Due to gravitational self-compression, the pressure in planetary interiors can reach millions of times the atmospheric pressure. Such high pressure has a significant influence on their rheology. In the present paper, we focus on how pressure in the range of the Earth's lower mantle may influence the structure of a MgO {310}/[001] tilt boundary. The defected structure of the grain boundary (GB) will be described through its dislocation, disclination, and generalized-disclination (g-disclination) density fields. At first, the strain and rotation fields in the boundary area at different pressures are derived from the discrete atomic positions simulated by first-principles calculations. For each pressure, the discontinuities of displacement, rotation, and strain in the boundary area are continuously rendered by dislocation, disclination, and g-disclination density fields, respectively. These density fields measured at different pressures are compared to provide understanding on how pressure does influence the GB structures in Earth materials.

We study an atomistic pair potential-energy E (n)(y) that describesthe elastic behavior of two-dimensional crystals with n atoms where $y \in {\mathbb R}^{2\times n}$ characterizes the particle positions. The mainfocus is the asymptotic analysis of the ground state energy as ntends to infinity. We show in a suitable scaling regime where theenergy is essentially quadratic that the energy minimum of E (n)admits an asymptotic expansion involving fractional powers of n:

${\rm min}_y E^{(n)}(y) = n \, E_{\mathrm{bulk}}+ \sqrt{n} \, E_\mathrm{surface} +o(\sqrt{n}), \qquad n \to \infty.$

The bulk energy density E bulk is given by an explicitexpression involving the interaction potentials. The surface energyE surface can be expressed as a surface integral where theintegrand depends only on the surface normal and the interactionpotentials. The evaluation of the integrand involves solving a discrete algebraic Riccati equation. Numerical simulations suggestthat the integrand is a continuous, but nowhere differentiable function ofthe surface normal.

Wilkins' method has been successfully used since early 60s for numericalsimulation of high velocity contact elastic-plastic flows. The present workproposes some effective modifications of this method including moresophisticated material model including the Baushinger effect; modification ofthe algorithm based on correction of the initial configuration of a solid; acontact algorithm based on the idea of a mild contact.

In order to describe a solid which deforms smoothly in some region, but non smoothly in some other region, many multiscale methods have recently been proposed. They aim at coupling an atomistic model (discrete mechanics) with a macroscopic model (continuum mechanics). We provide here a theoretical ground for such a coupling in a one-dimensional setting. We briefly study the general case of a convex energy, and next concentrate on a specific example of a nonconvex energy, the Lennard-Jones case. In the latter situation, we prove that the discretization needs to account in an adequate way for the coexistence of a discrete model and a continuous one. Otherwise, spurious discretization effects may appear. We provide a numerical analysis of the approach.

The set of the attainable structures of a continuous time homogeneous Markov system (HMS) with fixed size, is considered as a continuum and the evolution of the HMS in the Euclidean space corresponds to its motion. Taking account of the velocity field of the HMS, a suitable model of continuum–defined by its stress tensor–is proposed in order to explain the motion of the system. The adoption of this model (equivalently of its stress tensor) enables us to establish the concept of the energy of a structure of the HMS.