Post-buckling involving large deflection of micro-cantilever using the consistent couple stress theory

Free vibration analysis of functionally graded piezoelectric cylindrical nanoshell based on consistent couple stress theory

Development of size-dependent consistent couple stress theory of Timoshenko beams

In this paper, the free vibrations of a short cylindrical nanotube made of piezoelectric material are studied based on the consistent couple stress theory and using the shear deformable cylindrical theory. This new model has only one length scale parameter and can consider the size effects of nanostructures in nanoscale. To model size effects in nanoscale, and considering the nanotube material which is piezoelectric, the consistent couple stress theory is used. First, using Hamilton\'s principle, the equations of motion and boundary condition of the piezoelectric cylindrical nanoshell are developed. Afterwards, using Navier approach and extended Kantorovich method (EKM), the governing equations of the system with simple-simple (S-S) and clamped-clamped (C-C) supports are solved. Afterwards, the effects of size parameter, geometric parameters (nanoshell length and thickness), and mechanical and electric properties (piezoelectric effect) on nanoshell vibrations are investigated. Results demonstrate that the natural frequency on nanoshell in nanoscale is extremely dependent on nanoshell size. Increase in size parameter, thickness and flexoelectric effect of the material leads to increase in frequency of vibrations. Moreover, increased nanoshell length and diameter leads to decreased vibration frequency.

A unified size-dependent plate theory for static bending and free vibration analyses of micro- and nano-scale plates based on the consistent couple stress theory

This article is aimed at developing a nonclassical Mindlin rectangular functionally graded material (FGM) microplate based on the strain gradient theory (SGT) to study the thermal buckling behavior of microplates with different boundary conditions. This theory comprises material length scale parameters to interpret size effects. The developed model encompasses classical and modified couple stress Mindlin microplate models, if all the material length scale parameters or two of them are taken to be zero, respectively. The Mindlin rectangular FGM microplate is considered to be made of a mixture of metal and ceramic of which the volume fraction is described through a power low function. According to Hamilton's principle and the generalized differential quadrature (GDQ) method, the stability equations and associated boundary conditions are obtained and discretized, respectively. Current formulations provide a possibility to have all types of boundary conditions which herein, FGM microplates with three commonly used boundary conditions are considered. Three different types of thermal loads including uniform, linear and nonlinear temperature rises along the thickness of FGM microplates are considered. The dimensionless critical buckling temperature difference (DCBTD) predicted by SGT is compared with that of modified couple stress theory (CST) and classical theory (CT) which it is found that CST and CT underestimate the DCBTD. Also, effects of the boundary conditions, length scale parameter and material gradient index of FGM microplates on the DCBTD are judiciously investigated.

In this paper, pull-in behavior of cantilever micro/nano-beams made of functionally graded materials (FGM) with small-scale effects under electrostatic force is investigated. Consistent couple stress theory is employed to study the influence of small-scale on pull-in behavior. According to this theory, the couple tensor is skew-symmetric by adopting the skew-symmetric part of the rotation gradients. The material properties except Poisson’s ratio obey the power law distribution in the thickness direction. The approximate analytical solutions for the pull-in voltage and pull-in displacement of the microbeams are derived using the Rayleigh–Ritz method. Comparison between the results of the present work with Osterberg and Senturia’s article for pull-in behavior of microbeams made of isotropic material reveals the accuracy of this study. Numerical results explored the effects of material length scale parameter, inhomogeneity constant, gap distance and dimensionless thickness. Presented model has the ability to turn into the classical model if the material length scale parameter is taken to be zero. A comparison between classical and consistent couple stress theories is done which reveals the application of the consistent couple stress theory. As an important result of this study can be stated that a micro/nano-beams model based on the couple stress theory behaves stiffer and has larger pull-in voltages.

