Morphometric and mechanical characteristics of Equisetum hyemale stem enhance its vibration
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The order of the internodes, and their geometry and mechanical characteristics influence the capability of the Equisetum stem to vibrate, potentially stimulating spore liberation at the optimum stress setting along the stem.
Equisetum hyemale L. plants represent a special example of cellular solid construction with mechanical stability achieved by a high second moment of area and relatively high resistance against local buckling. We proposed the hypothesis that the order of E. hyemale L. stem internodes, their geometry and mechanical characteristics influence the capability of the stem to vibrate, stimulating spore liberation at the minimum stress setting value along the stem. An analysis of apex vibration was done based on videos presenting the behavior of an Equisetum clump filmed in a wind tunnel and also as a result of excitation by bending the stem by 20°. We compared these data with the vibrations of stems of the same size but deprived of the three topmost internodes. Also, we created a finite element model (FEM), upon which we have based the ‘natural’ stem vibration as a copy of the real object, ‘random’ with reshuffled internodes and ‘uniform’, created as one tube with the characters averaged from all internodes. The natural internode arrangement influences the frequency and amplitude of the apex vibration, maintaining an equal stress distribution in the stem, which may influence the capability for efficient spore spreading.
KeywordsMechanical properties Plant biomechanics Segmented structure Stem vibration Stress distribution Wind
The present-day representatives of the genus Equisetum are considered to be survivors from the Devonian period (Husby 2013). Therefore, it comes as no surprise that the 15 species which have survived to this day are subjects of interest for botanists of almost all specializations (Tschudy 1939; Spatz et al. 1998; Marais et al. 2003; Gierlinger et al. 2007; Channing et al. 2011). Equisetum hyemale L. is a unique representative of this plant group, primarily due to its special biomechanical features achieved with a simple body structure. It consists of an unbranched column—a stem which attains the height of 150 cm, while the base has a width of 4–6 mm and consists of characteristic ridged internodes and nodes (Niklas 1989b; Speck et al. 1998). The light construction is achieved by the large, central pith cavity and the vallecular and carinal canals. The strengthening tissue consisting of a thick-walled sclerenchyma occurs in the outer part of the perimeter. The central canal qualifies horsetails as a hollow tube structure, in which its geometry and the wall Young’s modulus will significantly determine the mechanical properties of the entire plant, and the transverse nodal septa will increase the resistance of the stem to bending and twisting (Niklas 1997a). Additionally, compared to other hollow tube plants (e.g., grasses) E. hyemale is characterized by special anatomical and physiological features that enable it to resist their mechanical collapse because of changes to its cellular water status during the vegetation period. One of these features is the ability to resist sub-freezing temperatures, due extracellular freezing phenomena (Niklas 1989a), whereby shoots remain erect after thawing processes. The second feature is almost independent of hydration state and relates to the ability of shoots to resist local buckling, because of the presence of a double layer of endodermis (Spatz et al. 1998; Speck et al. 1998).
The length of horsetail stem internodes, as well as their width and strength properties varies from the base to the plant apex. Internodes at the base and in the apical region of the stem are shorter and thinner than those in the middle part of the plant. It has also been demonstrated that E. hyemale L. internodal diameter and its wall thickness with respect to the internode radius increase from the base of shoots. The high second moment of inertia, from which the stem flexural rigidity results, is the outcome of the distribution of the area and strengthening of tissues in the outer perimeter, and the maximum of this value is located at 1/3 of the plant’s height (Niklas 1989b). Therefore, anatomy, as well as morphometric stem structure parameters, which are variable depending on the stem height, tends toward an increase in mechanical stability. The horsetail is a primitive plant and is not capable of dynamic mechanical adaptation via the formation of specialized tissues typical of specific biomechanical conditions (e.g., reaction wood), of which the higher plants are capable, but its reaction consists in an allometric change of the proportions of individual body parts (Niklas 1989c).
