scholarly journals The vibration of nanobeam resting on elastic foundation using modified couple stress theory

2018 ◽  
Vol 12 (4) ◽  
pp. 221-225 ◽  
Author(s):  
Necla Togun ◽  
Süleyman M. Bağdatli
Keyword(s):  
Couple Stress ◽  
Scale Parameter ◽  
Couple Stress Theory ◽  
Beam Theory ◽  
Modified Couple Stress Theory ◽  
Stress Theory ◽  
Material Length Scale Parameter ◽  
Modified Couple Stress ◽  
Material Length Scale ◽  
Material Length

In this paper, the vibration of nanobeams resting on the Winkler foundation is proposed using the modified couple stress theory. Hamilton’s principle is utilized to construct the governing equations. The size effect of the nanobeam cannot be captured by using classical Euler-Bernoulli beam theory, but the modified couple stress theory model can capture it because it includes material length scale parameter that a newly developed model has. Once the material length scale parameter is assumed to be zero, the classical Euler-Bernoulli beam theory equation is obtained. Multiple scale method is employed to obtain the result. Simply supported boundary condition is used to study natural frequencies. The influence of material length scale parameter and the Winkler elastic foundation parameter on the fundamental frequencies of the nanobeam is investigated and tabulated. Also, in the present study, Poisson’s ratio is taken as constant. Nanobeam resting on the Winkler foundation which is simply supported is analyzed to illustrate the size effects on the free vibration. Numerical results for the simply supported nanobeam indicate that the first fundamental frequency calculated by the presented model is higher than the classical one. Moreover, it is obtained that the size influence is more substantial for higher vibration modes. The results indicate that the significant importance of the size influences the analysis of nanobeams. The vibration of nanobeam exhibits a hardening spring behavior, and the newly developed models are the beams stiffer than according to the classical beam theory. Modified couple stress theory tends to be more helpful in describing the size-dependent mechanical properties of nanoelectromechanical systems (NEMS).

Size Dependent Pull-in Phenomena in Electro-Statically Actuated Micro-Beam Based on the Modified Couple Stress Theory

2011 ◽  
Vol 335-336 ◽  
pp. 633-640 ◽  
Author(s):  
Jun Feng Zhao ◽  
Shen Jie Zhou ◽  
Bing Lei Wang
Keyword(s):  
Couple Stress ◽  
Couple Stress Theory ◽  
Beam Theory ◽  
Modified Couple Stress Theory ◽  
Dynamic Responses ◽  
Euler Beam ◽  
Stress Theory ◽  
Modified Couple Stress ◽  
Material Length Scale ◽  
Material Length

A modified continuum model of electro-statically actuated micro-beam is presented based on the modified couple stress theory. The new model contains a material length scale parameter and can capture the size effect, unlike the classical Bernoulli-Euler beam theory. The governing equation of the micro-beam is derived based on the Hamilton’s principle, which accounts for the effects of the moderately large deflection, the residual stress and the fringing electrostatic field. The numerical analysis of mechanical characterization is performed by the Analog Equation Method (AEM). The effects of the couple stress on the static and dynamic responses, pull-in voltage and pull-in time are discussed.

Size-dependent free vibration analysis of a three-layered exponentially graded nano-/micro-plate with piezomagnetic face sheets resting on Pasternak’s foundation via MCST

2017 ◽  
Vol 22 (1) ◽  
pp. 55-86 ◽  
Author(s):  
Mohammad Arefi ◽  
Masoud Kiani ◽  
Ashraf M Zenkour
Keyword(s):  
Couple Stress ◽  
Couple Stress Theory ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Modified Couple Stress Theory ◽  
Stress Theory ◽  
Modified Couple Stress ◽  
Material Length Scale ◽  
Material Length ◽  
Exponentially Graded

The present work is devoted to the free vibration analysis of elastic three-layered nano-/micro-plate with exponentially graded core and piezomagnetic face-sheets using the modified couple stress theory. To capture size-dependency for a nano-/micro-sized rectangular plate, the couple stress theory is used as a non-classical continuum theory. The rectangular elastic three-layered nano-/micro-plate is resting on Pasternak’s foundation. The present model contains one material length scale parameter and can capture the size effect. Material properties of the core are supposed to vary along the thickness direction based on the exponential function. The governing equations of motion are derived from Hamilton’s principle based on the modified couple stress theory and first-order shear deformation theory. The analytical solution is presented to solve seven governing equations of motion using Navier’s solution. Eventually the natural frequency is scrutinized for different side length ratio, thickness ratio, inhomogeneity parameter, material length scale, and parameters of foundation numerically.

