Advertisement

Modified porosity model in analysis of functionally graded porous nanobeams

  • M. A. Eltaher
  • N. Fouda
  • Tawfik El-midany
  • A. M. Sadoun
Technical Paper

Abstract

This work studies the mechanical bending and vibration of functionally graded nanobeams using finite elements according to Euler beam theory. We implement a modified porosity model that represents porosity and Young’s modulus in an implicit form, where the density is assumed as a function of the porosity parameter, while Young’s modulus is assumed as a ratio of the mass density with porosity to that without porosity. The effect of nano-scale is described by the nonlocal continuum theory by adding the length scale into the constitutive equations as a material parameter comprising information about nanoscopic forces and its interactions. The material gradation of constituents is described by a power function through the thickness of nanobeam. The beam is simply supported and is assumed to be thin, and hence, the kinematic assumptions of Euler–Bernoulli beam theory are held. The mathematical model is solved numerically using the finite-element method. Numerical results show that the increase of porosity, material graduation, and nano-scale parameters tend to decrease the bending resistance as well as the fundamental frequencies of the nanobeam.

Keywords

Functionally graded material Modified porosity model Nanobeams Finite-element method 

References

  1. 1.
    Naebe M, Shirvanimoghaddam K (2016) Functionally graded materials: a review of fabrication and properties. Appl Mater Today 5:223–245CrossRefGoogle Scholar
  2. 2.
    Zhu J, Lai Z, Yin Z, Jeon J, Lee S (2001) Fabrication of ZrO2–NiCr functionally graded material by powder metallurgy. Mater Chem Phys 68(1):130–135CrossRefGoogle Scholar
  3. 3.
    Aqida S, Ghazali M, Hashim J (2004) Effects of porosity on mechanical properties of metal matrix composite: an overview. J Teknol 40(1):17–32Google Scholar
  4. 4.
    Detsi E, Sellès M, Onck P, De Hosson J (2013) Nanoporous silver as electrochemical actuator. Scr Mater 69(2):195–198CrossRefGoogle Scholar
  5. 5.
    Li J, Takagi K, Ono M, Pan W, Watanabe R, Almajid A, Taya M (2003) Fabrication and evaluation of porous piezoelectric ceramics and porosity-graded piezoelectric actuators. J Am Ceram Soc 86:1094–1098CrossRefGoogle Scholar
  6. 6.
    Kim H, Yang Y, Koh J, Lee K, Lee D, Lee K, Park S (2009) Fabrication and characterization of functionally graded nano-micro porous titanium surface by anodizing. J Biomed Mater Res B Appl Biomater 88(2):427–435CrossRefGoogle Scholar
  7. 7.
    Ji S, Gu Q, Xia B (2006) Porosity dependence of mechanical properties of solid materials. J Mater Sci 41(6):1757–1768CrossRefGoogle Scholar
  8. 8.
    Chen D, Yang J, Kitipornchai S (2015) Elastic buckling and static bending of shear deformable functionally graded porous beam. Compos Struct 133:54–61CrossRefGoogle Scholar
  9. 9.
    Kitipornchai S, Chen D, Yang J (2017) Free vibration and elastic buckling of functionally graded porous beams reinforced by graphene platelets. Mater Des 116:656–665CrossRefGoogle Scholar
  10. 10.
    Ebrahimi F, Zia M (2015) Large amplitude nonlinear vibration analysis of functionally graded Timoshenko beams with porosities. Acta Astronaut 116:117–125CrossRefGoogle Scholar
  11. 11.
    Ebrahimi F, Mokhtari M (2015) Transverse vibration analysis of rotating porous beam with functionally graded microstructure using the differential transform method. J Braz Soc Mech Sci Eng 37(4):1435–1444CrossRefGoogle Scholar
  12. 12.
    Wattanasakulpong N, Ungbhakorn V (2014) Linear and nonlinear vibration analysis of elastically restrained ends FGM beams with porosities. Aerosp Sci Technol 32(1):111–120CrossRefGoogle Scholar
  13. 13.
    Ebrahimi F, Jafari A (2016) A higher-order thermomechanical vibration analysis of temperature-dependent FGM beams with porosities. J Eng 2016 (Article ID 9561504)Google Scholar
  14. 14.
    Ebrahimi F, Barati M (2017) Porosity-dependent vibration analysis of piezo-magnetically actuated heterogeneous nanobeams. Mech Syst Signal Process 93:445–459CrossRefGoogle Scholar
  15. 15.
    Ebrahimi F, Barati M (2017) Wave propagation analysis of smart rotating porous heterogeneous piezo-electric nanobeams. Eur Phys J Plus 132(4):153CrossRefGoogle Scholar
  16. 16.
    Fouda N, Elmidany T, Sadoun AM (2017) Bending, buckling and vibration of a functionally graded porous beam using finite elements. J Appl Comput MechGoogle Scholar
  17. 17.
    Simsek M, Yurtku H (2013) Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory. Compos Struct 97(5):378–386CrossRefGoogle Scholar
