Modified Couple Stress Theory Research Articles

Thermoelastic damping in rectangular micro/nanoplate resonators by considering three-dimensional heat conduction and modified couple stress theory

Thermoelastic damping (TED) has been confirmed as an intrinsic energy loss and determines the upper limit of the quality factor of micro/nanoelectromechanical system (M/NEMS) resonators. A framework for determining the TED in the full clamped (C-C-C-C), clamped-free-clamped-free (C-F-C-F), and cantilever (C-F-F-F) rectangular microplate resonators operating in the fundamental mode by considering three-dimensional heat conduction in conjunction with modified couple stress theory (MCST) is proposed in this article. Then by using Rayleigh’s method, three analytical models expressed by Fourier series for TED are derived, respectively. The difference between the present TED model and the previous one-dimensional (1-D) MCST model and 3-D classical model are comprehensively investigated. The comparison results indicate that the size effect on TED decreases to be ignorable as the thickness increases to h/l ≥ 50. In the case of h/l ≤ 50, the size effect cannot be ignored. The corresponding critical thicknesses for TED peaks are independent of the size effect. TED behaviors vary significantly under different structural support types.

Nonlinear Vibration Movements of the Mid-Supported Micro-Beam

This study analyzes the vibration movements of multi-support micro beams placed in an electrically smooth area using the modified couple stress theory. It has been assumed that the potential voltage that creates the electrical field strength varies harmonically. Large number of experiments in recent years have indicated that classical continuum theory is unable to predict the mechanical behavior of microstructure with small size. However, nonclassical continuum theory should be used to accurately design and analyze the microstructures. Modified couple stress theory models the micro and nanomechanical systems with higher accuracy because they employ additional material parameters to the equation considering size dependent behavior. The most general nonlinear motion equations for multi-support microbeams have been obtained by considering the material size parameter, the number of support and support positions, damping effect, axial stresses, electrical field strength, and nonlinear effects resulting from elongations. The nonlinear equations of motion are obtained according to the Hamilton method using the modified couple stress theory (MCST). The resulting equations of motion are nondimensionalized. In this way, the mathematical model has been made independent of the type and geometric structure of the material. Approximate solutions of the obtained dimensionless motion equation are obtained by the multi-scale method, which is one of the perturbation methods. As a result, an increase occurs in the first mode frequencies ([Formula: see text]) and nonlinear correction effect parameters ([Formula: see text]) with the progress of the center support position gradually towards [Formula: see text] and the increase of the microbeam elasticity coefficient ([Formula: see text]).

A three-dimensional weak formulation for stress, deformation, and free vibration analyses of functionally graded microscale plates based on the consistent couple stress theory

A three-dimensional (3D) weak formulation based on the consistent couple stress theory (CCST) is developed for displacement, stress, and free vibration analyses of simply-supported, power-law-/exponential-type functionally graded (FG) microscale plates. In the formulation, the microscale plate is artificially divided into numerous finite microscale layers, where Fourier functions and Lagrange/Hermite polynomials are used to interpolate the in- and out-of-plane variations in the displacement variables for each individual layer, respectively. Based on the weak formulation, the authors develop layer-wise C0 quadratic and cubic finite element methods (FEMs), as well as layer-wise C1 two- and three-node FEMs to address the current issue. The accuracy and convergence of these layer-wise C0 and C1 FEMs are validated by comparing their solutions with the 3D exact and the quasi-3D results available in the literature. A 3D weak formulation based on the modified couple stress theory (MCST) is also developed for comparison purposes. The similarity and differences between the results obtained using the CCST- and MCST-based weak formulations are examined and discussed. Some key effects on stress, deformation, and free vibration characteristics of the FG microscale plate are examined, including the material length scale, the material-property gradient index, and the length-to-thickness ratio effects, which are shown to be significant for the FG microscale plate.

