Elastic networks with optimal mechanical properties

Various elastic systems can be modeled as networks of interconnected beams. Examples include polymer gels, protein networks, crystal atomic lattices, granular materials, wood, and bones. On a length scale much larger than the typical beam length (“macroscopic” scale), such a network can be viewed as a continuous and homogeneous medium characterized by spatially constant elastic moduli. The relationship between the mechanical properties of elastic networks on a macroscopic level, and the details of their microstructures is the key to optimization and design of lightweight, strong, and tough materials. The continuum modelling of such discrete structures has a long history, going as back as the pioneering work of A. Cauchy and S. Poisson. The stiffness of such a system clearly depends on its density , defined as the volume of beams per unit volume of material. But for a given value of , it is also dramatically affected by the specific spatial arrangement of the elastic phase within the material. On dimensional grounds, the volumetric density of strain energy associated with a stretch-dominated deformation varies linearly with , while it scales as for the deformation of a three-dimensional network dominated by the beam bending mode [1]. Thus, for the low-density materials considered here ( ), a structure deforming primarily through the beam stretching mode is usually much stiffer. However, the constant of proportionality between and still varies significantly among stretched-dominated networks.\\ In this talk, I will present our numerical and theoretical investigations on the relation between the macroscopic mechanical response of a network under small strain condition and the details of its microstructure. I will stress that this relation is not straightforward. For instance, networks with identical node connectivity can have different stiffness. Conversely, networks with very different microstructures can present identical mechanical response. Moreover, contrary to a common belief, stiffness is not directly related to the triangulation of the mesh, and a fully triangulated network may have a lower stiffness than a network with few triangulated units. I will then show the existence of a class of isotropic networks which are stiffer than any other ones with the same symmetry, density and elastic phase [2, 3]. The elastic moduli of these optimal elastic networks can be calculated explicitly. They constitute upper-bounds which compete or improve the well-known Hashin-Shtrikman bounds. I will establish a general set of criteria (which turn out to be necessary and sufficient conditions) that allow to identify these networks.\\ Relation between the mechanical and transport properties of these networks will also be discussed. Finally, examples of such networks with periodic arrangement are presented, in both two and three dimensions.