Abstract
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 twisttobend 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 crosssectional shape of the main loadbearing structures on the mechanical functionality. Three phase field simulations were carried out showing good agreement with the crosssections found in selected plants: (i) Ushaped crosssections comparable with those of Musa sp. petioles, (ii) starshaped crosssections with deep grooves as can be found in the lianoid wood of Condylocarpon guianense stems, and (iii) flat elliptic crosssections with one deep groove comparable with the crosssections of the climbing ribbonshaped stems of Bauhinia guianensis.
Introduction
Biological Materials Systems
During ontogeny and phylogeny, living organisms are confronted with the challenge of immediate individual response and evolutionary adaptation of populations in order to exist within changing environmental conditions^{1} and, simultaneously, have to ensure the survival of the species through reproduction. The German zoologist Günther Osche addressed this dilemma and pointed out that living beings cannot post a sign with the message “closed for reconstruction”^{2}. On the contrary, regardless of the respective time scale, responses and adaptations in living organisms must take place during “ongoing operation”, because the fulfilment of lifeensuring functions must be maintained permanently over the entire period. These responses and adaptations can include local or general changes in metabolism and changes in morphologicalanatomical characteristics and mechanical properties realised at various hierarchical levels^{3}. Because of their hierarchical structure from the molecular to the macroscopic level, a clear differentiation between “material” and “structure” is not possible in biology^{4}. On the basis of these smooth transitions Wegst et al.^{5} coined the term “structural materials” to describe the complex materials systems of living nature. In other words, plants and animals are materials systems that emerge characteristics far beyond those of their individual components^{6}.
From a mechanical point of view, biological materials systems are characterised, on the one hand, by anatomical heterogeneity through a specific threedimensional arrangement of various tissues and, on the other hand, by mechanical anisotropy through various mechanical properties of their individual tissues. With regard to the topic of this article, response and adaptation can therefore be considered as a consequence of successive or simultaneous changes in one or both of these aspects, which might occur during both ontogeny and phylogeny.
The TwisttoBend Ratio
In the context of response and adaptation to existing or changing environmental mechanical conditions, the dimensionless twisttobend ratio is particularly useful as it provides information about the ratio of flexural rigidity to torsional rigidity of materials systems determined by both material properties (bending modulus E and shear modulus G) and morphometric parameters (axial second moment of area I and polar second moment of area J). Additionally, it allows a comparison of bodies of different sizes because of its dimensionlessness^{7,8,9,10}. Flexural rigidity (=bending stiffness = EI) and torsional rigidity (=torsional stiffness = GJ) describe the resistance of a body to deformation caused by bending or torsion loading in the linearelastic range. Since both are composite variables that combine material properties and morphometric parameters, they are well suited for quantifying the mechanical functionality of biological and technical structures^{7}. On the one hand, sufficient flexural rigidity is relevant to counteract gravity. In plants, this prevents, for example, the sagging of the leaf blades or ensures an upright growth of the stems and thus an advantageous positioning of leaves, flowers and fruits. On the other hand, a low torsional rigidity may help for planar plant organs to streamline themselves under wind loads, e.g. by turning (large) leaves into the wind or by clustering compound leave blades and thus reducing their crosssectional area and thus ultimately the drag force^{7,11,12}.
In order to identify common patterns in the relationship between flexural rigidity and torsional rigidity, Etnier^{10} created a socalled stiffness mechanospace. By mapping the theoretical expectations of ideal beams based on a crosssectional shape (elliptic, circular) and various Poisson’s ratios varying from 0 to 0.5, biological beams are generally limited to particular regions of the mechanospace. Vogel^{7} reported that elongated biological structures can achieve higher values for EI/GJ than ideal isotropic and isovolumetric circular solid cylinders with a value of 1.5 (if E/G is set to 3.0), as natural structures are anatomically inhomogeneous and mechanically anisotropic. In addition to circular and elliptical crosssectional shapes, squareshaped, triangular and even Ushaped crosssections exist in biology. For instance, an average EI/GJ value of 13.3 ± 1.0 has been reported for the hollow and lenticular flower stalks of daffodils (Narcissus pseudonarcissus)^{13}. Furthermore, the values of the twisttobend ratio of the squareshaped stems of Leonurus cardiaca range on average between 15 and 19^{14} and the triangular flower stalks of the sedge Carex acutiformis lie in the range of 22 and 51^{15}. The Ushaped crosssections of banana petioles (Musa textilis) with EI/GJ values ranging from 40 to 100 show the highest values of any natural structures tested to date^{9,11}.
Aim of the Study
In principle, the dimensionless twisttobend ratio is a highly suitable parameter for the analysis and comparison of rodshaped biological and technical materials systems among and with each other. The aim of this study has been to investigate the development and interrelationship of flexural rigidity and torsional rigidity in relation to crosssectional shapes of the main loadbearing structural elements by using a mathematical model and suitable simulations. The results of these simulations have then been compared with the loadbearing structural elements (e.g., lignified strengthening tissues such as xylem, vascular bundles or sclerenchyma and collenchyma fibres) in crosssections of selected biological plant models. These biological models, which have previously been described in the literature, include the leaf stalks of banana plants (Musa sp.) with the highest twisttobend ratio known to date^{9,11} and the stems of two different lianas (Condylocarpon guianense, Bauhinia guianensis) with twisttobend ratios markedly changing during ontogeny^{16}.
The study is divided into three parts: (i) the mathematical model, which is based on a phase field description of the plant stem crosssection in the design space given by a unit square; (ii) the optimisation of the twisttobend ratio of the phase field with respect to its geometry by using a gradient flow method and by weighting flexural rigidity (maximal or minimal), torsional rigidity or both factors as objectives for maximisation and minimisation; (iii) a comparison of the phase field shapes and their mechanical properties acquired during the optimisation process with the selected biological plant models and an interpretation of the insights gained.
The three abovementioned weighting factors (maximal flexural rigidity, minimal flexural rigidity and torsional rigidity) theoretically allow for a large number of diverse simulations. In the context of this study, the authors have selected three exemplary simulations: (i) minimisation of the torsional rigidity and maximisation of the minimal flexural rigidity, (ii) only minimisation of the torsional rigidity and comparison with the case, whereby the maximal flexural rigidity is also minimised, (iii) minimisation of the torsional rigidity and minimisation of the minimal flexural rigidity and comparison with the case, whereby the maximal flexural rigidity is also maximised.
