Effect of Porosity on Nonlinear Tip Deflection of Functionally Graded Piezoelectric Actuators
Subject Areas : journal of New MaterialsMohammad Gholami 1 , Mansour Alizadeh 2 *
1 -
2 - School of Mechanical Engineering, Iran University of Science and Technology (IUST), Tehran, Iran
Keywords: Functionally graded piezoelectric actuator, Finite Element method, Nonlinear response, Porosity, FEniCS,
Abstract :
Introduction: Beam-shaped structures, especially bending-mode actuators, are more common and have attracted more attention from researchers due to flexibility in design and ease of fabrication.
Methods: In this study, the geometric nonlinear behavior of three-dimensional functionally graded piezoelectric porous actuators subjected to electro-mechanical loads is investigated usi th finite element method. Nonlinear von-Karman terms are included in the strain– displacement relation to capture the geometric nonlinear deformations. The governing equations and related boundary conditions were derived using the variational principle. The Newton-Raphson iteration procedure is adopted to solve nonlinear governing equations using a 10-node tetrahedral element via the robust open-source finite element FEniCS platform that exploits Python scripts.
Findings: The effects of different power law and porosity indexes, length to thickness ratios, and magnitude of applied loads are investigated on the tip deflection of the functionally graded piezoelectric porous actuator. The sensitivity of different patterns of porosity distribution along the thickness direction is explored on the dimensionless tip deflection of the actuator.
Conclusion: The uniform porosity distribution has the greatest effect on actuator deformation, whereas the central porosity distribution has the least effect. In addition, variation in the volume fraction index in the range 0–1 leads to the highest rate of change in actuator tip deflection. It has been found that the linear theory overestimates the deformations in the case of strong electromechanical loading. The findings of this research can be used to design and manufacture porous piezoelectric actuators.
1. Biot MA. Theory of Propagation of Elastic Waves in a Fluid‐Saturated Porous Solid. I. Low‐Frequency Range. The Journal of the Acoustical Society of America. 1956 Mar;28(2):168–78. doi.org/10.1121/1.1908239
2. Selvadurai AP, editor. Mechanics of poroelastic media. Springer Science & Business Media; 2013 Mar 14.
3. Busse A, Schanz M, Antes H. A Poroelastic Mindlin‐Plate. InPAMM: Proceedings in Applied Mathematics and Mechanics 2003 Dec (Vol. 3, No. 1, pp. 260-261). Berlin: WILEY‐VCH Verlag. doi.org/10.1002/pamm.200310402
4. Magnucki K, Stasiewicz P. Elastic bending of an isotropic porous beam. International Journal of Applied Mechanics and Engineering. 2004;9(2):351-60. ISSN : 1734-4492
5. Trinh MC, Kim SE. A three variable refined shear deformation theory for porous functionally graded doubly curved shell analysis. Aerospace Science and technology. 2019 Nov 1;94:105356. doi.org/10.1016/j.ast.2019.105356
6. Wang YQ, Zhao HL, Ye C, Zu JW. A porous microbeam model for bending and vibration analysis based on the sinusoidal beam theory and modified strain gradient theory. International Journal of Applied Mechanics. 2018 Jun 11;10(05):1850059. doi.org/10.1142/S175882511850059X
7. Babaei M, Asemi K, Safarpour P. Buckling and static analyses of functionally graded saturated porous thick beam resting on elastic foundation based on higher order beam theory. Iranian Journal of Mechanical Engineering Transactions of the ISME. 2019 Mar 1;20(1):94-112. 20.1001.1.16059727.2019.20.1.4.0
8. Phuong NT, Tu TM, Phuong HT, Van Long N. Bending analysis of functionally graded beam with porosities resting on elastic foundation based on neutral surface position. Journal of Science and Technology in Civil Engineering (JSTCE)-HUCE. 2019 Jan 31;13(1):33-45. doi.org/10.31814/stce.nuce2019-13(1)-04
9. Kiran MC, Kattimani SC, Vinyas M. Porosity influence on structural behaviour of skew functionally graded magneto-electro-elastic plate. Composite Structures. 2018 May 1;191:36-77. doi.org/10.1016/j.compstruct.2018.02.023
10. Taghian Dehaghani M, Ahmadian M. Fracture mechanism of CoCrMo porous nano-composite prepared by powder metallurgy route. International Journal of Engineering. 2018 Jan 1;31(1):19-24. Doi.org/10.5829/ije.2018.31.01a.03
11. Sarparast Z, Abdoli R, Rahbari A, Varmazyar M, Reza Kashyzadeh K. Experimental and numerical analysis of permeability in porous media. International Journal of Engineering. 2020 Nov 1;33(11):2408-15. doi.org/10.5829/ije.2020.33.11b.31
12. Gupta A, Talha M. Influence of initial geometric imperfections and porosity on the stability of functionally graded material plates. Mechanics Based Design of Structures and Machines. 2018 Nov 2;46(6):693-711. doi.org/10.1080/15397734.2018.1449656
13. Wang QM, Zhang Q, Xu B, Liu R, Cross LE. Nonlinear piezoelectric behavior of ceramic bending mode actuators under strong electric fields. Journal of Applied Physics. 1999 Sep 15;86(6):3352-60. doi.org/10.1063/1.371213
14. Rezazadeh G, Khanchehgardan A, Shah-Mohammadi-Azar A, Shabani R. Mechanical response of a piezoelectrically sandwiched nano-beam based on the non-local theory. International Journal of Engineering. 2013 Dec 1;26(12):1515-24. doi.org/10.5829/idosi.ije.2013.26.12c.12
