Effect of kerr foundation and in-plane forces on free vibration of fgm nanobeams with diverse distribution of porosity

• Autorzy: Jankowski Piotr

Identyfikatory

DOI

10.2478/ama-2020-0020

• Języki publikacji: EN

Abstrakty

EN

In the present paper, the effect of diverse distribution of functionally graded porous material and Kerr elastic foundation on natu-ral vibrations of nanobeams subjected to in-plane forces is investigated based on the nonlocal strain gradient theory. The displacement field of the nanobeam satisfies assumptions of Reddy higher-order shear deformation beam theory. All the displacements gradients are assumed to be small, then the components of the Green-Lagrange strain tensor are linear and infinitesimal. The constitutive relations for functionally graded (FG) porous material are expressed by nonlocal and length scale parameters and power-law variation of material pa-rameters in conjunction with cosine functions. It created possibility to investigate an effect of functionally graded materials with diverse dis-tribution of porosity and volume of voids on mechanics of structures in nano scale. The Hamilton’s variational principle is utilized to derive governing equations of motion of the FG porous nanobeam. Analytical solution to formulated boundary value problem is obtained in closed-form by using Navier solution technique. Validation of obtained results and parametric study are presented in tabular and graphical form. Influence of axial tensile/compressive forces and three different types of porosity distribution as well as stiffness of Kerr foundation on natural frequencies of functionally graded nanobeam is comprehensively studied.

Słowa kluczowe

EN

porosity distribution; nanobeam; reddy beam theory; free vibrations; nonlocal strain gradient theory; strain gradient theory

• Wydawca: Oficyna Wydawnicza Politechniki Białostockiej
• Czasopismo: Acta Mechanica et Automatica
• Rocznik: 2020
• Tom: Vol. 14, no. 3
• Strony: 135--143
• Opis fizyczny: Bibliogr. 40 poz., rys.

Twórcy

autor

Jankowski Piotr

• Faculty of Mechanical Engineering, Bialystok University of Technology, ul. Wiejska 45C, 15-351 Białystok, Poland

