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Recent Developments and Future Challenges in the Application of Nonlocal Elasticity Theory

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Computational Continuum Mechanics of Nanoscopic Structures

Abstract

So far, we have provided very comprehensive general formulations of the standard nonlocal continuum-based models and have considered their detailed applications in the investigation of various mechanical characteristics and functioning of a variety of nanoscopic structures. Recent developments in the nonlocal continuum mechanics have led to the emergence of some new ideas in the specialized literature. These ideas are different from the conventional ideas discussed so far in this book. To complement and complete our discussions about the nonlocal elasticity approaches, we include in this chapter some of the important concepts that have been proposed in these latest studies.

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References

  1. N. Challamel, C.M. Wang, The small length scale effect for a non-local cantilever beam: a paradox solved. Nanotechnology 19, 345703 (2008)

    Article  Google Scholar 

  2. J. Peddieson, G.R. Buchanan, R.P. McNitt, Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci. 41, 305–312 (2003)

    Article  Google Scholar 

  3. C. Li, L. Yao, W. Chen, S. Li, Comments on nonlocal effects in nano-cantilever beams. Int. J. Eng. Sci. 87, 47–57 (2015)

    Article  Google Scholar 

  4. M.A. Eltaher, A.E. Alshorbagy, F.F. Mahmoud, Vibration analysis of Euler-Bernoulli nanobeams by using finite element method. Appl. Math. Model. 37, 4787–4797 (2013)

    Article  MathSciNet  Google Scholar 

  5. M. Tashakorian, E. Ghavanloo, S.A. Fazelzadeh. D.H. Hodges, Nonlocal fully intrinsic equations for free vibration of Euler-Bernoulli beams with constitutive boundary conditions. Acta Mech. 229, 3279–3292 (2018)

    Article  MathSciNet  Google Scholar 

  6. E. Benvenuti, A. Simone, One-dimensional nonlocal and gradient elasticity: closed-form solution and size effect. Mech. Res. Commun. 48, 46–51 (2013)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. G. Romano, R. Barretta, M. Diaco, F. Marotti de Sciarra, Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams. Int. J. Mech. Sci. 121, 151–156 (2017)

    Article  Google Scholar 

  9. N. Challamel, C.M. Wang, I. Elishakoff, Discrete systems behave as nonlocal structural elements: bending, buckling and vibration analysis. Eur. J. Mech. A Solids 44, 125–135 (2014)

    Article  MathSciNet  Google Scholar 

  10. N. Challamel1, J. Lerbet, C.M. Wang, Z. Zhang, Analytical length scale calibration of nonlocal continuum from a microstructured buckling model. ZAMM-Z Angew. Math. Mech. 94, 402–413 (2014)

    Article  MathSciNet  Google Scholar 

  11. F. Hache, N. Challamel, I. Elishakoff, C.M. Wang, Comparison of nonlocal continualization schemes for lattice beams and plates. Arch. Appl. Mech. 87, 1105–1138 (2017)

    Article  Google Scholar 

  12. P. Khodabakhshi, J.N. Reddy, A unified integro-differential nonlocal model. Int. J. Eng. Sci. 95, 60–75 (2015)

    Article  MathSciNet  Google Scholar 

  13. C.C. Koutsoumaris, K.G. Eptaimeros, G.J. Tsamasphyros, A different approach to Eringen’s nonlocal integral stress model with applications for beams. Int. J. Solids Struct. 112, 222–238 (2017)

    Article  Google Scholar 

  14. G. Romano, R. Barretta, Stress-driven versus strain-driven nonlocal integral model for elastic nano-beams. Compos. Part B 114, 184–188 (2017)

    Article  Google Scholar 

  15. J.N. Reddy, Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45, 288–307 (2007)

    Article  Google Scholar 

  16. J.N. Reddy, Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates. Int. J. Eng. Sci. 48, 1507–1518 (2010)

    Article  MathSciNet  Google Scholar 

  17. S. Adali, Variational principles for multi-walled carbon nanotubes undergoing buckling based on nonlocal elasticity theory. Phys. Lett. A 372, 5701–5705 (2008)

    Article  Google Scholar 

  18. S. Adali, Variational principles for transversely vibrating multi-walled carbon nanotubes based on nonlocal Euler-Bernoulli beam models. Nano Lett. 9, 1737–1741 (2009)

    Article  Google Scholar 

  19. S. Adali, Variational principles for nonlocal continuum model of orthotropic graphene sheets embedded in an elastic medium. Acta Math. Sci. 32, 325–338 (2012)

