Analysis Of Microplates Research Articles

Hamiltonian System-Based Symplectic Framework for Analytical Vibration Analysis of Microplates

Abstract The classical elasticity is inadequate for the dynamic analysis of microplates due to the size effect. This study incorporates a higher-order strain gradient theory into the Hamiltonian system-based symplectic framework and derives new analytical solutions for the free vibration of microplates. The analytical solutions are obtained using rigorous mathematical techniques, including separation of variables, symplectic eigen expansion, and superposition, without relying on predetermined solution forms. Hence, they are not restricted to Lévy-type boundary conditions. Using these analytical solutions, we present comprehensive vibration results for microplates and perform detailed parametric studies to explore the impact of length scale parameters on the natural frequencies. Given the growing demand for microplates in advanced engineering applications, the obtained analytical solutions are expected to facilitate their design and performance optimization.

Nonlinear thermal buckling analysis of temperature-dependent porous annular and circular microplates reinforced by graphene platelets by using isogeometric analysis method

In this research, by using isogeometric analysis (IGA) method, nonlinear thermal buckling analysis of porous circular and annular microplates reinforced by graphene platelets (GPLs) under uniform temperature rise is performed by taking both the temperature dependence of the material properties and the size effects into account. The closed-cell metal foam scheme combined with the Halpin-Tsai micromechanics model and the rule of mixture is utilized to identify the effective material properties of the porous nanocomposites for three GPL dispersion patterns and three porosity distributions. With the aid of the virtual work principle, the classical and the non-classical motion equilibrium formulation based on the higher-order shear deformation theory in conjunction with the modified strain gradient (MSG) theory for the nonlinear temperature-dependent buckling behaviour are established. The higher-order NURBS basis functions capable of accomplishing easily the smoothness of arbitrary continuity order are successfully utilized to solve the nonlinear governing equations by satisfying the C2-continuity of the displacement field required by the MSG theory. The thermal buckling behaviour is assessed in detail by using various parameters such as the thickness-to-radius ratio, the porosity coefficient, the GPL weight fraction and the material length scale as well as the porosity distribution scheme and the GPL dispersion pattern. The results show the significance of incorporating the temperature dependency of the material properties when analyzing the thermal buckling behaviour and present the combination scheme of porosity and GPL distributions for enhancing the thermal buckling resistance.

Size-Dependent Frequency Analysis of Higher-Order Microplates with FGP Core and Polymeric CNTRC Faces Considering Piezoelectricity

The present study examines a microplate with a porous structure and two nanocomposite piezoelectric layers. All the layers’ properties are graded functionally, bonded to each other, and supported by an elastic foundation that can withstand both normal and shear loads. Additionally, carbon nanotubes are used to increase the electro-mechanical performance of the piezoelectric patches, which are exposed to an externally applied electric voltage. Using a higher-order trigonometric shear deformation theory and von Karman’s assumptions, the kinematic relations are demonstrated. The governing motion equations are derived using Hamilton’s principle and variational technique, and the modified couple stress theory is employed to take the scale effect into account. An analytical method based on Fourier series functions is used to solve the differential motion equations, and the impact of diverse factors such as porosity percentage, pore distribution patterns, carbon nanotubes distribution patterns, and other key parameters on the normalized frequencies of the model is analyzed after verifying the accuracy of the results. The findings of this research may aid in the development and production of smart structures and devices with increased efficiency.

Higher order finite element models for transient analysis of strain gradient functionally graded microplates

In the study, three different higher order finite element models (HOFEMs) are developed to perform the transient analysis of functionally graded (FG) microplates subjected to different dynamic loads, namely, sinusoidally distributed step and exponential blast loads. A normal and shear deformable plate theory with five unknowns is used to present displacement field and modified strain gradient theory is employed for small-scale effect. A rectangular four-noded element and Newmark's method is used to solve transient analysis of FG microplates. The effects of boundary condition, type of dynamic loading, thickness to material length parameter, aspect ratio, gradient index and CPU time are investigated. It is found that two of the HOFEMs developed based on the C1 and C2 continuity requirements produced almost same numerical results. However, the one with 36 unknows per element yields slightly different results than the others. The HOFEM satisfying the C1 continuity requirement consumes the minimum computational time.

