Dimensionless Length Scale Parameter Research Articles

Periodic motion of microscale cantilevered fluid-conveying pipes with symmetric breaking on the cross-section

A new theoretical model is developed for the three-dimensional (3D) nonlinear vibration analysis of cantilevered fluid-conveying micropipes with asymmetric cross-section. Fluid-conveying microscale pipes with ring-shaped cross-sections of O(2) symmetry are widely used in engineering. However, their cross-sections usually have a small symmetric breaking owing to manufacturing errors, which can be described by a small parameter. Hence, the dynamic behaviors of such geometrically imperfect micropipes are extremely required to be explored. This study mainly investigates the effects of the symmetric breaking parameter, flow velocity and dimensionless material length scale parameter on the dynamic behaviors of microscale cantilevered fluid-conveying micropipes. Based on the modified couple stress theory (MCST) and Hamilton's principle, the governing equation is derived, where the effects of the asymmetry of cross-section on the equation are reflected. Using the center manifold theory and normal form method, the original governing equation is rigorously reduced to a two-degree-of-freedom (2DOF) dynamical system, and the averaging equation of the reduced system is obtained by combining the averaging method. The existence and stability of the equilibrium point of the averaging equation are studied to obtain the results of the existence and stability of periodic motions of the original system. The results demonstrate that the types of periodic motions and stability when the pipe loses stability from different places on the curve of the critical flow velocity versus mass ratio (critical flow velocity curve) may differ. Meanwhile, the effects of various parameters on these periodic motions are presented. The increase in the dimensionless material length scale parameter increases the section corresponding to stable plane periodic motion and reduces the section corresponding to stable spatial periodic motion on the critical flow velocity curve. When dimensionless material length scale parameter is given, the existence and stability of various periodic motions of the system depend on the ratio of the increment of flow velocity around the critical value to the symmetric breaking parameter. In addition, some new phenomena are found in this paper. Stable 0 planar periodic motion of the macropipe can occur when pipe loses its stability from the hysteresis part of the critical flow velocity curve. Stable 0 planar periodic motion of microscale pipes can occur when pipe loses its stability from the non-hysteresis part of the critical flow velocity curve, and then stable 0 planar periodic motion and stable π planar periodic motion can coexist. Furthermore, stable planar and stable spatial periodic motions of pipes with symmetric breaking can coexist, and stable torus motion has also been observed; this is different from the dynamics of pipes with O(2) symmetry.

Finite element formulation for nano‐scaled beam elements

AbstractIn the present study, size‐dependent buckling and free vibration behaviors of single‐walled boron nitride nanotube (SWBNNT) are performed in conjunction with various size‐dependent elasticity theories. Modified couple stress theory (MCST) and Eringen's nonlocal elasticity theory are used for size‐dependent models of SWBNNT. Also, the buckling loads and frequencies are obtained by using local theory to emphasize the effects and differences of these size‐dependent theories. Consequently, three different elasticity theories (two non‐classical and one classical) are utilized to achieve the detailed buckling and vibration analyses of SWBNNT. In this study, the buckling loads and frequencies of SWBNNTs are obtained via presented finite element formulation. In the finite element procedures based on two different size‐dependent elasticity theories, matrices containing the small size parameter are derived. With these matrices containing the small size parameters, eigenvalue problems for buckling and free vibration analyses are formed. The buckling loads and frequency values of the SWBNNTs under the size effect are obtained. The influences of the dimensionless nonlocal parameter, dimensionless material length scale parameter, length‐to‐diameter ratio and boundary conditions on nanotube's buckling and vibration characteristics are investigated. In addition to these influences, the rotary inertia effect neglected in many other studies is also examined.