Mathematical Model
Plant stems as slender elastic rods
We describe a plant stem as a long thin elastic rod with domain \(B=A\times (0,L)\) of length L and constant crosssection A for an open bounded sufficiently regular domain \(A\subset {{\mathbb{R}}}^{2}\). We assume \(L\gg {\rm{diam}}A\) as well as material isotropy. It is, of course, possible to take into account heterogeneity and anisotropy of the material when optimising the rigidity properties of crosssectional shapes, see, e.g.^{17,18,19,20}. Here, however, we neglect such heterogeneity and anisotropic effects as well as viscosity and other timedependent processes, since we are only interested in the influence of the crosssectional shape on the mechanical properties of the stem.
Consider B fixed at \(z=0\) and bending of B to be due to an outer normal force on A at \(z=L\). Starting from 3D elasticity, in the limit of a slender rod, following Mora & Müller^{21}, the flexural (or bending) rigidity is given by the moment curvature relation
where M_{y}, M_{x} denote the bending moments on the end of the beam and \({\kappa }_{x}\), \({\kappa }_{y}\) denote the curvature in the direction of x and y, respectively, see Fig. 1. In our idealised case the bending modulus of elasticity is equivalent to the tensile modulus (Young’s modulus) or compressive modulus of elasticity. Thus the parameter E is just the Young’s modulus of the linearisation of the elastic energy and the moments of inertia D_{x}, D_{y} as well as the product of inertia D_{xy} are given by
where we have
The maximal and minimal flexural rigidities D_{max} and D_{min} along the principal axes are then given by the maximal and minimal eigenvalue of the matrix
after multiplying with the material Young’s modulus E, which leads to
For simplicity we write \({D}_{{\rm{\max }}/{\rm{\min }}}={D}_{{\rm{mean}}}\pm RM\).
Remark. Note that Mora & Müller^{21} adapt their coordinate axes to the domain such that \(x=\hat{x}\), \(y=\hat{y}\), and \({D}_{xy}=0\). Since we will shortly move to a phase field description of the stem crosssection, we will work with arbitrary coordinate axis and origin and thus carry those additional terms.
As for the flexural rigidity, the torsional rigidity for an elastic slender rod with domain B was rigorously derived by Mora & Müller^{21}. This rigorous derivation considers the limit of a very slender and long rod, for which it is shown that the requirements of St. Venant’s torsion theory^{22} are satisfied. We assume that torsion is due to a moment T at the top of B, see Fig. 1. Note that for very thinwalled structures (once the thinness of the walls becomes comparable to the lengthtocrosssection ratio), the assumptions of St. Venant’s theory of torsion are not applicable and in this case Vlasov’s theory of torsion should be applied. We argue, however, that the structures used in the comparison to plant morphology as displayed in section “Numerical results and comparison to plant morphology” are still within the realm where St. Venant’s theory can be justified.
In this framework the torsional rigidity may be expressed by Prandtl’s stress function. In Prandtl’s stress formulation the shear stress components are described by the derivatives of the stress function \(\varphi (x,y)\)
Assuming without loss of generality a constant unit twist rate, the stress function \(\varphi \) must then satisfy Poisson’s equation
with shear modulus G. Using that moments are only appearing on the top of B, the tractionfree beam wall condition leads to the boundary condition
where the boundary \(\partial A\) is given by a curve parameterised by s.
We restrict ourselves to simply connected plant stem crosssections and thus may assume, without loss of generality, that
The torsional rigidity D_{z} is then given by
A priori bounds on the twisttobend ratio
In a shape optimisation problem regarding the twisttobend ratio of a plant stem, we are considering the minimisation problem
with weighting factors \({\sigma }_{1},{\sigma }_{2},{\sigma }_{3}\in {\mathbb{R}}\). For example, if \({\sigma }_{1} > 0\), \({\sigma }_{2} < 0\), and \({\sigma }_{3}=0\), then solutions of Eq. (2) tend to minimise torsional rigidity and maximise minimal flexural rigidity.
Following Kim & Kim^{23} we deduce that D_{z} has the representation
where the socalled “warping” function \(\omega \) is given by the solution of Laplace’s equation with Neumann boundary condition \(\frac{\partial \omega }{\partial \eta }=\frac{1}{2}\frac{d}{ds}(x(s),y(s)){}^{2}\), where \((x(s),y(s))\) is an arclength parametrisation of the boundary curve of the crosssection and \(\eta \) is its outer unit normal. This leads to the observation that for a circular domain A torsional rigidity D_{z} is determined by D_{mean}.
We thus deduce, that, if for example the crosssection A is constrained to be a material distribution in a rotationally symmetric reference domain, merely maximising the flexural rigidity leads to nonsimply connected domains, such as symmetric hollow circular tubes (see Condylocarpon guianense). Therefore, as we are interested in simply connected crosssections having a high twisttobend ratio, a sole maximisation of the flexural rigidity is not useful for our purposes.
For domains with circular boundary curve (simply connected or not) and isotropic material as well as Poisson’s ratio \(\nu \in (0,0.5)\) we deduce from Eq. (3) the estimate
The theorem of St. Venant, see, e.g., Pólya^{24}, states, that circular domains lead to maximal torsional rigidities among simply connected domains. As a bound for D_{z}(A) we can furthermore use the radius \({\rho }_{A}\) of the largest inscribed circle in a domain A. This is due to a theorem of Makai^{25}, proving the inequality
for every simply connected domain \(A\subset {{\mathbb{R}}}^{2}\).
While it is instructive to consider such bounds on the functional above, the problem in Eq. (2) is ill posed even among simply connected domains, in the sense that no minimum exists. This can easily be seen due to the fact that thin fingers cause high flexural rigidity but no torsional rigidity. Thus, a sequence of thinner, but increasingly wide Ibeams will lead to larger and larger negative values of the functional in Eq. (2) for \({\sigma }_{1} > 0\), \({\sigma }_{2} < 0\), \({\sigma }_{3}=0\). We therefore restrict ourselves to a bounded domain for our designs and add a perimeter penalty. Such a perimeter penalisation is indeed also sensible in our application, as plants should not have arbitrary large surfaces of exposure. Furthermore, we impose a fixed crosssectional area (or, equivalently, mass).
