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Nonlocal strain gradient torsion of elastic beams: variational formulation and constitutive boundary conditions

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Abstract

Nonlocal strain gradient continuum mechanics is a methodology widely employed in the literature to assess size effects in nano-structures. Notwithstanding this, improper higher-order boundary conditions (HOBC) are prescribed to close the corresponding elastostatic problems. In the present study, it is proven that HOBC have to be replaced with univocally determined boundary conditions of constitutive type, established by a consistent variational formulation. The treatment, developed in the framework of torsion of elastic beams, provides an effective approach to evaluate scale phenomena in smaller and smaller devices of engineering interest. Both elastostatic torsional responses and torsional-free vibrations of nano-beams are investigated by applying a simple analytical method. It is also underlined that the nonlocal strain gradient model, if equipped with the inappropriate HOBC, can lead to torsional structural responses which unacceptably do not exhibit nonlocality. The presented variational strategy is instead able to characterize significantly peculiar softening and stiffening behaviors of structures involved in modern nano-electro-mechanical systems.

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References

  1. Marotti de Sciarra, F., Russo, P.: Experimental Characterization, Predictive Mechanical and Thermal Modeling of Nanostructures and Their Polymer Composites. Elsevier, Amsterdam (2019). https://doi.org/10.1016/C2016-0-00081-5

    Book  Google Scholar 

  2. Lam, J.K., Koay, S.C., Lim, C.H., Cheah, K.H.: A voice coil based electromagnetic system for calibration of a sub-micronewton torsional thrust stand. Measurement 131, 597–604 (2019). https://doi.org/10.1016/j.measurement.2018.09.029

    Article  Google Scholar 

  3. Liu, D., Tarakanova, A., Hsu, C.C., Yu, M., Zheng, S., Yu, L., Liu, J., He, Y., Dunstan, D.J., Buehler, M.J.: Spider dragline silk as torsional actuator driven by humidity. Sci. Adv. 5, eaau9183 (2019). https://doi.org/10.1126/sciadv.aau9183

    Article  Google Scholar 

  4. Xiao, D., Xia, D., Li, Q., Hou, Z., Liu, G., Wang, X., Chen, Z., Wu, X.: A double differential torsional accelerometer with improved temperature robustness. Sens. Actuators A 243, 43–51 (2016). https://doi.org/10.1016/j.sna.2016.03.011

    Article  Google Scholar 

  5. Heinisch, M., Voglhuber-Brunnmaier, T., Reichel, E.K., Dufour, I., Jakoby, B.: Electromagnetically driven torsional resonators for viscosity and mass density sensing applications. Sens. Actuators A 229, 182–191 (2015). https://doi.org/10.1016/j.sna.2015.03.033

    Article  Google Scholar 

  6. Yue, Y.M., Xu, K.Y., Tan, Z.Q., Wang, W.J., Wang, D.: The influence of surface stress and surface-induced internal residual stresses on the size-dependent behaviors of Kirchhoff microplate. Arch. Appl. Mech. (2019). https://doi.org/10.1007/s00419-018-01504-x

    Article  Google Scholar 

  7. Dineva, P., Marinov, M., Rangelov, T.: Dynamic fracture of a nano-cracked finite exponentially inhomogeneous piezoelectric solid. Arch. Appl. Mech. (2019). https://doi.org/10.1007/s00419-018-01505-w

    Article  MATH  Google Scholar 

  8. Schopphoven, C., Birster, K., Schweitzer, R., Lux, C., Huang, S., Kästner, M., Auernhammer, G., Tschöpe, A.: Elastic deformations in semi-dilute Ni nanorod/hydrogel composites. Arch. Appl. Mech. 89, 119–132 (2019). https://doi.org/10.1007/s00419-018-1461-z

    Article  Google Scholar 

  9. Ouakad, H.M., Sedighi, H.M.: Static response and free vibration of MEMS arches assuming out-of-plane actuation pattern. Int. J. Non Linear Mech. 110, 44–57 (2019). https://doi.org/10.1016/j.ijnonlinmec.2018.12.011

