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Exact nonlocal solutions of circular nanoplates subjected to uniformly distributed loads and nonlocal concentrated forces

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Abstract

The nonlocal solutions of the bending problem of the circular elastic nanoplates subjected to uniformly distributed loads by using the Eringen’s nonlocal theory are approximate. Seeking for an exact solution of the concerning problem by adding a concentrated force; a specific concentrated force which depends on the nonlocal parameter and, therefore, vanishes in the local case, is obtained. In this context; starting from the local bending solution of the simply supported circular nanoplates under uniformly distributed loads and evolving the solution step by step, exact nonlocal solutions of the bending problem of the simply supported and clamped circular nanoplates subjected to uniformly distributed loads and the concerning nonlocal concentrated forces acting at the centers of the circular nanoplates are obtained with the expectation that the exact nonlocal solution is not very different than the corresponding exact local solution. The closed-form solutions are checked to satisfy exactly the equations of equilibrium, constitutive equations and the boundary conditions of the relevant nonlocal problem. The application of the concerning specific concentrated force, in addition to the uniformly distributed load, is seen to make the radial bending moments for the simply supported circular nanoplate and the deflections for the clamped circular nanoplate free from the effects of the nonlocality.

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Notes

  1. For the acceptable values of the nondimensional nonlocal parameter \( \mu^{*} \), Ref. [51] can be referred.

References

  1. Eringen AC (1983) On differential-equations of nonlocal elasticity and solutions of screw dislocation and surface-waves. J Appl Phys 54:4703

    Google Scholar 

  2. Eringen AC (1992) Vistas of nonlocal continuum physics. Int J Eng Sci 30:1551–1565

    MathSciNet  MATH  Google Scholar 

  3. Eringen AC (2002) Nonlocal continuum field theories. Springer, New York

    MATH  Google Scholar 

  4. Ansari R, Rouhi H, Sahmani S (2011) Calibration of the analytical nonlocal shell model for vibrations of double-walled carbon nano-tubes with arbitrary boundary conditions using molecular dynamics. Int J Mech Sci 53:786–792

    Google Scholar 

  5. Kiani K (2011) Small-scale effect on the vibration of thin nanoplates subjected to a moving nanoparticle via nonlocal continuum theory. J Sound Vib 330:4896–5914

    Google Scholar 

  6. Romano G, Barretta R, Diaco M, Francesco MS (2017) Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams. Int J Mech Sci 121:151–156

    Google Scholar 

  7. Fernández-Sáeza J, Zaera R, Loya JA, Reddy JN (2016) Bending of Euler–Bernoulli beams using Eringen’s integral formulation: a paradox resolved. Int J Eng Sci 99:107–116

    MathSciNet  MATH  Google Scholar 

  8. Tuna M, Kirca M (2016) Exact solution of Eringen’s nonlocal integral model for bending of Euler–Bernoulli and Timoshenko beams. Int J Eng Sci 105:80–92

    MathSciNet  MATH  Google Scholar 

  9. Reddy JN, El-Borgi S (2014) Eringen’s nonlocal theories of beams accounting for moderate rotations. Int J Eng Sci 82:159–177

    MathSciNet  MATH  Google Scholar 

  10. Sciarra FM, Barretta R (2014) A new nonlocal bending model for Euler–Bernoulli nanobeams. Mech Res Commun 62:25–30

    Google Scholar 

  11. Li C (2014) A nonlocal analytical approach for torsion of cylindrical nanostructures and the existence of higher-order stress and geometric boundaries. Compos Struct 118:607–621

    Google Scholar 

  12. Challamel N, Kocsis A, Wang CM (2015) Discrete and non-local elastica. Int J Non Linear Mech 77:128–140

    Google Scholar 

  13. Ghavanlooa E, Fazelzadeha SA (2015) Nonlocal shell model for predicting axisymmetric vibration of spherical shell-like nanostructures. Mech Adv Mater Struct 22(7):597–603. https://doi.org/10.1080/15376494.2013.828816

    Article  Google Scholar 

  14. Sari MS (2015) Free vibration analysis of non-local annular sector Mindlin plates. Int J Mech Sci 96–97:25–35

    Google Scholar 

  15. Reddy JN (2016) Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates. Int J Eng Sci 48:1507–1518

    MathSciNet  MATH  Google Scholar 

  16. Khorshidi K, Fallah A (2016) Buckling analysis of functionally graded rectangular nano-plate based on nonlocal exponential shear deformation theory. Int J Mech Sci 113:94–104

