Abstract
This study investigates the size-dependent free axial vibration of a nanorod made of porous material. In this context, nonlocal elasticity theory for size dependence and Bishop rod theory are used in the study. The porous nanorod is considered in arbitrary boundary conditions and for this purpose, it is modeled with elastic springs at both ends. A method based on the combination of Fourier sine series and Stokes’ transform is presented to realize the solution. Thanks to the presented approach, an eigenvalue problem is established to find the frequencies of a porous Bishop nanorods in general boundary conditions. Finally, the axial vibration frequencies of the porous Bishop nanorod based on the nonlocal elasticity theory are obtained depending on various parameters and the effects of these parameters are discussed.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs42417-022-00610-z/MediaObjects/42417_2022_610_Fig1_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs42417-022-00610-z/MediaObjects/42417_2022_610_Fig2_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs42417-022-00610-z/MediaObjects/42417_2022_610_Fig3_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs42417-022-00610-z/MediaObjects/42417_2022_610_Fig4_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs42417-022-00610-z/MediaObjects/42417_2022_610_Fig5_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs42417-022-00610-z/MediaObjects/42417_2022_610_Fig6_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs42417-022-00610-z/MediaObjects/42417_2022_610_Fig7_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs42417-022-00610-z/MediaObjects/42417_2022_610_Fig8_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs42417-022-00610-z/MediaObjects/42417_2022_610_Fig9_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs42417-022-00610-z/MediaObjects/42417_2022_610_Fig10_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs42417-022-00610-z/MediaObjects/42417_2022_610_Fig11_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs42417-022-00610-z/MediaObjects/42417_2022_610_Fig12_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs42417-022-00610-z/MediaObjects/42417_2022_610_Fig13_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs42417-022-00610-z/MediaObjects/42417_2022_610_Fig14_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs42417-022-00610-z/MediaObjects/42417_2022_610_Fig15_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs42417-022-00610-z/MediaObjects/42417_2022_610_Fig16_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs42417-022-00610-z/MediaObjects/42417_2022_610_Fig17_HTML.png)
Similar content being viewed by others
Data availability
The authors confirm that the data supporting the findings of this study are available within the article (and/or) its supplementary. All data generated or analyzed during this study are included in this published article.
References
Patra JK, Das G, Fraceto LF, Campos EVR, Rodrigez-Torres MDP, Acosta-Torres LS et al (2018) Nano based drug delivery systems: recent developments and future prospects. NanoBiotechnology 16(1):1–33. https://doi.org/10.1186/S12951-018-0392-8
Li J, De Ávila BEF, Gao W, Zhang L, Wang J (2017) Micro/nanorobots for biomedicine: delivery, surgery, sensing, and detoxification. Sci Robot 2(4):1–9. https://doi.org/10.1126/SCIROBOTICS.AAM6431
Bhushan B (ed) (2017) Springer handbook of nanotechnology, 4th edn. Springer, Berlin
Zhu Y (ed) (2020) Micro and nano machined electrometers. Springer, Singapore
Demir C, Civalek O (2017) On the analysis of microbeams. Int J Eng Sci 121:14–33. https://doi.org/10.1016/j.ijengsci.2017.08.016
Numanoğlu HM, Akgöz B, Civalek O (2018) On dynamic analysis of nanorods. Int J Eng Sci 130:33–50. https://doi.org/10.1016/j.ijengsci.2018.05.001
Zhang N, Jiao B, Ye Y, Kong Y, Du X, Liu R, Cong B, Yu L, Jia S, Jia K (2022) Embedded cooling method with configurability and replaceability for multi-chip electronic devices. Energy Convers Manag 253:115124. https://doi.org/10.1016/j.enconman.2021.115124
