zbMATH — the first resource for mathematics

Numerical research on a three-dimensional solid element based on generalized elasticity theory. (English) Zbl 1435.74011
Summary: A finite element equation is established based on generalized elasticity theory by applying a virtual work principle. Then, a penalty function term is added to the virtual work equation by imposing rotation and displacement as independent variables. An 8-node element with full integration, an 8-node element with reduced integration, and a 20-node element with full integration are constructed using difference integration schemes and shape functions. The influences of structural degrees of freedom and the penalty parameter on convergence are analyzed via the three elements. It is shown that the 8-node element with reduction integration and the 20-node element with full integration are convergent, whereas the 8-node element with full integration is divergent. The scale effects of a slender beam, a short beam, a thin plate, and a medium-thick plate are numerically analyzed. Lastly, the scale effects of the frequencies that correspond to the bending mode, torsion mode, and tension-compression mode for a pretwisted plate are studied. It is found that the frequencies that correspond to the bending mode and torsion mode exert a scale effect, whereas the frequency that corresponds to the tension-compression mode does not. The essence of the scale effect is that the rotational deformation of the microstructure is amplified.
74B05 Classical linear elasticity
74S05 Finite element methods applied to problems in solid mechanics
Full Text: DOI
[1] Korayem, M. H.; Korayem, A. H., Modeling of AFM with a piezoelectric layer based on the modified couple stress theory with geometric discontinuities, Applied Mathematical Modelling, 45, 439-456 (2017) · Zbl 1446.74026
[2] Yang, W.; He, D.; Chen, W., A size-dependent zigzag model for composite laminated micro beams based on a modified couple stress theory, Composite Structures, 179, 646-654 (2017)
[3] Sun, Y.; Cheng, J.; Wang, Z.; Yu, Y.; Tian, L.; Lu, J., Analytical approximate solution for nonlinear behavior of cantilever FGM MEMS beam with thermal and size dependency, Mathematical Problems in Engineering, 2019 (2019)
[4] Ghavami, M.; Azizi, S.; Ghazavi, M. R., On the dynamics of a capacitive electret-based micro-cantilever for energy harvesting, Energy, 153, 967-976 (2018)
[5] Jabbari Behrouz, S.; Rahmani, O.; Amirhossein Hosseini, S., On nonlinear forced vibration of nano cantilever-based biosensor via couple stress theory, Mechanical Systems and Signal Processing, 128, 19-36 (2019)
[6] Ghayesh, M. H.; Farokhi, H., Nonsymmetric nonlinear dynamics of piezoelectrically actuated beams, Journal of Vibration and Acoustics, 141, 5 (2019)
[7] Salehipour, H.; Nahvi, H.; Shahidi, A. R., Exact closed-form free vibration analysis for functionally graded micro/nano plates based on modified couple stress and three-dimensional elasticity theories, Composite Structures, 124, 283-291 (2015)
[8] Askari, A. R.; Tahani, M., Size-dependent dynamic pull-in analysis of geometric non-linear micro-plates based on the modified couple stress theory, Physica E: Low-Dimensional Systems and Nanostructures, 86, 262-274 (2017)
[9] Khorasani, V. S.; Bayat, M., Bending analysis of FG plates using a general third-order plate theory with modified couple stress effect and MLPG method, Engineering Analysis with Boundary Elements, 94, 159-171 (2018) · Zbl 1403.74113
[10] Soni, S.; Jain, N. K.; Joshi, P. V., Analytical modeling for nonlinear vibration analysis of partially cracked thin magneto-electro-elastic plate coupled with fluid, Nonlinear Dynamics, 90, 1, 137-170 (2017)
