A standard experimental method for determining the material length scale based on modified couple stress theory

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.

Download Full-text

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

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.335-336.633 ◽

2011 ◽

Vol 335-336 ◽

pp. 633-640 ◽

Cited By ~ 1

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.

Download Full-text

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

Tehnički glasnik ◽

10.31803/tg-20180214212115 ◽

2018 ◽

Vol 12 (4) ◽

pp. 221-225 ◽

Cited By ~ 3

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).

Download Full-text

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

Journal of Vibration and Control ◽

10.1177/10775463211048272 ◽

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.

Download Full-text

Determination of vibration of single-layered graphene sheets using nonlocal modified couple stress theory

International Journal of Modern Physics C ◽

10.1142/s0129183122500115 ◽

2021 ◽

pp. 2250011

Author(s):

F. Attar ◽

R. Khordad ◽

A. Zarifi

Keyword(s):

Couple Stress ◽

Plate Theory ◽

Couple Stress Theory ◽

Nonlocal Parameter ◽

Modified Couple Stress Theory ◽

Length Scale ◽

Vibration Modes ◽

Classical Plate Theory ◽

Stress Theory ◽

Modified Couple Stress

The free vibration of single-layered graphene sheet (SLGS) has been studied by nonlocal modified couple stress theory (NMCS), analytically. Governing equation of motion for SLGS is obtained via thin plate theory in conjunction with Hamilton’s principle for two cases: (1) using nonlocal parameter only for stress tensor, (2) using nonlocal parameter for both stress and couple stress tensors. Navier’s approach has been used to solve the governing equations for simply supported boundary conditions. It is found that the frequency ratios decrease with increasing nonlocal parameter and also with enhancing vibration modes, but increase with raising length scale parameter. The nonlocal and length scale parameters are more prominent in higher vibration modes. The obtained results have been compared with the previous studies obtained by using classical plate theory, the modified couple stress theory and nonlocal elasticity theory, separately.

Download Full-text

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

Journal of Vibration and Control ◽

10.1177/1077546317706887 ◽

2017 ◽

Vol 24 (15) ◽

pp. 3471-3486 ◽

Cited By ~ 14

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).

Download Full-text

Flexural Vibration of an Atomic Force Microscope Cantilever Based on Modified Couple Stress Theory

International Journal of Structural Stability and Dynamics ◽

10.1142/s0219455415400258 ◽

2015 ◽

Vol 15 (07) ◽

pp. 1540025 ◽

Cited By ~ 21

Author(s):

Li-Na Liang ◽

Liao-Liang Ke ◽

Yue-Sheng Wang ◽

Jie Yang ◽

Sritawat Kitipornchai

Keyword(s):

Size Effect ◽

Couple Stress ◽

Scale Parameter ◽

Couple Stress Theory ◽

Flexural Vibration ◽

Modified Couple Stress Theory ◽

Length Scale ◽

Stress Theory ◽

Atomic Force ◽

Modified Couple Stress

This paper is concerned with the flexural vibration of an atomic force microscope (AFM) cantilever. The cantilever problem is formulated on the basis of the modified couple stress theory and the Timoshenko beam theory. The modified couple stress theory is a nonclassical continuum theory that includes one additional material parameter to describe the size effect. By using the Hamilton's principle, the governing equation of motion and the boundary conditions are derived for the AFM cantilevers. The equation is solved using the differential quadrature method for the natural frequencies and mode shapes. The effects of the sample surface contact stiffness, length scale parameter and location of the sensor tip on the flexural vibration characteristics of AFM cantilevers are discussed. Results show that the size effect on the frequency is significant when the thickness of the microcantilever has a similar value to the material length scale parameter.

