A Nonlinear Electro-Mechanical Analysis of Nanobeams Based on the Size-Dependent Piezoelectricity Theory

2016 ◽  
Vol 33 (3) ◽  
pp. 289-301 ◽  
Author(s):  
Y. T. Beni
Keyword(s):  
Continuum Theory ◽  
Mechanical Analysis ◽  
Free Vibrations ◽  
Geometrically Nonlinear ◽  
Static Deflection ◽  
Nonlinear Formulation ◽  
Size Dependent ◽  
Dependent Theory ◽  
Piezoelectric Nanobeam ◽  
Special Case

AbstractNonlinear formulation of isotropic piezoelectric Euler-Bernoulli nano-beam is developed based on consistent size-dependent piezoelectricity theory. By considering geometrically nonlinear and axial displacement of the centroid of beam sections, basic nonlinear equations of piezoelectric nanobeam are derived using Hamilton's principle and variational method. Afterwards, in the special case for the formulation derived, hinged-hinged piezoelectric nanobeam is studied, and static deflection as well as free vibrations of the nanobeam under mechanical loads is determined. In this case, results of the linear formulation of the size-dependent theory are compared to those of the linear and nonlinear classical continuum theory.

USE OF STRAIN GRADIENT THEORY FOR MODELING THE SIZE-DEPENDENT PULL-IN OF ROTATIONAL NANO-MIRROR IN THE PRESENCE OF MOLECULAR FORCE

2013 ◽  
Vol 27 (18) ◽  
pp. 1350083 ◽  
Author(s):  
Y. TADI BENI ◽  
M. ABADYAN
Keyword(s):  
Strain Gradient ◽  
Continuum Theory ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Cross Sectional ◽  
Size Dependent ◽  
Proposed Model ◽  
Molecular Force ◽  
Intrinsic Material Length ◽  
Material Length

Experiments reveal that mechanical behavior of nanostructures is size-dependent. Herein, the size dependent pull-in instability of torsional nano-mirror is investigated using strain gradient nonclassic continuum theory. The governing equation of the mirror is derived taking the effect of electrostatic Coulomb and molecular van der Waals (vdW) forces into account. Variation of the rotation angle of the mirror as a function of the applied voltage is obtained and the instability parameters i.e., pull-in voltage and pull-in angle are determined. Nano-mirrors with square and circular cross-sectional beams are investigated as case studies. It is found that when the thickness of the torsional nano-beam is comparable with the intrinsic material length scales, size effect can substantially increase the instability parameters of the rotational mirror. Moreover, the effect of vdW forces on the size-dependent pull-in instability of the system is discussed. The proposed model is able to predict the experimental results more accurately than the previous classic models and reduce the gap between experiment and previous theories.

scholarly journals Plasticity of crystals and interfaces: From discrete dislocations to size-dependent continuum theory

2010 ◽  
Vol 37 (4) ◽  
pp. 289-332 ◽  
Author(s):  
Sinisa Mesarovic
Keyword(s):  
Unsolved Problem ◽  
Continuum Theory ◽  
Elastic Interaction ◽  
Interface Energy ◽  
Crystalline Materials ◽  
Size Dependent ◽  
Dislocation Mechanics ◽  
Pile Up ◽  
Slip Planes ◽  
Discrete Dislocations

In this communication, we summarize the current advances in size-dependent continuum plasticity of crystals, specifically, the rate-independent (quasistatic) formulation, on the basis of dislocation mechanics. A particular emphasis is placed on relaxation of slip at interfaces. This unsolved problem is the current frontier of research in plasticity of crystalline materials. We outline a framework for further investigation, based on the developed theory for the bulk crystal. The bulk theory is based on the concept of geometrically necessary dislocations, specifically, on configurations where dislocations pile-up against interfaces. The average spacing of slip planes provides a characteristic length for the theory. The physical interpretation of the free energy includes the error in elastic interaction energies resulting from coarse representation of dislocation density fields. Continuum kinematics is determined by the fact that dislocation pile-ups have singular distribution, which allows us to represent the dense dislocation field at the boundary as a superdislocation, i.e., the jump in the slip filed. Associated with this jump is a slip-dependent interface energy, which in turn, makes this formulation suitable for analysis of interface relaxation mechanisms.

