Materials structure and behavior: From atomistics to continuum
Research Group Reina
June 16 - August 16, 2014
Organizers: Isaac V. Chenchiah, Kaushik Dayal, Gero Friesecke, Richard D. James, Celia Reina
Natural and engineered materials exhibit multiple length scales ranging from the subatomic to the continuum, and a broad range of time scales. These scales are distinguished not only by structure, but also in terms of their impact on macroscopic behavior.
Models and simulation techniques at certain scales are well-developed; e.g. quantum mechanics, molecular dynamics, statistical mechanics and continuum mechanics are widely used to understand the physical processes occurring at appropriate scales. Current – and even the most optimistic predictions of future – computational resources simply do not allow direct use of fine-scale models for macroscopic applications. As a result, multiscale methods that transfer information across scales and models, thereby combining physical fidelity with computational efficiency, present the possibility of predictive calculations.
However, rigorous connection between the scales with controllable approximations remains a central open problem. This is the key hurdle to the computational design of materials and structures for applications ranging from biomedical, energy production and storage, nanotechnology, aerospace, etc. This hurdle also presents scientifically fundamental and intellectually interesting problems to both the engineering/physics and mathematics research communities. Physically, the transition between scales enables a differentiation between the essential physics and the irrelevant degrees of freedom for a specific application. Mathematically, “essential physics” translates to a mathematically defined property and relevant degrees of freedom concerns the search for simplified descriptions of that property. The rigorous transition between scales requires tools from emerging areas, e.g. Γ-convergence and other asymptotic methods, combining PDEs and statistical mechanics descriptions, extending statistical mechanics to solid/crystalline systems and nonequilibrium situations, etc.
The aim of this research group was to attack three interrelated problems in the specific area of multiscale modeling for inelastic processes. First, the use of group theory and non-equilibrium statistical mechanics to probe the dynamic, far-from-equilibrium behavior of materials and nanostructures. Second, the application of differential geometry and group theory methods from crystals to geometrically characterize defects in non-crystalline, high-symmetry nanostructures such as nanotubes, nanobeams, monolayers, etc. Third, the use of rigorous scale-transition methods to obtain the atomistic-to-continuum limit for inelastic processes with mass transfer, e.g. growth, diffusion, crystal formation, etc.