Modified Couple Stress Theory Research Articles

Symmetric/Asymmetric Vibrational Characteristics of Bi-Directional Functionally Graded Annular Microplates

In this study, a free vibration model of bi-directional functionally graded (BD-FG) annular microplates is established based on the modified couple stress theory (MCST) and the first-order shear deformation theory (FSDT), and the corresponding governing equations are derived from Hamilton’s principle. The annular microplates are assumed to be composed of metal and ceramic materials, and the equivalent material properties are a combination of power-law function and exponential ones. The generalized differential quadrature method (GDQM) is used to obtain the vibration frequencies and mode displacements for the C–C, S–C, C–S, and S–S (S: simply supported; C: clamped) annular microplates. The convergence and validity of the proposed model are verified through selective numerical examples. Further, the effects of gradient index, material length scale parameter, boundary conditions, and geometrical dimensions on the symmetric/asymmetric vibration frequencies of the annular microplate are explored. The modal assurance criterion is applied to estimate the influence of the radial gradient index and material length scale parameter on different-order mode displacements of the annular microplate. The results show that: (1) the vibration frequencies of the annular microplate are sensitive to changes in the geometry and boundary conditions; (2) compared with the axial gradient index, the radial gradient index not only affects the vibration frequencies of the annular microplate, but also impacts the distribution of the highest and lowest points of the vibration mode displacements as well as the vibration mode nodal lines; (3) as the material length scale parameter increases, the vibration frequencies are significantly increased, and the higher-order vibration modes of the annular microplate also change.

Vibrational response of a sandwich microplate considering the impact of flexoelectricity and based on a novel porous-FGM formulation

Flexoelectricity indicates a unique characteristic of insulators and dielectric materials that results in electrical polarization induction in response to the non-uniform strain gradient. The impact of this property is bolder on the microscale and nanoscale rather than the macroscale. This article is conducted to derive the governing equations of motion for a flexoelectric-multilayer structure and evaluate its vibrational behavior in response to various variable variations. The model includes a functionally graded (FG) porous core which is confined between two flexoelectricity-based face sheets at both sides and a Vlasov type of foundation. To gain the governing equations, the new type of power law for porous material beside Hamilton’s principle (HP) and modified couple stress theory (MCST) is implemented. Capturing the effects of length scale parameter, different materials, aspect and thickness ratios, porosity index, foundation parameter and boundary conditions (B.C.s) on the normalized natural frequency (NNF) of such a multilayer model based on the novel FG porous material formulation serve as the main novelty of this article. According to the results, it is revealed that the lateral ratio enhancement causes NNF and stiffness reduction in the model. Furthermore, among all considered B.C.s, simply support (SSSS) and clamped support (CCCC) refer to the lowest and the highest NNF respectively. The content of the recent article serves as a good source to provide a better understanding of the vibrational response of high stiffness-to-weight ratio structures to different variable variations.

Vibration analysis of a smart viscoelastic porous sandwich micro-shell with magnetorheological fluid core using the modified couple stress theory

In the current study, for the first time, the size-dependent vibration of a functionally graded (FG) porous sandwich cylindrical microshell with magnetorheological fluid core is investigated using modified couple stress theory (MCST) and the first-order shear deformation theory (FSDT). The microshell is made of FG porous material and for investigating the material properties, two types of porous distributions are considered. Based on FSDT and the MCST, the governing equations are derived using Hamilton’s principle regarding the energy dissipation due to the shear deformation of the magnetorheological core layer and the interaction between all layers. Different assumptions, based on the continuity in transverse shear stresses and slope of in-plane displacements, are considered in the theoretical formulation. The natural frequency and loss factor of the microshell with sliding simply supported boundary conditions are determined using Navier’s solution method. Finally, the influence of some parameters such as material length scale(l/h: 0-1), types of porous distributions (symmetric-asymmetric), porosity volume fraction (e0: 0-1), magnetic field intensity (B: 0-800) and aspect ratio (L/R: 1-8, R/h: 10-100) on the vibration damping characteristics of microshell, is investigated and validity of the results is studied.

