Journal of Experimental Botany, Vol. 51, No. 352, pp. 1825-1842,
November 1, 2000
© 2000 Oxford University Press
Original Papers |
Biomechanical study of the effect of a controlled bending on tomato stem elongation: local strain sensing and spatial integration of the signal
1 INRA, Unité associée Bioclimatologie-PIAF, 234 av. du Brézet, 63039 Clermont-Ferrand cedex 02, France
2 INRA, Unité d'Ecophysiologie des Plantes Fourragères, 86600 Lusignan, France
Received 26 April 2000; Accepted 15 June 2000
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Abstract |
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In a previous paper it has been demonstrated that tomato stems, submitted to a controlled basal bending, had a reduced terminal primary elongation, indicating mechanosensing and intra plant signalling. The ‘intensity’ of the growth response, as measured by the time to recover an elongation rate similar to the control, varied hugely between plants. However, no relation was found between the intensity of this response and the mechanical variables characterizing the global mechanical state of the stem. In this paper, a local analysis of mechanical state of each bent stem is performed in the context of beam theory. The spatial distributions of local variables all along the stem (curvature, bending moment, strains and stresses) are established. The validity of hypotheses underlying the mechanical analysis is demonstrated. To investigate the relationships between the mechanical stimulus and the growth response, a novel biomechanical analysis based on spatial integration of the mechanical stimulus is presented. It revealed that the mechanosensing is local and scattered through the stem and that the variability of the growth response is only explained by the integrals of the longitudinal strain field.
Key words: Biomechanics, mecanoperception, strain, thigmo-morphogenesis, tomato stem.
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Introduction |
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TopAbstractIntroductionMaterials and methodsResultsDiscussionAppendix 1: Models of...Appendix 2: Mathematical...References |
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The effect of mechanical perturbations on plant growth has been reported in many studies and several efforts have been made to elucidate the physiology of the primary steps of mechanosensing (see Biddington, 1986
; Trewavas and Knight, 1994, for reviews). The identification and quantification of the mechanical stimulus and of the dose-responses have, however, seldom been studied (but see Jaffe et al., 1980; Knight et al., 1992; Hepworth and Vincent, 1999). Additionally, the relative contribution of the different tissues of the plant to the mechanosensing and the way local sensing is integrated within the whole plant to produce growth response are still unclear.
In a previous paper an in situ mechanical test was designed allowing a controlled and quantified bending of the basal part of a stem, while continuously monitoring its terminal elongative growth (Coutand et al., 2000). When applied to the basal part of adult tomato stems, this test has revealed a bending-induced reduction of primary growth, demonstrating clear mechanosensing by the basal, non-elongating tissues, and intra-plant signalling.
However, no significant correlation has been found between the intensity of the growth response (time to recover normal elongation rate) and any of the variables characterizing the global amount of bending (e.g. force, bending moment, stored mechanical energy, mean curvature or basal inclination of the stem). A possible reason for this lack of dose-response (or dose-dependence) is that the global mechanical variables do not encapsulate correctly the interaction between the load and the perceptive system of the plant. Indeed the mechanical fields induced by the bending (curvature, strain and stress fields) depend not only on the applied load but also on the geometry and anatomy of the stem (Niklas and Moon, 1988; Beusmans and Silk, 1988; Niklas, 1992; Moulia and Fournier, 1997). Therefore, the mechanical stimulus for a similar global load can vary from place to place within the stem, and also between stems. For instance, stems of different diameters submitted to similar global bending load could experience distinctly different stimulations.
In this work the spatial distribution of the mechanical field variables generated in each individual stem during the standardized bending test was studied, using curvature and bending moment calculations (Moulia, 1994) and a model of beam element with composite cross-section (Moulia and Fournier, 1997).
It was then assessed whether such spatial characterization of the mechanical field variables could bring new insights into the identification of the mechanical variable sensed by the plant. Essentially, mechanosensing should be local (Badot et al., 1992; Knight et al., 1992; Zandomeni and Schopfer, 1994; Hepworth and Vincent, 1999). As the bent part of the stem is entirely submitted to a mechanical stimulation, the effective signal for growth reduction should thus correspond to some spatial integral of the local perception along the stem. The strategy used here was to compute various spatial integrals based on the spatial mechanical fields addressed previously and on a set of alternative simple hypothetical models concerning (i) the mechanical variable that is sensed, (ii) possible sensory functions, and (iii) differences in sensitivity between tissues. The dose-response of elongation growth to these integrals was then studied to assess their capacity of explaining the variability of the growth response (dose-response). It is shown that a dose-response can only be established assuming scattered strain perception (not stress) and additive integration of the signal along the stem. The significance of these results is discussed.
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Materials and methods |
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This work is based on the data collected in the experiment described in the companion paper (Coutand et al., 2000
). The experiment is thus only briefly summarized here. More emphasis is given to the specific measurements necessary for spatial analysis and to the methods used to process the data.
Plants
Tomato plants (Lycopersicon esculentum Mill. var. VFN8) were grown in control environment in a growth chamber (L/O 14 h/10 h, thermoperiod 25/18±1 °C, HR 55%) on nutrient solution (Coutand et al., 2000). When the eighth internode (from the stem base) was 1 cm long, two similar plants (same stage, same stem geometry) were chosen for each bending experiment (one control and the other for the bending treatment).
Bending experiment
The experiment took place in the same controlled conditions. The two plants (control and treated) were set up with the instruments (LVDT, clamp, dynamometer). Then, stem elongation was measured continuously for 3 d. The similarity in the growth curves of the two plants was assessed at the end of the first day and, if any significant difference was detected, the plants were changed and the experiment was re-started. On the second day, a photograph of the basal part of the stem was taken. Then the bending test was conducted. The stem collar was displaced horizontally at a constant rate of 80 mm min-1, up to 2 cm from its initial position. At maximal displacement, the maximal force applied (F) was measured and a photograph of the basal part of the stem was taken. Then the stem was de-loaded. A photograph of the stems was also taken after load removal to analyse the possibility of irreversible strains. The overall operation lasted less than 1 min (see Materials and methods of the companion paper for more details).
