In the past 30 years a wealth of new solid materials has emerged which were designed to have specific properties in order to offer optimal solutions to engineering problems and to allow the realization of new engineering devices.
This advancement would not have been possible without fundamental contributions from the theoretical sciences, in particular the mathematical sciences, which offer both analytical and numerical tools for the solution of complex problems. In order to further advance this design process, a concerted effort of experts in both mathematics and engineering is needed. The aim of the proposed SPP is exactly to address this need.
Within this general framework, mathematical tools from the field of variational analysis have proven to be successful. They include the theories of homogenization, relaxation, Gamma convergence and variational time evolution. On the smallest scale, the atomistic scale, continuum methods may be complemented by quantum mechanical models including density functional theory (DFT) or molecular dynamics (MD). Applications involve models of nonlinear elasticity, finite plasticity and phase transformations in general and the analysis of fracture, damage, motion of dislocations and the formation of microstructure in particular.
Since most problems in engineering are characterized by a strong interaction between multiple of these phenomena, it is mandatory to broaden the scope of the analysis, to consider combinations of these effects, and to develop new mathematical tools.
Therefore, the research in the proposed SPP is grouped in three major research directions:
A: Coupling of dimensions: In many systems a strong interplay of effects on structures with different dimensionality is observed.
B: Coupling of processes: The overall response of many materials depends critically on interacting processes taking place at different scales ranging from atomistic or nanoscales to macroscopic ones.
C: Coupling of structure and evolution: A major challenge is the combination of prediction of structures based on energetic considerations and the evolution of these structures in response to dynamic loadings.