Size-dependent vibrations of porous functionally graded rotating microplates under thermal environment

A comprehensive model for vibration analysis of porous functionally graded (P-FG) rotating microplates under thermal environment is established within the framework of the first-order shear deformation theory (FSDT) and modified couple stress theory (MCST). Material properties of the microplates are temperature-dependent and alter through the thickness following a power-law function. The microplates with even porosity subjected to the nonlinear and uniform temperature rises are considered. The second-kind Lagrange's equations are employed for derivation of the governing equations of motion, which are numerically solved by the method of assumed modes constructing using the Chebyshev polynomials. The effectiveness and accuracy of the present approach are validated by the examples of convergence and comparison. A parametric study is performed to investigate the effect of angular velocity, material gradient index, material length scale parameter, temperature distribution, temperature rise, porosity index and thickness-to-length ratio on dimensionless nature frequencies for the rotating SSSS and CFFF P-FG microplates. Mode shapes for rotating P-FG microplates with different angular velocities are also presented. Numerical results show that the frequencies increase with increasing the angular velocity and material length scale parameter, while decrease with increasing the temperature rise and material gradient index. Results also indicate that the in-plane extension motions can be ignored in vibration analysis for the thin microplates, whereas play an important role for the moderately thick or thick microplates. The frequency loci veering phenomena and the mode shape conversions are displayed.

Modified couple stress and thickness-stretching included formulation of a sandwich micro shell subjected to electro-magnetic load resting on elastic foundation

This paper studies electromagnetoelastic static investigation of a sandwich doubly curved microshell subjected to multi-field loading based on a new thickness stretching included refined higher order shear/normal deformable model. Modified-couple-stress-theory (MCST) is used for accounting small-scale-dependency. The numerical results are derived using an analytical method. The effect of small scale parameter in micro scale, initial electric and magnetic potentials and foundation parameters is studied on the electromagnetoelastic bending results. It is confirmed an enhancing in stiffness of small scale shell with an increase in micro length scale parameter.

Third order nonlinear vibration of viscoelastic circular microplate based on softening and hardening nonlinear viscoelastic foundation under thermal loading

The nonlinear free vibration of the viscoelastic solid and annular circular microplate is investigated. Nonlinear viscoelastic symmetric and asymmetric foundations are considered. Thermal loading is assumed as constant value. The nonlinear equations of the microplate are derived by Hamilton principle based on third order shear deformation theory (TSDT) and modified couple stress theory (MCST). The finite element methods (FEM) and Newton repeated algorithm are employed to obtain the nonlinear natural frequency. The obtained results are validated with the literatures. Then, the effects of various parameters such as, length scale parameter, outer radius to thickness ratio, maximum deflection to thickness ratio, softening and hardening nonlinear foundation, symmetric and asymmetric nonlinear viscoelastic foundations, different boundary conditions, structure and foundation damping are conducted. The obtained results are compared with several literature and a close agreement are observed between them. The results show that the symmetric and asymmetric nonlinear viscoelastic foundations affect more the nondimensional nonlinear frequency ratio. Reduction effect of temperature difference on MCST nondimensional nonlinear frequency ratio is higher than that of classical theory. The results can be applied micro and nanomachines and smart systems.

Size-Dependent Elastic Buckling of Two-Variable Refined Microplates Embedded in Elastic Medium

This study applies a two-variable refined shear deformation theory (TVRSDT) and a modified couple stress theory (MCST) to develop a size-dependent elastic buckling model for microplates under combined axial compression and in-plane shear and resting on Winkler–Pasternak foundation. Our model incorporates two transverse displacement variables and one material length scale parameter (MLSP); it satisfies the zero-traction boundary conditions on the top and bottom surfaces of microplates, thereby circumventing the use of a shear correction factor. The equations of motion are determined via the Euler–Lagrange equation. A closed-form buckling solution is presented for microplates with four edges simply supported. To handle the nonsimply supported microplates, a thirty-two-degree-of-freedom (32-DOF) four-node differential quadrature (DQ) finite element with [Formula: see text]-continuity is proposed. The corresponding element matrices are obtained using the minimum potential energy principle. Comparison studies are conducted to exhibit the validity of the derived formulations. Finally, the present buckling model is employed for predicting the elastic buckling behaviors of microplates embedded in an elastic medium. The effects of in-plane loading patterns, boundary conditions, aspect ratio, length-to-thickness ratio, MLSP and elastic medium parameters on the buckling load and buckling mode are elucidated.

