Discrete-continuum-discrete approach for the modeling of the dynamic behavior of 2D lattice systems

This work presents a new methodology to model elastic lattice systems through a two-step approach that permits to reliably capture its dynamic behavior with a lower computational cost than modeling the lattice explicitly. The first step consists of a non-standard continualization accounting for scale effects. Several methods are explored to derive new continuum models, whose dispersive and vibrational behaviors are compared to that of the lattice, considered as a reference. Non-classical models with micro-inertia reveal high accuracy, not presenting physical inconsistencies. The second step follows a FEM spatial discretization of the developed continua, accounting for micro inertia terms in the mass matrix. Finally, the FEM formulation allows the use of element sizes larger (up to four times) than the physical length scale of the lattice system, thus significantly reducing the computational cost while maintaining accuracy and enabling a versatile application to materials, geometries and boundary conditions. The methodology is tested here for a 2D system with displacements in the plane, but can be extended to other lattice typologies as well.