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

TL;DR

The paper develops a four-dimensional Riemann-Cartan framework to model acoustic waves in microstructured solids, introducing two co-moving frame fields to capture macro and micro deformations and encoding their geometric incompatibilities via torsion. The governing equations form a first-order hyperbolic system with reversible energy exchange between scales, closed by a convex energy potential and accompanied by a detailed dispersion analysis that reveals a complete phononic band gap whose width can be controlled by relaxation parameters $\alpha$ and $\beta$. A key result is the identification of rotational (optical) and longitudinal/shear modes, with band-gap formation arising from the microstructure and the scale separation, and a Maxwell-like structural analogy that links phononic metamaterials to photonic counterparts. The work emphasizes thermodynamic consistency, symmetric-hyperbolic formulation, and potential extensions to curvature, anisotropy, and boundary-value problems, with implications for designing metamaterials and interpreting microstructure–macroscale coupling.

Abstract

Modeling acoustic wave fields in microstructured elastic solids is discussed in the context of the Riemann-Cartan geometry. We consider a scenario where microstructural deformations occur much faster than those of the bulk material. This time-scale separation creates apparent geometric incompatibilities at the macroscopic level, even without any permanent inelastic deformation (at micro or macro-scale) or damage. We formalize this phenomenon using the concept of a non-holonomic tetrads to represent the macroscopic elastic deformations and the associated torsion field to characterize the resulting geometric incompatibilities. The spatial components of the torsion tensor quantify the instantaneous geometric incompatibility of the macroscopic elastic deformations, while its time 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 the existence of a complete frequency band gap. Furthermore, the governing equations exhibit a notable mathematical analogy to Maxwell's equations which can link the modeling of phononic and photonic metamaterials.

Figures (4)

• Figure 1: Dispersion curves $\omega(k)$ of the proposed model for microstructured solids. The parameters used to generate these curves are given in Table \ref{['tab.model_parameters']}. The complete frequency band gap is shown as the gray shaded rectangle. The acoustic modes are shown in yellow, while the optical modes are shown in green (fast) and purple (slow).
• Figure 2: Phase velocities $V_{\text{ph}}(\omega) = \omega/k$ of the proposed model and the material parameters from Table \ref{['tab.model_parameters']}. The complete frequency band gap is shown as the gray shaded rectangle. The acoustic modes are shown in yellow, while the optical modes are shown in green (fast) and purple (slow).
• Figure 3: Group velocities $V_{\text{gr}}(\omega) = \mathrm{d}\omega/\mathrm{d}k$ of the proposed model and the material parameters from Table \ref{['tab.model_parameters']}. The complete frequency band gap is shown as the gray shaded rectangle. The acoustic modes are shown in yellow, while the optical modes are shown in green (fast) and purple (slow).
• Figure 4: Dispersion curves $\omega(k)$ of the proposed model for the case $\alpha = \infty$ (the relaxation terms $\frac{1}{\alpha} \Pi^k_{\ A}$ and $\frac{1}{\alpha} E^A_{\ k}$ are switched off) and other parameters as in Table \ref{['tab.model_parameters']}. The band gap is absent, i.e. for every $\omega$ there is a wave number $k$.