Acta Mechanica

, Volume 230, Issue 3, pp 1105–1128 | Cite as

On the nanoscale behaviour of single-wall C, BN and SiC nanotubes

  • Alessandra GenoeseEmail author
  • Andrea Genoese
  • Ginevra Salerno
Original Paper


The paper presents a numerical study of defect-free single-wall carbon, boron nitride and silicon carbide armchair and zigzag nanotubes, through a simple stick-and-spring model, based on Morse and cosine potential functions. The study investigates the relaxed configuration of the tubes and gives a comprehensive evaluation of their elastic constants, which is performed by framing tensile, torsional and radial tests within the membrane behaviour of a Donnell thin shell model. Extensive comparisons with reference ab-initio results are given and used to refine some parameters of the potential functions for hexagonal silicon carbide nanomaterials.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



  1. 1.
    Choi, W., Lahiri, I., Seelaboyina, R., Kang, Y.S.: Synthesis of graphene and its applications: a review. Crit. Rev. Solid State Mater. Sci. 35, 52–71 (2010)Google Scholar
  2. 2.
    De Volder, M.F.L., Tawfick, S.H., Baughman, R.H., Hart, A.J.: Carbon nanotubes: present and future commercial applications. Science 339, 535–539 (2013)Google Scholar
  3. 3.
    Sun, C., Wen, B., Bai, B.: Recent advances in nanoporous graphene membrane for gas separation and water purification. Sci. Bull. 60, 1807–1823 (2015)Google Scholar
  4. 4.
    Nguyen, B.H., Nguyen, V.H.: Promising applications of graphene and graphene-based nanostructures. Adv. Nat. Sci. Nanosci. Nanotechnol. 7, 023002 (2016)Google Scholar
  5. 5.
    Ates, M., Eker, A.A., Eker, B.: Carbon nanotube-based nanocomposites and their applications. J. Adhes. Sci. Technol. 31, 1977–1997 (2017)Google Scholar
  6. 6.
    Kumar, R., Singh, R., Hui, D., Feo, L., Fraternali, F.: Graphene as biomedical sensing element: state of art review and potential engineering applications. Compos. B Eng. 134, 193–206 (2018)Google Scholar
  7. 7.
    Mohan, V.B., Lau, K.-T., Hui, D., Bhattacharyya, D.: Graphene-based materials and their composites: a review on production, applications and product limitations. Compos. B Eng. 142, 200–220 (2018)Google Scholar
  8. 8.
    Chopra, N.G., Luyken, R.J., Cherry, K., Crespi, V.H., Cohen, M.L., Louie, S.G., et al.: Boron nitride nanotubes. Science 269, 966–967 (1995)Google Scholar
  9. 9.
    Sun, X.H., Li, C.P., Wong, W.K., Wong, N.B., Lee, C.S., Lee, S.T., et al.: Formation of silicon carbide nanotubes and nanowires via reaction of silicon (from disproportionation of silicon monoxide) with carbon nanotubes. J. Am. Chem. Soc. 124, 14464–14471 (2002)Google Scholar
  10. 10.
    Pacilé, D., Meyer, J.C., Girit, Ç.Ö., Zettl, A.: The two-dimensional phase of boron nitride: few-atomic-layer sheets and suspended membrane. Appl. Phys. Lett. 92, 133107 (2008)Google Scholar
  11. 11.
    Song, L., Ci, L., Lu, H., Sorokin, P.B., Jin, C., Ni, J., et al.: Large scale growth and characterization of atomic hexagonal boron nitride layers. Nano Lett. 10, 3209–3215 (2010)Google Scholar
  12. 12.
    Lin, S.S.: Light-emitting two-dimensional ultrathin silicon carbide. J. Phys. Chem. C 116, 3951–3955 (2012)Google Scholar
  13. 13.
    Casady, J.B., Johnson, R.W.: Status of silicon carbide (SiC) as a wide-bandgap semiconductor for high temperature applications: a review. Solid State Electron. 39, 1409–1422 (1996)Google Scholar
  14. 14.
    Suryavanshi, A.P., Yu, M.F., Wen, J., Tang, C., Bando, Y.: Elastic modulus and resonance behavior of boron nitride nanotubes. Appl. Phys. Lett. 84, 2527 (2004)Google Scholar
  15. 15.