More importantly, however, we are not really interested in the minimisers of the functional themselves. Instead, we consider a gradient flow dynamics of our functional using an artificial time variable, hereafter called pseudotime. Solutions of this gradient flow are driven towards the direction of maximal decline, i.e., the direction of the biggest change in rigidities for small changes in shape. We propose that shape change in such a direction can be observed in plant stem geometries.
Phase field approximation
In order to treat our shape optimisation problem numerically, we describe the material distribution in a given domain \(\Omega \) by a phase field variable u. The phase field u shall take values close to 0 in the void and values close to 1 in the areas where material is present. In a phase field approach the interface between material and void is given by a diffuse interface layer, whose thickness is proportional to a small length scale parameter \(\varepsilon \). At this interface the phase field smoothly but rapidly changes its value between 0 and 1. The aforementioned mass constraint now simply reads
Further, we assume that \(u=0\) on the boundary \(\partial \Omega \) of \(\Omega \). We use the common approach of Blank et al.^{26} to describe the phase transition from material to void in the Young’s modulus E, obtaining a Young’s modulus \(E(u)\). In our case we simply take
with material constant E. This way we obtain udependent flexural rigidities \({D}_{{\rm{\max }}/{\rm{\min }}}(u)\) and torsional rigidity \({D}_{z}(u)\). For simplicity, we do not explicitly denote the dependence of these quantities on the length scale \(\varepsilon \). Using the model from section “Plant stems as slender elastic rods” the moments of inertia and the product of inertia can now be expressed in terms of
with
and
In an analogous way we obtain the torsional rigidity \({D}_{z}(u)\) by
If u were simply the characteristic function \({\chi }_{A}\) of our plant stem crosssection A, then \(\varphi (u)\) is given as the solution of Poisson’s problem
In a phase field approach we instead introduce a penalty such that the function \(\varphi \) is required to be constant where u is close to zero. By choosing zero boundary conditions on \(\partial \Omega \), these then get propagated such that \(u=0\) on \(\Omega \backslash \{u\approx 1\}\). We thus solve
where \(0 < {\theta }_{0}\ll 1\) is a small parameter. As long as the set where \(u\approx 1\) is simply connected, we obtain \(\varphi \) as an approximation of Prandtl’s stress function in (1). We note that a similar approach, outside of the phase field context, was used by Kim & Kim^{23}.
As described before, shape optimisation problems of this kind are in general ill posed and it is necessary to add a perimeter penalisation for regularisation. In phase field approaches such a perimeter penalisation is modelled by the help of the GinzburgLandau (or ModicaMortola) energy^{27}
where the function F is given by
so that F has exactly two global minima in 0 and 1. The factor \(\frac{1}{{c}_{0}}\) is a normalising constant.
We note that with \(\varepsilon \) tending to 0 minimisers of \({{\rm{Per}}}_{\varepsilon }(u)\) develop interfaces separating regions in which u is nearly constant with values close to the minima of F. This is due to an argument by Modica, Theorem I in Modica^{28}, which also proves the \(\Gamma \)convergence of \({{\rm{Per}}}_{\varepsilon }(u)\) to the perimeter functional \({\rm{Per}}(\{u=1\})\). Thus, adding \({{\rm{Per}}}_{\varepsilon }(u)\) to our problem penalises the perimeter of the set \(\{u=1\}\) and hence the perimeter of our crosssection.
The shape optimisation problem is then to find a solution
of the following minimisation problem
with
The function space \({H}_{0}^{1}\) denotes all functions with square integrable derivatives and zero boundary conditions on the boundary of \(\Omega \). We note that Eq. (6) does indeed admit minimisers as long as \(\gamma > 0\) and \(\Omega \) is bounded.
L ^{2}Gradient flow and numerical implementation
To compute solutions of Eq. (6) numerically we use a steepest descent approach, i.e., we make small steps in u towards the direction of maximal negative change of \({I}_{\varepsilon }\). In other words, we compute a timediscrete L^{2}gradient flow of \({I}_{\varepsilon }\) until a stationary state has been reached using a discretisation of our reference domain by P1 triangular finite elements. Time discretisation uses a time step variable \(\tau \) and along with integer iteration steps \(n\ge 1\) this leads to an artificial time variable \(t=\tau \cdot n\), also called pseudotime. A thus computed stationary state of the gradient is usually a local solution of our minimisation problem. Furthermore, we can decouple the solution of Poisson’s problem in Eq. (5) from the gradient flow and calculate it separately using a P1 finite element approach. The mass constraint is imposed using a Lagrange multiplier. For an initial configuration u^{0} of the phase field variable we use a semiimplicit first order Euler scheme with only the linear highest gradient term being treated implicitly. We thus can compute the new material distribution u^{n} from the previous distribution u^{n−1} showing us the direction of maximal decline. As described above this gives us the direction of the biggest change in rigidities for small changes in shape. A more detailed description of the gradient flow, the finite element approximation and the implementation details are provided in the Appendices A and B, respectively, in the supplement.
Numerical Results and Comparison to Plant Morphology
To derive that appearing crosssectional shapes of plant stems or petioles play an important part in their mechanical behaviour, we will present three numerical experiments and a comparison to the crosssectional shapes of the aforementioned loadbearing elements of the selected plants. As we are solely interested in the contribution of the crosssectional shape of a plant axes to the twisttobend ratio, we assume a fixed ratio of bending modulus and shear modulus \(E/G\approx 2.7\) for all numerical experiments, which for isotropic materials corresponds to a Poisson’s ratio \(\nu \in [0.2,0.5]\), a value range, that is reasonable to assume for many plant axes^{29}. Note again that we only compare the main loadbearing element of the respective crosssection of the selected plant with the crosssections from our simulations. Detailed morphologicalanatomical descriptions of the biological models plants are derived in Appendix C in the supplement.
