An extended finite element/level set method to study surface effects on the mechanical behavior and properties of nanomaterials

Abstract

SUMMARYWe present a new approach based on coupling the extended finite element method (XFEM) and level sets to study surface and interface effects on the mechanical behavior of nanostructures. The coupled XFEM-level set approach enables a continuum solution to nanomechanical boundary value problems in which discontinuities in both strain and displacement due to surfaces and interfaces are easily handled, while simultaneously accounting for critical nanoscale surface effects, including surface energy, stress, elasticity and interface decohesion. We validate the proposed approach by studying the surface-stressdriven relaxation of homogeneous and bi-layer nanoplates as well as the contribution from the surface elasticity to the effective stiffness of nanobeams. For each case, we compare the numerical results with new analytical solutions that we have derived for these simple problems; for the problem involving the surface-stress-driven relaxation of a homogeneous nanoplate, we further validate the proposed approach by comparing the results with those obtained from both fully atomistic simulations and previous multiscale calculations based upon the surface Cauchy-Born model. These numerical results show that the proposed method can be used to gain critical insights into how surface effects impact the mechanical behavior and properties of homogeneous and composite nanobeams under generalized mechanical deformation.

An analysis of the boundary layer in the 1D surface Cauchy–Born model

“…Figure 1 suggests that, while the error in the displacement and displacement gradient is indeed of order O (1) in the boundary layer, the displacement error at finite element nodes is visually negligable, which would imply that the SCB model approximates the mean strain (and possibly other averaged quantities) to a much higher degree of accuracy. This was indeed observed in extensive numerical tests presented in [11,22,23].…”

A hybrid smoothed extended finite element/level set method for modeling equilibrium shapes of nano-inhomogeneities

“…ε (x) L ε (x) (28) g different element sizes h. The quantities with superscript 'exact' are the xact solutions. Figure 9 show the displacement and energy norms for the cases without and wit the interface stress, respectively.…”

“…Recently, Yvonnet et al [27] developed an XFEM/level set approach to study the size-dependent effective properties of nanocomposites based on a Gurtin-Murchoch model with the coherent interface assumption [1]. Moreover, Farsad et al [28] presented an XFEM approach allowing for strong discontinuities across the interface based on a Gurtin-Murdoch incoherent interface model [2] and they studied the surface/interface effects on the mechanical behaviors of nanoscale materials. However, they limited their studies to static models without considering the evolution of the interfaces.…”

Topology optimization of piezoelectric nanostructures

“…For surface elasticity, the seminal work was that of Gurtin, Murdoch and co-workers (Gurtin and Murdoch, 1975;Gurtin et al, 1998), who were the first to establish a surface or interface elasticity model to capture surface stress and elastic effects. The elastic properties of nanostructures with surface and interface effects using the extended finite element method (XFEM) were proposed by Yvonnet et al (2008), and later extended by Farsad et al (2010) to study the mechanical behavior of homogeneous and composite nanobeams. For surface piezoelectricity, various analytic models have been developed, including an explicit formula for the electromechanical coupling coefficient considering surface effects (Yan and Jiang, 2011a) for piezoelectric nanowires, an Euler-Bernoulli beam theory for the vibrational and buckling behavior of piezoelectric nanobeams (Yan and Jiang, 2011b), and the electroelastic response of thin piezoelectric places considering surface effects using Kirchoff plate theory .…”