Analysing the mechanical stimulus within the beam theory
The basal part of the stem is a beam-like structure (Fig. 1A) subjected to an end-deflection (cantilever beam). According to the beam theory (Laroze, 1988), it can thus be considered as a longitudinal pile of transversal cross-sectional slices (with infinitesimal height dh) (Fig. 1B). Contrasting with the typically squared cross-section of the upper stem, cambial activity in the basal part of the tomato stem under interest resulted in a rounded, approximately axisymmetric, shape (Fig. 1C). Five concentric tissues can be distinguished: the pith, the xylem, the phloem, the collenchyma, and the epidermis. The diameter of the cross-section displayed smooth and small tapering along the stem (Fig. 1D). During the loading, the local mechanical state of each successive cross-section can be described from two points of view: kinematically, amount of deflections and strains of the slice, and statically, forces and stresses.
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) along the basal part of the stem. The diameters vary continuously along the bent part with no abrupt variations. The diameter of the hypocotyl is smaller than that of the internodes'.From the kinematic point of view, the deflection of an end-loaded cantilever beam results from shearing and bending. Shearing can mainly be seen as an angular distortion of the slice (Fig. 2A
) while bending consists of successive rotations of the cross-sections composing the beam (Fig. 2B). Both processes contribute to the end-deflection. When the overall deformation is small, as it was the case in this experiment, the overall deflection can be estimated from the additive superposition of the ‘pure’ shear component of the deflection and of the ‘pure’ bending one (that is adding the deflections in Fig. 2A to the those in Fig. 2B). The relative contribution of shearing and bending to the total deflection depends on (i) the aspect ratio (length/diameter) of the beam, (ii) the tapering of the beam and (iii) the relative stiffness of the constitutive materials to shear strains and to longitudinal strains. In slender beams (high aspect ratio) with smooth and small tapering, it is usually assumed that bending deflections are much higher than shear deflection. In this experimental design the basal part of the stem displayed a small and smooth tapering (Fig. 1C
), but the aspect ratio of the bent part was around 1/10 (Fig. 1A), and was subjected to a maximal relative end-deflection of 2/10 of its length. This might be close to the limit of ‘warranted’ validity of the assumption of negligible shear deflection (Timoschenko, 1968) especially for highly anistropic composite beams.
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of the slice. The warping of the cross-section (due to coupling with longitudinal shear) has been neglected. (C) Case of an homogeneous un-tapered beam. (D) Case of an homogeneous taped beam. (E) Case of a tapered beam with longitudinally heterogeneous constitutive materials. 1 Pure bending; 2 pure shear (the effect of the boundary conditions have been neglected). In all the drawings, the infinitesimal deformations and displacements have been strongly exaggerated to make things visible, but the assumptions of small deformations are retained (see text for additional comments).From the static point of view, the bending load can be characterized by the bending moment (Mh) and the shear force (Th) in each cross-section (Fig. 2
). More precisely, each cross-section is subjected to a bending moment and to a shear force, which are balanced, respectively, by longitudinal bending stresses and by shear stresses. The bending moment expresses the effect of lever arms in the bending process and thus increases with distance from the loading point (Fig. 2, compare C1 and C2). Therefore, the bending moment field is heterogeneous along the stem. By contrast, in a cantilever beam bent by a punctual load (as performed in this experiment), the shear force Th is constant along the stem, and equal to the external force F.
It was shown in a previous paper that the amount of deflection of the stem base and its inclination were not per se relevant quantities for the mecanoperceptive process, nor was the total (shear) force (Coutand et al., 2000). Therefore, neither the shear deflection nor the shear force (Fig. 2, C2) were directly involved in the perception. Furthermore, cellular studies on mechanosensing have shown that the so-called ‘stretch-activated channels’ in the cell membrane are most probably involved (Cosgrove and Hedrich, 1991; Badot et al., 1992). As cell walls (and membranes) show mostly longitudinal and transverse orientations in tomato, shear can be disregarded as it induces angular deflections in the cells but no significant stretch (Fig. 2B; Niklas, 1992). The major source of longitudinal stretch in this experiment should result from the bending part of the deflection (Fig. 2A). Therefore, the bending was targeted in this study. The aim of this study was to extract the pure bending part of the mixed bending–shear deflection occurring in the bending test, and to analyse it in terms of longitudinal bending strains and stresses.
Analysis of the spatial distribution of curvature and bending moment
In the kinematic description of the pure bending component (Fig. 2, C1), the cross-section can be considered as remaining flat and only being submitted to a bending incremental rotation d
(Fig. 2A). The line joining the centre of rotations of the successive cross-sections is called the neutral line. The amount of rotation can change along the stem (longitudinal heterogeneity, see Fig. 2, C1). Quantitatively, this change can be characterized by the rate of lineic increase in bending angle d/dh. As in pure bending, each cross-section remains orthogonal to the neutral line, the neutral line gets curved and simple trigonometry shows that the amount of lineic increase in bending angle is equal to the curvature of the neutral line (Laroze, 1988). Note that this curvature is the inverse of the radius of a circle tangent to the neutral line at this point. It is thus a local variable that can change along the stem (non-uniform curvature field) and will accordingly be noted Ch. In axisymmetric stems, the neutral line can be confused with the central line of the stem, provided that the curvature is not too large (assumption of small curvatures and negligible eccentricity of the neutral line). Therefore, in pure bending, the measurement of the curvature of the central line characterizes the amount of bending rotation. But does curvature still characterize the amount of pure bending when applied to combined bending and shear deflections as in this experiment?
In cylindrical homogeneous beams (Fig. 2, C2), shear only produces an angular deflection, but no curvature (except close to the clamp). Shear deflection can, however, generate local curvature of the neutral line if the shear rigidity changes along the stem due either to changes in diameter (Fig. 2, D2) or in the shear stiffness of the constitutive materials (Fig. 2, E3). In this study, diametric changes were moderate, but a direct assessment of the negligibility of shear contribution to curvature was performed (see Section entitled Assessment of the hypotheses underlying the composite beam analysis).
The curvature field along the stem was computed by a numerical analysis of the central line (following Moulia et al., 1994). The central line was obtained by a manual digitizing of photographs using a 2D tablet (Summasketch Pro, Summasketch®, USA). The curvature field was then calculated using the COURB2D software (Moulia et al., 1994).