Stress and free vibration analysis of piezoelectric hollow circular FG-SWBNNTs reinforced nanocomposite plate based on modified couple stress theory subjected to thermo-mechanical loadings

2017 ◽  
Vol 24 (15) ◽  
pp. 3471-3486 ◽  
Author(s):  
Mehdi Mohammadimehr ◽  
S Javad Atifeh ◽  
Borhan Rousta Navi
Keyword(s):  
Natural Frequencies ◽  
Couple Stress Theory ◽  
Modified Couple Stress Theory ◽  
Functionally Graded ◽  
Stress Theory ◽  
Material Length Scale Parameter ◽  
Inner Radius ◽  
Modified Couple Stress ◽  
Material Length Scale ◽  
Material Length

In this article, stresses and free-vibration behaviors of annular circular piezoelectric nanocomposite plate reinforced by functionally graded single-walled boron nitride nanotubes (FG-SWBNNTs) embedded in an elastic foundation based on modified couple stress theory (MCST) are explored. The mechanical properties of FG-SWBNNT-reinforced nanocomposite plate are assumed to be graded in the direction of thickness and estimated through the micro-mechanical approach. The governing equations are obtained using the energy method. The natural frequencies and stresses of FG-SWBNNT-reinforced nanocomposite plate are computed using the differential quadrature method (DQM). An excellent agreement is observed between the obtained results and the results in the literature. Influences of the internal radius to the external radius, the thickness to the internal radius ratio, the material length scale parameter, the functionally graded parameter, temperature changes and elastic coefficients on the natural frequencies and stresses of the hollow circular nanocomposite plate are investigated. The results of this research show that the natural frequencies of the piezoelectric nanocomposite plate increase by increasing the material length scale parameter, the elastic foundation parameters, the ratio of the inner radius to the outer radius, the ratio of the thickness to the inner radius, and decreasing the power index and temperature change. The radial stress of the nanocomposite plate varies proportionally to its mode shape. The results can be employed to design smart structures in micro-electro-mechanical systems (MEMS).

A NONLOCAL CURVED BEAM MODEL BASED ON A MODIFIED COUPLE STRESS THEORY

2011 ◽  
Vol 11 (03) ◽  
pp. 495-512 ◽  
Author(s):  
Y. P. LIU ◽  
J. N. REDDY
Keyword(s):  
Couple Stress Theory ◽  
Classical Model ◽  
Modified Couple Stress Theory ◽  
Curved Beam ◽  
Beam Model ◽  
Stress Theory ◽  
Material Length Scale Parameter ◽  
Modified Couple Stress ◽  
Material Length Scale ◽  
Material Length

A nonlocal Timoshenko curved beam model is developed using a modified couple stress theory and Hamilton's principle. The model contains a material length scale parameter that can capture the size effect, unlike the classical Timoshenko beam theory. Both bending and axial deformations are considered, and the Poisson effect is incorporated in the model. The newly developed nonlocal model recovers the classical model when the material length scale parameter and Poisson's ratio are both taken to be zero and the straight beam model when the radius of curvature is set to infinity. In addition, the nonlocal Bernoulli–Euler curved beam model can be realized when the normal cross-section assumption is restated. To illustrate the new model, the static bending and free vibration problems of a simply supported curved beam are solved by directly applying the formulas derived. The numerical results for the static bending problem reveal that both the deflection and rotation of the simply supported beam predicted by the new model are smaller than those predicted by the classical Timoshenko curved beam model. Also, the differences in both the deflection and rotation predicted by the current and classical Timoshenko model are very large when the beam thickness is small, but they diminish with the increase of the beam height. Similar trends are observed for the free vibration problem, where it is shown that the natural frequency predicted by the nonlocal model is higher than that by the classical model, and the difference between them is significantly large only for very thin beams. These predicted trends of the size effect at the micron scale agree with those observed experimentally.

Buckling analysis of functionally graded sandwich cylindrical micro/nanoshells based on the couple stress theory

2017 ◽  
Vol 21 (3) ◽  
pp. 917-937 ◽  
Author(s):  
Hamid Zeighampour ◽  
Milad Shojaeian
Keyword(s):  
Couple Stress ◽  
Scale Parameter ◽  
Couple Stress Theory ◽  
Functionally Graded ◽  
Buckling Load ◽  
Critical Buckling Load ◽  
Stress Theory ◽  
Material Length Scale Parameter ◽  
Material Length Scale ◽  
Material Length

Buckling of functionally graded sandwich cylindrical microshell under axial load is investigated. For this purpose, Donnell shell theory as well as material length scale parameter as considered by the couple stress theory is used, and equations of motion of the functionally graded sandwich cylindrical microshell along with boundary conditions are developed using Hamilton’s principle. Finally, dimensionless critical buckling load is determined for three functionally graded sandwich cylindrical microshells using the Navier procedure. Results of the new model are compared with the classical theory. The results indicate that the rigidity of the functionally graded sandwich cylindrical microshell in the couple stress theory is higher than that in the classical theory, which leads to increased dimensionless critical buckling load. Besides, the effect of material length scale parameter on dimensionless critical buckling load of the functionally graded sandwich cylindrical microshell in different wavenumbers is considerable.