Nonlinear static bending analysis of microplates resting on imperfect two-parameter elastic foundations using modified couple stress theory

Nonlinear static bending analysis of microplates resting on imperfect two-parameter elastic foundations using modified couple stress theory

Porosity-dependent vibration analysis of FG microplates embedded by polymeric nanocomposite patches considering hygrothermal effect via an innovative plate theory

The sandwich structures contain three or more layers attached to the core. In the current research, a three-layered sandwich microplate containing functionally graded (FG) porous materials as core and piezoelectric nanocomposite materials as face sheets subjected to electric field resting on Pasternak foundation is chosen as a model to investigate its vibrational behavior. To make the face sheets stiffer, they are reinforced by carbon nanotubes (CNTs) via different distribution patterns which result in changing their properties along the thickness direction. An innovative quasi-3D shear deformation theory with five unknowns, Hamilton’s principle, and modified couple stress theory are hired to gain equations of motion related to the abovementioned microstructure. Eventually, the evaluation of materials’ properties, geometry specifications, foundation moduli, and hygrothermal environment on vibrational behavior of such structures became easier using the presented results of the current study in figure format. As an instance, it is revealed that CNTs’ volume fraction elevation causes mechanical properties improvement, and in the following, natural frequency increment. Besides, considering the hygrothermal environment causes significant effects on the results.

Generalized conforming Trefftz element for size-dependent analysis of thin microplates based on the modified couple stress theory

In this work, a new displacement-based Trefftz plate element is developed for size-dependent bending analysis of the thin plate structures in the context of the modified couple stress theory. This is achieved via two steps. First, the Trefftz functions, that are derived by introducing the thin plate bending assumptions into the three-dimensional governing equations of the modified couple stress elasticity, are adopted as the basis functions for designing the element's displacement interpolations. Second, the generalized conforming theory is employed to meet the interelement compatibility requirements in weak sense for ensuring the convergence property. The resulting 4-node displacement-based plate element performs like nonconforming models on coarse meshes and gradually converges into a conforming one with the mesh refinement. Numerical tests reveal that the new element can efficiently capture the size-dependent mechanical responses of thin microplates and exhibits satisfactory numerical accuracy and distortion tolerance. Moreover, as the element has only three degrees of freedom (DOF) per node, it can be easily incorporated into the commonly available finite element programs.

Vibration and thermal buckling analyses of three-layered centrosymmetric piezoelectric microplates based on the modified consistent couple stress theory

In this study, free and forced vibration investigations and thermal buckling analysis of three-layered centrosymmetric piezoelectric microplates are examined. To model the size effects, the size-dependent consistent couple stress theory is used. To be compatible with the modified coupled stress theory, a modification is proposed to apply to the consistent couple stress theory. Resorting to the Navier’s approach, the governing equations are treated in the case of simply supported boundary conditions to extract the free and forced vibration outcomes and the thermal buckling numerical results. The verifications demonstrate the effectiveness of the proposed modification. The effects of the material length scale parameter and the flexoelectricity coefficient on the findings are investigated. Moreover, the closed- and open-circuit condition impacts on the free and forced vibration and the thermal buckling analyses are studied.

Analysis of porous micro-plates reinforced with FG-GNPs based on Reddy plate theory

The present paper studies size-dependent deflection analysis of FG graphene nanoplatelets (GNPs) reinforced composite micro-plates with porosity subjected to transverse load. Third-order shear deformation (TOSD) theory of Reddy’s and principle of virtual work are employed for derivation of governing equations of static bending via modified couple stress theory (CST). Various models of distribution of porosity and GNPs are employed in our analysis. Halpin-Tsai micromechanical model and rule of mixture is used for effective material. After derivation of governing equations and presentation of solution procedure, the numerical results are presented in terms of significant input parameters. The numerical outputs including displacements and stress components are presented in terms of porosity coefficient, distribution of porosity, distribution and amount of GNPs and various dimensionless geometric parameters of the composite micro-plate. A comprehensive comparison with previous works have been performed for validation of the present formulation and corresponding numerical results.

