Graphene-based sp 2-carbon nanostructures such as carbon nanotubes and nanofibers can fail near their ideal strengths due to their exceedingly small dimensions. We have calculated the phonon spectra of graphene as a function of uniaxial tension by density functional perturbation theory to assess the first occurrence of phonon instability on the strain path, which controls the strength of a defect-free crystal at 0 K. Uniaxial tensile strain is applied in the x ͑nearest-neighbor͒ and y ͑second nearest-neighbor͒ directions, related to tensile deformation of zigzag and armchair nanotubes, respectively. The Young's modulus E = 1050 GPa and Poisson's ratio = 0.186 from our small-strain results are in good agreement with previous calculations. We find that in both x and y uniaxial tensions, phonon instabilities occur near the center of the Brillouin zone, at ͑ xx = 0.194, xx = 110 GPa, yy = −0.016͒ and ͑ yy = 0.266, yy = 121 GPa, xx = −0.027͒, respectively. Both soft phonons are longitudinal elastic waves in the pulling direction, suggesting that brittle cleavage fracture may be an inherent behavior of graphene and carbon nanotubes at low temperatures. We also predict that a phonon band gap will appear in highly stretched graphene, which could be a useful spectroscopic signature for highly stressed carbon nanotubes.

Abstract. The heterogeneous multiscale method (HMM) is applied to various parabolic problems with multiscale coefficients. These problems can be either linear or nonlinear. Optimal estimates are proved for the error between the HMM solution and the homogenized solution.

Abstract. The accuracy of the quasicontinuum method is analyzed using a series of models with increasing complexity. It is demonstrated that the existence of the ghost force may lead to large errors. It is also shown that the ghost force removal strategy proposed by E, Lu and Yang leads to a version of the quasicontinuum method with uniform accuracy.Key words. Quasicontinuum method, ghost force, geometrically consistent scheme AMS subject classifications. 65N12, 65N06, 74G20, 74G151. Introduction. The quasicontinuum (QC) method [33] is among the most successful multiscale methods for modeling the mechanical deformation of crystalline solids. It is designed to deal with situations when the crystal is undergoing mostly elastic deformation except at isolated regions with defects. The QC method is usually formulated as an adaptive finite element method. But instead of relying on a continuum model, the QC method is based on an atomistic model. Its main ingredients are: adaptive selection of representative atoms (rep-atoms), with fewer atoms selected in regions with smooth deformation; division of the whole sample into local and nonlocal regions, with the defects covered by the nonlocal regions; and the application of the Cauchy-Born (CB) approximation in the local region as a device for reducing the complexity involved in computing the total energy of the system.The quasicontinuum method has several distinct advantages. First of all, it has a reasonably simple formulation. In fact, it can be considered as a natural extension of adaptive finite element methods in which one simply uses the atomistic model where the mesh is refined to the atomic scale. Secondly, in the QC method, the treatment in different regions is based on the same model, the atomistic model, with the additional Cauchy-Born approximation used in the local region. For this reason, it is also considered to be more seamless than methods that are based on an explicit coupling between continuum and atomistic models. We refer to the review articles [6,21] for a discussion of methods that are based on explicitly coupling atomistic and continuum models.However, this does not mean that the QC method is free of the problems that one encounters when formulating coupled atomistic-continuum methodologies. In some sense, one may also regard the QC method as an example of such a strategy, with the local region playing the role of the continuum region, and the Cauchy-Born nonlinear elasticity model playing the role of the continuum model. In particular, the issue of consistency between the continuum and atomistic models across the coupling interface is very much manifested in the accuracy at the local-nonlocal interface for the QC method. This is the issue we will focus on in this paper. In fact, even though the atomistic models are used in both the local and the nonlocal regions, the CauchyBorn approximation made in the local regions means that the effective model in this

We propose an arbitrary-order discontinuous Galerkin method for secondorder elliptic problem on general polygonal mesh with only one degree of freedom per element. This is achieved by locally solving a discrete least-squares over a neighboring element patch. Under a geometrical condition on the element patch, we prove an optimal a priori error estimates for the energy norm and for the L 2 norm. The accuracy and the efficiency of the method up to order six on several polygonal meshes are illustrated by a set of benchmark problems.