Alessandro Fraccica

Enrique Romero

Thierry Fourcaud

Large cell triaxial tests of a partially saturated soil with vegetation

The use of vegetation roots as a nature-based solution against landslides and erosion requires the definition of sample preparation protocols and adoption of equipment that allows testing representative elementary volumes of the whole soil-root system. For this purpose, large cell triaxial compression tests were carried out on fallow and vegetated samples at different degrees of saturation. Samples were prepared by static compaction of a silty sand and seeded with Cynodon dactylon. The hydraulic state during plants growth was controlled and reproduced on bare soil samples. After isotropic compressions, the shearing phase was carried out at very low confining stresses (i.e., below 50 kPa). Tests were deemed to be comparable by assessing the normalised volume of roots with respect to soil, after shearing. For a given confining stress, soil samples with higher matric suction exhibited higher shear strength, furtherly increased by roots. The stress-strain behaviour observed in the vegetated soil systematically changed, when comparing tests at low and high matric suction values, due to the different mechanisms of vegetation reinforcement depending on the hydraulic state at the soil-root interface. The results were successfully interpreted within a failure criterion and skeleton stress framework for partially saturated soils, considering soil suction, degree of saturation, soil microstructure and the normalised volume of roots.

Introduction and background

Nature-based solutions, like the use of vegetation as reinforcement technique in slope stability or soil erosion problems are being commonly used by practitioners in natural areas and find increasing interest, given the actions against climate change and in favour of sustainable development. A fundamental step prior to important applications and the potential standardisation of such techniques is the quantification of the reinforcement and its correlation to given plant morphomechanical traits. In the literature, plant reinforcement on soils has been extensively evaluated through direct shear and root pull-out tests [1-6, among others], while the effects of: other stress paths that may lead to soil failure [7][START_REF] Potts | Finite Element Analysis in Geotechnical Engineering: Theory[END_REF][START_REF] Cui | [END_REF], hydraulic states and volumetric deformations upon shearing on soil reinforcement are still poorly investigated [10][11][12][13][START_REF]Fraccica Experimental Study and Numerical Modelling of Soil-Roots Hydro-Mechanical Interactions[END_REF]. Given that vegetation strongly affects the soil-atmosphere interaction and that natural hazards are often triggered by phenomena that induce changes in soil water content [START_REF] Alonso | [END_REF][16][17][18][19], knowledge and laws from soil mechanics for partially saturated soils are essential to make considerations, tests and interpretation of the results.

Finally, in the framework of the geotechnical laboratory testing and especially for vegetated soils, the choice of a representative volume element has a * Corresponding author: alessandro.fraccica@isprambiente.it prominent role on the significance and repeatability of tests. Morphological plant traits such the normalised volume of roots or root lengths with respect to soil volume should be assessed to both assess the representativeness of the in-situ conditions of the volume element and interpret the geotechnical results within a unified framework [20,21].

Considering all the above-mentioned, large cell triaxial compression tests will be carried out on fallow and vegetated samples at different degrees of saturation in this study. A protocol for soil compaction and root growth within soil samples will be defined and a shear strength criterion for partially saturated soils will be implemented with plant morpho-mechanical variables.

Material and experimental protocol 2.1 Soil used

Silty sand retrieved close the Llobregat river's delta in Barcelona was used. Its natural granulometric composition is: 41.2% of gravel, 28% of sand, 25.4% of silt and 5.4% of clay [START_REF]Fraccica Experimental Study and Numerical Modelling of Soil-Roots Hydro-Mechanical Interactions[END_REF], it is a low-plasticity soil (plasticity index between 9.6 and 13.5%) with a density of solid particles is 2.65 Mg/m 3 [START_REF]Fraccica Experimental Study and Numerical Modelling of Soil-Roots Hydro-Mechanical Interactions[END_REF]. The soil has been sieved through ASTM #4 (4.76 mm), to prepare the testing samples. The bare and vegetated soil water retention curves were fitted using the model of [START_REF] Romero | Retention curves of deformable clays[END_REF], based on a modified form of the van Genuchten expression [START_REF] Van Genuchten | [END_REF]:

Sr = C(s)(1/(1+(αs) n )) m 1 C(s)=1-(ln(1+s/a)/ln(2)) 2
Where s is matric suction, Sr the degree of saturation, C(s) a function used to adjust the higher suction range of the curve. The air entry value is approximately linked to 1/α, the slope of the curve to n and the residual degree of saturation to m. The maximum suction corresponding to the residual degree of saturation is indicated by parameter a. The calibrated values of these parameters are provided in Table 1.

Sample compaction and test preparation

Soil samples were prepared by static compaction in PVC moulds, in eight layers each of 50 mm and diameter 200 mm. Scarifications were created with a spatula on the surface of each layer to ensure continuity with the upper layer. The total height of each sample was of 400 mm. Soil gravimetric water content and dry density at compaction were w = 15% and ρd = 1.6 Mg/m 3 , respectively, as indicated by point A in Fig. 1, resulting in a degree of saturation Sr = 0.61, and a matric suction s = 40 kPa. The total vertical stress recorded on compaction was 100 kPa. The compaction state was chosen to be at dry of the optimum water content to create good physiological conditions for the plant to grow: a balanced proportion of air and water volumes in the voids and average macro-pore sizes consistent with root diameters. After compaction, the samples were seeded with Cynodon dactylon in 2-mm deep holes realized at a fixed spacing between each other (30 mm, Fig. 1) and at a given seeding density (i.e., 34 g/m 2 ). Finally, soil samples have been wetted to reach the hydraulic state indicated by point B in Fig. 1 (w = 21%, Sr = 0.84, matric suction s = 1 kPa) to induce plant growth, which lasted around eight months. Plant growth was deemed to be completed when observing roots at the bottom side of each specimen. A ceramic tip tensiometer (T5x UMS, München, Germany) allowed monitoring soil matric suction throughout the period of root development at 50 mm below the surface of the samples (Fig. 1).

Fig. 1 Compaction plane and hydraulic states at compaction (A) and during root growth in samples (B)

Before being tested, soil samples were placed in a controlled-atmosphere room (T = 20±1°C and RH = 50±5%) to dry up the desired gravimetric content (evaluated by oven check [START_REF] Astm D | Standard Test Methods for Laboratory Determination of Water (Moisture) Content of Soil and Rock by Mass www[END_REF], between points B and C in Fig. 1), then sealed in plastic sheet for 24 hours in darkness to equalise matric suction. At the end of this process, samples were extruded out of the PVC moulds by a piston moving at a constant rate (i.e., 1 mm/min) and pushed directly into the rubber membrane, whose internal surface was lubricated by oil to minimise the friction at the interface with soil. Matric and total suction, and the associated gravimetric water contents, were assessed before and after each test by a ceramic tip tensiometer, a chilled mirror dew point hygrometer (WP4, Decagon Devices, Pullman, WA, USA), and oven check, respectively, in four points of the samples. Two diametrically opposite measuring points were selected at the bottom and two at the upper surfaces of each sample, at 70 mm from its centre. Matric suction was not monitored continuously during the shearing phase.

Large cell triaxial equipment

The triaxial equipment used was developed in UPC geotechnical laboratory [START_REF] Alvarado De Menéndez | Comportamiento hidro-mecánico de agregados gruesos[END_REF] to allow the testing of large samples (height = 400 mm and diameter = 200 mm). Confinement is applied by humid air (RH = 100%) pressure, checked by a pressure transducer, while axial and radial displacements in the samples are recorded by one vertical and twelve horizontal displacement transducers (LVDTs). Horizontal transducers are placed in contact with the sample's membrane through a system of regulating nuts, rods, o-rings and springs that crosses the steel cell at twelve positions, along two orthogonal directions and at three different heights of the sample (90-180-270 mm). See Fig. 2 and [START_REF] Alvarado De Menéndez | Comportamiento hidro-mecánico de agregados gruesos[END_REF] for more details. Radial deformations were mathematically inferred as described in [START_REF]Fraccica Experimental Study and Numerical Modelling of Soil-Roots Hydro-Mechanical Interactions[END_REF]. Tests were carried out on samples at constant water content conditions. Samples were firstly subjected to an isotropic compression, until a volumetric strain rate lower than 0.1%/day was observed. Then, triaxial drained compressions were carried out at a vertical displacement rate of 0.016 mm/min.

