In this study,the frequency and vibration responses of a sandwich micro-beam are derived based on the modified couple stress theory(MCST).The face sheets are made of pure aluminum,and a magneto-rheological(MR)core is ...In this study,the frequency and vibration responses of a sandwich micro-beam are derived based on the modified couple stress theory(MCST).The face sheets are made of pure aluminum,and a magneto-rheological(MR)core is used to control vibrations.The displacement fields are assumed based on the classical beam theory(CBT)and modified classical beam theories.Based on Hamilton's principle,the governing equations of motion are obtained.To solve these temporal equations,the finite difference method(FDM)with an optimal number of nodes is applied.The effects of various parameters,including viscoelastic properties,magnetic fields,aspect ratio,core-to-face-sheet thickness ratio,MR materials,face sheets,and material length-scale parameters,on the frequency and vibrational response are investigated.In the literature,the effects of different parameters on the frequency response function(FRF)or vibration response are considered.The results obtained from the vibration response and FRF using the FDM show that the viscoelastic property of the MR core causes settling time in the vibration response and a decrease in the excitation frequency of the FRF.Increasing the magnetic field has a negligible effect on the excitation frequency in the FRF,but it increases the vibration response and settling time.The piezoelectric face sheets raise the FRF and suppress the vibration response of the MR sandwich micro-beam compared with the aluminum counterpart.展开更多
Peridynamics(PD)is a non-localmechanics theory that overcomes the limitations of classical continuummechanics(CCM)in predicting the initiation and propagation of cracks.However,the calculation efficiency of PDmodels i...Peridynamics(PD)is a non-localmechanics theory that overcomes the limitations of classical continuummechanics(CCM)in predicting the initiation and propagation of cracks.However,the calculation efficiency of PDmodels is generally lower than that of the traditional finite elementmethod(FEM).Structural idealization can greatly improve the calculation efficiency of PD models for complex structures.This study presents a PD shell model based on the micro-beam bond via the homogenization assumption.First,the deformations of each endpoint of themicro-beam bond are calculated through the interpolation method.Second,the micro-potential energy of the axial,torsional,and bending deformations of the bond can be established from the deformations of endpoints.Finally,the micro moduli of the shellmodel can be obtained via the equivalence principle of strain energy density(SED).In addition,a new fracture criterion based on the SED of the micro-beam bond is adopted for crack simulation.Numerical examples of crack propagation are provided,and the results demonstrate the effectiveness of the proposed PD shell model.展开更多
This paper focuses on the buckling behaviors of a micro-scaled bi-directional functionally graded (FG) beam with a rectangular cross-section, which is now widely used in fabricating components of micro-nano-electro-...This paper focuses on the buckling behaviors of a micro-scaled bi-directional functionally graded (FG) beam with a rectangular cross-section, which is now widely used in fabricating components of micro-nano-electro-mechanical systems (MEMS/NEMS) with a wide range of aspect ratios. Based on the modified couple stress theory and the principle of minimum potential energy, the governing equations and boundary conditions for a micro-structure-dependent beam theory are derived. The present beam theory incorporates different kinds of higher-order shear assumptions as well as the two familiar beam theories, namely, the Euler-Bernoulli and Timoshenko beam theories. A numerical solu- tion procedure, based on a generalized differential quadrature method (GDQM), is used to calculate the results of the bi-directional FG beams. The effects of the two exponential FG indexes, the higher-order shear deformations, the length scale parameter, the geomet- ric dimensions, and the different boundary conditions on the critical buckling loads are studied in detail, by assuming that Young's modulus obeys an exponential distribution function in both length and thickness directions. To reach the desired critical buckling load, the appropriate exponential FG indexes and geometric shape of micro-beams can be designed according to the proposed theory.展开更多
