Nonlocal large-amplitude vibration of embedded higher-order shear deformable multiferroic composite rectangular nanoplates with different edge conditions

Abstract

Based on the nonlocal elasticity theory, a unified nonlocal, nonlinear, higher-order shear deformable nanoplate model is developed to investigate the size-dependent, large-amplitude, nonlinear vibration of multiferroic composite rectangular nanoplates with different boundary conditions resting on an elastic foundation. By considering a unified displacement vector and using von Kármán’s strain tensor, the strain–displacement components are obtained. Using coupled nonlocal constitutive relations, the coupled ferroelastic, ferroelectric, ferromagnetic, and thermal properties of multiferroic composite materials and small-scale effect are taken into account. The electric and magnetic potential distributions in the nanoplate are calculated via Maxwell’s electromagnetic equations. Furthermore, Hamilton’s principle is utilized to obtain the mathematical formulation associated with the coupled governing equations of motions and boundary conditions. The developed model enables us to consider the effects of rotary inertia and transverse shear deformation without using any shear correction factor. Also, it can be degenerated to the models based on the Kirchhoff and existing shear deformation plate theories. To solve the large-amplitude vibration problem, an efficient multistep numerical solution approach is utilized. Effects of various important parameters such as the type of the plate theory, and parameters of nonlocality and coupled fields on the nonlinear frequency response are investigated.

Nonlinear vibrations of axially moving simply supported viscoelastic nanobeams based on nonlocal strain gradient theory

“…There are two main forms of non-classic continuum mechanics theory: nonlocal elastic theory and strain gradient elasticity theory. Compared with that of nonlocal elastic theory, the field of influence of strain gradient elastic theory is significantly smaller and is usually regarded as having 'weak nonlocality'; the nonlocal elastic and strain gradient elastic theories are applicable to explaining the scale effects of the 'softening [3] (i.e. the material stiffness decreases as the feature size decreases)' and 'hardening [4][5][6] (i.e.…”

Parametric vibrations of graphene sheets based on the double mode model and the nonlocal elasticity theory

“…It is also worth noting that in the case of an isotropic material within the framework of the classical theory (the nonlocal parameter μ = 0) and without taking into account the in-plane periodic load, we get system of nonlinear second-order ordinary differential equations where the obtained coefficients coincide with those presented in the work [24], which also confirms the results of our work. The numerical simulations are performed for mentioned material properties (20), they are assumed to be fixed as well as the damping parameter δ = 1. For studying parametrical resonant case we took the excitation frequency is equal to the double first natural frequency of plate ω = 2 √ α 0 .…”

“…There are many works in which linear vibrations and buckling of isotropic and orthotropic micro and nanoplates are studied [15][16][17][18][19]. A significantly smaller number of works is devoted to solving problems in a geometrically nonlinear formulation with various types of loading as well as under magnetic and electric excitation [20][21][22][23]. Moreover, in these works, periodic regimes are studied based on the single-mode approximation of the deflection.…”

“…Those few recent studies of micro nanostructures use the classical Bolotin method for dynamical stability analysis. Gholami et al studied axial buckling and dynamic stability of functionally graded microshells in [20]. The dynamic stability of Timoshenko nanobeams subjected to an axial load studied in the work of Behdad et al [30], Bolotin's approach applied to the investigation of dynamic stability of functionally graded microbeams by Ke and Wang [31], where authors used the modified couple stress theory.…”

Double mode model of size-dependent chaotic vibrations of nanoplates based on the nonlocal elasticity theory

“…Setoodeh et al [34] investigated the nonlinear vibrations of the orthotropic Mindlin plate by the differential quadrature method. Gholami et al [14] studied the nonlinear vibrations of multiferroic composite rectangular nanoplates resting on an elastic foundation by the generalized differential quadrature method. The nonlinear vibrations of viscoelastic double-layered plates with several types of boundary conditions were studied by Wang et al [41,43].…”