EXACT BOUNDARY CONDITIONS FOR BUCKLING ANALYSIS OF RECTANGULAR MICRO-PLATES BASED ON THE MODIFIED STRAIN GRADIENT THEORY

2015 ◽  
Vol 13 (3) ◽  
pp. 265-280 ◽  
Author(s):  
Meisam Mohammadi ◽  
Majid Fooladi ◽  
Hossein Darijani
Keyword(s):  
Boundary Conditions ◽  
Strain Gradient ◽  
Buckling Analysis ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Modified Strain Gradient Theory ◽  
Exact Boundary ◽  
Micro Plates

Free nonlinear vibration analysis of nano-truncated conical shells based on modified strain gradient theory

2021 ◽  
pp. 146442072110419
Author(s):  
Alireza Sheykhi ◽  
Shahrokh Hosseini-Hashemi ◽  
Adel Maghsoudpour ◽  
Shahram E Haghighi
Keyword(s):  
Boundary Conditions ◽  
Strain Gradient ◽  
Couple Stress ◽  
Equations Of Motion ◽  
Differential Quadrature ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Modified Strain Gradient Theory ◽  
Conical Shells ◽  
Modified Couple Stress

In this study, the nonlinear free vibrations behaviour of nano-truncated conical shells was analysed, using the first-order shear deformable shell model. The analysis took into account the structure size through modified strain gradient theory, and differential quadrature and Fréchet derivative methods in von Kármán-Donnell-type approach to kinematic nonlinearity. The governing equations were obtained, utilizing Hamilton's principle. Partial differential equations plus the non-classical and classical boundary conditions were used to obtain the shells’ equations of motion. Discretizing the boundary conditions and equations of motion were performed based on a generalized differential quadrature analogy. The eigenvalue system was considered based on the harmonic balance technique. The Galerkin and Fréchet derivative approaches were used to determine the nonlinear free vibration behaviour of the carbon nano-cone, which was modelled in the simply- and clamped-supported boundary conditions. Comparisons were made between the findings from the new model versus the couple and classical stress theories, indicating that the classical and modified couple stress theories are distinct representations of modified strain gradient theory. The results also revealed that the degree of hardening of nano-truncated conical shells in the modified strain gradient theory is less than that of modified couple stress and classical theories. This led to a rise in the non-dimensional amplitude and frequency ratios. This study investigated the effect of size on free nonlinear vibrations of nano-truncated conical shells for various apex angles and lengths. Finally, we evaluated and compared our findings versus those reported by previous studies, which confirmed the precision and accuracy of our results.

scholarly journals Strain Gradient Theory Based Dynamic Mindlin-Reissner and Kirchhoff Micro-Plates with Microstructural and Micro-Inertial Effects

Dynamics ◽  
2021 ◽  
Vol 1 (1) ◽  
pp. 49-94
Author(s):  
Stylianos Markolefas ◽  
Dimitrios Fafalis
Keyword(s):  
Boundary Conditions ◽  
Strain Gradient ◽  
Virtual Work ◽  
Plate Thickness ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Static Case ◽  
Cauchy Stress ◽  
Gradient Effect ◽  
Micro Plates

In this study, a dynamic Mindlin–Reissner-type plate is developed based on a simplified version of Mindlin’s form-II first-strain gradient elasticity theory. The governing equations of motion and the corresponding boundary conditions are derived using the general virtual work variational principle. The presented model contains, apart from the two classical Lame constants, one additional microstructure material parameter g for the static case and one micro-inertia parameter h for the dynamic case. The formal reduction of this model to a Kirchhoff-type plate model is also presented. Upon diminishing the microstructure parameters g and h, the classical Mindlin–Reissner and Kirchhoff plate theories are derived. Three points distinguish the present work from other similar published in the literature. First, the plane stress assumption, fundamental for the development of plate theories, is expressed by the vanishing of the z-component of the generalized true traction vector and not merely by the zz-component of the Cauchy stress tensor. Second, micro-inertia terms are included in the expression of the kinetic energy of the model. Finally, the detailed structure of classical and non-classical boundary conditions is presented for both Mindlin–Reissner and Kirchhoff micro-plates. An example of a simply supported rectangular plate is used to illustrate the proposed model and to compare it with results from the literature. The numerical results reveal the significance of the strain gradient effect on the bending and free vibration response of the micro-plate, when the plate thickness is at the micron-scale; in comparison to the classical theories for Mindlin–Reissner and Kirchhoff plates, the deflections, the rotations, and the shear-thickness frequencies are smaller, while the fundamental flexural frequency is higher. It is also observed that the micro-inertia effect should not be ignored in estimating the fundamental frequencies of micro-plates, primarily for thick plates, when plate thickness is at the micron scale (strain gradient effect).

