Two-phase elastic axisymmetric nanoplates

Abstract

In the present work, the two-phase integral theory of elasticity developed in Barretta et al. (Phys E 97:13–30, 2018) for nano-beams is generalized to model two-dimensional nano-continua. Notably, a well-posed mixture local/stress-driven nonlocal elasticity is proposed to accurately predict size effects in Kirchhoff axisymmetric nanoplates. The key idea is to express the elastic radial curvature as a convex combination of local and nonlocal integral responses, that is a coherent choice motivated by virtue of the plate axisymmetry. The relevant structural problem is shown to be governed by a set of integro-differential equations, whose solution is computationally onerous. Thus, Helmholtz’s averaging kernel is advantageously adopted, since it enables explicit inversion of the integral constitutive law by virtue of an equivalence property. Specifically, the elastostatic problem of axisymmetry nanoplates is equivalently formulated in a differential form whose solution in terms of transverse displacement field is governed by nonlocal and mixture parameters. A parametric study is performed for case studies of applicative interest, and numerical solutions are finally provided and discussed. The presented methodology can be adopted to design and optimization of plate-based nano-electro-mechanical-systems (NEMS).

Nonlocal Elasticity for Nanostructures: A Review of Recent Achievements

“…Modeling of two-dimensional nonlocal continua is a topic of current interest in the scientific literature, with a wide range of applications concerning smart ultra-small devices. A stress-driven nonlocal methodology is conceived in [56] to capture scale effects in nanoplates and then generalized in [35] on the basis of a two-phase elasticity theory. The nonlocal mechanics of two-dimensional continua is studied in [31], vibration and buckling analysis of composite nanoplates are carried out in [95], static and dynamic behaviors of nonlocal elastic plates are examined in [96], modeling of circular nanoplate actuators is addressed in [97], chemical sensing systems are proposed in [98], vibration of resonant nanoplate mass sensors is analyzed in [99], nonlinear dynamics of nanoplates is investigated in [100], magneto-electromechanical nanosensors are modeled in [101], thermoelastic damping models for rectangular micro-and nanoplate resonators are proposed in [102], free vibration of functionally graded porous nanoplates is addressed in [103], nonlinear mechanical behavior of porous sandwich nanoplates is characterized in [104], and dynamics of nanoplates is investigated in [99,105].…”

“…Stress-driven two-phase elasticity has been recently applied in [35] to capture sizedependent behaviors of two-dimensional continua modeled by the Kirchhoff plate theory. Notably, with reference to an axisymmetric annular plate of internal radius R i and external radius R e , a polar coordinate system r, θ, z is conveniently introduced.…”

“…Moreover, the mixture methodology based on the stress-driven approach is well posed for any local fraction. This theory has been successfully applied in recent contributions, such as in [33][34][35].…”