# Axial magnetic field effects on dynamic characteristics of embedded multiphase nanocrystalline nanobeams

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## Abstract

This article investigates vibrational behavior of a multi-phase nanocrystalline nanobeam resting on Winkler–Pasternak foundation and subjected to a longitudinal magnetic field in the framework of nonlocal couple stress and surface elasticity theories. In this model, the essential measures to describe the real material structure of nanocrystalline nanobeams and the size effects were incorporated. This non-classical nanobeam model contains couple stress effect to capture grains micro-rotations. Moreover, the nonlocal elasticity theory is employed to study the nonlocal and long-range interactions between the particles. The present model can degenerate into the classical model if the nonlocal parameter, couple stress and surface effects are omitted. Hamilton’s principle is employed to derive the governing equations and the related boundary conditions which are solved applying differential transform method. The frequencies are compared with those of nonlocal and couple stress based beams. It is showed that vibration frequencies of a nanocrystalline nanobeam depend on the grain size, grain rotations, porosities, interface, elastic foundation, magnetic field, surface effect, nonlocality and boundary conditions.

## References

- Abdelaziz HH, Meziane MAA, Bousahla AA, Tounsi A, Mahmoud SR, Alwabli AS (2017) An efficient hyperbolic shear deformation theory for bending, buckling and free vibration of FGM sandwich plates with various boundary conditions. Steel Compos Struct 25(6):693–704Google Scholar
- Ahouel et al (2016) Size-dependent mechanical behavior of functionally graded trigonometric shear deformable nanobeams including neutral surface position concept. Steel Compos Struct 20(5):963–981CrossRefGoogle Scholar
- Al-Basyouni KS, Tounsi A, Mahmoud SR (2015) Size dependent bending and vibration analysis of functionally graded micro beams based on modified couple stress theory and neutral surface position. Compos Struct 125:621–630CrossRefGoogle Scholar
- Ansari R, Mohammadi V, Shojaei MF, Gholami R, Rouhi H (2014a) Nonlinear vibration analysis of Timoshenko nanobeams based on surface stress elasticity theory. Eur J Mech A Solids 45:143–152MathSciNetCrossRefGoogle Scholar
- Ansari R, Mohammadi V, Shojaei MF, Gholami R, Sahmani S (2014b) Postbuckling analysis of Timoshenko nanobeams including surface stress effect. Int J Eng Sci 75:1–10CrossRefGoogle Scholar
- Ansari R, Oskouie MF, Gholami R (2016) Size-dependent geometrically nonlinear free vibration analysis of fractional viscoelastic nanobeams based on the nonlocal elasticity theory. Physica E 75:266–271CrossRefGoogle Scholar
- Attia MA, Mahmoud FF (2016) Modeling and analysis of nanobeams based on nonlocal-couple stress elasticity and surface energy theories. Int J Mech Sci 105:126–134CrossRefGoogle Scholar
- Aydogdu M (2009) A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Physica E 41(9):1651–1655CrossRefGoogle Scholar
- Beldjelili et al (2016) Hygro-thermo-mechanical bending of S-FGM plates resting on variable elastic foundations using a four-variable trigonometric plate theory. Smart Struct Syst 18(4):755–786CrossRefGoogle Scholar
- Bellifa et al (2017) A nonlocal zeroth-order shear deformation theory for nonlinear postbuckling of nanobeams. Struct Eng Mech 62(6):695–702Google Scholar
- Berrabah HM, Tounsi A, Semmah A, Adda B (2013) Comparison of various refined nonlocal beam theories for bending, vibration and buckling analysis of nanobeams. Struct Eng Mech 48(3):351–365CrossRefGoogle Scholar
- Bouafia et al (2017) A nonlocal quasi-3D theory for bending and free flexural vibration behaviors of functionally graded nanobeams. Smart Struct Syst 19(2):115–126CrossRefGoogle Scholar
- Bouderba et al (2016) Thermal stability of functionally graded sandwich plates using a simple shear deformation theory. Struct Eng Mech 58(3):397–422CrossRefGoogle Scholar
- Bounouara et al (2016) A nonlocal zeroth-order shear deformation theory for free vibration of functionally graded nanoscale plates resting on elastic foundation. Steel Compos Struct 20(2):227–249CrossRefGoogle Scholar
