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.

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).

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.

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 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.

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

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.

Size-Dependent Postbuckling of Piezoelectric Microbeams Based on a Modified Couple Stress Theory

2017 ◽  
Vol 09 (04) ◽  
pp. 1750053 ◽  
Author(s):  
Xingjia Li ◽  
Ying Luo
Keyword(s):  
Couple Stress ◽  
Couple Stress Theory ◽  
Classical Model ◽  
Modified Couple Stress Theory ◽  
Beam Model ◽  
Postbuckling Behavior ◽  
Stress Theory ◽  
Size Dependent ◽  
Critical Buckling Force ◽  
Modified Couple Stress

This paper aims to investigate the postbuckling behavior of piezoelectric microbeams (PMBs) using a modified couple stress theory (MCST) and a Euler–Bernoulli–von Kármán beam model. The critical buckling force, voltage and the deformation amplitude were calculated for the buckling of the axially compressed microbeams with a clamp–clamp boundary condition. It is found that the stiffness of microbeams considering the MCST is higher than that given by the classical model when the feature size decreases to the microscale. Moreover, the microscale size effect has a strong influence on the critical buckling loads and the amplitude of postbuckling deformation. This study brings an improved understanding of the postbuckling behavior of PMBs, and offers useful guidance for the design of piezobeam-based sensors, actuators and stretchable microelectronics.

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.

A Nonclassical Model for Circular Mindlin Plates Based on a Modified Couple Stress Theory

2014 ◽  
Vol 81 (5) ◽  
Author(s):  
S.-S. Zhou ◽  
X.-L. Gao
Keyword(s):  
Couple Stress Theory ◽  
Plate Thickness ◽  
Modified Couple Stress Theory ◽  
Mindlin Plate ◽  
Plate Model ◽  
Stress Theory ◽  
New Model ◽  
Modified Couple Stress ◽  
Material Length Scale ◽  
Material Length

A nonclassical model for circular Mindlin plates subjected to axisymmetric loading is developed using a modified couple stress theory. The equations of motion and boundary conditions are simultaneously obtained through a variational formulation based on Hamilton's principle. The new model contains a material length scale parameter and can capture the size effect, unlike existing circular Mindlin plate models based on classical elasticity. In addition, both the stretching and bending of the plate are considered in the formulation. The current plate model reduces to the classical elasticity-based Mindlin plate model when the material length scale parameter is set to be zero. Additionally, the new circular Mindlin plate model recovers the circular Kirchhoff plate model as a special case. To illustrate the new model, the static bending problem of a clamped solid circular Mindlin plate subjected to an axisymmetrically distributed normal pressure is analytically solved by directly applying the new model and using the Fourier–Bessel series. The numerical results show that the deflection and rotation angle predicted by the new model are smaller than those predicted by the classical Mindlin plate model. It is further seen that the differences between the two sets of predicted values are significantly large when the plate thickness is small, but they are diminishing with the increase of the plate thickness.

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