Exact variational nonlocal stress modeling with asymptotic higher-order strain gradients for nanobeams

Abstract

This article presents a complete and asymptotic representation of the one-dimensional nanobeam model with nonlocal stress via an exact variational principle approach. An asymptotic governing differential equation of infinite-order strain gradient model and the corresponding infinite number of boundary conditions are derived and discussed. For practical applications, it explores and presents a reduced higher-order solution to the asymptotic nonlocal model. It is also identified here and explained at length that most publications on this subject have inaccurately employed an excessively simplified lower-order model which furnishes intriguing solutions under certain loading and boundary conditions where the results become identical to the classical solution, i.e., without the small-scale effect at all. Various nanobeam examples are solved to demonstrate the difference between using the simplified lower-order nonlocal model and the asymptotic higher-order strain gradient nonlocal stress model. An important conclusion is the discovery of significant over-or underestimation of stress levels using the lower-order model, particularly at the vicinity of the clamped end of a cantilevered nanobeam under a tip point load. The consequence is that the design of a nanobeam based on the lower-order strain gradient model could be flawed in predicting the nonlocal stress at the clamped end where it could, depending on the magnitude of the small-scale parameter, significantly over-or underestimate the failure criteria of a nanobeam which are governed by the level of stress.

Wave propagation in carbon nanotubes: nonlocal elasticity-induced stiffness and velocity enhancement effects

“…In the presence of a nonlocal elastic stress field, it has been a common practice to directly replace the classical M c in the equation of motion above and in Figure 1 with the nonlocal moment M x x defined in (8) [Wang and Hu 2005;Wang et al 2006c;Lu et al 2007;Heireche et al 2008;Wang et al 2006b;Wang and Varadan 2007;Xie et al 2007a;Xie et al 2007b;Lim and Wang 2007]. Such models are termed the partial nonlocal models.…”

Analytical solutions for coupled tension-bending of nanobeam-columns considering nonlocal size effects

“…Exact equilibrium conditions and higher-order differential governing equations with the corresponding higher-order nonlocal boundary conditions were first derived. These equations and conditions involve essential higher-order terms that are missing in virtually all previous analyses and published works on statics and dynamics of nonlocal nanostructures based on the partial nonlocal models [15][16][17][18][19][20][21][22][23][24][25][26][27]. The negligence of higher-order terms in these works results in contradictory nanoscale effects with respect to the conclusion using the exact nonlocal stress model [28,29].…”

“…The nonlocal field theories consider the stress at a reference point to be a function of the strain field at every point in the body. It has been extensively applied to analyze bending, buckling, vibration and wave propagation of CNTs and other nanostructures [15][16][17][18][19][20][21][22][23][24][25][26][27]. With respect to the classical or local models, these early nonlocal models yielded a series of analytical solutions in nanomechanics of CNTs.…”

Scale effects on ultrasonic wave dispersion characteristics of monolayer graphene embedded in an elastic medium

“…It is also reported in [Chen et al 2004] that nonlocal continuum theory-based models are physically reasonable from the atomistic viewpoint of lattice dynamics and molecular dynamics simulations. Understanding the importance of employing nonlocal elasticity theory in small-scale structures, a number of research works have conducted static, dynamic, and stability analyses of micro/nanostructures [Yakobson et al 1997;Peddieson et al 2003;Wang and Hu 2005;Lu et al 2006;2007;Duan and Wang 2007;Duan et al 2007;Ece and Aydoydu 2007;Lim and Wang 2007;Reddy 2007;Wang and Liew 2007;Adali 2008;Kumar et al 2008;Reddy and Pang 2008;Tounsi et al 2008;Wang and Duan 2008;Yang et al 2008;Aydogdu 2009a;2009b;Murmu and Pradhan 2009;Narendar and Gopalakrishnan 2009a;2009b;2010a;2010b;2010c;2010d;.…”