On the variable length scale parameter in functionally graded non-porous and porous microplate/nanoplate

In this article, the solution for free vibration and static bending of nonporous and porous square microplate/nanoplate is studied by practicing the modified couple stress theory (MCST). The impact of incorporating the length scale parameter of MCST is confirmed by comparing the results to that of molecular dynamics simulations. This study is aimed at investigating the influence of implementing a variable length scale parameter on the static and dynamic behavior of the small-scaled plates. Results are obtained adopting the analytical technique and compared to cases of presupposing constant numerical values of material length scale throughout each plate. The novelty of this theoretical study is the approach employed to achieve the formulation of the material length scale parameter, which varies along the thickness of both porous and nonporous plates. The results indicate that the inclusion of the variable length scale parameter increases the fundamental frequency and stiffness of plates. When comparing two cases of variable length scale parameter, one approximating the scale parameter of each constituent based on the bending parameter and Poisson’s ratio of the material, and the other adopting the numerical values estimated by prior experimental studies, reveals that the latter gives higher results and is chosen correspondingly.

A modified couple stress model to analyze the effect of size-dependent on thermal interactions in rotating nanobeams whose properties change with temperature

Understanding the behavior of rotating materials and structures on small scales is crucial for many scientific and engineering fields, and such studies play an important role in this regard. This paper aims to propose a novel paradigm for analyzing the vibrational characteristics of thermoelastic nanobeams with diverse physical attributes. The incorporation of size effects in the structural and thermal constitutive relationships involves the consideration of nonlocal elasticity theory (NET) and the modified couple stress (MCS) model, together with the utilization of the Euler–Bernoulli assumptions for thin beams. The work also involves the development of a new non-Fourier thermoelasticity model that incorporates the Moore–Gibson–Thompson (MGT) equation. Furthermore, it was taken into account that the thermal conductivity of the flexible materials is not consistent but rather changes with temperature. Periodic pulse heating was applied to rotating nanobeams, and the behavior of the nanobeams was investigated with respect to thermal, rotational, and length-scale effects. To demonstrate the impact of the distinctive characteristics of the MCS and MGT thermoelastic models on the physical fields, a range of numerical data are presented. The study also investigated the propagation characteristics of thermo-mechanical waves, taking into account aspects such as thermal relaxation time and the influence of temperature change on physical properties. Based on the observed results, including the size impact in the structural and thermal equations can lead to significant disparities when compared to conventional models. The inclusion of the length-scale component in the MCS theory, which increases the rigidity and hardiness of the nanobeam structure, may help to explain the observed effect.

Size and Temperature Effects on Band Gap Analysis of a Defective Phononic Crystal Beam

In this paper, a new defective phononic crystal (PC) microbeam model in a thermal environment is developed with the application of modified couple stress theory (MCST). By using Hamilton’s principle, the wave equation and complete boundary conditions of a heated Bernoulli–Euler microbeam are obtained. The band structures of the perfect and defective heated PC microbeams are solved by employing the transfer matrix method and supercell technology. The accuracy of the new model is validated using the finite element model, and the parametric analysis is conducted to examine the influences of size and temperature effects, as well as defect segment length, on the band structures of current microbeams. The results indicate that the size effect induces microstructure hardening, while the increase in temperature has a softening impact, decreasing the band gap frequencies. The inclusion of defect cells leads to the localization of elastic waves. These findings have significant implications for the design of microdevices, including applications in micro-energy harvesters, energy absorbers, and micro-electro-mechanical systems (MEMS).