A size-dependent continuum-based model is developed for the functionally graded(FG)Timoshenko micro-beams with viscoelastic properties,in which material parameters vary according to the power law along its axial direc...A size-dependent continuum-based model is developed for the functionally graded(FG)Timoshenko micro-beams with viscoelastic properties,in which material parameters vary according to the power law along its axial direction.The size effect is incorporated by employing the modified couple stress theory and Kelvin-Voigt viscoelastic model,so that viscous components are included in the stress and the deviatoric segments of the symmetric couple stress tensors.The components of strain,curvature,stress and couple stress are formulated by combining them with the Timoshenko beam theory.Based on the Hamilton principle,the governing differential equations and boundary conditions for the micro-beam are expressed with arbitrary beam section shape and arbitrary type of loads.The size effect,FG effect,Poisson effect,and the influence of the beam section shape on the mechanical behaviors of viscoelastic FG micro-beams are investigated by taking the simply supported micro-beam subjected to point load as an example.Results show that the size effect on deflection,normal stress and couple stress are obvious when the size of the micro-beam is small enough,and the FG effects are obvious when the size of the micro-beam is large enough.Moreover,the Poisson ratio influences the size effect significantly and the beam section shape is also an important factor influencing the mechanical behavior of the micro-beam.展开更多
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 introdu...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.展开更多
Thermoelastic damping(TED)is one of the main internal energy dissipation mechanisms in microano-resonators.Accurate evaluation of TED is important in the design of micro-electromechanical systems and nano-electromecha...Thermoelastic damping(TED)is one of the main internal energy dissipation mechanisms in microano-resonators.Accurate evaluation of TED is important in the design of micro-electromechanical systems and nano-electromechanical systems.In this paper,a theoretical analysis on the TED in functionally graded material(FGM)micro-beam resonators is presented.Equations of motion and the heat conduction equation governing the thermodynamic coupling free vibration of non-homogenous micro-beams are established based on the Euler Bernoulli beam theory associated with the modified couple stress theory.Material properties of the FGM micro-beam are assumed to change in the depth direction as power-law functions.The layer-wise homogenization method is used for solving the heat conduction equation.By using the mathematical similarity of eigenvalue problem between the FGM beam and the reference homogeneous one,the complex natural frequency including TED is expressed in terms of the natural frequency of the isothermal homogenous beam.In the presented numerical results,influences of various characteristic parameters,such as beam thickness,material gradient index,structure size,vibration mode and boundary conditions,on TED are examined in detail.It shows that TED decreases with the increases in the values of length scale parameters because the latter lead to the increase in structural stiffness.展开更多
A buckling model of Timoshenko micro-beam with local thickness defects is established based on a modified gradient elasticity.By introducing the local thickness defects function of the micro-beam,the variable coeffici...A buckling model of Timoshenko micro-beam with local thickness defects is established based on a modified gradient elasticity.By introducing the local thickness defects function of the micro-beam,the variable coefficient differential equations of the buckling problem are obtained with the variational principle.Combining the eigensolution series of the complete micro-beam with the Galerkin method,we obtain the critical load and buckling modes of the micro-beam with defects.The results show that the depth and location of the defect are the main factors affecting the critical load,and the combined effect of boundary conditions and defects can significantly change the buckling mode of the micro-beam.The effect of defect location on buckling is related to the axial gradient of the rotation angle,and defects should be avoided at the maximum axial gradient of the rotation angle.The model and method are also applicable to the static deformation and vibration of the micro-beam.展开更多
