Twist-to-Bend Ratios and Cross-Sectional Shapes of Petioles and Stems

Two structural properties, resistance to twisting (torsionalrigidity or torsional stiffness, GJ) and resistance to bending(flexural rigidity or flexural stiffness, El), were measuredon a variety of herbaceous stems and petioles. Specimens withnon-circular cross-sections had higher values of the ratio ofEl to GJ that is, such specimens were relatively more flexiblein twisting than in bending. But both kinds had higher ratiosthan those that characterize simple, isotropic materials, andthus both structural and material factors contribute to achievinga high twist-to-bend ratio. The composite property expressedas the dimensionless variable EI/GJ appears to be a functionallyrelevant parameter in many biological situations. Key words: Stems, petioles, flexural rigidity, torsional rigidity, biomechanics     

Flexural and torsional stiffness in multi-jointed biological beams

Flexibility, the ability to deform in response to loads, is a common property of biological beams. This paper investigates the mechanical behavior of multi-jointed beams, which are characterized by a linear series of morphologically similar joints. Flexural stiffness and torsional stiffness were measured in two structurally distinct beams, crinoid arms (Echinodermata, Comatulida) and crustacean antennae (Arthropoda, Decapoda). Morphological data from these beams were used to determine the relative contributions of beam diameter and joint density (number of joints per millimeter of beam length) to the flexural and torsional stiffness of these two structures. As predicted by beam theory, beam diameter influenced stiffness in both crinoid arms and crustacean antennae. In crinoid arms, increases in joint density were associated with decreases in stiffness, but joint density had no significant influence on stiffness in crustacean antennae. In both crinoid arms and crustacean antennae, the magnitudes of flexural and torsional stiffness, as well as the ratio of these two variables, were similar to previously reported values for non-jointed biological beams. These results suggest that the structural design of a biological beam is not a limiting factor determining its mechanical properties.     

Development and Growth Form of the Neotropical Liana <Emphasis Type="Italic">Croton nuntians</Emphasis>: The Effect of Light and Mode of Attachment on the Biomechanics of the Stem

The neotropical liana Croton nuntians (Euphorbiaceae) can occur in a variety of different growth habits. Juvenile freestanding plants are mechanically stable without support and resemble morphologically young trees or shrubs, whereas adult plants are climbers. Ontogenetic variation of bending and torsion properties of different growth phases are analyzed by measurements of flexural stiffness, structural bending modulus, torsional stiffness and structural torsional modulus. Mechanical and anatomical data show two fundamentally different patterns for juvenile freestanding and adult climbing plants. In freestanding plants, mechanical properties and the contribution of cortex, wood, and pith to the stem cross-section vary only little during ontogeny as is typical for semi-self-supporting plants. In contrast, climbing plants become significantly more flexible during ontogeny, as is characteristic for lianas. This is accompanied by a transition to the formation of a less dense wood type with large diameter vessels and an increasing contribution of flexible tissues (less dense wood and cortex) to the cross-sectional area and the axial second moment of area of the stems. Depending on the environmental conditions, freestanding plants can differ considerably in their appearance due to differences in branching system or stem taper. Therefore the influence of light quantity, measured as percentage of canopy opening, on the mechanical properties and the stem anatomy was tested. Freestanding plants grown with strong shade are significantly more stiff in bending compared with plants grown with a moderate light environment.     

Mandibular stiffness in humans: numerical predictions

The chin is a feature unique to humans. This study evaluates the effect of mandibular symphyseal design on biomechanical masticatory effectiveness as determined by structural stiffness and stress developed under flexural and torsional loading. A simple model of three symphyseal shapes (chin, flat symphysis and lingual buttress), was built to represent human, Neanderthal and higher primate symphyses and these were subjected to wishboning and torsional forces. Additionally, an anatomically detailed reconstruction was made of the CT scan of an actual human mandible, which was then also morphed into a chinless model. The results of a 3-D finite element analysis show firstly, that none of the three different symphyseal shapes is biomechanically more advantageous than the others for the given loading condition. Secondly, we show in a CT-derived model, that the presence of a chin does not confer significantly improved stiffness to torsional or flexural loading. These results indicate that the acquisition of a chin in modern humans is not related to the functional demands placed upon the mandible during mastication, but suggest that it may have developed in response to other biomechanical demands.     

