Living in a physical world XI. To twist or bend when stressed

Charting the twist-to-bend ratio of plant axes

During the evolution of land plants many body plans have been developed. Differences in the cross-sectional geometry and tissue pattern of plant axes influence their flexural rigidity, torsional rigidity and the ratio of both of these rigidities, the so-called twist-to-bend ratio. For comparison, we have designed artificial cross-sections with various cross-sectional geometries and patterns of vascular bundles, collenchyma or sclerenchyma strands, but fixed percentages for these tissues. Our mathematical model allows the calculation of the twist-to-bend ratio by taking both cross-sectional geometry and tissue pattern into account. Each artificial cross-section was placed into a rigidity chart to provide information about its twist-to-bend ratio. In these charts, artificial cross-sections with the same geometry did not form clusters, whereas those with similar tissue patterns formed clusters characterized by vascular bundles, collenchyma or sclerenchyma arranged as one central strand, as a peripheral closed ring or as distributed individual strands. Generally, flexural rigidity increased the more the bundles or fibre strands were placed at the periphery. Torsional rigidity decreased the more the bundles or strands were separated and the less that they were arranged along a peripheral ring. The calculated twist-to-bend ratios ranged between 0.85 (ellipse with central vascular bundles) and 196 (triangle with individual peripheral sclerenchyma strands).

Twist-to-Bend Ratios and Safety Factors of Petioles Having Various Geometries, Sizes and Shapes

From a mechanical viewpoint, petioles of foliage leaves are subject to contradictory mechanical requirements. High flexural rigidity guarantees support of the lamina and low torsional rigidity ensures streamlining of the leaves in wind. This mechanical trade-off between flexural and torsional rigidity is described by the twist-to-bend ratio. The safety factor describes the maximum load capacity. We selected four herbaceous species with different body plans (monocotyledonous, dicotyledonous) and spatial configurations of petiole and lamina (2-dimensional, 3-dimensional) and carried out morphological-anatomical studies, two-point bending tests and torsional tests on the petioles to analyze the influence of geometry, size and shape on their twist-to-bend ratio and safety factor. The monocotyledons studied had significantly higher twist-to-bend ratios (23.7 and 39.2) than the dicotyledons (11.5 and 13.3). High twist-to-bend ratios can be geometry-based, which is true for the U-profile of Hosta x tardiana with a ratio of axial second moment of area to torsion constant of over 1.0. High twist-to-bend ratios can also be material-based, as found for the petioles of Caladium bicolor with a ratio of bending elastic modulus and torsional modulus of 64. The safety factors range between 1.7 and 2.9, meaning that each petiole can support about double to triple the leaf's weight.

Peak values of twist-to-bend ratio in triangular flower stalks of Carex pendula: a study on biomechanics and functional morphology

PREMISE

Because of their own weight and additional wind forces, plants are exposed to various bending and torsional loads that sometimes require contradictory structural characteristics and mechanical properties. The resulting trade-off between flexural and torsional rigidity can be quantified and compared using the dimensionless twist-to-bend ratio.

METHODS

The flexural rigidity of the stems of Carex pendula was determined by 2-point bending tests. Additionally, 4-point bending tests and torsional tests were carried out on segments of two internodes directly below the inflorescences to measure flexural and torsional rigidity. Anatomical investigations were performed to quantify the cross-sectional distribution of their tissues.

RESULTS

The flexural rigidity of the stems, segments of the apical internode 1, and the more basal internode 2 differed significantly from each other, whereas the bending elastic moduli were not significantly different. The torsional rigidity of segments of internode 2 was a factor of 3.3 higher than that of internode 1, whereas the torsional moduli did not differ significantly. The twist-to-bend ratios of segments of internode 1 and 2 reached values between 85 and 403. Light microscopic investigations revealed a triangular stem possessing individual sclerenchyma strands, with internode 2 having significantly more strands than internode 1.

CONCLUSIONS

In the case of Carex pendula, flexural and torsional rigidity are adapted to the given mechanical constraints by significant changes in morphometric variables (axial and polar second moment of area, number of sclerenchyma strands), whereas the material properties (bending and torsional modulus) do not change markedly along the stem.

