Size-dependent analysis of a functionally graded piezoelectric micro-cylinder based on the strain gradient theory with the consideration of flexoelectric effect: plane strain problem

2020 ◽  
Vol 42 (8) ◽  
Author(s):  
Ali Dini ◽  
Mahmoud Shariati ◽  
Fatemeh Zarghami ◽  
Mohammad Amin Nematollahi
Keyword(s):  
Plane Strain ◽  
Strain Gradient ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Plane Strain Problem ◽  
Size Dependent ◽  
Flexoelectric Effect ◽  
Functionally Graded Piezoelectric

Equilibrium equations and boundary conditions of strain gradient theory in arbitrary curvilinear coordinates

2014 ◽  
Vol 23 (5-6) ◽  
pp. 169-176
Author(s):  
Mikhail Guzev ◽  
Chengzhi Qi ◽  
Jiping Bai ◽  
Kairui Li
Keyword(s):  
Boundary Conditions ◽  
Plane Strain ◽  
Strain Gradient ◽  
Special Form ◽  
Curvilinear Coordinates ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Equilibrium Equations ◽  
Plane Strain Problem

AbstractEquilibrium equations and boundary conditions of the strain gradient theory in arbitrary curvilinear coordinates have been obtained. Their special form for an axisymmetric plane strain problem is also given.

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.

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.

scholarly journals An analytical study of vibration in functionally graded piezoelectric nanoplates: nonlocal strain gradient theory

2019 ◽  
Vol 40 (12) ◽  
pp. 1723-1740 ◽  
Author(s):  
Z. Sharifi ◽  
R. Khordad ◽  
A. Gharaati ◽  
G. Forozani
Keyword(s):  
Strain Gradient ◽  
Analytical Study ◽  
Natural Frequencies ◽  
Nonlocal Parameter ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Nonlocal Strain Gradient Theory ◽  
Functionally Graded Piezoelectric ◽  
Nonlocal Strain Gradient

AbstractIn this paper, we analytically study vibration of functionally graded piezoelectric (FGP) nanoplates based on the nonlocal strain gradient theory. The top and bottom surfaces of the nanoplate are made of PZT-5H and PZT-4, respectively. We employ Hamilton’s principle and derive the governing differential equations. Then, we use Navier’s solution to obtain the natural frequencies of the FGP nanoplate. In the first step, we compare our results with the obtained results for the piezoelectric nanoplates in the previous studies. In the second step, we neglect the piezoelectric effect and compare our results with those obtained for the functionally graded (FG) nanoplates. Finally, the effects of the FG power index, the nonlocal parameter, the aspect ratio, and the side-tothickness ratio, and the nanoplate shape on natural frequencies are investigated.

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