The Growth Form of Croton pullei (Euphorbiaceae) - Functional Morphology and Biomechanics of a Neotropical Liana

Abstract: Croton pullei (Euphorbiaceae) is a woody climber of the lowland rainforest in French Guyana and Surinam. During ontogeny, a shift from a juvenile free-standing growth phase to an older supported growth phase is observed. The following biomechanical parameters were studied: structural Young's modulus, structural torsional modulus, flexural stiffness and bend to twist ratios. Changes in anatomical development were also analysed for different stages of development of C. pullei which differ significantly in their mechanical properties. Free-standing plants show a nearly constant structural Young's modulus and structural torsional modulus during ontogeny, with flexural stiffness increasing proportionally with the axial second moment of area. These patterns are typical for “semi-self-supporting plants". In contrast, supported plants show a significant decrease in structural Young's modulus in older stem parts, as well as a decrease in structural torsional modulus. Due to the decrease in structural Young's modulus, flexural stiffness does not increase proportionally with the axial second moment of area. These patterns are typical for non-self-supporting lianas. In all supported plants, a sudden transition occurs from early dense wood to a wood type with a much higher proportion of large diameter vessels. In contrast, only the dense wood type is present in all tested free-standing plants. The data are compared with results from other climbing species of the same study area and discussed with reference to observed features characterizing the growth form and life history of C. pullei.     

Estimating the flexural rigidity of Arabidopsis inflorescence stems: Free-vibration test vs. three-point bending test

The mechanical strength of a plant stem (a load-bearing organ) helps the plant resist drooping, buckling and fracturing. We previously proposed a method for quickly evaluating the stiffness of an inflorescence stem in the model plant Arabidopsis thaliana based on measuring its natural frequency in a free-vibration test. However, the relationship between the stiffness and flexural rigidity of inflorescence stems was unclear. Here, we compared our previously described free-vibration test with the three-point bending test, the most popular method for calculating the flexural rigidity of A. thaliana stems, and examined the extent to which the results were correlated. Finally, to expand the application range, we present an example of a modified free-vibration test. Our results provide a reference for improving estimates of the flexural rigidity of A. thaliana inflorescence stems.     

Twisting and bending of biological beams: distribution of biological beams in a stiffness mechanospace

Most biological beams bend and twist relatively easily compared to human-made structures. This paper investigates flexibility in 57 diverse biological beams in an effort to identify common patterns in the relationship between flexural stiffness and torsional stiffness. The patterns are investigated by mapping both ideal and biological beams into a mechanospace defined by flexural and torsional stiffness. The distribution of biological beams is not random, but is generally limited to particular regions of the mechanospace. Biological beams that are stiff in bending are stiff in torsion, while those that bend easily also twist easily. Unoccupied regions of the mechanospace represent rare combinations of mechanical properties, without proving that they are impossible. The mechanical properties of biological beams closely resemble theoretical expectations for ideal beams. Both distributions are potentially being driven by the interdependence of the material and structural properties determining stiffness. The mechanospace can be used as a broadly comparative tool to highlight systems that fall outside the general pattern observed in this study. These outlying beams may be of particular interest to both biologists and engineers due to either material or structural innovations.     

The twist-to-bend compliance of the Rheum rhabarbarum petiole: integrated computations and experiments

Plant petioles can be considered as hierarchical cellular structures, displaying geometric features defined at multiple length scales. Their macroscopic mechanical properties are the cumulative outcome of structural properties attained at each level of the structural hierarchy. This work appraises the compliance of a rhubarb stalk by determining the stalk’s bending and torsional stiffness both computationally and experimentally. In our model, the irregular cross-sectional shape of the petiole and the layers of the constituent tissues are considered to evaluate the stiffness properties at the structural level. The arbitrary shape contour of the petiole is generated with reasonable accuracy by the Gielis superformula. The stiffness and architecture of the constituent layered tissues are modeled by using the concept of shape transformers so as to obtain the computational twist-to-bend ratio for the petiole. The rhubarb stalk exhibits a ratio of flexural to torsional stiffness 4.04 (computational) and 3.83 (experimental) in comparison with 1.5 for isotropic, incompressible, circular cylinders, values that demonstrate the relative structural compliance to flexure and torsion.     

