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

Influence of Interface Scattering on Shock Waves in Heterogeneous Solids

In heterogeneous media, the scattering due to interfaces between dissimilar materials play an important role in shock wave dissipation and dispersion. In this work the influence of interface scattering effect on shock waves was studied by impacting flyer plates onto periodically layered polycarbonate/6061 aluminum, polycarbonate/304 stainless steel and polycarbonate/glass composites. The experimental results (using VISAR and stress gauges) indicate that the rise time of the shock front decreases with increasing shock strength, and increases with increasing mechanical impedance mismatch between layers; the strain rate at the shock front increases by about the square of the shock stress. Experimental and numerical results also show that due to interface scattering effect the shock wave velocity in periodically layered composites decreases. In some cases the shock velocity of a layered heterogeneous composite can be lower than that of either of its components

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

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

A Minimal Mechanosensing Model Predicts Keratocyte Evolution on Flexible Substrates

A mathematical model is proposed for shape evolution and locomotion of fish epidermal keratocytes on elastic substrates. The model is based on mechanosensing concepts: cells apply contractile forces onto the elastic substrate, while cell shape evolution depends locally on the substrate stress generated by themselves or external mechanical stimuli acting on the substrate. We use the level set method to study the behaviour of the model numerically, and predict a number of distinct phenomena observed in experiments, such as (i) symmetry breaking from the stationary centrosymmetric to the well-known steadily propagating crescent shape, (ii) asymmetric bipedal oscillations and travelling waves in the lamellipodium leading edge, (iii) response to remote mechanical stress externally applied to the substrate (tensotaxis) and (iv) changing direction of motion towards an interface with a rigid substrate (durotaxis)