Nonlocal strain gradient shell theory for bending analysis of FG spherical nanoshells in thermal environment
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Nonlocal strain gradient shell theory for bending analysis of FG spherical nanoshells in thermal environment Mohammad Hassan Dindarloo1, Ashraf M. Zenkour2,3,a 1 Department of Mechanical Engineering, Tarbiat Modares University, Tehran, Iran 2 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia 3 Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh 33516, Egypt
Received: 9 May 2020 / Accepted: 22 September 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this study, we focus on the bending of the functionally graded spherical nanoshells using first-order shell theory in the thermal environment. Nonlocal strain gradient theory is applied to consider the small-scale impacts with considering both softening and stiffness enhancement effects of the spherical panel. The governing equations are deduced by applying Hamilton’s principle, and Navier’s series is used to solve the bending deflection of spherical nanoshells. The work provides a possibility that the bending behaviors of spherical shallow and deep spherical nanoshells can allow being investigated in a general framework. The simulations indicate that the temperature variation has a significant influence when the nonlocal parameter value is greater than 1 nm. Furthermore, the impacts of several parameters like a nonlocal parameter, strain gradient length scale parameter, and temperature variation are investigated on the deflection response of the spherical panel.
1 Introduction Functionally graded material (FGM), as a new class of composite materials, make a significant impact on modern engineering. FGMs with two or more constituents such as metals and ceramic have optimized properties compared with laminated composites. Recently, FGMs with their amazing mechanical properties have been attracted to the consideration of the many researchers [1–11]. The use of nanosize structures such as nanoshells, nanoplates, and nanorods has been increasing rapidly in nanoelectromechanical systems due to their stupendous electrical, thermal, and mechanical properties. Therefore, there have been multiple size-dependent continuum theories such as nonlocal and strain gradient theories to realize the mechanical behavior of the nanosize structures [12–19]. Nonlocal elasticity is one of the size-dependent continuum theories which reflects the interaction between atoms (Eringen [20]). Many types of research have been applied to Eringen’s nonlocal elasticity theory to analyze the nanosize structures [21–27]. Although Eringen’s nonlocal elasticity theory has been applied for analyzing the nanosize structures, it distinguishes only the softening effect. The impact of the stiffness enhancement, which is presented from the theoretical and experimental researches [28–30]
a e-mail: [email protected] (corresponding author)
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Eur. Phys. J. Plus
(2020) 135:785
cannot be considered by Eringen’s nonlocal elas
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