However, morphometric and biomechanical studies on E. hyemale do not provide an answer to the question of whether variable internode length and their strength parameters increase stem vibration capability, which, for a plant bearing the strobilus on its apex, may be important for the spore spreading process. Wind, and the oscillation motions induced by it, constitute an integral element creating thigmomorphogenesis in plants (Jaffe 1973; Doaré et al. 2004). This also applies to the type of anatomical adaptation influencing its mechanical characteristics (Niklas and Speck 2001; Gardiner et al. 2016). Vibration of anatomical elements enhancing pollen and spore dispersal with the wind is often a sign of plant adaptation for the optimal use of the physical characteristics of wind for the most efficient dispersal. As an example, it has been demonstrated that inflorescence vibration in Plantago is an indispensable factor for pollen release (Urzay et al. 2009; Timerman et al. 2014), similar to sporophyte vibration in certain bryophytes, which aids more efficient spore release (Lee 2010; Johansson et al. 2014); however, there are no studies on the vibration of entire stems in such phylogenetically important plants as Equisetum. In such small particles as spores, movement is mostly dependent on drag based on Stokes’s law of resistance (Stevens et al. 1975; Dickinson and Preece 1976; Murray 1986; de Langre 2008). Thus, the dominant factor influencing spore dispersal is wind.
In the horsetail, in which the strobilus is located at the top of the plant, the ejection force resulting from stem flexibility might not be the significant element in spore liberation, but rather that could be the duration vibration of the top portion of the stem, i.e., prolonged exposure to the effect of wind. Our study aimed to demonstrate that the distribution of internode sizes along the stem, as well as their strength properties increase the capability and duration of vibration, thus prolonging the duration of spore dispersal. To test this hypothesis, we performed experiments with static stem excitations of intact stems and stems where the uppermost internodes had been cut. We analyzed their stem vibrations in a wind tunnel, as well spore liberation efficiency. Based on mechanical tests we were able to obtain the stem physical characteristics needed for setting an finite element model (FEM) model that could be transformed to test the morphometric stem parameters essential for the most efficient vibration with the lowest possible mechanical stress.
Materials and methods
One-year-old E. hyemale L. plants with developed strobili were collected from a forest within the Białobrzegi Forest District in central Poland 51°39′46.8″N 20°40′06.3″E at two dates: 26 Sept. 2015 and 5 Oct. 2016. Plants were dug out with rhizomes and placed in pots. The plants were automatically hydrated and stored outdoors at the University Garden, at average daily temperatures of +15 °C (in 2015) and 13 °C (in 2016) according to the weather station at the Warsaw University of Life Sciences.
Vibration caused by static excitation
Twenty horsetail stems were deflected from the vertical by approximately 20°, which was chosen as the maximum deflection with no risk of shoot failure. Stem bases were mounted onto a laboratory stand with softly isolated clamp. The second uppermost node was bent to 20° (the angle was drawn on the background wall) and held by a grip mounted to the stand. After releasing the grip, the behavior of the stems was recorded using a Sony RX 100 IV camera (Sony, Tokyo, Japan) at 500 frames per second, adjusting the lens axis perpendicularly to the plant axis. Tests were conducted on intact stems and stems from which the last three internodes had been removed. Stems selected for cutting were initially longer than intact stems, by the length of approximately three internodes. The vibration frequency of the apex and amplitude changes during the vibration absorption were analyzed in Tracker software (https://www.cabrillo.edu/dbrown/tracker/) based on the Open Source Physics (OSP) Java framework. Data were analyzed with a periodogram which shows the main frequencies and their power expressed in units proportional to the square of the amplitudes of sinusoids present in the data (Press et al. 1992). For this analysis, PastProject (Hammer et al. 2001) software was used.