From the static point of view, the bending load can be characterized by the bending moment Mh. If the displacements are small, the field of bending moments along the stem can be estimated as the product of the force (F) by the lever arm in the unloaded state (here, distance between a given cross-section and the applied force).
 | (1) |
If larger displacements occur however, the bending moment field has to be computed calculating the lever arms in the loaded state (Moulia et al., 1994). The bending moment in this experiment was computed using both the small displacement approximation (equation 1) and a more accurate procedure for large displacement, using the MOMENT2D software and the digitized photographs of each test (Moulia et al., 1994).
The curvature and bending load fields only characterize the longitudinal heterogeneity of the bending along the stem. To get to local longitudinal strains and stresses, it is necessary to consider the behaviour of the cross-section.
Local model of the cross-section and calculations of longitudinal strains and stresses (transverse heterogeneity)
There are two ways to obtain the values of local longitudinal strains: (i) experimental measurements by strain gauges or (ii) local mechanical model. The local mechanical model proved particularly suitable here for three reasons. First, a local mechanical model makes it possible to compute the whole fields of the mechanical variables (strains and stresses) whereas strain gauges provide only discrete measurements of longitudinal strains (i.e. at the position they are struck with adhesive). Secondly, strain gauges are particularly difficult to adapt to a flexible material like herbaceous tissues. Lastly, composite beam models have already been tested by several authors (Niklas, 1991a; Moulia and Fournier, 1997) on plant organs. They were to prove reliable, provided the underlying hypotheses are respected (Moulia and Fournier, 1997). In this study, a general model of the flexure of heterogeneous composite beam (Moulia and Fournier, 1997) was used. However, the simplified version of the equations was used based on the approximate axisymmetry of the basal part of tomato stems.
In the composite beam model, under the assumption of negligible shear and of small curvatures, the local longitudinal strain due to bending (at a distance y from the neutral axis) is given directly by the product of the curvature and the distance (y) from the central line, independently of the components of the beam:
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The estimation of the longitudinal strains across the cross-section thus requires an anatomical description. Serial freehand cuts were performed every 1 cm along each stem, on fresh material just after the end of the test. To minimize changes in size due to cell disruption and dehydration, the cuts were intentionally thick (1 mm) and immediately placed in a drop of water, without any staining or fixing. The mean diameters of each tissue in the cross-section were measured as the mean of four radii. These data were then combined with the curvature field measured in the same stem and with equation (2) to compute longitudinal bending strains.
For example, in a given cross-section, the maximal longitudinal strain was estimated taking y equal to the external radius:
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According to the composite beam model of Moulia and Fournier, longitudinal stresses have to be calculated from the strain field and from the rheological characteristics of the constitutive tissues (Moulia and Fournier, 1997). Indeed, in contrast with homogeneous beams, the stress distribution in composite cross-sections is dependent on the rheology of the constitutive tissues. These tests were performed within the linear elastic domain of the stem tissues (Coutand et al., 2000). From the theory of linear elasticity, the longitudinal bending stress (
LL(h,y)) is thus given by:
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As five tissues could be distinguished in these anatomical cross-sections, at least five longitudinal Young's moduli would have been required. However, most of these tissues were very thin, making it very difficult to get good specimens and to perform direct rheological testing. The strategy used in this study was thus to extract as much information as possible about the constitutive stiffness of the tissues from the results of the bending test, using this composite beam model.
Based on the curvature and bending moment fields, it is possible to compute the bending rigidity K(h) all along the stem (Silk et al., 1982; Niklas, 1992; Moulia et al., 1994) as
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Now considering a beam composed of five tissues, the composite beam model expresses the total rigidity (K(h)) of a cross-section as the sum of the products of the longitudinal Young modulus and the second moment of area of each tissue i:
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However combining Young's moduli found in the literature for similar tissues and the anatomical data from the tests in this study, it is possible to assess the magnitude of the contributions of the tissues. Preliminary studies using this approach demonstrated that the pith parenchyma only accounted for 1–11% of the total bending rigidity, being central and very flexible. This was in accordance with previous studies which have demonstrated that the pith has a limited influence on the total flexural rigidity (e.g. for tobacco stems, Hepworth and Vincent, 1999), and bears very low longitudinal strain and stress levels. So that, in the first instance, the influence of the pith was not considered. In addition, the xylem—because of its lignification, is a stiffer material compared to the living tissues: phloem, collenchyma and parenchyma. Moreover, it is likely that non-living tissues greatly differ from living tissues in their sensitivity to mechanical stimuli. Thus, in this case, the problem can be restricted in first approximation to the case of a hollow composite beam with two materials: an outer ring of living ‘tissue’ composed by epidermis, collenchyma and parenchyma (noted OR) and a group of dead tissue, the xylem (noted xyl).
Assuming that the longitudinal variation of the longitudinal Young modulus (ELL)i of each tissue between two cross-sections close within the stem is negligible compared to the variation of second moment of area (which relates to radius to the fourth power), then these data enabled the values of (ELL)OR and (ELL)xyl along each stem to be computed by solving the system of two equations with two unknown quantities ((ELL)OR and (ELL)xyl):
 | (7|<|)|>| |<|(|>|8) |
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Knowing the second moment of area of a total cross-section (excepted the pith) I(h), the bending moment (M(h)) and the curvature (C(h)), it is also possible to define and compute the global equivalent Young modulus of the cross-section Eeq(h):
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LLeq(h) by using Eeq(h) in equation (4). LLeq(h) corresponds to the longitudinal bending stress that would exist if the two tissues were homogenized (mixture). As this quantity is often characterized in biomechanical studies (Niklas, 1992), the statistical relationship was examined between Eeq(h), (ELL)OR(h) and (ELL)xyl(h) and assessed to what extent the cruder estimation of longitudinal stresses could replace the more detailed one.
The study was completed with an anatomical study of the living tissues and the xylem, using classical double stain: iodur green and red carmine to analyse the density of cell walls visually (which is linked to the values of the equivalent moduli of elasticity) along stems (Niklas, 1992).