Gao N, Zhang Z, Deng J, Guo X, Cheng B, Hou H (2022) Acoustic metamaterials for noise reduction: a review. Adv Mater Technol. https://doi.org/10.1002/admt.202100698
Altenbach H, Öchsner A (eds) (2020) Encyclopedia of continuum mechanics. Springer, Berlin
Harik VM (2002) Mechanics of carbon nanotubes: applicability of the continuum-beam models. Comput Mater Sci 24(3):328–342. https://doi.org/10.1016/S0927-0256(01)00255-5
Manolis GD, Dineva PS, Rangelov T, Sfyris D (2021) Mechanical models and numerical simulations in nanomechanics: a review across the scales. Eng Anal Bound Elem 128:149–170. https://doi.org/10.1016/J.ENGANABOUND.2021.04.004
Sánchez-Portal D, Artacho E, Soler JM, Rubio A, Ordejón P (1999) Ab initio structural, elastic, and vibrational properties of carbon nanotubes. Phys Rev B 59(19):12678. https://doi.org/10.1103/PhysRevB.59.12678
Prathab B, Subramanian V, Aminabhavi TM (2007) Molecular dynamics simulations to investigate polymer–polymer and polymer–metal oxide interactions. Polymer 48(1):409–416. https://doi.org/10.1016/J.POLYMER.2006.11.014
Panin VE, Surikova NS, Smirnova AS, Pochivalov YI (2018) Mesoscopic structural states in plastically deformed nanostructured metal materials. Phys Mesomech 21(5):396–400. https://doi.org/10.1134/S102995991805003X
Xia C, Xu W, Nie G (2021) Dynamic quasi-continuum model for plate-type nano-materials and analysis of fundamental frequency. Appl Math Mech 42(1):85–94. https://doi.org/10.1007/S10483-021-2688-8
Budarapu PR, Zhuang X, Rabczuk T, Bordas SPA (2019) Multiscale modeling of material failure: theory and computational methods. Adv Appl Mech 52:1–103. https://doi.org/10.1016/BS.AAMS.2019.04.002
Tadmor EB, Miller RE (2011) Modeling materials: continuum, atomistic and multiscale techniques. Cambridge University Press, Cambridge
Wang J, Duan HL, Huang ZP, Karihaloo BL (2006) A scaling law for properties of nano-structured materials. Proc. R. Soc. A Math. Phys. Eng. Sci. 462:1355–1363. https://doi.org/10.1098/rspa.2005.1637
Wong EW, Sheehan PE, Lieber CM (1997) Nanobeam mechanics: Elasticity, strength, and toughness of nanorods and nanotubes. Science 277(5334):1971–1975. https://doi.org/10.1126/SCIENCE.277.5334.1971
Wu B, Heidelberg A, Boland JJ (2005) Mechanical properties of ultrahigh-strength gold nanowires. Nat Mater 4(7):525–529. https://doi.org/10.1038/nmat1403
Fleck NA, Muller GM, Ashby MF, Hutchinson JW (1994) Strain gradient plasticity: theory and experiment. Acta Metall Mater 42(2):475–487. https://doi.org/10.1016/0956-7151(94)90502-9
Toupin R (1962) Elastic materials with couple-stresses. Arch Ration Mech Anal 11(1):385–414. https://doi.org/10.1007/BF00253945
Yang F, Chong ACM, Lam DCC, Tong P (2002) Couple stress based strain gradient theory for elasticity. Int J Solids Struct 39(10):2731–2743. https://doi.org/10.1016/S0020-7683(02)00152-X
Mindlin RD (1964) Micro-structure in linear elasticity. Arch Ration Mech Anal 16:51–78. https://doi.org/10.1007/BF00248490
Mindlin RD (1965) Second gradient of strain and surface-tension in linear elasticity. Int J Solids Struct 1(4):417–438. https://doi.org/10.1016/0020-7683(65)90006-5
Lim CW, Zhang G, Reddy JN (2015) A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J Mech Phys Solids 78:298–313. https://doi.org/10.1016/J.JMPS.2015.02.001
Gurtin ME, Ian MA (1975) A continuum theory of elastic material surfaces. Arch Ration Mech Anal 57(4):291–323. https://doi.org/10.1007/BF00261375
Silling SA (2000) Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 48(1):175–209. https://doi.org/10.1016/S0022-5096(99)00029-0
Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54(9):4703–4710. https://doi.org/10.1063/1.332803
Eringen AC (1987) Theory of nonlocal elasticity and some applications. Res Mech 21(4):313–342. https://doi.org/10.21236/ada145201
Eringen AC (2002) Nonlocal continuum field theories. Springer, New York
Barretta R, Ali Faghidian S, de Sciarra FM, Pinnola FP (2021) Timoshenko nonlocal strain gradient nanobeams: variational consistency, exact solutions and carbon nanotube Young moduli. Mech Adv Mater Struct 28(15):1523–1536. https://doi.org/10.1080/15376494.2019.1683660
