Contribution of nonlocal integral elasticity to modified strain gradient theory

2021 ◽  
Vol 136 (5) ◽  
Author(s):  
S. Ali Faghidian
Author(s):  
Alireza Sheykhi ◽  
Shahrokh Hosseini-Hashemi ◽  
Adel Maghsoudpour ◽  
Shahram E Haghighi

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.


2019 ◽  
Vol 30 (4) ◽  
pp. 593-605 ◽  
Author(s):  
Mohammad Hosseini ◽  
Reza Bahaadini ◽  
Zahra Khalili-Parizi

This investigation aims to explore the non-conservative instability of a functionally graded material micro-beam subjected to a subtangential force. The functionally graded material micro-beam is integrated with piezoelectric layers on the lower and upper surfaces. To take size effect into account, the mathematical derivations are expanded in terms of three length scale parameters using the modified strain gradient theory in conjunction with the Euler–Bernoulli beam model. However, the modified strain gradient theory includes modified couple stress theory and classical theory as special cases. Applying extended Hamilton’s principle and Galerkin method, the governing equation and corresponding boundary conditions are obtained and then solved numerically by the eigenvalue analysis, respectively. The results illustrated effects of non-conservative parameter, length scale parameter, different material gradient index, and various values of piezoelectric voltage on the natural frequencies, flutter and divergence instabilities of a cantilever functionally graded material micro-beam. It is found that both the material gradient index and applied piezoelectric voltage have significant influence on the vibrational behaviors, divergence and flutter instability regions. Furthermore, a comparison between the various micro-beam theories on the basis of modified couple stress theory, modified strain gradient theory, and classical theory are presented.


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