The dynamic characteristics of a quartz crystal resonator(QCR) in thicknessshear modes(TSM) with the upper surface covered by an array of micro-beams immersed in liquid are studied. The liquid is assumed to be inv...The dynamic characteristics of a quartz crystal resonator(QCR) in thicknessshear modes(TSM) with the upper surface covered by an array of micro-beams immersed in liquid are studied. The liquid is assumed to be inviscid and incompressible for simplicity. Dynamic equations of the coupled system are established. The added mass effect of liquid on micro-beams is discussed in detail. Characteristics of frequency shift are clarified for different liquid depths. Modal analysis shows that a drag effect of liquid has resulted in the change of phase of interaction(surface shear force), thus changing the system resonant frequency. The obtained results are useful in resonator design and applications.展开更多
Based on the Modified Couple Stress Theory,a functionally graded micro-beam under electrostatic forces is studied.The FGM micro-beam is made of two materials and material properties vary continuously along the beam th...Based on the Modified Couple Stress Theory,a functionally graded micro-beam under electrostatic forces is studied.The FGM micro-beam is made of two materials and material properties vary continuously along the beam thickness according to a power-law.Dynamic and static pull-in voltages are obtained and it is shown that the static and dynamic pull-in voltages for some materials cannot be obtained using classic theories and components of couple stress must be taken into account.In addition,it is shown that the values of pull-in voltages depend on the variation through the thickness of the volume fractions of the two constituents.展开更多
In this research, vibration and wave propagation analysis of a twisted micro- beam on Pasternak foundation is investigated. The strain-displacement relations (kine-matic equations) are calculated by the displacement...In this research, vibration and wave propagation analysis of a twisted micro- beam on Pasternak foundation is investigated. The strain-displacement relations (kine-matic equations) are calculated by the displacement fields of the twisted micro-beam. The strain gradient theory (SGT) is used to implement the size dependent effect at micro-scale. Finally, using an energy method and Hamilton's principle, the governing equations of motion for the twisted micro-beam are derived. Natural frequencies and the wave prop- agation speed of the twisted micro-beam are calculated with an analytical method. Also, the natural frequency, the phase speed, the cut-off frequency, and the wave number of the twisted micro-beam are obtained by considering three material length scale parameters, the rate of twist angle, the thickness, the length of twisted micro-beam, and the elastic medium. The results of this work indicate that the phase speed in a twisted micro-beam increases with an increase in the rate of twist angle. Moreover, the wave number is in- versely related with the thickness of micro-beam. Meanwhile, it is directly related to the wave propagation frequency. Increasing the rate of twist angle causes the increase in the natural frequency especially with higher thickness. The effect of the twist angle rate on the group velocity is observed at a lower wave propagation frequency.展开更多
A quasi-three dimensional model is proposed for the vibration analysis of functionally graded(FG)micro-beams with general boundary conditions based on the modified strain gradient theory.To consider the effects of tra...A quasi-three dimensional model is proposed for the vibration analysis of functionally graded(FG)micro-beams with general boundary conditions based on the modified strain gradient theory.To consider the effects of transverse shear and nor-mal deformations,a general displacement field is achieved by relaxing the assumption of the constant transverse displacement through the thickness.The conventional beam theories including the classical beam theory,the first-order beam theory,and the higher-order beam theory are regarded as the special cases of this model.The material proper-ties changing gradually along the thickness direction are calculated by the Mori-Tanaka scheme.The energy-based formulation is derived by a variational method integrated with the penalty function method,where the Chebyshev orthogonal polynomials are used as the basis function of the displacement variables.The formulation is validated by some comparative examples,and then the parametric studies are conducted to investigate the effects of transverse shear and normal deformations on vibration behaviors.展开更多