Large Amplitude Free Vibration Analysis of Nanobeams Based on Modified Couple Stress Theory

In this study, the large amplitude free vibration of nanobeams based on the modified couple stress theory was developed by using Total Lagrangian finite element formulation. In this study, Timoshenko beam theory has been used in free vibration analysis of nanobeams. Minimal kinematic assumptions have been used to model nanobeams. With this model, free vibration of nanobeams with small to large amplitude and with arbitrary boundary conditions can be analyzed. The numerical results obtained for free vibration based on the modified couple stress theory with small amplitude and the results obtained for free vibration with large amplitude without considering the modified couple stress theory are in good agreement with the similar results reported in previous research. Effects of the dimensionless length scale parameter, slenderness ratio, vibration amplitude and different boundary conditions on the nonlinear frequency ratio of nanobeams have been investigated. The results show the importance of considering nonlinear and size effects in the free vibration analysis of nanobeams with large amplitude.

Impact of dimensionless length scale parameter on material dependent thermoelastic attenuation and study of frequency shifts of rectangular microplate resonators

Quality factor is one of the decisive performance parameters of micro/nano plate resonators based sensors and filters .Various energy dissipation mechanisms limit the maximum attainable quality factor, and in micro/nano scales thermoelastic damping is a critical energy loss mechanism. When the devices are scaled down, in order to accurately model micro/nano scale resonators, non-classical elasticity theories like Modified Couple Stress Theory (MCST) are valid. In this paper, by including a length scale parameter (l) the size effects were incorporated in the analysis and thermoelastic energy dissipation and related attenuation were found to be diminished by selecting a dimensionless length scale parameter i.e. l/h. Thermoelastic attenuation and frequency shifts related to thermoelastic damping in microplate rectangular resonators with five structural materials (SiC, polySi, diamond, Si and GaAs) were investigated with l/h =0.The attenuation related to thermoelastic energy dissipation and frequency shifts were found to be minimum for polySi material. With the inclusion of size effect, the impact of dimensionless length scale parameter on thermoelastic attenuation of polySi based microplate resonators was also explored. The conventional thermoelastic damping analysis in rectangular plate was modified by applying Modified Couple Stress Theory and the impact of dimensionless length scale (l/h) on thermoelastic energy attenuation was simulated numerically using MATLAB 2015.

Free torsional vibration of triangle microwire based on modified couple stress theory

In this article, the free torsional vibration of the noncircular microwire is obtained via modified couple stress theory. The noncircular cross section is considered to be equilateral triangle. The governing equations are derived using Hamilton’s principle, and the natural frequencies of the clamped-clamped and clamped-free microwires are obtained based on a Galerkin method. The natural frequency for analyzing of the results is considered to be dimensionless. The effects of the different parameters including the dimensionless material length scale parameter, the dimensionless edge length of the triangle, and mode number on the dimensionless natural frequency of the microwire are evaluated. Ultimately, the influence of the Poisson’s ratio on the natural frequency of the microwire under various boundary conditions is investigated.

Three-dimensional thermo-electro-mechanical buckling analysis of functionally graded piezoelectric micro/nano-shells based on modified couple stress theory considering thickness stretching effect

The present work uses a new three-dimensional framework for thermo-electro-mechanical buckling analysis of functionally graded piezoelectric cylindrical nano/micro-shells subjected to axial mechanical compression, an external applied voltage and uniform temperature rising. To account thickness stretching effect, three-dimensional shear and normal deformation theory is employed. Size dependency is modeled based on modified couple stress theory. Principle of virtual work is used to derive governing equations. The solution procedure is developed based on Navier’s technique for simply-supported boundary conditions. The importance of the present new framework is justified through comparison between present results and corresponding previous lower-order analysis. The full numerical results are presented in terms of significant parameters of the problem such as some dimensionless geometric parameters, temperature change, dimensionless length scale parameter and applied voltages.