Arefi M, Zenkour AM (2016) Free vibration, wave propagation and tension analyses of a sandwich micro/nano rod subjected to electric potential using strain gradient theory. Mater Res Express 3(11):115704. https://doi.org/10.1088/2053-1591/3/11/115704
Barretta R, Faghidian SA, de Sciarra F (2019) Aifantis versus Lam strain gradient models of Bishop elastic rods. Acta Mech 230(8):2799–2812. https://doi.org/10.1007/s00707-019-02431-w
Barretta R, de Sciarra FM (2018) Constitutive boundary conditions for nonlocal strain gradient elastic nano-beams. Int J Eng Sci 130:187–198. https://doi.org/10.1016/j.ijengsci.2018.05.009
Lian L, Li Z (2022) Dynamic and frequency responses of the FG nanopipe using deep neural network and nonlocal strain/stress gradient theory. Waves Random Complex Media. https://doi.org/10.1080/17455030.2022.2050438
Liu Z, Wu S, Jin S, Liu Q, Ji S, Lu S, Cheng L (2022) Investigating pose representations and motion contexts modeling for 3D motion prediction. IEEE Trans Pattern Anal Mach Intell. https://doi.org/10.1109/tpami.2021.3139918
Heidari M, Tadi Beni Y, Homaei H (2013) Estimation of static pull-in instability voltage of geometrically nonlinear Euler-Bernoulli microbeam based on modified couple stress theory by artificial neural network model. Adv Artif Neural Syst. https://doi.org/10.1155/2013/741896
Wang S, Guo H, Zhang S, Barton D, Brooks P (2022) Analysis and prediction of double-carriage train wheel wear based on SIMPACK and neural networks. Adv Mech Eng 14(3):16878132221078492. https://doi.org/10.1177/16878132221078491
Roudbari MA, Jorshari TD, Lü C, Ansari R, Kouzani AZ, Amabili M (2022) A review of size-dependent continuum mechanics models for micro- and nano-structures. Thin-Walled Struct 170:108562. https://doi.org/10.1016/j.tws.2021.108562
Yayli MÖ (2011) Stability analysis of a gradient elastic beam using finite element method. Int J Phys Sci 6(12):2844–2851. https://doi.org/10.5897/ıjps11.361
Akgöz B, Civalek O (2014) Longitudinal vibration analysis for microbars based on strain gradient elasticity theory. J Vib Control 20(4):606–616. https://doi.org/10.1177/1077546312463752
Yayli MÖ (2018) Torsional vibration analysis of nanorods with elastic torsional restraints using non-local elasticity theory. Micro Nano Lett 13(5):595–599. https://doi.org/10.1049/mnl.2017.0751
Numanoğlu HM, Civalek Ö (2019) On the torsional vibration of nanorods surrounded by elastic matrix via nonlocal FEM. Int J Mech Sci 161:105076. https://doi.org/10.1016/j.ijmecsci.2019.105076
Civalek O, Uzun B, Yaylı MO, Akgöz B (2020) Size-dependent transverse and longitudinal vibrations of embedded carbon and silica carbide nanotubes by nonlocal finite element method. Eur Phys J Plus 135:381. https://doi.org/10.1140/epjp/s13360-020-00385-w
Civalek Ö, Numanoğlu HM (2020) Nonlocal finite element analysis for axial vibration of embedded love–bishop nanorods. Int J Mech Sci 188:105939. https://doi.org/10.1016/j.ijmecsci.2020.105939
Civalek Ö, Uzun B, Yaylı MÖ (2020) Stability analysis of nanobeams placed in electromagnetic field using a finite element method. Arab J Geosci 13(21):1–9. https://doi.org/10.1007/s12517-020-06188-8
Uzun B, Kafkas U, Yaylı MÖ (2021) Free vibration analysis of nanotube based sensors including rotary inertia based on the Rayleigh beam and modified couple stress theories. Microsyst Technol 27(5):1913–1923. https://doi.org/10.1007/s00542-020-04961-z
Akbas SD, Ersoy H, Akgöz B, Civalek O (2021) Dynamic analysis of a fiber-reinforced composite beam under a moving load by the Ritz method. Mathematics 9:1048. https://doi.org/10.3390/math9091048
Tuna M, Kırca M, Trovalusci P (2019) Deformation of atomic models and their equivalent continuum counterparts using Eringen’s two-phase local/nonlocal model. Mech Res Commun 97:26–32. https://doi.org/10.1016/j.mechrescom.2019.04.004