With introduction of the first-order strain-gradient of surface micro-beams into the energy density function,we developed a two-dimensional dynamic model for a compound quartz crystal resonator(QCR) system,consistin...With introduction of the first-order strain-gradient of surface micro-beams into the energy density function,we developed a two-dimensional dynamic model for a compound quartz crystal resonator(QCR) system,consisting of a QCR and surface micro-beam arrays.The frequency shift that was induced by micro-beams with consideration of strain-gradients is discussed in detail and some useful results are obtained,which have important significance in resonator design and applications.展开更多
We study the dynamic behavior of a quartz crystal resonator(QCR)in thickness-shear vibrations with the upper surface covered by an array of micro-beams(MBs)under large deflection.Through taking into account the contin...We study the dynamic behavior of a quartz crystal resonator(QCR)in thickness-shear vibrations with the upper surface covered by an array of micro-beams(MBs)under large deflection.Through taking into account the continuous conditions of shear force and bending moment at the interface of MBsesonator,dependences of frequency shift of the compound QCR system versus material parameter and geometrical parameter are illustrated in detail for nonlinear and linear vibrations.It is found that the frequency shift produces a little right(left)translation for increasing elastic modulus(lengthadius ratio)of MBs.Moreover,the frequency right(left)translation distance caused by nonlinear deformation becomes more serious in the second-order mode than in the first-order one.展开更多
This work focus on the mechanical behaviors,which are related to the size effect,functionally graded(FG)effect and Poisson effect,of an axially functionally graded(AFG)micro-beam whose elastic modulus varies according...This work focus on the mechanical behaviors,which are related to the size effect,functionally graded(FG)effect and Poisson effect,of an axially functionally graded(AFG)micro-beam whose elastic modulus varies according to sinusoidal law along its axial direction.The displacement field of the AFG micro-beam is set according to the Bernoulli-Euler beam theory.Employing the modified couple stress theory(MCST),the components of strain,curvature,stress and couple stress are expressed by the second derivative of the deflection of the AFG micro-beam.A size-dependent model related to FG effect and Poisson effect,which includes the formulations of bending stiffness,deflection,normal stress and couple stress,is developed to predict the mechanical behaviors of the AFG microbeam by employing the principle of minimum potential energy.The mechanical behaviors of a simply supported AFG micro-beam are numerically investigated using the developed model for demonstrating the size effects,FG effects and Poisson effects of the AFG micro-beam.Results show that the mechanical behaviors of AFG micro-beams are distinctly size-dependent only when the ratio of micro-beam height to material length-scale parameter is small enough.The FG parameter is an important factor that determines and regulates the size-dependent behaviors of AFG micro-beams.The influences of Poisson’s ratio on the mechanical behaviors of AFG micro-beams are not negligible,and should be also considered in the design and analysis of an AFG micro-beam.This work supplies a theoretical basis and a technical reference for the design and analysis of AFG micro-beams in the related regions.展开更多
This paper focuses on the size-dependently mechanical behaviors of a micro-beam under forced vibration.Governing equations of a micro-beam under forced vibration are established by using the modified couple stress the...This paper focuses on the size-dependently mechanical behaviors of a micro-beam under forced vibration.Governing equations of a micro-beam under forced vibration are established by using the modified couple stress theory,Bernoulli-Euler beam theory and D’Alembert’s principle together.A simply supported micro-beam under forced vibration is solved according to the established governing equations and the method of separation of variables.The dimensionless deflection,amplitude mode and period mode are defined to investigate the size-dependently mechanical behaviors of a micro-beam under forced vibration.Results show that the performance of a micro-beams under forced vibration is distinctly size-dependent when the ratio of micro-beam height to material length-scale parameter is small enough.Both frequency ratio and loading location are the important factors that determine the sizedependent performance of a micro-beams under forced vibration.展开更多