Nonlinear resonant behaviors of bi-directional functionally graded material microbeams: One-/two-parameter bifurcation analyses

In this paper, the nonlinear resonance behaviors of bi-directional functionally graded (BDFG) microbeams are investigated. The material properties including the material length scale parameter vary along both thickness and axial directions. Employing Hamilton’s principle, the differential equations are derived based on von-Karman geometric nonlinearity and Timoshenko beam theory. The modified couple stress theory is adopted to capture the size effects. The continuous system dynamics model is discretized by method of Galerkin scheme along with appropriate eigenfunctions, resulting in the reduced order model which is a coupled large-dimension system of nonlinear ordinary differential equations. The nonlinear resonance behaviors of two type of BDFG microbeams are explored by performing one- and two-parameter bifurcation analyses. In one-parameter bifurcation analysis, the frequency- and force-response curves are constructed by tracing the period motion of microbeam using the pseudo-arclength continuation technique. Cyclic-fold bifurcation which indicates jump phenomenon is detected in the period motion. The trajectories of cyclic-fold bifurcation points are achieved by implementing the two-parameter bifurcation analysis. The cusp bifurcation of periodic motion implies the occurrence of CF bifurcation. Numerical simulations are performed to examine the influences of the system parameters, e.g. gradient indexes,dimensionless length scale parameter, damping coefficients and aspect ratio on the nonlinear resonance of BDFG microbeams.

Thermal effect on vibrational behaviors of rotating functionally graded microbeams

Thermal effect on size-dependent vibrational behavior of rotating microbeams made of functionally graded materials (FGMs) is investigated. It is assumed that the material properties of FGMs change smoothly through the thickness of the microbeam according to a power-law variation. Within the framework of Timoshenko beam theory (TBT) and Euler-Bernoulli beam theory (EBT), the governing equations and associated boundary conditions of rotating hinged-hinged, clamped-free, clamped-clamped and clamped-hinged microbeams are determined by using Hamilton's principle and modified couple stress theory. The axial nonlinear coupling deformation is taken into account to capture the centrifugal stiffening effect. An assumed-mode discretization approach is applied to solve these equations numerically. Convergence study and comparative examples are presented to verify the accuracy and reliability of the present model. A detailed numerical study is carried out to analyze the effects of dimensionless material length scale parameter, slenderness ratio, dimensionless angular velocity, temperature change and FG index on the vibrational behaviors of rotating FG microbeams. Numerical results indicate that the increasing value of temperature change leads to an increase in size dependency of rotating microbeams. The size dependency of rotating microbeams considerably differs from that of stationary microbeams. In addition, the gradient change in material properties greatly affects the size dependency of rotating FG microbeams.

Influence of homogenization schemes on vibration of functionally graded curved microbeams

The influence of various homogenization models is studied on the free vibration behavior of a functionally graded material (FGM) curved microbeam which is modeled using the modified strain gradient theory of elasticity as well as the first-order shear deformation theory. Different homogenization models (such as Voigt, Reuss, Hashin-Shtrikman bounds, and cubic local representative volume elements (LRVE) schemes) are used to predict the effective material properties of a two-phase particle composite as a function of the volume fraction of particles which are continuously varying along the thickness of a functionally graded microbeam. Employing Hamilton’s principle, the governing equations of motion and boundary conditions are derived. Finally, the numerical results are presented to determine the effect of homogenization models on the vibrational behavior of FGM curved microbeams corresponding to different continuum models for over a wide range of material composition, opening angle, dimensionless length scale parameter and aspect ratio. It is shown that the common used Voigt model overestimates frequencies and the Mori-Tanaka model also overestimates frequencies but provides smaller error in comparison with the Voigt model. In terms of theoretical analysis, LRVE model creates a good compromise between estimation accuracy and the easy of implement, especially for the FGM with a relative high stiffness ratio. The difference of various homogenization models can be raised when considering the decrement in aspect ratio or the increment of opening angle and mode number.

