An atomistic-based finite deformation membrane for single layer crystalline films

Abstract

A general methodology to develop hyper-elastic membrane models applicable to crystalline ÿlms one-atom thick is presented. In this method, an extension of the Born rule based on the exponential map is proposed. The exponential map accounts for the fact that the lattice vectors of the crystal lie along the chords of the curved membrane, and consequently a tangent map like the standard Born rule is inadequate. In order to obtain practical methods, the exponential map is locally approximated. The e ectiveness of our approach is demonstrated by numerical studies of carbon nanotubes. Deformed conÿgurations as well as equilibrium energies of atomistic simulations are compared with those provided by the continuum membrane resulting from this method discretized by ÿnite elements.

Shape control of active surfaces inspired by the movement of euglenids

“…(3). By N we denote a unit normal to and by K the pull-back of the second fundamental form b IJ = x ,IJ · N to 0 , which is the curvature measure compatible with C (Arroyo and Belytschko, 2002). For instance, the mean curvature is computed as H = K : C. The distinction between K and b is important in the numerical implementation, where {u, v} are the finite element coordinates.…”

Finite crystal elasticity of carbon nanotubes based on the exponential Cauchy-Born rule

“…44 Given the crystalline nature of carbon nanotubes, and the large elastic ͑reversible͒ deformations they exhibit, finite crystal elasticity appears to be appropriate for their mechanical analysis. As recently suggested, 13 the standard theories aimed at space-filling crystals do not capture the effects of the curvature of crystalline monolayers deforming in three dimensions such as nanotubes. The general idea behind standard finite crystal elasticity in the case of space-filling crystals is sketched below, its limitations for carbon nanotubes illustrated, and the extended theory briefly outlined.…”

“…It is postulated that the atoms lie on the surface, and therefore the lattice vectors are chords of the surface. 13 The appropriate framework to describe two-dimensional continua deforming in three dimensional Euclidean space is continuum mechanics on manifolds. 48 The undeformed body ⍀ 0 , which represents the planar ground configuration of graphene, is now a subset of R 2 .…”

Simple optimized Brenner potential for thermodynamic properties of diamond

“…In [3,4], a QC Monte Carlo (QCMC) method and a QC free energy minimization (QCFEM) method were proposed to study equilibrium properties of defects at Enite temperature. Arroyo and Belytschko [6], Zhang et al [7][8][9][10] and Jiang et al [11] have proposed nanoscale continuum theories for carbon nanotubes based on interatomic potentials for carbon. Based on the local harmonic approximation [12], Jiang et al [13] established a Enite-temperature continuum theory directly from the interatomic potential.…”