro-beams are by far larger than material internal length scale parameter. Therefore, the new model can capture size effect on buckling loads of slender columns and can provide a theoretical basis for design and control of similar buckling MEMS devices and products.
For pull-in characteristics of electrostatically actuated micro-structures commonly used in practical engineerings, a theoretical model for pull-in characteristics of micro-beams is established based on a modified couple stress theory. It should be emphasized that, in the new model, only one additional material length scale parameter is included besides two classical material constants.Taking into account size effect on mechanical properties of materials and the inherent nonlinear property of electrostatic force, approximate analytical solutions of the pull-in voltage and pull-in displacement of electrostatically actuated micro-beams are obtained by using the Rayleigh-Ritz method. In the light of feature size of cantilever micro-beams and fixed-fixed micro-beams, size effect on pull-in characteristics of micro-beams is assessed. It is shown that the normalized pull-in voltage of the micro-beams increases by a factor on 2.7 as the thickness of microbeams equals to the material internal length scale parameter and exhibits size dependent. However, size effect is almost diminishing as the thickness of microbeams is far greater than material internal length scale parameter. Moreover, the normalized pull-in displacement of microbeams is size independence and equals to 0.448 and 0.398 for cantilever beams and clamped-clamped beams, respectively. Therefore, the new model can capture size effect on pull-in characteristics of electrically actuated micro-beams and can give a theoretical guide for optimal design and experimental tests of electrostatically actuated MEMS structures.
The modified couple stress proposed by Yang et al. only includes the effect of strain tensor and rotation gradient tensor. Afterwards, Lam et al. presented a new strain gradient elasticity theory, which takes into account the effects of strain tensor, rotation gradient tensor and stretch gradient tensor. In order to analyze the bending behaviour of micro-beams, a bending model for micro-beams is established based on the strain gradient elasticity theory and governing equation and boundary conditions are derived. It can be seen that only 3 additional material length scale parameters are introduced besides 2 classical material constants in the new model. Moreover, boundary conditions consist of 2 classical boundary conditions and 1 additional higher-order boundary condition. In the light of geometries and boundary conditions of micro-beams, size effect on normalized bending rigidities of micro-beams and boundary layer effect are assessed. It is shown that bending rigidities of micro-beams based on strain gradient elasticity theory are about 12 times as that from classical theory model, when the thickness of retangular micro-beams become comparable to material internal length scale parameter and exhibit size dependent (h=20μm, l=17.6μm). However, bending rigidities of micro-beams based on starin gradient elasticity theory conform to that calculated from classical theory model when the feature sizes of micro-beams are far greater than material internal length scale parameters. Moreover, the higher-order boundary conditions results in boundary layer effect of deformed cantilever micro-beams at the fixed end. Therefore, the newly established model includes influences of rotation gradient tensor, deviatoric stretch gradient tensor and dilatation gradient tensor besides strain tensor, and can capture size effect on normalized bending rigidities and boundary layer effect of cantilever micro-beams at the fixed end.
Taking into account the effect of rotation gradient tensor and stretch gradient tensor on dynamic property of micro-beams, a theoretical model for flexural micro-beams is established based on the strain gradient elasticity theory and by using Hamilton principle. Governing equation, initial conditions and boundary conditions of flexural micro-beams motion are derived. It can be seen that only 3 additional material length scale parameters are introduced in the new model besides 2 classical material constants. Moreover, boundary conditions consist of 2 classical boundary conditions and 1 additional higher-order boundary condition. In the light of geometry and boundary conditions of cantilever and simply supported micro-beams, size effect on natural frequencis of micro-beams is assessed. It is shown that natural frequencies of micro-beams based on strain gradient elasticity theory are about 12 times as that from classical theory model, as the thickness of retangular micro-beams become comparable to material internal length scale parameter and exhibit size dependent (h=20μm, l=17.6μm). However, natural frequencies of beams based on starin gradient elasticity theory conform to that calculated from classical theory model when the feature sizes of micro-beams are far greater than material internal length scale parameters. Therefore, the newly established model includes influences of rotation gradient tensor, deviatoric stretch gradient tensor and dilatation gradient tensor besides strain tensor, and can capture size effect on normalized natural frequencies of micro-beams.
The above mechanical models can capture size effect on mechanical properties of micro-beams and conclusions can render theoretical guidelines for optimal design and experimental tests of MEMS devices.