Twist-to-bend ratio: an important selective factor for many rod-shaped biological structures

Mechanical optimisation plays a key role in living beings either as an immediate response of individuals or as an evolutionary adaptation of populations to changing environmental conditions. Since biological structures are the result of multifunctional evolutionary constraints, the dimensionless twist-to-bend ratio is particularly meaningful because it provides information about the ratio of flexural rigidity to torsional rigidity determined by both material properties (bending and shear modulus) and morphometric parameters (axial and polar second moment of area). The determination of the mutual contributions of material properties and structural arrangements (geometry) or their ontogenetic alteration to the overall mechanical functionality of biological structures is difficult. Numerical methods in the form of gradient flows of phase field functionals offer a means of addressing this question and of analysing the influence of the cross-sectional shape of the main load-bearing structures on the mechanical functionality. Three phase field simulations were carried out showing good agreement with the cross-sections found in selected plants: (i) U-shaped cross-sections comparable with those of Musa sp. petioles, (ii) star-shaped cross-sections with deep grooves as can be found in the lianoid wood of Condylocarpon guianense stems, and (iii) flat elliptic cross-sections with one deep groove comparable with the cross-sections of the climbing ribbon-shaped stems of Bauhinia guianensis.

Mechanics of a plant in fluid flow

Plants live in constantly moving fluid, whether air or water. In response to the loads associated with fluid motion, plants bend and twist, often with great amplitude. These large deformations are not found in traditional engineering application and thus necessitate new specialized scientific developments. Studying fluid-structure interaction (FSI) in botany, forestry, and agricultural science is crucial to the optimization of biomass production for food, energy, and construction materials. FSIs are also central in the study of the ecological adaptation of plants to their environment. This review paper surveys the mechanics of FSI on individual plants. I present a short refresher on fluid mechanics then dive into the statics and dynamics of plant-fluid interactions. For every phenomenon considered, I examine the appropriate dimensionless numbers to characterize the problem, discuss the implications of these phenomena on biological processes, and propose future research avenues. I cover the concept of reconfiguration while considering poroelasticity, torsion, chirality, buoyancy, and skin friction. I also assess the dynamical phenomena of wave action, flutter, and vortex-induced vibrations.

Biomimetic 3D printed lightweight constructions: a comparison of profiles with various geometries for efficient material usage inspired by square-shaped plant stems

Within a bottom-up approach in biomimetics, this study was inspired by the plant motherwort (Leonurus cardiaca), which has stems divided longitudinally into hollow internodes and solid nodes, a lightweight concept well-known in biology. The square cross-sections show a specific geometric arrangement of various tissues, each with different mechanical properties. We have used CAD software and selective laser sintering technology to produce (1) extruded hollow profiles with various cross-sections analogous to internodes, and (2) integrated additional elements mimicking nodes. The design of the individual profiles with their different geometries is based on an increasing degree of abstraction, starting with profile A, which comes closest to the plant model, through profile B in the form of a greatly simplified material distribution, to profile C with the simplest geometry of a square hollow profile. In the context of resource-saving constructions, we have determined the flexural and torsional stiffness, the twist to bend ratio, and the lightweight efficiency of each individual profile. In general, profiles A, B, and C and all profiles from the A- and C-family show higher torsional stiffness than flexural stiffness. However, the profiles of the B-family exhibit no such uniform mechanical behavior. Interestingly, profile A has a higher lightweight efficiency than profile B but a lower efficiency than the most abstracted profile C. This indicates that a simple blueprint of nature is not useful, because, in plants, the geometric designs of various tissues and of globally and locally adaptable material properties are coupled to optimize performance based on multifunctionality. In contrast, 3D laser sintered profiles consist of a single isotropic and homogeneous material with defined material properties and therefore show different flexural and torsional efficiency because of their diverse geometries alone. These results reveal the influences of the geometric arrangement on the bending and torsional stiffness of the plant without interference from variations in material properties (reverse biomimetics).

Adaptive spatiotemporal changes in morphology, anatomy, and mechanics during the ontogeny of subshrubs with square-shaped stems

PREMISE OF THE STUDY

Plant stems can be regarded as fiber-reinforced structures characterized by anatomical heterogeneity, mechanical anisotropy, and adaptability to changing internal and external constraints. Our study focused on adaptive spatiotemporal changes in morphology, anatomy, and mechanical properties during the ontogeny of Leonurus cardiaca L. (Lamiaceae) internodes, proving considerable functional adaptability.