    Article  Google Scholar 

  10. She, G.-L., Ren, Y.-R., Yuan, F.-G.: Hygro-thermal wave propagation in functionally graded double-layered nanotubes systems. Steel Compos. Struct. 31, 641–653 (2019). https://doi.org/10.12989/scs.2019.31.6.641

    Article  Google Scholar 

  11. Yang, W., Hu, T., Liang, X., Shen, S.: On band structures of layered phononic crystals with flexoelectricity. Arch. Appl. Mech. 88, 629–644 (2018). https://doi.org/10.1007/s00419-017-1332-z

    Article  Google Scholar 

  12. Numanoglu, H.M., Akgöz, B., Civalek, Ö.: On dynamic analysis of nanorods. Int. J. Eng. Sci. 130, 33–50 (2018). https://doi.org/10.1016/j.ijengsci.2018.05.001

    Article  Google Scholar 

  13. Li, X.B., Li, L., Hu, Y.J., Ding, Z., Deng, W.M.: Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory. Compos. Struct. 165, 250–265 (2017). https://doi.org/10.1016/j.compstruct.2017.01.032

    Article  Google Scholar 

  14. Hache, F., Challamel, N., Elishakoff, I., Wang, C.M.: Comparison of nonlocal continualization schemes for lattice beams and plates. Arch. Appl. Mech. 87, 1105–1138 (2017). https://doi.org/10.1007/s00419-017-1235-z

    Article  Google Scholar 

  15. Demir, Ç., Civalek, Ö.: On the analysis of microbeams. Int. J. Eng. Sci. 121, 14–33 (2017). https://doi.org/10.1016/j.ijengsci.2017.08.016

    Article  MathSciNet  MATH  Google Scholar 

  16. Demir, Ç., Civalek, Ö.: A new nonlocal FEM via Hermitian cubic shape functions for thermal vibration of nano beams surrounded by an elastic matrix. Compos. Struct. 168, 872–884 (2017). https://doi.org/10.1016/j.compstruct.2017.02.091

    Article  Google Scholar 

  17. Challamel, N., Kocsis, A., Wang, C.M., Lerbet, J.: From Ziegler to Beck’s column: a nonlocal approach. Arch. Appl. Mech. 86, 1095–1118 (2016). https://doi.org/10.1007/s00419-015-1081-9

    Article  Google Scholar 

  18. Barretta, R., Čanadija, M., Marotti de Sciarra, F.: A higher-order Eringen model for Bernoulli–Euler nanobeams. Arch. Appl. Mech. 86, 483–495 (2016). https://doi.org/10.1007/s00419-015-1037-0

    Article  MATH  Google Scholar 

  19. Mercan, K., Civalek, Ö.: DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix. Compos. Struct. 143, 300–309 (2016). https://doi.org/10.1016/j.compstruct.2016.02.040

    Article  Google Scholar 

  20. Barretta, R., Marotti de Sciarra, F.: Analogies between nonlocal and local Bernoulli–Euler nanobeams. Arch. Appl. Mech. 85, 89–99 (2015). https://doi.org/10.1007/s00419-014-0901-7

    Article  MATH  Google Scholar 

  21. Demir, Ç., Civalek, Ö.: Torsional and longitudinal frequency and wave response of microtubules based on the nonlocal continuum and nonlocal discrete models. Appl. Math. Model. 37, 9355–9367 (2013). https://doi.org/10.1016/j.apm.2013.04.050

    Article  MATH  Google Scholar 

  22. Li, C.: Torsional vibration of carbon nanotubes: comparison of two nonlocal models and a semi-continuum model. Int. J. Mech. Sci. 82, 25–31 (2014)

    Article  Google Scholar 

  23. Li, C.: A nonlocal analytical approach for torsion of cylindrical nanostructures and the existence of higher-order stress and geometric boundaries. Compos. Struct. 118, 607–621 (2014). https://doi.org/10.1016/j.compstruct.2014.08.008