The role of the epidermis as a stiffening agent in Tulipa (Liliaceae) stems

We investigated the hypothesis that the epidermis is a tension-stressed "skin' whose contribution to stem stiffness depends on the turgor pressure exerted on it by an hydrostatically inflated inner "core' of tissues. This hypothesis was tested by relying on the intensities of bending stresses due to stem flexure, which must reach their maximum levels at the outer surface of epidermis such that damage to the surface of the stem should produce the most significant decrease in overall flexural stiffness. We discerned whether the principal tension supporting members at the stem surface (cellulosic microfibrils) were oriented parallel or normal to stem length by comparing the bending stiffness of stems before and after their surface cells first received three parallel longitudinal incisions followed by one helical incision, and by comparing the bending stiffness of stems for which the sequence of cuts was reversed. The same protocol was also applied to stems with various water potentials to determine the effect of hydrostatic pressure on stem stiffness contributed by the surface. Based on the behavior of 82 turgid Tulipa stems, parallel cuts reduced, on average, stem stiffness by 8%, whereas a subsequent helical incision further reduced stiffness by 42%. In contrast, an initial helical incision reduced stem stiffness by 50%, while three subsequent parallel cuts through the same stems did not significantly further reduce stiffness. These results suggested that the net orientation of cellulose microfibrils in the outer epidermal walls was parallel to stem length. This was confirmed by microscopic observations of cells with dichroic staining and polarized light. The responses to surgical damage were directly proportional to stem water potential. We thus conclude that the epidermis, probably in conjunction with a single layer of subepidermal collenchyma cells, acts as a tension-stiffening agent that can contribute as much as 50% to overall stem stiffness We present a simple mechanical model that can account for all our observations.     

Stem biomechanics of three columnar cacti from the Sonoran Desert

The allometric relationship of stem length L with respect to mean stem diameter D was determined for 80 shoots of each of three columnar cactus species (Stenocereus thurberi, Lophocereus schottii, and S. gummosus) to determine whether this relationship accords with that predicted by each of three contending models purporting to describe the mechanical architecture of vertical shoots (i.e., geometric, stress, and elastic similitude, which predict L proportional to D(alpha), with alpha = 1/1, 1/2, and 2/3, respectively). In addition, anatomical, physical, and biomechanical stem properties were measured to determine how the stems of these three species maintain their elastic stability as they increase in size. Reduced major axis regression of L with respect to D showed that alpha = 2.82 ± 0.14 for S. thurberi, 2.32 ± 0.19 for L. schottii, and 4.21 ± 0.31 for S. gummosus. Thus, the scaling exponents for the allometry of L differed significantly from that predicted by each of the three biomechanical models. In contrast, these exponents were similar to that for the allometry previously reported for saguaro. Analyses of biomechanical data derived from bending tests performed on 30 stems selected from each of the three species indicated that the bulk stem tissue stiffness was roughly proportional to L2, while stem flexural rigidity (i.e., the ability to resist a bending force) scaled roughly as L3. Stem length was significantly and positively correlated with the volume fraction of wood, while regression analysis of the pooled data from the three species (i.e., 90 stems) indicated that bulk tissue stiffness scaled roughly as the 5/3-power of the volume fraction of wood in stems. These data were interpreted to indicate that wood served as the major stiffening agent in stems and that this tissue accumulates at a sufficient rate to afford unusually high scaling exponents tot stem length with respect to stem diameter (i.e., disproportionately large increments of stem length with respect to increments in stem diameter). Nevertheless, the safety factor against the elastic failure of stems (computed on the basis of the critical buckling height divided by actual stem length) decreased with increasing stem size tot each species, even though each species maintained an average safety factor equal to two. We speculate that the apparent upper limit to plant height calculated for each species may serve as a biomechanical mechanism for vegetative propagation and the establishment of dense plant colonies by means of extreme stem flexure and ultimate breakage, especially for S. gummosus.     

