Hermite finite elements for the vibrations and buckling of strain gradient nano plates in hygro-thermal environment

Nano structural components have been widely spreading recently due to their novel applications in several engineering fields. Such components have a nonlocal mechanical behavior that might force to employ generalized or higher-order elasticity theories to take into consideration effects on the nano scale. In this context, nonlocal strain gradient theory has been utilized for investigating the linear vibrations and buckling phenomena of nano plates where orthotropic mechanical properties are functionalized through the plate thickness. Reinforcing fibers of such orthotropic layers can be modelled by assuming a non-uniform distribution along each ply thickness. The homogenization of the orthotropic layers is performed by following the Halpin–Tsai approach starting from the two main constituents (fiber and matrix). Through-the-thickness functions are introduced to describe the variation of their volume fraction. As an evidence from industrial applications nano plate behavior depends on external stimuli such as hygro-thermal effects. For this reason, these effects have been included into the formulation in order to study their influence in the dynamic and linear buckling phenomena. The governing partial differential equations are solved via a finite element model where Hermitian shape functions are introduced due to the higher-order nature of the nonlocal theory selected which involved both in-plane and out-of-plane displacement parameters. Numerical applications are provided in order to show hygro-thermal effects on the vibration and buckling problems of nano plates, in particular, the effect of the fiber distribution along the ply thickness is underlined.