Assessment of the hypotheses underlying the composite beam analysis
The three major assumptions of the previous analysis are (i) the assumption of negligible effect of shear deflection on curvature (pure bending), (ii) the assumption of small curvatures (no eccentricity of the neutral line) and (iii) the assumption of small displacements. Preliminary studies (Coutand, 1999) demonstrated that the errors due to the assumptions of small curvature could be neglected, as expected. The assumption of small displacement was assessed as described previously (bending moment calculations).
The hypothesis of negligible effect of shear in this analysis of curvature is more central, due to the aspect ratio and the anisotropy of the basal part of the stem. It was tested by comparing directly the experimental deflection measured on photographs with an estimation of the deflection Dh due to shear force T, along the bent part. Dh was calculated by the equation from classical beam theory (Laroze, 1988) designed for smoothly and slowly tapering cantilever beams with circular cross-section:
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) with the value of the Poisson ratio v=0.3 taken from the literature. An estimate of the Geq value was computed from our experimental values of Eeq by using the ratio E/G=10 given by (Vogel, 1992) for tomato stems obtained from combined bending and twisting assays. It has been found in cellular plant tissues, that G and E variations along the stem and between plants are usually correlated (despite their independence from a theoretical rheological point of view in anisotropic materials). This is due to the fact that both depend on the stiffening of the cell wall (Niklas, 1991b). Although equation (11) is not based on an accurate modelling of the shearing of the composite cross-section (it does not take into account the warping of the cross-section related to the distribution of shear stresses and the phenomenon of longitudinal shear), it should give a good order of magnitude of shear deflection, as tapering is smooth and slow (Timoshenko, 1968), (Fig. 1C). Note also that this test includes both the effect of shear on angular deflection and on curvature, so it overestimates the effect of shear. If the focus is on curvature, then only the spatial derivative (slopes) of the deflection curves should be considered.
Statistical analyses
The procedure CORR (SAS Institute) was used to estimate the linear correlation between sets of variables. The Pearson correlation coefficient and its probability (Pr>F) to differ from zero were calculated. When significantly correlated (at the level 0.05), the sets of points were fitted by a linear or logarithmic regression using EXCEL software (MicrosoftTM). In each case, the best fit was obtained considering the coefficient of determination R2 and the distribution of the residuals.
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Results |
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Curvature and bending moment measurements and validity of the hypotheses underlying the cantilever beam analysis
The calculated deflection due to shear force was negligible compared to the deflection measured experimentally (Fig. 3A
), and the slopes differences were even bigger. So, the hypothesis of pure bending was suitable for these computations and the curvature measurements were sufficient to characterize the bending kinematics.
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) and deflection due to shear force (
). (B) Typical distributions of the central line curvatures. (C) Comparison of the bending moments computed at the non loaded state (Typical distributions of the central line curvature are given in Fig. 3B
. Curvature varies largely and non-linearly along the stem from values of 0 at the base of the stem to maximal values near the clamp. In most bending tests (89%) this maximal curvature was smaller than 0.008 mm-1. In two peculiar cases, the curving was higher with maximal curvatures up to 0.02 mm-1.
A typical distribution of the estimated bending moment using either the small displacement estimate (equation 1) or the more accurate computation in the loaded configuration is given in Fig. 3C. Bending moment varies almost linearly along the stem, from 0 at the base of the stem to typical values around 80 N mm near the clamp. The two sets of points are confused which means that no significant difference between the two methods of moment computation was found, in accordance with the hypothesis of small displacements.
Spatial distribution of the longitudinal bending strains and stresses and of the Young modulus of the tissues
The complex spatial distributions of longitudinal strains and stresses are illustrated in Fig. 4A, B and that of the equivalent Young modulus in Fig. 4C. Eeq was found to vary greatly and not monotonously along the stem, with typical values of 600 MPa in the highly lignified hypocotyl to 10 MPa in the upper part of the stem segment. Moreover the Eeq(h) distributions varied considerably from one stem to another. Considering the overall sample of stems in this study, highly significant correlations were found between (ELL)xyl and Eeq (r=0.54, Pr(>F)=0.001), and between (ELL)OR and Eeq (r=0.41, Pr(>F)=0.0001). Moreover, regression analysis revealed that the intercept was not significantly different from zero in both cases, so that both variables could be considered as proportional to Eeq. Therefore, the spatial distributions of the stresses based on the two tissue models just differed from those based on Eeq by a scale factor. For the sake of simplicity only the distribution of equivalent stresses LLeq(h) will be presented in the following.
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, two different stems).Both longitudinal strains and stresses displayed large spatial variations and their spatial distribution was complex (e.g. S-shaped in the case of strains). As the result of the heterogeneous distribution of Eeq(h), the strains and stresses distributions are not parallel. Moreover, rather large differences in the spatial distributions of longitudinal strains and stresses were found between stems. The question was then: could these differences in the distributions of the local mechanical variables, that is in the effective mechanical stimulus in the stem, explain the variability in the growth response? This raised the problem of sensory functions and of the spatial integration of the signal all along the stem.
Sensory functions and spatial integration of the signal
As stated in the Introduction, the effective signal inducing the growth response is likely to result from some spatial integral of the local mechanical variables. More precisely, the local perception p(h,y) of the mechanical variable X(h,y) is likely to sum up along the stem into a global signal P transported to the primary growing tissues. In the literature, three local mechanical variables have been proposed as possible candidates for mechanosensing: curvature (Wilson and Archer, 1979), stress (Mattheck, 1991) and strain (Wilson and Archer, 1979; Ingber, 1993; 1998a, b; Zandomeni and Schopfer, 1994; Ramahaleo et al., 1996; Heworth and Vincent, 1999). However no clear-cut evidence has been reported. Given the estimated spatial distribution of each of these variables, their integral could thus be computed numerically on each of the tested stems, and plotted versus the growth response. Before doing so, it is necessary, however, to specify three additional elements (i) the local sensory function (f), (ii) its distribution within the stem (i.e. the domain of perception), and (iii) the signal transport and the remote growth response.
