Boundary-layer analysis of a pile-up of walls of edge dislocations at a lock

In this paper we analyse the behaviour of a pile-up of vertically periodic walls of edge dislocations at an obstacle, represented by a locked dislocation wall. Starting from a continuum non-local energy $E_gamma$ modelling the interactions$-$at a typical length-scale of $1/gamma$$-$of the walls subjected to a constant shear stress, we derive a first-order approximation of the energy $E_gamma$ in powers of $1/gamma$ by $Gamma$-convergence, in the limit $gammatoinfty$. While the zero-order term in the expansion, the $Gamma$-limit of $E_gamma$, captures the `bulk profile of the density of dislocation walls in the pile-up domain, the first-order term in the expansion is a `boundary-layer energy that captures the profile of the density in the proximity of the lock. This study is a first step towards a rigorous understanding of the behaviour of dislocations at obstacles, defects, and grain boundaries.

Upscaling of the dynamics of dislocation walls

We perform the discrete-to-continuum limit passage for a microscopic model describing the time evolution of dislocations in a one dimensional setting. This answers the related open question raised by Geers et al. in [GPPS13]. The proof of the upscaling procedure (i.e. the discrete-to-continuum passage) relies on the gradient flow structure of both the discrete and continuous energies of dislocations set in a suitable evolutionary variational inequality framework. Moreover, the convexity and $Gamma$-convergence of the respective energies are properties of paramount importance for our arguments.

Is adding charcoal to soil a good method for CO2 sequestration? - Modeling a spatially homogeneous soil

Carbon sequestration is the process of capture and long-term storage of atmospheric carbon dioxide (CO2) with the aim to avoid dangerous climate change. In this paper, we propose a simple mathematical model (a coupled system of nonlinear ODEs) to capture some of the dynamical effects produced by adding charcoal to fertile soils. The main goal is to understand to which extent charcoal is able to lock up carbon in soils. Our results are preliminary in the sense that we do not solve the CO2 sequestration problem. Instead, we do set up a flexible modeling framework in which the interaction between charcoal and soil can be tackled by means of mathematical tools. We show that our model is well-posed and has interesting large-time behaviour. Depending on the reference parameter range (e.g. type of soil) and chosen time scale, numerical simulations suggest that adding charcoal typically postpones the release of CO2.