International Journal of Engineering Science 154 (2020) 103338 Contents lists available at ScienceDirect International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci Higher modes of buckling in shear deformable nanobeams Hossein Darban a , Raimondo Luciano a,∗ , Andrea Caporale b , Francesco Fabbrocino c a Department of Engineering, University of Naples Parthenope, 80133 Naples, Italy b Department of Civil and Mechanical Engineering, University of Cassino and Southern Lazio, 03043 Cassino, Italy c Department of Engineering, Pegaso Telematic University, 80132 Naples, Italy a r t i c l e i n f o Article history: Received 6 May 2020 Accepted 4 June 2020 Keywords: Nanobeam Buckling Elastic foundation Closed form solution Nonlocal elasticity Size effect a b s t r a c t The size-dependent buckling instability of shear deformable nanobeams rested on a two- parameter elastic foundation is studied through the stress-driven nonlocal theory of elas- ticity and the kinematic assumptions of the Timoshenko beam theory. The small-scale size effects are taken into account by nonlocal constitutive relationships, which define the strains at each point as integral convolutions in terms of the stresses in all the points and a kernel. In this manner, the nonlocal elasticity formulation is well-posed and does not include inconsistencies usually arising using other nonlocal models. The size-dependent governing differential equations in terms of the transverse displacement and the cross- sectional rotation are decoupled, and closed form solutions are presented for the displace- ment functions. Proper boundary conditions are imposed and the buckling problem is re- duced to finding roots of a determinant of a matrix, whose elements are given explicitly for different classical edge conditions. The closed form treatment of the problem avoids the numerical instabilities usually occurring within numerical techniques, and allows to find also higher buckling loads and shape modes. Several nanobeams rested on the Win- kler or Pasternak elastic foundations and characterized by different boundary conditions, shear deformability, and nonlocality are considered and the critical loads and shape modes are presented, including those for the higher modes of buckling. Excellent agreements are found with the available approximate numerical results in the literature and novel insight- ful findings are presented and discussed, which are in accordance with experimental ob- servations. © 2020 Elsevier Ltd. All rights reserved. 1. Introduction The enhanced performance and capabilities of nanostructures and nanomaterials make them an excellent choice for use in different modern technologies. Nanobeams, nanoplates, carbon nanotubes, and nanoparticles have currently vast applications in Engineering Science as, for instance, energy harvesters, fluid conveying nanotubes, nanoswitches, nanores- onators, nanosensors, and nanoactuators (Almagableh, Omari & Sevostianov, 2019; Basutkar, 2019; Farokhi & Ghayesh, 2018; Ghayesh, Farajpour & Farokhi, 2019, 2019; Govorov et al., 2018; Medina, Gilat & Krylov, 2018; SoltanRezaee & Afrashi, 2016; Tran, Ghayesh & Arjomandi, 2018; Wentzel & Sevostianov, 2018). Mechanical properties and response of materials and struc- ∗ Corresponding author E-mail addresses: hossein.darban@uniparthenope.it (H. Darban), raimondo.luciano@uniparthenope.it (R. Luciano), a.caporale@unicas.it (A. Caporale), francesco.fabbrocino@unipegaso.it (F. Fabbrocino). https://doi.org/10.1016/j.ijengsci.2020.103338 0020-7225/© 2020 Elsevier Ltd. All rights reserved.