Vibration analysis of size-dependent higher-order plates based on micropolar theory

Document Type : Research Article

Authors

1 Department of Civil Engineering, Shahid Bahonar University of Kerman, Kerman, Iran.

2 Faculty of Civil and Environmental Engineering, Tarbiat Modares University, Tehran, Iran.

10.24200/sci.2024.61589.7388

Abstract

By considering micro-rotational Degrees of Freedom (DOF), the Micro-Polar Continuum Theory (MPCT) can characterize the effect of micro-structures on the mechanical analysis of material particles, which Classical Theories (CT) of elasticity are unable to describe. The vibration behavior of the higher-order plates with a drilling DOF is discussed in this article to suggest a novel size-dependent rectangular element based on the micropolar elasticity theory. To do this, a new general formulation of the MPCT, which can be employed with ease in the Finite Element Method (FEM), is initially developed. The displacements and micro-rotations are therefore computed using quadratic shape functions on a rectangular plate element. In this element, the proper stiffness and mass matrices for the drilling DOF are derived, and to demonstrate the precision and application of the proposed element, several numerical examples of micropolar plates with various boundary conditions have been carried out. The current finite element formulation shown here is effectively used to take into account the micropolar efficiency for modeling microplates. This research contributes to advancing our understanding of the mechanical response of materials at the microscale.