    Falin, A., Cai, Q., Santos, E.J.G., Scullion, D., Qian, D., Zhang, R., et al.: Mechanical properties of atomically thin boron nitride and the role of interlayer interactions. Nat. Commun. 8, 15815 (2017)Google Scholar
  16. 16.
    Ouyang, T., Chen, Y., Xie, Y., Yang, K., Bao, Z., Zhong, J.: Thermal transport in hexagonal boron nitride nanoribbons. Nanotechnology 21, 245701 (2010)Google Scholar
  17. 17.
    Golberg, D., Bando, Y., Huang, Y., Terao, T., Mitome, M., Tang, C., et al.: Boron nitride nanotubes and nanosheets. ACS Nano 4, 2979–2993 (2010)Google Scholar
  18. 18.
    Kakay, S., Yilmaz, Z., Sen, O., Emanet, M., Kazanc, E., Çulha, M.: Synthesis of boron nitride nanotubes and their applications. Beilstein J. Nanotechnol. 6, 84–102 (2015)Google Scholar
  19. 19.
    Kumar, R., Parashar, A.: Atomistic modeling of BN nanofillers for mechanical and thermal properties: a review. Nanoscale 8, 22–49 (2016)Google Scholar
  20. 20.
    Li, Q., Liu, M., Zhang, Y., Liu, Z.: Hexagonal boron nitride–graphene heterostructures: synthesis and interfacial properties. Small 12, 32–50 (2016)Google Scholar
  21. 21.
    Mpourmpakis, G., Froudakis, G.E., Lithoxoos, G.P., Samios, J.: SiC nanotubes: a novel material for hydrogen storage. Nano Lett. 6, 1581–1583 (2006)Google Scholar
  22. 22.
    Zhang, G., Zhang, Y.-W.: Strain effects on thermoelectric properties of two-dimensional materials. Mech. Mater. 91, 382–398 (2015)Google Scholar
  23. 23.
    Shima, H.: Buckling of carbon nanotubes: a state of art review. Materials 5, 47–84 (2012)Google Scholar
  24. 24.
    Akinwande, D., Brennan, C.J., Scott Bunch, J., Egberts, P., Felts, J.R., Gao, H., et al.: A review on mechanics and mechanical properties of 2D materials-graphene and beyond. Extreme Mech. Lett. 13, 42–77 (2017)Google Scholar
  25. 25.
    Krishnan, A., Dujardin, E., Ebbesen, T.W., Yianilos, P.N., Treacy, M.M.J.: Young’s modulus of single-walled nanotubes. Phys. Rev. B 58, 14013–14019 (1998)Google Scholar
  26. 26.
    Sánchez-Portal, D., Artacho, E., Soler, J.M., Rubio, A., Ordejón, P.: Ab-initio structural, elastic, and vibrational properties of carbon nanotubes. Phys. Rev. B 59, 678–688 (1999)Google Scholar
  27. 27.
    Kudin, K.N., Scuseria, G.E., Yakobson, B.I.: \(\text{ C }_2\)F, BN, and C nanoshell elasticity from ab-initio computations. Phys. Rev. B 64, 235406 (2001)Google Scholar
  28. 28.
    Machón, M., Reich, S., Thomsen, C., Sánchez-Portal, D., Ordejón, P.: Ab-initio calculations of the optical properties of 4-Å-diameter single-walled nanotubes. Phys. Rev. B 66, 155410–5 (2002)Google Scholar
  29. 29.
    Cabria, I., Mintmire, J.W., White, C.T.: Metallic and semiconducting narrow carbon nanotubes. Phys. Rev. B 67, 121406(R) (2003)Google Scholar
  30. 30.
    Popov, V.N.: Curvature effects on the structural, electronic and optical properties of isolated single-walled nanotubes within a symmetry-adapted non-orthogonal tight-binding model. New J. Phys. 6, 1–18 (2004)MathSciNetGoogle Scholar
  31. 31.
    Bogár, F., Mintmire, J.W., Bartha, F., Mezö, T., Van Alsenoy, C.: Density-functional study of the mechanical and electronic properties of narrow carbon nanotubes under axial stress. Phys. Rev. B 72, 085452 (2005)Google Scholar
  32. 32.
    Jia, J.-F., Wu, H.-S., Jiao, H.: The structure and electronic property of BN nanotube. Physica B 381, 90–95 (2006)Google Scholar
  33. 33.