Roots volume and void ratio assessment

Void ratio was assessed at the end of each test by means of paraffin tests [START_REF] Astm D | Standard Test Method fo Laboratory Determination of Density (Unit Weigth) of Soil Specimens[END_REF] on two points of each sample. Soil volumes inferred by paraffin method were corrected from the root volume contribution, according to the mathematical procedure described in [START_REF]Fraccica Experimental Study and Numerical Modelling of Soil-Roots Hydro-Mechanical Interactions[END_REF]. Roots were retrieved after washing the soil samples and their volumes were inferred by image analysis and pycnometry. The root volume ratio, Rv (i.e., the normalised volume of roots by the unit volume of soil) observed in the tested samples ranged between 0.012 and 0.018.

Results

Fig. 3 presents an example of stress-strain (deviatoric stress q vs axial strain Ꜫax) and volumetric strain (Ꜫvol) responses of vegetated and bare samples at low confinement and low matric suction. Root volume ratio Rv observed in the samples is indicated in the legend. Soil strength increase in vegetated soil was systematically observed after large deformations (i.e., around axial strain Ꜫax = 4%) and without the possibility of reaching a stress plateau within the displacement range allowed by the equipment. Bare soil reached maximum strengths without observing a post-peak decrease. Volume deformations started to be higher in vegetated samples consistently with the trend described for the stress-strain curves.

Fig. 3 Comparison of triaxial compression results on bare (black lines) and vegetated soil (green lines) at low matric suction.

An increase in strength and stiffness was generally observed in samples tested at drier hydraulic states. However, a different stress-strain response was observed in drier vegetated samples with respect to what was observed in Fig. 3. Indeed, drier vegetated samples exhibited more pronounced reinforcement and volume compression at small strains (i.e., around Ꜫax = 1%), while almost reaching asymptotically the maximum stress level (Fig. 4). Void ratios at the end of the tests are reported in Table 2, jointly with the calculated average degree of saturation, the average of suction measurements and the root volume ratios. Despite higher void ratios were observed by [START_REF]Fraccica Experimental Study and Numerical Modelling of Soil-Roots Hydro-Mechanical Interactions[END_REF] along drying trajectories on unconfined vegetated samples of the same soil and roots, the void ratios observed after triaxial compressions were slightly smaller than those of bare samples at similar states and confinement, consistently with the higher volumetric deformations observed in the samples with roots. Root volume ratios were quite similar in all the vegetated samples, allowing a good confidence on the comparability of the results.

Reinforcement interpretation

The failure criterion adopted to interpret the results was the Mohr-Coulomb for partially saturated soils based on the following constitutive stress formulation (σ', see e.g. where p' and p'f are the mean and ultimate mean skeleton stresses, p the total mean stress, ua the air pressure, Sr,eff is the effective degree of saturation (i.e. the degree of saturation in the soil macropores and evaluated as Sr elevated by α, the model parameter [29]). σ1 and σ3 are the vertical and confining total stresses, q is the deviator stress and the slope of the failure envelope is Mc= 6 sinφ'/(3-sinφ'), where φ' is the friction angle, and c' the cohesion. This last term was implemented with the root reinforcement term, modified from the formulation presented by [31], in which γ is the model parameter, σtr is the roots tensile strength (negative according to geotechnical conventional signs and whose value of -2.35 MPa was measured through tensile tests on single roots by [START_REF]Fraccica Experimental Study and Numerical Modelling of Soil-Roots Hydro-Mechanical Interactions[END_REF] for Cynodon dactylon), and Rv is the root volume ratio. A calibration of two sets (for vegetated and bare soil) of α, MC and γ using the Microsoft Excel's solver was carried out to minimise the squared error between qf evaluated experimentally (as the ultimate stress at the end of test) and qf formulated in the constitutive framework (equations 3 to 6). The values obtained are presented in Table 3. The two failure envelopes and the points at failure (p'f, qf) for each test, are plotted in Fig. 5. As it is possible to see, minor effects of vegetation on soil friction angle were observed, while roots contributed to increase the cohesion term. Friction values were in line with those observed in the same soil subjected to other stress paths [START_REF]Fraccica Experimental Study and Numerical Modelling of Soil-Roots Hydro-Mechanical Interactions[END_REF]30]. The calibrated γ parameter was far below the 1.2 evaluated by [31] and inferred from backanalysis on landslide vegetated scarps (where roots have been potentially subjected to larger deformations than in triaxial compressions). The α parameter is in line with those presented by [29] for similar kinds of soil. Vegetated soil presented a slightly lower α. This decrease is linked to the changes in the soil microstructure that roots generate during their growth: macropores enhancement due to fissuring and grain displacements and partial clogging of micro-pores due to complex chemo-mechanical interactions. This change in soil structure is affecting the water retention properties and the volume change behaviour observed upon shearing.

Although the failure stress states of vegetated soil were successfully represented by a failure criterion, the stress-strain responses of soil with roots were strongly affected by the hydraulic states (Fig. 3 and Fig. 4): at low values of suction, the small confinement at the soilroot interaction made the roots stretch and align before generating some reinforcement in the soil due to the friction generated by fibres pull-out. Similar trends were observed in plant-and fibre-reinforced soils [5,[32][33][34]. On the contrary, shrinkage upon drying and higher suction values contributed to increase soil confinement around roots, and in turn impeded differential displacements at the soil-root interface. Hence the elastic properties and the tensile strength of roots were better exploited in reinforcing the soil. Fig. 6 reports the secant Young Modulus values calculated along the triaxial compression stress-strain curves, at axial strain Ꜫax = 1% as function of the confining skeleton stress σ'3. Vegetation roots slightly contributed to increase soil stiffness too. This observation could be in part explained because Cynodon dactylon has roots spreading in all the directions within the samples [START_REF]Fraccica Experimental Study and Numerical Modelling of Soil-Roots Hydro-Mechanical Interactions[END_REF]: tensile forces developed within those inclined towards the outer lateral surface, due to soil radial strains. In turn these forces had a component in the radial direction of the samples, generating an additional confining effect which made the vegetated soil response stiffer.

Conclusions

This study presented laboratory results and vegetation reinforcement interpretation on a partially saturated silty sand tested under triaxial compression in a large cell equipment.

A unique protocol was followed for the preparation of all the samples: soil was compacted at the same hydraulic state and dry density, then wetted under unconfined conditions to let the plants grow and finally dried to the desired matric suctions. Bare soil underwent the same hydro-mechanical history for the sake of comparison. Tests on vegetated soils were carried out after 8 months of plant growing periods. After each test, the normalised volume of roots by the unit volume of soil was assessed. Roots affected soil mechanical response in different ways depending on the suction level: close to saturation roots more likely were stretched and pulled out, generating friction forces at the soil interface, and inducing a constant increase in soil shear stresses. When soil was drier, the higher embedment of roots caused by soil shrinkage and higher suction at the soil-roots interface impeded large roots displacements within the matrix, hence exploiting their tensile strength and generating a root reinforcement mechanism typical of fibre breakage.

Soil matric/total suction was measured before and after the tests while the degree of saturation evaluated through water content and void ratio checks at the end of each test.

The results were successfully interpreted within a failure criterion and skeleton stress framework for partially saturated soils, considering soil suction, degree of saturation, and soil microstructure. The good result in representing vegetated samples at different confinements by a unique failure criterion is indicative that a good repeatability was ensured by the large size of the samples, given that they had very similar root volumes ratios. Hence the failure criterion was implemented to predict the effect of this morphological trait of roots and their tensile strength on soil shear strength increase.

As vegetated and fallow soil failure envelopes were traced considering the ultimate strength recorded in each test, it was observed that they differed essentially in the cohesion term. However, this shear strength parameter has been found to be strongly linked to soil shear deformation and in turn to the stretching of root fibres.

Fig. 2

 2 Fig. 2 Triaxial equipment with a vegetated sample and schematic operating principle of horizontal LVDTs of the equipment.

Fig. 4

 4 Fig. 4 Comparison of triaxial compression results on bare (black lines) and vegetated soil (green lines) at high matric suction.

  [START_REF] Vanapalli | [END_REF][28][29]): p' = p -ua+Sr,eff•s ; Sr,eff = Sr α 3 q = σ1 -σ3 4 qf = 6c'•cosφ'/(3-sinφ') + Mc•p'f 5 c' = c'bare + c'roots ; c'roots = -γσtrRv 6

Fig. 5 Fig. 6

 56 Fig. 5 Failure envelopes and qf / p'f end-of-test states for bare and vegetated soil

Table 1 .

