Structural Properties of Stiff Elastic Networks

Networks of elastic beams can deform either by stretching or bending of their members. The primary mode of deformation (bending or stretching) crucially depends on the specific details of the network architecture. In order to shed light on the relationship between microscopic geometry and macroscopic mechanics, we characterize the structural features of networks which deform uniformly, through the stretching of the beams only. We provide a convenient set of geometrical criteria to identify such networks, and derive the values of their effective elastic moduli. The analysis of these criteria elucidates the variability of mechanical response of elastic networks. In particular, our study rationalizes the difference in mechanical behavior of cellular and fiber networks. Introduction. – Various elastic systems can be understood as networks of interconnected rods which deform by a combination of bending, stretching, twisting and shearing mechanisms. Examples include polymer gels, protein networks and cytoskeletal structures [1–7], crystal atomic lattices and granular materials [8,9], paper [10,11], wood, foams, and bones [12–16], and even continuous elastic bodies under certain circumstances [17]. Moreover, the pairwise interaction potentials used in standard elastic percolation models can also be identified with the strain energy of elastic beams [1, 8, 18]. Despite extensive research [2–7,19–22], the connection between the mechanical properties of such networks on a macroscopic level and the description of their structures on a microscopic level has not been completely elucidated yet. Interestingly, under identical loading conditions, some structures appear to deform primarily through the local stretching of the beams, while in other structures the elastic energy is stored via local bending [12,13] (twisting and shearing contributions are usually neglected). For instance, “foam-like” cellular architectures tend to be bending-dominated [2, 3], while fibrous architectures exhibit a rich mechanical behavior: Head et al. [4, 5] and Wilhelm and Frey [6] simulated the two-dimensional elastic deformation of a network of crosslinked fibers and observed a transition from a nonaffine, bending-dominated regime to an affine, stretch-dominated regime with increasing density of fibers. Recent experimental studies [23, 24] and mean field theories [25, 26] have confirmed this transition. Buxton and Clarke [7] also characterized a similar bending-to-stretching transition in three-dimensional networks, in terms of the connectivity of nodes. Elucidating this variability of mechanical response is of interest to structural applications, as well as to our understanding of various biological systems. With this aim in view, we analyze in this letter the structural conditions under which a network of beams deforms uniformly (affinely) through the extension or compression of its members. Only some specific network geometries are compatible with such an affine, stretchdominated, deformation. Indeed, the network architecture must meet two requirements: the possible symmetries of the structure, and the mechanical equilibrium at every point of the network, respectively. The first requirement results in restrictions on the beam angular distribution. We will limit our analysis to isotropic structures, though the reasoning can be transposed without difficulty to materials with lower symmetries. The second requirement results in restrictions on the possible configurations of the junctions. The inspection of these requirements provides a convenient set of geometrical criteria to identify the structures that deform affinely. Moreover, the analysis of these geometrical criteria rationalizes the observations reported on the mechanical behavior of cellular and fiber networks, and clarifies how the microscopic structural parameters