Barretta R, Faghidian SA, de Sciarra F (2020) A consistent variational formulation of Bishop nonlocal rods. Continuum Mech Thermodyn 32(5):1311–1323. https://doi.org/10.1007/s00161-019-00843-6
Güçlü G, Artan R (2020) Large elastic deflections of bars based on nonlocal elasticity. ZAMM J Appl Math Mech/Zeitschrift für Angew Math und Mech 100(4):e201900108. https://doi.org/10.1002/zamm.201900108
Reddy JN (2010) Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates. Int J Eng Sci 48(11):1507–1518. https://doi.org/10.1016/J.IJENGSCI.2010.09.020
Pham QH, Tran TT, Tran VK, Nguyen PC, Nguyen-Thoi T, Zenkour AM (2021) Bending and hygro-thermo-mechanical vibration analysis of a functionally graded porous sandwich nanoshell resting on elastic foundation. Mech Adv Mater Struct. https://doi.org/10.1080/15376494.2021.1968549
Civalek Ö, Uzun B, Yaylı MÖ (2020) Frequency, bending and buckling loads of nanobeams with different cross sections. Adv Nano Res 9(2):91–104. https://doi.org/10.12989/anr.2020.9.2.091
Arefi M, Mohammad-Rezaei Bidgoli E, Dimitri R, Bacciocchi M, Tornabene F (2019) Nonlocal bending analysis of curved nanobeams reinforced by graphene nanoplatelets. Compos Part B Eng 166:1–12. https://doi.org/10.1016/j.compositesb.2018.11.092
Hamed MA, Sadoun AM, Eltaher MA (2019) Effects of porosity models on static behavior of size dependent functionally graded beam. Struct Eng Mech 71(1):89–98. https://doi.org/10.12989/sem.2019.71.1.089
de Sciarra FM (2008) Variational formulations and a consistent finite-element procedure for a class of nonlocal elastic continua. Int J Solids Struct 45(14):4184–4202. https://doi.org/10.1016/j.ijsolstr.2008.03.003
Daghigh H, Daghigh V, Milani A, Tannant D, Lacy TE, Reddy JN (2020) Nonlocal bending and buckling of agglomerated CNT-reinforced composite nanoplates. Compos Part B Eng 183:107716. https://doi.org/10.1016/j.compositesb.2019.107716
Soltani M, Atoufi F, Mohri F, Dimitri R, Tornabene F (2021) Nonlocal elasticity theory for lateral stability analysis of tapered thin-walled nanobeams with axially varying materials. Thin-Walled Struct 159:107268. https://doi.org/10.1016/j.tws.2020.107268
Li YD, Bao R, Chen W (2018) Buckling of a piezoelectric nanobeam with interfacial imperfection and van der Waals force: is nonlocal effect really always dominant? Compos Struct 194:357–364. https://doi.org/10.1016/j.compstruct.2018.04.031
Arefi M, Amabili M (2021) A comprehensive electro-magneto-elastic buckling and bending analyses of three-layered doubly curved nanoshell, based on nonlocal three-dimensional theory. Compos Struct 257:113100. https://doi.org/10.1016/j.compstruct.2020.113100
Murmu T, Pradhan SC (2009) Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM. Phys E Low-dimensional Syst Nanostruct 41(7):1232–1239. https://doi.org/10.1016/J.PHYSE.2009.02.004
Uzun B, Kafkas U, Yaylı MÖ (2020) Axial dynamic analysis of a Bishop nanorod with arbitrary boundary conditions. ZAMM J. Appl. Math. Mech./Zeitschrift für Angew Math und Mech 100(12):e202000039. https://doi.org/10.1002/ZAMM.202000039
Lim CW, Islam MZ, Zhang G (2015) A nonlocal finite element method for torsional statics and dynamics of circular nanostructures. Int J Mech Sci 94–95:232–243. https://doi.org/10.1016/J.IJMECSCI.2015.03.002
Farajpour A, Shahidi AR, Mohammadi M, Mahzoon M (2012) Buckling of orthotropic micro/nanoscale plates under linearly varying in-plane load via nonlocal continuum mechanics. Compos Struct 94(5):1605–1615. https://doi.org/10.1016/J.COMPSTRUCT.2011.12.032
Kolahchi R, Zarei MS, Hajmohammad MH, Naddaf OA (2017) Visco-nonlocal-refined Zigzag theories for dynamic buckling of laminated nanoplates using differential cubature-Bolotin methods. Thin-Walled Struct 113:162–169. https://doi.org/10.1016/J.TWS.2017.01.016
Najafzadeh M, Adeli MM, Zarezadeh E, Hadi A (2020) Torsional vibration of the porous nanotube with an arbitrary cross-section based on couple stress theory under magnetic field. Mech Based Des Struct Mach. https://doi.org/10.1080/15397734.2020.1733602