Spore liberation during stem static excitation
In this experiment the efficiency of spore release was compared between intact stems and stems with the three uppermost internodes cut. For the variant with a cut apical part longer stems were chosen, so that after cutting they had the same height as the intact stem. Due to the high variability of strobilus dimensions and the number of spores inside them, we used small plastic cups with a volume of 20 mm3 attached to the apical part of the stem with non-toxic soft modeling clay (Amos, Seoul, Korea) and we filled them full of spores that had been collected (22 Sept.–5 Oct. 2016) from plants growing at the test area (Bialobrzegi Forest Distinct). The cups with spores weighed 1.5 g which was similar to an average strobilus from the tested plants. This was applied for both types of examined stems. The intact stem (with all internodes) had to have its strobilus removed. Horsetail stems were mounted in laboratory stands with elastic and softly isolated clamps and then placed precisely 1 m under a green laser pointer (Wicked Lasers, Kowloon, Hong Kong) positioned parallel to the plant axis in a dark room. The laser pointer generated a multi pattern of light beams (100 mW power output and 532 nm wavelength) that created a pattern of 2 × 2 cm grid points on the base where stems had been placed. Stems were deflected from vertical orientation by 20°, in the same manner as in studies on stem vibration caused by static excitation (mentioned above). After releasing the grip, the behavior of the stems was photographed with 10-s exposure time using a Canon 5d mark IV camera (Canon, Tokyo, Japan) mounted on a tripod.
Three minutes after stem excitation the area about the stem was illuminated by a Blak-Ray B-100Y UV lamp (UVP, LLC, Upland, CA, USA), and the area where spores were visible was measured. The experiment was repeated 11 times for every variant.
Horsetail clumps in pots were subjected to a controlled wind speed effect in a wind tunnel at the Faculty of Power and Aeronautical Engineering at the Technical University in Warsaw. The open-circuit wind tunnel possesses a 1 × 1 m test chamber and generates a maximum flow velocity of 25 m/s. Our study was conducted in a wind velocity range from 0–6 m/s that is similar to the environmental conditions in which the studied horsetail lives (under the canopy in wet deciduous forests) (Geiger et al. 2003). Behavior in the wind was recorded using a Photron Fastcam SA-Z (Photron Limited, Tokyo, Japan) camera at a frame rate of 5000 frames per second. The camera was operated by PFV software (Photron Fastcam Viewer, bundled by Photron together with the camera).
maximum deviation in the X axis (perpendicular to main stem axis horizontal line in the image) of the five subsequent stem nodes from the original vertical stem axis (Y axis), depending on the wind velocity. Based on video observations, the first 5 uppermost internodes were chosen for analyses given that this zone was dynamically engaged in apex (strobilus) vibration. The analysis was performed on an image of a feature of interest (the boundary between the dark whorl of leaves and the light stem), then by searching each frame for a match to that template. The results are a set of X, Y coordinates that could be transformed into positional changes along X or Y axes. Ten stems were analyzed.
vibration frequency and amplitude of intact stems in comparison to three cut apex internodes (average 8 cm) with equal height. The amplitude was measured based on the maximum deflection of the measured apex along the X axis. During a manual mark, the apex point on the video timeline frame, when the vibration period had ended, and then based on the frame rate per second of the video, the real frequency of the apex was recalculated. Ten intact stems and ten cut stems were analyzed (Supplementary material Video S1). The results were analyzed using a periodogram showing main frequencies.
Finite element modeling (FEM)
- one stem called the ‘natural’ stem composed of ten segments (internodes) with connectors (nodes), with the same dimensions and properties (Young modulus and bending stiffness) as presented in Table 1,Table 1
Results of measurements of ten internodes and nine nodes of a stem which was the basis for the FEM model in the ‘natural’ version
D max (mm)
D wall (mm)
two virtual stems—the ‘random’ stem composed of the same segments as for ‘natural’ stems; however, the order of the segments was arbitrarily changed using a random numbers generation procedure. The following order of the internodes was generated: 1, 2, 7, 6, 5, 4, 8, 10, 9 (the internode No. 1 was located at the top and internode No. 9 at base of the virtual stem). The ‘uniform’ stem composed of one segment with the length of a ‘natural’ stem and other dimensions and material properties calculated as an average of ‘natural’ segments.
For the above three kinds of stem structure numerical simulations were performed where Hooke’s law had been assumed. The damping effect was included in the model that is viscoelastic material properties were assumed. The effect of damping was analyzed for two other kinds of stem that were investigated experimentally, as was described in the previous section: the ‘cut’ stem and the ‘intact’ stem. For the latter, the same geometry was assumed as for the ‘natural’ stem.
