Investigation of thermoelastic damping in rectangular microplate resonator using modified couple stress theory

Thermoelastic damping is a significant energy lost mechanism at room temperature in micro-scale resonators. Prediction of thermoelastic damping (TED) is crucial in the design of high quality MEMS resonators. In this study the governing equations of motion and the thermal couple equation of a microplate with an arbitrary rectangular shape are derived using the modified version of the couple stress theory. Analytical expressions are presented for calculating the quality factor (QF) of TED in a rectangular microplate considering the plane stress and plane strain conditions. As a case study, a rectangular microplate resonator is considered with material property of gold that has a considerably high value of length-scale parameter in comparison with silicon and the effect of the length-scale parameter on the QF of TED is discussed in detail. The relation between QF and temperature increment for microplates with clamped boundary conditions based on plane stress and plane strain models are studied and results obtained by considering classical and modified couple stress theory (MCST) are compared. The effect of thickness of the plate on the rigidity ratio is studied and the critical thickness which is an important design parameter is obtained using the MCST for three boundary conditions. Variations of TED versus the plate thickness for various boundary conditions according to the classical and the modified couple stress theories are investigated.     

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.     

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.     

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.     

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。      

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.     

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.     

Thin film hardness determination using indentation loading curve modelling

A methodology for determining the thin film hardness from a microindentation loading curve is proposed. The loading curve is modelled to compute the dynamic Martens hardness using the indentation depth reached during the test. Moreover, the indentation size effect is taken into account by applying the strain gradient plasticity theory. Then, the dynamic Martens hardness and the hardness length-scale factor are used to express the applied load as a function of the indentation depth. The proposed model involves three parameters: (i) the dynamic Martens macro-hardness, equivalent to the hardness obtained for an infinite applied load, (ii) the hardness length-scale factor, which represents the material resistance to plastic deformation under indentation and (iii) a corrective load, considering the rounded tip effect of the indenter and the zero shift. The model is validated on a 316L stainless steel which subsequently is used as a substrate material for two different Diamond Like-Carbon thin films. The coated systems involved both a hydrogen-free mostly amorphous carbon-chromium (a-C) film of ∼ 2.6 μm in thickness and a hydrogenated, amorphous carbon (a-C:H) solid lubricant of ∼ 2 μm.     

On the nonlinear dynamics of a piezoelectrically tuned micro-resonator based on non-classical elasticity theories

Size dependent static and dynamic behavior of a fully clamped micro beam under electrostatic and piezoelectric actuations is investigated. The microbeam is modeled under the assumptions of Euler–Bernoulli beam theory. Viscous damping and nonlinearities due to electrostatic actuation and mid-plane stretching are considered. Residual stress and fringing field effect are taken into account as well. Governing equation of motion is derived using Hamilton’s principle along with the strain gradient theory (SGT), which is a non-classical continuum theory capable of taking size effect of elastic materials into account. Reduced order model of the partial differential equations of the system is obtained using Galerkin method. Static deflection, pull-in voltage and the primary resonance of the microbeam are examined and the effect of piezoelectric voltage and its polarization on the size dependent static and dynamic response is studied. It is found that the piezoelectric voltage can effectively change the flexural rigidity of the system which in turn affects the pull-in instability regime. The effect of material length scale parameter is examined by comparing the results of the SGT with the modified couple stress (MCST) and classical theory (CT), both of which are special cases of the former. Comparison demonstrates that the CT underestimates the stiffness and consequently the pull-in voltage and overestimates the amplitude of periodic solutions. The difference between the results of classical and non-classical theories becomes more and more as the dimensions of the system gets close to the length scale parameter. Non-classical theories predict more realistic behaviors for the micro system. The results of this paper can be used in designing microbeam based MEMS devices.     

Size-dependent behaviour of electrically actuated microcantilever-based MEMS

In this paper, the nonlinear size-dependent static and dynamic behaviours of a microelectromechanical system under an electric excitation are investigated. A microcantilever is considered for the modelling of the deformable electrode of the MEMS. The governing equation of motion is derived based on the modified couple stress theory (MCST), a non-classical model capable of capturing small-size effects. With the aid of a high-dimensional Galerkin scheme, the nonlinear partial differential equation governing the motion of the deformable electrode is converted into a reduced-order model of the system. Then, the pseudo-arclength continuation technique is used to solve the governing equations. In order to investigate the static behaviour and static pull-in instabilities, the system is excited only by the electrostatic actuation (i.e., a DC voltage). The results obtained for the static pull-in instability predicted by both the classical theory and MCST are compared. In the second stage of analysis, the nonlinear dynamic behaviour of the deformable electrode due to the AC harmonic actuation is investigated around the deflected configuration, incorporating size dependence.     

