Spinodal decomposition-inspired metamaterial: tailored homogenized elastic properties via the dimensionless Cahn-Hilliard equation

Metamaterials are increasingly gaining attention due to their ability to exhibit exceptional properties through architectural design. Among these, spinodal topologies, inspired by minimal surface structures, represent a novel class derived from the spinodal decomposition process occurring in certain metal alloys. In metamaterials research, the spinodal decomposition process is typically approximated using statistical methods, such as the superposition of Gaussian random fields, which are only valid during the initial stages of the phenomenon. In this study, we advance the exploration of metamaterials inspired by spinodal decomposition by addressing the full nonlinear dynamic evolution governed by the Cahn-Hilliard partial differential equation (PDE). While solving this nonlinear PDE at the natural scales of spinodal decomposition usually demands ad hoc algorithms, our approach focuses on scales orders of magnitude larger—those relevant to the production of metamaterials. This scale adjustment allows us to employ a straightforward finite difference algorithm, which is computationally efficient and well-suited to the structural scales of interest. To design metamaterials with tailored elastic properties, we solve the Cahn-Hilliard equation in its dimensionless form, governed by two key dimensionless parameters. The solution fields generated are then transformed into CAD models via a dedicated algorithm, enabling subsequent finite element analysis (FEA) to extract the homogenized elastic properties of the resulting structures. Our analysis investigates how variations in the two dimensionless parameters influence the anisotropic elastic properties of the metamaterial unit-cell. By constructing response surfaces for these properties, we enable a reverse homogenization approach, allowing to design materials with targeted mechanical characteristics. This capability offering precise control over the elastic behavior of metamaterials, of interest in the field of material design. Finally, numerical and experimental validations demonstrate the accuracy of the proposed homogenization framework in predicting displacement fields, even for the complex topologies of spinodal decomposition-inspired metamaterial.