- Bousahla et al (2016a) On thermal stability of plates with functionally graded coefficient of thermal expansion. Struct Eng Mech 60(2):313–335CrossRefGoogle Scholar
- Bousahla et al (2016b) On thermal stability of plates with functionally graded coefficient of thermal expansion. Struct Eng Mech 60(2):313–335CrossRefGoogle Scholar
- Ebrahimi F, Barati MR (2016a) A nonlocal higher-order shear deformation beam theory for vibration analysis of size-dependent functionally graded nanobeams. Arab J Sci Eng 41(5):1679–1690MathSciNetCrossRefzbMATHGoogle Scholar
- Ebrahimi F, Barati MR (2016b) A nonlocal higher-order refined magneto-electro-viscoelastic beam model for dynamic analysis of smart nanostructures. Int J Eng Sci 107:183–196CrossRefGoogle Scholar
- Ebrahimi F, Barati MR (2016c) Dynamic modeling of a thermo–piezo-electrically actuated nanosize beam subjected to a magnetic field. Appl Phys A 122(4):1–18CrossRefGoogle Scholar
- Ebrahimi F, Barati MR (2016d) Vibration analysis of smart piezoelectrically actuated nanobeams subjected to magneto-electrical field in thermal environment. J Vib Control 24(3):549–564. https://doi.org/10.1177/1077546316646239 MathSciNetCrossRefGoogle Scholar
- Ebrahimi F, Barati MR (2016e) Magnetic field effects on buckling behavior of smart size-dependent graded nanoscale beams. Eur Phys J Plus 131(7):1–14CrossRefGoogle Scholar
- Ebrahimi F, Barati MR (2016f) Vibration analysis of nonlocal beams made of functionally graded material in thermal environment. Eur Phys J Plus 131(8):279CrossRefGoogle Scholar
- Ebrahimi F, Barati MR (2016g) A unified formulation for dynamic analysis of nonlocal heterogeneous nanobeams in hygro-thermal environment. Appl Phys A 122(9):792CrossRefGoogle Scholar
- Ebrahimi F, Barati MR (2017a) Buckling analysis of nonlocal third-order shear deformable functionally graded piezoelectric nanobeams embedded in elastic medium. J Braz Soc Mech Sci Eng 39(3):937–952CrossRefGoogle Scholar
- Ebrahimi F, Barati MR (2017b) Small-scale effects on hygro-thermo-mechanical vibration of temperature-dependent nonhomogeneous nanoscale beams. Mech Adv Mater Struct 24(11):924–936CrossRefGoogle Scholar
- Ebrahimi F, Boreiry M (2015) Investigating various surface effects on nonlocal vibrational behavior of nanobeams. Appl Phys A 121(3):1305–1316CrossRefGoogle Scholar
- Ebrahimi F, Ghadiri M, Salari E, Hoseini SAH, Shaghaghi GR (2015) Application of the differential transformation method for nonlocal vibration analysis of functionally graded nanobeams. J Mech Sci Technol 29(3):1207–1215CrossRefGoogle Scholar
- Ebrahimi F, Shaghaghi GR, Boreiry M (2016a) An investigation into the influence of thermal loading and surface effects on mechanical characteristics of nanotubes. Struct Eng Mech 57(1):179–200CrossRefzbMATHGoogle Scholar
- Ebrahimi F, Barati MR, Dabbagh A (2016b) A nonlocal strain gradient theory for wave propagation analysis in temperature-dependent inhomogeneous nanoplates. Int J Eng Sci 107:169–182CrossRefGoogle Scholar
- Eltaher MA, Mahmoud FF, Assie AE, Meletis EI (2013a) Coupling effects of nonlocal and surface energy on vibration analysis of nanobeams. Appl Math Comput 224:760–774MathSciNetzbMATHGoogle Scholar
- Eltaher MA, Alshorbagy AE, Mahmoud FF (2013b) Vibration analysis of Euler–Bernoulli nanobeams by using finite element method. Appl Math Model 37(7):4787–4797MathSciNetCrossRefGoogle Scholar
- Eringen AC (1972) Nonlocal polar elastic continua. Int J Eng Sci 10(1):1–16MathSciNetCrossRefzbMATHGoogle Scholar
- Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54(9):4703–4710CrossRefGoogle Scholar
- Gheshlaghi B, Hasheminejad SM (2011) Surface effects on nonlinear free vibration of nanobeams. Compos B Eng 42(4):934–937CrossRefGoogle Scholar
- Gleiter H (2000) Nanostructured materials: basic concepts and microstructure. Acta Mater 48(1):1–29CrossRefGoogle Scholar
- Guo JG, Zhao YP (2007) The size-dependent bending elastic properties of nanobeams with surface effects. Nanotechnology 18(29):295701CrossRefGoogle Scholar
- Gurtin ME, Murdoch AI (1975) A continuum theory of elastic material surfaces. Arch Ration Mech Anal 57(4):291–323MathSciNetCrossRefzbMATHGoogle Scholar