 1 Fitting parameters of the retention curves used in this study, from[START_REF]Fraccica Experimental Study and Numerical Modelling of Soil-Roots Hydro-Mechanical Interactions[END_REF]

	Soil	α (MPa -1 )	n (-)	m (-)	a (MPa)
	Bare	0.53	0.28	1.77	200
	Vegetated	0.15	0.19	2.38	200

Table 2

 2 Variables related to the hydraulic state and root volume ratios for vegetated (V) and bare (B) samples.

				Averag	Confin
		Final	Average	e	ing	Root
	#	void ratio,	degree of saturatio	matric/t otal	total stress,	volum e ratio,
		e (-)	n, Sr (-)	suction,	σ3	Rv (-)
				s (kPa)	(kPa)
	B2 0.567	0.301	2000	50	-
	B3 0.533	0.693	8	48	-
	B4 0.589	0.527	95	40	-
	B6 0.590	0.546	49	30	-
	B8 0.571	0.619	38	50	-
	B9 0.578	0.544	135	18	-
	B10 0.606	0.593	50	21	-
	B12 0.614	0.462	210	10	-
	B13 0.600	0.457	250	30	-
	B15 0.588	0.375	1780	25	-
	V1 0.570	0.536	90	11	0.016
	V2 0.658	0.242	3800	30	0.014
	V3 0.657	0.213	4000	5	0.013
	V4 0.512	0.779	12	10	0.014
	V5 0.549	0.311	2900	45	0.014
	V6 0.505	0.697	25	46	0.018
	V7 0.481	0.938	1	30	0.012
	V8 0.462	0.887	5	28	0.015

Table 3

 3 Calibrated parameters for the failure criterion framework used.

		Bare soil	Vegetated soil
	α (-)	3.67	3.20
	MC (-)	1.42	1.45
	γ (-)	-	0.30
	c' (kPa)	0	10.3
	φ' (°)	35.1	35.8

E3S Web of Conferences 382, 05005 (2023) https://doi.org/10.1051/e3sconf/202338205005 UNSAT 2023

The authors wish to acknowledge the support of the European Commission via the Marie Skłodowska-Curie Innovative Training Networks (ITN-ETN) project TERRE 'Training Engineers and Researchers to Rethink geotechnical Engineering for a low carbon future' (H2020-MSCA-ITN-2015-675762).


B Thibaut

S Caré

E Maurin

Oak beams in medieval frameworks: constraints and advantages for restoration

 Wood is always a top material for beams  40 million of slender young oak trees are available in French forest  Round-wood squared at the minimum keeps the high resistance of trunks  Using green wood is the fastest and most energy-saving method  High tech modern techniques can be used for the best sorting of trees

Technical know-how about timber frame is old in human societies and was improved during the Middle ages, but scientific knowledge about wood as an anisotropic material is very recent. The background in mechanics for such material has been developed in early 20 th century and observation tools for material structures at the nanometre scale appeared in late 20 th century. Wood can be considered as an archetypal high-tech material using polymer-with-additive technology combined to nanostructured fibre-composite within a honeycomb-like microstructured material. This explains why nature solution for tree beams is very efficient in terms of bending resistance per weight.

The mechanical behaviour of wooden beams is highly sensitive to the orientation of the main direction of anisotropy (grain direction) relative to bending forces. Natural beams in trees are built by a kind of 3D printing process so that grain orientation at beam periphery is always optimum. Mechanical bending performance of these beams (round-wood), in spite of branching, is equivalent to the performance of standard clear-wood specimens documented in databases. Squaring at the minimum this round-wood should be done while keeping grain angle closest to zero-degree value, which is performed by specific axe-hewn log squaring. The result Introduction

In April 2019 Notre-Dame de Paris was partially destroyed by fire. In particular, all wooden beams of the nave and choir timber frames dating from the Middle ages as well as spire timber frame dating from nineteenth century have been burnt. Following the destruction of Notre-Dame frameworks, three alternative solutions for restoration (identical, idealized or freeinspired) presented by the chief architects were discussed in a meeting of the National architecture and heritage commission, in march 2021. The commission vote in favour of identical reconstruction, chosen for its respect of Venice chart, the ability to attest and transfer carpenter's know-how and the information about history of the cathedral building. Final decision was taken by the highest French authority. Indeed, roughly half of these beams were still serving their required mechanical function since 800 years, half since 160 years. Moreover, wood material may be considered, as the sole material, apart from stone, to allow so long durability in building construction. But using material issued from forest trees raised questions about what is "identical" in the reconstruction process, what do we know about oak trees in the Middle ages forest and about artisanal knowledge and know-how 800 years ago in France? Here, we mainly focus on the selection and use of the long oak beams for the reconstruction of the Middle ages frame of Notre-Dame.

In the section 2, we propose a state of art on the evolution of French forest evolution, the technological knowledge in civil engineering and the scientific knowledge on wood. Section 3 deals with the mechanical characteristics of squared timber beams. In section 4, the use of oak species is justified for cathedral timber frame and section 5 analyses the impact of green wood use for the construction phase. Finally, the use of today's techniques and knowledge is quickly discussed in section 6 in order to assist the restoration of Notre-Dame.

2.

Major evolution in the French forest area and on technological and scientific knowledge

2.1.French forest evolution

We have a rather good estimation of forest area within hexagonal France territory (excluding overseas departments) during the last millennium [START_REF] Birot | Forests and people: what co-evolutions[END_REF][START_REF] Escurat | Histoire des forêts françaises[END_REF]. In the 12 th century, when the Middle ages timber frame of Notre-Dame was built, total forest area was very similar to 21 st century situation: around 16 million hectares (Fig. 1). But between these two periods there was first a strong decline down to 7 million hectares at the beginning of the 19 th century, when Viollet-le-Duc built the spire. Colbert issued a government decree on forest management in 1669 in order to restore wood production imposing severe restriction to tree felling and higher harvest age (> 150 years) for timber use.

Fig 1. French forest area (million ha) and population (million people) in the last 15000 years (From Escurrat 1995 [2])

The diocese of Paris was the owner of oak forests where Maurice de Sully, bishop of Paris at the beginning of Notre-Dame construction could harvest some 1300 trees converted into beams for the choir and nave timber frame. From dendrochronological studies [START_REF] Chevrier | La charpente de la cathédrale Notre-Dame de Paris à travers la dendrochronologie, mémoire de DEA en histoire de l'art et archéologie médiévale[END_REF][START_REF] Epaud | La Charpente De La Cathédrale De Bourges -De La Forêt Au Chantier[END_REF] on Notre-Dame beams and other cathedrals built in the same period, it is well documented that selected oak trees were young (50 -100 years old) and slender trees (ratio between total height and diameter above 70), with a small diameter at breast height (25 -40 cm). This was a deliberate decision because there were plenty of old (200 years old) big (60 cm diameter or more) oak trees available at that period. But it is not clear whether these trees come from young high forests or old coppices. The long, straight, and slender beams used in the frame were nearly crude trunks with pith at the centre, just squared, keeping the local small curves if needed.

For Viollet-le-Duc, for the construction of the spire timber frame, the situation was totally different. Oak trees available for harvest should be old enough by forest legislation and they were big trees coming from mature high forests. The long, perfectly straight slender beams were sawn.

Today French forest is always very rich in oak trees, both in high forests and coppices which are sometimes rather old. Looking in the data from the national French forest inventory (IFN) there are 44 million oak trees for the Middle ages category needed for the restoration of Notre-Dame (Table 1). There is a large place for a good choice but the question is: what was a good choice by the carpenter in the Middle ages forest?

Technological knowledge in civil engineering

Woody parts of a tree like trunks are slender beams resisting to compression and bending efforts, just like beams in a timber frame. In fact, inspiration from nature was an obvious and easy solution. Timber frame using round wood (eventually squared) is a very old and universal building method before the Middle ages period of Notre-Dame [START_REF] Wood | Strength and related properties of wood poles[END_REF][START_REF] Barnard | Roundwood -Its mechanical and physical characteristics[END_REF] and the transfer of technological know-how was done by carpenters themselves. There were surely many innovations in the field of connexions for example, and an increase of knowledge about construction and wood material during the very rich period of cathedral building in the Middle ages. But it is not well documented, apart from wooden heritage where scientific investigation could try to identify the main innovations that occurs.

