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

• A fast new procedure is described for calculating the effective elastic moduli of 3D solid–solid phononic crystals of arbitrary anisotropy.
• The method is related to the plane-wave expansion (PWE) approach, and compared with this.
• The main idea is to combine classical PWE techniques with a one-dimensional approach, similar to what is adopted in layered media, and which leads to relatively simple analytical expressions.
• The new method described here shares some of the analytical simplicity of the 1D problem.
• Furthermore, it is far more efficient as a numerical scheme than the PWE approach, being capable of handling closely spaced inclusions very well.

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.