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.

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.

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.

scholarly journals Effect of Kerr Foundation and in-Plane Forces on Free Vibration of FGM Nanobeams with Diverse Distribution of Porosity

2020 ◽  
Vol 14 (3) ◽  
pp. 135-143
Author(s):  
Piotr Jankowski
Keyword(s):  
Porous Material ◽  
Constitutive Relations ◽  
Equations Of Motion ◽  
Strain Tensor ◽  
Beam Theory ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Graphical Form ◽  
Solution Technique ◽  
Gradient Theory

Abstract In the present paper, the effect of diverse distribution of functionally graded porous material and Kerr elastic foundation on natural vibrations of nanobeams subjected to in-plane forces is investigated based on the nonlocal strain gradient theory. The displacement field of the nanobeam satisfies assumptions of Reddy higher-order shear deformation beam theory. All the displacements gradients are assumed to be small, then the components of the Green-Lagrange strain tensor are linear and infinitesimal. The constitutive relations for functionally graded (FG) porous material are expressed by nonlocal and length scale parameters and power-law variation of material parameters in conjunction with cosine functions. It created possibility to investigate an effect of functionally graded materials with diverse distribution of porosity and volume of voids on mechanics of structures in nano scale. The Hamilton’s variational principle is utilized to derive governing equations of motion of the FG porous nanobeam. Analytical solution to formulated boundary value problem is obtained in closed-form by using Navier solution technique. Validation of obtained results and parametric study are presented in tabular and graphical form. Influence of axial tensile/compressive forces and three different types of porosity distribution as well as stiffness of Kerr foundation on natural frequencies of functionally graded nanobeam is comprehensively studied.

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.

scholarly journals A Refined Theory for Bending Vibratory Analysis of Thick Functionally Graded Beams

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1422
Author(s):  
Youssef Boutahar ◽  
Nadhir Lebaal ◽  
David Bassir
Keyword(s):  
Beam Theory ◽  
Functionally Graded ◽  
Effective Characteristics ◽  
Hyperbolic Function ◽  
Transverse Displacement ◽  
Refined Theory ◽  
Distribution Law ◽  
Fg Beam ◽  
The Impact ◽  
Beam Theories

A refined beam theory that takes the thickness-stretching into account is presented in this study for the bending vibratory behavior analysis of thick functionally graded (FG) beams. In this theory, the number of unknowns is reduced to four instead of five in the other approaches. Transverse displacement is expressed through a hyperbolic function and subdivided into bending, shear, and thickness-stretching components. The number of unknowns is reduced, which involves a decrease in the number of the governing equation. The boundary conditions at the top and bottom FG beam faces are satisfied without any shear correction factor. According to a distribution law, effective characteristics of FG beam material change continuously in the thickness direction depending on the constituent’s volume proportion. Equations of motion are obtained from Hamilton’s principle and are solved by assuming the Navier’s solution type, for the case of a supported FG beam that is transversely loaded. The numerical results obtained are exposed and analyzed in detail to verify the validity of the current theory and prove the influence of the material composition, geometry, and shear deformation on the vibratory responses of FG beams, showing the impact of normal deformation on these responses which is neglected in most of the beam theories. The obtained results are compared with those predicted by other beam theories. It can be concluded that the present theory is not only accurate but also simple in predicting the bending and free vibration responses of FG beams.

A nonlocal strain gradient theory for rotating thermo-mechanical characteristics on magnetically actuated viscoelastic functionally graded nanoshell

2020 ◽  
Vol 31 (12) ◽  
pp. 1511-1523
Author(s):  
Mohammad Mahinzare ◽  
Hossein Akhavan ◽  
Majid Ghadiri
Keyword(s):  
Boundary Conditions ◽  
Strain Gradient ◽  
Equations Of Motion ◽  
Nonlocal Parameter ◽  
Functionally Graded ◽  
Smart Structure ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Nonlocal Strain Gradient Theory ◽  
Nonlocal Strain Gradient

In this article, a first-order shear deformable model is expanded based on the nonlocal strain gradient theory to vibration analysis of smart nanostructures under different boundary conditions. The governing equations of motion of rotating magneto-viscoelastic functionally graded cylindrical nanoshell in the magnetic field and corresponding boundary conditions are obtained using Hamilton’s principle. To discretize the equations of motion, the generalized differential quadrature method is applied. The aim of this work is to investigate the effects of the temperature changes, nonlocal parameter, material length scale, viscoelastic coefficient, various boundary conditions, and the rotational speed of this smart structure on natural frequencies of rotating cylindrical nanoshell made of magneto-viscoelastic functionally graded material.

Sign in / Sign up

Export Citation Format

Share Document