Thermoelastic damping in a micro-beam resonator using modified couple stress theory

In this paper, analytical expressions for the quality factor (QF) of thermoelastic damping (TED) applying modified couple stress theory (MCST) are presented for plane stress and strain conditions. For gold and nickel micro-beam resonators, which have a considerable length-scale parameter as case studies, the effect of the length-scale parameter on the QF of TED has been discussed in details. Results for the QF of TED show that when the beam thickness is close to the length-scale parameter of the material, the MCST diverges from the classical theory; otherwise, the two theories converge to each other. Also critical thickness variation due to applying MCST is studied.     

A comprehensive study on the size-dependent hygrothermal analysis of exponentially graded microplates on elastic foundations

Abstract

In this paper, the effects of hygrothermal conditions on various behaviors, such as bending, free vibration, mechanical and thermal buckling, of exponentially graded microplates lying on two-parameter elastic foundations are investigated. The trigonometric four-variable plate theory incorporated to the modified couple stress theory (MCST) is employed to derive the equations of motion. The present MCST contains an internal material length scale parameter, thus it can capture the size effect. The microplate is assumed to be subjected to a temperature rise and moisture concentration which are varied linearly through the thickness of the plate. Based on an exponential law, the material properties of the microplate are graded only in z direction. The equations of motion are solved analytically to obtain the displacements, stresses, eigenfrequencies and critical buckling load and temperature of the microplates. The present results are validated by comparing them with those previously published. The numerical examples reveal that considering the size effect and/or the elastic foundations leads to an increment in plate stiffness and thereby leads to a decrement in the deflection and an increment in eigenfrequency and buckling loads. It is also shown that the size effect is negligible for the thicker plate.     

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.     

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.     

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.     

Bending-stretching analysis of thick functionally graded micro-plates using higher-order shear and normal deformable plate theory

In the present article, higher-order shear and normal deformable plate theory together with modified couple stress theory are developed to study the bending analysis of thick functionally graded rectangular micro-plates. One material length scale parameter is used for capturing the size effects. Utilizing the variational approach and also a principle of virtual displacement, a new form of equilibrium equations and the corresponding boundary conditions are derived. It is assumed that material properties vary through the thickness according to the power law function. Finally, an analytical solution for the bending problem of a simply supported FG rectangular micro-plate is presented.     

Size-dependent generalized thermoelasticity model for Timoshenko microbeams

A size-dependent, explicit formulation for coupled thermoelasticity addressing a Timoshenko microbeam is derived in this study. This novel model combines modified couple stresses and non-Fourier heat conduction to capture size effects in the microscale. To this purpose, a length-scale parameter as square root of the ratio of curvature modulus to shear modulus and a thermal relaxation time as the phase lag of heat flux vector are considered for predicting the thermomechanical behavior in a microscale device accurately. Governing equations and boundary conditions of motion are obtained simultaneously through variational formulation based on Hamilton’s principle. As for case study, the model is utilized for simply supported microbeams subjected to a constant impulsive force per unit length. A comparison of the results with those obtained by the classical elasticity and Fourier heat conduction theories is carried out. Findings indicate that simultaneous considering the length-scale parameter and thermal relaxation time has strong influence on the thermoelastic behavior of microbeams. In dynamic thermoelastic analysis of the microbeam, while the non-Fourier heat conduction model is employed, the modified couple stress theory predicts larger deflection compared with the classical theory.     

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.     

A Size-Dependent Functionally Graded Higher Order Plate Analysis Based on Modified Couple Stress Theory and Moving Kriging Meshfree Method

A size-dependent computational approach for bending, free vibration and buckling analyses of isotropic and sandwich functionally graded (FG) microplates is in this study presented. We consider both shear deformation and small scale effects through the generalized higher order shear deformation theory and modified couple stress theory (MCST). The present model only retains a single material length scale parameter for capturing properly size effects. A rule of mixture is used to model material properties varying through the thickness of plates. The principle of virtual work is used to derive the discrete system equations which are approximated by moving Kriging interpolation (MKI) meshfree method. Numerical examples consider the inclusions of geometrical parameters, volume fraction, boundary conditions and material length scale parameter. Reliability and effectiveness of the present method are confirmed through numerical results.     

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.     

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.     

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.     

Constrained thickness optimization of rectangular orthotropic fiber-reinforced plate for fundamental frequency maximization

This paper addresses the thickness optimization of rectangular orthotropic fiber-reinforced composite plates by using mathematical programming theory of nonlinear constrained optimization. We optimize the thickness distribution of various composite plates for reaching a maximum in the plate natural frequency, subject to an equality constraint in a plate volume and a number of inequality constraints on the lower and upper bounds of allowable range of thicknesses. For the modeling part of our optimization algorithm, we formulate the kinematical hypothesis of the classical plate theory (CPT), reduced plane stress theory of linearized elasticity, and Hamilton’s principle. For the solution part, we implement the Rayleigh-Ritz semi-analytical solution technique to spatially discretize the weak form of the plate partial differential equation (PDE), and transforming it to a generalized algebraic eigen-problem. For the design optimization part, we use a conjugate Rosen’s gradient projection method. The design variables of the optimization design space are thicknesses of an arbitrary but rectangular partition of the plate into constant-thickness subdivisions. After simulating our algorithm, we noticed many interesting and impressive results, and tried to justify them. The most impressive finding is that, as the fiber orientation of the composite plate rotates, the plate thickness distribution rotates also in the same way, but with different manifestations for different boundary conditions. For parametric studies, we analyze the sensitivity of the algorithm against parameters, like, the degree of fineness in the number of plate partitions into constant-thickness sub-plates, the degree of plate anisotropy, the fiber orientation, the rectangular plate aspect ratio, the upper to lower bounds of allowable thickness ratio, and boundary conditions.     