微束斑电子源是实现高空间分辨率精细电子束的核心部件,在微纳真空电子器件、电子束成像、曝光和精密制造设备中具有关键应用价值。文章采用碳纳米管纤维冷阴极,利用聚焦离子束技术对碳纳米管纤维顶端形貌进行锐化微纳加工,获得端面直径...微束斑电子源是实现高空间分辨率精细电子束的核心部件,在微纳真空电子器件、电子束成像、曝光和精密制造设备中具有关键应用价值。文章采用碳纳米管纤维冷阴极,利用聚焦离子束技术对碳纳米管纤维顶端形貌进行锐化微纳加工,获得端面直径7.5 m m的柱形尖端结构,在直流场发射工作模式下,器件最大场发射电流为1305.4 m A、电流密度为2954.8 A·cm-2、场增强因子达7169.9。设计并研制出基于碳纳米管纤维冷阴极和双膜孔静电聚焦透镜的微束斑电子源,测试结果表明,在阳极电压11 kV、阳极电流0.7 m A条件下,电子束斑尺寸为8.7 m m,达到亚10 m m尺度。该研究表明碳纳米管纤维冷阴极具有优异的电子发射性能,可用于制备微束斑电子源,在真空电子器件与设备应用中具有广阔前景。展开更多
基金the Iranian Nanotechnology Development Committee for supporting this researchthe University of Kashan under Grant No.1392194/7funded by the University of Kashan。
摘要In this study,the frequency and vibration responses of a sandwich micro-beam are derived based on the modified couple stress theory(MCST).The face sheets are made of pure aluminum,and a magneto-rheological(MR)core is used to control vibrations.The displacement fields are assumed based on the classical beam theory(CBT)and modified classical beam theories.Based on Hamilton's principle,the governing equations of motion are obtained.To solve these temporal equations,the finite difference method(FDM)with an optimal number of nodes is applied.The effects of various parameters,including viscoelastic properties,magnetic fields,aspect ratio,core-to-face-sheet thickness ratio,MR materials,face sheets,and material length-scale parameters,on the frequency and vibrational response are investigated.In the literature,the effects of different parameters on the frequency response function(FRF)or vibration response are considered.The results obtained from the vibration response and FRF using the FDM show that the viscoelastic property of the MR core causes settling time in the vibration response and a decrease in the excitation frequency of the FRF.Increasing the magnetic field has a negligible effect on the excitation frequency in the FRF,but it increases the vibration response and settling time.The piezoelectric face sheets raise the FRF and suppress the vibration response of the MR sandwich micro-beam compared with the aluminum counterpart.
基金funded by Project of the National Natural Science Foundation of China(Grant No.11872017).
摘要Peridynamics(PD)is a non-localmechanics theory that overcomes the limitations of classical continuummechanics(CCM)in predicting the initiation and propagation of cracks.However,the calculation efficiency of PDmodels is generally lower than that of the traditional finite elementmethod(FEM).Structural idealization can greatly improve the calculation efficiency of PD models for complex structures.This study presents a PD shell model based on the micro-beam bond via the homogenization assumption.First,the deformations of each endpoint of themicro-beam bond are calculated through the interpolation method.Second,the micro-potential energy of the axial,torsional,and bending deformations of the bond can be established from the deformations of endpoints.Finally,the micro moduli of the shellmodel can be obtained via the equivalence principle of strain energy density(SED).In addition,a new fracture criterion based on the SED of the micro-beam bond is adopted for crack simulation.Numerical examples of crack propagation are provided,and the results demonstrate the effectiveness of the proposed PD shell model.
基金supported by the National Natural Science Foundation of China(Nos.51375184 and 51605172)the Fundamental Research Funds for the Central Universities(No.2015MS014)
摘要This paper focuses on the buckling behaviors of a micro-scaled bi-directional functionally graded (FG) beam with a rectangular cross-section, which is now widely used in fabricating components of micro-nano-electro-mechanical systems (MEMS/NEMS) with a wide range of aspect ratios. Based on the modified couple stress theory and the principle of minimum potential energy, the governing equations and boundary conditions for a micro-structure-dependent beam theory are derived. The present beam theory incorporates different kinds of higher-order shear assumptions as well as the two familiar beam theories, namely, the Euler-Bernoulli and Timoshenko beam theories. A numerical solu- tion procedure, based on a generalized differential quadrature method (GDQM), is used to calculate the results of the bi-directional FG beams. The effects of the two exponential FG indexes, the higher-order shear deformations, the length scale parameter, the geomet- ric dimensions, and the different boundary conditions on the critical buckling loads are studied in detail, by assuming that Young's modulus obeys an exponential distribution function in both length and thickness directions. To reach the desired critical buckling load, the appropriate exponential FG indexes and geometric shape of micro-beams can be designed according to the proposed theory.