Bending, buckling and vibration analysis of Bi-directional functionally graded Circular/Annular microplate based on MCST

The modified couple stress theory (MCST) is applied to analyze axisymmetric static bending, elastic buckling, and free vibration behaviors of circular/annular microplates made of bi-directionally functionally graded (BiFG) materials. The material properties of functionally graded (FG) microplates are assumed to vary along both through-the-thickness and through-the-radius directions. The differential governing equation and boundary conditions are derived through the Hamilton's principle, and expressed in nominal form with the introduced nominal variables. The generalized differential quadrature method is applied to discretize the differential governing equation and boundary conditions. The bending deflection can be obtained through solving a set of linear equations, while buckling loads and vibration frequencies can be determined through solving general eigenvalue problems. The current model is verified with those existed in literature. Selected numerical results are illustrated to indicate the effects of material length scale parameter, FG material indices and inner-to-outer radius ratio on the bending deflection, buckling loads and free vibration frequency of BiFG circular/annular microplates subjected to different boundary conditions.

Modified couple stress theory application to analyze mechanical buckling behavior of three-layer rectangular microplates with honeycomb core and piezoelectric face sheets

The current study is aimed to analyze the mechanical buckling of a three-layered microplate made from honeycomb structures sandwiched between two piezoelectric face sheets. The microplate is located on an elastic foundation that stands both normal and shearing loads. Minghui and Jiuren relations were employed to determine the effective mechanical properties of the honeycomb core. The sinusoidal shear deformation theory (SSDT) was also employed as a higher-order theory with trigonometric shear deformation function to describe the displacement components. The governing equations were derived using the virtual displacement principle and modified couple stress theory (MCST) was employed to predict the results in micro dimension. Navier’s solution method was chosen to analytically extract the critical buckling loads for a simply supported edges microplate. The effect of the most important parameters such as geometrical specifications of the honeycomb core on the results was also considered and discussed. Nowadays the honeycomb structures are widely used in different areas of sciences and technologies due to their lightweight and high stiffness. Also, the piezoelectric materials are applied as sensors and actuators in smart engineering structures. Therefore, the outcomes of this study may help in designing and manufacturing more applicable structures and smart devices capable of tolerating loading conditions, especially in small dimensions.

Selection of size dependency theory effects on the wave's dispersions of magneto-electro-thermo-elastic nano-beam resting on visco-elastic foundation

Current research develops a comprehensive wave propagation analysis of a magneto-electro-thermo-elastic (METE) nano-beam (NB) resting on the visco-elastic medium. To model the size dependency effects, modified couple stress (MCS) and Eringen's nonlocal (ENL) theories are employed to analyze and describe the wave propagation behaviors for those nano-beams. These theories were the most used in the literature due to the inclusion of one additional size-dependent length scale parameter. Those theories are compared side by side in this investigation and their impacts/differences on the wave propagation of specific materials are explored. Sinusoidal shear deformation beam model with Hamilton's principle is adopted to develop the governing equations of motion. Then, an analytical solution is executed to extract numerical results for transverse wave propagation in both elastic and METE configurations of the nano-beam. The effects of size-dependent length scale of both theories, thickness of NB, Winkler-Pasternak coefficients, thermal gradient, and magnetic potential and external electric voltage are illustrated and discussed in details. It is concluded that wave frequency decreases with increment of nonlocal parameter for ENL model. On the other hand, a stiffening effect takes place for the wave frequency when the MCS model is considered. Hence, the results indicate that there is a significant difference between ENL and MCS theories in the estimation of the behavior of wave propagation in small-scale structures. One of the main results of this investigation indicates that the MCS theory has similar nonlocal effects as the Eringen's theory for specific conditions.

