On the quasistatic effective elastic moduli for elastic waves in three-dimensional phononic crystals

TL;DR

This paper introduces two analytical methods, plane-wave expansion and monodromy-matrix, to accurately compute the effective elastic moduli of 3D phononic crystals, with the MM method showing faster convergence and higher efficiency.

Contribution

It develops a new monodromy-matrix approach that improves accuracy and computational efficiency in determining elastic properties of complex 3D phononic crystals.

Findings

MM method converges faster than PWE for effective moduli

MM provides explicit formulas and bounds for the Christoffel tensor

Enhanced accuracy in high-contrast and closely spaced inclusion configurations

Abstract

Effective elastic moduli for 3D solid-solid phononic crystals of arbitrary anisotropy and oblique lattice structure are formulated analytically using the plane-wave expansion (PWE) method and the recently proposed monodromy-matrix (MM) method. The latter approach employs Fourier series in two dimensions with direct numerical integration along the third direction. As a result, the MM method converges much quicker to the exact moduli in comparison with the PWE as the number of Fourier coefficients increases. The MM method yields a more explicit formula than previous results, enabling a closed-form upper bound on the effective Christoffel tensor. The MM approach significantly improves the efficiency and accuracy of evaluating effective wave speeds for high-contrast composites and for configurations of closely spaced inclusions, as demonstrated by three-dimensional examples.