A quasi-3D hyperbolic formulation for the buckling study of metal foam microplates layered with graphene nanoplatelets-embedded nanocomposite patches with temperature fluctuations

The work analyzes the buckling behavior of size-dependent microplates with a metal foam core, covered by graphene nanoplatelets (GNPs)-embedded nanocomposite patches. Microplates rest on a bi-parameter elastic substrate, and they are immersed in a thermal environment to observe the effect of temperature fluctuations on their elastic buckling performance. All material properties for each layer of the microstructure are thickness-dependent. A novel quasi-3D shear and normal (Q-3D S-N) hyperbolic theory is here proposed to describe the kinematic relations, accounting for the transverse normal strain. At the same time, a modified couple stress theory (MCST) is employed to account for the size dependence of the mechanical behavior due to the presence of a material length-scale parameter. Using the energy method and virtual work principle, the differential equilibrium equations are derived and solved analytically, where solutions are verified against the existing literature. The study focuses on the impact of different parameters on the normalized critical buckling load (NCBL). Based on the results from the systematic investigation, it is found that the addition of GNPs to the microplate enhances its stiffness, leading to increased values of NCBL, which in turn reduce for an increased imperfection ratio.

Three-dimensional analysis of thermoelastic damping in couple stress-based rectangular plates with nonlocal dual-phase-lag heat conduction

Since thermoelastic damping (TED) plays a decisive role in the functioning of micro/nanoresonators, accurate assessment of its magnitude in these tiny systems is imperative. Successfully simulating the thermomechanical behavior of various mechanical elements requires consideration of heat transfer in all three principal directions. Furthermore, there is no denying the impact of size on the mechanical and thermal domains. In light of these reasons, the current study strives to create a three-dimensional (3D) size-dependent TED model for rectangular micro/nanoplates via the use of the modified couple stress theory (MCST) and nonlocal dual-phase-lag (NDPL) heat transfer model concurrently for the first time. To accomplish this, by applying the Galerkin method to the 3D NDPL-based heat equation, the relation of temperature field is derived. Moreover, the coupled thermoelastic constitutive relations are extracted by including the couple stress effect. Lastly, by exploiting the obtained relations in the energy dissipation approach, an analytical TED formula in the form of infinite trigonometric series is attained. To appraise the rightness of derived formulation, the method of model reduction is applied. Moreover, an elaborate convergence analysis is implemented to realize the number of sufficient terms of the provided solution that result in well-grounded outcomes. A rigorous simulation study is also accomplished on the role of 3D heat conduction, scale parameters of MCST and NDPL model, boundary constraints, geometry and material in the variations of TED. Findings imply the definite impact of heat conduction dimension on TED value in rectangular plates with a large ratio of thickness to length and width.

BENDING ANALYSIS OF A PERFORATED MICROBEAM WITH LAPLACE TRANSFORM

In recent years, the analysis of materials and elements with dimensions at nano/micro levels has gained momentum. While analyzing these small-scale materials and elements, higher-order elasticity theories have started to be used instead of classical elasticity theories (CETs). One of these theories, which includes a small-scale parameter in its constitutive equations, is the modified couple stress theory (MCST). In this study, the bending analysis of a cantilever perforated microbeam is investigated by MCST and Euler-Bernoulli (EB) beam theory. First, the perforation characteristics of the microbeam are described and incorporated into the equation governing the bending problem based on the modified couple stress theory found in the literature. Then, the Laplace transform is applied to the governing equation. The known boundary conditions of the cantilever microbeam are substituted into the equation and the inverse Laplace transform is applied to obtain the deflection equation.

Scale-Dependent Generalized Thermoelastic Damping in Vibrations of Small-Sized Rectangular Plate Resonators by Considering Three-Dimensional Heat Conduction

Given that thermoelastic damping (TED) is one of the main causes of energy dissipation in miniaturized structures, calculation of its exact amount is of particular importance in the design of such elements. Considering three-dimensional (3D) heat transfer along with the scale effect on both mechanical and thermal areas is one of the decisive factors in the more rigorous modeling of TED in small-sized resonators. This paper exploits the modified couple stress theory (MCST) and dual-phase-lag (DPL) heat conduction model to establish an analytical TED relation for rectangular micro/nanoplate resonators with 3D heat conduction. At the start, the non-Fourier heat equation corresponding to 3D DPL model is derived. Then, the solution of temperature distribution is determined by employing infinite trigonometric series. Moreover, the scale-dependent frequency of the system is presented in the context of MCST. Eventually, with the help of entropy generation (EG) method, an analytical TED expression considering 3D heat conduction is established. The precision of the presented solution is examined by comparing it with a simpler model available in the literature. By choosing two common types of boundary conditions, i.e. fully-clamped (CCCC) and fully-simply (SSSS) supported plates, the influence of various factors on TED changes is comprehensively investigated in simulation section. According to the outcomes, in plates with a lower ratio of length and width to thickness, the dimension considered for heat conduction model has a noticeable impact on the variations of TED.