Nonlinear random vibrations of micro-beams with fractional viscoelastic core

This work investigates the nonlinear vibration of a fractional viscoelastic micro-beam exposed to random excitation. For the micro-scale effect, the micro-beam model is established based on the Modified Couple Stress Theory (MCST). The fractional nonlinear partial differential equation is obtained using Von-Karman’s nonlinear strains and the Kelvin–Voigt fractional viscoelastic model. The micro-beam is exposed to a random force which its amplitude is considered as the Gaussian white noise. The Finite Difference Method (FDM) and the Galerkin method are applied to discretize the stochastic governing equation of motion. To solve the stochastic partial differential equation, a numerical discretization for the white noise excitation is applied. Moreover, the Statistical Linearization (SL) method and the Spectral Representation (SR) method are also employed and a comparison is done between these methods. The simulations show good agreement between the numerical method and other methods. In addition, it is shown that the proposed method can be used for different boundary conditions of the micro-beam. The impacts of the fractional derivative order, white noise intensity, viscoelastic model, and length-scale parameter on the statistical properties of the micro-beam are shown. The numerical simulations indicate that the high values of fractional-order can significantly affect the variance and the Root Mean Square (RMS) of the transverse amplitude, However, this influence for small values of the fractional-order is negligible; the impact of the fractional-order derivative on the statistical characteristics of the micro-beam is very different from 0<α<0.6, and 0.6<α<1.

Vibrations of axially excited rotating micro-beams heated by a high-intensity laser in light of a thermo-elastic model including the memory-dependent derivative

The introduction of the memory effect into thermal and mechanical models makes them more adaptive than the fractional effect. Actually, in this concept, not only the time delay but also the kernel function can be adjusted arbitrarily to meet the requirements of different dynamic processes. Furthermore, because of the remarkable thermomechanical characteristics of the microstructures, research into the rotating beam material may effectively improve the mechanical behavior of rotating systems. This paper addresses the thermoelastic vibration of spinning microbeams using a heat transfer model with a memory-dependent derivative (MDD). The influence of the centrifugal tensile force owing to rotation is taken into account, and the length-scale effect is addressed using modified couple stress theory (MCST). The equations controlling axially excited spinning microbeams have been established using the Euler–Bernoulli assumptions and Hamilton’s approach. Time-dependent variable temperature and laser pulses are responsible for thermoelastic vibrations in the microbeam. The Laplace transform approach is applied to establish a general solution for the analyzed fields. Some graphs are shown to demonstrate the effects of the length-scale, angular velocity of rotation, time-delay factor, and various types of kernel function on all studied fields.

Size-dependent behavior of a MEMS microbeam under electrostatic actuation

The size-dependent behavior of a silicon microbeam with an axial force in MEMS is studied using a nonlinear finite element procedure. Based on a refined third-order shear deformation theory and the modified couple stress theory (MCST), nonlinear differential equations of motion for the beam are derived from Hamilton’s principle, and they are transferred to a discretized form using a two-node beam element. Newton-Raphson based iterative procedure is used in conjunction with Newmark method to obtain the pull-in voltages and deflections of a clamped-clamped microbeam under electrostatic actuation. The influence of the axial force, applied voltage and material length scale parameter on the behavior of the beam is studied in detail and highlighted.

