Mechanical properties of silicon nanobeams with an undercut evaluated by combining the dynamic resonance test and finite element analysis

Mechanical properties of silicon nanobeams are of prime importance in nanoelectromechanical system applications.A numerical experimental method of determining resonant frequencies and Young’s modulus of nanobeams by combining finite element analysis and frequency response tests based on an electrostatic excitation and visual detection by using a laser Doppler vibrometer is presented in this paper.Silicon nanobeam test structures are fabricated from silicon-oninsulator wafers by using a standard lithography and anisotropic wet etching release process,which inevitably generates the undercut of the nanobeam clamping.In conjunction with three-dimensional finite element numerical simulations incorporating the geometric undercut,dynamic resonance tests reveal that the undercut significantly reduces resonant frequencies of nanobeams due to the fact that it effectively increases the nanobeam length by a correct value △L,which is a key parameter that is correlated with deviations in the resonant frequencies predicted from the ideal Euler-Bernoulli beam theory and experimentally measured data.By using a least-square fit expression including △L,we finally extract Young’s modulus from the measured resonance frequency versus effective length dependency and find that Young’s modulus of a silicon nanobeam with 200-nm thickness is close to that of bulk silicon.This result supports that the finite size effect due to the surface effect does not play a role in the mechanical elastic behaviour of silicon nanobeams with thickness larger than 200 nm.     

Mechanical properties an undercut evaluated resonance test and of silicon nanobeams with by combining the dynamic finite element analysis

Mechanical properties of silicon nanobeams are of prime importance in nanoelectromechanical system applications. A numerical experimental method of determining resonant frequencies and Young＇s modulus of nanobeams by combining finite element analysis and frequency response tests based on an electrostatic excitation and visual detection by using a laser Doppler vibrometer is presented in this paper. Silicon nanobeam test structures are fabricated from silicon-oninsulator wafers by using a standard lithography and anisotropic wet etching release process, which inevitably generates the undercut of the nanobeam clamping. In conjunction with three-dimensional finite element numerical simulations incorporating the geometric undercut, dynamic resonance tests reveal that the undercut significantly reduces resonant frequencies of nanobeams due to the fact that it effectively increases the nanobeam length by a correct value △L, which is a key parameter that is correlated with deviations in the resonant frequencies predicted from the ideal Euler-Bernoulli beam theory and experimentally measured data. By using a least-square fit expression including △L, we finally extract Young＇s modulus from the measured resonance frequency versus effective length dependency and find that Young＇s modulus of a silicon nanobeam with 200-nm thickness is close to that of bulk silicon. This result supports that the finite size effect due to the surface effect does not play a role in the mechanical elastic behaviour of silicon nanobeams with thickness larger than 200 nm.     

黏弹性材料的动态力学特性及其对覆盖层吸声性能的影响

本文利用标准化动态力学测量手段获得了某种高分子聚合物的动态杨氏模量,并根据时温等效原理对动态杨氏模量与声学测量在频段上的差异加以分析和转换,得到了500—7500 Hz频率范围内该黏弹性材料杨氏模量随频率变化的特性.基于所测得动态杨氏模量,采用有限元方法分析了均匀黏弹材料的吸声性能,并将仿真结果与样品声管实验数据进行对比,验证了测试所得参数的准确性.进一步仿真分析了含有局域共振结构的声学覆盖层吸声性能,并讨论了黏弹性材料的动态特性对其吸声性能的影响,提出了改进水声覆盖层低频宽带吸声特性的建议.     

Size effect of the elastic modulus of rectangular nanobeams:Surface elasticity effect

The size-dependent elastic property of rectangular nanobeams （nanowires or nanoplates） induced by the surface elas- ticity effect is investigated by using a developed modified core-shell model. The effect of surface elasticity on the elastic modulus of nanobeams can be characterized by two surface related parameters, i.e., inhomogeneous degree constant and surface layer thickness. The analytical results show that the elastic modulus of the rectangular nanobeam exhibits a distinct size effect when its characteristic size reduces below 1 O0 nm. It is also found that the theoretical results calculated by a mod- ified core-shell model have more obvious advantages than those by other models （core-shell model and core-surface model） by comparing them with relevant experimental measurements and computational results, especially when the dimensions of nanostructures reduce to a few tens of nanometers.     

