Size-Dependent Pull-In Instability of Hydrostatically and Electrostatically Actuated Circular Microplates

Modeling of viscoelastic behavior can be useful to accurate study of micro-shell vibration. In this study, influence of viscoelastic coefficient and material length scale parameter on frequency of a cylindrical micro-shell with/without conveying fluid are investigated. Considering trapezoidal shape factor via Kirchhoff-Love’s hypotheses and modified couple stress theory (MCST), governing equations of motion are derived using Hamilton’s principle. Viscoelastic properties are modeled according to Kelvin-Voigt viscoelasticity. The novelty of the current study is the consideration of viscoelastic effect, trapezoidal shape factor and size effect based on shell theory as well as influence of conveying viscous fluid on the frequency of micro-shells in thermal environment using MCST. Equations of motion are solved using Fourier series expansion along the axial and circumferential coordinates with Navier procedure as an analytical solution of frequency for a simply-supported micro-shell. As a numerical solution, considering conveying fluid with changing temperature, generalized differential quadrature (GDQ) method along the axial direction for different boundary conditions are utilized. Influences of length-to-mid-radius, thickness-to-mid-radius, length-to-thickness ratio and the number of the mode shapes of the micro-shell on the natural frequencies are examined. Considering fluid flow, effects of length-to-outer-radius, viscoelastic coefficient, fluid viscosity, Knudsen number with the effect of slip/no-slip boundary condition and temperature changes on critical flow velocity versus fundamental frequency of the micro-shell are investigated. Numerical results reveal that structural damping effect of viscoelastic coefficients and damping effect of fluid flow on frequency of the linear vibration are more significant to detect the stability domain of the micro-shell.

Coupled longitudinal-transverse-rotational free vibration of post-buckled functionally graded first-order shear deformable micro- and nano-beams based on the Mindlin's strain gradient theory

In present study, vibration analysis of functionally graded (FG) porous microbeam under thermal effects is investigated considering modified strain gradient theory (MSGT) of elasticity. As the main novelty of this study, in order to effectively acquire the effects of size of size-dependent porous structures, MSGT is employed (considers three material length scale parameters) rather than modified couple stress theory (MCST) (considers one material length scale parameter). The nonlinear governing equation of microbeam based on MSGT is derived from Hamilton’s principle and to determine the natural frequency of the system under simply supports, Navier solution is employed. Numerical results presented for different beam models and theories and are compared with those available in previous studies. The obtained results show that frequencies from MSGT are higher than those from MCST and also classical theory (CT), especially when the microbeam thickness is comparable to the microbeam length scale parameter. Also, increases in the temperature of microbeam working environment, lead to decrease of microbeam natural frequency. Finally, parametric study is performed on natural frequency of FG microbeams to indicate the effects of: temperature changes, length scale parameter, slender ratio, and gradient index and porosity volume fraction.

Size-dependent nonlinear bending and postbuckling of functionally graded Mindlin rectangular microplates considering the physical neutral plane position

A size-dependent quasi-3D isogeometric model for functionally graded graphene platelet-reinforced composite microplates based on the modified couple stress theory

An efficient size-dependent computational approach for functionally graded isotropic and sandwich microplates based on modified couple stress theory and moving Kriging-based meshfree method

On the static and dynamic responses of smart piezoelectric functionally graded graphene platelet-reinforced microplates

Consideration of spatial variation of the length scale parameter in static and dynamic analyses of functionally graded annular and circular micro-plates

The aim of the current study is to investigate torsional vibrations of bi-directional functionally graded (FG) small-scale tubes. The size effect is captured using the modified couple stress theory (MCST). Through-the-thickness distribution profile of material properties are characterized by employing a power-law function, which incorporates a gradation index, β, and longitudinal varation in material properties is featured by an exponential function which employs a gradient parameter, α. Similar to other material properties, continuous spatial variation of the length scale parameter is also incorporated into the analysis of two-dimensionally FG tubes. The model, including governing equations of motion and boundary conditions, is developed through the utilization of Hamilton’s principle. By employing differential quadrature method (DQM) numerical results regarding torsional vibrations are provided. Accuracy of the proposed model and procedures are verified through verification comparisons made to limiting cases available in the literature. Numerical results reveal effects of various material and geometric parameters on natural frequencies and indicate that torsional free vibration characteristics of small-sized tubes are sensitive to distribution profile of constituents, geometric aspect ratios and effective length scale parameter. Results also demonstrate that length scale parameter variation has significant influence on natural frequencies which justifies the necessity of its consideration in analyses.

