A Bloch analysis extended to weakly disordered periodic media

Prediction of dispersion properties is crucial for metamaterials and phononic crystals when advantageous wave propagation characteristics are sought, such as negative refractive index, frequency band gaps in which waves are filtered, and vibration isolation. The study of wave propagation phenomena in a periodic system can be reduced to a primitive cell by Floquet-Bloch analysis[1]. However, due to the manufacturing process, exact periodicity is generally not achieved in periodic materials used in engineering[2]. Periodicity defects throughout the structure can deteriorate those characteristics, and their effects on wave propagation phenomena need to be investigated. This contribution presents a semi-analytical method based on asymptotic expansion to predict the dispersion curves of a weakly disordered periodic medium, whose properties are obtained by weakly deforming the field of periodic coefficients[3]. By a change of variables, the acoustic wave equation in the perturbed medium is transformed to the one with perturbed coefficients expressed in the periodic medium[4]. Then an asymptotic expansion with respect to a small parameter measuring the perturbation is applied, and a series of equations of different orders of this parameter is obtained. It can be shown that the 0- and 1st-order problems can be analyzed using the Floquet-Bloch theory. The proposed approach consists in post-processing the Bloch eigenfrequencies and eigenmodes of the non-disturbed periodic medium to obtain the dispersion curves of the disordered periodic medium. Several 1D and 2D examples are considered to perform parametric analysis and discuss the validation domain of our approach. The fact that only Bloch eigenvalues and eigenmodes of the periodic medium are needed to estimate the dispersion characteristics, namely the band gaps, of a disturbed periodic medium makes the proposed method attractive. Comparison between dispersion relations obtained numerically by solving a large-scale perturbed model and our approach are provided. References [1] Floquet, G. Sur les équations différentielles linéaires à coefficients périodiques. in Annales scientifiques de l'École normale supérieure. 1883. [2] Suo, Y., et al., The effect of fabrication defects on the mechanical behaviors of metal matrix composites. Materials Today Communications, 2020. 25: p. 101663. [3] Blanc, X., C. Le Bris, and P.-L. Lions, Stochastic homogenization and random lattices. Journal de mathématiques pures et appliquées, 2007. 88(1): p. 34-63. [4] Milton, G.W., M. Briane, and J.R. Willis, On cloaking for elasticity and physical equations with a transformation invariant form. New Journal of Physics, 2006. 8(10): p. 248.