- Hanifi Hachemi Amar L, Kaci A, Tounsi A (2017) On the size-dependent behavior of functionally graded micro-beams with porosities. Struct Eng Mech 64(5):527–541Google Scholar
- Houari et al (2016) A new simple three-unknown sinusoidal shear deformation theory for functionally graded plates. Steel Compos Struct 22(2):257–276CrossRefGoogle Scholar
- Huang Y, Hu KX, Wei X, Chandra A (1994) A generalized self-consistent mechanics method for composite materials with multiphase inclusions. J Mech Phys Solids 42(3):491–504CrossRefzbMATHGoogle Scholar
- Karami et al (2017) Effects of triaxial magnetic field on the anisotropic nanoplates. Steel Compos Struct 25(3):361–374Google Scholar
- Ke LL, Wang YS, Yang J, Kitipornchai S (2012) Nonlinear free vibration of size-dependent functionally graded microbeams. Int J Eng Sci 50(1):256–267MathSciNetCrossRefGoogle Scholar
- Khetir et al (2017) A new nonlocal trigonometric shear deformation theory for thermal buckling analysis of embedded nanosize FG plates. Struct Eng Mech 64(4):391–402Google Scholar
- Kim HS, Bush MB (1999) The effects of grain size and porosity on the elastic modulus of nanocrystalline materials. Nanostruct Mater 11(3):361–367CrossRefGoogle Scholar
- Meziane AA et al (2014) An efficient and simple refined theory for buckling and free vibration of exponentially graded sandwich plates under various boundary conditions. J Sandw Struct Mater 16(3):293–318MathSciNetCrossRefGoogle Scholar
- Mouffoki et al (2017) Vibration analysis of nonlocal advanced nanobeams in hygro-thermal environment using a new two-unknown trigonometric shear deformation beam theory. Smart Struct Syst 20(3):369–383Google Scholar
- Murmu T, Adhikari S (2012) Nonlocal elasticity based vibration of initially pre-stressed coupled nanobeam systems. Eur J Mech A Solids 34:52–62MathSciNetCrossRefzbMATHGoogle Scholar
- Reddy JN (2007) Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci 45(2):288–307CrossRefzbMATHGoogle Scholar
- Sahmani S, Bahrami M, Ansari R (2014) Surface energy effects on the free vibration characteristics of postbuckled third-order shear deformable nanobeams. Compos Struct 116:552–561CrossRefGoogle Scholar
- Shaat M (2015) Effects of grain size and microstructure rigid rotations on the bending behavior of nanocrystalline material beams. Int J Mech Sci 94:27–35CrossRefGoogle Scholar
- Shaat M, Abdelkefi A (2015a) Modeling the material structure and couple stress effects of nanocrystalline silicon beams for pull-in and bio-mass sensing applications. Int J Mech Sci 101:280–291CrossRefGoogle Scholar
- Shaat M, Abdelkefi A (2015b) Pull-in instability of multi-phase nanocrystalline silicon beams under distributed electrostatic force. Int J Eng Sci 90:58–75MathSciNetCrossRefGoogle Scholar
- Shaat M, Abdelkefi A (2016) Modeling of mechanical resonators used for nanocrystalline materials characterization and disease diagnosis of HIVs. Microsyst Technol 22(2):305–318CrossRefGoogle Scholar
- Shaat M, Khorshidi MA, Abdelkefi A, Shariati M (2016) Modeling and vibration characteristics of cracked nano-beams made of nanocrystalline materials. Int J Mech Sci 115:574–585CrossRefGoogle Scholar
- Şimşek M (2014) Large amplitude free vibration of nanobeams with various boundary conditions based on the nonlocal elasticity theory. Compos B Eng 56:621–628CrossRefGoogle Scholar
- Tounsi A, Semmah A, Bousahla AA (2013) Thermal buckling behavior of nanobeams using an efficient higher-order nonlocal beam theory. J Nanomech Micromech 3(3):37–42CrossRefGoogle Scholar
- Wang GF, Feng XQ, Yu SW, Nan CW (2003) Interface effects on effective elastic moduli of nanocrystalline materials. Mater Sci Eng A 363(1):1–8Google Scholar
- Yang FACM, Chong ACM, Lam DCC, Tong P (2002) Couple stress based strain gradient theory for elasticity. Int J Solids Struct 39(10):2731–2743CrossRefzbMATHGoogle Scholar
- Zemri et al (2015) “A mechanical response of functionally graded nanoscale beam: an assessment of a refined nonlocal shear deformation theory beam theory. Struct Eng Mech Int J 54(4):693–710CrossRefGoogle Scholar
- Zenkour AM, Abouelregal AE, Alnefaie KA, Abu-Hamdeh NH, Aljinaidi AA, Aifantis EC (2015) State space approach for the vibration of nanobeams based on the nonlocal thermoelasticity theory without energy dissipation. J Mech Sci Technol 29(7):2921–2931CrossRefGoogle Scholar