This article reports on the thermal instability of functionally graded (FG) annular microplates with different boundary conditions. The modified strain gradient elasticity theory is employed to capture size effects. The non-linear governing equations and boundary conditions are derived based on the first-order shear deformation theory (FSDT) and virtual displacements principle. The generalized differential quadrature technique is implemented so as to discretize. To obtain the critical buckling temperature, the set of linear discretized governing equations is solved as an eigenvalue problem. Also, the non-linear problem of thermal postbuckling is solved by the pseudo arc-length continuation method. The effects of boundary conditions, length scale parameter, and the variation of material through the thickness and geometrical properties on both critical buckling temperature and thermal postbuckling behavior are studied.

In this paper, the static and vibration responses of a microresonator by considering distributed proof mass are investigated based on the modified couple stress theory (MCST). The microresonator has a proof mass placed on the end of micro-cantilever beam and the proof mass actuated by electrostatic actuation. Due to the large length of proof mass compared to beam length, the proof mass is considered as a lengthy mass in the modeling. Hence, the resultant force and moment acting on the proof mass can be calculated by integrating the density of the electrostatic force along the mass length. Considering the length effects of the proof mass, MCST and electrostatic actuation, the equation of dynamic motion is derived using the extended Hamilton principle. Using a model approximation, i.e., Galerkin decomposition method, the governing equations of static and oscillatory motions are reduced and the resultant equation is solved by analytical (multiple scale method) and numerical methods. The result shows that the developed model, that includes distributed proof mass and size dependency effects, improves the results of pull-in instability voltage, natural frequency and amplitude of vibration.

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 nonlinear analysis and damping responses of FG-CNTRC micro-plates

In this article, the vibrational analysis of temperature-dependent cylindrical functionally graded (FG) microshells surrounded by viscoelastic a foundation is investigated by means of the modified couple stress theory (MCST). MCST is applied to this model to be productive in design and analysis of micro actuators and micro sensors. The modeled cylindrical FG microshell, its equations of motion and boundary conditions are derived by Hamilton’s principle and the first-order shear deformation theory (FSDT). For the first time, in the present study, functionally graded length scale parameter which changes along the thickness has been considered in the temperature-dependent cylindrical FG microshell. The accuracy of the present model is verified with previous studies and also with those obtained by analytical Navier method. The novelty of the current study is consideration of viscoelastic foundation, various thermal loadings and size effect as well as satisfying various boundary conditions implemented on the temperature-dependent cylindrical FG microshell using MCST. Generalized differential quadrature method (GDQM) is applied to discretize the equations of motion. Then, some factors are investigated such as the influence of length to radius ratio, damping, Winkler and Pasternak foundations, different temperature changes, circumferential wave numbers, and boundary conditions on natural frequency of the cylindrical FG microshell. The results have many applications such as modeling of microrobots and biomedical microsystems.

The primary objective of this study is to examine the bending behavior of graphene nanoplates under asymmetric multi-part conditions. Efforts focus on distinguishing the commonly used Modified Couple Stress Theory (MCST) and the newly proposed Nonlocal Modified Couple Stress Theory (NMCST) in capturing size dependency, including a detailed investigation of the influences of material length scale parameters (MLSPs). Constitutive equations for nonlinear bending analysis of orthotropic rectangular micro/nanoplates are derived using three-dimensional elasticity theory and NMCST. Unlike MCST, which uses a single MLSP and suits isotropic materials, NMCST incorporates three MLSPs, better suited to the non-isotropic nature of graphene nanoplates. This enables studying distinct effects of each MLSP, particularly under asymmetric conditions. Furthermore, NMCST offers deeper insight into the influence and material dependency of MLSP values. Employing 3D elasticity theory, the study avoids typical approximations in displacement fields defined by conventional plate theories. Consequently, governing equations based on 3D elasticity theory and NMCST are developed in Cartesian coordinates for the first time. These equations are numerically solved using a newly introduced Semi-Analytical Polynomial Method (SAPM), tailored to graphene’s orthotropic properties. The method addresses complex asymmetric conditions such as multi-part boundary scenarios, loading distributions, and elastic foundations—by segmenting the nanoplate and applying distinct conditions to each part. SAPM is presented as a robust technique for solving various partial differential equations under symmetric and asymmetric configurations. The numerical solution explores behavior and interaction of MLSPs across diverse conditions, emphasizing their dependency on the base material. Due to scarce data on bending behavior under multi-part conditions, validation is performed using Molecular Mechanics Method (MMM) simulations of graphene nanoplates. Results obtained via MM, FE, and SAPM approaches are compared and verified for reliability and consistency.