Stability of Vibrating Functionally Graded Nanoplates with Axial Motion Based on the Nonlocal Strain Gradient Theory

2020 ◽  
Vol 20 (08) ◽  
pp. 2050088 ◽  
Author(s):  
J. P. Shen ◽  
P. Y. Wang ◽  
W. T. Gan ◽  
C. Li
Keyword(s):  
Strain Gradient ◽  
Critical Speed ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Length Scale ◽  
Gradient Index ◽  
Axial Motion ◽  
Material Length Scale ◽  
Material Length ◽  
Nonlocal Strain Gradient

Considering both the nonlocal scale and material length scale effects, we investigate vibration and stability behaviors of functionally graded nanoplates with axial motion in order to model the two-dimensional nanobelt in nanoengineering. The nonlocal strain gradient theory is applied and the differential nonlocal strain gradient constitutive relation is adopted. Using the physical neutral plane of a functionally graded thin plate, we derive the governing equation of motion for the functionally graded nanoplate with axial motion via Hamilton’s principle, where the kinematic characteristics are introduced into the dynamic behaviors. The governing equation is numerically solved using the Galerkin method. Effects of the nonlocal scale and material length scale parameters, axial velocity, gradient index, biaxial pre-tensions and aspect ratio are discussed. The results demonstrate that complex frequencies of the functionally graded nanoplate with axial motion decrease with an increase of axial velocity in the subcritical region, while the moving nanoplate experiences a divergent instability or flutter instability in the supercritical region. Natural frequency and critical speed decrease with the increase of the nonlocal scale parameter while increase with the increase of the material length scale parameter, reflecting the nonlocal softening and strain gradient hardening mechanisms, respectively. Besides, natural frequency and critical speed increase with the increase of the biaxial pre-tensions and aspect ratio, but decrease with the increase of the gradient index. In particular, the influences of the gradient index and size or weight of functionally graded nanoplates on the critical speed are explored.

scholarly journals A nonlocal strain gradient shell model incorporating surface effects for vibration analysis of functionally graded cylindrical nanoshells

2019 ◽  
Vol 40 (12) ◽  
pp. 1695-1722 ◽  
Author(s):  
Lu Lu ◽  
Li Zhu ◽  
Xingming Guo ◽  
Jianzhong Zhao ◽  
Guanzhong Liu
Keyword(s):  
Strain Gradient ◽  
Scale Parameter ◽  
Nonlocal Parameter ◽  
Functionally Graded ◽  
Length Scale ◽  
Proposed Model ◽  
Material Length Scale Parameter ◽  
Material Length Scale ◽  
Material Length ◽  
Nonlocal Strain Gradient

AbstractIn this paper, a novel size-dependent functionally graded (FG) cylindrical shell model is developed based on the nonlocal strain gradient theory in conjunction with the Gurtin-Murdoch surface elasticity theory. The new model containing a nonlocal parameter, a material length scale parameter, and several surface elastic constants can capture three typical types of size effects simultaneously, which are the nonlocal stress effect, the strain gradient effect, and the surface energy effects. With the help of Hamilton’s principle and first-order shear deformation theory, the non-classical governing equations and related boundary conditions are derived. By using the proposed model, the free vibration problem of FG cylindrical nanoshells with material properties varying continuously through the thickness according to a power-law distribution is analytically solved, and the closed-form solutions for natural frequencies under various boundary conditions are obtained. After verifying the reliability of the proposed model and analytical method by comparing the degenerated results with those available in the literature, the influences of nonlocal parameter, material length scale parameter, power-law index, radius-to-thickness ratio, length-to-radius ratio, and surface effects on the vibration characteristic of functionally graded cylindrical nanoshells are examined in detail.

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.

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.