Ushapes
In a first experiment we consider the shape optimisation problem (Eq. (2)) with weighting factors \({\sigma }_{1}=1,\,{\sigma }_{2}=\,1\) and \({\sigma }_{3}=0\). This corresponds to a minimisation of the torsional rigidity D_{z} and a maximisation of the minimal flexural rigidity D_{min}. The small weighting factor for the perimeter regularisation is set to \(\gamma \approx 1.4\cdot {10}^{2}E\).
Description of the simulation
The evolution of the phase field is shown in Fig. 2. During the evolution, the circular initial shape of the plant petioles changes noticeably. After a short time period, small grooves form on the outer boundary of the phase. Such grooves are known to facilitate the twisting of a geometry as described by Vogel^{7}. The results of the experiment confirm his finding and indicate that groove formation is the first dominant mode to reduce torsional rigidity starting from a circular disc as rod crosssection. The flexural rigidity barely changes in this initial phase.
During the further optimisation steps this trend becomes clearer (Fig. 2a). The central groove deepens further and leads to an even greater reduction in torsional rigidity, which results in a major increase in the twisttobend ratio (Fig. 2b). After half of the optimisation pseudotime has passed, the flexural rigidity increases perceptibly for the first time. This effect can be attributed to the widening of the central groove, which causes the phase to shift outwards, building a Ushaped domain. This process continues until the phase has reached the boundary of the reference domain \(\Omega \) (\(t=0.35\)). As soon as the phase contacts the boundary, the resulting shapes of course cannot be compared to plant morphology anymore. A numerical steady state of the gradient flow is obtained at pseudotime \(t\approx 0.8\).
Comparison with the leaf stalk of bananas (Musa sp.)
The shape change of the phase in this simulation shows great similarities with the crosssections of banana leaf stalks (Musa sp.) (Fig. 3a and Fig. C1 in the supplement). Generally, leaf stalks (=petioles) should resist static loads such as bending caused by the leaf weight in order to hold the large leaf blade (=lamina) in place and to ensure its orientation towards the sun. Additionally, they have to withstand high dynamic loads caused, in particular, by wind forces acting on the lamina. These drag forces can be reduced by streamlining in the wind in terms of twisting the petiole^{7,9,11,12,30,31}. This is extremely important for the integrity of the herbaceous banana plant, which consists of a pseudostem of densely packed leaf sheaths at the base of the petioles and leaf laminae with large surfaces, the latter being especially susceptible to damage from wind forces.
Morphologicalanatomical studies of the Ushaped crosssection of banana petioles have revealed an inner and an outer shell comprising an epidermis and fibrereinforced parenchyma with radial (sometimes branched) parenchymatous strands lying in between^{11,32,33,34} (Fig. 4a and Fig. C1 in the supplement). This crosssectional arrangement is associated with an increase in flexural rigidity, since its special structure prevents the petiole from bending downwards (and finally collapsing) by converting bending forces into tensile forces. In addition, the combination of shape and inner structure also reduces torsional rigidity and thus supports streamlining by torsion. These characteristics lead to extremely high twisttobend ratios of petioles of banana plants with values ranging from 40 to 100^{9,11}, compared with petioles of various tree species with values between 1.6 and 9^{7,12}. A comparable Ushape is visible in the phase field simulation alongside the trend towards an increasingly higher twisttobend ratio (Fig. 2).
The shape of the phases at various pseudotimes shows striking similarities with individual crosssections along the longitudinal axis of the banana petiole^{33,34}. The phase shapes at pseudotimes \(t=0.3438\), \(t=0.2475\) and \(t=0.1512\) correspond well to the crosssectional shapes found in the basal, middle and apical parts of the petioles (Fig. 2b, as well as Figs. C1 and C2 in the supplement). Figure 2b clearly shows that, with increasing pseudotime, the torsional rigidity decreases, whereas the bending rigidity increases, and the resulting twisttobend ratio increases strongly in the relevant period between \(t=0.1512\) and \(t=0.3438\). From the viewpoint of functional morphology and biomechanics, this shapedependent increase of the twisttobend ratio is strongly related to the mechanical loading of banana leaves. As a result of the increased leverage, because of the own weight of the leaf itself, the flexural rigidity in the basal part of the petiole has to be larger than that in the apical parts. For the torsional rigidity, on the other hand, it is advantageous to be uniformly low over the entire petiole in order to allow easy twisting under wind loads and thus to protect the leaf stalk from damage^{11,31}.
Deep grooves
In a second experiment we consider Eq. (2) with \({\sigma }_{2}={\sigma }_{3}=0\) and \({\sigma }_{1}=1\), which leads to a minimisation of torsional rigidity only. The weighting factor for the perimeter regularisation is \(\gamma =1\cdot {10}^{2}\,G\). We note that the bending modulus does not play a role here since the objective function does not include bending.
Description of the simulation
The evolution of the phase field is shown in Fig. 5. Similar to the first simulation, grooves appear again at the boundary of the phase field shape, which reduce the torsional rigidity. In contrast to the first experiment, however, these grooves appear uniformly distributed along the boundary (Fig. 5a).
The sole weighting of the torsional rigidity then leads to an even deepening of some grooves. By this, the twisttobend ratio is increased strongly after a comparatively short period of time and a characteristic cloverleafshaped crosssection is formed (Fig. 5b). A widening of the grooves, as in the experiment described above, only occurs to a lesser extent. The following smoothening of the phase boundary has no major influence on the torsional and flexural rigidity, respectively. A numerical steady state is reached at \(t\approx 0.1435\). The flexural rigidity increases only marginally during the simulation and has no considerable influence on the twisttobend ratio. Compared to the first simulation the running pseudotime needed to reach the steady state and thus the optimum under the given boundary conditions, is much lower.
Comparison with the stem of the liana Condylocarpon guianense
In this case, the shape change of the phase field has similarities with the ontogenetic developments of the crosssectional shape of the woody part and thus the main loadbearing tissues of the stems of the liana Condylocarpon guianense (Fig. 3b). The rodshaped stems of this species respond to the typical mechanical loads to which they are subjected in a certain ontogenetic phase by changes in their internal structure and in the material properties of the tissues involved. In young selfsupporting C. guianense shoots, which are still searching for a support and are therefore mainly exposed to bending loads, a dense and stiff type of wood (secondary xylem) is formed. It is arranged in a centripetal pattern, with a central pith and an adjacent ring of dense wood (secondary xylem consisting of narrowdiameter vessels and small wood rays). This wood type is responsible for the relatively high flexural rigidity of young C. guianense stems, enabling them to bridge the distance to potential supports. As soon as the stems are securely attached to a support, a different type of wood is built, which is significantly less dense and mechanically more flexible (secondary xylem comprising widediameter vessels and broad wood rays forming grooves in the wood cylinder), contributing to the pronounced flexibility of old lianescent stems. Because of the formation of the two wood types in subsequent ontogenetic phases, young “searchers” form a dense and stiff wood cylinder, which is surrounded by lianoid nondense wood built during older phases^{16,35,36,37,38} (Fig. 4b and Fig. C2 in the supplement).