Bibliografia

• 1. Ashoori A.R, Salari E., Sadough Vanini S.A., (2017), A Thermo-Electro-Mechanical Vibration Analysis of Size-Dependent Functionally Graded Piezoelectric Nanobeams, Advances in High Temperature Ceramic Matrix Composites and Materials for Sustainable Development; Ceramic Transactions, Vol. 263, 547–558.
• 2. Aydogdu M., (2009), A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration, Physica E: Low-dimensional Systems and Nanostructures, Vol. 41(9), 1651–1655.
• 3. Bhushan B. (Ed), (2004), Springer Handbook of Nanotechnology, Springer Verlag, Berlin.
• 4. El-Borgi S., Fernandes R., Reddy J.N., (2015), Non-local free and forced vibrations of graded nanobeams resting on a non-linear elastic foundation, International Journal of Non-Linear Mechanics, Vol. 77, 348–363.
• 5. Eltaher M.A., Emam S.A., Mahmoud F.F., (2012), Free vibration analysis of functionally graded size-dependent nanobeams. Applied Mathematics and Computation, Vol. 218(14), 7406–7420.
• 6. Eltaher M.A., Emam S.A., Mahmoud F.F., (2013), Static and stability analysis of nonlocal functionally graded nanobeams, Composite Structures, Vol. 96, 82–88.
• 7. Eltaher M.A., Fouda N., El-midany, T., Sadoun, A.M., (2018), Modified porosity model in analysis of functionally graded porous nanobeams. Journal of the Brazilian Society of Mechanical Sciences and Engineering, Vol. 40, 141.
• 8. Eringen A.C., (1972), Nonlocal polar elastic continua. International Journal of Engineering Science, Vol. 10(1), 1–16.
• 9. Eringen A.C., Edelen D.G.B., (1972), On nonlocal elasticity. International Journal of Engineering Science, Vol. 10(3), 233–248.
• 10. Ghadiri M., Rajabpour A., Akbarshahi A., (2017), Non-linear forced vibration analysis of nanobeams subjected to moving concentrated load resting on a viscoelastic foundation considering thermal and surface effects, Applied Mathematical Modelling, Vol. 50, 676–694.
• 11. Karami B., Janghorban M., (2019), A new size-dependent shear deformation theory for free vibration analysis of functionally graded/anisotropic nanobeams, Thin-Walled Structures, Vol. 143, 106227.
• 12. Kerr A.D., (1965), A study of a new foundation model. Acta Mechanica, Vol. 1(2), 135–147.
• 13. Kim J., Żur K.K., Reddy, J.N., (2019), Bending, free vibration, and buckling of modified couples stress-based functionally graded porous micro-plates, Composite Structures, Vol. 209, 879–888.
• 14. Lam D.C.C., Yang F., Chong A.C.M., Wang J., Tong P., (2003), Experiments and theory in strain gradient elasticity, Journal of the Mechanics and Physics of Solids, Vol. 51(8), 1477–1508.
• 15. Leondes C.T. (Ed), (2006), MEMS/NEMS Handbook Techniques and Applications, Springer, New York.
• 16. Li L., Hu Y., (2015), Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory, International Journal of Engineering Science, Vol. 97, 84–94.
• 17. Lim C.W., Li C., Yu J.-L., (2010), Free vibration of pre-tensioned nanobeams based on nonlocal stress theory, Journal of Zhejiang University-SCIENCE A, Vol. 11, 34–42.
• 18. Lim C.W., Zhang G., Reddy J.N., (2015), A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. Journal of the Mechanics and Physics of Solids, Vol. 78, 298–313.
• 19. Lu L., Guo X., Zhao J., (2017), Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory, International Journal of Engineering Science, Vol. 116, 12–24.
• 20. Lv Z., Qiu Z., Zhu J., Zhu B., Yang W., (2018), Nonlinear free vibration analysis of defective FG nanobeams embedded in elastic medium, Composite Structures, Vol. 202, 675–685.
• 21. Lyshevski S.E., (2002), MEMS and NEMS: System, Devices and Structures, CRC Press, New York.
• 22. Mindlin R.D., (1964), Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis, Vol. 16, 51–78.
• 23. Mindlin R.D., (1965), Second gradient of strain and surface-tension in linear elasticity. International Journal of Solids and Structures Vol. 1(4), 417–438.
• 24. Nazemnezhad R., Hosseini-Hashemi S., (2014), Nonlocal nonlinear free vibration of functionally graded nanobeams, Composite Structures, Vol. 110, 192–199.
• 25. Pasternak P.L., (1954), On a New method of Analysis of an Elastic Foundation by Means of Two Foundation Constants,
• Gosudarstvennoe Izdatelstvo Literaturi po Stroitelstvu I Arkhitekture, Moscow.
• 26. Rahmani O., Jandaghian A.A., (2015), Buckling analysis of functionally graded nanobeams based on a nonlocal third-order shear deformation theory. Applied Physics A, Vol. 119, 1019–1032.
• 27. Reddy J.N., (2017), Energy principles and variational methods in applied mechanics, John Wiley & Sons, New York.
• 28. Reza Barati M., (2017), Investigating dynamic response of porous inhomogeneous nanobeams on hybrid Kerr foundation under hygro-thermal loading, Applied Physics A, Vol. 123, 332.
• 29. Saffari S., Hashemian M., Toghraie D., (2017), Dynamic stability of functionally graded nanobeam based on nonlocal Timoshenko theory considering surface effects, Physica B: Condensed Matter, Vol. 520, 97–105.
• 30. Sahmani S., Ansari R., (2011), Nonlocal beam models for buckling of nanobeams using state-space method regarding different boundary conditions, Journal of Mechanical Science and Technology, Vol. 25, 2365.
• 31. Shafiei N., Mirjavadi S.S., Afshari B.M., Rabby S., Kazemi, M., (2017), Vibration of two-dimensional imperfect functionally graded (2D-FG) porous nano-/micro-beams, Computer Methods in Applied Mechanics and Engineering, Vol. 322, 615–632.
• 32. Şimşek M., (2014), Large amplitude free vibration of nanobeams with various boundary conditions based on the nonlocal elasticity theory, Composites Part B: Engineering, Vol. 56, 621–628.
• 33. Şimşek M., (2016), Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach, International Journal of Engineering Science, Vol. 105, 12–27.
• 34. Thai H.T., (2012), A nonlocal beam theory for bending, buckling, and vibration of nanobeams, International Journal of Engineering Science, Vol. 52, 56–64.
• 35. Thai H.T., Vo T.P., (2012a), A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams. International Journal of Engineering Science, Vol. 54, 58–66.
• 36. Thai H.T., Vo, T.P., (2012b), Bending and free vibration of functionally graded beams using various higher order shear deformation beam theories. International Journal of Mechanical Sciences, Vol. 62(1), 57–66.
• 37. Timoshenko S., Woinowsky-Krieger S., (1959), Theory of plates and shells, McGraw-Hill Book Company, New York.
• 38. Toupin R,A., (1962), Elastic materials with couple-stresses. Archive for Rational Mechanics and Analysis, Vol. 11, 385–414.
• 39. Yang F., Chong A.C.M., Lam D.C.C., Tong P., (2002), Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures, Vol. 39(10), 2731–2743.
• 40. Zhang K, Ge M.-H., Zhao C., Deng Z-C., Lu, X-J., (2019), Free vibration of nonlocal Timoshenko beams made of functionally graded materials by Symplectic method, Composites Part B: Engineering, 156, 174–184.

Uwagi

1. The work has been conducted within W/WM-IIM/3/2020 project and was financed by the funds of the Ministry of Science and Higher Education, Poland.

2. Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).