基金The National Science and Technology Major Project(No.2017ZX05009-003)the National Key Research and Development Program of China(No.2017YFC0307604)the Talent Foundation of China University of Petroleum(No.Y1215042)。
摘要A size-dependent continuum-based model is developed for the functionally graded(FG)Timoshenko micro-beams with viscoelastic properties,in which material parameters vary according to the power law along its axial direction.The size effect is incorporated by employing the modified couple stress theory and Kelvin-Voigt viscoelastic model,so that viscous components are included in the stress and the deviatoric segments of the symmetric couple stress tensors.The components of strain,curvature,stress and couple stress are formulated by combining them with the Timoshenko beam theory.Based on the Hamilton principle,the governing differential equations and boundary conditions for the micro-beam are expressed with arbitrary beam section shape and arbitrary type of loads.The size effect,FG effect,Poisson effect,and the influence of the beam section shape on the mechanical behaviors of viscoelastic FG micro-beams are investigated by taking the simply supported micro-beam subjected to point load as an example.Results show that the size effect on deflection,normal stress and couple stress are obvious when the size of the micro-beam is small enough,and the FG effects are obvious when the size of the micro-beam is large enough.Moreover,the Poisson ratio influences the size effect significantly and the beam section shape is also an important factor influencing the mechanical behavior of the micro-beam.
摘要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.
基金Project supported by the National Natural Science Foundation of China(No.11672260)the Natural Science Foundation of Jiangsu(No.BK20180894).
摘要Thermoelastic damping(TED)is one of the main internal energy dissipation mechanisms in microano-resonators.Accurate evaluation of TED is important in the design of micro-electromechanical systems and nano-electromechanical systems.In this paper,a theoretical analysis on the TED in functionally graded material(FGM)micro-beam resonators is presented.Equations of motion and the heat conduction equation governing the thermodynamic coupling free vibration of non-homogenous micro-beams are established based on the Euler Bernoulli beam theory associated with the modified couple stress theory.Material properties of the FGM micro-beam are assumed to change in the depth direction as power-law functions.The layer-wise homogenization method is used for solving the heat conduction equation.By using the mathematical similarity of eigenvalue problem between the FGM beam and the reference homogeneous one,the complex natural frequency including TED is expressed in terms of the natural frequency of the isothermal homogenous beam.In the presented numerical results,influences of various characteristic parameters,such as beam thickness,material gradient index,structure size,vibration mode and boundary conditions,on TED are examined in detail.It shows that TED decreases with the increases in the values of length scale parameters because the latter lead to the increase in structural stiffness.
基金Project supported by the Young Core Instructor and Domestic Visitor Foundation from the Education Commission of Hunan Province(No.21B0315)。
摘要A buckling model of Timoshenko micro-beam with local thickness defects is established based on a modified gradient elasticity.By introducing the local thickness defects function of the micro-beam,the variable coefficient differential equations of the buckling problem are obtained with the variational principle.Combining the eigensolution series of the complete micro-beam with the Galerkin method,we obtain the critical load and buckling modes of the micro-beam with defects.The results show that the depth and location of the defect are the main factors affecting the critical load,and the combined effect of boundary conditions and defects can significantly change the buckling mode of the micro-beam.The effect of defect location on buckling is related to the axial gradient of the rotation angle,and defects should be avoided at the maximum axial gradient of the rotation angle.The model and method are also applicable to the static deformation and vibration of the micro-beam.
基金Project supported by the National Natural Science Foundation of China(Nos.11272127 and51425006)the Research Fund for the Doctoral Program of Higher Education of China(No.20130142110022)the Grant from the Impact and Safety of Coastal Engineering Initiative Program of Zhejiang Provincial Government at Ningbo University(No.zj1213)
摘要The dynamic characteristics of a quartz crystal resonator(QCR) in thicknessshear modes(TSM) with the upper surface covered by an array of micro-beams immersed in liquid are studied. The liquid is assumed to be inviscid and incompressible for simplicity. Dynamic equations of the coupled system are established. The added mass effect of liquid on micro-beams is discussed in detail. Characteristics of frequency shift are clarified for different liquid depths. Modal analysis shows that a drag effect of liquid has resulted in the change of phase of interaction(surface shear force), thus changing the system resonant frequency. The obtained results are useful in resonator design and applications.