Keywords

Main Subjects

References

References:
1. Hou, D., Wang, L., and Yan, J. “Vibration analysis of a cylindrical shell by using strain gradient theory via a moving Kriging interpolation-based meshfree method”, Thin-Walled Structures, 184, 110466 (2023). https://doi.org/10.1016/j.tws.2022.110466
2. Ahmad, F., Nazeer, M., Ali, W., et al. “Analytical study on couple stress fluid in an inclined channel”, Scientia Iranica, 28(4), pp. 2164-2175 (2021). https://doi.org/10.24200/sci.2021.55579.4291
3. Gholami, M. and Alizadeh, M. “A quasi-3D modified strain gradient formulation for static bending of functionally graded micro beams resting on Winkler-Pasternak elastic foundation”, Scientia Iranica, 29(1), pp. 26-40 (2022). https://doi.org/10.24200/sci.2021.55000.4019
4. Cosserat, E. and Cosserat, F. Theorie des corps deformables, Nature 81(67) (1909). https://doi.org/10.1038/081067a0
5. Eringen, A.C. and Suhubi, E.S. “Nonlinear theory of simple microelastic solid, I and II”, International Journal of Engineering Science, 2(2), pp. 189-203 (1964). https://doi.org/10.1016/0020-7225(64)90004-7
6. Eringen, A.C., Microcontinuum Field Theories: I. Foundations and Solids, Springer Science and Business Media, New York (2012). https://doi.org/10.1007/978-1-4612-0555-5
7. Sargsyan, A.H. and Sargsyan, S.H. “Dynamic model of micropolar elastic thin plates with independent fields of displacements and rotations”, Journal of Sound and Vibration, 333, pp. 4354-4375 (2014). https://doi.org/10.1016/j.jsv.2014.04.048
8. Afsar Khan, A., Batool, R., and Kousar, N. “Examining the behavior of MHD micropolar fluid over curved stretching surface based on the modified Fourier law”, Scientia Iranica, 28(1), pp. 223-230 (2021). https://doi.org/10.24200/sci.2019.51472.2199
9. Nadeem, S., Amin, A., Abbas, N., et al. “Effects of heat and mass transfer on stagnation point flow of micropolar Maxwell fluid over Riga plate”, Scientia Iranica, 28(6), pp. 3753-3766 (2021). https://doi.org/10.24200/sci.2021.53858.3454
10. Eringen, A.C., Theory of Micropolar Elasticity, Liebowitz, Academic Press, New York, pp. 621-729 (1968). https://doi.org/10.1007/978-1-4612-0555-5_5
11. Daraei, B., Shojaee, S., and Hamzehei-Javaran, S. “Analysis of stationary and axially moving beams considering functionally graded material using micropolar theory and Carrera unified formulation”, Composite Structures, 271, 114054 (2021). https://doi.org/10.1016/j.compstruct.2021.114054
12. Aganovic´, I., Tambacˇa, J., and Tutek, Z. “Derivation and justification of the models of rods and plates from linearized three-dimensional micropolar elasticity”, Journal of Elasticity, 84(2), pp. 131-152 (2006). https://doi.org/10.1007/s10659-006-9060-6
13. Jeong, J. and Neff, P. “Existence, uniqueness and stability in linear Cosserat elasticity for weakest curvature conditions”, Mathematics and Mechanics of Solids, 15(1), pp. 78-95 (2008). https://doi.org/10.1177/1081286508093581
14. Bhattacharyya, A. and Mukhopadhyay, B. “Study of linear isotropic micro-polar plate in an asymptotic approach”, Computers and Mathematics with Applications, 66(6), pp. 1047-1057 (2013). https://doi.org/10.1016/j.camwa.2013.07.007
15. Altenbach, H. and Eremeyev, V.A. “Strain rate tensors and constitutive equations of inelastic micropolar materials”, International Journal of Plasticity, 63, pp. 3-17 (2014). https://doi.org/10.1016/j.ijplas.2014.05.009
16. Eremeyev, V.A. and Konopińska-Zmysłowska, V. “On dynamic extension of a local material symmetry group for micropolar media”, Symmetry, 12(10), 1632 (2020). https://doi.org/10.3390/sym12101632
17. Allman, D.J. “Evaluation of the constant strain triangular with drilling rotations”, International Journal for Numerical Methods in Engineering, 26, pp. 2645-2655 (1988). https://doi.org/10.1002/nme.1620261205
18. MacNeal, R.H. and Harder, R.L. “A refined four-noded membrane element with rotational degrees of freedom”, Computers and Structures, 28, pp. 75-84 (1988). https://doi.org/10.1016/0045-7949(88)90094-6
19. Hughes, T.J.R. and Brezzi, F. “On drilling degrees of freedom”, Computer Methods in Applied Mechanics and Engineering, 72, pp. 105-121 (1989). https://doi.org/10.1016/0045-7825(89)90124-2
20. Ibrahimbegovic, A., Taylor, R.L., and Wilson, E.L. “A robust quadrilateral membrane finite element with drilling degrees of freedom”, International Journal for Numerical Methods in Engineering, 30, pp. 445-457 (1990). https://doi.org/10.1002/nme.1620300305
21. Sangtarash, H., Arab, H.G., Sohrabi, M.R., et al. “A high-performance four-node flat shell element with drilling degrees of freedom”, Engineering with Computers, 37, pp. 2837-2852 (2021). https://doi.org/10.1007/s00366-020-00974-4
22. Altenbach, J., Altenbach, H., and Eremeyev, V.A. “On generalized Cosserat-type theories of plates and shells: A short review and bibliography”, Archive of Applied Mechanics, 80(1), pp. 73-92 (2010). https://doi.org/10.1007/s00419-009-0365-3
23. Yeh, J.T. and Chen, W.H. “Shell elements with drilling degree of freedoms based on micropolar elasticity theory”, International Journal for Numerical Methods in Engineering, 36, pp. 1145-1159 (1993). https://doi.org/10.1002/nme.1620360705
24. Huang, F.Y., Yan, B.H., Yan, J.L., et al. “Bending analysis of micropolar elastic beam using a 3-d finite element method”, International Journal of Engineering Science, 38, pp. 275-286 (2000). https://doi.org/10.1016/S0020-7225(99)00041-5
25. Ansari, R., Norouzzadeh, A., Shakouri, A.H., et al. “Finite element analysis of vibrating micro-beams and plates using three-dimensional micropolar element”, Thin-Walled Structures, 124. pp. 489-500 (2018). https://doi.org/10.1016/j.tws.2017.12.036
26. Kohansal-Vajargah, M. and Ansari, R. “Quadratic tetrahedral micropolar element for the vibration analysis of three-dimensional micro-structures”, Thin-Walled Structures, 167, 108152 (2021). https://doi.org/10.1016/j.tws.2021.108152
27. Kohansal Vajargah, M. and Ansari, R. “Vibration analysis of two-dimensional micromorphic structures using quadrilateral and triangular elements”, Engineering Computations, 39, pp. 1922-1946 (2022). https://doi.org/10.1108/EC-12-2020-0758
28. Natarajan, S., Ferreira, A., Bordas, S., et al. “Analysis of functionally graded material plates using triangular elements with cell-based smoothed discrete shear gap method”, Mathematical Problems in Engineering (2014). https://doi.org/10.1155/2014/247932
29. Lakes, R. “Experimental microelasticity of two porous solids”, International Journal of Solids and Structures, 22, pp. 55-63 (1986). https://doi.org/10.1016/0020-7683(86)90103-4
30. Owen, D.R.J. and Hinton, E. Finite Elements in Plasticity: Theory and Practice, Pineridge Press, Swansea, U.K. (1980).
Scientia Iranica
Volume 32, Issue 11
Transactions on Mechanical Engineering
May and June 2026 Article ID:7388

Files

History

  • Receive Date: 23 December 2022
  • Revise Date: 19 January 2024
  • Accept Date: 22 June 2024

Share

How to cite

Statistics