Reduced-Order Models: Numerical Homogenization for the Elastic Response of Material Microstructures

Many crystalline solids whose structures contain multiple components (where a component is typically an element but could be more complex, such as a molecule) undergo phase transitions that lead to complex microstructure. In many instances, the phase transitions are mechanochemical, where structural changes of the crystal are coupled with compositional rearrangements, resulting in dynamically evolving microstructures. (Changes in ordering can also be coupled in.) Rapid computation of the macroscopic response based on detailed microstructures is of importance for high-throughput discovery and design of materials. In this first chapter on a continuum materials physics problem of broad interest, we concern ourselves with the macroscopic, nonlinear elastic response of solids that develop microstructure. Microstructural patterns can form with considerable complexity. There is therefore interest in taking a purely computational approach to predict their macroscopic response. However, the obvious approach: evaluation of macroscopic, nonlinear elastic properties from direct numerical simulations (DNS) comes at a computational expense that renders it unviable for material design when a large number of microstructures need to be tested.