摘要Based on the Modified Couple Stress Theory,a functionally graded micro-beam under electrostatic forces is studied.The FGM micro-beam is made of two materials and material properties vary continuously along the beam thickness according to a power-law.Dynamic and static pull-in voltages are obtained and it is shown that the static and dynamic pull-in voltages for some materials cannot be obtained using classic theories and components of couple stress must be taken into account.In addition,it is shown that the values of pull-in voltages depend on the variation through the thickness of the volume fractions of the two constituents.
基金Project supported by the Iranian Nanotechnology Development Committee and the University of Kashan(No.463855/11)
摘要In this research, vibration and wave propagation analysis of a twisted micro- beam on Pasternak foundation is investigated. The strain-displacement relations (kine-matic equations) are calculated by the displacement fields of the twisted micro-beam. The strain gradient theory (SGT) is used to implement the size dependent effect at micro-scale. Finally, using an energy method and Hamilton's principle, the governing equations of motion for the twisted micro-beam are derived. Natural frequencies and the wave prop- agation speed of the twisted micro-beam are calculated with an analytical method. Also, the natural frequency, the phase speed, the cut-off frequency, and the wave number of the twisted micro-beam are obtained by considering three material length scale parameters, the rate of twist angle, the thickness, the length of twisted micro-beam, and the elastic medium. The results of this work indicate that the phase speed in a twisted micro-beam increases with an increase in the rate of twist angle. Moreover, the wave number is in- versely related with the thickness of micro-beam. Meanwhile, it is directly related to the wave propagation frequency. Increasing the rate of twist angle causes the increase in the natural frequency especially with higher thickness. The effect of the twist angle rate on the group velocity is observed at a lower wave propagation frequency.
基金Project supported by the National Natural Science Foundation of China(Nos.51805250 and 11602145)the Natural Science Foundation of Jiangsu Province of China(No.BK20180429)+1 种基金the China Postdoctoral Science Foundation(No.2019M660114)the Jiangsu Planned Projects for Postdoctoral Research Funds of China(No.2019K054)。
摘要A quasi-three dimensional model is proposed for the vibration analysis of functionally graded(FG)micro-beams with general boundary conditions based on the modified strain gradient theory.To consider the effects of transverse shear and nor-mal deformations,a general displacement field is achieved by relaxing the assumption of the constant transverse displacement through the thickness.The conventional beam theories including the classical beam theory,the first-order beam theory,and the higher-order beam theory are regarded as the special cases of this model.The material proper-ties changing gradually along the thickness direction are calculated by the Mori-Tanaka scheme.The energy-based formulation is derived by a variational method integrated with the penalty function method,where the Chebyshev orthogonal polynomials are used as the basis function of the displacement variables.The formulation is validated by some comparative examples,and then the parametric studies are conducted to investigate the effects of transverse shear and normal deformations on vibration behaviors.
基金supported by the National Science Foundation of China(Grants 11272127 and 51435006)Research Fund for the Doctoral Program of Higher Education of China(Grant 20130142110022)the Grant from the Impact and Safety of Coastal Engineering Initiative Program of Zhejiang Provincial Government at Ningbo University(Grant zj1213)
摘要With introduction of the first-order strain-gradient of surface micro-beams into the energy density function,we developed a two-dimensional dynamic model for a compound quartz crystal resonator(QCR) system,consisting of a QCR and surface micro-beam arrays.The frequency shift that was induced by micro-beams with consideration of strain-gradients is discussed in detail and some useful results are obtained,which have important significance in resonator design and applications.