In general, the local sensory function (i.e. the relationship between the sensed mechanical variable and the local signal p generated by the perception) can be defined as:
 | (12) |
; Badot et al., 1992; Knight et al., 1992; Malone, 1993; Malone et al., 1994; Ingber, 1998b) and very few quantitative data can be found in the literature. Moreover, when experimental data about the stretch dependence of candidate reactions are given, the characterization of the mechanical stimulus cannot readily be linked to the characterization of the local mechanical state considered in the experiment (e.g. pipette suction force on a plasmalemma patch as in Cosgrove and Hedrich, 1991). To start with simple cases, two extreme hypothetical sensory functions have been assumed, as illustrated in Fig. 5.
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In the first scenario the sensory function (f0) is supposed to be linear, and the threshold is assumed to be zero.
 | (13) |
In the second case (fmax), only the highest value of the mechanical variable is sensed
 | (14) |
In equation (14) it is assumed that the plant is able to compare the values of X within a domain in order to retain the maximal value finally. It has been suggested (McMahon, 1975) that the domain for such differential perception could be the cross-section (or radial rays of cells within the cross-section). Therefore this case will be noted fmax-CS.
Given the local perception field p(h,y), the second step is to integrate it into the global signal P produced along the bent stem and transported to the growth zone. In this experiment, the time-course of signal generation and transport was found to be very short compared to the duration of the response (Coutand et al., 2000). If it is assumed that it could be thus neglected (likewise any signal attenuation during transport), the global signal P produced along the bent stem then comes simply as the spatial sum of the local signals p(s) along the stem
 | (15) |
To compute P using equation (15) from the fields X(h,y) and the sensory functions equations (13) and (14), it is necessary to define the spatial distribution of the sensivity (i.e. the capacity to sense), namely the distribution of k(h,y) within the stem. There is very limited information in the literature about the distribution of sensitivity within plants. The most likely differences in sensitivity have an anatomical basis. It is likely that lignified cells that lack membranes and protoplasm also lack sensitivity. Therefore, based on the anatomy of the cross-section (Fig. 1C), three simple all-or-none hypothetical cases concerning tissue sensitivity were retained: (i) equal sensitivity on all the tissues excepting the pith (for the reasons detailed previously), noted AT-P sensitivity, (ii) sensitivity restricted to the outer living ring, noted OR sensitivity, (iii) sensitivity restricted to the surface of the stem, noted EP sensitivity (for the epidermis). Obviously these tissue-based sensitivities could only apply to the variables that display cross-sectional heterogeneity, namely bending longitudinal stresses and strains.
Differences in sensitivity along the stem could also exist, as (i) there is a graded variation of age and differentiation of the cross-sections along the stem, and (ii) there might be some signal attenuation along transport. To start with simple extreme cases, a strategy based on an extension of the fmax function defined in equation (14) was used. It was assumed that only the highest values within the whole bent part of the stem are perceived (as also suggested by Goodman and Ennos, 1997), an hypothesis noted fmax-WS. As in bending, the maximal values sit on the upper part of the stem, they also correspond to the younger (possibly more sensitive) tissues, and to the places close to the growth zones (i.e. where the least signal attenuation could take place).
Combining the hypotheses for the sensory function, the hypotheses for the tissue based sensitivities (AT-P, OR or EP), the longitudinal distribution of sensitivity and the three variables candidates for sensing yield a total of 16 possible models of signal perception and integration, summarized in Fig. 5B. The formulae used for the numerical computation of the corresponding spatial integrals from the experimental data are explained in Appendix 1 and their mathematical derivation is presented in Appendix 2. Depending on the variable and on the type of perceptive function, the integral equation (15) can be a linear integral (e.g. curvature with f0, or all the fmax-CS cases), a surface integral (integral over the epidermis f0-EP) or a volume integral (all the other cases). Due to the distributions of the mechanical variables in bending, some of the scenarios yield to the same mathematical expression so that only 10 distinct models were really studied, as presented in Table 1. Note that the assumptions about the domains of sensitivity induce changes in the power law exponent and in the values affected to the stem radii in the integral formula. Therefore clear differences in the numerical outputs for these different scenarios were obtained.
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Given the models for the total signal P arriving at the remote growth zone, and the measured growth response G, plots of experimental G versus modelled P were established for all the modelled scenarios (each point corresponding to a tested stem). The closest statistical relation between the response and the different models should reveal the sensed variable and the best model. Moreover, the regression equation
 | (16) |
Relation between spatial integrals of local mechanical variables and the growth response
Figure 6A and B corresponds to the scenario with f0 sensory function and whole stem sensitivity (AT-P-f0). The correlation coefficient between the longitudinal stresses integrated over the stem and the growth response is 0.115 and does not differ significantly from 0 (Pr(>F)=0.65) (Fig. 6A). On the contrary, there was a high correlation coefficient (0.84, Pr(>F)=0.0001) between longitudinal strains integrated on the whole stem and the growth response. The data were best fitted by a logarithmic curve with the determination coefficient of R2=0.72 (Fig. 6B). The results for the curvature integral are not presented. It equals the inclination angle of the basis of the stem (see Appendix 2), a variable which has been previously found not to be related to the growth response duration (Coutand et al., 2000).
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The results obtained for the scenarios of a sensitivity limited to the outer ring of living tissues (OR-f0, Fig. 7A
) or to the epidermis (EP-f0, Fig. 7B) applied to the strains fields are shown in Fig. 7. For the OR-f0 scenario applied to the strain field (Fig. 8A), the correlation coefficient was 0.81 (Pr(>F)=0.001). This set of points fitted a logarithmic curve well (R2=0.67). On the contrary, the correlation coefficient between the growth response and the longitudinal stresses integrated spatially on the living tissues was very low (0.01) and did not differ significantly from 0 (Pr(>F)=0.72) (not shown). For the EP-f0, scenario (Fig. 7B) on strains, the correlation was also high (0.83, Pr(>F)=0.0001) and the set of point fitted a logarithmic curve with a determination coefficient R2=0.68. Here again, the correlation with the stress integral was not significant (not shown).
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Figure 8
corresponds to the fmax scenarios with AT-P sensitivity (i.e. assuming that only the highest values within the whole bent part of the stem are perceived). No significant relation was found with maximal curvature (Fig. 8A, correlation coefficient R=0.31, Pr(>F)=0.24), nor with the maximal longitudinal stress (Fig. 8B, R=0.004, Pr(>F)=0.99). The correlation between the duration of the growth response and the maximal longitudinal strain (Fig. 8C) was somewhat higher (0.53) and differs significantly from 0 (Pr(>F)=0.035). This last set of points was best fitted using linear regression, but the determination coefficient remains rather low (R2=0.28).