The scientific revolution, with documented technological processes and basic knowledge (mechanics of materials and structure for example) comes in the late 18 th century [START_REF] Navier | L'application de la mécanique à l'établissement des constructions et des machines[END_REF][START_REF] Chevandier | Mémoire sur les propriétés mécaniques du bois[END_REF]. When Viollet-le-Duc built the spire of Notre-Dame, basics in mechanics and civil engineering (including timber frames) were taught by Navier [START_REF] Navier | L'application de la mécanique à l'établissement des constructions et des machines[END_REF]. More or less, the same basics are always behind timber engineering today.

In Europe, since 2004, standard rules for timber building conception are in use under the title: Eurocode 5: Design of timber structures [START_REF]Eurocode 5: Design of timber structures -Part 1-1: General -Common rules and rules for buildings [Authority: The European Union Per Regulation 305/2011[END_REF]. They are based on limit state design considering both the deformation of the structure which shall remain within appropriate limits and the resistance to rupture under expected loads. Moreover, timber beams should be sorted by lower limit in stiffness and strength. Unusual forms of construction like Notre-Dame are not covered and expert consideration is required.

Scientific knowledge on wood

Timber was studied as a structural material in civil engineering at the same time as other materials, from the beginning of 19 th century and was an important structural material in the early ages of airplane manufacturing. In particular, military aeronautics research was very active in wood mechanics in Europe and the USA [START_REF] Hankinson | Investigation of crushing strength of spruce at varying angles of grain[END_REF][START_REF] Monnin | Essais physiques, statiques et dynamiques des bois[END_REF][START_REF] Markwardt | Strength and related properties of woods grown in the United States[END_REF].

In the late 18 th and early 19 th the making of wood within the living tree was a controversy subject between botanists [START_REF] Du | La physique des arbres -Première partie[END_REF] until the discovery of cambium as a thin layer of stem cells producing both xylem and phloem [START_REF] Nägeli | Beiträge zur wissenschaftlichen botanic[END_REF]. At the same time technical progress in optical microscopy prove that wood is a cellular material similar to honeycomb with very small cell diameter (20 -40 µm).

It was only during the 20 th century and the invention of electronic microscopy that the fibre cell wall can be observed as a multi-layered fibre-composite at the nanometre level. X-ray diffraction made it possible to measure cellulose nano-fibre dimensions and orientation. In the second part of 20 th century, chemical composition of wood as a mixture of bio-polymers with a whole cocktail of additives (small organic molecules) was elucidated with the help of mass spectroscopy and nuclear magnetic resonance spectroscopy. In the same period theoretical tools in mechanical anisotropy which is typical of both honeycomb and fibre composites was developed [START_REF] Ricci | Méthodes de calcul différentiel absolu et leurs applications[END_REF] and applied to wood for airplane manufacturing [START_REF] Carrington | Elastic constant of spruce as influenced by moisture[END_REF].

"Squared timber" as a beam for building

Round-wood is a portion of trunk where connected branches are cut close to the trunk. It is approximately a long straight cylinder with some tiny curves from place to place. It was used directly as beam in many domestic building in the past. But round wood connexions are difficult to achieve [START_REF] Barnard | Roundwood -Its mechanical and physical characteristics[END_REF] and squaring, i.e. cutting operation on the round wood in order to obtain plane parallel faces is a solution. We will use here the term "squared timber" for axe-hewn logs including the pith (bois de brin in French). This should be done with respect of Richards recommendations [START_REF] Richards | A treatise on the construction and operation of wood-working machines[END_REF]: "Cutting wood consists of two distinct operations: cross severing the fibre and splitting the fibre parallel with its lamination" (Fig. 2). This technique ensures the perfect parallelism of wood grain direction with force direction at beam periphery, keeping the mechanical efficiency of the trunk: these squared timber beams have globally the same mechanical properties as than round-wood beams. But tiny local curves are still present, needing dunnage pieces for precise adjustments in framework building process.

Fig. 2 Squaring an oak log by severing fibre and splitting (Photo Florian Carpentier)

Severing the fibre is done first every 50 cm by two axe cutting Splitting a long 50 cm chips parallel to wood grain in then achieved

Clear-wood and "squared timber" mechanical properties

A tree trunk is a beam subjected to high flexure forces under wind action. This beam is a succession of clear parts and connecting zones, with branch departures. In order to resist wind forces, connecting zones should be as resistant as clear parts. This is the result of tree growth by a kind of additive construction, optimizing fibres orientation in every part of the tree [START_REF] Thibaut | Mechanics of wood and trees: some new highlights for an old story[END_REF][START_REF] Thibaut | Three-dimensional printing, muscles and skeleton: mechanical functions of living wood[END_REF]. The global mechanical properties of a sound trunk (without rot occurring sometimes within the trunk, often in connection zones with dead branches): flexure stiffness and strength, is equivalent to clear wood parts properties [START_REF] Wood | Strength and related properties of wood poles[END_REF]. Clear-wood specimens used in standards for measuring wood mechanical properties are issued from these trunk clear parts, keeping the parallel orientation of fibres in the rods [START_REF] Hankinson | Investigation of crushing strength of spruce at varying angles of grain[END_REF].

Both for tree and for engineering purpose, minimizing weight for a given stiffness (elastic deformation under given forces) is important for bending beams [START_REF] Ashby | Materials Selection in Mechanical Design -Fourth Edition[END_REF]. This can be measured by the efficiency index EI = MOE 1/2 /D where MOE is the modulus of elasticity (MOE) in the beam direction and D the material density.

Two basic indicators control the elastic modulus (MOE) in the fibre direction for wood: its density (kg/m 3 ) signature of the honeycomb structure, and its specific modulus (SM) signature of the elastic behaviour of the cell wall. SM is the square value of sound speed in both cell wall and wood itself. It can be measured by acoustic methods [START_REF] Brémaud | Characterization and categorization of the diversity in vibrational properties between 98 wood types[END_REF]. The elastic modulus is the product of the two values D and SM: 𝑀𝑂𝐸 = 𝐷. 𝑆𝑀. The efficiency index can be expressed through these two independent factors: 𝐸𝐼 = √(𝑆𝑀/𝐷).

Wood is a good material for music instrument partly because the speed of sound (4000 to 6000 m/s) is as high as for steel blade in a vibraphone. It is also the building material with the smallest density thus it is always efficient as bending beam. The best efficiency will be found for higher values of SM together with lower density, with is often the case for softwoods as spruce or fir.

Moreover, for a building material, stress at rupture in bending (MOR) and maximum strain (MS) before rupture are also important criteria. In structural mechanics, MOE is the key for calculation of local stresses and strains in the elastic domain, MOR or MS are "red" signals for overloading.

Fig. 3 Standard bending test for a clear wood rod (length 380 mm, section 20 mm x 20 mm)

The results on bending tests for clear-wood show that MS is high (Fig. 3). In a building, there are always maximum allowable beam deflexions for comfort reasons. The maximum deflection of a clear-wood beam as a trunk is very high (see deflexions of tall trees in strong winds) and not allowable in a building framework (or for a floor). Thus for round-wood beams, the use of MOE values is often sufficient in mechanical calculation in order to guarantee good building performance, provided that there are no big defects as internal rot zones within the trunk.

3.2.Eurocode 5 and timber mechanical properties

Due to its structure, wood is a highly anisotropic material [START_REF] Hankinson | Investigation of crushing strength of spruce at varying angles of grain[END_REF][START_REF] Markwardt | Strength and related properties of woods grown in the United States[END_REF]: mechanical properties as stiffness (MOE) or strength (MOR) are ten times higher along the grain direction than along the transversal directions (Fig. 4). It should be noted that, apart from the first few degrees from grain direction (angle of fibre direction with beam direction), the mechanical properties drop off very quickly in the first 15 degrees.

Rupture

Before rupture

During bending test Eurocode 5 [START_REF]Eurocode 5: Design of timber structures -Part 1-1: General -Common rules and rules for buildings [Authority: The European Union Per Regulation 305/2011[END_REF] is basically based on dry sawn timber. Sawing is the more common method of trunk deconstruction in order to produce straight beams. Unlike round-wood splitting, used to produce clear-wood boards for high value wooden products as dowel for barrels or soundboards for musical instrument, basic sawing process cannot keep the integrity of branch connexions producing knots in the board and leads to local grain angle that can be rather high.

