On the size-dependent flexural vibration characteristics of unbalanced couple stress-based micro-spinning beams

On the basis of the modified couple stress theory, some analytical results are obtained for vibrational parameters of micro-spinning Rayleigh beams with an axial mass-eccentricity distribution. The modified couple stress theory is a nonclassical continuum theory capable to capture the size effects in small-scale structures. After writing the governing equations of motion, they are transformed to the complex form. Then by utilizing the Galerkin method, analytical expressions for natural frequencies of the micro-spinning beam in forward and backward whirl motions are obtained. Critical speeds are also analytically presented in the two whirl motions for different modes. Moreover, the vibrational amplitude of the micro-spinning beam with axial mass eccentricity distribution is determined. Some numerical results are presented to study the effect of the length scale parameter on the vibrational characteristics.     

Effects of thermo-magneto-electro nonlinearity characteristics on the stability of functionally graded piezoelectric beam

Due to the increasing interests in using functionally graded piezoelectric materials(FGPMs) in the design of advanced micro-electro-mechanical systems, it is important to understand the stability behaviors of the FGPM beams. In this study, considering the effects of geometrical nonlinearity, temperature, and electricity in the constitutive relations and the effect of the magnetic field on the FGPM beam, the Euler-Bernoulli beam model is adopted, and the nonlinear governing equation of motion is ...     

Size-dependent effect on biaxial and shear nonlinear buckling analysis of nonlocal isotropic and orthotropic micro-plate based on surface stress and modified couple stress theories using differential quadrature method

The size-dependent effect on the biaxial and shear nonlinear buckling analysis of an isotropic and orthotropic micro-plate based on the surface stress,the modified couple stress theory(MCST),and the nonlocal elasticity theories using the differential quadrature method(DQM)is presented.Main advantages of the MCST over the classical theory(CT)are the inclusion of the asymmetric couple stress tensor and the consideration of only one material length scale parameter.Based on the nonlinear von K′arm′an assumption,the governing equations of equilibrium for the micro-classical plate considering midplane displacements are derived based on the minimum principle of potential energy.Using the DQM,the biaxial and shear critical buckling loads of the micro-plate for various boundary conditions are obtained.Accuracy of the obtained results is validated by comparing the solutions with those reported in the literature.A parametric study is conducted to show the effects of the aspect ratio,the side-to-thickness ratio,Eringen’s nonlocal parameter,the material length scale parameter,Young’s modulus of the surface layer,the surface residual stress,the polymer matrix coefficients,and various boundary conditions on the dimensionless uniaxial,biaxial,and shear critical buckling loads.The results indicate that the critical buckling loads are strongly sensitive to Eringen’s nonlocal parameter,the material length scale parameter,and the surface residual stress effects,while the effect of Young’s modulus of the surface layer on the critical buckling load is negligible.Also,considering the size dependent effect causes the increase in the stiffness of the orthotropic micro-plate.The results show that the critical biaxial buckling load increases with an increase in G12/E2and vice versa for E1/E2.It is shown that the nonlinear biaxial buckling ratio decreases as the aspect ratio increases and vice versa for the buckling amplitude.Because of the most lightweight micro-composite materials with high strength/weight and stiffness/weight ratios,it is anticipated that the results of the present work are useful in experimental characterization of the mechanical properties of micro-composite plates in the aircraft industry and other engineering applications.     

新修正偶应力理论Reddy型层合板稳定分析

基于新修正偶应力理论建立了一个Reddy型复合材料层合板稳定性模型。该理论中曲率张量不对称,而偶应力矩张量对称。Reddy型层合板模型能够满足横向剪切应力为0的自由表面条件,而且横向剪切为二次函数,避免了常剪力一阶理论需要引入的剪力修正系数。为了便于工程应用,通过虚功原理推导了只含纤维材料尺度参数正交铺设的Reddy型层合板偶应力模型的稳定性方程,并以微尺度正交铺设四边简支层合方板为例,分析了不同铺设角和轴向载荷作用时临界载荷的细观尺度效应,并且与一阶剪切变形和Kirchhoff板理论结果对比。结果表明,本文建立的新修正偶应力Reddy型层合板模型更适合分析较厚的复合材料层合板稳定性的尺度效应。     

The analysis of crack problems with non-local elasticity

IntroductionTheclassicalconhnuummechanicshasbeenusedtosolvemanyproblemsinmacrofracturemechanics,butencountersdifficulheswhentheeffectofITilcrocharacteristicdimensionshouldbetakenintoaccount.Thestressfieldverynearthecracktipisstillnotclear.Somephenomenaofshortcrackscannotbeexplained["']andsomemechanismoffracturehasnotbeensolvedyet.Thenon-localelashcitytheoryseemsattractivetotheseproblems.Thetheoryofnon-localelasticity,establishedanddevelopedbyEringenetal[3),connectstheclassicalcontinuummechan…     

