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.

Modified strain gradient theory for nonlinear vibration analysis of functionally graded piezoelectric doubly curved microshells

2021 ◽  
pp. 095440622110458
Author(s):  
Vahid Movahedfar ◽  
Mohammad M Kheirikhah ◽  
Younes Mohammadi ◽  
Farzad Ebrahimi
Keyword(s):  
Nonlinear Vibration ◽  
Material Properties ◽  
Vibration Analysis ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Modified Strain Gradient Theory ◽  
Scale Parameters ◽  
Functionally Graded Piezoelectric ◽  
Doubly Curved

Based on modified strain gradient theory, nonlinear vibration analysis of a functionally graded piezoelectric doubly curved microshell in thermal environment has been performed in this research. Three scale parameters have been included in the modeling of thin doubly curved microshell in order to capture micro-size effects. Graded material properties between the top and bottom surfaces of functionally graded piezoelectric doubly curved microshell have been considered via incorporating power-law model. It is also assumed that the microshell is exposed to a temperature field of uniform type and the material properties are temperature-dependent. By analytically solving the governing equations based on the harmonic balance method, the closed form of nonlinear vibration frequency has been achieved. Obtained results indicate the relevance of calculated frequencies to three scale parameters, material gradation, electrical voltage, curvature radius, and temperature changes.

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.

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.

scholarly journals Transverse shear and normal deformation effects on vibration behaviors of functionally graded micro-beams

2020 ◽  
Vol 41 (9) ◽  
pp. 1303-1320
Author(s):  
Zhu Su ◽  
Lifeng Wang ◽  
Kaipeng Sun ◽  
Jie Sun
Keyword(s):  
Transverse Shear ◽  
Three Dimensional ◽  
Beam Theory ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Penalty Function Method ◽  
Transverse Displacement ◽  
Three Dimensional Model ◽  
Special Cases

Abstract A quasi-three dimensional model is proposed for the vibration analysis of functionally graded (FG) micro-beams with general boundary conditions based on the modified strain gradient theory. To consider the effects of transverse shear and normal deformations, a general displacement field is achieved by relaxing the assumption of the constant transverse displacement through the thickness. The conventional beam theories including the classical beam theory, the first-order beam theory, and the higherorder beam theory are regarded as the special cases of this model. The material properties changing gradually along the thickness direction are calculated by the Mori-Tanaka scheme. The energy-based formulation is derived by a variational method integrated with the penalty function method, where the Chebyshev orthogonal polynomials are used as the basis function of the displacement variables. The formulation is validated by some comparative examples, and then the parametric studies are conducted to investigate the effects of transverse shear and normal deformations on vibration behaviors.

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.

On size-dependent bending behaviors of shape memory alloy microbeams via nonlocal strain gradient theory

2021 ◽  
pp. 1045389X2098699
Author(s):  
Bo Zhou ◽  
Zetian Kang ◽  
Xiao Ma ◽  
Shifeng Xue
Keyword(s):  
Shape Memory Alloy ◽  
Shape Memory ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Size Dependent ◽  
Material Length Scale Parameter ◽  
Material Length Scale ◽  
Material Length ◽  
Nonlocal Strain Gradient

This paper focuses on the size-dependent behaviors of functionally graded shape memory alloy (FG-SMA) microbeams based on the Bernoulli-Euler beam theory. It is taken into consideration that material properties, such as austenitic elastic modulus, martensitic elastic modulus and critical transformation stresses vary continuously along the longitudinal direction. According to the simplified linear shape memory alloy (SMA) constitutive equations and nonlocal strain gradient theory, the mechanical model was established via the principle of virtual work. Employing the Galerkin method, the governing differential equations were numerically solved. The functionally graded effect, nonlocal effect and size effect of the mechanical behaviors of the FG-SMA microbeam were numerically simulated and discussed. Results indicate that the mechanical behaviors of FG-SMA microbeams are distinctly size-dependent only when the ratio of material length scale parameter to the microbeam height is small enough. Both the increments of material nonlocal parameter and ratio of material length-scale parameter to the microbeam height all make the FG-SMA microbeam become softer. However, the stiffness increases with the increment of FG parameter. The FG parameter plays an important role in controlling the transverse deformation of the FG-SMA microbeam. This work can provide a theoretical basis for the design and application of FG-SMA microstructures.

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