  It is well known that the classical energetically consistent micropolar model
has limits in simulating the frequency band structure of packed granular
materials (see Merkel et al., 2011). It is here shown that if a standard
continualization of the difference equation of motion of the discrete model is
carried out, an equivalent micropolar continuum is obtained which is able to
accurately simulate the optical branches of the discrete model. Nevertheless,
this homogenized continuum presents non-positive defined elastic potential
energy, a deficiency that limits the reliability of the model and implies
instability phenomena (destabilizing effects) in the acoustic branches. This
drawback is circumvented here through an high-frequency dynamic homogenization
scheme which is based on an enhanced continualization of the discrete governing
equations into pseudo-differential equations. Through a formal Taylor expansion
of the pseudo-differential operators a higher order differential equation
corresponding to the governing equation of a non-local continuum
thermodynamically consistent are obtained. The resulting approach allows
obtaining an equivalent micropolar continuum characterized by inertial
non-locality. Moreover, higher order continua with non-local constitutive and
inertial terms may be derived. The proposed continuum models are proved to be
able to accurately describe both the static and dynamic behavior of the
discrete granular model. Finally, the convergence to the response of the
discrete system is shown when increasing the order of the higher order
continuum.