Similar to the woody part of a young stem of C. guianense, the phase field simulation starts with a circular crosssection (\(t=0.0000\)) (Fig. 5 and Fig. D3 in the supplement). The twisttobend ratio of the phase field is very low at this point. However, within a very short period of time, especially compared with the first simulation, the twisttobend ratio of the simulated phase shape increases strongly (Fig. 5b). This increase in the twisttobend ratio can be attributed to the formation of the above mentioned grooves that strongly reduce the torsional rigidity while maintaining the flexural rigidity (Fig. 5a). Figuratively this can be imagined such that, because of the grooves, the largest possible resulting circular area of the crosssection is reduced, an event that is ultimately responsible for the torsional rigidity.
The resulting deeply grooved shape of the phase field is similar to the crosssectional shape of the wood in older C. guianense stems. These older stages, which are by now attached to a support, are highly flexible in both bending and twisting and therefore allow the slender liana stem passively to follow the movement of the host tree under wind loading and even to survive the breakage of branches of even the entire stem of supporting host tree^{35,36,39} (Fig. 3b). The shape and threedimensional arrangement of the wood within the older crosssections differ markedly from those of the younger stages. Only a small circular ring of the dense wood remains around the central pith, while the lianoid lessdense wood is arranged in a deeply grooved (starlike) crosssectional shape, analogous to the phase field, and fills most of the crosssectional area^{35}. We can thus assume that the decrease in torsional rigidity in older stages of C. guianense is (mainly) attributable to the different shape and arrangement of wood within the crosssections of young and old stems.
If, in addition to the reduction of the torsional rigidity, a minimisation of the flexural rigidity is also included in the simulation, as is the case during the ontogeny of C. guianense, the resulting shape phase differs markedly from the real shape of the xylem in C. guianense stems (Fig. 6b). Consequently, we can reasonably assume that the decrease in flexural rigidity in C. guianense is primarily determined by the modified material properties of the wood formed in the lianescent phase of growth^{16} and not by the shape and 3D arrangement of the tissues involved.
In general, the selfsupporting early stages of lianas show the typical values for flexural rigidity, as they are also known for other selfsupporting woody stems. In contrast, the nonselfsupporting older ontogenetic stages of lianas that are attached to a host support have considerably lower values (reduction of up to an order of magnitude). Although fewer data exist concerning torsional rigidity in the literature, the values of torsional rigidity of the secondary wood do not seem to decrease in the same way as the values of flexural rigidity^{9}. To summarise, all lianas tested to date develop especially low \({D}_{{\rm{\min }}}/{D}_{z}\) ratios after “giving up” selfsupport^{9}.
Ribbons
In a third experiment we consider Eq. (2) with \({\sigma }_{3}=0\) and \({\sigma }_{1},{\sigma }_{2}=1\), which leads to a minimisation of both torsional rigidity and minimal flexural rigidity. The perimeter penalty is set to \(\gamma \approx 1.4\cdot {10}^{2}\), as in the first experiment.
Description of the simulation
The evolution of the phase field is shown in Fig. 7. A minimisation of both torsional and flexural rigidity allows the phase to form a nearly elliptic shape, leading to a noticeable decrease in both the torsional and the minimal flexural rigidity (in direction of the short axis), whereas the maximal flexural rigidity in the direction of the long axis is increased. A formation of grooves also occurs here. This is shown by a large groove in the middle of the shape, which like in the previous experiments deepens further, in this case leading to an even greater reduction in both torsional and minimal flexural rigidity. Before the phase touches the boundary the characteristic shape appears at \(t=0.1040\). A numerical steady state is reached at pseudotime \(t\approx 0.3960\). For more details see Fig. D4 in the supplement.
Considering the twisttobend ratio of the shapes, it is important to say that contrary to the previous experiments, only the ratio \({D}_{{\rm{\max }}}/{D}_{z}\) is markedly increased (Fig. D1 in the supplement), where the ratio \({D}_{{\rm{\min }}}/{D}_{z}\) is reduced (Fig. 7b). The crosssectional shapes that occur thus only show a high twisttobend ratio in the direction of the maximum flexural rigidity. In the previous experiments these two ratios were almost identical due to the near symmetry of the crosssectional shapes.
The mentioned groove in the middle of the shape is the decisive characteristic, which distinguishes this model from the model with an additional maximisation of the maximum flexural rigidity D_{max}. The formation of such a groove slows the increase in maximum flexural rigidity D_{max} and is thus suppressed when the maximum flexural rigidity is included as an objective to be maximised (Fig. 8).
Comparison with the stem of the monkey ladder liana (Bauhinia guianense)
Like the stems of Condylocarpon guianense, the stems of the liana Bauhinia guianensis change their mechanical properties, wood type and wood shape markedly during ontogeny^{16}. Young selfsupporting stems and young apical axes of B. guianensis have a circular crosssectional shape composed of a central pith and an adjacent ring of dense stiff wood (secondary xylem consisting of narrowdiameter vessels and small wood rays) (Fig. 4c and Fig. C3 in the supplement). These young axes are stiff in both bending and torsion^{16,37,39}. As in C. guianense the young shoots act as “searchers”, spanning gaps between the host supports (Fig. 3c) and therefore rely on high values of flexural and torsional rigidity^{36,37,39}.