Static and dynamic analysis of the postbuckling of bi-directional functionally graded material microbeams

In this paper, the static and dynamic responses of bi-directional functionally graded (BDFG) microbeams are investigated. The material properties vary along both thickness and axial directions. Employing Hamilton’s principle, the differential equations are derived based on von-Karman geometric nonlinearity and third-order shear deformation beam theory. The modified couple stress theory is adopted to capture the size effects. The material length scale parameter of microbeam is considered as a function of spatial coordinates and varies with the material gradient parameters. The differential equations and boundary conditions are discretized using differential quadrature method. Subsequently, static bifurcation of microbeams is calculated utilizing pseudo-arc length continuation technique. The free vibration around the postbuckling configuration of microbeams is studied by solving the associated linear eigenvalue problem. The influences of the functional gradient parameters, dimensionless length scale parameter and aspect ratio on the static response and vibration characteristics of buckled BDFG microbeams with various boundary conditions are investigated. It is demonstrated that the buckling of BDFG microbeam occurs through a transcritical bifurcation. The static stable responses of BDFG microbeam are asymmetric due to the stretching-bending coupling that results from the asymmetry of the material distribution in thickness direction of beam. The mode veering phenomenon is detected in postbuckling domain of microbeams. It is examined that the longitudinal dynamic displacement plays a pivotal role in the mode veering phenomenon.

Dynamic bifurcations analysis of a micro rotating shaft considering non-classical theory and internal damping

In this study, the dynamic bifurcation of a viscoelastic micro rotating shaft is investigated. The non-classical theory (the modified couple stress theory) and the Kelvin Voigt model are used for modeling the viscoelastic micro shaft. The transverse equations of motion are derived using the variational approach. The reduced order model of the system is obtained by the Galerkin method. Using the Routh–Hurwitz criteria the stability regions of the system are extracted in which the effect of the length scale parameter is significant. Using the center manifold theory and the normal form method the double zero eigenvalue bifurcation is analyzed. The results show that the internal and external damping coefficients, the rotational speed and the material length scale parameter influence the critical speed, amplitude, and phase of a non-trivial solution, and radius of limit cycle (periodic solution). Also, it is seen that by increasing the dimensionless length scale parameter (material length scale per radius of the shaft) the radius of the limit cycle is decreased, whereas the critical rotational speed and the rate of the phase are increased. However, the radius of the limit cycle concerning the classical theory is higher than that of regarding the modified couple stress theory. Furthermore, with an increase of the external damping coefficient the radius of the limit cycle is linearly decreased; however, the critical speed of the system is increased. Additionally, by decreasing length scale parameter the results of the modified couple stress theory approach the classical theory ones.

Thermal buckling of functionally graded triangular microplates

The thermal buckling behavior of thin to moderately thick functionally graded isosceles triangular microplates with temperature-dependent material properties is investigated. The governing equations are derived based on the modified strain gradient theory (MSGT) in conjunction with the first-order shear deformation theory. The adjacent equilibrium criterion and Chebyshev–Ritz method are employed to derive the nonlinear thermal buckling eigenvalue equations, which are solved by a direct iterative method. The fast rate of convergence and accuracy of the method are demonstrated numerically. Then, the effects of length scale parameters, material gradient index, different boundary conditions, apex angle and ratio of width to thickness on the critical temperature rises of the triangular microplates are studied. In addition, comparisons between the results of MSGT and modified couple stress theory and classical theory (CT) are performed. The results show that by increasing the apex angle, the critical temperature rise increases, but increase in the material gradient index and the dimensionless length scale parameter decreases the critical temperature rise. In addition, it is observed that by considering the temperature dependence of material properties, the critical temperature rises decrease significantly. Also, the MSGT and CT yield the highest and the lowest critical temperature rise, respectively.