掺杂单晶硅纳米薄膜杨氏模量的多尺度理论模型

硅纳米材料物理性能的研究对其在半导体技术中的应用是十分重要的. 而掺杂有利于改善硅纳米材料的物理特性, 提高应用价值, 所以本文基于半连续体模型运用Keating形变势, 通过模型计算, 研究了不同位置及不同掺杂浓度的单晶硅纳米薄膜[100]方向的杨氏模量, 分析了掺杂浓度及掺杂位置不同时硅膜杨氏模量与膜厚关系, 结果表明, 与纯硅膜杨氏模量相比, 不同位置的掺杂对硅膜杨氏模量的影响并不明显, 不同浓度的掺杂对硅膜杨氏模量的影响较小. 而随着硅膜厚度的不断增加, 掺杂硅膜杨氏模量与纯硅膜杨氏模量的变化趋势一致, 特别是较小尺寸时的硅膜杨氏模量变化较大. 说明影响硅膜杨氏模量的主要因素是硅膜厚度. 该计算结果对研究硅纳米材料的其他力学特性有一定的参考价值, 也为进一步研究掺杂对纳米硅材料力学性能的影响提供一种全新思路.     

Scale-dependent effects on wave propagation in magnetically affected single/double-layered compositionally graded nanosize beams

This article deals with the wave propagation analysis of single/double layered functionally graded (FG) size-dependent nanobeams in elastic medium and subjected to a longitudinal magnetic field employing nonlocal elasticity theory. Material properties of nanobeam change gradually according to the sigmoid function. Applying an analytical solution, the acoustical and optical dispersion relations are explored for various wave number, nonlocality parameter, material composition, elastic foundation constants, and magnetic field intensity. It is found that frequency and phase velocity of waves propagating in S-FGM nanobeam are significantly affected by these parameters. Also, presence of cut-off and escape frequencies in wave propagation analysis of embedded S-FGM nanobeams is investigated.     

Measurement of the Elasticity of Biological Soft Tissue of Finite Thickness

Elasticity is of profound significance to evaluating the function of a biological soft tissue.When the elasticity of a tissue is macroscopically changed,it means that the biological function of the tissue is abnormal and some disease or injury may occur.In the present work,an elastometer is developed to measure the elasticity of biological soft tissues.The measurement is based on the indentation method and the force is measured by the bending of the cantilever.The force-indentation data of the soft tissue is experimentally measured by this elastometer and Young's modulus of the tissue is calculated using the Hertz-Sneddon model.For comparison,a numerical model for the indentation method is established using the finite element method.The difference between the actual modulus and the measured modulus is discussed.The effect of the thickness of the specimen on the measurement is investigated.Young's moduli of beef,porcine liver and porcine kidney are experimentally measured.The results indicate that our elastometer is effective in measuring Young's modulus of a soft tissue quantitatively.     

Mechanical properties of nanotubes of polyelectrolyte multilayers

The elastic properties of nanotubes fabricated by layer-by-layer (LbL) assembly of polyelectrolytes in the nanopores of polycarbonate track-etched membranes have been investigated by resonant contact Atomic Force Microscopy (AFM), for nanotube diameters in the range of 100 to 200 nm. The elastic modulus of the nanotubes was computed from the resonance frequencies of a cantilever resting on freely suspended LbL nanotubes. An average value of 115 MPa was found in air for Young's modulus of these nanostructures, well below the values reported for dry, flat multilayers, but in the range of values reported for water-swollen flat multilayers. These low values are most probably due to the lower degree of ionic cross-linking of LbL nanotubes and their consequently higher water content in air, resulting from the peculiar mode of growth of nanoconfined polyelectrolyte multilayers.     

Wave propagation in magneto-electro-elastic nanobeams via two nonlocal beam models

This paper makes the first attempt to investigate the dispersion behavior of waves in magneto-electro-elastic (MEE) nanobeams. The Euler nanobeam model and Timoshenko nanobeam model are developed in the formulation based on the nonlocal theory. By using the Hamilton’s principle, we derive the governing equations which are then solved analytically to obtain the dispersion relations of MEE nanobeams. Results are presented to highlight the influences of the thermo-electro-magnetic loadings and nonlocal parameter on the wave propagation characteristics of MEE nanobeams. It is found that the thermo-electro-magnetic loadings can lead to the occurrence of the cut-off wave number below which the wave can’t propagate in MEE nanobeams.     

Propagation of waves in nonlocal porous multi-phase nanocrystalline nanobeams under longitudinal magnetic field

ABSTRACT

This article investigates wave propagation behavior of a multi-phase nanocrystalline nanobeam subjected to a longitudinal magnetic field in the framework of nonlocal couple stress and surface elasticity theories. In this model, the essential measures to describe the real material structure of nanocrystalline nanobeams and the size effects were incorporated. This non-classical nanobeam model contains couple stress effect to capture grains micro-rotations. Moreover, the nonlocal elasticity theory is employed to study the nonlocal and long-range interactions between the particles. The present model can degenerate into the classical model if the nonlocal parameter, couple stress and surface effects are omitted. Hamilton’s principle is employed to derive the governing equations which are solved by applying an analytical method. The frequencies are compared with those of nonlocal and couple stress-based beams. It is showed that wave frequencies and phase velocities of a nanocrystalline nanobeam depend on the grain size, grain rotations, porosities, interface, magnetic field, surface effect and nonlocality.     

