Size effect on vibration properties of axially moving nanoplates under different boundary conditions

The nonlocal strain gradient theory is employed to investigate the transverse free vibration characteristics of two-dimensional nano-plates with axial velocities. A generalized Hamiltonian principle has been used to establish the vibration governing equations for the system as well as the corresponding boundary conditions. By applying complex modal analysis to three boundary conditions, the plate’s natural frequency is determined, including four-end simply supported, four-end clamped, and opposite-edge simply supported and clamped, and comparing the effect of the size parameters on the natural frequency in relation to the boundary conditions; based on different theories, the effects of changing boundary conditions on natural frequencies are systematically compared. In the numerical study, it is demonstrated that the size effect significantly influences only the self-oscillation frequency at the nanoscale, whereas the nonlocal parameter as well as the material characteristic parameter have “softening” and “hardening” effects on the equivalent stiffness of the nanoplates, respectively, which are directly related to their natural frequencies. Compared to simple supports, clamped boundary conditions are more significantly affected by size parameters. In addition, higher order frequencies exhibit greater sensitivity and are susceptible to changes in boundary conditions and size parameters.


Introduction
As a result of the discovery of carbon nanotubes in 1991, the area of nanomechanics has attracted considerable research attention.In many experiments and numerical simulations, the size effect of material deformation makes the applicability of classical continuum theory doubtful.Therefore, it is inevitable that the classical theory requires revision.Therefore, several nonclassical continuum theories have been developed, including nonlocal theory, couple stress theory, strain gradient theory, velocity gradient theory, high-order shear deformation theory, etc.It is widely accepted that non-local and strain gradient theories use "hardening" and "softening" size parameters to describe the effects of size at the microscopic level, respectively.The combination of these two types of parameters has been proposed by Lim [1] in 2015 as a theoretical model of nonlocal strain gradients.In Lim's study, by exploring the dispersion relation in the wave propagation problem of carbon nanotubes, it was found that the phenomena of "softening" and "hardening" of the size parameters can exist simultaneously.Since the theory was proposed, scholars have applied it to the study of mechanical behaviors such as bending, buckling, vibration, and wave propagation of a variety of micro-nanostructures and have concluded that mechanical behavior of the structures governed by the theory differs from the mechanical behavior of a single theory in that it is dependent on the dimensions, making it one of the most exciting research topics of today.Nanomedicine, microchips, sensors, and the military are some of the many engineering fields that require axial motion systems.In order to design and optimize related devices, it is essential to analyze the mechanical behavior and properties of these structures.By using the generalized differential integral method and molecular dynamics simulations, Ansari et al. [2] developed a nonlocal elastic model of free vibration of a single-layer graphene sheet.Under unidirectional prestressing conditions, Murmu et al. [3] assessed the vibration response of nonlocal nanoplates.Hosseini and Jamalpoor [4] investigated the dynamics of non-localized bilayer functional gradient viscoelastic nanoplates considering surface effects (surface elasticity, tension, and density) under temperature variation.Lu et al [5] explored the free vibration of a functional gradient plate and shell model by combining nonlocal effects, strain gradient effects and surface energy effects.Chen et al [6][7][8] established a general theory of modified coupled stresses applicable to anisotropic materials and investigated the vibration, stability and large deformation of plates under the size effect based on a composite laminate model.
This paper proposes a class of two-dimensional nano-plate models with axial velocities based on nonlocal strain gradient theory.A generalized Hamiltonian principle is used to establish the governing equation, and the natural frequencies under three different boundary conditions of four-end simply supported (SSSS), four-end clamped (CCCC), and opposite-edge simply supported and clamped (SCSC) are analyzed using complex modal analysis.Using this method, the influences of the two types of size parameters, nonclassical continuum theories, and boundary conditions are analyzed in terms of their influence laws.

Axially moving nonlocal strain gradient nano-plate model
Consider an axially moving homogeneous rectangular nanoplate as shown in Figure 1.The length is a L , the width b L , the thickness h , the material density is , the modulus of elasticity E ,and the ends are subjected to tension forces x f and y f that do not vary with the thickness, and move with a velocity v at a uniform speed along the x -direction.In accordance with the displacement field, the expressions for the plane strain in the directions under the nanoplate can be produced: u uu x y t v vv x y t w ww x y t are the displacements of the mid-surface of the nanoplate along the , , x y z directions.In accordance with the nonlocal strain gradient theory proposed by Lim, the total stress in a nanostructured model consists of two main components, a conventional nonlocal stress and a higher order nonlocal stress: (1) By introducing a linear nonlocal operator e e e , Equation ( 2) can be simplified to: Equation ( 3) can be used to derive the non-local constitutive relationship, which includes two types of size parameters: In dimensionless form, the dynamic equation is as follows:

The influence of boundary conditions on natural frequencies
To determine how each parameter affects the natural frequency of axially moving nanoplates under various boundary conditions, the nanoplates with dimensionless boundary conditions and vibration mode functions are discussed as follows with three boundary conditions of SSSS, CCCC, SCSC: One of the vibration properties of free linear vibration is that the modes are harmonic in time, and therefore the displacement with time can be expressed as follows: where mn is the natural frequency of the nanoplate.According to Galerkin's principle can get: As an example, for a four-end simply supported when both M and N are taken as 2, the corresponding dimensionless deflection functions are: ( , ) sin( )sin( ) sin( )sin( 2) sin( 2)sin( ) sin( 2)sin( 2) Equation ( 11) can be reduced to the following equation by substituting it into Equation (12): where M is the mass matrix, G is the damping matrix, K is the stiffness matrix.
where , 1,2 m n . The characteristic equation of the system's natural frequency can be derived by satisfying a sufficient necessary condition that Equation( 14) has a non-zero solution.As a result of considering the matrix M , G , K ,which contains the parameters of an axially moving nanoplate, then can calculate the natural frequency of the nanoplate at each order based on the value of each parameter.
According to Figures 2 and 3, nonlocal parameters and material characteristics were used to investigate the effect of SSSS, CCCC, and SCSC on the first four orders of the natural frequency.Taking the parameter as 1 2 3, 100, 1 P P Figure 2 describes the influence of non-local parameters on the natural frequency under different boundary conditions.The natural frequency of the nanoplate can be influenced by non-local parameters and boundary conditions, and CCCC boundary conditions reduce the plate's stiffness and increase its frequency.As the order increases, the size effect gradually increases; however, the CCCC boundary conditions are most prone to size effects.When all the other parameters are consistent, the boundary conditions are characterized by a greater transverse vibration frequency of the clamped nanoplate compared to a simply supported nanoplate.As a result, when the non-local parameter is zero, the natural frequency reaches its maximum value, i.e., the presence of the non-local parameter causes the equivalent stiffness of the nanoplate to decrease, which in turn decreases the natural frequency, and the larger the value of the non-local parameter is, the greater the decrease of the natural frequency will be.In the presence of the material length characteristic parameter, the nanoplate stiffness increases, which in turn increases the natural frequency.The influence of the size effect becomes more evident when the material length characteristic parameter is increased as well as the order is increased.CCCC boundary conditions are the most sensitive to the size effect, and as the order increases, the size effect is accentuated, and the change amplitude is greatest under CCCC boundary conditions.