Variational principles and finite element Bloch analysis in couple stress elastodynamics

We address the numerical simulation of periodic solids (phononic crystals) within the framework of couple stress elasticity. The additional terms in the elastic potential energy lead to dispersive behavior in shear waves, even in the absence of material periodicity. To study the bulk waves in these materials, we establish an action principle in the frequency domain and present a finite element formulation for the wave propagation problem related to couple stress theory subject to an extended set of Bloch-periodic boundary conditions. A major difference from the traditional finite element formulation for phononic crystals is the appearance of higher-order derivatives. We solve this problem with the use of a Lagrange-multiplier approach. After presenting the variational principle and general finite element treatment, we particularize it to the problem of finding dispersion relations in elastic bodies with periodic material properties. The resulting implementation is used to determine the dispersion curves for homogeneous and porous couple stress solids, in which the latter is found to exhibit an interesting bandgap structure.