Mechanical Gripper Design and Force Analysis of Microplates for Automated High-Throughput Nucleic Acid Detection System.

In the automated high-throughput nucleic acid detection system, we need to grip and transfer consumables frequently when carrying out multichannel nucleic acid detection. In order to ensure the efficiency of experiments and solve problems of the deflection and drop when transferring microplates, we design a self-locking mechanical gripper which consists of a rotary positioning module and a gripping module. The absolute position encoder fixed on the top of the stepper motor can collect the position data of the mechanical gripper in real time and send them to the master control board based on STM32 for processing, which ensures the accuracy of the movement of the mechanical gripper. We used SolidWorks to build models of the mechanical gripper and different microplates, and we carried on finite element analysis of microplates to find the suitable gripping position. Through the force analysis, we obtained the pressure distribution and the deformation of different microplates, and defined the effective gripping areas, which is important to the grip and transfer of microplates.

Nonlinear analysis of size-dependent annular sector and rectangular microplates under transverse loading and resting on foundations based on the modified couple stress theory

Abstract The nonlinear behavior of annular sector microplates remains poorly understood. In this study, a non-classical model for nonlinear bending analysis of annular sector microplates under transverse mechanical loading is developed based on the modified couple stress theory and the first-order shear deformation theory with von Karman nonlinearity. The model captures the size effect by a material length scale parameter. Using the principle of minimum total potential energy, the equilibrium equations and boundary conditions are obtained. The equilibrium equations are discretized according to the generalized differential quadrature method and are solved using the Newton's technique. By a proper change in the geometric parameters of annular sector plates, the results are extended to nonlinear analysis of square microplates under transverse loading. It is assumed that the microplates are resting on the nonlinear elastic foundations. The present results are verified by comparison with existing results in the literature. The effects of nonlinearity, length scale parameter, foundation parameters and geometric parameters on deflections of annular sector and square microplates subjected to distributed transverse loading are studied.

The effect of multidimensional temperature distribution on the vibrational characteristics of a size-dependent thick bi-directional functionally graded microplate

This article develops the modified couple stress theory to study the free vibration of bi-directional functionally graded microplates subjected to multidimensional temperature distribution. Third-order shear deformation and classical theories of plates are adapted for free vibration analysis of thick and thin microplates, respectively. Employing the third-order shear deformation theory, both normal and shear deformations are considered without the need for shear correction factor. Material of the bi-directional functionally graded microplate is graded smoothly through the length and thickness of the microplate. Gradient of the material is assumed to obey from the power law in terms of the volume fraction of the constituents. Assuming the uniform and nonuniform temperature distributions, the effect of thermal environment on dynamic behavior of the microplate is discussed in detail. Applying the Ritz method, the displacement field is expanded by admissible functions which satisfy the essential boundary conditions, and Hamilton principle is employed to determine the natural frequencies of the microplate. Developed model has been applied to determine the natural frequencies in problems of thin/thick, one-directional/bi-directional functionally graded, and homogeneous/nonhomogeneous microplates. Effects of parameters such as the thermal environment, power law indexes [Formula: see text] and length scale parameter on free vibration of these problems are studied in detail. The results show that higher values of length scale parameter and temperature rise decrease the natural frequency of the bi-directional functionally graded microplate. According to results obtained by classical and third-order shear deformation theories, the third-order shear deformation theory is proposed for vibration analysis of microplates with thickness-to-length ratio less than five.