    Article  MathSciNet  Google Scholar 

  20. N. Challamel, Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams. Compos. Struct. 105, 351–368 (2013)

    Article  Google Scholar 

  21. N. Challamel, Z. Zhang, C.M. Wang, J.N. Reddy, Q. Wang, T. Michelitsch, B. Collet, On nonconservativeness of Eringen’s nonlocal elasticity in beam mechanics: correction from a discrete-based approach. Arch. Appl. Mech. 84, 1275–1292 (2014)

    Article  Google Scholar 

  22. M. Taghizadeh, H.R. Ovesy, S.A.M. Ghannadpour, Beam buckling analysis by nonlocal integral elasticity finite element method. Int. J. Struct. Stab. Dyn. 16, 1550015 (2016)

    Article  MathSciNet  Google Scholar 

  23. A.C. Eringen, Nonlocal Continuum Field Theories (Springer, New York, 2002)

    MATH  Google Scholar 

  24. D.H. Hodges, Geometrically exact, intrinsic theory for dynamics of curved and twisted anisotropic beams. AIAA J. 41, 1131–1137 (2003)

    Article  Google Scholar 

  25. H. Hencky, ber die angenherte losung von stabilittsproblemen im raum mittels der elastischen gelenkkette. Der Eisenbau 11, 437–452 (1920)

    Google Scholar 

  26. Z. Zhang, C.M. Wang, N. Challamel, I. Elishakoff, Obtaining Eringen’s length scale coefficient for vibrating nonlocal beams via continualization method. J. Sound Vib. 333, 4977–4990 (2014)

    Article  Google Scholar 

  27. C.M. Wang, Z. Zhang, N. Challamel, W.H. Duan, Calibration of Eringen’s small length scale coefficient for initially stressed vibrating nonlocal Euler beams based on microstructured beam model. J. Phys. D Appl. Phys. 46, 345501 (2013)

    Article  Google Scholar 

  28. F. Hache, N. Challamel, I. Elishakoff, Nonlocal approaches for the vibration of lattice plates including both shear and bending interactions. Int. J. Struct. Stab. Dyn. 18, 1850094 (2018)

    Article  MathSciNet  Google Scholar 

  29. B. Hérisson, N. Challamel, V. Picandet, A. Perrot, C.M. Wang, Static and dynamic behaviors of microstructured membranes within nonlocal mechanics. J. Eng. Mech. 144, 04017155 (2018)

    Article  Google Scholar 

  30. A.C. Eringen, Linear theory of nonlocal elasticity and dispersion of plane waves. Int. J. Eng. Sci. 10, 425–435 (1972)

    Article  Google Scholar 

  31. H. Rafii-Tabar, E. Ghavanloo, S.A. Fazelzadeh, Nonlocal continuum-based modeling of mechanical characteristics of nanoscopic structures. Phys. Rep. 638, 1–97 (2016)

    Article  MathSciNet  Google Scholar 

  32. G. Romano, R. Barretta, Nonlocal elasticity in nanobeams: the stress-driven integral model. Int. J. Eng. Sci. 115, 14–27 (2017)

    Article  MathSciNet  Google Scholar 

  33. A. Apuzzo, R. Barretta, R. Luciano, F. Marotti de Sciarra, R. Penna, Free vibrations of Bernoulli-Euler nano-beams by the stress-driven nonlocal integral model. Compos. Part B 123, 105–111 (2017)

    Article  Google Scholar 

  34. R. Barretta, M. Diaco, L. Feo, R. Luciano, F. Marotti de Sciarra, R. Penna, Stress-driven integral elastic theory for torsion of nano-beams. Mech. Res. Commun. 87, 35–41 (2018)

    Article  Google Scholar 

  35. R. Barretta, S.A. Fazelzadeh, L. Feo, E. Ghavanloo, R. Luciano, Nonlocal inflected nano-beams: A stress-driven approach of bi-Helmholtz type. Comp. Struct. 200, 239–245 (2018)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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Correspondence to Esmaeal Ghavanloo .

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Ghavanloo, E., Rafii-Tabar, H., Fazelzadeh, S.A. (2019). Recent Developments and Future Challenges in the Application of Nonlocal Elasticity Theory. In: Computational Continuum Mechanics of Nanoscopic Structures. Springer Tracts in Mechanical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-11650-7_12

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  • DOI: https://doi.org/10.1007/978-3-030-11650-7_12

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