    Zhao, M., Xia, Y., Li, F., Zhang, R.Q., Lee, S.-T.: Strain energy and electronic structures of silicon carbide nanotubes: density functional calculations. Phys. Rev. B 71, 085312 (2005)Google Scholar
  34. 34.
    Baumeier, B., Krüger, P., Pollmann, J.: Structural, elastic, and electronic properties of SiC, BN, and BeO nanotubes. Phys. Rev. B 76, 085407 (2007)Google Scholar
  35. 35.
    Alam, K.M., Ray, A.K.: Hybrid density functional study of armchair SiC nanotubes. Phys. Rev. B 77, 035436 (2008)Google Scholar
  36. 36.
    Şahin, H., Cahangirov, S., Topsakal, M., Bekaroglu, E., Akturk, E., Senger, R.T., et al.: Monolayer honeycomb structures of group-IV elements and III–V binary compounds: first-principles calculations. Phys. Rev. B 80, 155453 (2009)Google Scholar
  37. 37.
    Domínguez-Rodríguez, G.: An assessment of finite element analysis to predict the elastic modulus and Poisson’s ratio of single wall carbon nanotubes. Comput. Mater. Sci. 82, 257–263 (2014)Google Scholar
  38. 38.
    Chandraseker, K., Mukherjee, S.: Atomistic-continuum and ab-initio estimation of the elastic moduli of single-walled carbon nanotubes. Comput. Mater. Sci. 40, 147–158 (2007)Google Scholar
  39. 39.
    Hung, N.T., Truong, D.V., Thanh, V.V., Saito, R.: Intrinsic strength and failure behaviors of ultra-small single-walled carbon nanotubes. Comput. Mater. Sci. 114, 167–171 (2016)Google Scholar
  40. 40.
    Peng, Y.-J., Zhang, L.-Y., Jin, Q.-H., Li, B.-H., Ding, D.-T.: Ab-initio studies of elastic properties and electronic structures of C and BN nanotubes. Physica E 33, 155–159 (2006)Google Scholar
  41. 41.
    Chang, T., Gao, H.: Size-dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model. J. Mech. Phys. Solids 51, 1059–1074 (2003)zbMATHGoogle Scholar
  42. 42.
    Meo, M., Rossi, M.: Prediction of Young’s modulus of single wall carbon nanotubes by molecular-mechanics based finite element modelling. Compos. Sci. Technol. 66, 1597–1605 (2006)Google Scholar
  43. 43.
    Verma, V., Jindal, V.K., Dharamvir, K.: Elastic moduli of a boron nitride nanotube. Nanotechnology 18, 435711 (2007)Google Scholar
  44. 44.
    Jindal, V.K., Imtani, A.N.: Bond lengths of armchair single-walled carbon nanotubes and their pressure dependence. Comput. Mater. Sci. 44, 156–162 (2008)Google Scholar
  45. 45.
    Meo, M., Rossi, M.: On the estimation of mechanical properties of single-walled carbon nanotubes by using a molecular-mechanics based FE approach. Compos. Sci. Technol. 69, 1394–1398 (2009)Google Scholar
  46. 46.
    Xiao, J.R., Staniszewski, J., Gillespie Jr., J.W.: Fracture and progressive failure of defective graphene sheets and carbon nanotubes. Compos. Struct. 88, 602–609 (2009)Google Scholar
  47. 47.
    Wernik, J.M., Meguid, S.A.: Atomistic-based continuum modeling of the nonlinear behavior of carbon nanotubes. Acta Mech. 212, 167–179 (2010)zbMATHGoogle Scholar
  48. 48.
    Berinskii, I.E., Krivtsov, A.M.: On using many-particle interatomic potentials to compute elastic properties of graphene and diamonds. Mech. Solut. 45, 815–883 (2010)Google Scholar
  49. 49.
    Oh, E.-S.: Elastic properties of boron-nitride nanotubes through the continuum lattice approach. Mater. Lett. 64, 859–862 (2010)Google Scholar
  50. 50.
    Jiang, L., Guo, W.: A molecular mechanics study on size-dependent elastic properties of single-walled boron nitride nanotubes. J. Mech. Phys. Solids 59, 1204–1213 (2011)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Oh, E.-S.: Elastic properties of various boron-nitride structures. Met. Mater. Int. 17, 21–27 (2011)Google Scholar
  52. 52.
