Improving the accuracy of analytical relationships for mechanical properties of permeable metamaterials

Permeable porous implants must satisfy several physical and biological requirements in order to be promising materials for orthopaedic application: they should have the proper levels of stiffness, permeability, and fatigue resistance approximately matching the corresponding levels in bone tissues. This can be achieved using designer materials, which exhibit exotic properties, commonly known as metamaterials. In recent years, several experimental, numerical, and analytical studies have been carried out on the influence of unit cell micro-architecture on the mechanical and physical properties of metamaterials. Even though experimental and numerical approaches can study and predict the behaviour of different micro-structures effectively, they lack the ease and quickness provided by analytical relationships in predicting the answer. Although it is well known that Timoshenko beam theory is much more accurate in predicting the deformation of a beam (and as a result lattice structures), many of the already-existing relationships in the literature have been derived based on Euler–Bernoulli beam theory. The question that arises here is whether or not there exists a convenient way to convert the already-existing analytical relationships based on Euler–Bernoulli theory to relationships based on Timoshenko beam theory without the need to rewrite all the derivations from the start point. In this paper, this question is addressed and answered, and a handy and easy-to-use approach is presented. This technique is applied to six unit cell types (body-centred cubic (BCC), hexagonal packing, rhombicuboctahedron, diamond, truncated cube, and truncated octahedron) for which Euler–Bernoulli analytical relationships already exist in the literature while Timoshenko theory-based relationships could not be found. The results of this study demonstrated that converting analytical relationships based on Euler–Bernoulli to equivalent Timoshenko ones can decrease the difference between the analytical and numerical values for one order of magnitude, which is a significant improvement in accuracy of the analytical formulas. The methodology presented in this study is not only beneficial for improving the already-existing analytical relationships, but it also facilitates derivation of accurate analytical relationships for other, yet unexplored, unit cell types.