Modeling and analysis of a thermoviscoelastic rotating micro-scale beam under pulsed laser heat supply using multiple models of thermoelasticity

In recent decades, microelectromechanical system (MEMS) technology has aroused great interest from researchers in the fields of science and technology, electromechanics, chemistry, and bioengineering. Rotary systems are also used in a variety of sectors, from a space station with a spinning wheel to a small turbine in smartphones. In this paper, the thermoviscoelastic interactions of rotating microbeams were investigated using the modified couple stress theory (MCST) to examine the size-dependent effect. Also, the governing equations were derived within the framework of the Lord and Shulman generalized thermoelastic theory and the Euler–Bernoulli beam model. In addition, the Kelvin–Voigt approach is used to calculate the properties of viscoelastic materials. By applying the proposed model, an axially compressed microbeam that rotates at a constant angular velocity and is exposed to a femtosecond laser pulsed heat source was studied To solve the equations of motion in the space domain, the well-known Laplace transform technique is utilized. The inverse Laplace transform is calculated numerically using a well-established and reliable approximation approach. The effect of the length scale parameter, laser pulse duration, and angular velocity on the heat and elastic waves of a rotating thin beam is explored and discussed. The effect of the viscous damping factor on different physical fields of the microbeam was also studied.

On size-dependent nonlinear forced dynamics of MRE-cored sandwich micro-pipes in presence of moving flow and harmonic excitation

This paper is dedicated to examine the size-dependent nonlinear forced dynamics of MRE-cored sandwich micro-pipes in existence of external distributed load and internal flow. For this aim, Euler-Bernoulli beam theory, considering von-Karman nonlinearity, together with the modified couple stress theory (MCST) are exploited to attain a mathematical formulation of the problem. Employing Galerkin’s approach with consideration of two axial and transverse modes, the obtained nonlinear partial differential equations are discretized, from which nonlinear ordinary equations are achieved. Thereafter, static condensation method is utilized in order to reduce the number of the obtained ordinary equations. The acquired nonlinear equations are then solved according to the method of multiple scales, which leads to nonlinear natural frequencies and frequency-response curves of the system. Throughout the numerical analysis, the effect of a smart MRE core existence on the system’s response, considering various boundary conditions, is explored. Additionally, a numerical investigation is implemented to highlight the influences of MRE-related intrinsic properties namely the intensity of the applied magnetic potential and MRE core thickness on the natural frequencies, frequency-response and load-response curves of the system. Totally, the substantial role of the MRE core and its intrinsic characteristics on the dynamic response of the system is explored, which implies that, this smart core could be a good candidate in such sandwich structures in order to control and boost the dynamic properties and response of the mechanical systems.

On the mechanics of shear deformable micro beams under thermo-mechanical loads using finite element analysis and deep learning neural network

In this paper, the mechanics of shear deformable micro beams is investigated via finite element analysis, and a machine learning technique (Deep Learning Neural Network (DLNN)). DLNN is provided for the first time to accurately model and predict this behavior. Using the Finite Element Method (FEM), the obtained coupled equations of motion for shear deformable micro-scale beams are solved using a combination of Timoshenko beam theory and the modified couple stress theory (MCST). The results are compared with literature for each analysis to show the accuracy and the proposed finite element model. After presenting the mechanical model, by using Finite Element Analysis (FEA), a deep learning neural network model is developed for small-scale structures and the capability of this model in predicting different mechanical behavior under thermo-mechanical loading is investigated. It is shown that the presented model has great accuracy in predicting both static and dynamic behavior of small-scale structures with a significant reduction in the computational cost. The application of DLNN for modeling the machines of small- scale structures is a step forward for predicting the behavior of Nano/Micro Electro Mechanical Systems (NEMS/MEMS).

Dynamic response of porous E-FGM thick microplate resting on elastic foundation subjected to moving load with acceleration