Similar to the crosssections of young B. guianensis stems, the phase field simulation starts from a round shape (\(t=0.00\)) (Fig. 4c and Fig. D4 in the supplement), which also features high minimal flexural rigidity and high torsional rigidity and thus the highest twisttobend ratio within this simulation (Fig. 7). In contrast, adult nonselfsupporting lianescent stems of B. guianensis are much more flexible and have a markedly lower modulus of elasticity and their crosssectional shape differs considerably from that of young stems^{16}. These changes in the mechanical properties of the stem have been correlated with changes in the contribution of the various wood types (small amounts of dense stiff secondary xylem built in the young selfsupporting stage and large amounts of nondense flexible secondary xylem with widediameter vessels and broad wood rays formed after attachment in the lianescent stage) to the axial second moment of area of the stems^{16,37}. This also becomes apparent with regard to the change in the crosssection of the stem from a circular to a ribbon shape during the ontogeny of the plant^{39}. The ribbon shape, which gives B. guianensis its vernacular name of “monkey ladder”, can also be seen in the phase field simulation and is associated with reductions in minimal flexural rigidity, torsional rigidity and twisttobend ratio. Moreover, as described above, the phase field shape exhibits a midline groove, which further decreases the minimal flexural rigidity and the torsional rigidity. This groove is also present in B. guianensis but has a slightly different shape. In the actual plant, the groove is much wider towards the outside than the groove in the phase field simulation.
Regardless of their individual shape, grooves have a similar effect on the mechanics of the overall structure. Figuratively speaking, the grooves reduce the largest possible resulting circular area within the crosssection, which ultimately leads to a decrease of the torsional rigidity. Another similarity with the phase field simulation is the crosssectional stem shape of B. guianensis in the transition phase from young to adult stages. Since additional largelumen wood is only formed on two opposite sides of the young circular stem, the crosssection shows more and more similarities with the elliptical shape of the phase field shortly after the start of the simulation (Fig. 4c). The simulation reveals that this change in the crosssectional shape results in a simultaneous decrease of the torsional rigidity and of the minimal flexural rigidity (Fig. 7a).
Discussion
In summary, some similarities, but also some differences, exist between the three experiments, namely the simulations of “Ushapes”, “Deep grooves” and “Ribbons”. The simulations of “Ushapes” and “Deep grooves” are readily comparable insofar that, dependent on the various weighting factors, both lead to an increase of the twisttobend ratio. Comparison of these two simulations demonstrates clear differences with regard to the increase and the maximum values of the twisttobend ratio. With the exclusive minimisation of torsional rigidity, as performed in “Deep grooves”, a twisttobend ratio of \({D}_{{\rm{\min }}}/{D}_{z}\approx 4\) can be achieved even after a pseudotime of \(t\approx 0.025\), whereas with the minimisation of the torsional rigidity and a simultaneous maximisation of the minimum flexural rigidity, as was carried out in “Ushapes”, a twisttobend ratio of \({D}_{{\rm{\min }}}/{D}_{z}\approx 4\) could only be reached after a pseudotime of \(t\approx 0.26\). On the other hand, the overall twisttobend ratio is higher if the minimal flexural rigidity is additionally maximised, as conducted in “Ushapes”, with twisttobend ratios of \({D}_{{\rm{\min }}}/{D}_{z}\approx 20\), instead of just the minimisation of the torsional rigidity as performed in “Deep grooves”, where the maximal twisttobend ratio only has values of \({D}_{{\rm{\min }}}/{D}_{z}\approx 4.5\). Interestingly, in the simulation of “Ribbons”, the twisttobend ratio \({D}_{{\rm{\min }}}/{D}_{z}\) decreases over time, whereby the \({D}_{{\rm{\max }}}/{D}_{z}\) increases and achieves after a pseudotime of \(t\approx 0.1\), a twisttobend ratio of \({D}_{{\rm{\max }}}/{D}_{z}\approx 4\) and maximum values of \({D}_{{\rm{\max }}}/{D}_{z}\approx 5.5\).
Apart from these differences, a common shaperelated characteristic noticeably occurs in all three simulation, namely the formation of grooves. Figuratively, these grooves reduce the largest possible circular area that can be placed in the phase field shapes, whose size corresponds to the torsional rigidity and thus ultimately leads to a reduction in the torsional rigidity of the overall structure. Since all simulations are at least partly aimed at minimising the torsional rigidity, we can expect that these grooves will occur in all three simulations. Only the design of these grooves varies depending on the additional optimisation requirements.
What conclusions can be drawn from these findings for the selected plant models? Since plants as biological structures are generally the result of multifunctional requirements and, moreover, can only respond or adapt within the framework of their respective bauplan, the influence of the shape of a structure on the overall performance in terms of flexural and torsional rigidity cannot be derived from the plant models. With the simulations presented here, this assignment is possible for the first time, although the twisttobend ratio is clearly a measure for a compromise of various mechanical functions. Possible deviations of plant axes from the optimised shape are indications for further functions that are vital for the survival of the respective plant species. Precisely for this reason and because the twisttobend ratio is a dimensionless parameter, it is particularly suitable for comparing biological structures not only with each other, but also with technical structures.
Experimental investigations on the petiole of the banana leaf have shown a twisttobend ratio ranging from 40 to 100^{11,32}. Analogous to the various phase field shapes found in the simulation of “Ushapes”, the banana petiole also displays various crosssectional shapes along its longitudinal axis and thus a change in mechanical functionality. In addition to this spatial resolution based on the various crosssectional shapes, a difference exists between the theoretically achievable maximum value of \({D}_{{\rm{\min }}}/{D}_{z}\approx 20\), as determined in the simulation “Ushapes” and purely resulting from the respective shape, and the values determined experimentally. This difference can only be explained by the special inner structure of the petiole. The fact that the banana petiole is up to 100 times stiffer in bending than in torsion represents a selective advantage with regard to the alignment of the leaf blade to sunlight in the sense of efficient photosynthesis and simultaneously avoids damage to the leaf blade, as the leaves are streamlined under wind load.
In contrast to the banana leaf, which represents a spatial resolution of various crosssectional shapes, the two selected liana species have a temporal resolution of the different crosssectional shapes as a function of ontogenetic development from the young and old ontogenetic stages. First of all, the various stages differ mainly in their mechanical properties: young stages are selfsupporting and are stiff “searchers”, whereas the older stages are safely attached to the host support and are nonselfsupporting and characterised by high flexural and torsional flexibility. The reduction in flexural and torsional rigidity takes place via rapid transitions from dense stiff wood built in the early stages to lessdense flexible wood developed in the older stages. Later shifts in development include the change in the crosssectional shape by the formation of woody lobes and resulting grooves as described in simulations “Deep grooves” and “Ribbons”.
