Mechanical Behavior Analysis of Micro-Scaled Functionally Graded Timoshenko Beams by the Strain Gradient Theory

2012 ◽  
Author(s):  
S. A. Tajalli ◽  
M. H. Kahrobaiyan ◽  
M. Rahaeifard ◽  
M. T. Ahmadian ◽  
M. R. Movahhedy ◽  
...  
Keyword(s):  
Strain Gradient ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Length Scale ◽  
Timoshenko Beams ◽  
New Model ◽  
Scale Parameters ◽  
Size Dependent ◽  
Special Cases

In this paper, a size-dependent formulation is developed for Timoshenko beams made of functionally graded materials (FGM). The developed formulation is based on the strain gradient theory; a non-classical continuum theory able to capture the size-effect in micro-scaled structures. Considering the material length scale parameters of the FG beams vary through the thickness, the new equivalent length scale parameters are proposed as functions of the constituents’ length scale parameters to describe the size-dependent static and dynamic behavior of FG microbeams. The governing differential equations of equilibrium and both classical and non-classical sets of boundary conditions are derived for the proposed strain gradient FG Timoshenko beam using variational approach. As case studies, the static bending deformation of the new model is investigated for an FG simply supported beam made of tungsten/copper (W/Cu) in which properties are varying through the thickness according to a power law. The results of the new model are compared to those of the modified couple stress and the classical theories where the two latter theories are special cases of the strain gradient theory.

Study of Small Scale Effects on the Nonlinear Vibration Response of Functionally Graded Timoshenko Microbeams Based on the Strain Gradient Theory

2012 ◽  
Vol 7 (3) ◽  
Author(s):  
R. Ansari ◽  
R. Gholami ◽  
S. Sahmani
Keyword(s):  
Strain Gradient ◽  
Beam Theory ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Length Scale ◽  
Scale Parameters ◽  
Linear And Nonlinear ◽  
Material Length Scale ◽  
Material Length

In the current study, the nonlinear free vibration behavior of microbeams made of functionally graded materials (FGMs) is investigated based on the strain gradient elasticity theory and von Karman geometric nonlinearity. The nonclassical beam model is developed in the context of the Timoshenko beam theory which contains material length scale parameters to take the size effect into account. The model can reduce to the beam models based on the modified couple stress theory (MCST) and the classical beam theory (CBT) if two or all material length scale parameters are taken to be zero, respectively. The power low function is considered to describe the volume fraction of the ceramic and metal phases of the FGM microbeams. On the basis of Hamilton’s principle, the higher-order governing differential equations are obtained which are discretized along with different boundary conditions using the generalized differential quadrature method. The dimensionless linear and nonlinear frequencies of microbeams with various values of material property gradient index are calculated and compared with those obtained based on the MCST and an excellent agreement is found. Moreover, comparisons between the various beam models on the basis of linear and nonlinear types of strain gradient theory (SGT) and MCST are presented and it is observed that the difference between the frequencies obtained by the SGT and MCST is more significant for lower values of dimensionless length scale parameter.

Nonlinear Pull-In Instability of Strain Gradient Microplates Made of Functionally Graded Materials

2019 ◽  
Vol 19 (02) ◽  
pp. 1950007 ◽  
Author(s):  
R. Gholami ◽  
R. Ansari ◽  
H. Rouhi
Keyword(s):  
Functionally Graded Materials ◽  
Strain Gradient ◽  
Scale Effects ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Small Scale ◽  
Gradient Theory ◽  
Graded Materials ◽  
First Order ◽  
Size Dependent

In this paper, the size-dependent nonlinear pull-in behavior of rectangular microplates made from functionally graded materials (FGMs) subjected to electrostatic actuation is numerically studied using a novel approach. The small scale effects are taken into account according to Mindlin’s first-order strain gradient theory (SGT). The plate model is formulated based on the first-order shear deformation theory (FSDT) using the virtual work principle. The size-dependent relations are derived in general form, which can be reduced to those based on different elasticity theories, including the modified strain gradient, modified couple stress and classical theories (MSGT, MCST and CT). The solution of the problem is arrived at by employing an efficient matrix-based method called the variational differential quadrature (VDQ). First, the quadratic form of the energy functional including the size effects is obtained. Then, it is discretized by the VDQ method using a set of matrix differential and integral operators. Finally, the achieved discretized nonlinear equations are solved by the pseudo arc-length continuation method. In the numerical results, the effects of material length scale parameters, side length-to-thickness ratio and FGM’s material gradient index on the nonlinear pull-in instability of microplates with different boundary conditions are investigated. A comparison is also made between the predictions by the MSGT, MCST and CT.

