Finite element implementation of novel low-order continuum models for the dynamics of a mass-in-mass lattice with long-range interactions: A discrete-continuum-discrete approach

This work presents a finite element (FE) methodology to model a mass-in-mass lattice system with long-range interactions through a two-step discrete–continuum–discrete approach. In the first step, standard and non-standard continualization procedures are employed to derive non-classical continuum models from the discrete lattice. Non-classical models with micro-inertia reveal high accuracy, not presenting physical inconsistencies while avoiding higher-order spatial derivatives that require extra boundary conditions whose physical meaning is unclear. The predictive capabilities of the new continuum models are assessed by comparing their dispersion relations and natural frequencies with those of the discrete system. In the second step, the novel continuum models are spatially discretized using FE models. Special emphasis is placed on the treatment of micro-inertia terms in the mass matrix, which enables a separation between the microstructural length scale and the numerical element size. As a result, element sizes larger than the characteristic length can be used without compromising accuracy, leading to a substantial reduction in degrees of freedom and computational cost. Therefore, the proposed methodology reliably reproduces the dynamic behaviour of the mass-in-mass lattice while providing a versatile framework suitable for heterogeneous materials, complex geometries, and arbitrary boundary conditions. Although demonstrated here for a mass-in-mass system, the approach can be extended to other lattice typologies.