The size-dependent natural frequency of Bernoulli-Euler micro-beams

The dynamic problems of Bernoulli-Euler beams are solved analytically on the basis of modified couple stress theory. The governing equations of equilibrium, initial conditions and boundary conditions are obtained by a combination of the basic equations of modified couple stress theory and Hamilton’s principle. Two boundary value problems (one for simply supported beam and another for cantilever beam) are solved and the size effect on the beam’s natural frequencies for two kinds of boundary conditions are assessed. It is found that the natural frequencies of the beams predicted by the new model are size-dependent. The difference between the natural frequencies predicted by the newly established model and classical beam model is very significant when the ratio of characteristic sizes to internal material length scale parameter is approximately equal to one, but is diminishing with the increase of the ratio.     

Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory

Investigated herein is the free vibration characteristics of microbeams made of functionally graded materials (FGMs) based on the strain gradient Timoshenko beam theory. The material properties of the functionally graded beams are assumed to be graded in the thickness direction according to the Mori–Tanaka scheme. Using Hamilton’s principle, the equations of motion together with corresponding boundary conditions are obtained for the free vibration analysis of FGM microbeams including size effect. A detailed parametric study is performed to indicate the influences of beam thickness, dimensionless length scale parameter, and slenderness ratio on the natural frequencies of FGM microbeams. Moreover, a comparison between the various beam models on the basis of the classical theory (CT), modified couple stress theory (MCST), and strain gradient theory (SGT) is presented for different values of material property gradient index. It is observed that the value of gradient index play an important role in the vibrational response of the microbeams of lower slenderness ratios. It is further observed that by increasing the length-to-thickness ratio of the microbeam, the value of dimensionless natural frequency tends to decrease for all amounts of the gradient index.     

Consideration of spatial variation of the length scale parameter in static and dynamic analyses of functionally graded annular and circular micro-plates

This article introduces new methods for static and free vibration analyses of functionally graded annular and circular micro-plates, which can take into account spatial variation of the length scale parameter. The underlying higher order continuum theory behind the proposed approaches is the modified couple stress theory. A unified way of expressing the displacement field is adopted so as to produce numerical results for three different plate theories, which are Kirchhoff plate theory (KPT), Mindlin plate theory (MPT), and third-order shear deformation theory (TSDT). Governing partial differential equations and corresponding boundary conditions are obtained following the variational approach and the Hamilton's principle. Derived systems of differential equations are solved numerically by utilizing the differential quadrature method (DQM). Comparisons to the results available in the literature demonstrate the high level of accuracy of the numerical results generated through the developed methods. Extensive analyses are presented in order to illustrate the influences of various geometric and material parameters upon static deformation profiles, stresses, and natural vibration frequencies. In particular, the length scale parameter ratio -which defines the length scale parameter variation profile-is shown to possess a profound impact on both static and dynamic behaviors of functionally graded annular and circular micro-plates.     

Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory

Investigated herein is the free vibration characteristics of microbeams made of functionally graded materials (FGMs) based on the strain gradient Timoshenko beam theory. The material properties of the functionally graded beams are assumed to be graded in the thickness direction according to the Mori–Tanaka scheme. Using Hamilton’s principle, the equations of motion together with corresponding boundary conditions are obtained for the free vibration analysis of FGM microbeams including size effect. A detailed parametric study is performed to indicate the influences of beam thickness, dimensionless length scale parameter, and slenderness ratio on the natural frequencies of FGM microbeams. Moreover, a comparison between the various beam models on the basis of the classical theory (CT), modified couple stress theory (MCST), and strain gradient theory (SGT) is presented for different values of material property gradient index. It is observed that the value of gradient index play an important role in the vibrational response of the microbeams of lower slenderness ratios. It is further observed that by increasing the length-to-thickness ratio of the microbeam, the value of dimensionless natural frequency tends to decrease for all amounts of the gradient index.     