Outcrossing rates in the gynomonoecious-gynodioecious species Dianthus sylvestris (Caryophyllaceae)

Some species described as gynodioecious are truly gynomonoecious-gynodioecious. Three distinct phenotypes may be found in their natural populations-female and hermaphrodite pure-sexed plants bearing either only pistillate or perfect flowers, respectively, and mixed plants bearing both types of flowers. In one such species, Dianthus sylvestris, we investigated mating system parameters using allozyme data. Outcrossing rates and correlations of outcrossed paternity were calculated for the three types of plants and separately for pistillate and perfect flowers on mixed plants. The mean outcrossing rate for the population was t(m) ± SD = 0.885 ± 0.032. Females were more outcrossed than hermaphrodites (0.987 ± 0.112 and 0.790 ± 0.076, respectively), whereas mixed plants were not significantly more or less outcrossed than hermaphrodites (0.840 ± 0.060). Within mixed plants, perfect flowers showed an intermediate outcrossing rate (0.898 ± 0.057), whereas pistillate flowers were as selfed as perfect flowers on hermaphrodite plants (0.782 ± 0.111). Family estimates of outcrossing rates were highly variable. Globally, no biparental inbreeding was detected in this species, and there was a mean of 61.5 ± 19.9% of full-sibs within families. Floral dimorphism between small pistillate and large perfect flowers together with pollinator preference for larger flowers could explain the observed patterns for both mating parameters. The advantages of gynomonoecy-gynodioecy are discussed. We conclude that mixed plants do not reduce selfing for all flowers on a plant, but perfect flowers on these plants seem to have an outcrossing advantage.     

Shade induced changes in biomechanical petiole properties in the stoloniferous herb <Emphasis Type="Italic">Trifolium repens</Emphasis>

Increased cell number and cell length both contribute to shade induced elongation of petioles which enables stoloniferous plants to place their leaf lamina higher up in the canopy. Although petiole elongation is assumed to be beneficial, it may also imply costs in terms of decreased biomechanical stability. We test the hypothesis that shade induced elongation changes the biomechanical properties of petioles and that the underlying mechanisms, cell division and cell elongation, differentially affect biomechanical properties. This was done by subjecting 14 genotypes differing in the relative contribution of cell size and cell number to shade induced elongation responses to high light conditions and to simulated canopy shade. Developmental traits (cell size and cell number), morphological traits characterizing the petioles, as well as biomechanical characteristics were measured. Our results show that, comparable to stems of non-clonal plants, the rigidity of a petiole’s tissue (the Young’s modulus) increases, leading to increased flexural stiffness of petioles subjected to shading. Increased flexural stiffness proved to be associated with increased performance under shaded conditions. Our results also indicate that cell number affected the material properties and the flexural stiffness of petioles. However, the degree and pattern of the effects differed between light environments. Shade induced increase in cell number translated into shade induced increase of Young’s modulus and flexural stiffness. Genotypes producing relatively larger cells under shaded conditions experienced a decrease in tissue rigidity. In concert our results indicate that the pattern of selection on flexural stiffness, and thereby also on shade induced changes of cell number and cell size differs among light environments. An erratum to this article can be found at     

Multi-axial spine biomechanical testing system with speckle displacement instrumentation

This paper reports on the design and development of a multi-axis (up to 6 axes) mechanical tester for spinal research and testing. The developed spine tester allowed true motion to be simulated on a specimen in pure or combined modes. To demonstrate the capability of the new tester flexural stiffness properties of sheep lumbar motion segments were evaluated together wiith a non-contact speckle displacement measurement system. The flexural stiffness of the specimens was measured and compared under constrained and non-constrained testing conditions; with relieving of shear forces (non-constrained), it was found that the specimen behaved in a 'stiffer' manner.     