The dispersion of MOR values for sawn wood is very large and probability approaches are required by Eurocode 5. A very large quantity (1000) of representative sawn-boards from same species distributed on the whole supply zone are tested in bending until rupture and the MOR distribution is characterized by the 5th MOR percentile, which means that only 5% of the boards have a lower MOR. 5% is considered as an acceptable risk in building rules. Using the same probability approach for clear-wood and sawn-wood from the same trees shows that the 5 th percentile can be nearly 3 times lower for sawn-wood (Table 2). Clear wood specimen and boards were sawn in the same trees In brackets: respectively, number of rod and of boards MOR: rupture resistance; MOE: modulus of elasticity Both knots and local grain angle above 5° are mechanical defects for the beam and will lower rigidity (MOE), resistance (MOR) and maximum strain (MS) at the same time. This is why the relationship between MOR and MOE for sawn-wood is strongly different from that for clearwood coming from the same oak tree (Fig. 5).

Fig. 5 Relationship between MOE and MOR for oak

Results coming from Faydi PhD thesis [START_REF] Faydi | Classement pour la résistance mécanique du chêne par méthodes vibratoires et par mesure des orientations des fibres[END_REF] Clear wood: 139 rods coming from the 139 boards Regressions for clear wood: percentile 5% (lower curve); percentile 95% (higher curve)

Eurocode 5 values for sawn-wood cannot be used for trunk timber [START_REF] Ranta-Maunus | Round small-timber for construction[END_REF]. Moreover, it seems unrealistic to test until bending rupture thousand trees of the same provenance. The relationship between MOE and MOR is similar (power law) for round wood (Fig. 6) and for clear wood (Fig. 5). Considering that clear wood and round wood properties, are the same, results on clear wood tests can be used to build the 5 th lower percentile curve for MOR from known values of MOE, and this curve can be used to predict round wood MOR from its measured MOE.

Fig. 6 Relationship between MOE and MOR for pine round wood

MOR vs MOE

Results coming from Barnard PhD thesis [START_REF] Barnard | Roundwood -Its mechanical and physical characteristics[END_REF] on 100 scots pine trees (poles) Regressions for pine round wood: percentile 5% (lower curve); percentile 95% (higher curve)

A detailed 3D description of Middle ages framework before the fire is available for building information modelling (BIM) and simulation of the mechanical behaviour of the framework. A 3D description of each squared oak trunk together with values for their mean elastic modulus is enough to make the calculation and verify the maximum deflections or stresses in these beams.

An efficient procedure can be as follow: i) possible occurrence of big defects of the trunk is verified by man expertise or by X-ray tomography, ii) beam geometry is digitized by photogrammetry or Lidar, iii) mean modulus of elasticity for the beam is directly assessed by a bending test on the work site, iv) use a power law derived from clear wood tests for an estimation of a 5 th percentile value of MOR . Each squared trunk beam gets an identifier (bar code or QR code) for all successive operations.

Oak timber choice

For today building, timber frame is often used but the rule is spruce dry sawn timber. Oak alone accounts for one fourth of the total standing tree volume in the French forest. Behind, fir and spruce account for 15%, pines for 13% and beech for 10%. Oak is abundant in nearly all French regions and was the natural frame material until the 18 th century. Spruce and fir, abundant in the mountain regions, were used in these regions.

Mechanical properties

We have data on the mechanical properties for oak, spruce and fir species in the USA (Table 3). Spruce is a better choice for bending, when looking at the stiffness index, due both to a lower density and a higher specific modulus (Table 3). Oak have better transverse properties useful for good connexions, but this is not enough to explain permanent oak choice in French cathedrals.

Oak durability

The biggest difference between oak and spruce is the durability against insects and fungal attack. Oak wood is rich in additives of tannin family which can stop the insect risk and slow down the rotting process. Furthermore, if the timber is kept under shelter (roof) from the rain, it will be air dry and fungi can no longer be active in wood eating. Oak beams are everlasting under these conditions which is not so true for spruce and fir. As oak was very abundant at Cathedral periphery and Bishop want everlasting monuments, oak was a good choice.

We know today that rot resistance (measured by the mass loss during a standard test against fungi) of oak heartwood can be very different from tree to tree [START_REF] Guilley | Decay resistance against Coriolus versicolor in Sessile oak (Quercus petraea Liebl.): analysis of the between-tree variability and correlations with extractives, tree growth and other basic wood properties[END_REF], from poorly durable (a minority of trees) to very durable (a majority of trees). Measures on 86 oak trees (160 years old at a mean) [START_REF] Guilley | Decay resistance against Coriolus versicolor in Sessile oak (Quercus petraea Liebl.): analysis of the between-tree variability and correlations with extractives, tree growth and other basic wood properties[END_REF] Nb spec.: number of oak specimen tested; ML: mass loss due to fungal attack E. Tannin: indirect measure of ellagitannins content Very durable: % of specimens in the highest class of natural durability in European standard Poorly durable: % of specimens in the two (out of 5) lowest classes of natural durability First to fourth quarter: 4 successive portions along a ray from sapwood limit to pith.

In the experiments, wood age varies from around 40 to around 140 years old from first to fourth quarter (Table 4). At the same time, ellagitannins content decreases strongly from high durability to poor durability. Durability decrease for the older wood rings, near the pith is due to chemical ageing of the tannins. Globally, young trees have a better mean resistance to rot risk. However, it can be useful to assess the rot resistance of the oak beams, knowing that a small percentage of trees have a low level of durability (and we do not have the empirical knowledge of Middle ages carpenters regarding the durability of oak provenance). Today, wood oak is still widely used for wine or alcohol barrels. Tannin content is also important for barrel makers and they have developed a tool based on near infra-red spectroscopy (NIRS) in order to control their clear wood staves [START_REF] Michel | Influence of Wood Barrels Classified by NIRS on the Ellagitannin Content/Composition and on the Organoleptic Properties of Wine[END_REF]. NIRS technique proves to be an efficient tool in order to get a good estimation of ellagitannins in oak wood, hence to grade oak timber for rot resistance.

Tree choice

In order to find 10 to 12 meters long beams, total height of the trees should be above 20 meters. For small diameter beams, the slenderness of the tree (total height divided by diameter at breast height) should be high (more than 70). This is not unusual (Table 1) but most of the oak trees have lower slenderness (below 50). Slender trees have higher values for sawn wood MOR and MOE (Fig. 7) and a lower taper which can be a reason for such choice by Middle ages carpenters.

Fig. 7 Tree slenderness and timber properties for French oak sawn wood [START_REF] Lanvin | Caractérisation du chêne sessile et pédonculé de France en vue de son utilisation en structure[END_REF] Values in the bracket are the maximum values (1 in relative value) Relative value is the value divided by the maximum Beams equivalent to Middle ages choice are the lowest diameter DBH: diameter of the standing tree at breast height

Green round wood use

Using green wood is interesting for two main reasons: i) quick installation of beams after tree falling, ii) no energy input for wood drying.

Shrinkage of green round-wood

Green wood will dry during years following cathedral building, until moisture content equilibrium (ratio between mass of water and wood anhydrous mass) with the surrounding climate (standard dry wood has a moisture content around 12%). During drying there will be changes in timber dimension (shrinkage) with different values for each direction (Fig. 8). Tangential shrinkage is roughly twice the radial shrinkage while longitudinal shrinkage is generally considered as negligible compared to the shrinkage of the two other directions. Hence surface shrinkage is equal to volumetric shrinkage. The value for longitudinal shrinkage in long oak beams, around 15 mm from green to dry state should be taken into account as the thermal dilatations for steel or concrete beams. The great difference between radial and tangential shrinkage leads to a geometrical impossibility for round wood (Fig. 9). If at is the tangential shrinkage, ar the radial shrinkage, R the log radius and C log perimeter at green state (C = 2..R), after shrinkage, the new radius is R1=R.(1-ar) while the new perimeter is C1=C.(1-at). But this perimeter is smaller than C2=2..R1 and the perimeter length difference is C2-C1=2..R.(at-ar). The relative (to initial perimeter) length difference is the value of the difference between tangential and radial shrinkage. In the case of oak (Fig. 8) drying from green state to 12% moisture content the values of shrinkage are 3% in radial direction, 6% in tangential direction and 9% in surface. The relative perimeter length difference will be 3% which is much higher than radial crushing strain. So, radial cracks from pith to periphery will occurs during shrinkage from green to dry wood (Fig. 9). Fortunately, such longitudinal cracks do not influence the rigidity or the strength of the beam issued from straight grained trees [START_REF] Ranta-Maunus | Round small-timber for construction[END_REF][START_REF] Arriaga | The effect of fissures on the strength of structural timber[END_REF]. There is a 6% reduction of section area (surface shrinkage) meaning a 6% reduction in compression for a column. The second moment of inertia varies as the fourth power of the radius; this means a 12% reduction. Adding the void in the crack (3%), the total reduction in stiffness will be around 16% due to shrinkage until 12% moisture content. For the flexure strength the reduction will be a little lower (around 12%).