For the first time, dynamic analysis of exponentially functionally graded material (E-FGM) including porosities is investigated. As a first endeavor, forced vibration response of thick microplates made of porous E-FGM when subjected to moving loads with an acceleration in speed is studied in this research study. The role of a two-parameter elastic foundation is also included. A powerful mathematical formulation supported by the assumptions of the general third-order shear deformation theory (GTSDT) as well as the modified couple stress theory (MCST) is developed to find an accurate model for thick size-dependent microplates. The effective material properties of imperfect FGM, which change exponentially along the z-direction, are established via developing a modified rule of mixture which includes the porosity imperfection. The governing equations for simply-supported microplate is found using a virtual work of Hamilton principle, and afterwards they are solved by developing a state space method (SSM) in conjunction with a set of mathematical series. The numerical results attempt to indicate the influence of variation in volume fraction index, porosity index, elastic foundations, and small-scale parameter, highlighted by different load speeds, acceleration ratios, and time histories of moving load under upper surface of E-FGM thick microplate. The parametrical studies can be used in better designing of micro/nanostructures made of porous FGMs under accelerated moving loads.

On wave propagation in piezoelectric-auxetic honeycomb-2D-FGM micro-sandwich beams based on modified couple stress and refined zigzag theories

In the present paper, wave propagation of micro-sandwich beams is investigated. The sandwich beam is consisted of the core, top, and bottom layers which are corresponding to auxetic honeycomb, piezoelectric, and Two-Dimensional Functionally Graded Material (2D-FGM), respectively. The material properties of the 2D-FGM layer differ in both thickness and axial directions. The structure is under the electric field with respect to the existence of piezoelectric material. Giving the hypothesis of modified couple stress theory (MCST) for size effects and the refined zigzag theory (RZT) for mathematical modeling, the equations of motion are gained based on Hamilton’s principle. The wave velocity, escaping frequency, and cut-off frequency of the structure are computed through the analytical method. The influences of different components (e.g. geometric parameters, electric field, auxetic honeycombs core, electric field, size parameter, and gradient index) of 2D-FGM layer on the wave velocity of the micro-sandwich beam are evaluated. The results illustrate that the wave velocity which is calculated by the modified couple stress theory increases by about 2.5 times than that in classic theory. Moreover, the phase velocity increases as the geometrical components of auxetic honeycombs core are enhanced.

Free vibration analysis of functionally graded porous elliptical microshells using modified couple stress theory

In this study, the free vibration analysis of functionally graded porous (FGP) elliptical microshells was studied for the first time. The microshell was modeled based on Donnell’s shell theory along with modified coupled stress theory (MCST). The set of the governing equations were derived using Hamilton’s principle and were solved via the Galerkin's method. The accuracy of the presented model was confirmed and the effects on the natural frequencies of the microshell are investigated such as porosity parameter, porosity distribution pattern, length scale parameter, geometrical parameters of the microshell, and the circumferential wave number (n). Numerical results show that for n = 1, the length scale parameter has no considerable effect on the natural frequencies, but for n > 1, the natural frequencies grow by increasing the length scale parameter. Also, it is shown that the effect of length scale parameter on the natural frequencies grows the higher values of elliptical ratio (large-to-small radii ratio).

Free vibration and buckling characteristics of porous functionally graded materials (FGMs) micro‐beams based on the modified couple stress theory

AbstractBased on the modified couple stress theory (MCST), the free vibration and buckling characteristics of porous functionally graded materials micro‐beams (P‐FGMs and E‐FGMs) are studied. Firstly, two types of pore distribution are considered for functionally graded materials (FGMs) micro‐beams. The governing differential equations of free vibration and buckling of FGMs micro‐beams are derived by Hamilton's principle and dimensionless. Then, the dimensionless governing differential equations and their boundary conditions are transformed by differential transformation method (DTM), then the equivalent algebraic characteristic equations with natural frequencies are obtained. Finally, the dimensionless fundamental frequency and critical buckling load of the free vibration of the porous FGMs micro‐beam under four different boundary conditions of clamped‐clamped (C‐C), clamped‐simply supported (C‐S), simply supported‐simply supported (S‐S) and clamped‐free (C‐F) are calculated. The results are compared with the existing literature data, and the correctness is verified. The effects of porosity, scale parameter, stiffness ratio, axial load and gradient index on the dimensionless natural frequency and critical buckling load of FGMs micro‐beams are discussed.