考虑尺度依赖的平面正交各向异性功能梯度微梁的自由振动分析

基于一种新修正偶应力理论建立了微尺度平面正交各向异性功能梯度梁的自由振动模型。模型中包含两个材料尺度参数,能够分别描述两个正交方向上不同程度的尺度效应。当梁的几何尺寸远大于材料尺度参数时,本文模型亦可自动退化为相应的传统宏观模型。基于哈密顿原理推导了运动控制方程并以简支梁的自由振动为例分析了几何尺寸、功能梯度变化指数等对尺度效应产生的影响。算例结果表明:采用本文模型所预测的梁自振频率总是大于传统理论的结果,即捕捉到了尺度效应。尺度效应会随着梁几何尺寸的增大而逐渐减弱并在几何尺寸远大于尺度参数时消失;高阶自振频率所体现出的尺度效应较低阶自振频率更加明显。此外,功能梯度变化指数对尺度效应也有一定的影响。     

A modified nonlocal couple stress-based beam model for vibration analysis of higher-order FG nanobeams

In the present paper, nonlocal couple stress theory is developed to investigate free vibration characteristics of functionally graded (FG) nanobeams considering exact position of neutral axis. The theory introduces two parameters based on nonlocal elasticity theory and modified couple stress theory to capture the size effects much accurately. Therefore, a nonlocal stress field parameter and a material length scale parameter are used to involve both stiffness-softening and stiffness-hardening effects on responses of FG nanobeams. The FG nanobeam is modeled via a higher-order refined beam theory in which shear deformation effect is verified needless of shear correction factor. A power-law distribution is used to describe the graded material properties. The governing equations and the related boundary conditions are derived by Hamilton's principle and they are solved applying Galerkin's method, which satisfies various boundary conditions. A comparison study is performed to verify the present formulation with the provided data in the literature and a good agreement is observed. The parametric study covered in this paper includes several parameters, such as nonlocal and length scale parameters, power-law exponent, slenderness ratio, shear deformation, and various boundary conditions on natural frequencies of FG nanobeams in detail.     

Bending,buckling and vibration of size-dependent functionally graded annular microplates

The bending, buckling and free vibration of annular microplates made of functionally graded materials (FGMs) are investigated in this paper based on the modified couple stress theory and Mindlin plate theory. This microplate model incorporates the material length scale parameter that can capture the size effect in FGMs. The material properties of the FGM microplates are assumed to vary in the thickness direction and are estimated through the Mori–Tanaka homogenization technique. The higher-order governing equations and boundary conditions are derived by using Hamilton’s principle. The differential quadrature (DQ) method is employed to discretize the governing equations and to determine the deflection, critical buckling load and natural frequencies of FGM microplates. A parametric study is then conducted to investigate the influences of the length scale parameter, gradient index and inner-to-outer radius ratio on the bending, buckling and vibration characteristics of FGM microplates with hinged–hinged and clamped–clamped supports. The results show that the size effect on the bending, buckling and vibration characteristics is significant when the ratio of the microplate thickness to the material length scale parameter is smaller than 10.     

Nonlinear non-classical microscale beams: Static bending, postbuckling and free vibration

This paper initiates the theoretical analysis of nonlinear microbeams and investigates the static bending, postbuckling and free vibration. The nonlinear model is conducted within the context of non-classical continuum mechanics, by introducing a material length scale parameter. The nonlinear equation of motion, in which the nonlinear term is associated with the mean axial extension of the beam, is derived by using a combination of the modified couple stress theory and Hamilton’s principle. Based on this newly developed model, calculations have been performed for microbeams simply supported between two immobile supports. The static deflections of a bending beam subjected to transverse force, the critical buckling loads and buckled configurations of an axially loaded beam, and the nonlinear frequencies of a beam with initial lateral displacement are discussed. It is shown that the size effect is significant when the ratio of characteristic thickness to internal material length scale parameter is approximately equal to one, but is diminishing with the increase of the ratio. Our results also indicate that the nonlinearity has a great effect on the static and dynamic behaviors of microscale beams. To attain accurate and reliable characterization of the static and dynamic properties of microscale beams, therefore, both the microstructure-dependent parameters and the nonlinearities have to be incorporated in the design of microscale beam devices and systems.     