Dynamic stability analysis of microcomposite annular sandwich plate with carbon nanotube reinforced composite facesheets based on modified strain gradient theory

In this article, dynamic stability of annular sandwich plate with carbon nanotubes reinforced composite facesheets and an isotropic homogeneous core are presented based on first-order shear deformation theory and modified strain gradient theory. The generalized rule of mixture is employed to predict mechanical properties of microcomposite sandwich plate. The equations of motion are derived from Hamilton’s principle and solved by differential quadrature method. The fast rate of convergence of the method is shown and the results are compared against existing results in the literature. The results indicate that volume fraction of carbon nanotubes in facesheets and dimensionless length scale parameter has significant effects on the dynamic stability region and the parametric resonance. Dynamic stability region increases with considering of dimensionless length scale parameter, increasing of volume fraction of carbon nanotubes, and static load factor. Also, the influence of inner-to-outer radius ratios, radius-to-thickness ratios, and core-to-facesheets ratios are considered. The results can be employed for design of materials science, in junction high pressure micropipe connections, solid-state physics, micro-electro-mechanical systems, and nano electromechanical systems such as microactuators and microsensor.

Three-dimensional vibration of cantilevered fluid-conveying micropipes—Types of periodic motions and small-scale effect

A new theoretical model is developed for the three-dimensional (3D) nonlinear vibration analysis of fluid-conveying cantilevered micropipes. Particular attention is given on the derivation and analysis of the reduced equations, and the small-scale effect on the periodic motions. Based on the modified couple stress theory (MCST), the governing equations are derived by using Hamilton’s principle. The material length scale parameter and large-deflection-induced geometric nonlinearities given by the Lagrangian strain tensor are incorporated into the governing equations. Utilizing the center manifold theory, normal form method and O(2) symmetry, the original governing equations can be rigorously reduced to a two-degree-of-freedom (2DOF) dynamical system. Then two possible types of periodic motions, i.e. planar periodic and spatial periodic motions, together with their stabilities are investigated by means of averaging methods and numerical simulations. Results show that the larger the dimensionless material length scale parameter is, the wider the region of mass ratio for stable planar periodic motion is. Particularly, the presence of small length scale parameter makes micropipes be more likely to oscillate in a plane. It is also shown that for mass ratio corresponding to the hysteresis of the curves of critical flow velocity versus mass ratio, the stabilities for bifurcating periodic motions at lower, moderate and higher critical flow velocities may be different.

Thermal-mechanical-electrical buckling behavior of functionally graded micro-beams based on modified couple stress theory

This paper presents thermal-mechanical-electrical buckling analysis of micro-beams that are made of functionally graded materials(FGMs) with temperature-dependent thermo-elastic properties. The material properties of the micro-beam are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The governing equations are based on the principle of the minimum total potential energy, von Kármán geometric nonlinearity theory, and modified couple stress theory is employed to consider the size effect. A differential quadrature (DQ) method combined with an iteration process is used to predict the critical buckling external axial force and buckling temperature increment. The effects of the volume fraction profile parameter, dimensionless length scale parameter, initial gap ratio, ground electrode shape parameter, the applied voltage, slenderness ratio and the axial residual stress on the thermal-mechanical-electrical buckling behavior are evaluated in detail through parametric studies.

Non-classical plate model for FGMs

In this work, we developed a non-classical plate model for the analysis of the behavior of microplates prepared from functionally graded materials (FGMs). It was assumed that the material properties of FGM microplates vary along the direction of thickness and could be estimated using Mori–Tanaka homogenization technique. A variational approach based on Hamilton’s principle was employed to achieve motion equations and corresponding boundary conditions simultaneously. Unlike classical plate theory, the proposed plate model contained three material length scale parameters which allowed size effect to be taken into account effectively. Moreover, the proposed non-classical plate model could be divided into classical and modified couple stress plate models when all material length scale parameters were set at zero. To approve the applicability of the proposed size-dependent plate model, analytical solution for free vibration problem of a simply supported FGM microplate was obtained according to the Navier solution, where generalized displacements were stated as multiplication of undetermined functions with known trigonometric functions to identically satisfy simply supported boundary conditions for all edges. The results obtained by the proposed non-classical plate model were compared with those obtained from reduced ones relevant to different values of material property gradient index and dimensionless length scale parameter. It was found that the differences observed between the sets of predicted natural frequencies obtained from different types of plate models were more significant for lower values of dimensionless length scale parameter.

