Figure 1: The structure of natural materials are characterized by adaptations to mechanical loading. (a) The trabecular struts within bone grow directionally in response to complex loading conditions to provide mechanical support while being light enough to allow mobility. (b) Norwegian spruce trees can grow spiraling grain to increase torsional strength in the direction of spirality. Spiral handedness can differ from tree to tree depending on the prevailing wind direction. (c) The hollow culm and radial density gradient of bamboo maximize its moment of inertia and beam-bending stiffness, allowing rapid and material-efficient growth. (c) Cholla cacti grow in harsh, windy deserts of the North American southwest. The fleshy, succulent tissue is supported by an internal lignified skeleton of wood with helical porosity. Previous work that examined the tubercle pores hypothesized they help optimize spine-packing density or thermal stability of the cactus.

Figure 2: (a) Manual measurements of the tubercle dimensions, pitch, and angle of orientation with respect to the radial-tangential plane were obtained for each cholla sample. (b) Before testing, all /specimens were fully hydrated to mimic in vivo conditions. (c) A torsional adaptor from previous work [39] was fitted to the Instron 3367 Dual Column Testing Systems device (Norwood, MA). It comprises a rack and a pinion that translates the vertical displacement of the Instron crosshead to torque on the sample. (d) A hydrated cholla sample with epoxy reinforcements at each end, fitted into the torsion tester.}

Figure 3: (a) Though useful for creating 3D models that capture the general shape of a biological structure, the laser scanner was unable to capture the fiber complexity of the pencil cholla skeleton imaged. (b) The photogrammetric model successfully captured the fiber complexity of the staghorn cholla. By unrolling the model in MATLAB, a more accurate assessment of overall tubercle arrangement could be obtained than by hand measurements: lines can be drawn (red, dashed) through the centers of tubercle pores to examine the overall growth pattern (i.e. angle of orientation with respect to the radial-tangential plane) rather than neighbor-to-neighbor comparisons as was done manually (Figure 2a).}

Figure 4: (a) The endgrain of the cholla cactus as observed in optical microscopy features aggregate rays and a diffuse-porous arrangement of vessel elements. (b) The relatively small but numerous vessels allow maximum axial conduction of water during intermittent rain while preventing cavitation during dry seasons \cite{ogburn2010ecological}. The similar ray cells conduct water to fleshy succulent tissue for storage, and can store water themselves.

Figure 5: µ-CT of a specimen revealed (a) banding of latewood and earlywood in the radial-transverse plane, corresponding to slow and fast growth periods respectively. (b) Fibers were observed to weave in between, divert around, or split off into smaller fibers at tubercle pores. (c) Slowly raising the density threshold progressively from (i—iii), radially continuous regions of increased lignification were observed, mainly at the tubercle ligaments.

Figure 6: (a) Compared to the density-normalized performance of bamboo , balsa, and trabecular bone, cholla exhibited a far greater combination of stiffness, strength, and toughness. (b) Its behavior was characterized by three regimes of deformation: linear elastic deformation of the whole structure (i) from initially elliptical tubercle pores, (ii) delamination of the fibers and crack propagation in the axial direction (marked with arrows) initiated at the axial points of the tubercle pores, (iii) and ultimately, hinging of the tubercle ligaments in a spiraling band (circled in blue) and fiber rupture. These numerous and progressive failure modes contrast with the brittle behavior of hydrated bamboo and bone.

Figure 7: (a) Micro-computed tomography images of a specimen (i) before and (ii) after torsion revealed several extrinsic toughening mechanisms. (b) (i) Cracks typically initiated at the axial ends of tubercle pores near the center of the specimen and propagated axially. The tubercle ligaments near the center would still be loaded in tension, however, and eventually fail by rupture with extensive fiber pullout. (ii) On encountering a neighboring tubercle pore, crack blunting would occur and slow its advance. (iii, iv) Alternatively, the weaving and criss-crossing fibers would deflect the crack on a still more tortuous path.

Figure 8: (a) Finite element analysis (FEA) was used to model a unit cell of the cholla wall using homogeneous, isotropic material parameters to isolate the geometry contribution to mechanical behavior. By taking the density-normalized specific, effective moduli and comparing it against that of the input material, we observed that the cholla geometry was highly optimized for compression in the axial direction and shear. Observing the stress concentration in (b) axial compression and (c) pure shear indicated that the lignification of the tubercle ligaments (Figure 5c, i, ii, iii) was primarily a plant response to shear forces.

Figure 9:  Finite element analysis (FEA) was used to model unit cells of identical pore angles and pitch but varying roundness (left to right: 1.0, 0.98, 0.78, 0.54) to understand how the pore shape contributes to mechanical behavior. At approximately the geometry of the original staghorn cholla (roundness = 0.78), there is a triple point of high specific axial and shear stiffnesses, reinforcing the hypothesis that the cholla pore geometry is a unique adaptation to help resist torsion and axial compression.

Figure 10: A topology optimized hollow tube in torsion yielded a somewhat similar structure to the cholla wood skeleton. The angular, rhombohedral geometry of the pores contrasted against the smooth ellipses observed in staghorn cholla (and circular pores observed in other species). Discrepancies may be attributed to the fact that the topology optimization, unlike nature, did not simultaneously account for other biological functions. (b) Even though the topology optimized result did not completely match that of the cholla, it may still be combined with other architectural features of the cholla (e.g. fiber-crossing) to inspire light and torsionally stiff composites. Applications could include prosthetics or next-generation composite driveshafts.