MC (%) 60%

Change in mechanical properties between green and dry wood

There is always a very strong proportional relationship between green and air dry properties with coefficient of determination R² around 80% [START_REF] Thibaut | Tree growth forces and wood properties[END_REF]. We can use data on the mechanical properties for oak species in the US oaks at green and dry state [START_REF]Wood handbook-Wood as an engineering material[END_REF] in order to have a good estimate of the changes in mechanical properties from green to dry (12% moisture content) wood (Table 5).

Green wood Anhydrous wood

Except for transverse tensile strength (and of course density), all the mechanical properties are higher for dry wood, and for each case, the increase is greater than the reduction in geometry due to shrinkage: -16% for inertia compared to +30% in MOE; -6% for section area compared to + 94% in compressive strength; -12% in inertia contribution compared to + 67% in MOR.

The reduction in density brings also a positive contribution by a decrease of the beam dead weight.

Globally the riskiest period is at the beginning of full loading for the green wood frame and high beam deflexion should be a rapid signal of misconception. This may explain how carpenter find a very good fitting of the frame without calculation.

Long term creep

Wood as a viscoelastic material will present a long term increasing of the initial elastic deformation called creep. The creep is known to be higher for green wood and even higher for a constant stress associated to moisture content changes. The coupling action between a permanent load and moisture content variation during the whole life of the frame (including initial drying), called mechanosorptive creep [START_REF] Thibaut | Mechanics of wood and trees: some new highlights for an old story[END_REF] should lead to a more important beam deflection. The subject is still a matter of active basic research and there are not yet results for oak beams. Information can be derived from the existing heritage for oak beams. Besides large variability in observed old beam deflections, they do not seem to be so large.

6.

How new scientific knowledge may help for the restoration of Notre-Dame?

It seems quite clear that Middle ages carpenters had a keen knowledge of the timber they can harvest from well-known forests. They were probably aware of both mechanical, rot and insect resistance of the chosen timbers. We do not have this experience-based knowledge, but we have new scientific knowledge and modern tools available for sorting and grading.

We have, for Notre-Dame de Paris a full 3D description of the existing timber frame giving geometric parameters for sorting: length, diameter, slenderness, straightness. These parameters are registered in the national French forest inventory (https://inventaire-forestier.ign.fr/) helping the search for forest plots with the desired oak trees. Terrestrial or airborne LIDAR, if necessary can be used to help the final tree choice within the forest plot [START_REF] Windrim | Detection, Segmentation, and Model Fitting of Individual Tree Stems from Airborne Laser Scanning of Forests Using Deep Learning[END_REF].

For the physical, mechanical and chemical pertinent parameters, the weight of wood (density), the sound of wood (specific modulus) and the taste of wood (ellagitannins content) are often use by artisans. We can use modern techniques as X-Ray tomography [START_REF] Jacquin | X-ray microdensitometry of wood: A review of existing principles and devices[END_REF], vibration or ultrasound techniques [START_REF] Olsson | Performance of grading methods based on fibre orientation and axial resonance frequency applied to Norway spruce (Picea abies L.), Douglas fir (Pseudotsuga menziesii (Mirb.) Franco) and European oak (Quercus petraea (Matt.) Liebl. /Quercus robur L.)[END_REF][START_REF] Newton | Acoustic-Based Non-Destructive Estimation of Wood -Quality Attributes within Standing Red Pine Trees[END_REF] and NIRS tools [START_REF] Michel | Influence of Wood Barrels Classified by NIRS on the Ellagitannin Content/Composition and on the Organoleptic Properties of Wine[END_REF] so that we can have a good prediction of properties for modern computer aided building conception.

Moreover, rigidity and resistance of a wooden frame is also determined by connexions' resistance. As shown in [START_REF] Fromont | Relevé des charpentes médiévales de la cathédrale Notre-Dame de Paris, Mémoire de DSA Architecture et Patrimoine de l'Ecole de Chaillot[END_REF] in 2014, some disorders have been observed in the frame of Notre-Dame with restorations probably undertaken during the Middle ages. The origin of these disorders are not well understood at the scale of the frame structure. It has been supposed that the local mechanical properties were weak, maybe due to high water content and fungi attack.

Innovations in pure wooden assembly were very active during cathedral building period [START_REF] Epaud | La Charpente De La Cathédrale De Bourges -De La Forêt Au Chantier[END_REF][START_REF] Calame | Pour une étude de la charpente rurale en France : l'exemple de l'Oise[END_REF]. Mechanical behaviour of oak assembly during and after shrinkage is very poorly documented apart from the observations on cultural heritage [START_REF] Maurin | Villeréal -24, Lot et Garonne (Nouvelle Aquitaine)[END_REF]. Furthermore, many structural damages in wooden frame heritage come from connexions either due to high long-term moisture levels or to mechanical disorder [START_REF] Epaud | La Charpente De La Cathédrale De Bourges -De La Forêt Au Chantier[END_REF] issuing sometime from shrinkage differences between wooden pieces (and possibly due to longitudinal shrinkage [START_REF] Maurin | Villeréal -24, Lot et Garonne (Nouvelle Aquitaine)[END_REF]). Anyway, the Middle ages best know-how, documented by heritage, is well maintained by today carpenters keeping alive the Middle ages know-how as with the association Carpenters without borders [37] (Fig. 10).

Studies combining experimental and numerical approaches are clearly needed in order to provide the basis for computer aided wood framing construction using green wood, in accordance with European standards and with the same efficiency as 800 years ago. This is part of the joint efforts piloted by CNRS task force in charge of coordinating the various research efforts and compiling the available data (https://news.cnrs.fr/articles/notre-dame-researchsteps-in).

Conclusion

The choice of identical reconstruction of the timber frame of Notre-Dame de Paris, in accordance with Venice charter [START_REF]International Charter for the Conservation and Restoration of Monuments and Sites (Venice Charter)[END_REF][START_REF] Icomos | Principles for the conservation of wooden built heritage[END_REF], is relevant, considering the high level of description of the burned frame, the maintenance of Middle ages carpentry competence and the level of today scientific knowledge on timber properties. The frame reconstruction allows to conserve a witness of the architectural story (the cathedrals grow during de middle age). This worksite induced the conservation of trades and, it is also a way to rediscover "buried" human knowledge and nature building know-how. Wood and tree beams are common materials for building since the very beginning of humanity but the most recent scientific investigation attests their modernity as complex multi-scaled structural material highly competitive for light stiff structural members as in light airplane manufacturing [START_REF] Laa | Aircraft wood information[END_REF].

Using tree-length oak material just squared at minimum and rapidly placed in the timber frame at green state does not only keep tree part performance. It is the highest possible environmentfriendly choice: no pollution, very small energy use, full carbon storage as long or even longer, in the case of old cathedrals, as the forest carbon storage, easy low-cost recycling. It seems worth to re-start from Middle ages timber building knowledge in order to build innovations for better use of the huge availability and diversity and of oak trees for tomorrow material needs.