The modified couple stress functionally graded Timoshenko beam formulation

In this paper, a size-dependent formulation is presented for Timoshenko beams made of a functionally graded material (FGM). The formulation is developed on the basis of the modified couple stress theory. The modified couple stress theory is a non-classic continuum theory capable to capture the small-scale size effects in the mechanical behavior of structures. The beam properties are assumed to vary through the thickness of the beam. The governing differential equations of motion are derived for the proposed modified couple-stress FG Timoshenko beam. The generally valid closed-form analytic expressions are obtained for the static response parameters. As case studies, the static and free vibration of the new model are respectively investigated for FG cantilever and FG simply supported beams in which properties are varying according to a power law. The results indicate that modeling beams on the basis of the couple stress theory causes more stiffness than modeling based on the classical continuum theory, such that for beams with small thickness, a significant difference between the results of these two theories is observed.     

Buckling analysis of functionally graded microbeams based on the strain gradient theory

The buckling behavior of size-dependent microbeams made of functionally graded materials (FGMs) for different boundary conditions is investigated on the basis of Bernoulli–Euler beam and modified strain gradient theory. The higher-order governing differential equation for buckling with all possible classical and non-classical boundary conditions is obtained by a variational statement. The effects of the power of the material property variation function, boundary conditions, slenderness ratio, ratio of additional material length scale parameters for two constituents, beam thickness-to-additional material length scale parameter ratio on the buckling response of FGM microbeams are investigated. Some comparative results are presented in tabular and graphical form in order to show the differences between the results obtained by the present model and those predicted by modified couple stress and classical continuum models.     

Investigation of the size-dependent dynamic characteristics of atomic force microscope microcantilevers based on the modified couple stress theory

In this paper, the resonant frequency and sensitivity of atomic force microscope (AFM) microcantilevers are studied using the modified couple stress theory. The classical continuum mechanics is incapable of interpreting micro-structure-dependent size effects when the size of structures is in micron- and sub-micron scales. However, this dependency can be well treated by using non-classical continuum theories. The modified couple stress theory is a non-classic continuum theory which employs additional material parameters besides those appearing in classical continuum theory to treat the size-dependent behavior. In this work, writing differential equations of motion of AFM cantilevers together with appropriate boundary conditions based on the couple stress theory, the analytical expressions are derived for the natural frequency and sensitivity. According to the numerical results, when the ratio of beam thickness to the material length scale parameter is less than 10, the difference between the classical based and the couple stress based results of resonance frequencies and sensitivities is considerable. The results show the significant importance of the size effects in behavior of AFM microcantilevers.     

A non-classical Mindlin plate model based on a modified couple stress theory

A non-classical Mindlin plate model is developed using a modified couple stress theory. The equations of motion and boundary conditions are obtained simultaneously through a variational formulation based on Hamilton??s principle. The new model contains a material length scale parameter and can capture the size effect, unlike the classical Mindlin plate theory. In addition, the current model considers both stretching and bending of the plate, which differs from the classical Mindlin plate model. It is shown that the newly developed Mindlin plate model recovers the non-classical Timoshenko beam model based on the modified couple stress theory as a special case. Also, the current non-classical plate model reduces to the Mindlin plate model based on classical elasticity when the material length scale parameter is set to be zero. To illustrate the new Mindlin plate model, analytical solutions for the static bending and free vibration problems of a simply supported plate are obtained by directly applying the general forms of the governing equations and boundary conditions of the model. The numerical results show that the deflection and rotations predicted by the new model are smaller than those predicted by the classical Mindlin plate model, while the natural frequency of the plate predicted by the former is higher than that by the latter. It is further seen that the differences between the two sets of predicted values are significantly large when the plate thickness is small, but they are diminishing with increasing plate thickness.     