    Google Scholar 

  17. Shafiei N, Ebrahimi F, Kazemi M, Abdollahi SMM (2016) Thermo-mechanical vibration analysis of rotating nonlocal nanoplates applying generalized differential quadrature method. Mech Adv Mater Struct. https://doi.org/10.1080/15376494.2016.1227499

    Article  Google Scholar 

  18. Wu CP, Liou JY (2016) RMVT-based nonlocal Timoshenko beam theory for stability analysis of embedded single-walled carbon nanotube with various boundary conditions. Int J Struct Stab Dyn 16(10) Article Number: 1550068

    MathSciNet  MATH  Google Scholar 

  19. Golmakani ME, Vahabi H (2016) Nonlocal buckling analysis of functionally graded annular nanoplates in an elastic medium with various boundary conditions. Microsyst Technol Micro Nanosyst Inf Storage Process Syst 23(8):3613–3628

    Google Scholar 

  20. Khaniki HB, Hosseini-Hashemi Sh (2017) Dynamic response of biaxially loaded double-layer viscoelastic orthotropic nanoplate system under a moving nanoparticle. Int J Eng Sci 115:51–72

    MathSciNet  MATH  Google Scholar 

  21. Tsiatas GC, Babouskos NG (2017) Linear and geometrically nonlinear analysis of non-uniform shallow arches under a central concentrated force. Int J Non Linear Mech 92:92–101

    Google Scholar 

  22. Mortazavi P, Mirdamadi HR, Shahidi AR (2017) Post-buckling, limit point, and bifurcation analyses of shallow nano-arches by generalized displacement control and finite difference considering small-scale effects. Int J Struct Stab Dyn 18(1):1850014

    MathSciNet  Google Scholar 

  23. Mirjavadi SS, Afshari BM, Khezel M, et al. (2018) Nonlinear vibration and buckling of functionally graded porous nanoscaled beams. J Braz Soc Mech Sci Eng 40(7), Article Number: UNSP 352

  24. Shiva K, Raghu P, Rajagopal A, et al. (2019) Nonlocal buckling analysis of laminated composite plates considering surface stress effects. Comp Struct 226, Article Number: UNSP 111216

  25. Shasavari D, Karami B, Janghorban M (2019) Size-dependent vibration analysis of laminated composite plates. Adv Nano Search 7(5):337–349

    Google Scholar 

  26. Aria AI, Rabczuk T, Friswell MI (2019) A finite element model for the thermo-elastic analysis of functionally graded porous nanobeams. Eur J Mech A Solids 77, Article Number: UNSP 103767

  27. Karami B, Shahsavari D, Janghormn M, et al. (2019) Elastic guided waves in fully-clamped functionally graded carbon nanotube-reinforced composite plates. Mater Res Express 6 (9), Article Number: 0950a9

    Google Scholar 

  28. Arefi M (2019) Magneto-electro-mechanical size-dependent vibration analysis of three-layered nanobeam with initial curvature considering thickness stretching. Int J Nano Dimens 10(1):48–61

    Google Scholar 

  29. Thai BN, Reddy JN, Rungamornrat J, et al. (2019) Nonlinear analysis for bending, buckling and post-buckling of nano-beams with nonlocal and surface energy effects. Int J Struct Stab Dyn 19 (11), Article Number: 1950130

  30. Wu CP, Yu JJ (2019) A review of mechanical analyses of rectangular nanobeams and single-, double-, and multi-walled carbon nanotubes using Eringen’s nonlocal elasticity theory. Arch Appl Mech 89(9):1761–1792

    Google Scholar 

  31. Karami B, Janghorban M, Shahsavari D et al (2019) Nonlocal buckling analysis of composite curved beams reinforced with functionally graded carbon nanotubes. Molecules 24(15), Article Number: 2750

  32. Dastjerdi S, Abbasi M, Yazdanparast L (2017) A new modified higher-order shear deformation theory for nonlinear analysis of macro- and nano- annular sector plates using the extended Kantorovich method in conjunction with SAPM. Acta Mech 228(10):3381–3401

    MathSciNet  MATH  Google Scholar 

  33. Yang WD, Kang WB, Wang X (2017) Thermal and surface effects on the pull-in characteristics of circular nanoplate NEMS actuator based on nonlocal elasticity. Appl Math Model 63:321–336