Wang J, Zhou W, Huang Y, Lyu C, Chen W, Zhu W (2018) Controllable wave propagation in a weakly nonlinear monoatomic lattice chain with nonlocal interaction and active control. Appl Math Mech 39(8):1059–1070. https://doi.org/10.1007/s10483-018-2360-6
Eltaher MA, Khater ME, Emam SA (2016) A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams. Appl Math Model 40(5–6):4109–4128. https://doi.org/10.1016/J.APM.2015.11.026
Wang YQ, Liang C (2019) Wave propagation characteristics in nanoporous metal foam nanobeams. Results Phys 12:287–297. https://doi.org/10.1016/J.RINP.2018.11.080
Arefi M (2016) Surface effect and non-local elasticity in wave propagation of functionally graded piezoelectric nano-rod excited to applied voltage. Appl Math Mech 37(3):289–302. https://doi.org/10.1007/s10483-016-2039-6
Romano G, Barretta R, Diaco M, de Sciarra FM (2017) Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams. Int J Mech Sci 121:151–156. https://doi.org/10.1016/j.ijmecsci.2016.10.036
Uzun B, Yaylı MÖ (2020) A solution method for longitudinal vibrations of functionally graded nanorods. Int J Eng Appl Sci 12(2):78–87. https://doi.org/10.24107/ijeas.782419
Uzun B, Yaylı MÖ (2020) Nonlocal vibration analysis of Ti-6Al-4V/ZrO2 functionally graded nanobeam on elastic matrix. Arab J Geosci 13(4):1–10. https://doi.org/10.1007/s12517-020-5168-4
Berghouti H, Bedia EA, Benkhedda A, Tounsi A (2019) Vibration analysis of nonlocal porous nanobeams made of functionally graded material. Adv Nano Res 7:351–364. https://doi.org/10.12989/anr.2019.7.5.351
Shafiei N, Mirjavadi SS, MohaselAfshari B, Rabby S, Kazemi M (2017) Vibration of two-dimensional imperfect functionally graded (2D-FG) porous nano-/micro-beams. Comput Methods Appl Mech Eng 322:615–632. https://doi.org/10.1016/J.CMA.2017.05.007
Al-Maliki AF, Faleh NM, Alasadi AA (2019) Finite element formulation and vibration of nonlocal refined metal foam beams with symmetric and non-symmetric porosities. Struct Monit Maint 6(2):147–159. https://doi.org/10.12989/smm.2019.6.2.147
Ehyaei J, Akbarshahi A, Shafiei N (2017) Influence of porosity and axial preload on vibration behavior of rotating FG nanobeam. Adv Nano Res 5(2):141–169. https://doi.org/10.12989/anr.2017.5.2.141
Ghandourah EE, Abdraboh AM (2020) Dynamic analysis of functionally graded nonlocal nanobeam with different porosity models. Steel Compos Struct 36(3):293–305. https://doi.org/10.12989/SCS.2020.36.3.293
Rahmani A, Faroughi S, Friswell MI (2020) The vibration of two-dimensional imperfect functionally graded (2D-FG) porous rotating nanobeams based on general nonlocal theory. Mech Syst Signal Process 144:106854. https://doi.org/10.1016/J.YMSSP.2020.106854
Rastehkenari SF, Ghadiri M (2021) Nonlinear random vibrations of functionally graded porous nanobeams using equivalent linearization method. Appl Math Model 89:1847–1859. https://doi.org/10.1016/J.APM.2020.08.049
Hadji L, Avcar M (2021) Nonlocal free vibration analysis of porous FG nanobeams using hyperbolic shear deformation beam theory. Adv Nano Res 10(3):281–293. https://doi.org/10.12989/ANR.2021.10.3.281
Alasadi AA, Ahmed RA, Faleh NM (2019) Analyzing nonlinear vibrations of metal foam nanobeams with symmetric and non-symmetric porosities. Adv Aircr Spacecr Sci 6(4):273–282. https://doi.org/10.12989/aas.2019.6.4.273
Jalaei M, Civalek O (2019) On dynamic instability of magnetically embedded viscoelastic porous FG nanobeam. Int J Eng Sci 143:14–32. https://doi.org/10.1016/j.ijengsci.2019.06.013
Karličić DZ, Ayed S, Flaieh E (2019) Nonlocal axial vibration of the multiple Bishop nanorod system. Math Mech Solids 24(6):1668–1691. https://doi.org/10.1177/1081286518766577
Reddy JN (2002) Energy principles and variational methods in applied mechanics, 2nd edn. Wiley, New York
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Uzun, B., Kafkas, U., Deliktaş, B. et al. Size-Dependent Vibration of Porous Bishop Nanorod with Arbitrary Boundary Conditions and Nonlocal Elasticity Effects. J. Vib. Eng. Technol. 11, 809–826 (2023). https://doi.org/10.1007/s42417-022-00610-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s42417-022-00610-z