Comprehensive investigations of crystalline systems often require methods bridging atomistic and continuum scales. In this context, coarse-grained mesoscale approaches are of particular interest as they allow the examination of large systems and time scales while retaining some microscopic details. The so-called phase-field crystal (PFC) model conveniently describes crystals at diffusive time scales through a continuous periodic field which varies on atomic scales and is related to the atomic number density. To go beyond the restrictive atomic length scales of the PFC model, a complex amplitude formulation was first developed by Goldenfeld et al (2005 Phys. Rev. E 72 020601). While focusing on length scales larger than the lattice parameter, this approach can describe crystalline defects, interfaces, and lattice deformations. It has been used to examine many phenomena including liquid/solid fronts, grain boundary energies, and strained films. This topical review focuses on this amplitude expansion of the PFC model and its developments. An overview of the derivation, connection to the continuum limit, representative applications, and extensions is presented. A few practical aspects, such as suitable numerical methods and examples, are illustrated as well. Finally, the capabilities and bounds of the model, current challenges, and future perspectives are addressed.

Nickel-based alloys are used in components such gas turbines within the aerospace industry and electric power generation due to its high tensile, rapture and creep strength. Increasing the efficiency of gas turbines are crucial to reduce emissions within the aerospace industry and increasing power gain for electric power generation. Innovation to increase the efficiency relies in part on the development of new nickel-based alloys with beneficial material properties. But also on stable and predictable material behavior during processing and post-processing of the components in the gas turbine. In two prominent material processing fields of precipitation hardened nickel-based alloys, additive manufacturing and welding, strain-age cracking (SAC) is a common phenomenon. SAC is a solid state phenomenon that generally occurs in alloys strengthened with 𝛾′, L12(Pm3m), or 𝛾′′, D022(I4/mmm), phase precipitates during post weld heat treatment or reheating where it manifests as intergranular cracking. Even though the existence of SAC has been known for several decades, its dominant mechanisms are still under considerable debate and the undertaken modeling efforts to gain insight on the phenomenon are virtually non-existent. This study aims to clarify the dominant mechanisms behind strain-age cracking. Breaching this gap would allow for new development for nickel-based alloys within both additive manufacturing and welding. To that extent the goal of this study is to provide tools to aid in clarifying the dominant mechanisms behind strain-age cracking. This is done by implementing the recently developed structural phase-field crystal (XPFC) model and examining the capabilities to model a precipitation event during reheating for a reference binary alloy in two dimensions. To evaluate the strain because of precipitation, a simple method based on the principles of neutron and synchrotron strain scanning is outlined and tested on the limited precipitation event achieved within the study. The XPFC model is capable of modeling precipitation with some restrictions that need further development with extended computational recourses. Lastly, the possibilities to extend the implemented XPFC model to cover nickel-based alloys is discussed.