Vol.:(0123456789) 1 3 Iranian Journal of Science and Technology, Transactions of Mechanical Engineering https://doi.org/10.1007/s40997-019-00325-9 RESEARCH PAPER Nonlinear Forced Vibration of Thermally Postbuckled Double‑Layered Triangular Graphene Sheet with Clamped Boundary Conditions G. A. Varzandian 1  · S. Ziaee 1  · M. Farid 2  · A. Niknejad 1 Received: 6 April 2019 / Accepted: 11 October 2019 © Shiraz University 2019 Abstract In the present research, vibration behavior is presented for a thermally postbuckled double-layered triangular graphene sheet (DLTGS). The DLTGS is modeled as a nonlocal orthotropic plate and contains small-scale efects. The formulations are based on the Kirchhof’s plate theory, and a nonlinearity of von Karman-type is considered in strain–displacement rela- tions. The thermal efects and van der Waals forces between layers are also included and some of the material properties are assumed to be temperature-dependent. A semi-analytical solution is obtained using the multiple time scales method. The efects of variation of small-scale parameter to the natural frequencies, defections and response curve of DLTGS are analyzed, and the numerical results are obtained from the nonlocal plate model. Numerical results are compared with those of similar researches. Efects of various parameters on the postbuckled vibration of DLTGS in thermal environments such as scale parameter and thermal load are presented. The stability and occurrence of the internal resonance between vibration modes around a stable buckled confguration is investigated. Keywords Double-layered triangular graphene sheet · Thermal postbuckling · Nonlocal plate model · Internal resonance · Multiple scale method 1 Introduction Graphene structures with diferent shapes and confgurations have superior properties and they are introduced as one of the fundamental carbon forms, and graphene is the base of many other confgurations such as graphite, carbon nano- tubes and fullerenes; therefore, studying single- and dou- ble-layered graphene sheets is very important in nanoscale studies (Gholami and Ansari 2019). Recently, nanoscale structures such as nanobeams and nanoplates have gained noticeable attention from both the experimental and theoreti- cal researchers (Gholami et al. 2018; Setoodeh et al. 2011; Setoodeh and Rezae 2017; Varzandian and Ziaee 2017). This is because nanostructures possess superior electri- cal, mechanical and thermal properties as compared to the conventional materials (Ansari and Gholami 2016; Ansari et al. 2016). In nanoscale plate problems, solutions due to classical theories such as Kirchhof’s plate theory and shear deforma- tion plate theory usually have remarkable errors because in these theories, the relation between stress and strain is point wise and the size efects are not considered (Eringen 2002). In nonlocal elasticity theory, the stress at a reference point is assumed to be a functional of the strain feld at every point in the body and the efects of scale are considered by a new quantity called scale parameter (Eringen 1983). Based on nonlocal elasticity theory, size efects are taken into account by integrating a scale parameter into classi- cal continuum models (Eringen 1983, 2002). Pradhan and Phadikar (2009) developed classical plate theory (CLPT) and frst-order shear deformation theory (FSDT) of plates using the nonlocal differential constitutive relations of Eringen. Ansari and Gholami (2016) studied the nonlinear postbuckling and free vibration of a rectangular nanoplate using the Gurtin–Murdoch surface stress elasticity theory and showed that the surface efects are important. Nazemn- ezhad (2015) investigated the effects of van der Waals (vdWs) interactions on free vibration of multilayer graphene * S. Ziaee ziaee@yu.ac.ir 1 Mechanical Engineering Department, Yasouj University, Yasouj, Iran 2 School of Mechanical Engineering, Shiraz University, Shiraz, Iran