A general finite element based non-local theory for the medium-long-range correlation of metallic glasses

Mean-field theory is extensively used in statistical physics, solid-state physics, and biophysics to compress a large number of interacting multi-body problems into an effective single-body problem, providing some insight into the system's behavior at a lower computing cost. Nonethe-less, the mean-field approximation always suppresses the non-local effects, hence obscuring the underlying mechanisms involving unit interactions between microscopic events, such as competition, co-evolution, and self-organized criticality. This paper presents a general, non-local mean-field approximation for the medium-long-range correlation of materials based on the finite element method, which has excellent compatibility and scalability with existing theories. It ties the evolution of a deformation unit to the status of other spatial domain elements, which is analogous to the fundamental principle of cellular automata. When applied to metallic glasses, the model demonstrates excellent spatial-temporal agreement between the shear band evolution measurements and simulation findings. Excitingly, the self-organized criticality of amorphous systems and the self-adaptive evolution mechanism of shear bands are realized in both two and three dimensions for the first time. Without restriction to amorphous systems, this work explores the application of mean-field theories to non-local phenomena. Future research into new mathematical types of medium-long range interaction is conceivable through multiscale simulation techniques.

Automated summary

Mean-field theory is widely used to simplify complex problems, but it tends to overlook the non-local effects between interacting particles. This paper presents a non-local mean-field approximation based on the finite element method, which captures the spatial correlation of materials and demonstrates good agreement with experimental measurements. The model also reveals the self-organization and evolution mechanism in two and three dimensions, opening up possibilities for applying mean-field theories to non-local phenomena.