Through the last decades several nonlocal models of linear elasticity have been introduced as enhancements of the Cauchy-elastic model, often with the purpose of providing an improved mechanical description of solids at the microscale level. Although many efforts have been devoted to the analytical formulation of these advanced constitutive models, a definitive interpretation of the relevant static quantities is still incomplete and Finite Element (FE) solvers are practically unavailable. In this thesis, after providing a mechanical interpretation to the static quantities involved in strain gradient (of Mindlin type) elastic materials, an overview on the possible quadrilateral Hermitian finite elements is given to treat quasi-static plane problems. Beside the classical finite elements inspired by those adopted for modeling Kirchhoff plates, an alternative quadrilateral self-constrained finite element formulated through Lagrange multipliers is also proposed. With reference to a hexagonal lattice structure, for which the equivalent constitutive tensors have been recently derived as closed-form expressions, the developed FE codes are exploited to assess the reliability of modelling lattices through higher-order constitutive equations. These analyses are developed for one-dimensional and two dimensional problems, where the former are considered for both homogeneous layers (with a finite size in one direction) and rod-type structures (with a finite uniform cross section along one direction). It is confirmed that higher-order modelling improves the mechanical description. In particular, the macroscale response is shown to be strongly affected by higher-order contributions in the presence of extreme elastic contrast between microstructural elements. Indeed, in this last case, only higher-order modeling captures a non-null residual stiffness, which vanishes in the framework of classical models. Therefore, higher-order modeling becomes important not only to describe the mechanical response at a microlevel, but also for macrolevel modelling, when extreme mechanical properties are addressed. The presented results pave the way to a refined modelling of architected materials leading to improved design of microstructures displaying innovative mechanical features.