Couple stress Rayleigh–Bénard convection in a square cavity

Based on the consistent couple stress theory (CCST), we develop a size-dependent Hermitian C 2 finite layer method (FLM) for carrying out the three-dimensional (3D) static bending analysis of a simply-supported, functionally graded (FG) piezoelectric microplate which is placed under closed-circuit surface conditions. The microplate of interest is assumed to be resting on a Winkler-Pasternak foundation and subjected to either sinusoidal or uniformly distributed electro-mechanical loads. By setting the material length scale parameter at zero and ignoring the piezoelectric and flexoelectric effects, we reduce the formulation of the Hermitian C 2 FLM for analyzing FG piezoelectric microplates to that for analyzing FG piezoelectric macroplates and FG elastic microplates, respectively. The accuracy and the convergence rate of the Hermitian C 2 FLM are assessed by comparing the solutions it produces with the exact and approximate 3D solutions of FG piezoelectric macroplates and FG elastic microplates reported in the literature. Because the Hermitian C 2 FLM requires that the first-order and second-order derivatives of the primary variables must be continuous at each nodal plane, which in turn leads to their solutions converging rapidly and being able to obtain the accurate results of the electric and elastic variables induced in the microplates, especially for the transverse shear and normal stresses and the electric displacements. The effects of piezoelectricity, flexoelectricity, and the material length scale parameter on the deformations and stresses induced in the FG piezoelectric microplates are significant. Highlights A CCST-based Hermitian C 2 finite layer method (FLM) is developed for analyzing functionally graded (FG) piezoelectric microplates. The current FLM can be reduced to that for analyzing FG elastic microplates by ignoring the piezoelectric and flexoelectric effects. The current FLM can be reduced to that for analyzing FG piezoelectric macroplates by setting the material length scale parameter at zero. Implementing the current FLM reveals that its solutions converge rapidly and are in excellent agreement with those obtained using the exact 3D models. The effects of piezoelectricity, flexoelectricity, and the material length scale parameter on the static bending behavior of FG piezoelectric microplates are significant.

Size dependent electro-mechanical vibration of single-walled piezoelectric nanotubes using thin shell model

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.

Based on the consistent couple stress theory (CCST), we develop a Hermite-family [Formula: see text] finite layer method (FLM) for the three-dimensional (3D) free vibration analysis of a simply-supported, exponentially graded (EG) piezoelectric microplate under open- and closed-circuit surface conditions. In the formulation of the FLM, the microplate is artificially divided into a number of finite microlayers, and Fourier functions and Hermite polynomials are used to interpolate the in-plane and out-of-plane variations of a number of primary variables, respectively, including elastic displacement components and the electric potential variable for each individual layer. The Hermite-family [Formula: see text] FLM for analyzing EG piezoelectric microplates is reduced to the Hermite-family [Formula: see text] FLM for analyzing EG piezoelectric macroscale plates and functionally graded (FG) elastic microplates by assigning a value of zero to the material length scale parameter and by ignoring the piezoelectric and flexoelectric effects in the formulation, respectively. The accuracy and convergence rate of the FLM are assessed by comparing their solutions with the benchmark solutions of both the EG piezoelectric macroplates and the power-law-type FG elastic microplates that are available in the relevant literature. We examine and discuss some key effects on the free vibration characteristics of an EG piezoelectric microplate, including the impact of the material length scale parameter, the material-property gradient index, the length-to-thickness ratio, the piezoelectric effect, and the flexoelectric effect.

This article is concerned with the development of a distributed model based on the modified strain gradient elasticity theory (MSGT), which enables us to investigate the size-dependent pull-in instability of circular microplates subjected to the uniform hydrostatic and nonuniform electrostatic actuations. The model developed herein accommodates models based on the classical theory (CT) and modified couple stress theory (MCST), when all or two material length scale parameters are set equal to zero, respectively. On the basis of Hamilton's principle, the higher-order nonlinear governing equation and corresponding boundary conditions are obtained. In order to linearize the nonlinear equation, a step-by-step linearization scheme is implemented, and then the linear governing equation is discretized along with different boundary conditions using the generalized differential quadrature (GDQ) method. In the case of CT, it is indicated that the presented results are in good agreement with the existing data in the literature. Effects of the length scale parameters, hydrostatic and electrostatic pressures, and various boundary conditions on the pull-in voltage and pull-in hydrostatic pressure of circular microplates are thoroughly investigated. Moreover, the results generated from the MSGT are compared with those predicted by MCST and CT. It is shown that the difference between the results from the MSGT and those of MCST and CT is considerable when the thickness of the circular microplate is on the order of length scale parameter.

Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory

Size-dependent bending, buckling and free vibration of functionally graded Timoshenko microbeams based on the most general strain gradient theory

Classical elasticity theory failed to produce accurate results given in agreement with the experimental results. As this theory fails at stress concentration near the gaps or holes, or materials with significant microstructure contributions as soil, composite material, and polymer material generally in aerospace materials. In order to improve accuracy, the consistent couple stress theory is used. In the consistent couple stress theory, the microrotation and macro rotation are equal, where the macro rotation is half the curl of displacement, and the couple stress tensor is skew-symmetric. In this paper, the fundamental solutions for couple stress elasticity are derived in a systematic way via the Hörmander technique. The used technique is characterized by its ease and also develops the Galerkin tensor used in many applications, for example, the transformation of domain integral, computing the body force in the domain without discretization. All necessary kernels for displacements and tractions are derived and given in terms of a generalized Galerkin tensor for further use.