Refined Nonconforming Finite Element Methods in Couple Stress/Strain Gradient Theory

2011 ◽  
Vol 5 (2) ◽  
pp. 139-154
Author(s):  
Chen Wanji ◽  
Ma Xu ◽  
Li Li
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Strain Gradient ◽  
Couple Stress ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Nonconforming Finite Element ◽  
Stress Strain ◽  
Nonconforming Finite Element Methods

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.

Resonant Frequency and Sensitivity of a Caliper Formed With Assembled Cantilever Probes Based on the Modified Strain Gradient Theory

2014 ◽  
Vol 20 (6) ◽  
pp. 1672-1681 ◽  
Author(s):  
Mohammad Abbasi ◽  
Seyed E. Afkhami
Keyword(s):  
Resonant Frequency ◽  
Strain Gradient ◽  
Couple Stress ◽  
Contact Stiffness ◽  
Current Model ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
The Difference ◽  
Material Length Scale ◽  
Beam Theories

AbstractThe resonant frequency and sensitivity of an atomic force microscope (AFM) with an assembled cantilever probe (ACP) is analyzed utilizing strain gradient theory, and then the governing equation and boundary conditions are derived by a combination of the basic equations of strain gradient theory and Hamilton’s principle. The resonant frequency and sensitivity of the proposed AFM microcantilever are then obtained numerically. The proposed ACP includes a horizontal cantilever, two vertical extensions, and two tips located at the free ends of the extensions that form a caliper. As one of the extensions is located between the clamped and free ends of the AFM microcantilever, the cantilever is modeled as two beams. The results of the current model are compared with those evaluated by both modified couple stress and classical beam theories. The difference in results evaluated by the strain gradient theory and those predicted by the couple stress and classical beam theories is significant, especially when the microcantilever thickness is approximately the same as the material length-scale parameters. The results also indicate that at the low values of contact stiffness, scanning in the higher cantilever modes decrease the accuracy of the proposed AFM ACP.

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