MODELING THE SIZE DEPENDENT STATIC AND DYNAMIC PULL-IN INSTABILITY OF CANTILEVER NANOACTUATOR BASED ON STRAIN GRADIENT THEORY

2014 ◽  
Vol 06 (05) ◽  
pp. 1450055 ◽  
Author(s):  
HAMID M. SEDIGHI ◽  
A. KOOCHI ◽  
M. ABADYAN
Keyword(s):  
Strain Gradient ◽  
Time History ◽  
Gradient Elasticity ◽  
Beam Theory ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Nanoscale Systems ◽  
Scale Parameters ◽  
Size Dependent ◽  
Electromechanical Instability

It is well-established that mechanical behavior of nanoscale systems is size dependent. In this paper, strain gradient elasticity theory is used for mathematical modeling of size dependent electromechanical instability of cantilever nanoactuator. The nanoactuator is modeled using Euler–Bernoulli beam theory and equation of motion is derived using Hamilton's principle. In order to solve the nonlinear governing equation, reduced order method (ROM) is employed. The dynamic pull-in instability of the nanoactuator is investigated through plotting the time history and phase portrait of the system. Static and dynamic pull-in voltage of nanoactuator as a function of dimensionless length scale parameters is determined. The obtained results show that when thickness of the nanoactuator is comparable with the intrinsic material length scales, size effect can substantially influence the pull-in behavior of the system.

Thermo-Mechanical Vibration Analysis of Size-Dependent Functionally Graded Micro-Beams with General Boundary Conditions

2018 ◽  
Vol 10 (08) ◽  
pp. 1850088 ◽  
Author(s):  
Zhu Su ◽  
Guoyong Jin ◽  
Lifeng Wang ◽  
Dan Wang
Keyword(s):  
Vibration Analysis ◽  
Mechanical Vibration ◽  
Beam Theory ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Penalty Function Method ◽  
Cosine Series ◽  
Scale Parameters ◽  
Size Dependent

A unified formulation for thermo-mechanical vibration analysis of size-dependent Timoshenko micro-beams comprised of functionally graded materials (FGMs) with general restraints is presented. The size effect is considered by incorporating the modified strain gradient theory into Timoshenko beam theory. The thermal and mechanical properties of FGMs are related to temperature and are assumed as continuous variation along the thickness. The Mori–Tanaka estimate is used for calculation of the material properties of FGM micro-beam. The formulation is deduced on the basis of the variational principle combined with penalty function method. The displacements and rotation of the FGM micro-beam are uniformly expanded by a modified Fourier series composed of traditional cosine series and some appropriate supplementary functions. Several comparisons of the present solutions with those from existing literature confirm the validity of the current formulation. In addition, a parametric study is given to demonstrate the influence of length scale parameters, gradient indices, end restraints and temperature changes on vibration characteristic of functionally graded micro-beam.

Size-dependent dynamic modeling of inhomogeneous curved nanobeams embedded in elastic medium based on nonlocal strain gradient theory

2016 ◽  
Vol 231 (23) ◽  
pp. 4457-4469 ◽  
Author(s):  
Farzad Ebrahimi ◽  
Mohammad R Barati
Keyword(s):  
Stress Field ◽  
Elastic Medium ◽  
Strain Gradient ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Beam Model ◽  
Simply Supported ◽  
Size Dependent ◽  
Nonlocal Strain Gradient

In this paper, size-dependent free vibration analysis of curved functionally graded nanobeams embedded in Winkler–Pasternak elastic medium is carried out via an analytical solution method. Three kinds of boundary condition namely, simply supported-simply supported, simply supported-clamped and clamped-clamped are investigated. Material properties of curved functionally graded beam change in thickness direction according to the Mori–Tanaka model. Nonlocal strain gradient elasticity theory is adopted to capture the size effects in which the stress is considered for not only the nonlocal stress field but also the strain gradients stress field. Nonlocal governing equations of curved functionally graded nanobeam are obtained from Hamilton’s principle based on Euler–Bernoulli beam model. Finally, the influences of length scale parameter, nonlocal parameter, opening angle, elastic medium, material composition, slenderness ratio and boundary conditions on the vibrational characteristics of nanosize curved functionally graded beams are explored.

Size-Dependent elastic response in functionally graded microbeams considering generalized first strain gradient elasticity

2019 ◽  
Vol 72 (3) ◽  
pp. 273-304 ◽  
Author(s):  
S Sidhardh ◽  
M C Ray
Keyword(s):  
Exact Solutions ◽  
Power Law ◽  
Strain Gradient ◽  
Size Effects ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Elastic Response ◽  
Gradient Theory ◽  
Size Dependent ◽  
Exponentially Graded

Summary In this article, the size-dependent mechanical response of an isotropic functionally graded (FG) microbeam has been investigated. The size-effects over the elastic response have been modeled by the Mindlin–Toupin strain gradient theory, with the coefficients evaluated from the generalized first strain gradient theory of elasticity. In order to facilitate the derivation of the exact solutions to the governing differential equations of equilibrium, an exponentially graded FG beam is chosen. These exact solutions are derived for a simply supported beam subjected to a sinusoidally distributed mechanical load. Following this, an element-free Galerkin (EFG) model involving moving least squares interpolations across the domain is also developed here. The EFG model is validated with the exact solutions for the exponentially graded beam. Finally, the EFG model is extended to the more general case of a power law-graded beam. The mechanical responses for the power law-graded beams under various loading and boundary conditions are presented here. These results may serve as benchmark for further studies over size-effects in FG beams.

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