    Article  Google Scholar 

  24. Lim, C.W., Islam, M.Z., Zhang, G.: A nonlocal finite element method for torsional statics and dynamics of circular nanostructures. Int. J. Mech. Sci. 94–95, 232–243 (2015). https://doi.org/10.1016/j.ijmecsci.2015.03.002

    Article  Google Scholar 

  25. Ieşan, D.: On the torsion of chiral bars in gradient elasticity. Int. J. Solids Struct. 50, 588–594 (2013). https://doi.org/10.1016/j.ijsolstr.2012.10.023

    Article  Google Scholar 

  26. Polyzos, D., Huber, G., Mylonakis, G., Triantafyllidis, T., Papargyri-Beskou, S., Beskos, D.: Torsional vibrations of a column of fine-grained material: a gradient elastic approach. J. Mech. Phys. Solids 76, 338–58 (2015). https://doi.org/10.1016/j.jmps.2014.11.012

    Article  MathSciNet  MATH  Google Scholar 

  27. Lazopoulos, K.A., Lazopoulos, A.K.: On the torsion problem of strain gradient elastic bars. Mech. Res. Commun. 45, 42–47 (2012). https://doi.org/10.1016/j.mechrescom.2012.06.007

    Article  Google Scholar 

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

    Article  Google Scholar 

  29. Barretta, R., Faghidian, S.A., Luciano, R., Medaglia, C.M., Penna, R.: Stress-driven two-phase integral elasticity for torsion of nano-beams. Compos. Part B 145, 62–69 (2018)

    Article  Google Scholar 

  30. Zhu, X., Li, L.: Twisting statics of functionally graded nanotubes using Eringen’s nonlocal integral model. Compos. Struct. 178, 87–96 (2017)

    Article  Google Scholar 

  31. Guo, S., He, Y., Liu, D., Lei, J., Shen, L., Li, Z.: Torsional vibration of carbon nanotube with axial velocity and velocity gradient effect. Int. J. Mech. Sci. 119, 88–96 (2016)

    Article  Google Scholar 

  32. Shen, Y., Chen, Y., Li, L.: Torsion of a functionally graded material. Int. J. Eng. Sci. 109, 14–28 (2016). https://doi.org/10.1016/j.ijengsci.2016.09.003

    Article  MathSciNet  MATH  Google Scholar 

  33. Farajpour, A., Ghayesh, M.H., Farokhi, H.: A review on the mechanics of nanostructures. Int. J. Eng. Sci. 133, 231–263 (2018). https://doi.org/10.1016/j.ijengsci.2018.09.006

    Article  MathSciNet  MATH  Google Scholar 

  34. Ghayesh, M.H., Farajpour, A.: A review on the mechanics of functionally graded nanoscale and microscale structures. Int. J. Eng. Sci. 137, 8–36 (2019). https://doi.org/10.1016/j.ijengsci.2018.12.001

    Article  MathSciNet  MATH  Google Scholar 

  35. Eringen, A.: Nonlocal Continuum Field Theories. Springer, New York (2002)

    MATH  Google Scholar 

  36. Romano, G., Luciano, R., Barretta, R., Diaco, M.: Nonlocal integral elasticity in nanostructures, mixtures, boundary effects and limit behaviours. Contin. Mech. Thermodyn. 30(3), 641–655 (2018). https://doi.org/10.1007/s00161-018-0631-0

    Article  MathSciNet  MATH  Google Scholar 

  37. Romano, G., Barretta, R.: Nonlocal elasticity in nanobeams: the stress-driven integral model. Int. J. Eng. Sci. 115, 14–27 (2017). https://doi.org/10.1016/j.ijengsci.2017.03.002

    Article  MathSciNet  MATH  Google Scholar 

  38. Barretta, R., Faghidian, S.A., Marotti de Sciarra, F.: Stress-driven nonlocal integral elasticity for axisymmetric nano-plates. Int. J. Eng. Sci. 136, 38–52 (2019). https://doi.org/10.1016/j.ijengsci.2019.01.003

