scholarly journals Bending, buckling and Vibrations Analysis of the graphene nanoplate using the modified couple stress theory

Mechanika ◽  
2021 ◽  
Vol 27 (5) ◽  
pp. 376-384
Author(s):  
Majid Eskandari Shahraki ◽  
Mahmoud Shariati ◽  
Mohsen Heydari Beni ◽  
Jafar Eskandari Jam
Keyword(s):  
Stress Tensor ◽  
Couple Stress ◽  
Strain Energy ◽  
Couple Stress Theory ◽  
Strain Tensor ◽  
Modified Couple Stress Theory ◽  
External Work ◽  
Stress Theory ◽  
Modified Couple Stress ◽  
Graphene Nanoplates

In this paper using the modified couple stress theory, to study the bending, buckling and vibration characteristics of the rectangular Mindlin's nanoplates with graphene material was investigated. With the aim of considering the effects of small scales, the modified couple stress theory, which has only one parameter of length scale and also was presented by Yang in 2002, was used. In the modified couple stress theory; the strain energy density is a function of the components of the strain tensor, curvature tensor, stress tensor and the symmetric part of the couple stress tensor. After obtaining the strain energy, external work, and buckling equation and placing them in the Hamilton's equation, the basic and auxiliary equations of the nanoplates were obtained. Then, by applying boundary and force conditions in the governing equations, the bending, buckling and vibration of the rectangular graphene nanoplates with thickness h and simply-supported conditions were explored. Also, the solution method was the Navier's solution.

scholarly journals An Assessment of Couple Stress Theories

2018 ◽  
Author(s):  
Ali R. Hadjesfandiari ◽  
Gary F. Dargush
Keyword(s):  
Stress Tensor ◽  
Continuum Mechanics ◽  
Couple Stress ◽  
Couple Stress Theory ◽  
Modified Couple Stress Theory ◽  
Two Dimensional ◽  
Stress Theory ◽  
Size Dependent ◽  
Modified Couple Stress ◽  
Beam Bending

In this paper, we examine the mathematical and physical consistencies of the three primary couple stress theories: original Mindlin-Tiersten-Koiter couple stress theory (MTK-CST), modified couple stress theory (M-CST) and consistent couple stress theory (C-CST).  As has been known for many years, MTK-CST suffers from some fundamental inconsistencies, such as the indeterminacy of the couple-stress tensor.  Therefore, despite the fact that MTK-CST has a fundamental position in the evolution of size-dependent continuum mechanics, it is not a reliable theory within continuum mechanics, for example, in developing new size-dependent multi-physics formulations.  We also observe that M-CST not only inherits all inconsistences from the original MTK-CST, but also suffers from new additional inconsistencies, such as the introduction of a new non-physical governing equation.  These inconsistencies refute the claim of those who state that the couple-stress tensor may be chosen symmetric.  Therefore, the apparent success of MTK-CST and M-CST in describing a size-effect for some problems, such as two-dimensional plate and beam bending, is not enough to justify these theories as suitable for general cases.  In fact, the symmetric couple-stresses in M-CST create torsional or anticlastic deformation, not bending.  On the other hand, C-CST, with a skew-symmetric couple-stress tensor, is the consistent continuum mechanics suitable for solving different size-dependent solid, fluid and multi-physics problems. 

Size-dependent free vibration analysis of a three-layered exponentially graded nano-/micro-plate with piezomagnetic face sheets resting on Pasternak’s foundation via MCST

2017 ◽  
Vol 22 (1) ◽  
pp. 55-86 ◽  
Author(s):  
Mohammad Arefi ◽  
Masoud Kiani ◽  
Ashraf M Zenkour
Keyword(s):  
Couple Stress ◽  
Couple Stress Theory ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Modified Couple Stress Theory ◽  
Stress Theory ◽  
Modified Couple Stress ◽  
Material Length Scale ◽  
Material Length ◽  
Exponentially Graded

The present work is devoted to the free vibration analysis of elastic three-layered nano-/micro-plate with exponentially graded core and piezomagnetic face-sheets using the modified couple stress theory. To capture size-dependency for a nano-/micro-sized rectangular plate, the couple stress theory is used as a non-classical continuum theory. The rectangular elastic three-layered nano-/micro-plate is resting on Pasternak’s foundation. The present model contains one material length scale parameter and can capture the size effect. Material properties of the core are supposed to vary along the thickness direction based on the exponential function. The governing equations of motion are derived from Hamilton’s principle based on the modified couple stress theory and first-order shear deformation theory. The analytical solution is presented to solve seven governing equations of motion using Navier’s solution. Eventually the natural frequency is scrutinized for different side length ratio, thickness ratio, inhomogeneity parameter, material length scale, and parameters of foundation numerically.

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