Influence of structural reinforcements on the twist-to-bend ratio of plant axes: a case study on Carex pendula

Abstract

During biological evolution, plants have developed a wide variety of body plans and concepts that enable them to adapt to changing environmental conditions. The trade-off between flexural and torsional rigidity is an important example of sometimes conflicting mechanical requirements, the adaptation to which can be quantified by the dimensionless twist-to-bend ratio. Our study considers the triangular flower stalk of Carex pendula, which shows the highest twist-to-bend ratios ever measured for herbaceous plant axes. For an in-depth understanding of this peak value, we have developed geometric models reflecting the 2D setting of triangular cross-sections comprised of a parenchymatous matrix with vascular bundles surrounded by an epidermis. We analysed the mathematical models (using finite elements) to measure the effect of either reinforcements of the epidermal tissue or fibre reinforcements such as collenchyma and sclerenchyma on the twist-to-bend ratio. The change from an epidermis to a covering tissue of corky periderm increases both the flexural and the torsional rigidity and decreases the twist-to-bend ratio. Furthermore, additional individual fibre reinforcement strands located in the periphery of the cross-section and embedded in a parenchymatous ground tissue lead to a strong increase of the flexural and a weaker increase of the torsional rigidity and thus resulted in a marked increase of the twist-to-bend ratio. Within the developed model, a reinforcement by 49 sclerenchyma fibre strands or 24 collenchyma fibre strands is optimal in order to achieve high twist-to-bend ratios. Dependent on the mechanical quality of the fibres, the twist-to-bend ratio of collenchyma-reinforced axes is noticeably smaller, with collenchyma having an elastic modulus that is approximately 20 times smaller than that of sclerenchyma. Based on our mathematical models, we can thus draw conclusions regarding the influence of mechanical requirements on the development of plant axis geometry, in particular the placement of reinforcements.

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

“…The low torsional rigidity can be attributed to the arrangement of the strengthening tissues (vascular and collenchyma bundles) in separate individual not connected strands. This tissue arrangement is quite flexible in torsion, as mentioned earlier, compared to closed rings of strengthening tissues as found in H. alternata ( Niklas, 1999 ; Ennos et al, 2000 ; Wolff-Vorbeck et al, 2021 ).…”

“…This holds true especially for strengthening tissues like vascular bundles, collenchyma, and sclerenchyma. In the case of the C. pendula flower stalks the peripheral arrangement of the individual sclerenchyma strands, in combination with the high elastic modulus of sclerenchyma, is predominantly responsible for the high flexural rigidity of the flower stalks ( Niklas, 1999 ; Wolff-Vorbeck et al, 2021 ). On the other hand, according to Wolff-Vorbeck et al (2021) , the reinforcement by not connected individual sclerenchyma strands only moderately increases the torsional rigidity of the entire stalk.…”

“…In the case of the C. pendula flower stalks the peripheral arrangement of the individual sclerenchyma strands, in combination with the high elastic modulus of sclerenchyma, is predominantly responsible for the high flexural rigidity of the flower stalks ( Niklas, 1999 ; Wolff-Vorbeck et al, 2021 ). On the other hand, according to Wolff-Vorbeck et al (2021) , the reinforcement by not connected individual sclerenchyma strands only moderately increases the torsional rigidity of the entire stalk. While the bending stiffness remains almost constant with an increasing number of peripheral sclerenchyma strands, the torsional rigidity shows a minimum and the twist-to-bend ratio a maximum at 49 sclerenchyma strands.…”

Acclimation to wind loads and/or contact stimuli? A biomechanical study of peltate leaves of Pilea peperomioides

“…The difference between the tensile elastic moduli and the bending elastic moduli is therefore probably attributable to the anatomical inhomogeneity (constructed of different materials) and the mechanical anisotropy (different mechanical properties in the main directions) of the biological samples. While in tension it is mainly important which and how much of the respective materials are present in the cross-section of the structure (and their arrangement is of minor importance), in bending it is also crucial where and how these materials are arranged in the cross-section ( Wolff-Vorbeck et al , 2021 ). Under tensile loading, only tensile stresses act rather uniformly across the cross-section.…”

Charting the twist-to-bend ratio of plant axes

“…As in previous work [ 8 , 9 ], we use methods from linearized elasticity and Saint-Venant's theory of pure torsion of non-homogeneous elastic beams. We therefore describe a plant stem as a long thin elastic rod with domain of length L and simply connected cross-section remaining constant along the longitudinal axis.…”