本文以微梁为研究对象,基于应变梯度理论(包括偶应力理论和全应变梯度弹性理论)对其力学性能的尺寸效应进行了研究,建立了相应的理论模型,并对其力学性能的变化规律进行了深入分析。研究工作的主要内容如下:
考虑微结构力学性能的尺寸效应对微梁动态特性的影响,基于偶应力理论并利用哈密尔顿原理,建立了微梁动态性能的理论模型。该模型中除了2个传统的材料参数之外,只包含1个材料内禀特征尺寸参数。结合悬臂梁和简支梁的特征尺寸分析了微梁固有频率的变化规律,结果表明,微梁的固有频率存在尺寸效应现象,所得结论可以为微梁的结构设计和实验测试提供理论基础。
针对细长压杆受到轴向压缩载荷时的屈曲问题,基于偶应力理论并利用能量变分法建立了细长压杆屈曲特性的理论模型。分析细长压杆屈曲特性的变化规律并给出了几种典型边界条件下细长压杆的屈曲载荷和屈曲模态的解析表达式,结果表明,细长压杆的屈曲载荷存在尺寸效应现象,而其屈曲模态与特征尺寸无关,所得结论可以为屈曲型微构件的设计和优化以及相关的实验研究提供理论依据。
考虑微结构尺寸效应对力电耦合环境下微梁Pull-in特性的影响,基于偶应力理论建立了静电驱动微梁Pull-in特性的理论模型。考虑到静电载荷固有的非线性特性和静电驱动微梁的力电耦合特性,采用瑞恩里茨法来近似处理静电载荷,利用能量变分法给出了微梁Pull-in特性的近似解析表达式,分析了静电驱动微梁Pull-in电压和Pull-in位移的变化规律。结果表明:当微梁的厚度h等于材料内禀特征尺寸参数l时,其Pull-in电压增大2.7倍,呈现出明显的尺寸效应现象;而当微梁的厚度h远大于材料内禀特征尺寸参数l时,其Pull-in电压与传统理论值基本相等。另外,静电驱动微梁的无量纲Pull-in位移与材料内禀特征尺寸参数l无关。
基于全应变梯度弹性理论,建立了微梁静动态弯曲特性的理论模型,给出其控制方程和边界条件。该模型中除了2个传统的材料常数之外,还包括3个分别对应于拉伸梯度张量偏斜分量、膨胀梯度张量和旋转梯度张量的材料内禀特征尺寸参数;而边界条件是由传统边界条件和高阶边界条件所组成。结合微梁的特征尺寸和边界条件,分析了微梁弯曲刚度和固有频率的变化规律。结果表明,微梁的无量纲弯曲刚度和无量纲固有频率存在尺寸效应。所建模型不仅包含了应变张量和旋转梯度张量的影响,而且还包含了拉伸梯度张量偏斜分量和膨胀梯度张量的影响,能全面地反映出微梁静动态弯曲特性的尺寸效应现象。
本文所建立的微梁力学模型能够反映出其弯曲、动态、屈曲和在力电耦合环境下的Pull-in特性等力学性能的尺寸效应,研究结果可以为MEMS产品的结构设计和实验研究提供理论依据。
With the advances of Micro-electro-mechanical systems (MEMS), a wide range of MEMS devices and products have been fabricated. According to geometry and loaded forms of those devices and products, main structures can be simplified to be some typical structural forms. Micro-beam is one of most used structural forms in MEMS devices and products. Due to small size and special processing technology, mechanical properties of micro-beams are very different from that in the macro-scale range. Therefore, thoroughly understanding and accurate characterization of mechanical properties of micro-beams is critical for optimal design and reliability of MEMS devices and products. Nowadays, many micro-scale experiments have verified that mechanical properties of some materials at the micron scale are size dependent. No material length scale parameter is introduced into the constitutive relation of conventional theory, so it cannot characterize and interpret size effect on the mechanical properties of materials and structures at the micron scale. Strain gradient theories, which include effects of strain gradient tensor and possess material length scale parameter, can characterize and interpret size effect on the mechanical properties of micro-structures at the micron scale.
Based on strain gradient theories (including couple stress theory and strain gradient elasticity theory), size effect on the mechanical properties of micro-beams are studied in this work. Theoretical models are established and influence of size effect on mechanical properties of micro-beams is analyzed thoroughly. Conclusions can render basic guidelines for optimal design and experimental tests of MEMS devices and products. Main contents of this work are as follows:
In order to analyze influence of size effect of materials on the dynamic property of micro-beams, a theoretical model of dynamic property of micro-beams is established on the basis of a modified couple stress theory. It can be seen that in the new model, only one additional material length scale parameter is included besides two classical material constants. In the light of feature sizes of simply supported micro-beams and cantilever micro-beams, the governing equations of micro-beams are solved and size effect on natural frequencies of micro-beams are assessed. It is found that natural frequencies predicted by the newly established model are 2.6 and 3 times as that calculated from classical beam model when feature sizes of micro-beams (for rectangular and circular micro-beams, feature sizes are thickness and diameter, respectively) equal to material internal length scale. However, Natural frequencies predicted by the newly established model are approximate to that calculated from classical beam model when feature sizes of micro-beams are by far larger than material internal length scale parameter. Therefore, the new model can capture size effect on natural frequencies of micro-beams and can provide the theoretical basis for optimal design and experimental test of dynamic properties of MEMS products and devices.
