Surface Energy Effects on the Nonlinear Free Vibration of Functionally Graded Timoshenko Nanobeams Based on Modified Couple Stress Theory

Abstract

An integrated nonlinear couple stress-surface energy continuum model is developed to study the nonlinear vibration characteristics of size-dependent functionally graded nanobeams for the first time. The nanobeam theory is formulated based on the Timoshenko kinematics, augmented by von Kármán’s geometric nonlinearity. The modified couple stress and Gurtin–Murdoch surface elasticity theories are incorporated to capture the long-range interaction and surface energy, respectively. Unlike existing Timoshenko nanobeam models, the effects of surface elasticity, residual surface stress, surface mass density and Poisson’s ratio, in addition to bending and axial deformations, are incorporated in the newly developed model. A power law function is used to model the material distribution through the thickness of the beam, considering the gradation of bulk and surface material parameters. A variational formulation of the nonlinear nonclassical governing equations and associated nonclassical boundary conditions is established by employing Hamilton’s principle. The generalized differential quadrature method is exploited in conjunction with either the Pseudo-arclength continuation or Runge–Kutta method to solve the problem with an exact implementation of the nonclassical boundary conditions. The formulation and solution procedure presented are verified by comparing the obtained results with available ones. Based on the parametric study, it is concluded that the nonclassical boundary conditions, material length scale parameter, residual surface stress, surface elasticity, bulk elasticity modulus, gradient index, nonlinear amplitude and thickness have important influences on the linear and nonlinear vibration responses of functionally graded Timoshenko nanobeams.

On the dynamic response of bi-directional functionally graded nanobeams under moving harmonic load accounting for surface effect

“…Gurtin and Murdoch ([23,24]) proposed a surface elasticity theory to approximate the contribution of surface energy by assuming an elastic body with elastic surface layers of zero thickness that is perfectly bonded to the surface of the bulk continuum. The Gurtin and Murdoch surface elasticity theory (GM-SET) has been successfully employed in conjunction with the other nonclassical continuum theories in the analysis of homogeneous and one-dimensional FG structures, i.e., GM-SET combined with the DNET ( [47][48][49]), GM-SET combined with the SGT ( [50]), GM-SET combined with the DNSGT ( [51]), GM-SET combined with the MCST ( [52][53][54][55][56][57][58][59][60][61][62][63][64]), and GM-SET combined with the MCST in the framework of DNET ( [65,66]). Based on these studies, it has been shown that incorporating the influence of surface energy may show a stiffness-hardening or stiffness-softening of the studied nanostructures depending on whether the signs of the surface elastic constants are positive or negative.…”

A uniform framework for the dynamic behavior of linearized anisotropic elastic rods

“…where we have used (16) 2, 3 . From equations (18)(19)(20), one can see that the O(R 2 0 ) terms in (14) 1À3 (or ( 17) and (14) 2À3 ) become the leading-order ones; thus, they cannot be simply dropped. Actually, the terms on the left-hand side of (18)(19)(20) represent the bending along Y -axis, X -axis, and the torsion of the cross section, respectively.…”

“…The use of variational Hamilton's principle leads to determination of the equations of motion and complete boundary conditions for this beam model. Timoshenko kinematics and von Ka´rma´n's geometric nonlinearity are combined to construct a new nano-beam theory [18]. A variational formulation with nonclassical boundary conditions is obtained through Hamilton's principle.…”

Effective numerical technique applied for Burgers' equation of (1 + 1)‐, (2 + 1)‐dimensional, and coupled forms

“…Finite difference and other modifications, 18–24 finite element and B‐spline finite element, 25–27 spectral least squares method, 28–30 variational iteration method, 31,32 Adomian–Pade technique, 33 homotopy analysis, 34 and automatic differentiation method 35 are examples of such numerical techniques. Moreover, miscellaneous numerical techniques have been employed either for Burgers' equation or other engineering applications such as boundary element techniques for cavitation of hydrofoils, 36,37 step cubic polynomial, 38 technique of modified diffusion coefficient for studying convection diffusion equation, 39 and differential quadrature for functionally graded nanobeams 40,41 . Majeed et al, 42 presented a technique for solving time fractional Burgers' and Fisher's equations via cubic B‐spline approximation.…”