Static and dynamic analysis of micro beams based on strain gradient elasticity theory

The static and dynamic problems of Bernoulli-Euler beams are solved analytically on the basis of strain gradient elasticity theory due to Lam et al. The governing equations of equilibrium and all boundary conditions for static and dynamic analysis are obtained by a combination of the basic equations and a variational statement. Two boundary value problems for cantilever beams are solved and the size effects on the beam bending response and its natural frequencies are assessed for both cases. Two numerical examples of cantilever beams are presented respectively for static and dynamic analysis. It is found that beam deflections decrease and natural frequencies increase remarkably when the thickness of the beam becomes comparable to the material length scale parameter. The size effects are almost diminishing as the thickness of the beam is far greater than the material length scale parameter.     

Nonlinear free vibration of size-dependent functionally graded microbeams

Nonlinear free vibration of microbeams made of functionally graded materials (FGMs) is investigated in this paper based on the modified couple stress theory and von Kármán geometric nonlinearity. The non-classical beam model is developed within the framework of Timoshenko beam theory which contains a material length scale parameter related to the material microstructures. The material properties of FGMs are assumed to be graded in the thickness direction according to the power law function and are determined by Mori-Tanaka homogenization technique. The higher-order nonlinear governing equations and boundary conditions are derived by using the Hamilton principle. A numerical method that makes use of the differential quadrature method together with an iterative algorithm is employed to determine the nonlinear vibration frequencies of the FGM microbeams with different boundary conditions. The influences of the length scale parameter, material property gradient index, slenderness ratio, and end supports on the nonlinear free vibration characteristics of the FGM microbeams are discussed in detail. It is found that both the linear and nonlinear frequencies increase significantly when the thickness of the FGM microbeam is comparable to the material length scale parameter.     

Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory

Dynamic stability of microbeams made of functionally graded materials (FGMs) is investigated in this paper based on the modified couple stress theory and Timoshenko beam theory. This non-classical Timoshenko beam model contains a material length scale parameter and can interpret the size effect. The material properties of FGM microbeams are assumed to vary in the thickness direction and are estimated though Mori–Tanaka homogenization technique. The higher-order governing equations and boundary conditions are derived by using the Hamilton’s principle. The differential quadrature (DQ) method is employed to convert the governing differential equations into a linear system of Mathieu–Hill equations from which the boundary points on the unstable regions are determined by Bolotin’s method. Free vibration and static buckling are also discussed as subset problems. A parametric study is conducted to investigate the influences of the length scale parameter, gradient index and length-to-thickness ratio on the dynamic stability characteristics of FGM microbeams with hinged–hinged and clamped–clamped end supports. Results show that the size effect on the dynamic stability characteristics is significant only when the thickness of beam has a similar value to the material length scale parameter.     

A non-classical third-order shear deformation plate model based on a modified couple stress theory

A non-classical third-order shear deformation plate model is developed using a modified couple stress theory and Hamilton’s principle. The equations of motion and boundary conditions are simultaneously obtained through a variational formulation. This newly developed plate model contains one material length scale parameter and can capture both the size effect and the quadratic variation of shear strains and shear stresses along the plate thickness direction. It is shown that the new third-order shear deformation plate model recovers the non-classical Reddy-Levinson beam model and Mindlin plate model based on the modified couple stress theory as special cases. Also, the current non-classical plate model reduces to the classical elasticity-based third-order shear deformation plate model when the material length scale parameter is taken to be zero. To illustrate the new model, analytical solutions for the static bending and free vibration problems of a simply supported plate are obtained by directly applying the general forms of the governing equations and boundary conditions of the model. The numerical results show that the deflection and rotations predicted by the new plate model are smaller than those predicted by its classical elasticity-based counterpart, while the natural frequency of the plate predicted by the former is higher than that by the latter. It is further seen that the differences between the two sets of predicted values are significant when the plate thickness is small, but they are diminishing with increasing plate thickness.     