Size-Dependent Green’s Function for Bending of Circular Micro Plates Under Eccentric Load

In this paper, a Green’s function is developed for bending analysis of micro plates under an asymmetric load. In order to consider the length scale effect, the modified couple stress theory is used. This theory can accurately predict the behavior of micro structures. A thin micro plate is considered and therefore the classical plate theory is utilized. The size dependent governing equilibrium equation of a circular micro plate under an eccentric load is obtained by using the minimum total potential energy principle. This equation is a partial differential equation and it is hard to solve it for an arbitrary loading. A transformation of the coordinate system is introduced to obtain the asymmetric exact solution for deflection of circular micro-plates. By using the obtained size dependent Green’s function, the bending behavior of microplates under arbitrary loads can be easily defined. The results are presented for different asymmetric loads. Also, it is concluded that the length scale has a significant effect on bending of micro plates.

Vibration and Buckling of Functionally Graded Sandwich Micro-Plates Based on a New Size-Dependent Model

Within the framework of re-modified couple stress theory, the Refined Zigzag Theory is added to the vibration and buckling analysis of sandwich micro-plates embedding functionally graded layers. The disparity between the scale effects along two orthogonal directions is considered through two orthogonal material length scale parameters (MLSPs). Meanwhile, the solutions of natural frequencies and buckling loads show an improved predictive capability through comparing the results with exact and quasi-3D solutions. Two types of functionally graded sandwich micro-plates with simply supported boundary conditions are taken as the illustrative examples, namely, an isotropic functionally grade sandwich micro-plate with a power law and an orthotropic one with an exponential law. The numerical results indicate that the present model can capture the varying scale effects along two orthogonal directions, particularly when the geometric size of the micro-plates is comparable to the MLSPs. When microscopic isotropy is observed, the present model can also make accurate predictions on those kinds of micro-structures by setting the two orthogonal MLSPs equal to each other. In addition, the scale effects are less obvious as the functionally graded sandwich micro-plate is getting thinner and harder; the grading index also has an influence on the scale effects, but this influence is simultaneously depending on the side-to-thickness ratio of the micro-plate.

The size-dependent analysis of microplates via a newly developed shear deformation theory

This work deals with considering the small-scale effects on the static bending, vibrational behavior and buckling analysis of a simply supported microplate based on the newly developed shear deformation plate theory. This theory includes two unknown functions and meets the shear and couple-free conditions on the top and bottom surfaces of the plate without any shear correction factor. Hamilton’s principle and the modified couple stress theory are applied to obtain the governing equations and corresponding boundary conditions. Navier’s approach is used to analytically obtain the deflections, natural frequencies and critical buckling loads of the microplate. The reliability of the presented formulation in this paper is studied through comparison with previous existing data. It is revealed that the presented theory is comparable to the other shear deformation theories. Also, numerical results of this work demonstrate that the proposed theory predicts lower natural frequencies and critical buckling loads compared to the other theories because of the lower stiffness of the plate.

Size-dependent analysis of FG microplates with temperature-dependent material properties using modified strain gradient theory and isogeometric approach

This paper investigates the static bending and buckling behaviors of functionally graded microplates under mechanical and thermal loads by using isogeometric analysis (IGA) and modified strain gradient theory (MSGT). The material properties are assumed to be temperature-dependent and three different temperature rise patterns including uniform, linear and non-linear are considered. The material properties vary through the plate thickness based on the rule of mixture scheme. A refined hyperbolic shear deformation theory with four independent unknowns is used for analysis, which doesn't need a shear correction factor. For analysis the IGA using B-Spline or Non-Uniform Rational B-Spline (NURBS) functions can easily meet the C2 continuity requirement. Various parameters are dealt with including power index, material length scale parameter, temperature rise patterns and material combination. The MSGT results are also compared with those obtained by modified couple stress and classical theories. The obtained results indicate that the two studied types of functionally graded microplates show different behaviors under thermal load and the type of functionally graded material is an important parameter for thermal analysis.