[11] Soni, S.; Jain, N. K.; Joshi, P. V., Vibration and deflection analysis of thin cracked and submerged orthotropic plate under thermal environment using strain gradient theory, Nonlinear Dynamics, 96, 2, 1575-1604 (2019)
[12] Soni, S.; Jain, N. K.; Joshi, P. V., Stability and dynamic analysis of partially cracked thin orthotropic microplates under thermal environment: an analytical approach, Mechanics Based Design of Structures and Machines (2019)
[13] McFarland, A. W.; Colton, J. S., Role of material microstructure in plate stiffness with relevance to microcantilever sensors, Journal of Micromechanics and Microengineering, 15, 5, 1060-1067 (2005)
[14] Hassanpour, S.; Heppler, G. R., Micropolar elasticity theory: a survey of linear isotropic equations, representative notations, and experimental investigations, Mathematics and Mechanics of Solids, 22, 224-242 (2015) · Zbl 1371.74012
[15] Mindlin, R. D.; Tiersten, H. F., Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis, 11, 1, 415-448 (1962) · Zbl 0112.38906
[16] Mindlin, R. D., Influence of couple-stresses on stress concentrations, Experimental Mechanics, 3, 1, 307-308 (1963)
[17] Yang, F.; Chong, A. C. M.; Lam, D. C. C.; Tong, P., Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures, 39, 10, 2731-2743 (2002) · Zbl 1037.74006
[18] Hadjesfandiari, A. R.; Dargush, G. F., Couple stress theory for solids, International Journal of Solids and Structures, 48, 18, 2496-2510 (2011)
[19] Huang, K. Z.; Huang, Y. G., in Chinese, Beijing, China: Tsinghua University Press, Beijing, China
[20] Liu, Z.; Yan, S.; Fu, Z., Dynamic analysis on generalized linear elastic body subjected to large scale rigid rotations, Applied Mathematics and Mechanics, 34, 1001-1016 (2013) · Zbl 1286.74006
[21] Chakravarty, S.; Hadjesfandiari, A. R.; Dargush, G. F., A penalty-based finite element framework for couple stress elasticity, Finite Elements in Analysis and Design, 130, 65-79 (2017)
[22] Darrall, B. T.; Dargush, G. F.; Hadjesfandiari, A. R., Finite element Lagrange multiplier formulation for size-dependent skew-symmetric couple-stress planar elasticity, Acta Mechanica, 225, 1, 195-212 (2014) · Zbl 1401.74269
[23] Ma, X.; Chen, W., Refined 18-DOF triangular hybrid stress element for couple stress theory, Finite Elements in Analysis and Design, 75, 8-18 (2013) · Zbl 1368.74006
[24] Ma, X.; Chen, W., 24-DOF quadrilateral hybrid stress element for couple stress theory, Computational Mechanics, 53, 1, 159-172 (2014) · Zbl 1398.74365
[25] Kandaz, M.; Dal, H., A comparative study of modified strain gradient theory and modified couple stress theory for gold microbeams, Archive of Applied Mechanics, 88, 11, 2051-2070 (2018)
[26] Karttunen, A. T.; Romanoff, J.; Reddy, J. N., Exact microstructure-dependent Timoshenko beam element, International Journal of Mechanical Sciences, 111-112, 112, 35-42 (2016)
[27] Zhang, B.; He, Y.; Liu, D.; Gan, Z.; Shen, L., A non-classical Mindlin plate finite element based on a modified couple stress theory, European Journal of Mechanics—A/Solids, 42, 63-80 (2013) · Zbl 1406.74458
[28] Tahani, M.; Askari, A. R.; Mohandes, Y.; Hassani, B., Size-dependent free vibration analysis of electrostatically pre-deformed rectangular micro-plates based on the modified couple stress theory, International Journal of Mechanical Sciences, 94-95, 95, 185-198 (2015)
[29] Lobontiu, N.; Garcia, E., Mechanics of Microelectromechanical Systems (2005), New York, NY, USA: Kluwer Academic Publishers, New York, NY, USA
[30] Zienkiewicz, O.; Taylor, R.; Zhu, J. Z., The Finite Element Method: Its Basis and Fundamentals (2015), Beijing, China: Beijing Word Publishing Corporation, Beijing, China
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.