Free Vibration of Edge Cracked Functionally Graded Microscale Beams Based on the Modified Couple Stress Theory

2017 ◽  
Vol 17 (03) ◽  
pp. 1750033 ◽  
Author(s):  
Şeref Doğuşcan Akbaş
Keyword(s):  
Couple Stress ◽  
Scale Parameter ◽  
Couple Stress Theory ◽  
Beam Theory ◽  
Modified Couple Stress Theory ◽  
Functionally Graded ◽  
Beam Model ◽  
Cracked Beam ◽  
Stress Theory ◽  
Modified Couple Stress

In this study, the free vibration analysis of edge cracked cantilever microscale beams composed of functionally graded material (FGM) is investigated based on the modified couple stress theory (MCST). The material properties of the beam are assumed to change in the height direction according to the exponential distribution. The cracked beam is modeled as a modification of the classical cracked-beam theory consisting of two sub-beams connected by a massless elastic rotational spring. The inclusion of an additional material parameter enables the new beam model to capture the size effect. The new nonclassical beam model reduces to the classical one when the length scale parameter is zero. The problem considered is investigated using the Euler–Bernoulli beam theory by the finite element method. The system of equations of motion is derived by Lagrange’s equations. To verify the accuracy of the present formulation and results, the frequencies obtained are compared with the results available in the literature, for which good agreement is observed. Numerical results are presented to investigate the effect of crack position, beam length, length scale parameter, crack depth, and material distribution on the natural frequencies of the edge cracked FG microbeam. Also, the difference between the classical beam theory (CBT) and MCST is investigated for the vibration characteristics of the beam of concern. It is believed that the results obtained herein serve as a useful reference for research of similar nature.

scholarly journals Fluid-Induced Nonlinear Vibration of a Cantilevered Microtube with Symmetric Motion Constraints

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Yong Guo
Keyword(s):  
Couple Stress Theory ◽  
Modified Couple Stress Theory ◽  
Critical Flow Velocity ◽  
Stress Theory ◽  
Motion Constraints ◽  
Material Length Scale Parameter ◽  
Modified Couple Stress ◽  
Material Length Scale ◽  
The Galerkin Method ◽  
Material Length

This paper investigates the dynamic behavior of a cantilevered microtube conveying fluid, undergoing large motions and subjected to motion-limiting constraints. Based on the modified couple stress theory and the von Kármán relationship, the strain energy of the microtube can be deduced and then the governing equation of motion is derived by using the Hamilton principle. The Galerkin method is applied to produce a set of ordinary differential equations. The effect of the internal material length scale parameter on the critical flow velocity is investigated. By using the projection method, the Hopf bifurcation is demonstrated. The results show that size effect on the vibration properties is significant.

Dynamic stiffness formulation for a micro beam using Timoshenko–Ehrenfest and modified couple stress theories with applications

2021 ◽  
pp. 107754632110482
Author(s):  
J Ranjan Banerjee ◽  
Stanislav O Papkov ◽  
Thuc P Vo ◽  
Isaac Elishakoff
Keyword(s):  
Free Vibration ◽  
Couple Stress ◽  
Scale Parameter ◽  
Couple Stress Theory ◽  
Dynamic Stiffness ◽  
Modified Couple Stress Theory ◽  
Length Scale Parameter ◽  
Length Scale ◽  
Stress Theory ◽  
Modified Couple Stress

Several models within the framework of continuum mechanics have been proposed over the years to solve the free vibration problem of micro beams. Foremost amongst these are those based on non-local elasticity, classical couple stress, gradient elasticity and modified couple stress theories. Many of these models retain the basic features of the Bernoulli–Euler or Timoshenko–Ehrenfest theories, but they introduce one or more material scale length parameters to tackle the problem. The work described in this paper deals with the free vibration problems of micro beams based on the dynamic stiffness method, through the implementation of the modified couple stress theory in conjunction with the Timoshenko–Ehrenfest theory. The main advantage of the modified couple stress theory is that unlike other models, it uses only one material length scale parameter to account for the smallness of the structure. The current research is accomplished first by solving the governing differential equations of motion of a Timoshenko–Ehrenfest micro beam in free vibration in closed analytical form. The dynamic stiffness matrix of the beam is then formulated by relating the amplitudes of the forces to those of the corresponding displacements at the ends of the beam. The theory is applied using the Wittrick–Williams algorithm as solution technique to investigate the free vibration characteristics of Timoshenko–Ehrenfest micro beams. Natural frequencies and mode shapes of several examples are presented, and the effects of the length scale parameter on the free vibration characteristics of Timoshenko–Ehrenfest micro beams are demonstrated.

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