The result of the scenarios assuming (i) cross-sectional differential perception fmax-CS and (ii) an integration along the stem with equal sensitivity are shown in Fig. 9. The correlation between growth response and the maximal longitudinal strains integrated along the stem (Fig. 9A) is very high (0.86, (Pr(>F)=0.0001). The set of points for this scenario was best fitted by a logarithmic curve (R2=0.75). Fig. 9B shows the plot of the growth response versus the maximal longitudinal stresses integrated along the stem. The correlation coefficient was very low (0.119) and did not differ significantly from 0 (Pr(>F)=0.66).
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Discussion |
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TopAbstractIntroductionMaterials and methodsResultsDiscussionAppendix 1: Models of...Appendix 2: Mathematical...References |
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The analysis of the mechanical fields induced by bending made it possible to estimate quantitatively the large spatial heterogeneities of mechanical field variables (curvatures, longitudinal stresses and strains) along the stem. These heterogeneities were due to bending itself, but also to the shape and material properties of the stem. Moreover, it was shown that these fields were not parallel due to tapering and to changes in the equivalent longitudinal Young modulus along the stem. A large amount of variation between stems was also clearly revealed for each of the mechanical variable under consideration, despite the standardized bending test (same relative deflection). When an apparent equal global stimulus is applied to different plants at the boundaries of the stem, distinct mechanical fields and hence distinct real stimulation can be generated by pure physics, due to the interactions with the structure of the stem. Additionally, such differences cannot be characterized by the global variables such as maximal force, or bending moment even though they reflect inter-stem variability (Coutand et al., 2000
), because they do not capture the relevant spatial integral for mechanosensing. For example, the variability in the maximal force in the test used in this study is directly linked to the variability of the deflection rigidity of the stem, which integrates the bending rigidities of the cross-sections along the stem (Moulia et al., 1994), but which is not the relevant ‘biomechanical’ integral for mechanosensing. The plant can hardly be considered as a ‘black-box’ if quantitative dose-responses are sought, and a physical analysis of its internal state, together with considerations about the sensing function and the spatial integration of the signal are necessary. Considering the spatial integrals, it has been seen that they necessarily embody some assumptions about the sensory function and its spatial distribution (domains of sensitivity). Using models based on a simple set of hypothetical biomechanical scenarios, it was found that the growth response was well explained by the integrals based on a mechanosensing of the longitudinal strain field all along the stem, and not by the maximal values of any field variable, nor by any of the integrations based on stresses. The conclusions here are that (i) the sensing capacity is distributed along the stem, (ii) the mechanosensing is local but the growth response is triggered by the sum of the local mechanosensing, (iii) longitudinal strains are the obvious best candidate for the sensed variable and (iv) a log response seems to work well.
Unfortunately, this data set did not allow different sensing capacities between the plant tissues to be distinguished, despite the fact that the various integral over the whole stem (AT-P-f0, Fig. 6A), the outer living ring (OR-f0, Fig. 7A) or the epidermis (EP-f0, Fig. 7B) yielded distinct and different dependence on diameters (different power exponent, involvement of different ‘internal’ diameters: Rxyl, Rpith). Although it would seem logical that only living tissues are competent for mecanosensing, limiting the domain of sensitivity to the outer living tissues or to the epidermis (as suggested by the results of Zandomeni and Schopfer, 1994), did not result in sufficient differences between the distributions of these integrals, which remained highly correlated. Therefore, it was not possible to assess previous arguments in the bibliography. Similar conclusions also hold for the shape of the local sensory function. The comparison between f0 and fmax-CS resulted in very little variation in the prediction of the growth response. The striking conclusion here is that the major part of the variability generated in this experiment is merely linked to the variation in the mechanical structure producing variability in strain field. This is not to mean that the present approach is not suitable for the further analysis of these aspects of tissue sensitivity distribution. But it would require further experimental analysis, using various relative deflections instead of one or using another type of plant (i.e. another tomato variety, or another species) which have different proportions of tissues than the cultivar VFN8, to attempt to break the correlations between the various integrals.
The situation for strain versus stresses is a little bit more complex. Many authors have assumed stresses to be the obvious perceived mechanical variables (Mattheck, 1991, 1995; Goodmann and Ennos, 1997 amongst many). In this study, what is clear is that strain-based integrals lead to very good predictions of the variation of the growth response whereas the integrals of stresses did not. Moreover, these estimates of longitudinal bending strains are well substantiated because only based on a set of assumptions (pure bending, small curvature) that were directly tested. The estimates of longitudinal strains are geometrical in essence and not dependent on the material heterogeneity of the cross-section (Moulia and Fournier, 1997). Lastly they involved only two direct measurements, curvature and radius, and therefore minimized errors.
However, these stress calculations were based on the equivalent stresses which are not equal to the ‘real stresses’ in each tissue. When considering two tissues (OR and xyl) the conclusion is unchanged. The only restriction which could be made is that the external living ring is itself a composite material made of four tissues: epidermis, collenchyma, parenchyma, and phloem. From the anatomical study, gradients of cell wall thickness and cell size have been found in epidermis and collenchyma so fields of stresses in epidermis and collenchyma probably do not parallel the field of strains and the conclusion on the sensed variable probably remains unchanged. However, neither gradients in cell wall thickness nor obvious differences between stems were observed in parenchyma and phloem (data not shown). Similar restrictions also apply to the cambium, which has been proposed as a possible site of perception for the mechanical control of secondary growth (Mattheck, 1991).
The cellular and ultrastructural scales are now considered. It should be remembered that the putative mechanisms for mechanosensing involve stretch-activated ion channels (Sachs, 1986; Cosgrove and Hedrich, 1991; Badot et al., 1992), or perception through cytoskeletal deformation (Ingber, 1998b). At the cellular scale, there is a heterogeneous structure again, with a stiff and insensitive cell wall in parallel with a much flexible and sensitive structure. Clearly most of the load is borne by the cell wall, and the plasmalemma will follow the straining of the cell wall. It is thus likely that at the cellular level again, cell deformation (as measured by cell wall strain) is going to be a better index of the local mechanical stimulus to the cell than stresses that will be dominated by cell wall stresses. These results are thus consistent with cytological data. Finally it is only at the level of the plasmalemma or the cytoskeleton itself, and assuming that sensitive and insensitive zones are serially connected (Sachs, 1986) that the stresses might be relevant. However, if the focus is put on the sensitive zones themselves then the deformation is again the relevant variable (e.g. the opening of an ionic channel).