Vibration Bandgap Analysis of Periodic Composite Microplates using Isogeometric Approach

For periodic composite microplates, the microstructure dependent effect has great influence on the elastic wave vibration band gaps. In this paper, an effective isogeometric analysis method based on the modified couple stress theory (MCST) is proposed to calculate the vibration band gaps in periodic composite microplates. The MCST is employed to capture the microstructure effect of vibration band gaps in periodic composite microplates. The high order continuity requirement of MCST can be easily satisfied by applying the NURBS based isogeometric analysis approach. The elasto-dynamics and discrete equations of isogeometric analysis are obtained under the Mindlin kinematics assumptions. The present approach is validated by comparison with the analytical solution. And the influence of microstructure effect, volume fraction and unit cell length on the band gaps are investigated in details.

Quasi-3D free and forced vibrations of poroelastic microplates in the framework of modified couple stress theory

In this study, a comprehensive study on the vibration behaviour of functionally graded thick microplates with material imperfections is presented for free and forced vibrations within a quasi-3D model and the modified couple stress (MCS) theory. Axial, transverse, rotation, and stretching motions are considered for modelling the thick microplate in the framework of the modified power-law scheme and quasi-3D and MCS theories. The microplate deformation is assumed to be infinitesimal and is modelled using the linear strain–displacement relationships. Using a virtual work method, the governing motion equations are derived. For fourfold coupled (axial-transverse-rotation-stretching) characteristics, the partial differential equations components are discretised via trigonometric expressions and the corresponding natural frequencies and time histories (time-dependent deflections) are determined numerically. The methodology and model are initially validated through a comparative analysis between the natural frequencies of the macrolevel simplified structure and those obtained using finite element software. Additionally, the simulation is compared, for validation purposes, with other simplified versions from the literature. Once model's validity is confirmed, an investigation into the quasi-3D free and forced vibrations of the structure is conducted for the new model. The results demonstrate the considerable effect of thickness stretching and material imperfection on the vibration characteristics of the poroelastic thick microplates.

Application of multi-directional FG material to improve natural frequencies of perovskite solar cells under mechanical shock validated by deep-learning approach

In order to generate electricity from sunlight, spacecraft that operate within the inner solar system often utilize photovoltaic solar panels that are controlled by power electronics. For this purpose, the emerging technologies of perovskite solar cells (PSCs) have attracted plenty of attention recently. Improving the stability of this kind of cell is the main purpose of this research. In the current work, for the first time, dynamic stability analysis of the PSCs, including the multi-directional functionally graded (MD-FG) layer as the host layer, is presented. Also, due to the complex situation of solar cells, this kind of system may be under mechanical shock loading. Various types of PSCs, including cylindrical, spherical, and hyperbolic paraboloid solar cells, by changing curvature factors are modeled. For considering the size-dependency of the PSCs, modified couple stress theory (MCST) with one length scale factor on the strain energy is presented. The forced vibration caused by mechanical shock loading is illustrated by inverting the Laplace transform using the fully modified representation of Abate and Dubner’s solution. A deep neural network (DNN) with independent variables, hidden, input, and output layers, intrinsic neural network parameters, activation function, hidden neurons, and loss function is offered to train the current appropriate structure once the results of mathematical modeling have been obtained. Finally, by applying the two-dimensional differential quadrature method (2D-DQM) as an accurate integral operator, the discretized motion equations can be derived from the weak form of the equations. Finally, in the results section, some suggestion for improving the stability and frequency of the PSCs is discussed.

