Beam formulation and FE framework for architected structures under finite deformations

The breakthrough in additive manufacturing (AM) techniques is opening new routes into the conceptualisation of novel architected materials. However, there are still important roadblocks impeding the full implementation of these technologies in different application fields such as soft robotics or bioengineering. One of the main bottlenecks is the difficulty to perform topological optimisation of the structures and their functional design. To help this endeavour, computational models are essential. Although 3D formulations provide the most reliable tools, these usually present very high computational costs. Beam models based on 1D formulations may overcome this limitation but they need to incorporate all the relevant mechanical features of the 3D problem. Here, we propose a mixed formulation for Timoshenko-type beams to consistently account for axial, shear and bending contributions under finite deformation theory. The framework is formulated on general bases and is suitable for most types of materials, allowing for the straightforward particularisation of the constitutive description. To prove validity of the model, we provide original experimental data on a 3D printed elastomeric material. We first validate the computational framework using a benchmark problem and compare the beam formulation predictions with numerical results from an equivalent 3D model. Then, we further validate the framework and illustrate its flexibility to predict the mechanical response of beam-based structures. To this end, we perform original experiments and numerical simulations on two types of relevant structures: a rhomboid lattice and a bi-stable beam structure. In both cases, the numerical results provide a very good agreement with the experiments by means of both quantitative and qualitative results. Overall, the proposed formulation provides a useful tool to help at designing new architected materials and metamaterial structures based on beam components. The framework presented may open new opportunities to guide AM techniques by feeding machine learning optimisation algorithms.