The conclusion for this discussion on strain versus stress is that it is an intricate problem that requires detailed consideration and a clear definition of the scales. But whenever there is a parallel arrangement between stiff insensitive structures and flexible sensitive structures and spatial heterogeneity in stiffnesses, strains-based indices should be more relevant than their stress counterpart, as they do not incorporate the irrelevant effects of the stiffness of the insensitive parts. In consequence, they should also be useful at more scales than stress indices. Coming back to the tissue–whole stem scales, which are the ones this paper is focused on, it means that the bending stiffness changes along the stem are not relevant to the stimulus perception. For a given bending moment thus, a given cross-section would be less stimulated (submitted to less longitudinal bending strains) if it has high bending stiffness (i.e. large amount of lignified secondary tissues) than if it is more flexible.
The last point concerns the shape of the sensory function. Jaffe et al. found log shaped relations between global stimulus indices and the growth response for various mechanical stimulations, and argued that this probably reflected a log shaped (or a saturating) sensory function (Jaffe et al., 1980). Indeed, in the results of this study, a roughly log-shaped dependence of the growth response with a threshold was found, confirming the results of Jaffe et al. (Jaffe et al., 1980). However, it must be recalled that our integrals were based on linear (f0) or ‘concave’ (from f0 to fmax) sensory functions and that only the growth response r is log shaped. The log shape is thus more likely to result from the response function to the global integrated signal (r as defined in equation (14)) rather than from the local sensory function f as claimed by Jaffe et al. (Jaffe et al., 1980). In the authors' opinion, this demonstrates again the interest of a clear biomechanical modelling of the plant stimulation and response instead of a more global stimulus-response approach.
As a general conclusion, it is argued that a mechanical modelling of the stem during loading and a biomechanical modelling of the perception are central tools of elucidate the relation between external mechanical stimuli and whole plant responses. As stated by Paul Green, the usual experimental tactic or direct co-variation may not be completely tractable whenever integrated behaviour is studied (Green, 1996; Moulia, 2000). Intermediate formalization and modelling are required. The models in this study are simple and more detailed biomechanical studies involving a larger range of loading conditions, direct rheological studies and improved mechanical modelling would be interesting to confirm the analysis and extend it to a more accurate identification of the sensing tissues. Of particular interest would be to use this spatial biomechanical approach in combination with a spatial characterization of the first transduction steps (as detailed by Trewavas and Knight, 1994). This approach would uphold the hypothesis on the sensory function and make direct comparison of the spatial distributions of the mechanical variables and of the local primary reaction steps (for example, calcium ions delocalization: Thonat et al., 1993; Trewavas and Knight, 1994, or patterns of expression of TCH genes: Braam et al., 1997), presumably opening new avenues for the quantitative study of mecanoperceptive patterns in situ and their integration at the whole stem level.
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Appendix 1: Models of spatial integration of the mechanosensing |
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TopAbstractIntroductionMaterials and methodsResultsDiscussionAppendix 1: Models of...Appendix 2: Mathematical...References |
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A set of spatial integrals of the possible candidate variables for mechanosensing were calculated, under the previously defined set of hypothetical scenarios. In the following, the equations used to calculate these integrals numerically are given. The details of the demonstration of these formula are given in Appendix 2.
Notice that in this case bending leads to symmetrical distribution of strain and stresses from each side of the bending plane. The integrations were therefore processed on one half of the stem (except the central pith).
(1) Linear sensory function: f0 hypothesis
(1.1) f0 and a domain of sensitivity distributed through the whole stem (f0-AT-P hypothesis)
The computations for the three candidates of the sensed variable: strain, stress, curvature (noted 
, and C, respectively
- (a) Sum of longitudinal strains on the whole bent part
with
where H is the length of the bent part, Rext(h) and Rpith(h) are, respectively, the external and pith radius of an infinitesimal slide of beam and C(h) the curvature.
- (b) Sum of longitudinal stresses on the whole bent part
- (c) Sum of curvature along the whole bent part
The curvature of the central line is a one-dimension variable characterizing each infinitesimal cross-section. Moreover, the sum of the curvatures along the stem is by definition equal to the inclination at the stem base. This result has already been presented in the companion paper (Coutand et al., 2000) and did not explain the variability of the growth response.
- (b) Sum of longitudinal stresses on the whole bent part
(1.2) f0 and a domain of sensitivity limited to the outer living ring: OR
- (a) The sensed variable is assumed to be the longitudinal strain
- (b) The sensed variable is assumed to be the longitudinal stress
- (b) The sensed variable is assumed to be the longitudinal stress
(1.3) f0 and a domain of sensitivity limited to the epidermiss:EP
- (a) The sensed variable is assumed to be the longitudinal strain
- (b) The sensed variable is assumed to be the longitudinal stress
- (b) The sensed variable is assumed to be the longitudinal stress
(2) fmax hypothesis for the sensory function
(2.1)fmax-AT-P
In that case, Xmax corresponds to the maximal value of the field X within the whole stem (X being one of the three types of mechanical fields: strain, stress and curvature noted, , and C, respectively).
(2.2) fmax-CS
In that case, Xmax corresponds to the maximal value of the field X within the cross-section (X being either longitudinal strain or stress). Under fmax, the integral is computed as the sum of the maximal value of X within each cross-section for each putative candidate of sensed variable.
- (a) Longitudinal strain
- (b) Longitudinal stress
- (c) Curvature
It has no sense to compute the integral with the curvature under fmax-CS hypothesis because, the curvature of the central line characterizes the overall bending of the cross-section, and there is no basis for intra-cross-section perception of differences in curvature in these conditions.