Size-dependent analysis of FG-CNTRC microplates based on modified strain gradient elasticity theory

We present in this study a size-dependent computational approach based on the modified strain gradient theory (MSGT) and higher-order shear deformation theory for static bending and free vibration analyses of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) microplates. Three material length scale parameters (MLSPs) are taken into account in MSGT to capture size effects of microplate behavior. The effective material properties of FG-CNTRC microplates are obtained by an extended rule of mixture. Four types of carbon nanotube distributions, which are either uniform or functionally graded (FG) through the plate thickness, are considered. The governing equations are derived from the principle of virtual work and are then solved by isogeometric analysis (IGA). The IGA is suitable for a numerical implementation of the size-dependent models since it requires higher-order gradients in the weak form. The inclusion of geometrical parameters, boundary conditions, distributed types of carbon nanotube and material length scale parameters are studied to evaluate the displacement and natural frequency of FG-CNTRC microplates. In addition, the present size-dependent model can be retrieved into the modified couple stress model or classical model when a few MLSPs are ignored.

Vibration Analysis of Thick Functionally Graded Micro-plates Using HOSNDPT and Modified Couple Stress Theory

This paper develops a micro-scale vibration analysis of micro-plates using higher-order shear and normal deformable plate theory in conjunction with modified couple stress theory. The present model includes one material length scale parameter which takes into account the size effects. The equations of motions and boundary conditions are derived using Hamilton’s principle. Analytical solutions for the free vibration problem of simply supported rectangular micro-plates are obtained. Numerical results are presented to illustrate the effect of small scale on the dynamic response of functionally graded micro-plates. The results show that the size-dependent effect increases the stiffness of the micro-plate and consequently increases the natural frequencies.

Nonlinear Dynamics of Multilayered Microplates

This paper, for the first time, investigates the nonlinear forced dynamics of a three-layered microplate taking into account all the in-plane and out-of-plane motions. The Kirchhoff's plate theory, along with von Kármán nonlinear strains, is employed to derive the nonlinear size-dependent transverse and in-plane equations of motion in the modified couple stress theory (MCST) framework, based on Hamilton's energy principle. A nonconservative damping force of viscous type as well as an external excitation load consisting of a harmonic term is considered in the model. All the transverse and in-plane displacements and inertia are accounted for in both the theoretical modeling and numerical simulations; this leads to further complexities in the nonlinear model and simulations. These complexities arising in the theoretical model are overcome through the use of a well-optimized numerical scheme. The effects of different layer arrangements and different layer material percentages on the force–amplitude and frequency–amplitude curves of the microsystem are investigated. The results of this study shed light in the nonlinear resonant behavior of multilayered microplates and could be helpful in design and analysis of multilayered microplates in microelectromechanical systems (MEMS) applications.

Nonlinear size-dependent pull-in instability and stress analysis of thin plate actuator based on enhanced continuum theories including nonlinear effects and surface energy

In recent years, size dependent continuum theories have been commonly used to simulate material discontinuities in micro/nano-scales. In the present article, modified strain gradient theory (MSGT) and Gurtin–Murdoch surface elasticity, are used to investigate the size-dependent nonlinear pull-in instability and conduct a stress analysis of thin microplates. This paper modifies the classical equations and surface elasticity theory as a non-classical method which is able to account for the surface effect of nanomaterials with sufficient precision and lower calculation cost in comparison to atomic methods. The effects of van der Waals and Casimir forces are incorporated into the nonlinear governing equation of the system. A finite difference method (FDM) is employed to solve the nonlinear differential equation, and the pull-in parameters of the microplate are extracted. The effect of dispersion attractions and size-dependency on instability as well as the importance of the coupling between them is discussed. The most common types of thin-plates, including fully-clamped, simply-supported, and plates with mixed boundary conditions, are investigated. It is found that the significant difference between the pull-in instability parameters in the MSGT and the classical theory for thin microplates is merely due to the consideration of the size effect parameter in the modified strain gradient theory. The effect of surface/interface is also accounted for in the numerical FDM approach. Then, the stress analysis of the microplate is studied numerically by examining the surface/interface effect on the microplate with a square/rectangular section. The results obtained through this numerical method are plotted graphically.