偏置磁场对磁致伸缩/弹性/压电层合材料磁电效应的影响

根据磁致伸缩材料的非线性本构关系得到其动态杨氏弹性模量和动态压磁系数,结合等效电路法得到磁致伸缩/弹性/压电层合材料的磁电效应与磁致伸缩材料的动态杨氏弹性模量和动态压磁系数的关系,讨论了偏置磁场对这种层合材料的谐振频率和谐振磁电电压转换系数的影响.理论推导和实验结果均表明,存在最佳偏置磁场使磁致伸缩/弹性/压电层合材料的谐振磁电电压转换系数最大. 关键词： 磁致伸缩/弹性/压电层合材料 磁电效应 偏置磁场 非线性本构关系     

掺杂硅纳米梁谐振频率的理论模型及分子动力学模拟

通过理论计算与模拟,研究分析了P元素替代掺杂单晶硅纳米梁的谐振频率.计算模拟了两端固支单晶硅纳米梁的谐振频率随尺寸、掺杂浓度与温度的变化.通过对计算结果与模拟结果的分析得到:单晶硅纳米梁的谐振频率随着硅纳米梁长度尺寸的增大而减小;硅纳米梁的谐振频率随着掺杂浓度的增大而增大,但变化趋势并不明显;最后考虑了温度效应,发现掺杂硅纳米梁的谐振频率随着温度的增大而减小,但从谐振频率的数值来看,硅梁的谐振频率随温度的变化趋势并不明显,即温度对硅梁谐振频率基本无影响.由此得出结论:掺杂浓度与温度对硅纳米梁谐振频率的影响很小,影响单晶硅纳米梁谐振频率的主要因素是尺寸大小,掺杂单晶硅纳米梁的谐振频率具有尺寸效应.     

Experimental measurements on transverse vibration characteristics of piezoceramic rectangular plates by optical methods

This study provides two non-contact optical techniques to investigate the transverse vibration characteristics of piezoceramic rectangular plates in resonance. These methods, including the amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) and laser Doppler vibrometer (LDV), are full-field measurement for AF-ESPI and point-wise displacement measurement for LDV, respectively. The edges of these piezoceramic rectangular plates may either be fixed or free. Both resonant frequencies and mode shapes of vibrating piezoceramic plates can be obtained simultaneously by AF-ESPI. Excellent quality of the interferometric fringe patterns for the mode shapes is obtained. In the LDV system, a built-in dynamic signal analyzer (DSA) composed of DSA software and a plug-in waveform generator board can provide the piezoceramic plates with the swept-sine excitation signal, whose gain at corresponding frequencies is analyzed by the DSA software. The peaks appeared in the frequency response curve are resonant frequencies. In addition to these optical methods, the numerical computation based on the finite element analysis is used to verify the experimental results. Good agreements of the mode shapes and resonant frequencies are obtained for experimental and numerical results.     

Analysis of Single-Walled Carbon Nanotubes Using a Chemical Bond Element Model

提出了一种纳米尺度的有限元方法,碳纳米管中的碳-碳化学键被模拟为键单元．按照平衡关系,根据有限元理论,作用于每个碳原子上的作用力可以写成键单元的刚度矩阵与每个碳原子位移的乘积．在分子力学的基本假设下,键单元刚度矩阵的每个元素可以写为分子力学中力场常数的函数,这样建立起了宏观力学方法(有限元)与纳米尺度力学方法(分子力学)之间的联系．应用该方法模拟了扶椅型与锯齿型单壁碳纳米管的力学行为从而验证了该方法的有效性．分析结果说明单壁碳纳米管的弹性模量与管厚度的选取直接相关．此外,弹性模量对所选取的分子力学中的力场常数非常敏感,管的弹性模量显示出对半径的尺度依赖性,但是管长度对弹性模量的影响小到可以被忽略．     

New Insights on the Deflection and Internal Forces of a Bending Nanobeam

Nanowires, nanofibers and nanotubes have been widely used as the building blocks in micro/nano-electromechanical systems,energy harvesting or storage devices,and small-scaled measurement equipment. We report that the surface effects of these nanobeams have a great impact on their deflection and internal forces. A simply supported nanobeam is taken as an example. For the displacement and shear force of the nanobeam, its dangerous sections are different from those predicted by the conventional beam theory, but for the bending moment, the dangerous section is the same. Moreover, the values of these three quantities for the nanobeam are all distinct from those calculated from the conventional beam model. These analyses shed new light on the stiffness and strength check of nanobeams, which are beneficial to engineer new-types of nano-materials and nano-devices.     