- (b) Longitudinal stress
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Appendix 2: Mathematical calculation of the integrals |
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TopAbstractIntroductionMaterials and methodsResultsDiscussionAppendix 1: Models of...Appendix 2: Mathematical...References |
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Let us consider again the case of a cross-sectional slice of the stem, at position h and with infinitesimal thickness dh (see text). In bending, the rotation of the cross-section compared to initial position generates a cylindrical wedge (Fig. 10
). The volume of this wedge, dV, represents the sum over the cross-section of the displacements of the cross-section due to the straining of the slice. It also corresponds to the transverse integration of the strain field over the cross-section times the thickness of the slice (it can also be interpreted as the volume change of the upper half of the cylindrical stem slice, neglecting the transverse straining due to Poisson ratios). If the pith is now ignored (AT-P hypothesis) or the outer ring (OR), then the integral of strains over the pith or the OR area will have to be removed from dV (i.e. to consider the integral to the wedge volume restricted to the area of sensitivity).
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Given the local dV, it is now necessary to integrate along the stem. Due to the longitudinal heterogeneity of bending along the stem, these volume increments can vary along the stem dV(h). The calculation of the integrals over strain field under the f0-WS or f0-OR hypotheses is just the calculation of the volume of the wedge dV(h) and then its summation along the whole stem.
If longitudinal stresses are now considered, it is seen that they are equal to the longitudinal strain times the local Young's modulus. Thus the integrated perception under the f0-hypothesis applied to stresses were calculated by piecewise integrals of strains times local Young's moduli.
The cases of f0-EP and fmax-CS are slightly different. At the scale of the whole stem, the thickness of the epidermis can be neglected. Thus dV(h) under f0-EP is reduced to increment of the surface area of the wedge dS(h). And these surface increments are summed along the stem. In the case of fmax-CS, the maximal strain is limited to the most external location of the wedge and the perception p is equal to kxmaxxdh that is to kxdl. The integration along the stem thus corresponds to the calculation of the global increment of the outer line within the stem. These differences in the dimensions of the integrals explain the differences in the power of the radii found in the formulas.
There are various mathematical ways to compute such volume, area or line length increments and to sum them along the stem. Details of the classical methods can be found in many mathematical textbooks (such as Edwards and Penney, 1994 or Boas, 1983).
As an illustration, the calculus for the case of volume increments is detailed. Cylindrical coordinates are used in the following. Considering a truncated cylinder, which diameter equals R, located at distance h from the applied force, the angle of rotation of the slide around the Z axis is 
(for practical convenience the infinitesimal notation for the bending-rotation angle of the cross-section is not used here, and to avoid confusion, the symbol is also changed). The angle between the radius and the Y axis is . The counter-clockwise is set positive (Fig. 10).
(1) Computations of sum of mechanical variables on the whole stem or on the living ring
- (a) General case: computation of the volume of a wedge over a truncated cylinder
The volume of the wedge is calculated by sum of elementary triangular slices (Fig. 10A).
The area of a triangle (A) is given by:
Replacing dl and ry by their expression dl=ry()tan and ry()=Rcos it gives 
The volume of a triangular slice is then
The volume of the cylindrical wedge equals the sum of the triangular slices.
The calculation of the integral
is given by: 
Finally,
with tan=dl/r=constant.
and
and 
so 
Finally, it gives
Given dV(h) (transverse integral over the cross-section at position h), the overall change of volume along the stem can be obtained. In computation, the volume can be approximated numerically using Riemann's sum:
- (b) Sum of longitudinal strain and stresses under AT-P and OR hypotheses
That general formula was used to compute the integral of the longitudinal strains on:- (i) the whole stem: as the pith was neglected, R3 was replaced by (
) which gives: 
- (ii) the external living ring: R3 was replaced by
which gives: 
- (iii) The computations of the integrals of the longitudinal stresses derived directly from these two equations because of the linear elasticity of the stems which gives:
so the sum of longitudinal stresses on the whole stem is obtained by:
and the sum of longitudinal stresses on the external living ring is given by:
- (ii) the external living ring: R3 was replaced by
- (b) Sum of longitudinal strain and stresses under AT-P and OR hypotheses
(2) Computation of the sum of strain and stresses at the stem periphery (EP hypothesis)
The calculations of sum of longitudinal strain and stresses at the stem periphery derived from the general case of the computation of the external surface (Fig. 10A) of the wedge over a truncated cylinder (Fig. 10B).
- (a) General case
The surface
A equals the sum of infinitesimal surfaces (dA): dA=dlryd with dl=rytan and ry=Rcos (Fig. 10B)
Replacing dl and dr by their expressions gives:
As explained earlier, tan
=C(h)dS So,
Finally the external area of the wedge (corresponding to the area increment of the ‘epidermis’ of the cross-sectional slice during bending) is given by:
The overall sum of longitudinal strain at the stem periphery can thus be obtained by summing the area increment of the successive wedges along the stem:
In computation, A can be approximated by:
- (b) Sum of longitudinal strain and stresses at the stem periphery
- (i) The sum of the longitudinal strain under the f0 and EP hypotheses were obtained by replacing R2 by (
) which gives: 
- (ii) The sum of longitudinal stresses at the periphery of the stem was obtained directly from this formula by:
- (ii) The sum of longitudinal stresses at the periphery of the stem was obtained directly from this formula by:
- (b) Sum of longitudinal strain and stresses at the stem periphery
(3) Sum of the maximal longitudinal strain on the whole stem (fmax hypothesis and CS hypothesis)
The sum of the maximal longitudinal strain derived from the general case.
dlmax=Rexttan or as explained earlier tan=C(h)dh, which gives:
 |
LLmax(h)=RextC(h) so that the sum of the maximal longitudinal strain along the stem is given by:
 |
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Acknowledgments |
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This work was funded by INRA (Institut National de la Recherche Agronomique, France), department of Bioclimatologie and by the Conseil Général d'Auvergne, France. We greatly thank Dr Jean-Louis Julien (PIAF–University Blaise Pascal, Clermont-Ferrand, France) for reading the manuscript and Dr Jill Walcroft and Christopher Baraloto (University of Michigan) for helping with the English version.
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Notes |
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3 To whom correspondence should be addressed. Fax: +33 05 49 55 6068. E-mail: moulia{at}lusignan.inra.fr
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