  18. 18.
    Eltaher M, Emam S, Mahmoud F (2013) Static and stability analysis of nonlocal functionally graded nanobeams. Compos Struct 96:82–88CrossRefGoogle Scholar
  19. 19.
    Eltaher M, Alshorbagy A, Mahmoud F (2013) Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/nanobeams. Compos Struct 99:193–201CrossRefGoogle Scholar
  20. 20.
    Eltaher M, Khairy A, Sadoun A, Omar F (2014) Static and buckling analysis of functionally graded Timoshenko nanobeams. Appl Math Comput 229:283–295MathSciNetzbMATHGoogle Scholar
  21. 21.
    Ebrahimi F, Ghadiri M, Salarai E, Hoseini S, Shaghaghi G (2014) Application of the differential transformation method for nonlocal vibration analysis of functionally graded nanobeams. J Mech Sci Technol 29(3):1207–1215CrossRefGoogle Scholar
  22. 22.
    Ebrahimi F, Shafiei N (2016) Application of Eringen’s nonlocal elasticity theory for vibration analysis of rotating functionally graded nanobeams. Smart Struct Syst 17:837–857CrossRefGoogle Scholar
  23. 23.
    Ebrahimi F, Barati M (2017) Buckling analysis of nonlocal third-order shear deformable functionally graded piezoelectric nanobeams embedded in elastic medium. J Braz Soc Mech Sci Eng 39(3):937–952Google Scholar
  24. 24.
    Ebrahimi F, Barati M (2016a) Vibration analysis of smart piezoelectrically actuated nanobeams subjected to magneto-electrical field in thermal environment. J Vib Control 1077546316646239Google Scholar
  25. 25.
    Ebrahimi F, Barati M (2016b) Dynamic modeling of a thermo-piezo-electrically actuated nanosize beam subjected to a magnetic field. Appl Phys A 122(4):1–18CrossRefGoogle Scholar
  26. 26.
    Ebrahimi F, Salari E, Hoseini SA (2015) Thermomechanical vibration behavior of FG nanobeams subjected to linear and non-linear temperature distributions. J Therm Stress 38(12):1360–1386CrossRefGoogle Scholar
  27. 27.
    Ebrahimi F, Salari E (2015) Size-dependent thermo-electrical buckling analysis of functionally graded piezoelectric nanobeams. Smart Mater Struct 24(12):125007CrossRefGoogle Scholar
  28. 28.
    Xie X, Zheng H, Zou X (2016) An integrated spectral collocation approach for the static and free vibration analyses of axially functionally graded nonuniform beams. Proc Inst Mech Eng Part C J Mech Eng Sci 0954406216634393Google Scholar
  29. 29.
    Rouhi S, Pour Reza T, Ramzani B, Mehran S (2017) Investigation of the vibration and buckling of graphynes: a molecular dynamics-based finite element model. Proc Inst Mech Eng Part C J Mech Eng Sci 231(6):1162–1178CrossRefGoogle Scholar
  30. 30.
    Eltaher M, Khater M, Emam S (2016) A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams. Appl Math Model 40(5):4109–4128MathSciNetCrossRefGoogle Scholar
  31. 31.
    Hamed M, Eltaher M, Sadoun A, Almitani K (2016) Free vibration of symmetric and sigmoid functionally graded nanobeams. Appl Phys A 122(9):829CrossRefGoogle Scholar
  32. 32.
    Komijani M, Esfahani S, Reddy J, Liu Y, Eslami M (2014) Nonlinear thermal stability and vibration of pre/post-buckled temperature-and microstructure-dependent functionally graded beams resting on elastic foundation. Compos Struct 112:292–307CrossRefGoogle Scholar
  33. 33.
    Bert C (1985) Prediction of elastic moduli of solids with oriented porosity. J Mater Sci 20(6):2220–2224CrossRefGoogle Scholar
  34. 34.
    Zok F, Levi C (2001) Mechanical properties of porous-matrix ceramic composites. Adv Eng Mater 3(1–2):15–23CrossRefGoogle Scholar
  35. 35.
    Revel GM (2007) Measurement of the apparent density of green ceramic tiles by a non-contact ultrasonic method. Exp Mech 47(5):637–648CrossRefGoogle Scholar
  36. 36.
    Schaffler M, Burr D (1988) Stiffness of compact bone: effects of porosity and density. J Biomech 21:13–16CrossRefGoogle Scholar
  37. 37.
    Eltaher M, Agwa M (2016) Analysis of size-dependent mechanical properties of CNTs mass sensor using energy equivalent model. Sens Actuators A 246:9–17CrossRefGoogle Scholar
  38. 38.
    Alshorbagy A, Eltaher M, Mahmoud F (2011) Free vibration characteristics of a functionally graded beam by finite element method. Appl Math Model 35:412–425MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  • M. A. Eltaher
    • 1
    • 2
  • N. Fouda
    • 3
  • Tawfik El-midany
    • 3
  • A. M. Sadoun
    • 2
  1. 1.Department of Mechanical Engineering, Faculty of EngineeringKing Abdulaziz UniversityJeddahSaudi Arabia
  2. 2.Mechanical Design and Production Department, Faculty of EngineeringZagazig UniversityZagazigEgypt
  3. 3.Production Engineering and Mechanical Design Department, Faculty of EngineeringMansoura UniversityMansouraEgypt

Personalised recommendations