  The main purpose of the present paper is to solve the thermodynamic
inconsistencies that result when deriving equivalent micropolar models of
periodic beam-lattice materials through standard continualization schemes. In
fact, this technique identifies higher order micropolar continua characterized
by non-positive defined elastic potential energy. Despite this, such models are
capable of accurately simulating the optical branches of the discrete
Lagrangian model, a property lacking in the thermodynamically consistent
standard micropolar continuum. To overcome these thermodynamic inconsistencies
while preserving good simulations of the frequency band structure, a dynamic
high-frequency consistent continualization is proposed. This continualization
scheme is based on a first order regularization approach coupled with a
suitable transformation of the difference equation of motion of the discrete
Lagrangian system into pseudo-differential equations. A formal Taylor expansion
of the pseudo-differential operators allows to obtain differential field
equations at various orders according to the continualization order.
Thermodynamically consistent higher order micropolar continua with non-local
elasticity and inertia are obtained. Finally, the convergence of the frequency
band structure of the higher order micropolar models to that of the discrete
Lagrangian system is shown as the continualization order increases.
