A dynamic high-frequency consistent continualization of beam-lattice materials

The main purpose of the present paper is to solve the thermodynamic inconsistencies that result when deriving equivalent micropolar and/or higher order micropolar models of periodic beam-lattice materials through standard continualization schemes which provide a 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 energetic 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 having non-local positive defined elastic and kinetic energy 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.