Specifically, the bending modulus E of C. guianense axes decreases from a mean of 2722 MPa during early ontogenetic stages to a mean of 306 MPa during older stages, whereas the percentage contribution of the widelumen wood to the crosssectional area increases from 0 to 30%^{16,35,36}. From the simulation of “Deep grooves”, we learn that the minimisation of the torsional rigidity of C. guianense axes with almost constant bending rigidity is controlled by the increasing lobation of the crosssectional shape of the wood. We can conclude from this observation that additional flexural flexibility is controlled by the formation of flexible lianoid wood having other material properties.
Similar to C. guianense, changes in the crosssectional shape and mechanical behaviour of B. guianensis stems are linked to the ontogenetic stage of the plant. Early stages with a circular crosssection producing dense stiff wood are 2–3 metres long and occur as selfsupporting “searchers” that can bridge the gap to potential host supports or selfsupporting young saplings^{39}. As soon as the stem is attached to a supporting tree, rapid transitions to compliant wood take place. Interestingly, the cambial growth is highly modified producing a ribbonshaped stem attributable to the formation of lianoid wood at two opposite sides of the young circular stem and changes into an elliptical crosssection^{39}. During the period between wood built in young stages to wood built in older stages, the bending modulus E decreases from 24 GPa to 3.75 GPa and the torsional modulus G decreases from 0.91 GPa to 0.42 GPa^{16}. The simulation of “Ribbons” shows that the torsional flexibility at almost constant minimal flexural rigidity (\({D}_{{\rm{\min }}}/{D}_{z}\)) is controlled by the shape change from circular to elliptical and the additional formation of one groove at the centre of the major axis. This abovementioned rapid transition from one stage to the other is mirrored in the phase field simulation of “Ribbons” in which a relatively short pseudotime is required to optimise the twisttobend ratio (\(t\approx 0.2\)) for reaching the lowest value \({D}_{{\rm{\min }}}/{D}_{z}\approx 0.1\). This is different when the twisttobend ratio \({D}_{{\rm{\max }}}/{D}_{z}\) is considered. Here, the flexural rigidity can be 6 times as high as the torsional rigidity.
Conclusion
The use of gradient flow functions in the form of phase field simulations has proved to be a novel and appropriate approach that helps us to understand optimisation processes during evolution and ontogeny within biology. In the framework of this study, the gradient flow has been used to illustrate the fastest/largest possible changes in rigidity with the smallest possible change in the crosssectional shape of the loadbearing structures. A comparison with selected plant species suggests that evolution also follows this principle, as small changes in crosssectional shape are “easy to implement” at little “costs”, but still offer a large selective advantage. This approach can probably also be used to aid our understanding of other evolutionary or ontogenetic optimisation processes.
Data availability
This work does not have any experimental data. The shapeoptimisation C++code is made available as supplementary material.
References
 1.
Lambers, H., Chapin III, F. S. & Pons, T. L. Plant physiological ecology, 4–6 (Springer Science & Business Media, 2008).
 2.
Horn, R., Gantner, J., Widmer, L., Sedlbauer, K. P. & Speck, O. Bioinspired sustainability assessment–a conceptual framework. In Knippers, J., Nickel, K. & Speck, T. (eds) Biomimetic research for architecture and building construction, 361–377 (Springer, 2016).
 3.
Fratzl, P. & Weinkamer, R. Nature’s hierarchical materials. Prog. Mater. Sci. 52, 1263–1334, https://doi.org/10.1016/j.pmatsci.2007.06.001 (2007).
 4.
VDI. Bionik: Bionische Materialien, Strukturen und Bauteile; Biomimetics: Biomimetic materials, structures and components. VDI 6223 (2013).
 5.
Wegst, U. G., Bai, H., Saiz, E., Tomsia, A. P. & Ritchie, R. O. Bioinspired structural materials. Nat. Mater. 14, 23, https://doi.org/10.1038/NMAT4089 (2015).
 6.
Speck, T. & Speck, O. Emergence in biomimetic materials systems. In Wegner, L. H. & Lüttge, U. (eds) Emergence and modularity in life sciences, 97–115, https://doi.org/10.1007/9783030061289_5 (Springer, 2019).
 7.
Vogel, S. Twisttobend ratios and crosssectional shapes of petioles and stems. J. Exp. Bot. 43, 1527–1532, https://doi.org/10.1093/jxb/43.11.1527 (1992).
 8.
Vogel, S. Twisttobend ratios of woody structures. J. Exp. Bot. 46, 981–985, https://doi.org/10.1093/jxb/46.8.981 (1995).
 9.
Vogel, S. Living in a physical world xi. to twist or bend when stressed. J. Biosci. 32, 643–655 (2007).
 10.
Etnier, S. A. Twisting and bending of biological beams: distribution of biological beams in a stiffness mechanospace. The Biol. Bull. 205, 36–46, https://doi.org/10.2307/1543443 (2003).
 11.
Ennos, A. R., Spatz, H. & Speck, T. The functional morphology of the petioles of the banana, Musa textilis. J Exp Bot 51, 2085–2093, https://doi.org/10.1093/jexbot/51.353.2085 (2000).
 12.
Louf, J.F. et al. How wind drives the correlation between leaf shape and mechanical properties. Sci. Reports 8, 16314, https://doi.org/10.1038/s41598018345880 (2018).
 13.
Etnier, S. A. & Vogel, S. Reorientation of daffodil (Narcissus: Amaryllidaceae) flowers in wind: drag reduction and torsional flexibility. Am. J. Bot. 87, 29–32, https://doi.org/10.2307/2656682 (2000).
 14.
Kaminski, R., Speck, T. & Speck, O. Adaptive spatiotemporal changes in morphology, anatomy, and mechanics during the ontogeny of subshrubs with squareshaped stems. Am. J. Bot. 104, 1157–1167, https://doi.org/10.3732/ajb.1700110 (2017).
 15.
Ennos, A. R. The mechanics of the flower stem of the sedge Carex acutiformis. Annals Bot 72, 123–127, https://doi.org/10.1006/anbo.1993.1089 (1993).