Fig. 4

 4 Fig. 4 Evolution of relative property with grain angle. Graph featuring Hankinson formula [10] 1: property for 0° grain angle, MOR: modulus of elasticity, MOR: bending strength

Fig. 8

 8 Fig.8Schematic diagram for oak shrinkage from green wood to anhydrous wood (MC=0%) FSP: fibre saturation point; MC: wood moisture content (60% in the living tree at a mean for oak trees); VS: total volumetric shrinkage, TS: total tangential shrinkage, RS: total radial shrinkage; LS: total longitudinal shrinkage,

Fig. 9

 9 Fig. 9 Shrinkage of a log portion from green to anhydrous state (oak) X-Ray transmission image (darker means lower density), Images from F. Mothe, INRAE Nancy)

Fig. 10 A

 10 Fig. 10 A copy of N° 7 frame, from Notre-Dame timber frame, built on site on the occasion of French cultural heritage day (September 2020) Photography: S. Caré

Table 1

 1 Number of available oak trees in French forest

	Diameter class Total trees	Height	Slenderness
	cm	M trees	m	-
	22.5 to 27.5	14	24 to 31	90 to 130

Table 2

 2 Mechanical characteristics for mountain pine wood

	Clear wood (183)	MOR (MPa)	MOE (GPa)
	Mean	91.5	11.9
	Percentile 5%	70.3	8.4
	Percentile 95%	116.9	15.5
	Boards (945)	MOR (MPa)	MOE (GPa)
	Mean	48.0	9.1
	Percentile 5%	24.4	6.1
	Percentile 95%	73.8	12.5

Table 3

 3 Mean mechanical properties for oak, spruce and fir.

	Clear wood property (dry)	Oak	Spruce + Fir Oak/Spruce+fir
	Density (Kg/m 3)	670	380	1.76
	MOR (MPa)	99.8	68.2	1.46
	MOE (GPa)	12.0	10.4	1.16
	MOR/MOE	8.31	6.58	1.26
	Specific modulus (10 6 m²/s²)	18.1	27.1	0.67
	Efficiency index	5.2	8.5	0.61
	Compressive strength parallel (MPa)	49.4	37.8	1.31
	Comp. strength perpendicular (MPa)	8.1	3.3	2.42
	Shear strength (MPa)	13.1	7.5	1.74
	Transverse tensile strength (MPa)	5.6	2.1	2.62

Data are mean values for different species of oak and different species of spruce + fir in the US Forest

[START_REF]Wood handbook-Wood as an engineering material[END_REF]

Table 4

 4 Rot resistance of oak wood and ellagitannins content

	Position	Nb spec. ML E. Tannin Very durable Poorly durable
	Fourth quarter (close to the pith) 156	16% 69	44.2%	9.0%
	Third quarter	139	13% 129	54.7%	4.3%
	Second quarter	158	10% 220	65.8%	1.9%
	First quarter (close to sapwood) 161	6% 414	84.5%	1.2%

Table 5

 5 Changes in oak clear-wood mechanical properties from green to dry

	Clear wood property (oak)	Green	Dry	D/G
	Density (kg/m 3)	944	670	0.71
	MOR (MPa)	59.6	99.8	1.67
	MOE (GPa)	9.2	12.0	1.30
	MOR/MOE	6.5	8.3	1.29
	Specific modulus (10 6 m²/s²)	15.6	18.1	1.16
	Compressive strength parallel (MPa)	25.5	49.4	1.94
	Comp. strength perpendicular (MPa)	5.1	8.1	1.58
	Shear strength (MPa)	9.0	13.1	1.46
	Transverse tensile strength (MPa)	5.1	5.6	1.08
	Mean values for different American oaks.
	Green moisture content is 60%, dry moisture content is 12%
	D/G is the ratio between dry and wood state for each property


Précisions sur les contours des droits réels I -Les droits réels dans l'espace

A -La nature des droits réels clarifiée B -La portée des droits réels clarifiée II -Les droits réels dans le temps A -La liberté contractuelle contrainte par la perpétuité B -La clarification de la durée des droits réels de jouissance spéciale L'autonomie du droit réel de jouissance spéciale affirmée par l'arrêt Maison Poésie du 31 octobre 2012 1 a intensifié le vent libéral qui soufflait sur les droits réels. L'engouement suscité par ce droit réel innommé a contribué à son expansion au point de semer le trouble entre les différents droits réels. Cet arrêt de la troisième chambre civile en date du 6 juin 2019 établit une certaine clarification tant par le rappel des limites propres au régime de la servitude, qu'en établissant une restriction certaine à l'emploi du doit réel de jouissance spéciale.

Il était question en l'espèce de la vente d'une propriété surplombée par une excroissance de l'immeuble voisin appartenant au vendeur. Une clause réservant à ce dernier un droit exclusif de jouissance de cet étage avait été prévue à l'acte de vente par la création d'une servitude puisqu'il s'agissait pour le fonds de l'acquéreur de fournir une charge au service de celui du vendeur. Cette clause précisait que ce dernier devait ainsi assurer seul les frais d'entretien du premier étage de l'immeuble, que « les présentes [étaient] stipulées à titre de servitude, sans aucune copropriété, ni indivision » et que seuls le vendeur et ses ayants droit pourraient mettre fin à cette servitude.

Néanmoins, l'immeuble a, par la suite, été vendu et soumis au statut de copropriété. Les acquéreurs, subissant les travaux des nouveaux propriétaires, ont alors demandé la nullité de cette servitude ainsi que la démolition de ladite terrasse. Le tribunal ainsi que la cour d'appel ont fait droit à leur demande, considérant que « l'intention des parties était de constituer une servitude réservant la jouissance de surplomb au seul profit du vendeur, laquelle servitude devait être annulée car elle vidait de sa substance le droit de propriété de l'acquéreur sur ce surplomb ». La configuration des lieux, en l'espèce, confirme cette analyse des juges puisque les acquéreurs n'avaient pas accès à ce premier étage.

Seul le premier moyen du pourvoi nous intéressera puisqu'il concerne la qualification et la définition des droits réels. Ce moyen se décline en une multiplicité de branches parfois redondantes faisant ressortir trois arguments essentiels : la première branche reprend les fondements de l'arrêt Maison Poésie en estimant que la cour d'appel aurait méconnu la volonté des parties, violant ainsi les articles 544 et 1134 (ancien) du Code civil2 . Le deuxième argument reprochait aux juges du fond, sur le fondement de l'article 12 du Code de procédure civile, de ne pas avoir restitué aux faits leur exacte qualification. Enfin, le troisième argument, développé sous différentes branches, reprochait à la cour d'appel d'anéantir la clause au lieu de l'interpréter au-delà du vocabulaire employé par les parties 3 . La Cour de cassation confirme le raisonnement des juges du fond et rejette le pourvoi en retenant que d'une part « les parties avaient entendu créer de manière expresse et non équivoque un service à la charge du fonds acquis (…) au profit du fonds voisin initialement conservé par [le vendeur] », d'autre part, « la convention interdisait, compte tenu de la configuration des lieux, toute jouissance de la pièce objet de la clause par son propriétaire ».

Si la liberté contractuelle permet de moduler les droits réels, elle doit toutefois respecter la destination prévue à chacun de ces droits par le Code civil ainsi que par l'ordre public des biens. La Cour de cassation clarifie la nature des droits réels en vue de les caractériser dans l'espace (I), mais aussi de les réguler dans le temps (II).

I -Les droits réels dans l'espace

L'arrêt Caquelard du 13 février 1834 4 a mis fin au mythe du numerus clausus en permettant de « créer ou de modifier des droits réels par convention, voire même par un acte unilatéral, puisqu'aucun texte n'enlève ce pouvoir créateur aux particuliers ». Si la liberté contractuelle permet donc d'établir et de créer des droits réels, le démembrement se trouve limité par la destination et la définition propres à chacun de ces droits mais aussi par l'ordre public des biens. Ainsi, tant la servitude que le droit réel de jouissance spéciale n'ont pas vocation à priver totalement la jouissance du propriétaire du fonds servant. Les limites imposées à ces droits réels permettent de saisir plus clairement leur nature respective. Le refus de requalifier la clause litigieuse semble ainsi établir deux constats majeurs : le premier consiste à faire primer les droits réels nommés (A), le second vise à protéger le droit de propriété en limitant la portée des droits réels, il s'agit de faire primer la propriété sur la liberté contractuelle (B).

A -La nature des droits réels clarifiée La volonté certaine d'une servitude. L'autonomie du droit réel de jouissance spéciale a élargi considérablement la création des droits réels. Néanmoins, aucune définition claire n'est fournie par le juge, permettant ainsi de l'envisager de manière protéiforme. Cette potentialité confirme l'importance de la liberté contractuelle dans la constitution de droits réels, vue avec méfiance par certains auteurs5 , mais assurant ainsi une réelle adaptabilité aux situations éloignées de celles régies par les droits réels nommés6 . Cependant, en l'espèce, le silence gardé par la Haute juridiction sur la demande de requalification de cette servitude en droit réel de jouissance spéciale invite à s'interroger non seulement sur la possibilité de requalifier un droit réel établi par voie conventionnelle, mais aussi sur la portée que la Haute juridiction entend donner aux droits réels innommés. Dès la première partie de la solution, les juges confirment qu'il était question d'une qualification « expresse et non équivoque » entre les parties. Acceptant la qualification de la servitude, elle exclut d'office toute subsidiarité du droit réel de jouissance spéciale. Néanmoins, il n'est pas question d'y entrevoir un quelconque irrespect de l'article 12 du Code de procédure civile7 , mais plutôt une application respectueuse des droits réels nommés, évitant ainsi d'appréhender le droit réel de jouissance spéciale comme une « bouée de secours » des droits réels frappés de nullité.