    Article  MathSciNet  MATH  Google Scholar 

  39. Barretta, R., Canadija, M., Feo, L., Luciano, R., Marotti de Sciarra, F., Penna, R.: Exact solutions of inflected functionally graded nano-beams in integral elasticity. Compos. Part B 142, 273–286 (2018). https://doi.org/10.1016/j.compositesb.2017.12.022

    Article  Google Scholar 

  40. Barretta, R., Canadija, M., Luciano, R., Marotti de Sciarra, F.: Stress-driven modeling of nonlocal thermoelastic behavior of nanobeams. Int. J. Eng. Sci. 126, 53–67 (2018). https://doi.org/10.1016/j.ijengsci.2018.02.012

    Article  MathSciNet  MATH  Google Scholar 

  41. Barretta, R., Luciano, R., Marotti de Sciarra, F., Ruta, G.: Stress-driven nonlocal integral model for Timoshenko elastic nano-beams. Eur. J. Mech. A Solids 72, 275–286 (2018). https://doi.org/10.1016/j.euromechsol.2018.04.012

    Article  MathSciNet  MATH  Google Scholar 

  42. Barretta, R., Fabbrocino, F., Luciano, R., Marotti de Sciarra, F.: Closed-form solutions in stress-driven two-phase integral elasticity for bending of functionally graded nano-beams. Physica E 97, 13–30 (2018). https://doi.org/10.1016/j.physe.2017.09.026

    Article  Google Scholar 

  43. Barretta, R., Caporale, A., Faghidian, S.A., Luciano, R., Marotti de Sciarra, F., Medaglia, C.M.: A stress-driven local-nonlocal mixture model for Timoshenko nano-beams. Compos. Part B 164, 590–598 (2019). https://doi.org/10.1016/j.compositesb.2019.01.012

    Article  Google Scholar 

  44. Apuzzo, A., Barretta, R., Luciano, R., Marotti de Sciarra, F., Penna, R.: Free vibrations of Bernoulli–Euler nano-beams by the stress-driven nonlocal integral model. Compos. Part B 123, 105–111 (2017). https://doi.org/10.1016/j.compositesb.2017.03.057

    Article  Google Scholar 

  45. Apuzzo, A., Barretta, R., Fabbrocino, F., Faghidian, S.A., Luciano, R., Marotti de Sciarra, F.: Axial and torsional free vibrations of elastic nano-beams by stress-driven two-phase elasticity. Appl. Comput. Mech. 5, 402–413 (2019). https://doi.org/10.22055/jacm.2018.26552.1338

    Article  Google Scholar 

  46. Barretta, R., Faghidian, S.A., Luciano, R.: Longitudinal vibrations of nano-rods by stress-driven integral elasticity. Mech. Adv. Mater. Struct. 26, 1307–1315 (2019). https://doi.org/10.1080/15376494.2018.1432806

    Article  Google Scholar 

  47. Barretta, R., Faghidian, S.A., Luciano, R., Medaglia, C.M., Penna, R.: Free vibrations of FG elastic Timoshenko nano-beams by strain gradient and stress driven nonlocal models. Compos. Part B 154, 20–32 (2018). https://doi.org/10.1016/j.compositesb.2018.07.036

    Article  Google Scholar 

  48. Mahmoudpour, E., Hosseini-Hashemi, S.H., Faghidian, S.A.: Non-linear vibration analysis of FG nano-beams resting on elastic foundation in thermal environment using stress-driven nonlocal integral model. Appl. Math. Model. 57, 302–315 (2018). https://doi.org/10.1016/j.apm.2018.01.021

    Article  MathSciNet  MATH  Google Scholar 

  49. Barretta, R., Fabbrocino, F., Luciano, R., Marotti de Sciarra, F., Ruta, G.: Buckling loads of nano-beams in stress-driven nonlocal elasticity. Mech. Adv. Mater. Struct. (2019). https://doi.org/10.1080/15376494.2018.1501523