    Parvaneh, V., Shariati, M.: Effect of defects and loading on prediction of Young’s modulus of SWCNTs. Acta Mech. 216, 281–289 (2011)zbMATHGoogle Scholar
  53. 53.
    Silvestre, N., Faria, B., Canongia Lopes, J.N.: A molecular dynamics study on the thickness and post-critical strength of carbon nanotubes. Compos. Struct. 94, 1352–1358 (2012)Google Scholar
  54. 54.
    Ansari, R., Mirnezhad, M., Sahmani, S.: An accurate molecular mechanics model for computation of size-dependent elastic properties of armchair and zigzag single-walled carbon nanotubes. Meccanica 48, 1355–1367 (2013)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Berinskii, I.E., Borodich, F.M.: Elastic in-plane properties of 2D linearized models of graphene. Mech. Mater. 62, 60–68 (2013)Google Scholar
  56. 56.
    Davini, C.: Homogenization of a graphene sheet. Contin. Mech. Thermodyn. 26, 95–113 (2014)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Korobeynikov, S.N., Alyokhin, V.V., Babichev, A.V.: Application of the molecular mechanics method to simulation of buckling of single-walled carbon nanotubes. Eng. Fract. Mech. 130, 83–95 (2014)Google Scholar
  58. 58.
    Le, M.-Q.: Atomistic study on the tensile properties of hexagonal AlN, BN, GaN, InN and SiC sheets. J. Comput. Theor. Nanosci. 11, 1458–1464 (2014)Google Scholar
  59. 59.
    Le, M.-Q., Nguyen, D.-T.: Atomistic simulations of pristine and defective hexagonal BN and SiC sheets under uniaxial tension. Mater. Sci. Eng. A 615, 481–488 (2014)Google Scholar
  60. 60.
    Le, M.-Q.: Young’s modulus prediction of hexagonal nanosheets and nanotubes based on dimensional analysis and atomistic simulations. Meccanica 49, 1709–1719 (2014)zbMATHGoogle Scholar
  61. 61.
    Le, M.-Q.: Prediction of the Young’s modulus of hexagonal monolayer sheets based on molecular mechanics. Int. J. Mech. Mater. Des. 11, 15–24 (2015)Google Scholar
  62. 62.
    Korobeynikov, S.N., Alyokhin, V.V., Annin, B.D., Babichev, A.V.: Quasi-static buckling simulation of single-layer graphene sheets by the molecular mechanics method. Math. Mech. Solids 20, 836–870 (2015)MathSciNetzbMATHGoogle Scholar
  63. 63.
    Gamboa, A., Vignoles, G.L., Leyssale, J.-M.: On the prediction of graphene’s elastic properties with reactive empirical bond order potential. Carbon 89, 176–187 (2015)Google Scholar
  64. 64.
    Tao, J., Xu, G., Sun, Y.: Elastic properties of boron-nitride nanotubes through an atomic simulation method. Math. Probl. Eng. 2015, 240547 (2015)Google Scholar
  65. 65.
    Yan, J.W., Liew, K.M.: Predicting elastic properties of single-walled boron nitride nanotubes and nanocones using an atomistic-continuum approach. Compos. Struct. 125, 489–498 (2015)Google Scholar
  66. 66.
    Favata, A., Micheletti, A., Podio-Guidugli, P., Pugno, N.: Geometry and self-stress of single-wall carbon nanotubes and graphene via a discrete model based on a 2nd-generation REBO potential. J. Elast. 125, 1–37 (2016)MathSciNetzbMATHGoogle Scholar
  67. 67.
    Genoese, A., Genoese, A., Rizzi, N.L., Salerno, G.: On the derivation of the elastic properties of lattice nanostructures: the case of graphene sheets. Compos. B Eng. 115, 316–329 (2017)Google Scholar
  68. 68.
    Merli, R., Lázaro, C., Monleón, S., Domingo, A.: Energy approach to the unstressed geometry of single walled carbon nanotubes. Meccanica 52, 213–230 (2017)MathSciNetzbMATHGoogle Scholar
  69. 69.