Cofilin increases the bending flexibility of actin filaments: implications for severing and cell mechanics

We determined the flexural (bending) rigidities of actin and cofilactin filaments from a cosine correlation function analysis of their thermally driven, two-dimensional fluctuations in shape. The persistence length of actin filaments is 9.8 μm, corresponding to a flexural rigidity of 0.040 pN μm². Cofilin binding lowers the persistence length ∼5-fold to a value of 2.2 μm and the filament flexural rigidity to 0.0091 pN μm². That cofilin-decorated filaments are more flexible than native filaments despite an increased mass indicates that cofilin binding weakens and redistributes stabilizing subunit interactions of filaments. We favor a mechanism in which the increased flexibility of cofilin-decorated filaments results from the linked dissociation of filament-stabilizing ions and reorganization of actin subdomain 2 and as a consequence promotes severing due to a mechanical asymmetry. Knowledge of the effects of cofilin on actin filament bending mechanics, together with our previous analysis of torsional stiffness, provide a quantitative measure of the mechanical changes in actin filaments associated with cofilin binding, and suggest that the overall mechanical and force-producing properties of cells can be modulated by cofilin activity.     

The extremely long-tongued neotropical butterfly Eurybia lycisca (Riodinidae): proboscis morphology and flower handling

Few species of true butterflies (Lepidoptera: Papilionoidea) have evolved a proboscis that greatly exceeds the length of the body. This study is the first to examine the morphology of an extremely long butterfly proboscis and to describe how it is used to obtain nectar from flowers with very deep corolla tubes. The proboscis of Eurybia lycisca (Riodinidae) is approximately twice as long as the body. It has a maximal length of 45.6?mm (mean length 36.5?mm?±?4.1?S.D., N?=?20) and is extremely thin, measuring only about 0.26?mm at its maximum diameter. The proboscis has a unique arrangement of short sensilla at the tip, and its musculature arrangement is derived. The flower handling times on the preferred nectar plant, Calathea crotalifera (Marantaceae), were exceptionally long (mean 54.5?sec?±?28.5?S.D., N?=?26). When feeding on the deep flowers remarkably few proboscis movements occur. The relationship between Eurybia lycisca and its preferred nectar plant and larval host plant, Calathea crotalifera, is not mutualistic since the butterfly exploits the flowers without contributing to their pollination. We hypothesize that the extraordinarily long proboscis of Eurybia lycisca is an adaptation for capitalizing on the pre-existing mutualistic interaction of the host plant with its pollinating long-tongued nectar feeding insects.     

Mechanical loading of orthodontic miniscrews - significance and problems: an experimental study

Orthodontic miniscrews are exposed to three mechanical loading phases during clinical use: torsional loading upon insertion, flexural loading during anchorage function, and torsional loading upon removal. The aim of this study was to simulate clinical loading conditions for different types of orthodontic miniscrews in vitro to quantify the effects of combined torsional and bending stress. Various orthodontic miniscrew systems (Lomas, Dual-top, Aarhus anchorage, Tomas-pin and T.I.T.A.N.-pin) comprising 10 samples each were subjected to the following loading sequences in vitro: a torsional load corresponding to manual insertion with limited torque; and flexural loading at two different insertion depths. For all screw systems with torsional pre-loading (simulating insertion), subsequent flexural loading (simulating anchorage) yielded permanent deformations of approximately 0.15-0.25 mm, depending on the insertion depth. Since EDX analysis revealed comparable elemental compositions for the different screw systems, the differences in mechanical properties are attributed to screw design. Torsional loading during screw insertion may cause premature mechanical weakening and needs to be minimized. Unless fully inserted, screws show pronounced plastic deformation and hence fracture risk under subsequent flexural loading.     