Modified couple stress theory (MCST) and refined zigzag theory (RZT) for wave propagation of embedded smart sandwich plates with magnetorheological fluid core and viscoelastic FG-CNTRC magnetostrictive face sheets

This paper discusses the wave propagation of a magnetorheological fluid (MRF) micro sandwich plate with magnetostrictive face sheets embedded in the Kerr foundation. The face sheets are reinforced with functionally graded carbon nanotubes (FG-CNT). It was hypothesized that the distribution of carbon nanotubes would be functionally graded along the thickness of face sheets in various patterns such as uniform distribution (UD), and three type of functionally graded (FG), that is, FG-O, FG-X, and FG-AV. The properties of the magnetostrictive layer are considered viscoelastic based on the Kelvin-Voigt model. The refined zigzag theory (RZT) is utilized to simulate displacements of the plate due to the sandwich nature of the system. The modified couple stress theory (MCST) is utilized to predict the influence of small-scale parameter. The equations of motion for the micro sandwich plate are derived by Hamilton's principle, and analytical solutions are utilized to acquire the phase speed, escape, and cut-off frequencies. After model verification, extensive analytical results are presented in detail. They demonstrated that the wave characteristics of the system are influenced by the micro sandwich plate parameters. Consequently, the dimensionless phase speed rises 66% via increasing the magnetic field magnitude from 12.5 to 17.5 G.

Nonlinear vibration of microbeams subjected to a uniform magnetic field and rested on nonlinear elastic foundation

Abstract This study investigates the nonlinear vibration motions of the Euler–Bernoulli microbeam on a nonlinear elastic foundation in a uniform magnetic field based on Modified Couple Stress Theory (MCST). The effect of size, foundation, and magnetic field on the nonlinear vibration motion of microbeam has been examined. The governing equations related to the nonlinear vibration motions of the microbeam are obtained by using Hamilton’s Principle, and the Multiple Time Scale Method was used to obtain the solutions for the governing equations. The linear natural frequencies of microbeam are presented in the table according to nonlinear parameters and boundary conditions. The linear and nonlinear natural frequency ratio graphs are shown. The present study results are also compared with previous work for validation. It is observed that length scale parameters and magnetic force have a more significant effect on the natural frequency of microbeams. It is seen that when the linear elastic foundation coefficient, the Pasternak foundation and the magnetic force effects increase, the ratio of nonlinear and linear natural frequency decreases.

Relative frequencies of perovskite solar cells strengthened by composite materials: Application of deep neural networks for solving the vibration problem

Perovskite solar cells have shown competitive power conversion efficiencies (PCE) with the potential for better performance when compared to the most widely used photovoltaic technology. Perovskites may deteriorate when exposed for an extended amount of time to light, heat, or electrical current, as well as when they interact with moisture and oxygen. To increase stability, researchers are examining deterioration in both the perovskite material itself and the surrounding device layers. Cell durability has to be increased in order to produce perovskite solar goods for sale. Improved cell durability is critical for the development of commercial perovskite solar products. So, improving the dynamic properties of this kind of green energy device is a key issue in engineering. In the current report, the stability zone and relative frequency change of the perovskite solar cells reinforced by three-dimensional functionally graded material (3D-FGM) as the last layer is presented. For exact modeling of the perovskite solar cell structure due to its size (micro size), it is crucial to take the size effect into account. Modified couple stress theory (MCST) as a size-dependent theory for micro-scale systems with curvature derivate parameters is offered. Once the study’s findings have mastered the mathematical discoveries, they are instructed to produce the machine learning technique’s outputs. In actuality, the fast-converging numerical approach is used to derive the characteristics of the design points. The momentum-based optimizer is used to find the optimal values of the parameters influencing how the fully connected neural network functions. The method used in this research is efficient since it quickly and properly trains the multi-layered network. The funding now available may be used to promote future research on the importance of the dynamic stability of perovskite solar cells employing 3D-FG material as the constructed layer.