Localization-Induced Band and Cohesive Model1

2000 ◽  
Vol 67 (4) ◽  
pp. 803-812 ◽  
Author(s):  
S. Hao ◽  
W. K. Liu ◽  
D. Qian
Keyword(s):  
Strain Gradient ◽  
Multiple Scale ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Length Scale ◽  
Finite Width ◽  
Opening Displacement ◽  
One Dimensional ◽  
Material Length Scale ◽  
Material Length

A localization-induced cohesive model has been proposed for shear band evolution, crack growth, and fracture. Strain gradient theory has been applied to establish the criterion of the onset of localization and the governing equation in the post-bifurcation stage. Analytical solutions in one-dimensional case are used to establish the “traction-separation” law, in which strain gradient and material intrinsic length scale present strong effects. In addition, the solution predicts a finite width for the localization-induced band. It is observed that a larger length scale contributes to the growth of a larger width of localization region and separation for softening materials. The proposed model provides a procedure to establish the fracture toughness analytically since the material length scale is taken into account. From the traction-separation analysis, it is found that damage decreases separation, whereas an increase in material length scale increases the opening displacement; however, the traction-normalized opening displacement curves (with respect to the material length scale) are identical. Based on the methodology of multiple scale analysis in meshfree method, a computational approach has been proposed to enrich the one-dimensional traction-separation law to define fracture. [S0021-8936(00)01104-1]

scholarly journals Flow-induced vibration and stability analysis of carbon nanotubes based on the nonlocal strain gradient Timoshenko beam theory

2018 ◽  
Vol 25 (1) ◽  
pp. 203-218 ◽  
Author(s):  
Reza Bahaadini ◽  
Ali Reza Saidi ◽  
Mohammad Hosseini
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Timoshenko Beam ◽  
Strain Gradient ◽  
Size Effects ◽  
Length Scale ◽  
Governing Equations ◽  
Material Length Scale ◽  
Material Length ◽  
Nonlocal Strain Gradient

A nonlocal strain gradient Timoshenko beam model is developed to study the vibration and instability analysis of the carbon nanotubes conveying nanoflow. The governing equations of motion and boundary conditions are derived by employing Hamilton’s principle, including the effects of moving fluid, material length scale and nonlocal parameters, Knudsen number and gravity force. The material length scale and nonlocal parameters are considered, in order to take into account the size effects. Also, to consider the small-size effects on the flow field, the Knudsen number is used as a discriminant parameter. The Galerkin approach is chosen to analyze the governing equations under clamped–clamped, clamped–hinged and hinged–hinged boundary conditions. It is found that the natural frequency and critical fluid velocity can be decreased by increasing the nonlocal parameter or decreasing the material length scale parameter. Furthermore, it is revealed that the critical flow velocity does not affected by two size-dependent parameters and various boundary conditions in the free molecular flow regime.

A simple size-dependent isogeometric approach for bending analysis of functionally graded microplates using the modified strain gradient elasticity theory

2020 ◽  
Vol 42 (3) ◽  
pp. 255-267
Author(s):  
Chien H. Thai ◽  
H. Nguyen-Xuan
Keyword(s):  
Strain Gradient ◽  
Isogeometric Analysis ◽  
Deformation Theory ◽  
Shear Deformation Theory ◽  
Functionally Graded ◽  
Length Scale ◽  
First Order ◽  
Size Dependent ◽  
Material Length Scale ◽  
Material Length

In this study, a simple size-dependent isogeometric approach for bending analysis of functionally graded (FG) microplates using the modified strain gradient theory (MSGT), simple first-order shear deformation theory (sFSDT) and isogeometric analysis is presented for the first time. The present approach reduces one variable when comparing with the original first-order shear deformation theory (FSDT) within five variables and only considers three material length scale parameters (MLSPs) to capture size effects. Effective material properties as Young’s modulus, Poisson’s ratio and density mass are computed by a rule of mixture. Thanks to the principle of virtual work, the essential equations which are solved by the isogeometric analysis method, are derived. Rectangular and circular FG microplates with different boundary conditions, volume fraction and material length scale parameter are exampled to evaluate the deflections of FG microplates.

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.

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