    Article  MATH  Google Scholar 

  50. Ansari, R., Gholami, R., Shojaei, M.F., Mohammadi, V., Sahmani, S.: Size-dependent bending, buckling and free vibration of functionally graded Timoshenko microbeams based on the most general strain gradient theory. Compos. Struct. 100, 385–397 (2013). https://doi.org/10.1016/j.compstruct.2012.12.048

    Article  Google Scholar 

  51. Nguyen, B.H., Zhuang, X., Rabczuk, T.: NURBS-based formulation for nonlinear electro-gradient elasticity in semiconductors. Comput. Methods Appl. Mech. Eng. 346, 1074–1095 (2019). https://doi.org/10.1016/j.cma.2018.08.026

    Article  MathSciNet  Google Scholar 

  52. Rouhi, H., Ebrahimi, F., Ansari, R., Torabi, J.: Nonlinear free and forced vibration analysis of Timoshenko nanobeams based on Mindlin’s second strain gradient theory. Eur. J. Mech. A Solids 73, 268–281 (2019). https://doi.org/10.1016/j.euromechsol.2018.09.005

    Article  MathSciNet  MATH  Google Scholar 

  53. Mercan, K., Civalek, Ö.: Buckling analysis of silicon carbide nanotubes (SiCNTs) with surface effect and nonlocal elasticity using the method of HDQ. Compos. Part B 114, 34–45 (2017). https://doi.org/10.1016/j.compositesb.2017.01.067

    Article  Google Scholar 

  54. Mercan, K., Numanoglu, H.M., Akgöz, B., Demir, C., Civalek, Ö.: Higher-order continuum theories for buckling response of silicon carbide nanowires (SiCNWs) on elastic matrix. Arch. Appl. Mech. 87, 1797–1814 (2017). https://doi.org/10.1007/s00419-017-1288-z

    Article  Google Scholar 

  55. Zhao, B., Liu, T., Chen, J., Peng, X., Song, Z.: A new Bernoulli–Euler beam model based on modified gradient elasticity. Arch. Appl. Mech. 89, 277–289 (2019). https://doi.org/10.1007/s00419-018-1464-9

    Article  Google Scholar 

  56. Barretta, R., Faghidian, S.A., Marotti de Sciarra, F.: Aifantis versus Lam strain gradient models of Bishop elastic rods. Acta Mech. 230, 2799–2812 (2019). https://doi.org/10.1007/s00707-019-02431-w

    Article  MathSciNet  MATH  Google Scholar 

  57. Thai, C.H., Ferreira, A.J.M., Rabczuk, T., Nguyen-Xuan, H.: Size-dependent analysis of FG-CNTRC microplates based on modified strain gradient elasticity theory. Eur. J. Mech. A Solids 72, 521–538 (2018). https://doi.org/10.1016/j.euromechsol.2018.07.012

    Article  MathSciNet  MATH  Google Scholar 

  58. Zhang, B., He, Y., Liu, D., Gan, Z., Shen, L.: Non-classical Timoshenko beam element based on the strain gradient elasticity theory. Finite Elem. Anal. Des. 79, 22–39 (2014). https://doi.org/10.1016/j.finel.2013.10.004

    Article  MathSciNet  Google Scholar 

  59. Kahrobaiyan, M.H., Asghari, M., Ahmadian, M.T.: Strain gradient beam element. Finite Elem. Anal. Des. 68, 63–75 (2013). https://doi.org/10.1016/j.finel.2012.12.006

    Article  MathSciNet  MATH  Google Scholar 

  60. Kandaz, M., Dal, H.: A comparative study of modified strain gradient theory and modified couple stress theory for gold microbeams. Arch. Appl. Mech. 88, 2051–2070 (2018). https://doi.org/10.1007/s00419-018-1436-0

