Double groupoids of composites: Applications to uniformity

In this paper, we introduce a novel geometric framework to analyze the uniformity properties of composite materials by means of material double groupoids. This structure allows for the systematic study of the interaction between different constitutive responses within a composite body. Building upon the classical notion of uniformity in simple materials (characterized by the transitivity of the material groupoid), we extend this concept by introducing a rich taxonomy of uniformity conditions, derived from the various types of transitivity and “filling conditions” inherent in double groupoids. These include horizontal, vertical, weak and strong uniformities, each corresponding to physically meaningful compatibility properties between the constituents. Our approach not only generalizes existing theories but also provides a flexible tool to model and compare different configurations of composites, with potential applications to the theory of metamaterials and structured continua. Several examples, including crystalline solids with minimal symmetry, illustrate the scope and depth of the proposed framework.