Uniformly Gamma-equivalent theories for discrete-to-continuum limits
Project management at the University of Würzburg:
A trend in engineering is to design smaller and smaller devices. In the modeling of the physical properties of such devices, In doing so, classical continuum theories reach their limit of applicability, while purely atomistic models are still too complex.
The aim of the proposed project is to model physical effects of such small devices on the level of energy functionals and to analyze the obtained models as well as their energy minimizers. To this end we start from an energy functional in a discrete system and derive an appropriate continuum model which still contains a small-scale parameter, such as the interatomic distance, and thus can describe physical effects of such small devices.
n conventional discrete-to-continuum methods, as e.g. Gamma-convergence, small-scale parameters are lost in the limit. We will apply novel mathematical methods recently proposed by Braides and Truskinovsky and develop these methods further. The resulting energy functionals will be asymptotic expansions of the original energy in terms of a small-scale parameter; in particular they will be uniform with respect to a secondary parameter (as e.g.\ the macroscopic displacement of a material). These energy functionals are called uniformly Gamma-equivalent theories. A further task is to analyze these theories mathematically and to characterize local minimizers.
We will firstly derive such uniformly Gamma-equivalent theories for fracture mechanics and for materials with defects leading to rupture of a material. Secondly we will apply the methods developed also to other areas of continuum
mechanics and magnetism.
Projekt period: from 04.2011 to 06.2014