Modeling phononic band gap in microstructured solids using the Riemann-Cartan geometric framework

Abstract

This paper discusses the modeling of acoustic wave fields in microstructured elastic solids within the framework of Riemann-Cartan geometry. We consider a scenario in which microstructural deformations occur significantly faster than those of the bulk material. This time-scale separation creates apparent geometric incompatibilities at the macroscopic level, even in the absence of permanent inelastic deformation or damage. We formalize this phenomenon by using a non-holonomic frame field to represent macroscopic elastic deformations and an associated torsion field to characterize the resulting geometric incompatibilities. The spatial components of the torsion tensor quantify the instantaneous geometric incompatibility of the macroscopic deformations, while its temporal components capture the inertial effects arising from the reversible energy exchange between the micro- and macro-scales. A key finding is that the model's dispersion relation predicts a complete frequency band gap. Furthermore, the governing equations exhibit a mathematical analogy to Maxwell's equations, potentially bridging the modeling of phononic and photonic metamaterials.