In viscoelastic materials, the constitutive relations involve quite general, the stress rates and the strain rates. The “VISCOELASTIC, TIME = PRONY” option in Abaqus specifies dissipative behavior for use with elasticity and describes isotropic rate-dependent material behavior for materials in which dissipative losses are primarily caused by “viscous” (internal damping) effects. Abaqus assumes that the viscoelastic material is defined by a Prony series expansion of the dimensionless relaxation.
A numerical simulation was applied to evaluate local stress distribution along the stems and to find potential differences in vibration between these stems, following the hypothesis that frequency is important for spore dispersal, because it influences the velocity and acceleration of stem motion.
Finite elements mesh
Volumetric meshing uses the natural discretization of an 8-node cubic element. The element type is C3D8 from the Abaqus commercial code (Simulia 2013). This is an eight-node brick element with linear interpolation. Such first-order elements capture stress concentrations and are effective in bending-dominated problems. Each stem is composed of ten cylindrical segments and nine cylindrical connectors. Each segment has a different diameter, length and thickness. The connectors have the form of short (2 mm) cylinders with variable thickness, and the thickness at each end of a connector equals the thickness of the adjacent segment. The stiffness of connectors corresponds to the stiffness of nodes specified in the experiment. At the interface between internode and node, the “tie” option included in the Abaqus software was used. This makes the translational and rotational motion as well as all other active degrees of freedom equal for a pair of surfaces in contact. In numerical simulations, the total number of elements and the total number of nodes in the ‘natural’ stem model is 4224 and 9313, in random stem models 3776 and 8768 and in uniform stem models 3200 and 6464, respectively.
Boundary and initial conditions
In all considered models the bottom end of the sample is clamped, and the upper end is constrained so that only a motion in one (X–Y) plane is possible. We assumed that this simplification of the actual motion of stems observed in the wind tunnel still enabled a fair comparison of the motion of different stems and did not influence the conclusions presented in the paper. We also assumed free lateral faces. The initial angular velocity on the upper five segments was 0.34 rad/s, the assumed velocity enabled in the first half-cycle, at the beginning of the process, the same displacement amplitude at the top of the stem as in the mechanical test (deflection from the vertical by approximately 20°).
The surfaces of the horsetail stems, as well as their cross sections were visualized using scanning electron microscopy (Fei Quanta 200; Thermo Fisher Scientific, Waltham, MA, USA) at 25 kV. Microscopic observations of stem anatomy were performed on transverse and tangential sectional samples of stems in an Olympus BX-61 (Tokyo, Japan) optical microscope using UV light. Specimens were fixed in formaldehyde:acetic acid:50% ethanol (FAA; 1:1:18, by vol.), dehydrated with ethanol, embedded in epoxy resin (Epon 812, Serva, Sigma-Aldrich, St. Louis, MO, USA), then cut in sections of 5–8 μm using a Leica UC7 ultramicrotome (Leica Microsystems, Wetzlar, Germany) and stained with an aqueous 1% safranin solution.
Vibration caused by static excitation
Spore liberation during stem static excitation
The measured area of spores distributed around the stem showed that an intact stem released spores over an area two times greater (mean 1179 cm2, standard deviation 316 cm2) than the cut stems (mean 507 cm2, standard deviation 260 cm2).
Simulation of local equivalent stress maxima during stem deflection
Simulation of stem vibration
The absorption phenomenon was considered in the numerical analysis of the ‘intact’ and ‘cut’ stems (Fig. 11b, c). In both cases, the same viscoelastic material parameters were assumed. For ‘intact’ stems, a close agreement between theoretical and experimental results can be observed (Fig. 11b).
Equisetum hyemale lives in clumps, which causes the stems to hit against each other in the wind, which we have observed in wind tunnel experiments which among others showed that during wind higher deflection is noted in the uppermost internodes. Such a phenomenon may be significant for spore spread, because of changes in the kinetic energy of stems during their collision. Hypothetically, it can be assumed that spores in the vibrating strobilus have the same speed as the strobilus; at the moment of collision with the second stem, the spores are liberated from the strobilus and they may have a slightly lower speed than the speed just before collision.