METHODS

Four-point bending tests and torsional tests were carried out on the same internodes to measure flexural and torsional stiffness. Axial and polar second moments of area for entire cross sections and for individual tissues were determined from transverse stem sections immediately after testing. Based on these data, additional relevant mechanical parameters such as bending elastic modulus, torsional modulus and twist to bend ratio were calculated.

KEY RESULTS

Leonurus cardiaca is characterized by a square-shaped hollow stem in transverse section with an outer frame of various strengthening tissues and an inner ring of parenchyma. Statistical analyses of axial and polar second moment of area, flexural stiffness, torsional stiffness, bending elastic modulus, and torsional modulus revealed significant differences for all comparisons with respect to spatial resolution (two adjacent internodes) and temporal resolution (in June before flowering and in September after fruit formation). The twist to bend ratios of the internodes, however, always remain in the same range.

CONCLUSIONS

With respect to spatiotemporal development, stems of the subshrub L. cardiaca show a marked increase in flexural and torsional stiffness during ontogeny. Strikingly, changes in stem mechanics are more influenced by variations in mechanical tissue properties than by changes in relative proportion of different tissue types.

Chirality-dependent flutter of Typha blades in wind

Cattail or Typha, an emergent aquatic macrophyte widely distributed in lakes and other shallow water areas, has slender blades with a chiral morphology. The wind-resilient Typha blades can produce distinct hydraulic resistance for ecosystem functions. However, their stem may rupture and dislodge in excessive wind drag. In this paper, we combine fluid dynamics simulations and experimental measurements to investigate the aeroelastic behavior of Typha blades in wind. It is found that the chirality-dependent flutter, including wind-induced rotation and torsion, is a crucial strategy for Typha blades to accommodate wind forces. Flow visualization demonstrates that the twisting morphology of blades provides advantages over the flat one in the context of two integrated functions: improving wind resistance and mitigating vortex-induced vibration. The unusual dynamic responses and superior mechanical properties of Typha blades are closely related to their biological/ecosystem functions and macro/micro structures. This work decodes the physical mechanisms of chirality-dependent flutter in Typha blades and holds potential applications in vortex-induced vibration suppression and the design of, e.g., bioinspired flight vehicles.

Biomechanics and functional morphology of a climbing monocot

Plants with a climbing growth habit possess unique biomechanical properties arising from adaptations to changing loading conditions connected with close attachment to mechanical supports. In monocot climbers, mechanical adaptation is restricted by the absence of a bifacial vascular cambium. Flagellaria indica was used to investigate the mechanical properties and adaptations of a monocot climber that, uniquely, attaches to the surrounding vegetation via leaf tendrils. Biomechanical methods such as three-point bending and torsion tests were used together with anatomical studies on tissue development, modification and distribution. In general, the torsional modulus was lower than the bending modulus; hence, torsional stiffness was less than flexural stiffness. Basal parts of mature stems showed the greatest stiffness while that of more apical stem segments levelled off. Mechanical properties were modulated via tissue maturation processes mainly affecting the peripheral region of the stem. Peripheral vascular bundles showed a reduction in the amount of conducting tissue while the proportion and density of the bundle sheath increased. Furthermore, adjacent bundle sheaths merged resulting in a dense ring of fibrous tissue. Although F. indica lacks secondary cambial growth, the climbing habit is facilitated by a complex interaction of tissue maturation and attachment.

Biomechanical tactics of chiral growth in emergent aquatic macrophytes

Through natural selection, many plant organs have evolved optimal morphologies at different length scales. However, the biomechanical strategies for different plant species to optimize their organ structures remain unclear. Here, we investigate several species of aquatic macrophytes living in the same natural environment but adopting distinctly different twisting chiral morphologies. To reveal the principle of chiral growth in these plants, we performed systematic observations and measurements of morphologies, multiscale structures, and mechanical properties of their slender emergent stalks or leaves. Theoretical modeling of pre-twisted beams in bending and buckling indicates that the different growth tactics of the plants can be strongly correlated with their biomechanical functions. It is shown that the twisting chirality of aquatic macrophytes can significantly improve their survivability against failure under both internal and external loads. The theoretical predictions for different chiral configurations are in excellent agreement with experimental measurements.