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Mechanical model and parameters
Similarly to the existing studies on elastic reconguration mentioned in Section 1.2, we consider throughout this Chapter a cross ow over the entire structure. The ow is uniform, as the dependence of the stress on the wind velocity prole was shown to be small (Niklas & Spatz 2000). Also, only static loads are taken into account, and the where U is the free stream velocity, the uid density, D the local branch diameter and CD the drag coecient (de Langre 2008, Madin & Connolly 2006). The direction is assumed to be that of the ow velocity. In order to isolate the eect of the branched
geometry on the occurrence of breakage, the uid load is here computed on lea ess branches, and the in uence of leaves will be discussed in Section 2.5. This load is applied on the whole branched system, which is held by a perfect clamping at the base. The structure is brittle, as dened in Section 1.3. More precisely, the brittle behavior is introduced as follows: (i) the deformations are assumed to be negligible, so the stress state is computed on the initial conguration, without elastic recon guration; (ii) when increasing the ow velocity U, breakage occurs when and where the local skin stress reaches the critical yield stress, c; (iii) the broken branch is then removed, and this results in a new ow-induced stress state. Flow velocity may then be further increased until a new breaking event occurs.
In the absence of deformations, and because of the high slenderness of the system, we use a standard linear beam theory to derive the stress state, essentially the bending moment M. The maximum stress in the cross-section resulting from this bending moment is the skin stress, dened as = 32M=D3 (Niklas 1992, Gere & Timoshenko 1990).
By analogy with the non-dimensional uid-load introduced in Section 1.2 for the analysis of ow-induced elastic deformations, the dimensionless number to scale the uid eect on the structure is the Cauchy number, dened here as.
Flow-induced pruning of a walnut tree
The geometry of the branched system is expected to have a large in uence on the stress state and thus on the location and timing of breaking events. We therefore rst apply the procedure described above using the digitized geometry of an actual 20-yr-old walnut tree (Juglans Regia L.) described in Sinoquet et al. (1997) (Figure 2.1a). This tree is 7.9 m high, 18 cm in diameter at breast height (dbh); it has a sympodial branching pattern (Barthelemy & Caraglio 2007) and about eight orders of branching. Note that for this actual geometry, the dierent orientations of the branches with respect to the ow are taken into account; the eective uid velocity for the uid force is the velocity normal to the branch axis. The stress state under ow is computed using a standard nite element software (CASTEM v. 3M, Verpeaux et al. (1988)), and is presented in Figure 2.2a for four dierent branching paths.
We observe that the stress level is not uniform but shows a maximum located in the branches, which is consistent with the results of Niklas & Spatz (2000) (see Figure 1.10). Note that since the bending stress varies linearly with the uid-loading CY , one needs only to focus on the critical situation where = 1 is rst reached in the structure. In this tree, the criterion for breakage is satised rst in a branch and not in the trunk. This corresponds to the mechanism of branch breakage, as dened in Section 1.3. If the uid loading is further increased after removal of the broken parts, successive breaking events are observed, in a ow-induced pruning sequence: Figure 2.3a shows three states of the tree at increasing Cauchy number with branches progressively removed as they break o.
During the sequence of breakage, the bending moment at the base of the tree, mB, evolves signicantly with the Cauchy number, Figure 2.3b. Up to the rst breakage, the moment is proportional to the uid-loading CY (zone I in Figure 2.3b). Then, in a small range of load increase (zone II), all large branches are broken at an intermediate level, resulting in a signicant decrease of the bending moment. Breakage then continues but to a much smaller extent (zone III), while the moment increases almost linearly up to the value mB = 1 when the trunk breaks. Note that the benet of this sequence of breaking events is that the critical value of the base moment mB = 1 is reached only at Clast Y ‘ 10 instead of 1 if there was no branch breakage. This corresponds to more than a factor of 3 on the acceptable uid velocity. For instance, for a critical stress c = 30 MPa, which is the order of magnitude of maximum acceptable bending stresses measured in trees (Beismann et al. 2000, Lundstrom et al. 2008), the maximum sustainable uid velocity before trunk breakage is increased from U ‘ 20 m.s1 without branch breakage to U ‘ 70 m.s1 with branch breakage. To summarize, this set of computations clearly shows that branch breakage can occur prior to trunk breakage, and that the sequence of ow-induced pruning results in a signicant reduction in the load applied on the base of the tree, or equivalently, an increase in the sustainable uid velocity. To further analyze this process, we turn to a simple model in the next section.
Innite branched tree
To establish the relation between the parameters of the system and the ow-induced pruning process, we simplify the problem to its essential elements: the branched geometry and the slenderness of branches; we disregard now the eect of branch orientation relative to the ow. Similarly to Rodriguez et al. (2008), we consider rst an innitely 101 100 10 10 1 110010.
Table of contents :
1 Introduction
1.1 Flow eects on vegetation { A general overview
1.1.1 A stressful environment
1.1.2 Growth modication: thigmomorphogenesis
1.1.3 Flow-induced oscillations
1.2 Static response to externalow
1.2.1 Elastic reconguration
1.2.2 Load reduction by porosity eect: clumping
1.3 Flow-induced breakage: pruning
1.4 Models for the mechanical behavior of plants
1.5 Aim of the present work
2 Flow-induced pruning of branched systems
2.1 Mechanical model and parameters
2.2 Flow-induced pruning of a walnut tree
2.3 The ideal tree model
2.3.1 Innite branched tree
2.3.2 Finite branched tree
2.4 The slender cone model
2.4.1 Flow-induced stress
2.4.2 Sequence of breaking events
2.5 Discussion and conclusions
3 Combination of bending and pruning reconguration strategies
3.1 Model
3.1.1 Mechanical model
3.1.2 Non-dimensional parameters
3.2 Bending and pruning of a slender cone
3.2.1 Particular cases
3.2.2 The scaling of drag reduction for the limit strategies
3.2.3 Reconguration through bending and pruning
3.3 Generalization to tree-like geometries
3.3.1 Branching eect
3.3.2 Angle eect
3.4 Discussion and conclusions
4 Application: homogenization of tree-like structures underow
4.1 Introduction
4.2 Model construction
4.2.1 Derivation of the volume equations
4.2.2 Equations of the homogenized model
4.3 Model validation on ow-induced pruning
4.3.1 Mechanical analysis
4.3.2 Comparison with direct computations on idealized trees
4.4 Direct construction of the equivalent homogeneous domain
4.5 Discussion and conclusions
5 Conclusion
5.1 Main contributions
5.1.1 Static reconguration of plants
5.1.2 Continuous model for tree-like structures
5.2 Perspectives
5.2.1 Experimental validation
5.2.2 Extension of the models
5.2.3 Survival strategy: biological and biomimetic applications
A Ideal tree model and equivalent beam
A.1 Finite ideal tree model
A.2 Stress derivation in nite branched tree model
A.3 Equivalent slender beam model
B Derivation of the beam equations
B.1 General equations
B.2 Non-dimensional equations
B.2.1 Reference parameters and Cauchy number
B.2.2 Particular case of low deformations
B.3 Role of the truncation in the slender beam model
B.3.1 Reference length scale for non-dimensional equations
B.3.2 Inuence of the truncation at nite critical strain
C Additional material to Chapter 3
C.1 Scaling of the maximum acceptable Cauchy number
C.2 Evolution of the gain in Cauchy number
C.3 Scaling of drag reduction for the cone bundle in the pruning limit
D Homogenization: technical points
E Publication