Analysis and optimization of trimorph ring transducers

The Kirchoff-Love plate theory and electroelasticity theory are combined to simulate the dynamic behaviors of the trimorph ring transducers under different boundary conditions. The transducer consists of an isotropic elastic ring laminated between two identical piezoelectric rings. Their electric current response, resonant frequencies, antiresonant frequencies and electromechanical coupling coefficients (EMCCs) are theoretically formulated and studied by numerical simulation. Also, the resonant frequencies and their corresponding mode shape are simulated by the finite element modelling to verify the theoretical results. Finally, to obtain the maximum energy conversion efficiency, the dynamic EMCC is optimized by varying the proportion of piezoelectric and elastic parts. It is shown that the dynamic EMCC depends on geometric thickness and radii ratios. Optimum settings for a particular transducer to reach the maximum dynamic EMCC are found for different boundary conditions. The trimorph ring transducer for the fixed inner and free outer surfaces boundary condition has slightly lower resonant and anti-resonant frequencies, and larger EMCCs than that for the free inner and fixed outer surfaces boundary condition does.     

Keating改良模型的硅纳米梁动态特性

提出一种用于分析硅纳米梁动态特性的改良型半连续体模型,对比传统的连续体理论,这种新模型使用了Keating势,并考虑了纳米梁在宽厚两个维度的分立特性。依据Sun-Zhang模型思想和能量守恒定律,建立了改良型Keating模型,并进行了双端固支梁的基频计算。在这个过程中,对一些表面效应也进行系统分析。结果表明,该改良模型一方面在纳米尺度下与Material StudioTM软件仿真结果较为符合,另一方面也能在微观尺度下较好的接近连续体模型的计算数据。同时,该模型还反映了基频随纳米梁宽度变化的特性,这也符合一些实际实验。     

Surface effect on buckling configuration of nanobeams containing internal flowing fluid: A nonlinear analysis

In this paper, the post-buckling behavior of supported nanobeams containing internal flowing fluid with two surface layers is studied based on a nonlinear theoretical model. The nonlinear governing equation, in which the surface effect and stretching-related nonlinearity are accounted for, is analytically solved for both clamped-clamped and pinned-pinned systems. The effects of nanobeam length, bulk thickness and several dimensionless parameters on the post-buckling behavior are analyzed. It is found that, the nanobeam with low flow velocity is stable at its original static equilibrium position and then undergoes a buckling instability at a critical flow velocity, which depends on the system parameters. When buckled, in all cases, the amplitude of the resultant buckling increases with the increasing flow velocity. Typically, the surface effect is explored by considering different nanobeam lengths and bulk thicknesses. The buckling amplitude is found to be length-dependent and thickness-dependent, showing that the effect of surface layers is considerably strong.     

Ultrasonic characterization of ceramic fibres at ambient and elevated temperatures

A non-destructive laser-generated ultrasonic inspection system has been developed to evaluate the elastic properties of ceramic fibres. The approach uses a pulsed Nd:YAG laser to excite ultrasonic signals in fibres. The signal is detected by a piezoelectric acoustic emission transducer to obtain the appropriate frequency response suitable for an elastically one-dimensional sample. By using a differential time-of-flight system, a very accurate measure of the velocity can be obtained in the fibre, with a total scatter of less than 0.5%. This approach has been used to investigate the Young's modulus of polycrystalline carbon and boron fibres as a function of stress. Both types of fibres were found to have a Young's modulus increase as greater applied loads were imposed. The carbon and boron fibres, along with silicon carbide fibres, were evaluated at elevated temperatures up to 700 °C. The carbon fibres were found to have an immediate decrease in the Young's modulus as the temperature was increased, due to oxidation of the carbon. The Young's modulus of the boron fibres decreased only at temperatures higher than 200 °C, probably the result of a microstructural transformation or relaxation. The silicon carbide fibres were found to have no significant change in the elastic properties up to 700 °C. The ultrasonic technique was also applied to polycrystalline alumina fibres and fibre tows between ambient temperature and 1200 °C in a specially designed furnace. Using this technique, it was possible to distinguish the changes in the elasticity of the alumina fibres as they were processed into -alumina. The change in the Young's modulus was readily apparent during phase transformations to -alumina. In addition, the ultrasonic velocity can be used to infer information concerning any coatings that were applied to the alumina fibres. This can be used to aid in the quantification of the coating thickness and uniformity. The application of the ultrasonic inspection system has demonstrated the ability to determine rapidly and non-destructively the elastic properties in ceramic fibres. The information gained from the measurements can be used as a quality assurance technique, or can be modified to be a real-time process control/process monitoring system.