The open section effect in a long bone with a longitudinal defect - a theoretical modeling study

A longitudinal defect dramatically alters the stress distribution within a long bone. The altered stress distribution can influence the structural properties of the bone and the stimulus for repair and remodeling of the defect and the surrounding bone. For applied torsion, the defect interrupts the normal shear flow around the bone. Reversal of the shear flow along the inner cortex of the bone is the primary characteristic of the "open-section" effect. Stress concentration effects also produce large stresses at the defect corners. A finite element model of a femur mid-diaphysis with a rectangular defect in the posterior cortex was developed to quantify the femur stress distribution and torsional stiffness for defect widths ranging from one-tenth of the femur outer diameter (0.1 OD) to 0.3 OD, and defect lengths ranging from 0.5 to 5 OD. Defects with a length of 1 OD or shorter had little influence on the femur torsional stiffness or the femur shear-stress distribution. The torsional stiffness decreased most dramatically as the defect length increased from 2 to 3 OD, but began to approach an asymptote near 5 OD. Shear flow reversal peaked at the center of the defect for defects longer than 1 OD, and the magnitude of the reversal began to approach an asymptote near 5 OD. For each defect, the largest stresses within the bone, developed at the defect corners. The results indicate that the open-section effect decreases the torsional stiffness and stress concentration effects decrease the torsional strength of a long bone with a longitudinal defect.     

Mechanics and Growth Form of the MossHylocomium splendens

The mossHylocomium splendenshas two different growth forms.Sympodial growth occurs where the apical meristem ceases activityannually and growth is continued by a lateral bud. Sympodialplants are vertical and self-supporting. Monopodial growth occurswhen the apex continues growth. Monopodial plants are prostrate.The aims of the study were to examine stem mechanics of thedifferent growth forms and to compare mechanical propertiesalong stems. Stems of annual segments were subjected to threepoint bending tests. In sympodial plants the stiffness of thestem material increased significantly with segment age. Flexuralrigidity increased significantly with age in segments from 1to 4 years old, and then declined. Segment diameter decreasedsignificantly with age in sympodial plants. Monopodial plantsshowed no significant effects of segment age on the diameter,material stiffness or flexural rigidity of stems. Sympodialsegments were significantly wider, stronger, more rigid andcomposed of stiffer material with a higher stress at yield thanmonopodial segments, but did not have a larger proportion ofstrengthening material. Sympodial stems had significantly morecellulose than monopodial stems. The mossHylocomium splendensshowsa range of mechanical adaptations, as a self-supporting or aprostrate plant, which suit it to life in very different environments.Copyright1998 Annals of Botany Company. Hylocomium splendens, growth form, mechanics, cellulose, stiffness, flexural rigidity, bending, monopodial, sympodial, adaptation, anatomy, stems, plant.     

Pollen dispersal in Yucca filamentosa (Agavaceae): the paradox of self-pollination behavior by Tegeticula yuccasella (Prodoxidae)

We investigated pollen dispersal in an obligate pollination mutualism between Yucca filamentosa and Tegeticula yuccasella. Yucca moths are the only documented pollinator of yuccas, and moth larvae feed solely on developing yucca seeds. The quality of pollination by a female moth affects larval survival because flowers receiving small amounts of pollen or self-pollen have a high abscission probability, and larvae die in abscised flowers. We tested the prediction that yucca moths primarily perform outcross pollinations by using fluorescent dye to track pollen dispersal in five populations of Y. filamentosa. Dye transfers within plants were common in all populations (mean ± 1 SE, 55 ± 3.0%), indicating that moths frequently deposit self-pollen. Distance of dye transfers ranged from 0 to 50 m, and the mean number of flowering plants between the pollen donor and recipient was 5 (median = 0), suggesting that most pollen was transferred among near neighbors. A multilocus genetic estimate of outcrossing based on seedlings matured from open-pollinated fruits at one site was 94 ± 6% (mean ± 1 SD). We discuss why moths frequently deposit self-pollen to the detriment of their offspring and compare the yucca-yucca moth interaction with other obligate pollinator mutualisms in which neither pollinator nor plant benefit from self-pollination.