    Lee, H.-L., Wang, S.-W., Yang, Y.-C., Chang, W.-J.: Effect of porosity on the mechanical properties of a nanoporous graphene membrane using the atomic-scale finite element method. Acta Mech. 228, 2623–2629 (2017)Google Scholar
  70. 70.
    Budarapu, P.R., Javvaji, B., Sutrakar, V.K., Roy Mahapatra, D., Paggi, M., Zi, G., Rabczuk, T.: Lattice orientation and crack size effect on the mechanical properties of Graphene. Int. J. Fract. 203, 81–98 (2017)Google Scholar
  71. 71.
    Li, N., Ding, N., Qu, S., Liu, L., Guo, W., Wu, C.-H.L.: Mechanical properties and failure behavior of hexagonal boron nitride sheets with nano-cracks. Comput. Mater. Sci. 140, 356–366 (2017)Google Scholar
  72. 72.
    Natsuki, T., Natsuki, J.: Prediction of mechanical properties for hexagonal boron nitride nanosheets using molecular mechanics model. Appl. Phys. A 123, 283 (2017)Google Scholar
  73. 73.
    Tapia, A., Cab, C., Hernández-Pérez, A., Villanueva, C., Peñuñuri, F., Avilés, F.: The bond force constants and elastic properties of boron nitride nanosheets and nanoribbons using a hierarchical modelling approach. Physica E 89, 183–193 (2017)Google Scholar
  74. 74.
    Genoese, A., Genoese, A., Rizzi, N.L., Salerno, G.: Force constants of BN, SiC, AlN and GaN sheets through discrete homogenization. Meccanica 53, 593–611 (2018)MathSciNetGoogle Scholar
  75. 75.
    Hossain, M.Z., Hao, T., Silverman, B.: Stillinger-Weber potential for elastic and fracture properties in graphene and carbon nanotubes. J. Phys. Condens. Matter. 30, 055901 (2018)Google Scholar
  76. 76.
    Vijayaraghavan, V., Zhang, L.: Effective mechanical properties and thickness determination of boron nitride nanosheets using molecular dynamics simulation. Nanomaterials 8, 546 (2018)Google Scholar
  77. 77.
    Korobeynikov, S.N., Alyokhin, V.V., Babichev, A.V.: Simulation of mechanical parameters of graphene using the DREIDING force field. Acta Mech. 229, 2343–2378 (2018)MathSciNetGoogle Scholar
  78. 78.
    Genoese, A., Genoese, A., Salerno, G.: Elastic constants of achiral single-wall CNTs: analytical expressions and a focus on size and small scale effects. Compos. B Eng. 147, 207–226 (2018)Google Scholar
  79. 79.
    Wan, H., Delale, F.: A structural mechanics approach for predicting the mechanical properties of carbon nanotubes. Meccanica 45, 43–51 (2010)zbMATHGoogle Scholar
  80. 80.
    Tserpes, K.I.: Strength of graphenes containing randomly dispersed vacancies. Acta Mech. 223, 669–678 (2012)zbMATHGoogle Scholar
  81. 81.
    Torabi, H., Shariati, M., Sedaghat, E., Zadeh, A.L.: Buckling behavior of perfect and defective DWCNTs under axial, bending and torsional loadings via a structural mechanics approach. Meccanica 48, 1959–1974 (2013)zbMATHGoogle Scholar
  82. 82.
    Sakharova, N.A., Pereira, A.F.G., Antunes, J.M., Brett, C.M.A., Fernandes, J.V.: Mechanical characterization of single-walled carbon nanotubes: numerical simulation study. Compos. B Eng. 75, 73–85 (2015)Google Scholar
  83. 83.
    Giannopoulos, G.I., Kontoni, D.-P.N., Georgantzinos, S.K.: Efficient FEM simulation of static and free vibration behavior of single walled boron nitride nanotubes. Superlattice Microstruct. 96, 111–120 (2016)Google Scholar
  84. 84.
    Rafiee, R., Eskandariyun, A.: Comparative study on predicting Young’s modulus of graphene sheets using nano-scale continuum mechanics approach. Physica E 90, 42–48 (2017)Google Scholar
  85. 85.
    Giannopoulos, G.I.: On the buckling of hexagonal boron nitride nanoribbons via structural mechanics. Superlattices Microstruct. 115, 1–9 (2018)Google Scholar
  86. 86.