    Article  Google Scholar 

  61. Akgöz, B., Civalek, Ö.: Effects of thermal and shear deformation on vibration response of functionally graded thick composite microbeams. Compos. Part B 129, 77–87 (2017). https://doi.org/10.1016/j.compositesb.2017.07.024

    Article  Google Scholar 

  62. Akgöz, B., Civalek, Ö.: Buckling analysis of cantilever carbon nanotubes using the strain gradient elasticity and modified couple stress theories. J. Comput. Theor. Nanosci. 8, 1821–1827 (2011). https://doi.org/10.1166/jctn.2011.1888

    Article  MATH  Google Scholar 

  63. Fuschi, P., Pisano, A.A., Polizzotto, C.: Size effects of small-scale beams in bending addressed with a strain-difference based nonlocal elasticity theory. Int. J. Mech. Sci. 151, 661–671 (2019). https://doi.org/10.1016/j.ijmecsci.2018.12.024

    Article  Google Scholar 

  64. Aifantis, E.C.: Update on a class of gradient theories. Mech. Mater. 35, 259–280 (2003). https://doi.org/10.1016/S0167-6636(02)00278-8

    Article  Google Scholar 

  65. Aifantis, E.C.: On the gradient approach—relation to Eringen’s nonlocal theory. Int. J. Eng. Sci. 49, 1367–1377 (2011). https://doi.org/10.1016/j.ijengsci.2011.03.016

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  67. Barretta, R., Marotti de Sciarra, F.: Constitutive boundary conditions for nonlocal strain gradient elastic nano-beams. Int. J. Eng. Sci. 130, 187–198 (2018)

    Article  MathSciNet  Google Scholar 

  68. Barretta, R., Čanadija, M., Marotti de Sciarra, F.: Modified nonlocal strain gradient elasticity for nano-rods and application to carbon nanotubes. Appl. Sci. 9, 514 (2019). https://doi.org/10.3390/app9030514

    Article  Google Scholar 

  69. Apuzzo, A., Barretta, R., Faghidian, S.A., Luciano, R., Marotti de Sciarra, F.: Nonlocal strain gradient exact solutions for functionally graded inflected nano-beams. Compos. Part B 164, 667–674 (2019). https://doi.org/10.1016/j.compositesb.2018.12.112

    Article  Google Scholar 

  70. Apuzzo, A., Barretta, R., Faghidian, S.A., Luciano, R., Marotti de Sciarra, F.: Free vibrations of elastic beams by modified nonlocal strain gradient theory. Int. J. Eng. Sci. 133, 99–108 (2018). https://doi.org/10.1016/j.ijengsci.2018.09.002

    Article  MathSciNet  MATH  Google Scholar 

  71. Zaera, R., Serrano, Ó., Fernández-Sáez, J.: On the consistency of the nonlocal strain gradient elasticity. Int. J. Eng. Sci. 138, 65–81 (2019). https://doi.org/10.1016/j.ijengsci.2019.02.004

    Article  MathSciNet  MATH  Google Scholar 

  72. Romano, G., Barretta, A., Barretta, R.: On torsion and shear of Saint–Venant beams. Eur. J. Mech. Solids 35, 47–60 (2012). https://doi.org/10.1016/j.euromechsol.2012.01.007

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Financial supports from the Italian Ministry of Education, University and Research (MIUR) in the framework of the Project PRIN 2015 “COAN 5.50.16.01”—code 2015JW9NJT—and from the research program ReLUIS 2019 are gratefully acknowledged.

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  1. Department of Structures for Engineering and Architecture, University of Naples Federico II, Via Claudio 21, 80125, Naples, Italy

    R. Barretta, S. Ali Faghidian, Francesco Marotti de Sciarra & M. S. Vaccaro

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  1. R. Barretta
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Barretta, R., Faghidian, S.A., Marotti de Sciarra, F. et al. Nonlocal strain gradient torsion of elastic beams: variational formulation and constitutive boundary conditions. Arch Appl Mech 90, 691–706 (2020). https://doi.org/10.1007/s00419-019-01634-w

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