Download Full-text

Thermal Postbuckling and Free Vibration of Extensible Microscale Beams Based on Modified Couple Stress Theory

Journal of Mechanics ◽

10.1017/jmech.2014.47 ◽

2014 ◽

Vol 31 (1) ◽

pp. 37-46 ◽

Cited By ~ 9

Author(s):

Y.-G. Wang ◽

W.-H. Lin ◽

C.-L. Zhou ◽

R.-X. Liu

Keyword(s):

Free Vibration ◽

Temperature Rise ◽

Couple Stress ◽

Couple Stress Theory ◽

Beam Theory ◽

Modified Couple Stress Theory ◽

Length Scale ◽

Modified Couple Stress ◽

Thermal Postbuckling ◽

Material Length Scale

AbstractThis paper presents a mathematical model and a computational approach for the thermal post-buckling and free vibration in the vicinity of the buckled equilibrium position of microbeams based on the modified couple stress Euler-Bernoulli beam theory and geometrically accurate relation. The governing equations for the whole analysis are established with a intrinsic material length scale parameter to capture the size effect. These equations, in conjunction with the corresponding boundary conditions, are decomposed into two two-point boundary value problems, which are solved using the shooting method. For static deformation, the geometric nonlinearity is involved and the size dependent postbuckling configuration is obtained as a function of temperature rise. For dynamic one, the small amplitude free vibration about the prebuckled position is developed following an assumed harmonic time mode, and the length scale and temperature rise dependent fundamental natural frequency is presented. Numerical computations are executed to illustrate the size dependency in the thermal postbuckling behaviors and fundamental frequency of the vibration around the buckled configuration of microbeams.

Download Full-text

Size-dependent thermoelastic initially stressed micro-beam due to a varying temperature in the light of the modified couple stress theory

Applied Mathematics and Mechanics ◽

10.1007/s10483-020-2676-5 ◽

2020 ◽

Vol 41 (12) ◽

pp. 1805-1820

Author(s):

A. E. Abouelregal

Keyword(s):

Couple Stress ◽

Scale Effects ◽

Couple Stress Theory ◽

Generalized Thermoelasticity ◽

Modified Couple Stress Theory ◽

Stress Theory ◽

Temperature Pulse ◽

Modified Couple Stress ◽

Material Length Scale ◽

Improved Model

AbstractThe bending of the Euler-Bernoulli micro-beam has been extensively modeled based on the modified couple stress (MCS) theory. Although many models have been incorporated into the literature, there is still room for introducing an improved model in this context. In this work, we investigate the thermoelastic vibration of a micro-beam exposed to a varying temperature due to the application of the initial stress employing the MCS theory and generalized thermoelasticity. The MCS theory is used to investigate the material length scale effects. Using the Laplace transform, the temperature, deflection, displacement, flexure moment, and stress field variables of the micro-beam are derived. The effects of the temperature pulse and couple stress on the field distributions of the micro-beam are obtained numerically and graphically introduced. The numerical results indicate that the temperature pulse and couple stress have a significant effect on all field variables.

Download Full-text

Nonlinear free vibration and damping analysis of a microbeam with pseudoelastic shape memory alloy layer based on the modified couple stress theory

Journal of Vibration and Control ◽

10.1177/1077546320935284 ◽

2020 ◽

pp. 107754632093528

Author(s):

Mohammad Javad Ashrafi ◽

Iman Ghaffari ◽

Mohammad Elahinia ◽

Mohammad Reza Nematollahi

Keyword(s):

Shape Memory Alloy ◽

Shape Memory ◽

Couple Stress Theory ◽

Modified Couple Stress Theory ◽

Alloy Layer ◽

Sandwich Composite ◽

Stress Theory ◽

Modified Couple Stress ◽

Material Length Scale ◽

Material Length

The aim of this investigation is to evaluate the size effects on the nonlinear free vibration of the sandwich composite microbeam with an extensible shape memory alloy layer in the midplane. A one-dimensional constitutive model is considered to simulate the pseudoelastic behavior of the shape memory alloy layer. The governing equation of motion is derived using the Euler–Bernoulli beam theory together with the modified couple stress theory through the Hamilton’s principle. Midplane stretching and phase transformation of the shape memory alloy which are sources of nonlinearity were considered, and a numerical solution method is presented. A damped response of the sandwich composite microbeam is observed because of the hysteresis behavior of the extensible shape memory alloy layer in the midplane. Results are appraised by comparing with the available literature. The influence of material length scale, temperature, and initial velocity on the loss factor and other pivotal vibrational behavior is evaluated. Results show that increasing the material length scale to thickness ratio has a decreasing effect on damping capacity.

Download Full-text