Natural vibrations of micro beams with nonrigid supports

2016 ◽  
Vol 23 (19) ◽  
pp. 3233-3246 ◽  
Author(s):  
Diana V Bambill ◽  
Graciela I Guerrero ◽  
Daniel H Felix
Keyword(s):  
Boundary Conditions ◽  
Couple Stress Theory ◽  
Continuum Theory ◽  
Modified Couple Stress Theory ◽  
Free Vibrations ◽  
Small Scale ◽  
Stress Theory ◽  
Edge Conditions ◽  
Micro Systems ◽  
Micro Structures

The present study aims to provide some new information for the design of micro systems. It deals with free vibrations of Bernoulli–Euler micro beams with nonrigid supports. The study is based on the formulation of the modified couple stress theory. This theory is a nonclassical continuum theory that allows one to capture the small-scale size effects in the vibrational behavior of micro structures. More realistic boundary conditions are represented with elastic edge conditions. The effect of Poisson’s ratio on the micro beam characteristics is also analyzed. The present results revealed that the characterization of real boundary conditions is much more important for micro beams than for macro beams, and this is an assessment that cannot be ignored.

scholarly journals Calculation of plates in a geometrically nonlinear setting with the use of generalized equations of finite difference method

2018 ◽  
Vol 196 ◽  
pp. 01024 ◽  
Author(s):  
Natalia Uvarova ◽  
Radek Gabbasov
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Geometrically Nonlinear ◽  
Generalized Equations ◽  
Nonlinear Formulation ◽  
First Derivative ◽  
Equation Solving ◽  
Flexible Plates ◽  
Minimum Number ◽  
The Right

The article proposes a numerical method and an algorithm for analysis rectangular flexible plates in a geometrically nonlinear formulation. The generalized equations of the method of finite differences (MD) are used to solve the problem within the integrable region taking into account the discontinuities of the desired function, its first derivative and the right part of the original differential equation. Solving differential equations of the problem, composed with respect to the desired functions of deflection and stress are reduced to the 4th differential equations of the second order, which are solved numerically. As an example, a square plate loaded with a uniformly distributed load is considered. The results of the calculation with a minimum number of partitions are compared with the known analytical solution of A. S. Volmir [1] and indicate the possibility of using the numerical method for solving problems in a nonlinear formulation.

A Nonlinear Shear Deformable Nanoplate Model Including Surface Effects for Large Amplitude Vibrations of Rectangular Nanoplates with Various Boundary Conditions

2015 ◽  
Vol 07 (05) ◽  
pp. 1550076 ◽  
Author(s):  
Reza Ansari ◽  
Mostafa Faghih Shojaei ◽  
Vahid Mohammadi ◽  
Raheb Gholami ◽  
Mohammad Ali Darabi
Keyword(s):  
Boundary Conditions ◽  
Surface Stress ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Free Vibrations ◽  
Model Parameters ◽  
Geometrically Nonlinear ◽  
Residual Surface Stress ◽  
Various Boundary Conditions ◽  
Shear Deformable

In this paper, a geometrically nonlinear first-order shear deformable nanoplate model is developed to investigate the size-dependent geometrically nonlinear free vibrations of rectangular nanoplates considering surface stress effects. For this purpose, according to the Gurtin–Murdoch elasticity theory and Hamilton's principle, the governing equations of motion and associated boundary conditions of nanoplates are derived first. Afterwards, the set of obtained nonlinear equations is discretized using the generalized differential quadrature (GDQ) method and then solved by a numerical Galerkin scheme and pseudo arc-length continuation method. Finally, the effects of important model parameters including surface elastic modulus, residual surface stress, surface density, thickness and boundary conditions on the vibration characteristics of rectangular nanoplates are thoroughly investigated. It is found that with the increase of the thickness, nanoplates can experience different vibrational behavior depending on the type of boundary conditions.

Web-Stiffened Sandwich Structures

1971 ◽  
Vol 38 (4) ◽  
pp. 964-970 ◽  
Author(s):  
Y. N. Chen ◽  
D. Ranlet ◽  
J. Kempner
Keyword(s):  
Eigenfunction Expansion ◽  
Sandwich Beam ◽  
Continuum Theory ◽  
Free Vibrations ◽  
The Core ◽  
Rigid Frame ◽  
Discrete Nature ◽  
Comparison Of The Results ◽  
Energy Principles

A model representation of a web-cored sandwich structure, which is based upon the energy principles of continuum theory while retaining the discrete nature of the webs, is studied. The determination of a second-order shear modulus, which measures local bending effects, accounts for the transverse shear deformation of the core. This notion is exemplified through consideration of the flexure and buckling of a web-stiffened sandwich beam, and by comparison of the results calculated from rigid frame analysis with those based on the sandwich solution. Additionally, the technique for employing classical Bernoulli-Euler modes in the eigenfunction expansion of displacement functions is extended and applied to the problems of free vibrations of a free-free web-stiffened sandwich beam.

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