 16.
Hoffmann, B., Chabbert, B., Monties, B. & Speck, T. Mechanical, chemical and xray analysis of wood in the two tropical lianas Bauhinia guianensis and Condylocarpon guianense: variations during ontogeny. Planta 217, 32–40, https://doi.org/10.1007/s0042500209672 (2003).
 17.
Ashby, M. Overview no. 92: materials and shape. Acta metallurgica et materialia 39, 1025–1039 (1991).
 18.
Ashby, M. & Bréchet, Y. Designing hybrid materials. Acta materialia 51, 5801–5821 (2003).
 19.
Estrin, Y., Beygelzimer, Y. & Kulagin, R. Design of architectured materials based on mechanically driven structural and compositional patterning. Adv. Eng. Mater. 1900487 (2019).
 20.
Estrin, Y., Bréchet, Y., Dunlop, J. & Fratzl, P. Architectured Materials in Nature and Engineering (Springer, 2019).
 21.
Mora, M. G. & Müller, S. Derivation of the nonlinear bendingtorsion theory for inextensible rods by Γconvergence. Calc. Var. Partial. Differ. Equations 18, 287–305, https://doi.org/10.1007/s0052600302042 (2003).
 22.
Timoshenko, S. P. & Gere, J. M. Theory of elastic stability (Courier Corporation, 2009).
 23.
Kim, Y. Y. & Kim, T. S. Topology optimization of beam cross sections. Int. J. Solids Struct. 37, 477–493, https://doi.org/10.1016/S00207683(99)000153 (2000).
 24.
Pólya, G. Torsional rigidity, principal frequency, electrostatic capacity and symmetrization. Q. Appl. Math. 6, 267–277 (1948).
 25.
Makai, E. A proof of SaintVenant’s theorem on torsional rigidity. Acta Math. Hungarica 17, 419–422 (1966).
 26.
Blank, L. et al. Phasefield approaches to structural topology optimization. In Constrained optimization and optimal control for partial differential equations, 245–256, https://doi.org/10.1007/9783034801331_13 (Springer, Basel, 2012).
 27.
Modica, L. & Mortola, S. Un esempio di γconvergenza. Boll Unione Mat. Ital. Sez. B 14, 285–299 (1977).
 28.
Modica, L. The gradient theory of phase transitions and the minimal interface criterion. Arch. for Ration. Mech. Analysis 98, 123–142, https://doi.org/10.1007/BF00251230 (1987).
 29.
Niklas, K. J. Plant biomechanics: an engineering approach to plant form and function (University of Chicago Press, 1992).
 30.
Niklas, K. J. A mechanical perspective on foliage leaf form and function. The New Phytol. 143, 19–31 (1999).
 31.
Vogel, S. Drag and reconfiguration of broad leaves in high winds. J. Exp. Bot. 40, 941–948 (1989).
 32.
Ahlquist, S., Kampowski, T., Torghabehi, O. O., Menges, A. & Speck, T. Development of a digital framework for the computation of complex material and morphological behavior of biological and technological systems. Comput. Des. 60, 84–104, https://doi.org/10.1016/j.cad.2014.01.013 (2015).
 33.
Mattheck, C. Thinking tools after nature (Karlsruher Inst. of TechnologyCampus North, 2011).
 34.
Mattheck, C., Kappel, R., Bethge, K. & Kraft, O. Lernen vom Bananenblatt  der verrammelte Notausgang. Konstruktionspraxis spezial, Novemb. 50–52 (2005).
 35.
Rowe, N., Isnard, S. & Speck, T. Diversity of mechanical architectures in climbing plants: an evolutionary perspective. J. Plant Growth Regul. 23, 108–128 (2004).
 36.
Rowe, N. P. & Speck, T. Biomechanical characteristics of the ontogeny and growth habit of the tropical liana Condylocarpon guianense (Apocynaceae). Int. J. Plant Sci. 157, 406–417 (1996).
 37.
Speck, T. & Rowe, N. P. A quantitative approach for analytically defining size, growth form and habit in living and fossil plants. In Kurmann, M. H. & Hemsley, A. R. (eds) The evolution of plant architecture, 447–479 (Royal Botanic Gardens Kew, 1999).
 38.
Speck, T. et al. The potential of plant biomechanics in functional biology and systematics. In Stuessy, T. F., Mayer, V. & Hörandl, E. (eds) Deep morphology: Toward a renaissance of morphology in plant systematics, 241–271 (Koeltz, Königstein, 2004).
 39.
Rowe, N. & Speck, T. Plant growth forms: an ecological and evolutionary perspective. New Phytol. 166, 61–72, https://doi.org/10.1111/j.14698137.2004.01309.x (2005).
Acknowledgements
P.D. acknowledges partial support by the German Scholars Organization/CarlZeissStiftung in the form of the “WissenschaftlerRückkehrprogramm”. M.L. was funded by the German Research Foundation within the CRCTransregio 141 and by the Ministry of Science, Research and the Arts BadenWürttemberg within the framework of “BioElast”. T.S. and O.S. acknowledge the support of the German Research Foundation within the Cluster of Excellence “livMatS”. Our thanks are also extended to Dr. R. Theresa Jones for improving the English.
Author information
Affiliations
Contributions
S.W.V. conducted the mathematical experiments and wrote the first draft of the manuscript. M.L. contributed to the description of the model plants and the compilation of the diagrams and wrote the first draft of the manuscript. O.S. initiated the study and contributed to the improvement of the first draft of the manuscript. T.S. and P.D. initiated the study. All authors contributed to the data interpretation and reviewed and improved the final draft of the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
WolffVorbeck, S., Langer, M., Speck, O. et al. Twisttobend ratio: an important selective factor for many rodshaped biological structures. Sci Rep 9, 17182 (2019). https://doi.org/10.1038/s4159801952878z
Received:
Accepted:
Published:
Further reading

Laser powder bed fusion of bioinspired honeycomb structures: Effect of twist angle on compressive behaviors
ThinWalled Structures (2020)

Peak values of twist‐to‐bend ratio in triangular flower stalks of Carex pendula : a study on biomechanics and functional morphology
American Journal of Botany (2020)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.