RESEARCH PAPER Investigating Instability Regions of Harmonically Loaded Refined Shear Deformable Inhomogeneous Nanoplates Mohammad Reza Barati 1 • Ashraf Zenkour 2,3 Received: 21 February 2017 / Accepted: 18 June 2018 Ó Shiraz University 2018 Abstract Dynamic stability of nonlocal sigmoid functionally graded (S-FGM) nanoplates on elastic medium under bi-axial harmonic loads is studied via four-variable plate theory. A two power-law-based rule of mixture is adopted to describe graded material properties. Hamilton’s principle is employed to derive the governing equations. These equations are expressed in the form of Mathieu–Hill equations and Bolotin’s approach is implemented to evaluate the instability regions. Exactness of present method is verified by comparing obtained results with those provided in literature. The effects of static and dynamic load factors, nonlocal parameter, elastic foundation parameters, gradient index and boundary conditions on instability regions of S-FGM nanoplates are examined. Keywords Dynamic stability  Refined plate theory  S-FGM nanoplate  Nonlocal elasticity  Neutral surface 1 Introduction Functionally graded materials (FGMs) are known as microscopically inhomogeneous spatial composites and provide broad potential applications for various machineries which are subjected to mechanical loads. According to this point that FG materials have been placed in the classification of composite materials, the volume fractions of material phases are assumed to vary smoothly and continuously throughout the gradient directions. The FGM materials are made to take advantage of desirable features of its constituent phases, for example, in a thermal protection system; the ceramic phase is capable of enduring intense temperature fields due to having preferable thermal resistance characteristics, while the metal phase introduces better mechanical performance. Hence, presenting modern physical characteristics, FGMs have gained its applicability in several engineering fields, such as biomedical engi- neering, nuclear engineering and mechanical engineering (Ebrahimi and Barati 2016; Barati 2017a, b, c). For mod- eling FGM structures, several functions are suggested, such as power-law, exponential and sigmoid functions (Ak- barzadeh et al. 2015). It is known that S-FGM model is proposed to improve the stress concentrations created in one of the interfaces in which the material is continuously but rapidly varying in both power-law and exponential models (Han et al. 2008; Atmane et al. 2011; Lee et al. 2015; Prakash et al. 2009). Moreover, considerable progression in the utilization of structural elements such as beams and plates with micro and nanoscales in micro/nano electro-mechanical systems (MEMS/NEMS), due to providing outstanding mechanical, chemical, and electronic characteristics, led to a sudden momentum in modeling of micro and nanoscale structures (Zenkour 2016). In these applications, size effects become prominent. The problem in using the classical theory is that the classical continuum mechanics theory does not take into account the size influence in nanosize structures. The classical continuum mechanics over predicts the responses of micro/nano structures. So a new form of continuum mechanics that captures small scale effect is required. The & Mohammad Reza Barati mrb.barati@ymail.com; mrb.barati@aut.ac.ir 1 Aerospace Engineering Department and Center of Excellence in Computational Aerospace, AmirKabir University of Technology, Tehran, Iran 2 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 3 Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafr El-Sheikh 33516, Egypt 123 Iran J Sci Technol Trans Mech Eng https://doi.org/10.1007/s40997-018-0215-4