    Silvestre, N., Wang, C.M., Zhang, Y.Y., Xiang, Y.: Sanders shell model for buckling of single-walled carbon nanotubes with small aspect ratio. Compos. Struct. 93, 1683–1691 (2011)Google Scholar
  87. 87.
    Silvestre, N.: On the accuracy of shell models for torsional buckling of carbon nanotubes. Eur. J. Mech. A Solids 32, 103–108 (2012)zbMATHGoogle Scholar
  88. 88.
    Hollerer, S., Celigoj, C.C.: Buckling analysis of carbon nanotubes by a mixed atomistic and continuum model. Comput. Mech. 51, 765–789 (2013)MathSciNetzbMATHGoogle Scholar
  89. 89.
    Allahbakhshi, A., Allahbakhshi, M.: Vibration analysis of nano-structure multilayered graphene sheets using modified strain gradient theory. Front. Mech. Eng. 10, 187–197 (2015)Google Scholar
  90. 90.
    Fadaee, M., Ilkhani, M.R.: Study on the effect of an eccentric hole on the vibrational behavior of a graphene sheet using an analytical approach. Acta Mech. 226, 1395–1407 (2015)MathSciNetzbMATHGoogle Scholar
  91. 91.
    Aminpour, H., Rizzi, N.L.: A one-dimensional continuum with microstructure for single-wall carbon nanotubes bifurcation analysis. Math. Mech. Solids 21, 168–181 (2016)MathSciNetzbMATHGoogle Scholar
  92. 92.
    Jandaghian, A.A., Rahmani, O.: Buckling analysis of multi-layered graphene sheets based on a continuum mechanics model. Appl. Phys. A 123, 324 (2017)Google Scholar
  93. 93.
    Sahmani, S., Fattahi, A.M.: Development of efficient size-dependent plate models for axial buckling of single-layered graphene nanosheets using molecular dynamics simulation. Microsyst. Technol. 24, 1265–1277 (2018)Google Scholar
  94. 94.
    Singh, S., Patel, B.P.: A computationally efficient multiscale finite element formulation for dynamic and postbuckling analyses of carbon nanotubes. Comput. Struct. 195, 126–144 (2018)Google Scholar
  95. 95.
    Erhart, P., Albe, K.: Analytical potential for atomistic simulations of silicon, carbon and silicon carbide. Phys. Rev. B 71, 035211 (2005)Google Scholar
  96. 96.
    Rappé, A.K., Casewit, C.J., Colwell, K.S., Goddardlll, W.A., Skiff, W.M.: UFF, a full periodic table force field for molecular mechanics and molecular dynamics simulations. J. Am. Chem. Soc. 114, 10024–10035 (1992)Google Scholar
  97. 97.
    Donnell, L. H.: Stability of Thin-Walled Tubes Under Torsion. NACA 479 (1935)Google Scholar
  98. 98.
    Geuzaine, C., Remacle, J.-F.: Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng. 79, 1309–1331 (2009)zbMATHGoogle Scholar
  99. 99.
    Nozaki, H., Itoh, S.: Structural stability of \(\text{ BC }_2\)N. J. Phys. Chem. Solids 57, 41–49 (1996)Google Scholar
  100. 100.
    Malagù, M., Benvenuti, E., Simone, A.: One-dimensional nonlocal elasticity for tensile single-walled carbon nanotubes: a molecular structural mechanics characterization. Eur. J. Mech. A Solids 54, 160–170 (2015)Google Scholar
  101. 101.
    Barretta, R., Brčić, M., Čanađija, M., Luciano, R., Marotti-de-Sciarra, F.: Application of gradient elasticity to armchair carbon nanotubes: size effects and constitutive parameters assessment. Eur. J. Mech. A Solids 65, 1–13 (2017)MathSciNetzbMATHGoogle Scholar
  102. 102.
    Genoese, A., Genoese, A., Bilotta, A., Garcea, G.: Buckling analysis through a generalized beam model including section distortions. Thin Walled Struct. 85, 125–141 (2014)Google Scholar
  103. 103.
    Gabriele, S., Rizzi, N.L., Varano, V.: A 1D nonlinear TWB model accounting for in plane cross-section deformation. Int. J. Solids Struct. 94–95, 170–178 (2016)Google Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.LiMES, Dipartimento di ArchitetturaUniversità Roma TreRomaItaly

Personalised recommendations