Plasticity and topological defects in cellular structures: Extra matter, folds and crab moulting

This paper is concerned with elasticity and plasticity of two-dimensional cellular structures. The deformation of continuous media is defined by a mapping from the actual, deformed state of the material, into a reference (natural) state, where all elastic deformations have been relaxed. In two dimensions, the two states can be represented by a complex variable each, and the map, by a meromorphic function. Topological defects correspond to the singularity of the map: disclinations, dislocations and extra-matter, which appears as a fold or as a bulge. We show that these abstract concepts find a direct, geometric illustration in foams. The dual of a two-dimensional foam is a triangulation. The triangles are the finite elements for the two states of the material, with the vertices carrying the elementary defects. The meromorphic function can be analytically continued outside the basic triangle, and finite elements joined together in a natural fashion. The mapping enables us to define topological defects geometrically, and to compute their effect by contour integrals, and their image in the reference state. Disclinations alone can be defined intrinsically in the actual state of the material, without mapping into the reference state. For dislocation and extra matter, the mapping determines the (Burgers) contour integral, in its non-closure or its integral content, respectively. Before moulting, the crab has prepared its new shell that lies under the old one, folded, but metrically perfect and with all its topological intricacies. It stretches just after moulting through ingestion of salt water. Successive cellular divisions create the necessary folds.