15. Nosier A, Rouhi M. Three-dimensional analysis of laminated cylindrical panels with piezoelectric layers.
16. Zhang Z, Fu G, Xu D. A Dynamic Model for Laminated Piezoelectric Microbeam. International Journal of Engineering. 2023 Jun 1;36(6):1143-9. doi.org/10.5829/ije.2023.36.06c.13
17. Dardel M, Khavvajia A, Akbari Alashti R, Pashaei MH. Large Deflection Analysis of Compliant Beams of Variable Thickness and Non-Homogenous Material under Combined Load and Multiple Boundary Conditions. International Journal of Engineering. 2012 Dec 1;25(4):353-62. doi.org/10.5829/idosi.ije.2013.25.04c.10
18. Fotros F, Pashaei MH, Alashti RA. Effects of geometric nonlinearity on stress analysis in large amplitude vibration of thin circular functionally graded plates with rigid core. doi.org/10.5829/idosi.ije.2011.24.03a.07
19. Hecht F. New development in FreeFem++. Journal of numerical mathematics. 2012 Dec;20(3-4):251-66. doi.org/10.1515/jnum-2012-0013
20. Rathgeber F, Ham DA, Mitchell L, Lange M, Luporini F, McRae AT, Bercea GT, Markall GR, Kelly PH. Firedrake: automating the finite element method by composing abstractions. ACM Transactions on Mathematical Software (TOMS). 2016 Dec 21;43(3):1-27. doi.org/10.1145/2998441
21. Langtangen HP. A FEniCS tutorial. InAutomated solution of differential equations by the finite element method: The FEniCS book 2012 Feb 4 (pp. 1-73). Berlin, Heidelberg: Springer Berlin Heidelberg. ISSN 1439-7358
22. Rodriguez MA, Augustin CM, Shadden SC. FEniCS mechanics: A package for continuum mechanics simulations. SoftwareX. 2019 Jan 1;9:107-11. doi.org/10.1016/j.softx.2018.10.005
23. Shekarchizadeh N, Bersani AM. Developing an automatized optimization problem in FEniCS for parameter determination of metamaterials. InProceedings of FEniCS 2021 2021 (pp. 660-679). doi.org/10.6084/m9.figshare.14495607
24. Phunpeng V, Baiz P. Mixed finite element formulations for strain-gradient elasticity problems using the FEniCS environment. Finite Elements in Analysis and Design. 2015 Apr 1;96:23-40. doi.org/10.1016/j.finel.2014.11.002
25. Reddy JN. Mechanics of laminated composite plates and shells: theory and analysis. CRC press; 2003 Nov 24. ISBN 9780429210693
26. Abali BE, Müller WH, Eremeyev VA. Strain gradient elasticity with geometric nonlinearities and its computational evaluation. Mechanics of Advanced Materials and Modern Processes. 2015 Dec;1:1-1. doi.org/10.1186/s40759-015-0004-3
27. Logg A, Mardal KA, Wells G, editors. Automated solution of differential equations by the finite element method: The FEniCS book. Springer Science & Business Media; 2012 Feb 24. doi.org/10.1007/978-3-642-23099-8
28. Arnold DN, Logg A. Periodic table of the finite elements. Siam News. 2014 Nov 3;47(9):212.
29. Alnæs M, Blechta J, Hake J, Johansson A, Kehlet B, Logg A, Richardson C, Ring J, Rognes ME, Wells GN. The FEniCS project version 1.5. Archive of numerical software. 2015 Dec 7;3(100).
30. Doroushi A, Eslami MR, Komeili A. Vibration analysis and transient response of an FGPM beam under thermo-electro-mechanical loads using higher-order shear deformation theory. Journal of Intelligent Material Systems and Structures. 2011 Feb;22(3):231-43. doi.org/10.1177/1045389X11398162
31. Pandey VB, Parashar SK. Static bending and dynamic analysis of functionally graded piezoelectric beam subjected to electromechanical loads. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science. 2016 Dec;230(19):3457-69. doi.org/10.1177/0954406215596359