Continuum Surface Energy from a Lattice Model

We investigate some connections between the continuum and atomistic descriptions of de- formable crystals, using some interesting results from number theory. The energy of a deformed crystal is calculated in the context of a lattice model with binary interactions in two dimensions. A new bond counting approach is used, which reduces the problem to the lattice point problem of number theory. When the crystal shape is a lattice polygon, we show that the energy equals the bulk elastic energy, plus the boundary integral of a surface energy density, plus the sum over the vertices of a corner energy function. This is an exact result when the interatomic potential has finite range; for infinite-range potentials it is asymptotically valid as the lattice parameter zero. The surface energy density is obtained explicitly as a function of the deformation gradient and boundary normal. The corner energy is found as an explicit function of the deformation gradient and the normals of the two facets meeting at the corner. For more general convex domains with possibly curved boundary, the surface energy density depends on the unit normal in a striking way. It is continuous at irrational directions, discontinuous at rational ones and nowhere differ- entiable. This pathology is alarming since it renders the surface energy minimization problem (under domain variations) ill-posed. An alternative approach of defining the continuum region is introduced, that restores continuity of the surface energy density function

New Solutions for Slow Moving Kinks in a Forced Frenkel-Kontorova Chain

We construct new traveling wave solutions of moving kink type for a modified, driven, dynamic Frenkel-Kontorova model, representing dislocation motion under stress. Formal solutions known so far are inadmissible for velocities below a thresh- old value. The new solutions fill the gap left by this loss of admissibility. Analytical and numerical evidence is presented for their existence; however, dynamic simula- tions suggest that they are probably unstable

A Model for Compression-Weakening Materials and the Elastic Fields due to Contractile Cells

We construct a homogeneous, nonlinear elastic constitutive law, that models aspects of the mechanical behavior of inhomogeneous fibrin networks. Fibers in such networks buckle when in compression. We model this as a loss of stiffness in compression in the stress-strain relations of the homogeneous constitutive model. Problems that model a contracting biological cell in a finite matrix are solved. It is found that matrix displacements and stresses induced by cell contraction decay slower (with distance from the cell) in a compression weakening material, than linear elasticity would predict. This points toward a mechanism for long-range cell mechanosensing. In contrast, an expanding cell would induce displacements that decay faster than in a linear elastic matrix.Comment: 18 pages, 2 figure

On atomistic-to-continuum couplings without ghost forces in three dimensions

In this paper we construct energy based numerical methods free of ghost forces in three dimen- sional lattices arising in crystalline materials. The analysis hinges on establishing a connection of the coupled system to conforming finite elements. Key ingredients are: (i) a new representation of discrete derivatives related to long range interactions of atoms as volume integrals of gradients of piecewise linear functions over bond volumes, and (ii) the construction of an underlying globally continuous function representing the coupled modeling method

Identification and Kullback Information in the GLSEM

In this paper we employ the Kullback Information apparatus in (a) obtaining the strong consistency of the maximum likelihood (ML) estimator in the standard version of the general linear structural econometric model (GLSEM); (b) deriving very succinctly the necessary and sufficient (nas) conditions for identification by the use of exclusion restrictions. The arguments given in (a), however, are equally applicable to a wide class of nonlinear models and the arguments in (b) are equally applicable in the context of more general types of restrictions

A Note on Heteroskedasticity Issues

The purpose of this paper is to clarify certain issues related to the incidence of heteroskadisticity in the General Linear Model (GLM); to provide simpler and more accessible proofs for a number of propositions, and to allow the results to stand under conditions considerably less stringent that those hitherto available in the literature