Free Bloch Wave Propagation in Periodic Cauchy Materials: Analytical and Computational Strategies

The mechanical behaviour of periodic three-dimensional nonhomogeneous solids can be functionally designed by optimizing their cellular composite microstructure. The present contribution aims at establishing a theoretical and methodological framework to determine the dispersion properties of periodicCauchy materials, characterized by a generic microstructured repetitive cell. First, two conceptually alternative strategies are formulated to state the continuous complex-valued eigenproblem, governing the free undamped propagation ofBlochwaves, in the finite domain of the periodic cell. Second, a consistent mathematical procedure to convert the governing field equations into an equivalent discrete eigenproblem, suited to be computationally attacked through the finite element method, is outlined.