考虑尺度效应的微梁静态弯曲数值分析

基于修正偶应力理论及表面弹性理论,本文提出了一种新的双曲线剪切变形梁模型,用于均匀微尺度梁的静态弯曲分析。该理论可以直接利用本构关系获得横向剪切应力,满足梁顶部和底部的无应力边界条件,避免了引入剪切修正因子。根据广义Young-Laplace方程建立了梁的内部与表面层的应力连续性条件,单一的变量场可以描述梁的位移模式。通过在位移场中考虑表面层厚度以及表面层的应力连续条件,可以使新模型能够更准确地预测微尺寸和表面能相关的尺度效应。通过Hamilton原理推导出了梁的控制方程和边界条件。应变能除了考虑经典弹性理论,还要考虑微结构效应和表面能。Navier-type的解析解适用于简支边界条件,而基于拉格朗日插值的微分求积法(DQEM)可以研究在不同边界条件下的力学响应。把该数值解与Navier方法得出的解析解作了对比,得出:微尺度梁在考虑表面能或微尺寸效应、不同载荷和梁高变化下的响应一致;当不考虑微结构相关性和表面能效应时,该模型退化为经典的欧拉梁模型。     

Pull-in voltage analysis of electrostatically actuated MEMS with piezoelectric layers: A size-dependent model

A size-dependent model for electrostatically actuated microbeam-based MEMS (micro-electro-mechanical systems) with piezoelectric layers attached is developed based on a modified couple stress theory. By using Hamilton's principle, the nonlinear differential governing equation and boundary conditions of the MEM structure are derived. In the newly developed model, the residual stresses, fringing-field and axial stress effects are considered for the fixed–fixed microbeam with piezoelectric layers. The results of the present model are compared with those from the classical model. The results show the size effect becomes prominent if the beam dimension is comparable to the material length scale parameter (MLSP). The effects of MLSP, the residual stresses and axial stress on the pull-in voltage are also studied. The study may be helpful to characterize the mechanical and electrostatic properties of small size MEMS, or guide the design of microbeam-based devices for a wide range of potential applications.     

Nonlinear elasticity modeling of biogels

A second-order nonlinear elastic model is developed for biogels, whereby the elastic constants are estimated from the free energy densities for polymer networks and polymer-water mixtures. For typical values of the constants, important physical insights are revealed: a biogel stiffens under tension, contracts longitudinally under torsion (inverse Poynting effect), and experiences a negative normal stress under simple shear. This approach of extracting nonlinear elastic constants works for any constitutive law, and has the advantage of exploring physical phenomena without recourse to computationally intensive simulations.     

Three-dimensional elasticity based on quaternion-valued potentials

One of the most fruitful and elegant approach (known as Kolosov–Muskhelishvili formulas) for plane isotropic elastic problems is to use two complex-valued holomorphic potentials. In this paper, the algebra of real quaternions is used in order to propose in three dimensions, an extension of the classical Muskhelishvili formulas. The starting point is the classical harmonic potential representation due to Papkovich and Neuber. Alike the classical complex formulation, two monogenic functions very similar to holomorphic functions in 2D and conserving many of interesting properties, are used in this contribution. The completeness of the potential formulation is demonstrated rigorously. Moreover, body forces, residual stress and thermal strain are taken into account as a left side term. The obtained monogenic representation is compact and a straightforward calculation shows that classical complex representation for plane problems is embedded in the presented extended formulas. Finally the classical uniqueness problem of the Papkovich–Neuber solutions is overcome for polynomial solutions by fixing explicitly linear dependencies.     

基于辛弹性力学解析本征函数的有限元应力磨平方法

在实际工程结构的结构强度与优化等力学数值分析中,应力计算结果的精度是非常重要的。有限元法是得到最广泛应用的一类数值方法,并形成了众多通用的有限元程序系统。这些程序系统采用的几乎都是基于最小总势能的位移法,虽然其分析给出的有限元位移场具有较高的精度,但所得到的有限元应力场的精度较位移场大大降低。基于极坐标辛对偶体系所提供的平面弹性力学的解析辛本征展开解,并借用有限元程序系统所得到的节点位移,本文提出了一个应力分析的改进方法。数值结果表明,本方法给出的应力分析精度得到大幅提高,并具有良好的数值稳定性,可用于有限元程序系统的后处理,以提高应力尤其是关键区域应力的分析精度。     

The effect of transverse pressure on the stability of a plate

The stability problem of a centrally compressed infinite plate is solved with allowance for the transverse normal deformation caused by uniform load for various boundary conditions at the edges. The linearized nonlinear equations of elastic deformation of thin plates taking into account transverse shear and transverse normal deformation are used. The obtained critical loads are compared with existing solutions.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 170–178, March–April, 2005.     

气泡弹性对同心筒水下发射影响研究

潜射导弹自同心简装置发射过程中在筒口处会形成燃气泡,由于气泡在水下的弹性效应,水气耦合作用导致气泡周期性地膨胀压缩,会对导弹弹道参数、受载特性产生一定影响.以CFD为技术手段,建立了导弹水下发射动态仿真模型,通过数值模拟获得了发射过程中多相流场、导弹弹道参数及受力的时间历程曲线,分析了气体弹性效应对参数变化的影响;并针对发射深度及其发射艇速对气泡弹性效应的影响进行了仿真分析,给出了其影响规律.仿真方法和结果对工程研究有一定参考价值.     