双层微梁固有特性的尺寸效应

微尺寸梁存在明显尺寸效应,应变梯度理论可以描述这种尺寸效应。该文基于修正偶应力理论,应用双层梁与单层梁的等效关系,给出了双层微梁的动力学模型,具体求解了简支双层微梁的固有频率,并分析了微梁特征尺寸及双材料参数对双层微梁固有特性的影响规律。结果表明,当双层微梁的厚度接近材料内秉特征尺寸参数时,其固有频率值明显大于传统理论下的值;当双层微梁的厚度远大于材料内秉特征尺寸时,其固有频率值与传统理论下的值基本一致。双层微梁无量纲固有频率表现出明显尺寸效应,并随双材料参数的改变表现出一定的差异。当一层梁厚度远大于另一层厚度时,双层微梁可简化为单层微梁。     

Dynamic stability analysis of functionally graded higher-order shear deformable microshells based on the modified couple stress elasticity theory

Size-dependent dynamic stability response of higher-order shear deformable cylindrical microshells made of functionally graded materials (FGMs) and subjected to simply supported end supports is investigated. Material properties of the microshells vary in the thickness direction according to the Mori–Tanaka scheme. The modified couple stress elasticity theory in conjunction with the classical higher-order shear deformation shell theory is utilized to develop non-classical shell model containing additional internal length scale parameter to interpret size effect. The differential equations of motion and boundary conditions are derived by using Hamilton’s principle. The governing equations are then written in the form of Mathieu–Hill equations and then Bolotin’s method is employed to determine the instability regions. Selected numerical results are given to indicate the influences of internal length scale parameter, material property gradient index, static load factor and axial wave number on the dynamic stability behavior of FGM microshells. It is found that the width of the instability region for an FGM microshell increases with the decrease of the value of dimensionless length scale parameter. Moreover, it is shown that the classical shell model has an overestimated prediction for the width of instability region corresponding to the FGM microshells especially with lower values of material property gradient index.     

On the stability of a microbeam conveying fluid considering modified couple stress theory

In this paper, the size-dependent vibrational behavior of a microbeam conveying fluid was investigated using the Modified Couple Stress Theory. For cantilever and clamped-clamped microbeams, the small amplitude vibration equation of the micro-beams was solved using a Galerkin based reduced order model and the effects of material length-scale parameter on its natural frequencies were evaluated. It was found that for the both cantilever and clamped-clamped conditions, the critical fluid velocities predicted by the modified couple stress theory are higher than those predicted by the classical beam theory. In addition, the differences between the eigen-frequencies and the critical fluid velocities predicted by the modified couple stress theory and classical beam theory depends on the ratio of the material length-scale parameter to the beam height. In addition an unexpected result in the difference between the first eigen-frequency of the cantilever micro-beam obtained by the classical and the modified couple stress theory has been achieved.     

The modified couple stress model for bending of normal deformable viscoelastic nanobeams resting on visco-Pasternak foundations

ABSTRACT

The modified couple stress theory (MCST) is utilized to investigate the bending of viscoelastic nanobeams laying on visco-Pasternak elastic foundations based on a new shear and normal deformations beam theory. This model consists of the material length scale coefficient that captures the size impact on small-scale beams. The simply supported beam is made of viscoelastic material, subjected to time harmonic transverse load. The nanobeam is presumed to be laying on double layers of foundations. The first layer is modeled as Kelvin–Voigt viscoelastic model and the second is taken as a shear layer. Based on the proposed beam theory and MCST, the differential motion equations are deduced using Hamilton’s principle. To check the validity of the obtained formulations, the predicted results are compared with those available in the open literature. In addition, the influences of various parameters such as the material length scale parameter, length-to-depth ratio, viscoelastic damping structure, the stiffness and damping coefficients of the viscoelastic substrate, and shear and normal strains on the deflection and stresses are illustrated.