Dynamic analysis of an embedded microbeam carrying a moving microparticle based on the modified couple stress theory

In this study, analytical and numerical solution procedures are proposed for vibration of an embedded microbeam under action of a moving microparticle based on the modified couple stress theory (MCST) within the framework of Euler–Bernoulli beam theory. The governing equation and the related boundary conditions are derived by using Hamilton’s principle. The closed form solution of the transverse deflections of the embedded microbeam is obtained using the finite Fourier sine transformation. In the numerical solution, the dynamic deflections are computed by using the Lagrange’s equations in conjunction with the direct integration method of Newmark. The static deflections are also obtained analytically. A detailed parametric study is conducted to study the influences of the material length scale parameter, the Poisson’s ratio, the velocity of the microparticle and the elastic medium constant as well as the solution procedures on the dynamic responses of the microbeam. For comparison purpose, static deflections and free vibration frequencies of the microbeam are obtained and compared with previously published studies. Good agreement is observed. The results show that the above mentioned effects play an important role on the dynamic behavior of the microbeam.     

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.     

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.     

Large-amplitude dynamics of a functionally graded microcantilever with an intermediate spring-support and a point-mass

Numerical modelling and simulations are conducted on the large-amplitude dynamics of a functionally graded microcantilever with a tip-mass, additionally supported by an intermediate spring; the functionally graded microsystem is subject to a base excitation. Since one end of the microsystem is free to move, it undergoes large deformation; curvature-related nonlinearities play an important role. Taking into account this type of nonlinearity, using the Mori–Tanaka homogenisation scheme, as well as the modified couple stress theory, an energy technique is employed to derive the nonlinearly coupled equations for the longitudinal and transverse motions. An inextensibility assumption is applied for the functionally graded microcantilever, and hence, a nonlinear equation of motion for the transverse motion involving inertial (apart from stiffness) nonlinearity is obtained. For the functionally graded microsystem considered, effects of the length-scale parameter, the material gradient index, the tip-mass, and the stiffness of the spring-support on the nonlinear resonant responses are highlighted by means of a Houbolt’s finite difference scheme together with Newton–Raphson method.     

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.     

Stress concentration factors and stress-gradients due to circular holes in radially functionally graded panels subjected to anti-plane shear loading

Closed-form expressions for the elastic fields (stresses, stress-gradients, strains, and displacements) in radially graded continua without and with circular holes are derived. The conventional definition of the stress concentration factor, which is the ratio of the maximum tangential stress on the circumference of the hole to the far-field applied traction, leads to physically unrealistic interpretations in functionally graded materials. A novel definition is derived for the stress concentration factor in graded panels. The stress concentration factor increases when the rigidity modulus progressively decreases away from the center of the hole (softening materials). The stress concentration desirably reduces in hardening materials in which the rigidity modulus progressively increases away from the hole. The gradient of the tangential stress ahead of the circular hole introduces an affected zone that is size-dependent (depends on the size of the hole, even in homogeneous media). The affected zone size decreases, while the stress-gradients increase with increase in the inhomogeneity length-scale and results in the increasing propensity of damage in softening materials. In hardening materials, the propensity of damage reduces with increase in the inhomogeneity length-scale due to the stress dilution.     

复合材料层合厚圆柱壳高阶理论的改进及其应用

建立了一个改进的LCW型的精化高阶理论,以分析厚圆柱壳的振动。提出u,v为三次多项式、w为二次多项式的位移模式,并利用上、下自由表面横向剪应力为零的边界条件,对所假定的位移场作了化简,将三阶剪切变形理论的未知数缩减为7个,在此基础上建立了相应的有限元列式。通过一个典型算例,与Soldatos和Lam的高阶剪切变形理论的解析解作了比较,说明笔者的精化高阶理论是可行的,而且具有较高的精确性,比LCW高阶理论更具有实用性。还通过频率参数随长度半径比L/R的变化,说明由于考虑了法向应力和法向应变,本文方法更适用于长度半径比较小的结构。     

Optimization of a tablet containing chlorthalidone

In pharmaceutical technological research, optimization studies generally deal with the search for the formulation that is as effective and as functional as possible. The effect of a formulation parameter (the amount of lactose in the composition of the tablets) and of a technological parameter (compression pressure) on four physical characteristics (tablet thickness, friability, hardness, and drug dissolution rate) of tablets containing the antihypertensive drug chlorthalidone were studied. The results obtained indicate that, in the development of a tablet formulation, it is possible to identify the most suitable formulation by applying a simple optimization method. The effect of the microclimatic stress (temperature and humidity) was also evaluated, and it was found that the optimized tablets were no longer within limits that had been established for them. This may indicate that it is opportune to keep the storage conditions of the excipients under control before their use.