基金supported by the National Natural Science Foundation of China(11272127 and 51435006)the Research Fund for the Doctoral Program of Higher Education of China(20130142110022)
摘要We study the dynamic behavior of a quartz crystal resonator(QCR)in thickness-shear vibrations with the upper surface covered by an array of micro-beams(MBs)under large deflection.Through taking into account the continuous conditions of shear force and bending moment at the interface of MBsesonator,dependences of frequency shift of the compound QCR system versus material parameter and geometrical parameter are illustrated in detail for nonlinear and linear vibrations.It is found that the frequency shift produces a little right(left)translation for increasing elastic modulus(lengthadius ratio)of MBs.Moreover,the frequency right(left)translation distance caused by nonlinear deformation becomes more serious in the second-order mode than in the first-order one.
基金The authors of this paper acknowledge the supports from the National Key Research and Development Program of China(Grant No.2017YFC0307604)the Talent Foundation of China University of Petroleum(Grant No.Y1215042).
摘要This work focus on the mechanical behaviors,which are related to the size effect,functionally graded(FG)effect and Poisson effect,of an axially functionally graded(AFG)micro-beam whose elastic modulus varies according to sinusoidal law along its axial direction.The displacement field of the AFG micro-beam is set according to the Bernoulli-Euler beam theory.Employing the modified couple stress theory(MCST),the components of strain,curvature,stress and couple stress are expressed by the second derivative of the deflection of the AFG micro-beam.A size-dependent model related to FG effect and Poisson effect,which includes the formulations of bending stiffness,deflection,normal stress and couple stress,is developed to predict the mechanical behaviors of the AFG microbeam by employing the principle of minimum potential energy.The mechanical behaviors of a simply supported AFG micro-beam are numerically investigated using the developed model for demonstrating the size effects,FG effects and Poisson effects of the AFG micro-beam.Results show that the mechanical behaviors of AFG micro-beams are distinctly size-dependent only when the ratio of micro-beam height to material length-scale parameter is small enough.The FG parameter is an important factor that determines and regulates the size-dependent behaviors of AFG micro-beams.The influences of Poisson’s ratio on the mechanical behaviors of AFG micro-beams are not negligible,and should be also considered in the design and analysis of an AFG micro-beam.This work supplies a theoretical basis and a technical reference for the design and analysis of AFG micro-beams in the related regions.
基金The authors of this paper acknowledge the supports from the National Key Research and Development Program of China(Grant No.2017YFC0307604)the Talent Foundation of China University of Petroleum(Grant No.Y1215042).
摘要This paper focuses on the size-dependently mechanical behaviors of a micro-beam under forced vibration.Governing equations of a micro-beam under forced vibration are established by using the modified couple stress theory,Bernoulli-Euler beam theory and D’Alembert’s principle together.A simply supported micro-beam under forced vibration is solved according to the established governing equations and the method of separation of variables.The dimensionless deflection,amplitude mode and period mode are defined to investigate the size-dependently mechanical behaviors of a micro-beam under forced vibration.Results show that the performance of a micro-beams under forced vibration is distinctly size-dependent when the ratio of micro-beam height to material length-scale parameter is small enough.Both frequency ratio and loading location are the important factors that determine the sizedependent performance of a micro-beams under forced vibration.
摘要微束斑电子源是实现高空间分辨率精细电子束的核心部件,在微纳真空电子器件、电子束成像、曝光和精密制造设备中具有关键应用价值。文章采用碳纳米管纤维冷阴极,利用聚焦离子束技术对碳纳米管纤维顶端形貌进行锐化微纳加工,获得端面直径7.5 m m的柱形尖端结构,在直流场发射工作模式下,器件最大场发射电流为1305.4 m A、电流密度为2954.8 A·cm-2、场增强因子达7169.9。设计并研制出基于碳纳米管纤维冷阴极和双膜孔静电聚焦透镜的微束斑电子源,测试结果表明,在阳极电压11 kV、阳极电流0.7 m A条件下,电子束斑尺寸为8.7 m m,达到亚10 m m尺度。该研究表明碳纳米管纤维冷阴极具有优异的电子发射性能,可用于制备微束斑电子源,在真空电子器件与设备应用中具有广阔前景。