Bending, Buckling and Free Vibration Analyses of Nanobeam-Substrate Medium Systems

Abstract

This study presents a newly developed size-dependent beam-substrate medium model for bending, buckling, and free-vibration analyses of nanobeams resting on elastic substrate media. The Euler-Bernoulli beam theory describes the beam-section kinematics and the Winkler-foundation model represents interaction between the beam and its underlying substrate medium. The reformulated strain-gradient elasticity theory possessing three non-classical material constants is employed to address the beam-bulk material small-scale effect. The first and second constants is associated with the strain-gradient and couple-stress effects, respectively while the third constant is related to the velocity-gradient effect. The Gurtin-Murdoch surface elasticity theory is adopted to account for the surface-free energy. To obtain the system governing equation as well as corresponding boundary conditions, Hamilton’s principle is called for. Three numerical simulations are presented to characterize the influences of the material small-scale effect, the surface-energy effect, and the surrounding substrate medium on bending, buckling, and free vibration responses of nanobeam-substrate medium systems. The first simulation focuses on the bending response and shows the ability of the proposed model to eliminate the paradoxical characteristic inherent to nanobeam models proposed in the literature. The second and third simulations perform the sensitivity investigation of the system parameters on the buckling load and the natural frequency, respectively. All analytical results reveal that both material small-scale and surface-energy effects consistently stiffen the system response while the velocity-gradient effect weakens the system response. Furthermore, these sized-scale effects are more pronounced when the underlying substrate medium becomes softer.

Pull-down plateau of a Toda-like fractal-fractional oscillator and its application in grass carp’s roes’ agglomeration

“…The vibration of roes' agglomeration can be described as a particle system with nonlinear spring connections as shown in Figure 3, similar to a nanobeam system. 8,[29][30][31] It consists of a roe with mass m, which is connected to other roes by springs. It is also adsorbed by the entire system.…”

Size-Scale Effects on Bending Behaviour of Nanobeam on an Elastic Substrate Medium

“…Based on the Euler-Bernoulli beam theory [16], the deformed section of the nanobeam remains plane and normal to the longitudinal axis, as presented in Fig. 1.…”

“…2. Under these hypotheses, the in-plane and out-of-plane surface stresses for the planer Euler-Bernoulli beam system are given by Limkatanyu et al [16] Based on the kinematics of Euler-Bernoulli beam of Eq. ( 1), the surface compatibility equations can be expressed as [16]:…”

“…5. The geometric and material properties of the nanobeam are taken from Limkatanyu et al [16] 6 depicts the vertical displacement pattern along the length of the nanobeam. When compared to the classical model (without the sizedependent and small-scale effects), the results show that both size-dependent and small-scale effects increase system stiffness, particularly the smallscale effect.…”

“…To represent the size-dependent effect due to the surface stress and its residual, Gurtin and Murdoch [13,14] proposed the surface model. Due to its simplicity and capability, several works have used the Gurtin-Murdoch surface model to study the size-dependent effect inherent in micro-and nanoscale structures [15,16].…”