Exact solution for bending analysis of functionally graded micro-plates based on strain gradient theory

2018 ◽  
Vol 25 (3) ◽  
pp. 439-451
Author(s):  
Meisam Mohammadi ◽  
Afshin Iranmanesh ◽  
Seyed Sadegh Naseralavi ◽  
Hamed Farahmand
Keyword(s):  
Boundary Conditions ◽  
Strain Gradient ◽  
Material Properties ◽  
Plate Theory ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Governing Equations ◽  
Micro Plates ◽  
Micro Structures

Abstract In the present article, static analysis of thin functionally graded micro-plates, based on Kirchhoff plate theory, is investigated. Utilizing the strain gradient theory and principle of minimum total potential energy, governing equations of rectangular micro-plates, subjected to distributed load, are explored. In accordance with functionally graded distribution of material properties through the thickness, higher-order governing equations are coupled in terms of displacement fields. Introducing a novel methodology, governing equations are decoupled, with special privilege of solving analytically. These new equations are solved for micro-plates with Levy boundary conditions. It is shown that neutral plane in functionally graded micro-plate is moved from midplane to a new coordinate in thickness direction. It is shown that considering micro-structures effects affects the governing equations and boundary conditions. Finally, the effects of material properties, micro-structures, boundary conditions and dimensions are expounded on the static response of micro-plate. Results show that increasing the length scale parameter and FGM index increases the rigidity of micro-plate. In addition, it is concluded that using classical theories for study of micro-structures leads to inaccurate results.

On the size-dependent vibration of the embedded double-walled carbon nanotube conveying fluid using shell model

2018 ◽  
pp. 66-77
Author(s):  
Yaghoub Tadi Beni
Keyword(s):  
Carbon Nanotube ◽  
Size Effect ◽  
Boundary Conditions ◽  
Strain Gradient ◽  
Shell Theory ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Pasternak Foundation ◽  
Modified Strain Gradient Theory ◽  
Conveying Fluid

In this paper, the vibration and instability of double-walled carbon nanotube (DWCNT) conveying fluid were investigated by using the modified strain gradient theory. The Donnell's shell theory was used by taking into consideration the three size effects and simply-supported boundary conditions. The effect of van der Waals force between the two intended walls and the surroundings of the DWCNT was modelled as visco-Pasternak foundation. The governing equations of the problem and boundary conditions were derived from Hamilton’s principle. Also, Navier procedure was used to solve the vibration problem. To verify the results, a comparison was drawn between the results of this study and those of the references. According to the findings, the effects of fluid velocity, stiffness and damping of visco-Pasternak foundation, length of DWCNT and size effect are more considerable in the modified strain gradient theory than in the modified couple stress theory and classical theory. Keywords: Modified strain gradient theory, Donnell’s shell theory, DWCNT, size effect, shell vibration.

Free transverse vibration analysis of orthotropic multi-viscoelastic microplate system embedded in visco-Pasternak medium via modified strain gradient theory

2017 ◽  
Vol 21 (1) ◽  
pp. 175-210 ◽  
Author(s):  
A Jamalpoor ◽  
M Bahreman ◽  
M Hosseini
Keyword(s):  
Boundary Conditions ◽  
Strain Gradient ◽  
Natural Frequencies ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Modified Strain Gradient Theory ◽  
Free Transverse Vibration ◽  
Foundation Parameters

In this paper, an analytical process is proposed to investigate the size-dependent free vibration of orthotropic multi-viscoelastic microplate systems (OMVMPS) embedded in Kelvin–Voigt visco-Pasternak medium according to the modified strain gradient theory. Governing equations of motion in the partial form and the related boundary conditions are derived by utilizing the Kirchhoff plate theory and Hamilton’s variational principle. The two different sorts of “chain” boundary conditions like “clamped Chain” and “free chain” systems are considered for the ends of microplate system. Navier’s method, which convinces that the simply supported boundary conditions and trigonometric methods are applied to analytically investigate the size effect of the natural frequencies of OMVMPS. The numerical outcomes are offered to report the variation of OMVMPS natural frequencies with the numerous amounts of the microplate numbers, the length scale parameter, aspect ratio, visco-Pasternak foundation parameters, the thickness of microplate, and higher modes number. Several numerical outcomes of this research depict that when the number of microplates is low, there is a significant distinction between natural frequencies achieved for “clamped chain” and “free chain” systems. Also, it is demonstrated that by increasing the number of microplates, the effect of the visco-Pasternak substrate on the natural frequency of system vibration decreases.

A novel model and solution on the bending problem of arbitrary shaped magnetoelectroelastic plates based on the modified strain gradient theory

2021 ◽  
pp. 1045389X2110411
Author(s):  
Yating Han ◽  
Zhen Yan ◽  
Ji Lin ◽  
Wenjie Feng
Keyword(s):  
Boundary Conditions ◽  
Strain Gradient ◽  
Magnetic Coupling ◽  
Current Method ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
General Applicability ◽  
Modified Strain Gradient Theory ◽  
Size Dependent ◽  
Novel Model

A size-dependent magnetoelectroelastic (MEE) plate bending model is established where the governing equations and concrete forms of three different mechanical boundary conditions under modified strain gradient theory are derived by the variational principle. Then, the meshless method of polynomial particular solutions is further developed to solve this bending problem. Finally, the influences of size effect, mechanical-electric-magnetic coupling loads, and Pasternak foundation on the bending properties of MEE plates are detailed discussed by some typical numerical examples. Of importance, by virtue of the general applicability and superior flexibility of current method, the bending analyses of MEE plates under different mechanical boundary conditions and geometrical shapes can be carried out, and some novel conclusions are concluded.

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