Nonlocal chaotic analysis of double layered viscoelastic nanoplates embedded in Winkler–Pasternak elastic foundation and under in‐plane biaxial loads

Abstract

In this paper, based on the extended Melnikov method the homoclinic analysis for chaotic vibration of a double layered viscoelastic nanoplates (DLNPs) embedded in Winkler–Pasternak elastic foundation and under in‐plane excited biaxial loads using the Eringen's nonlocal elasticity theory has been investigated. The governing nonlinear coupled partial differential equations (PDEs) of motion for DLNP considering viscoelastic Kelvin–Voigt model in the Kirchhoff's thin plate theory in conjunction with the von‐Karman geometrical nonlinear strain–displacement relations have been derived. Then, applying the Galerkin method the governing nonlocal coupled PDEs of motion of DLNP have been converted into the nonlinear coupled ordinary differential equations (ODEs) using mode summation technique and discretized in terms of time. The criteria for the homoclinic transverse intersection for synchronous and asynchronous types of buckling are studied to predict the homoclinic orbits and chaotic motions applying the generalized Melnikov method. Besides, the critical buckling load has been determined for asynchronous and synchronous types of buckling. The obtained results are validated with comparing them with the results presented in the literature for a special case. Then, the effect of changes of different parameters including nonlocal small‐scale parameter, foundation coefficients, biaxial load ratio, geometric aspect ratio on the critical buckling load and ratio of nonlocal to local natural frequency have been studied. The critical threshold surface for the synchronous and asynchronous manifolds to predict transversely condition for the chaotic motion are discussed on the parametric space.

Size‐dependent Levinson beam theory for thermal vibration of a nanobeam with deformable boundary conditions

“…[25][26][27][28][29][30][31][32] modified couple stress theory have been considered to show the responses of scale-dependent nano/micro elements. Nonlocal elasticity theory has been used by researchers [33][34][35][36][37][38][39][40][41][42] to investigate the different effects on the nano/micro-scaled beams, rods, etc. Strain gradient elasticity has been adopted the buckling of nano/microbeams in refs.…”