  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.
