Optimal Elasticity of Biological Networks

Reinforced elastic sheets surround us in daily life, from concrete shell buildings to biological structures such as the arthropod exoskeleton or the venation network of dicotyledonous plant leaves. Natural structures are often highly optimized through evolution and natural selection, leading to the biologically and practically relevant problem of understanding and applying the principles of their design. Inspired by the hierarchically organized scaffolding networks found in plant leaves, here we model networks of bending beams that capture the discrete and nonuniform nature of natural materials. Using the principle of maximal rigidity under natural resource constraints, we show that optimal discrete beam networks reproduce the structural features of real leaf venation. Thus, in addition to its ability to efficiently transport water and nutrients, the venation network also optimizes leaf rigidity using the same hierarchical reticulated network topology. We study the phase space of optimal mechanical networks, providing concrete guidelines for the construction of elastic structures. We implement these natural design rules by fabricating efficient, biologically inspired metamaterials.

Figure 1

Leaf venation as a discrete beam network. (a) Abaxial surface of a leaf of Magnolia sp., showing the hierarchically organized reticulate venation network keeping the lamina flat and rigid, and transporting water and nutrients. (b) Adaxial surface of the same leaf, emphasizing the venation network embedded in the lamina. (c) Discrete model of beam bending. Dashed orange arrows correspond to the local reference frame $\{{\mathbf{e}}_{1,2}\}$ used to construct the elastic energy Eq. (2) with ${\sin }^2{\phi }_{1,2}=\parallel {\mathbf{e}}_{1,2}\times {\mathbf{b}}_{1,2}{\parallel }^2$. The reference frame $\{R_i{\mathbf{e}}_{1,2}\}$ compensating overall rigid rotations is shown in blue. (d) Plant leaf venation subject to gravitational load $\mathbf{g}$ as prototypical example of a natural DBN. One large vein branches off into three smaller veins that all bend under the load. (e) DBN model of the node from (d). Each discrete beam joining at the node is depicted with its bending constant by line thickness and color. Deviations from the local reference (blue) are penalized by Eq. (3).

Figure 2

Compliance-optimized flat DBNs resemble real leaf venation. We optimized triangular DBNs with $N=217$ nodes and $E_{\text{triang}}=600$ edges. (a)–(c) For $0<\gamma <1$, optimal networks are sparse and show hierarchical organization and anastomosing reticulation. (d) At the transition $\gamma =1$, the network becomes highly reticulate and less hierarchically organized. The networks (a)–(d) were subject to a uniform downward load, the petiole was modeled as one additional node the position of which was fixed, and overall twists of the petiole were removed. The lamina stiffness was ${\kappa }_0={10}^{-6}$. (e)–(g) Optimal networks reduce to just the main veins as the lamina stiffness ${\kappa }_0$ is increased. (h) An optimal network with the petiole at the center and subject to a uniform upward load. The cost parameter in (e)–(h) was $\gamma =1/2$, and the lamina stiffness in (h) was ${\kappa }_0={10}^{-6}$. Fixed nodes are shown as red dots, each triangle is colored by the average nodal compliance ${\mathbf{f}}_i^{\top }{\mathbf{u}}_i$ of the adjacent nodes normalized by the maximum, and the line thicknesses are proportional to ${\kappa }_b^{\gamma /2}$.

Figure 3

Topological transition and phase space of optimal DBNs with leaf boundary conditions. Each pixel in the $25\times 25$ images (a)–(c) corresponds to a mean over 10 annealed triangular networks with $N=92$ nodes and $E_{\max }=241$ edges. (a) Network topology is encoded in the loop density $L/L_{\max }$, where $L$ is the number of loops and $L_{\max }=150$ is the maximum number of loops in the triangular grid. Gray pixels correspond to $L=0$. The dashed and solid lines approximately mark the transitions to maximally loopy and tree topologies, respectively. (b) Network structure as measured by the number of nonzero bending constant edges $E$ normalized by the maximum number $E_{\max }$ of edges in the triangular grid. (c) The compliance $c$ of the optimized networks, normalized by the compliance $\overline{c}$ of a uniform network with identical cost $K$. The results in (a)–(c) remain qualitatively valid for larger networks [58]. (d) Optimal networks △, □, ◯, and a uniform network shown with their relative displacements under the same load. The optimal networks are also marked in panels (a)–(c). Displacements are measured relative to the tip of network ◯.

Figure 4

Biologically inspired metamaterials for flatness and rigidity. (a) Additively manufactured metamaterial based on an optimized DBN topology with $\gamma =1/2$. The vertical size is 11 cm, the material is thermoplastic polyurethane [58]. (b) Metamaterial based on a uniform DBN topology with equal size and total volume. Beam radii in (a), (b) are proportional to ${\kappa }_b^{1/4}$, and ${\kappa }_0=0$. (c),(d) Side view of the same networks clamped at the petiole. The optimized network (c) remained flat. The uniform network (d) showed a tip displacement of approximately 5 cm.