基于正弦剪切变形理论的功能梯度材料三明治微梁的静动态特性

基于修正的偶应力理论和正弦剪切变形梁理论,研究了功能梯度材料三明治微梁的静态弯曲和自由振动行为。考虑两种不同类型的功能梯度材料三明治微梁,根据哈密顿变分原理建立其静动态力学行为的控制方程,应用Navier解法,得到了简支边界条件下弯曲变形和振动频率的解析解,同时,给出了固支等边界条件时的里兹法求解过程。数值算例表明,功能梯度三明治微梁的静动态力学行为具有明显的尺度效应,微梁的无量纲厚度、功能梯度指数、长厚比和结构形式等因素对其静动态响应有很大影响,相关结果和规律对功能梯度材料三明治微梁的结构设计和性能优化等实际工程应用具有一定的指导意义。      

基于新修正偶应力理论的Mindlin层合板自由振动分析

该文基于各向异性修正偶应力理论建立一个Mindlin层合板（跨厚比10~20的中厚板）自由振动模型。该理论偶应力曲率张量不对称,但偶应力弯矩对称。利用Hamilton原理推导振动微分方程和边界条件。新模型可退化为修正偶应力层合薄板振动模型和经典Mindlin层合板振动模型。以正交铺设简支方板为例计算了偶应力模型的自振频率,分析偶应力Mindlin层合板的自由振动尺度效应。算例表明,该文建立的新修正偶应力层合板模型能够用于分析细观尺度下Mindlin层合板的自由振动及尺度效应。     

A nonlinear thick plate formulation based on the modified strain gradient theory

A geometric nonlinear first-order shear deformation theory-based formulation is presented to analyze microplates. The formulations derived herein are based on a modified strain gradient theory and the von Karman nonlinear strains. The modified strain gradient theory includes five material length scale parameters capable to capture the size effects in small scales. The governing equations of motion and the most general form of boundary conditions of an arbitrary-shaped plate are derived using the principle of virtual displacements. The analysis is general and can be reduced to the modified couple stress plate model or the classical plate model.     

Levy Solution for Buckling Analysis of Functionally Graded Rectangular Plates

In this article, an analytical method for buckling analysis of thin functionally graded (FG) rectangular plates is presented. It is assumed that the material properties of the plate vary through the thickness of the plate as a power function. Based on the classical plate theory (Kirchhoff theory), the governing equations are obtained for functionally graded rectangular plates using the principle of minimum total potential energy. The resulting equations are decoupled and solved for rectangular plate with different loading conditions. It is assumed that the plate is simply supported along two opposite edges and has arbitrary boundary conditions along the other edges. The critical buckling loads are presented for a rectangular plate with different boundary conditions, various powers of FGM and some aspect ratios.     

Analytical solutions for an infinite transversely isotropic functionally graded sectorial plate subjected to a concentrated force or couple at the tip

This paper considers the elastic responses of an infinite sectorial plate made of transversely isotropic functionally graded material (FGM), which is subjected to a concentrated force or couple at the tip. There is no load acting on the upper and lower surfaces, and the elastic coefficients can vary arbitrarily through the plate thickness. No constraint is required on the symmetry of the plate in the thickness direction. Based on the displacement assumption for the bending of an FGM plate and by using the complex variable method, this paper presents the general solutions to the basic equations governing transversely isotropic FGM plates, which are expressed in terms of four analytical functions (or complex potentials). The boundary conditions are a combination of those from the plane elasticity and those from the classical plate theory. For a particular boundary value problem, such as the ones considered here for a sectorial plate, with the specific conditions for determining solutions, the four analytic functions can be assumed in appropriate forms, which contain only some unknown constants. Once these constants are determined from the specific conditions, the complete solutions are readily derived too. Among the solutions presented here, the solutions for the infinite FGM sectorial plate under a concentrated couple are absolutely new to the literature, and they are also applicable to isotropic FGM sectorial plates. The solutions degenerate into the ones for a homogeneous sectorial plate, which coincide with the available solutions from the plane elasticity theory. There are three-dimensional correction terms in the mid-plane displacements.     

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.     

A nonlinear modified couple stress-based third-order theory of functionally graded plates

In this paper a general nonlinear third-order plate theory that accounts for (a) geometric nonlinearity, (b) microstructure-dependent size effects, and (c) two-constituent material variation through the plate thickness (i.e., functionally graded material plates) is presented using the principle of virtual displacements. A detailed derivation of the equations of motion, using Hamilton’s principle, is presented, and it is based on a modified couple stress theory, power-law variation of the material through the thickness, and the von Kármán nonlinear strains. The modified couple stress theory includes a material length scale parameter that can capture the size effect in a functionally graded material. The governing equations of motion derived herein for a general third-order theory with geometric nonlinearity, microstructure dependent size effect, and material gradation through the thickness are specialized to classical and shear deformation plate theories available in the literature. The theory presented herein also can be used to develop finite element models and determine the effect of the geometric nonlinearity, microstructure-dependent size effects, and material grading through the thickness on bending and postbuckling response of elastic plates.