In structures with hollow tube stems, their nodes operate as elastic springs, which store elastic energy during stem vibration and internode deformation (Niklas 1997b). This energy is immediately released, directly influencing the vibration frequency. This may be an important feature for the horsetail, where there is a clear node stiffness gradient. Indeed, stored elastic energy increases with increasing stiffness; it is higher in the nodes of the middle and lower parts of the stem, and it is released and capable of increasing vibration of the top, light and flexible portions.
The results of our experiments demonstrate that if the stem is deprived of the three top nodes, thus exposing to deflection only internodes connected with nodes of high stiffness, the stem exhibits considerably lower vibration, which is rapidly absorbed (Figs. 4, 7).
Moreover, it appears that a threshold force exists, which is necessary for initiation of the elastic spring phenomenon in nodes. This phenomenon assumes that nodes are able to store and release energy on the stem. The study in the wind tunnel demonstrated that clear deviations of the stem apex during vibrations appear at the velocity of 4 m/s (Fig. 6).
In our study, we created an finite element model (FEM) simulation for the behavior of a ‘natural’ stem, reflecting both morphometric as well as mechanical properties of the natural plant, and a stem with randomly shuffled internodes, as well as a uniform model, i.e., a straight hollow tube without nodes, consisting of one segment being the mean of the dimensions from all internodes. The vibration frequencies were highest in the ‘natural’ stem and, simultaneously, local stresses in the entire stem were the lowest. It should be noted that the lowest value for the local stress maximum is observed for a ‘natural’ stem. Thus, the ‘natural’ stem exhibits an optimal configuration from the point of view of reduction of stress levels after dynamic excitation. This may be important for enhancement of the fatigue life of stem structures when a high cycle loading in vibration process is considered. It should be noted that in the ‘uniform’ stem, with dimensions calculated as a mean value of all ‘natural’ internodes, twofold greater stress values are observed than in the ‘natural’ stem after the same excitation.
In the FEM simulation, the propagation of local stress during vibration can be observed. For the ‘random’ model, accumulation of stress was noted in the inner, thin internode (Fig. 9). Such a strong accumulation of local stress remaining in one zone of the stem is not observed in ‘natural’ stems. These are moved with the structures to the upper parts, and become dispersed to the level of the fourth internode, i.e., where the internode bending stiffness rapidly decreases (Fig. 3). During the entire vibration period, the three uppermost internodes of ‘natural’ stems are vibrating not only with the highest amplitudes, but also entirely free of clear stress gradients. Even though the FEM model was consistent with the general rule that an intact stem vibrates with higher frequency and its damping process occurs for a longer period, we could not obtain results that reflect experimental data (frequency rate and damping period). This is another proof that FEM models should be based on structural complexity and on precise anatomical and micro-biomechanical records.
Author contribution statement
UZ conceived and designed the research, conducted microscopic observations, designed and performed spore liberation experiments, analyzed all data, and wrote the manuscript. SK and UZ conducted measurements and analyzed indentation test data. NZ prepared numerical simulations. DG and UZ performed wind tunnel experiments and analyzed stem vibration data. All authors read and approved the manuscript.
We are grateful to Mr. K. Gumowski (Institute of Aeronautics and Applied Mechanics, Warsaw University of Technology) for sharing his laboratory. My special thanks to Professor Karl J. Niklas (Cornell University) for helpful comments and suggestions (UZ). Many thanks are due to unknown reviewers for substantial improvements of the manuscript.
Video S3 The dynamics of reduced stress (Mises) during oscillations of the modeled ‘natural’ stems (MOV 2002 kb)
Video S4 The dynamics of reduced stress (Mises) during oscillations of the modeled ‘random’ stems (MOV 1837 kb)
Video S5 The dynamics of reduced stress (Mises) during oscillations of the modeled ‘uniform’ stems (MOV 1864 kb)
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