Optimal material and geometry of 2D-FGM tapered microbeams under dynamic and static constraints

The present work applies meta-heuristic multi-objective particle swarm optimization (MOPSO) and non-dominated sorting genetic algorithm (NSGA)-II algorithms to predict optimal shape and material distribution of two-directional functionally graded tapered (2DFG-T) microbeams. The cross-section of the microbeam changes linearly along its length in both height and width directions. The rates of these changes are taken as the geometric design variables. Rather than the simple power law, two different trigonometric functions are introduced to provide various effective and flexible 2D- distributions of metal and ceramic phases. Using Hamilton’s principle, the mathematical model is derived based on the third order shear deformation theory (TSDT) and modified couple stress theory (MCST). Developing a computationally efficient and time saving semi-analytical approach, the governing set of variable-coefficients partial differential equations is solved for the frequency and critical buckling under different boundary conditions. Several constrained single- and multi-objective optimization problems are formulated and solved to determine optimal material and geometry parameters of 2DFG-T microbeams to achieve maximization of fundamental frequency and/or critical buckling load while minimizing the structural mass. Various optimization results demonstrate that including shape parameters in addition to material distribution parameters as design variables results in more economical and flexible profiles that is critical for the manufacture of 2DFG-T microbeams. The following model can be exploited in the optimum design of FG microscale nanobeams including their shapes and material distribution.

Bending, vibration and buckling isogeometric analysis of functionally graded porous microplates based on the TSDT incorporating size and surface effects

This paper presents a new microplate model for the static bending, free vibration, and buckling analysis of functionally graded (FG) porous plates. The model is based on the third-order shear deformation theory (TSDT) and incorporates the microstructural effect using the modified couple stress theory (MCST), as well as the surface stress effect using the Gurtin–Murdoch theory. The accuracy and reliability of the proposed model are evaluated using NURBS-based isogeometric analysis (IGA), which demonstrates excellent validation results. Furthermore, this study provides new insights into the mechanical behavior of FG porous microplates, investigating the influences of boundary conditions, surface effects, length scale parameters, porosity coefficients, plate aspect ratios, and buckling load conditions. Novel conclusions are drawn based on the analysis of deflection, stress, natural frequency, and buckling load.

Vibration analysis of functionally graded microbeams with a moving mass based on Timoshenko beam theory

Vibration analysis of functionally graded (FG) microbeams carrying a moving mass is carried out in the framework of Timoshenko beam theory. The beam material properties are considered to be graded in the thickness by a power-law function, and they are estimated by Mori-Tanaka scheme. The influence of the microsize effect is captured with the aid of the modified couple stress theory (MCST). A finite beam element is formulated and used to establish the discretized equation of motion for the beams. Vibration characteristics, including the natural frequencies, the time histories for mid-span deflection, the dynamic magnification factor and the stress distribution, are computed for a simply supported beam. The obtained result reveals that both the dynamic deflection and dynamic magnification factor are overestimated by ignoring the microsize effect. The effects of the material distribution, the moving mass velocity and the length scale parameter are studied in detail and highlighted.

Thermoelastic vibration of Timoshenko beam under the modified couple stress theory and the Moore–Gibson–Thompson heat conduction model

In modeling of micro/nanodevices, numerous approaches have been developed that take into account material length scale parameters to capture size effects. In this work, the variational formulation of the microstructure-dependent Timoshenko beam model is developed considering Hamilton’s principle and modified couple stress theory (MCST) which involves the material length scale parameter. The Moore–Gibson–Thompson (MGT) thermoelastic model is employed to present the governing equations of motion and boundary conditions. The numerical outcomes illustrated by classical and non-classical approaches reveal the behavior of both the deflection and thermal moment of the beam. In addition, the dissipation in the deflection and the thermal moment is visually depicted in standardized curves. Several peaks in graphical plots are then detected as a function of the dimensionless time, which bring to light some interesting features of MGT model in the prediction of thermoelastic vibration of the microstructure model of Timoshenko beam.