Size-dependent sinusoidal beam model for dynamic instability of single-walled carbon nanotubes

In this study, a model for dynamic instability of embedded single-walled carbon nanotubes (SWCNTs) is presented. SWCNTs are modeled by the sinusoidal shear deformation beam theory (SSDBT). The modified couple stress theory (MCST) is considered in order to capture the size effects. The surrounding elastic medium is described by a visco-Pasternak foundation model, which accounts for normal, transverse shear, and damping loads. The motion equations are derived based on Hamilton’s principle. The differential quadrature method (DQM) in conjunction with the Bolotin method is used in order to calculate the dynamic instability region (DIR) of SWCNTs. The effects of different parameters, such as nonlocal parameter, visco-Pasternak foundation, mode numbers, and geometrical parameters, are shown on the dynamic instability of SWCNTs. The results depict that increasing the nonlocal parameter shifts the DIR to right. The results presented in this paper would be helpful in design and manufacturing of nano-electromechanical system (NEMS) and micro-electro-mechanical system (MEMS).     

光源稳定性对光纤陀螺性能的影响及解决方案

分析了光源出纤功率和波长波动对陀螺性能的影响，并理论研究了双程后向结构掺铒光纤超荧光光源的特性。结果表明，在任意反射镜参数下，只要选取适当的掺铒光纤长度，该结构光源总能实现不依赖于泵浦功率的平均波长高稳定性运行；在高稳定性的前提下，反射镜参数优化后的该结构光源有较宽的频带宽度和较高的输出功率。得出了该结构光源可以满足高精度惯导级光纤陀螺对光源稳定性要求的结论。     

Quasi-optimal complexity of adaptive finite element method for linear elasticity problems in two dimensions

This paper introduces an adaptive finite element method (AFEM) using the newest vertex bisection and marking exclusively according to the error estimator without special treatment of oscillation. By the combination of the global lower bound and the localized upper bound of the posteriori error estimator, perturbation of oscillation, and cardinality of the marked element set, it is proved that the AFEM is quasi-optimal for linear elasticity problems in two dimensions, and this conclusion is verified by the numerical examples.     

Resonance frequencies and stability of a current-carrying suspended nanobeam in a longitudinal magnetic field

The resonance frequencies and stability of a nanobeam in a longitudinal magnetic field are investigated. To this aim, a three dimensional beam model is used in which the small-scale effect is taken into account based on the nonlocal elasticity theory. The Lorentz forces are obtained in terms of the local elastic rotations of the beam and the thermal stress due to current is modeled as an axial compressive force. Using the Galerkin method, the governing equations of motion are solved and the stability boundary of the nanobeam is determined.     

Size-dependent thermoelastic initially stressed micro-beam due to a varying temperature in the light of the modified couple stress theory

The bending of the Euler-Bernoulli micro-beam has been extensively modeled based on the modified couple stress (MCS) theory. Although many models have been incorporated into the literature, there is still room for introducing an improved model in this context. In this work, we investigate the thermoelastic vibration of a micro-beam exposed to a varying temperature due to the application of the initial stress employing the MCS theory and generalized thermoelasticity. The MCS theory is used to investigate the material length scale effects. Using the Laplace transform, the temperature, deflection, displacement, flexure moment, and stress field variables of the micro-beam are derived. The effects of the temperature pulse and couple stress on the field distributions of the micro-beam are obtained numerically and graphically introduced. The numerical results indicate that the temperature pulse and couple stress have a significant effect on all field variables.     

Stress concentration in the moment theory of elasticity

A multimoment theory of elasticity of inhomogeneous media is used to develop algorithms for determining a sequence of essentially independent fundamental parameters associated with the structure, composition, and characteristics of the medium components. The general form of stress concentration factors is established by analyzing specific models of media. It is found that the mean tangential stresses in a fibrous medium with a transverse stress state are asymmetric. As an example, the methods are used to solve specific problems for plates and shells __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 1, pp. 66–76, January 2007.     

Model of bending of a hydrostatically compressed shell near its stability threshold

A simplified model of bending dynamics of a hydrostatically compressed thin shell near the threshold of stability of its form is constructed within the framework of the nonlinear theory of elasticity. Conditions of existence and explicit expressions for spatially localized perturbations and patterns composed of dents on the shell surface, which are “precursors” of subsequent changes in the shell form, are found. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 6, pp. 124–134, November–December, 2007.     

基于新修正偶应力理论的Mindlin层合板热稳定性分析

以新修正偶应力理论为基础,首次提出了机械载荷与热载荷共同作用下的微尺度Mindlin层合板热稳定性模型,该模型只引入一个材料尺度参数,通过虚功原理推导出了控制方程和边界条件,以四边简支方板为例,进行了热稳定性分析,应用纳维叶解法得到解析解。结果表明,所建模型可以捕捉到尺度效应。材料尺度参数值越大,屈曲临界温度越高;当跨厚比增大时,屈曲临界温度下降;随着板几何参数的增大,模型将退化为宏观模型;温度变化量越大,考虑热载荷作用下的屈曲临界载荷越大,尺度效应体现越显著。