Free vibration analysis of functionally graded stiffened micro-cylinder based on the modified couple stress theory

In this paper, free vibration of the micro-cylinder made by functionally graded material that is stiffened in circumferential direction, has been investigated based on the modified couple stress and first order shear deformation theories. Modified couple stress theory (MCST) has been used to catch size effects in micro scales. By using first order shear deformation theory and Hamilton principle, general equations of motion and corresponding boundary conditions have been derived. Free vibration of the structure has been investigated by implementing simply supported boundary condition as a common case. The effects of different parameters such as dimensionless length scale parameter, distribution of FGM properties, number of stiffeners, thickness and length on the natural frequencies were calculated and compared with classical continuum theory. Results show that effects of the size are considerable and also using stiffeners lead to increase in natural frequencies which is because of increase in stiffness of the cylinder.

Thermo-electro-mechanical buckling analysis of cylindrical nanoshell on the basis of modified couple stress theory

In the present study, the buckling of piezoelectric cylindrical nanoshell subjected to an axial compression, an applied voltage and uniform temperature change resting on Winkler-Pasternak foundation is studied analytically. The modified couple stress theory combined with the geometrical nonlinear shell model is employed to derive the equilibrium equations and boundary conditions. The numerical results are proposed for the buckling of simply supported cylindrical nanoshell using the Navier-type solution. Thus, the effects of different parameters such as dimensionless length scale parameter, length and thickness to radius ratio, temperature change, external electric voltage and Winkler and Pasternak foundation stiffness on critical buckling load are illustrated. It is shown that increase in dimensionless length scale parameter results in increasing critical buckling load and even intensifying the influence of other parameters, such as length and thickness, on critical buckling load.

Buckling and free vibration of shallow curved micro/nano-beam based on strain gradient theory under thermal loading with temperature-dependent properties

In this study, influences of a uniform thermomechanical loading in buckling and free vibration of a curved FG microbeam have been investigated, based on strain gradient theory (SGT) theory and Timoshenko beam model. Distribution of structural materials varies continuously in thickness direction due to power-law exponent. Unlike classical models, this novel model employs three length scale parameters which can capture the size effect. This work is based on SGT theory and Timoshenko beam model. Governing equation of motion and associated boundary condition have been developed based on Hamilton’s principle, which is the specified case of virtual work theorem. In continuance, final differential equations were solved by Navier’s solution method and the results have been presented. Moreover, influences of dimensionless length-to-thickness ratio (aspect ratio), dimensionless length scale parameter, power-law exponent, temperature difference and arc angle for various values of mode numbers on natural frequency and critical temperature by considering temperature-dependent material properties have been investigated. In order to validate accomplished study, some of the results were compared with those of previous works. It has been concluded that applying a thermomechanical loading on a FG microbeam causes the natural frequency to become more sensitive about variations of geometrical, physical and mechanical properties and characteristics.

Size-dependent pull-in voltage and nonlinear dynamics of electrically actuated microcantilever-based MEMS: A full nonlinear analysis

This paper aims to develop a size-dependent nonlinear model for electrically actuated microcantilever-based MEMS based on the modified couple stress theory. The pull-in instability and nonlinear dynamics of the microcantilever are explored considering the full nonlinear equation of motion. The discretized equations are obtained using Galerkin method. It has been demonstrated that the material's small length scale parameter and geometric nonlinearities significantly influence the static and dynamic pull-in behaviors of the microcantilever-based MEMS. In the presence of the length-scale parameter, the pull-in voltage is found to be size-dependent. If a time-dependent harmonic component is superposed on the DC voltage, the primary resonances of the micro-cantilever are observed. When the dimensionless length-scale parameter is relatively large, the frequency-response curves indicate that the dynamic responses of the microbeam can evolve from softening-type to hardening-type nonlinear behaviors; the combined effects of length-scale parameter and geometric nonlinearities on the pull-in band of frequencies may be remarkable.