For buckling problems of slender columns under the action of axial compressive loads, a theoretical model for buckling characteristics of slender columns is established by a combination of the basic equations of the modified couple stress theory and the energy variational method. It can be seen that in the new model, only one additional material length scale parameter is included besides two classical material constants. Solutions of corresponding boundary value problems for buckling are presented and size effect on normalized buckling loads of slender columns with retangular and circular cross sectional shapes are analyzed. It is shown that the buckling load of slender columns predicted by the newly established model are 7 and 9 times as that calculated from classical beam model when feature sizes (for rectangular and circular micro-beams, feature sizes are thickness and diameter, respectively) of slender columns equal to material internal length scale parameter. However, the buckling load of slender columns predicted by the newly established model conforms to that calculated from classical beam model when feature sizes of micro-beams are by far larger than material internal length scale parameter. Therefore, the new model can capture size effect on buckling loads of slender columns and can provide a theoretical basis for design and control of similar buckling MEMS devices and products.
For pull-in characteristics of electrostatically actuated micro-structures commonly used in practical engineerings, a theoretical model for pull-in characteristics of micro-beams is established based on a modified couple stress theory. It should be emphasized that, in the new model, only one additional material length scale parameter is included besides two classical material constants.Taking into account size effect on mechanical properties of materials and the inherent nonlinear property of electrostatic force, approximate analytical solutions of the pull-in voltage and pull-in displacement of electrostatically actuated micro-beams are obtained by using the Rayleigh-Ritz method. In the light of feature size of cantilever micro-beams and fixed-fixed micro-beams, size effect on pull-in characteristics of micro-beams is assessed. It is shown that the normalized pull-in voltage of the micro-beams increases by a factor on 2.7 as the thickness of microbeams equals to the material internal length scale parameter and exhibits size dependent. However, size effect is almost diminishing as the thickness of microbeams is far greater than material internal length scale parameter. Moreover, the normalized pull-in displacement of microbeams is size independence and equals to 0.448 and 0.398 for cantilever beams and clamped-clamped beams, respectively. Therefore, the new model can capture size effect on pull-in characteristics of electrically actuated micro-beams and can give a theoretical guide for optimal design and experimental tests of electrostatically actuated MEMS structures.
The modified couple stress proposed by Yang et al. only includes the effect of strain tensor and rotation gradient tensor. Afterwards, Lam et al. presented a new strain gradient elasticity theory, which takes into account the effects of strain tensor, rotation gradient tensor and stretch gradient tensor. In order to analyze the bending behaviour of micro-beams, a bending model for micro-beams is established based on the strain gradient elasticity theory and governing equation and boundary conditions are derived. It can be seen that only 3 additional material length scale parameters are introduced besides 2 classical material constants in the new model. Moreover, boundary conditions consist of 2 classical boundary conditions and 1 additional higher-order boundary condition. In the light of geometries and boundary conditions of micro-beams, size effect on normalized bending rigidities of micro-beams and boundary layer effect are assessed. It is shown that bending rigidities of micro-beams based on strain gradient elasticity theory are about 12 times as that from classical theory model, when the thickness of retangular micro-beams become comparable to material internal length scale parameter and exhibit size dependent (h=20μm, l=17.6μm). However, bending rigidities of micro-beams based on starin gradient elasticity theory conform to that calculated from classical theory model when the feature sizes of micro-beams are far greater than material internal length scale parameters. Moreover, the higher-order boundary conditions results in boundary layer effect of deformed cantilever micro-beams at the fixed end. Therefore, the newly established model includes influences of rotation gradient tensor, deviatoric stretch gradient tensor and dilatation gradient tensor besides strain tensor, and can capture size effect on normalized bending rigidities and boundary layer effect of cantilever micro-beams at the fixed end.
Taking into account the effect of rotation gradient tensor and stretch gradient tensor on dynamic property of micro-beams, a theoretical model for flexural micro-beams is established based on the strain gradient elasticity theory and by using Hamilton principle. Governing equation, initial conditions and boundary conditions of flexural micro-beams motion are derived. It can be seen that only 3 additional material length scale parameters are introduced in the new model besides 2 classical material constants. Moreover, boundary conditions consist of 2 classical boundary conditions and 1 additional higher-order boundary condition. In the light of geometry and boundary conditions of cantilever and simply supported micro-beams, size effect on natural frequencis of micro-beams is assessed. It is shown that natural frequencies of micro-beams based on strain gradient elasticity theory are about 12 times as that from classical theory model, as the thickness of retangular micro-beams become comparable to material internal length scale parameter and exhibit size dependent (h=20μm, l=17.6μm). However, natural frequencies of beams based on starin gradient elasticity theory conform to that calculated from classical theory model when the feature sizes of micro-beams are far greater than material internal length scale parameters. Therefore, the newly established model includes influences of rotation gradient tensor, deviatoric stretch gradient tensor and dilatation gradient tensor besides strain tensor, and can capture size effect on normalized natural frequencies of micro-beams.
The above mechanical models can capture size effect on mechanical properties of micro-beams and conclusions can render theoretical guidelines for optimal design and experimental tests of MEMS devices.
引文
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