    MathSciNet  Google Scholar 

  34. Heydari A (2018) Size-dependent damped vibrations and buckling analyses of bidirectional functionally graded solid circular nano-plate with arbitrary thickness variation. Struct Eng Mech 68(2):171–182

    Google Scholar 

  35. Allahyari E, Asgari M (2019) Effect of magnetic-thermal field on nonlinear wave propagation of circular nanoplates. J Electromagn Waves Appl 33(17):2296–2316

    Google Scholar 

  36. Rajabi F, Ramezani S (2012) A nonlinear microbeam model based on strain gradient elasticity theory with surface energy. Arch Appl Mech 82:363–376

    MATH  Google Scholar 

  37. Sidhardh S, Ray MC (2018) Exact solutions for elastic response in micro- and nano-beams considering strain gradient elasticity. Math Mech Solids. https://doi.org/10.1177/1081286518761182

    Article  MATH  Google Scholar 

  38. Shen J, Wang H, Zheng S (2018) Size dependent pull-in analysis of a composite laminated micro-beam actuated by electrostatic and piezoelectric forces: generalized differential quadrature method. Int J Mech Sci 135:353–361

    Google Scholar 

  39. Tuna M, Kirca M (2019) Unification of Eringen’s nonlocal parameter through an optimization-based approach. Mech Adv Mater Struct. https://doi.org/10.1080/15376494.2019.1601312

    Article  Google Scholar 

  40. Barretta R, Marotti de Sciarra F (2019) Stress-driven nonlocal integral elasticity for axisymmetric nano-plates. Int J Eng Sci 136:38–52

    MathSciNet  MATH  Google Scholar 

  41. Lim CW, Reddy JN, Zhang G (2015) A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J Mech Phys Solids 78:298–313

    MathSciNet  MATH  Google Scholar 

  42. Barretta R, Luciano R, Marotti de Sciarra F, Ruta G (2018) Stress-driven nonlocal integral model for Timoshenko elastic nano-beams. Eur J Mech A Solids 72:275–286

    MathSciNet  MATH  Google Scholar 

  43. Marotti de Sciarra F (2009) On non-local and non-homogeneous elastic continua. Int J Solids Struct 46(3–4):651–676

    MathSciNet  MATH  Google Scholar 

  44. Zhang N, Yan JW, Li C, Zhou JX (2019) Combined bending-tension/compression deformation of micro-bars accounting for strain-driven long-range interactions. Arch Mech 71(1):3–21

    MathSciNet  MATH  Google Scholar 

  45. Liu JJ, Chen L, Xie F, Fan XL, Li C (2016) On bending, buckling and vibration of graphene nanosheets based on the nonlocal theory. Smart Struct Syst 17(2):257–274

    Google Scholar 

  46. Li C, Yao LQ, Chen WQ, Li S (2015) Comments on nonlocal effects in nano-cantilever beams. Int J Eng Sci 87:47–57

    Google Scholar 

  47. Duan WH, Wang CM (2007) Exact solutions for axisymmetric bending of micro/nanoscale circular plates based on nonlocal plate theory. Nanotechnology. https://doi.org/10.1088/0957-4484/18/38/385704

    Article  Google Scholar 

  48. Yan JW, Tong LH, Li C, Zhu Y, Wang ZW (2015) Exact solutions of bending deflections for nano-beams and nano-plates based on nonlocal elasticity theory. Compos Struct 125:304–313

    Google Scholar 

  49. Timoshenko S, Woinowsky-Krieger S (1955) Theory of plates and shells, 2nd edn. McGraw-Hill Book Company, Hamburg, pp 51–72

    MATH  Google Scholar 

  50. Kreyszig E (In collaboration with H. Kreyszig, and E. J. Norminton) (2011) Advanced engineering mathematics, 10th edn. Wiley, New York, pp 180–185

    Google Scholar 

  51. Li C, Lai SK, Yang X (2019) On the nano-structural dependence of nonlocal dynamics and its relationship to the upper limit of nonlocal scale parameter. Appl Math Model 69:127–141

    MathSciNet  Google Scholar 

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  1. Department of Civil Engineering, Yıldız Teknik Üniversitesi, Davutpasa Campus, 34220, Esenler, Istanbul, Turkey

    R. Faruk Yükseler

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Yükseler, R.F. Exact nonlocal solutions of circular nanoplates subjected to uniformly distributed loads and nonlocal concentrated forces. J Braz. Soc. Mech. Sci. Eng. 42, 61 (2020). https://doi.org/10.1007/s40430-019-2144-6

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