Modelling size-dependent bending, buckling and vibrations of 2D triangular lattices by strain gradient elasticity models: Applications to sandwich beams and auxetics

The present work is devoted to the modelling of strongly size-dependent bending, buckling and vibration phenomena of 2D triangular lattices with the aid of a simplified first strain gradient elasticity continuum theory. As a start, the corresponding generalized Bernoulli-Euler and Timoshenko sandwich beam models are derived. The effective elastic moduli corresponding to the classical theory of elasticity are defined by means of a computational homogenization technique. The two additional length scale parameters involved in the models, in turn, are validated by matching the lattice response in benchmark problems for static bending and free vibrations calibrating the strain energy and inertia gradient parameters, respectively. It is demonstrated as well that the higher-order material parameters do not depend on the problem type, boundary conditions or the specific beam formulation. From the application point of view, it is first shown that the bending rigidity, critical buckling load and eigenfrequencies strongly depend on the lattice microstructure and these dependencies are captured by the generalized Bernoulli-Euler beam model. The relevance of the Timoshenko beam model is then addressed in the context of thick beams and sandwich beams. Applications to auxetic strut lattices demonstrate a significant increase in the stiffness of the metamaterial combined with a clear decrease in mass. Furthermore, with the introduced generalized beam finite elements, essential savings in the computational costs in computational structural analysis can be achieved. For engineering applications of architectured materials or structures with a microstructure utilizing triangular lattices, generalized mechanical properties are finally provided in a form of a design table for a wide range of mass densities.