C'est en cela que le respect de l'ordre public s'impose : il s'agit pour le juge d'éviter que des volontés contractuelles contreviennent à des dispositions légales, par intérêt personnel8 . Le droit réel de jouissance spéciale serait alors une voie pertinente lorsque le Code civil ne prévoit pas la réalité envisagée.

Le droit réel de jouissance spéciale, ersatz des lacunes du Code civil. L'étendue de ce droit s'avère particulièrement délicate et soulève bien des questionnements depuis l'arrêt Maison Poésie. L'avocat général souligne, dans ses conclusions, la nécessité de reconnaître le caractère spécial à ce droit réel innommé de manière substantielle9 . Pourtant, l'arrêt Maison Poésie semait un certain trouble en ce qu'il n'octroyait ni plus, ni moins qu'un droit d'usage général à la fondation10 , ce qui l'éloigne de tout caractère spécial et l'assimile davantage à l'usufruit. L'arrêt à commenter marquerait-il un frein à la reconnaissance de ce droit réel innommé ? Il semblerait qu'une réponse positive soit ici envisageable, invitant à appréhender ce droit comme un succédané de droit réel nommé. Ainsi, lorsque la volonté des parties désigne, à juste titre, un droit réel nommé, aucune substitution ne semble envisageable, sous peine de s'apparenter à une fraude ou de contrevenir à l'ordre public des biens. Il semblerait que la priorité soit à l'absence de confusion, ce qui n'était pas le cas de jurisprudence antérieure. L'arrêt ERDF témoignait d'une possible assimilation entre la servitude et le réel de jouissance spéciale. Dans cet arrêt, la reconnaissance d'une charge, de fonds servant et dominant invitait à établir une servitude de surplomb plutôt qu'un droit réel de jouissance spéciale comme l'a retenu la Haute juridiction. Le présent arrêt met fin à cette potentielle confusion en clarifiant la nature de chacun des droits réels. Cette distinction permet d'envisager le droit réel de jouissance spéciale en dehors des cas prévus par le Code civil. En matière de copropriété, il se pourrait que le droit réel de jouissance spéciale soit plus approprié que la servitude, qui se trouve parfois reconnue en l'absence de fonds dominant11 . Suivant l'esprit de l'arrêt Caquelard12 , le droit réel de jouissance spéciale constituerait une voie efficace pour organiser la copropriété, en ce qu'il permet une adaptabilité certaine à la spécificité du domaine. Pourquoi distordre la définition de la servitude prévue par le Code civil, alors qu'un droit réel innommé peut constituer une meilleure appréhension de l'organisation foncière de la copropriété ? Néanmoins, si les droits réels innommés permettent de saisir des réalités non prévues par le Code civil, ils ne doivent pour autant pas entraver ni l'ordre public des biens, ni le droit de propriété.

B -La portée des droits réels clarifiée

Le droit de propriété privilégiée. En l'espèce, la vérification de la juste qualification par la Haute juridiction confirme que le surplomb constituait une charge supportée par le fonds servant de l'acquéreur au bénéfice du fonds dominant du vendeur 13 . Pourtant, sur le fond, la Cour de cassation réaffirme sa jurisprudence selon laquelle une « servitude ne peut être constituée par un droit exclusif interdisant au propriétaire du fonds servant toute jouissance de sa propriété » 14 . Puisqu'une servitude n'est pas la totale du droit de jouissance pour le fonds servant, ce précepte permet de conserver une cohérence au droit de propriété afin qu'il n'en reste pas que le titre. Une servitude de surplomb présente alors le risque de priver de jouissance perpétuellement le propriétaire du fonds servant, ce qui constitue d'une certaine manière, une « expropriation » 15 . C'est pourquoi l'avocat général souligne qu' « un tel effacement perpétuel du droit de propriété, un tel dépouillement irréversible du propriétaire de la maîtrise de la valeur d'usage de son bien nous semblent être un résultat proscrit dans le cadre de la libre contractualisation des droits réels sui generis » 16 . Cette solution semble s'imposer afin d'enrayer l'établissement de conventions contraires à l'utilisation normale de son droit de propriété. En matière de servitude, la protection du droit de propriété a été récemment confirmée par la Cour de cassation. Elle précise que la renonciation à une servitude conventionnelle n'interdit pas la reconnaissance d'une servitude légale 17 : l'intérêt foncier prime lorsque l'enclave n'est pas volontairement réalisée par le propriétaire.

Les limites au démembrement. Un des apports majeurs de cet arrêt réside dans la limite établie quant à l'emploi du droit réel de jouissance spéciale. Une analogie doit être ici établie, justifiant le refus de requalifier la clause prévue par les parties. En effet, si la servitude est en l'espèce frappée de nullité, c'est parce qu'elle « vide de toute substance » la propriété de l'acquéreur. La requalification en droit réel spécial de jouissance aurait ici provoqué le même effet puisque la configuration des lieux ne permettrait pas à l'acquéreur de bénéficier de cette terrasse. Ainsi, quelle que soit l'organisation des droits sur une propriété l'on ne peut réserver l'exclusivité des jouissances 13 Ces caractéristiques confirment bien l'application de l'article 637 du Code civil définissant la servitude comme « une charge imposée sur un héritage pour l'usage et l'utilité d'un héritage appartenant à un autre propriétaire ». 14 Cass. 3 e civ., 24 civ., 12 mars 2008, n°07-10.164, Bull. civ. III, n°47 : « une privation perpétuelle autorisée de la jouissance d'une fraction d'un bien, fût-elle minime, est une expropriation ». L'auteur critiquait ici la solution rendue par la troisième chambre civile reconnaissant la validité de la servitude de surplomb alors que l'acquéreur était privé de « toute jouissance de la pièce ». 16 Conclusions de B. STURLESE, précité. 17 Cass. 3 e civ., 24 octobre 2019, n°18-20.119 : « l'acquéreur d'une parcelle enclavée ne peut se voir opposer la renonciation d'un précédent propriétaire de la servitude légale de passage conventionnellement aménagée ». à un bénéficiaire unique. La formulation de la solution condamne l'exclusion de la jouissance du droit de propriété, néanmoins, aucune précision n'est fournie quant à l'acceptation d'un tel aménagement des droits réels dans le temps. Est-ce à dire que l'exclusion de toute jouissance de sa propriété pour une durée déterminée pourrait-être tolérée ? Rien n'en est moins sûr 18 .

II -Les droits réels dans le temps

Clarifier la nature des droits réels permet d'appréhender la durée envisageable du droit réel de jouissance spéciale. S'il est présenté comme n'ayant pas vocation à être perpétuel (A), le respect de son caractère spécial permettrait-il d'envisager un tel caractère perpétuel (B) ?

A -La liberté contractuelle contrainte par la perpétuité Une requalification limitée par l'ordre public. L'avocat général précise dans ses conclusions que c'est notamment le caractère perpétuel du droit constitué entre les parties qui annihile tout intérêt de requalification de la clause : « une limite incontournable à la libre création de droits réels de jouissance innommés doit tenir à l'impossibilité de priver perpétuellement de toutes les utilités liées à la valeur d'usage du fonds grevé » 19 . L'argument la jurisprudence de la Cour de cassation qui, jusqu'alors déterminait la durée du droit réel de jouissance spéciale. Ainsi, dans l'affaire ERDF, face à l'absence de durée conventionnelle prévue par les parties, les juges ont alors appliqué les mêmes conditions prévues par la réglementation de l'usufruit 20 . Ce qui signifie qu'au mieux, si le bénéficiaire est une personne physique, son droit sera viager 21 ; s'il s'agit d'une personne morale la limite trentenaire s'applique en vertu de l'article 625 du Code civil. La question propre à la durée de ce droit réel innommé n'est donc pas nouvelle. Pourtant, elle n'est ici même pas mentionnée par la Cour de cassation. Si les juges précisent ainsi la définition du droit réel de jouissance spéciale, il convient de s'interroger sur sa durée. 18 Néanmoins, des interrogations persistent. Si le démembrement n'a pas vocation à anéantir les jouissances attachées au fonds, le droit de superficie interpelle puisque le tréfoncier peut voir la jouissance de sa propriété réduite à peau de chagrin par le superficiaire. Serait-ce d'ailleurs le seul droit permettant de telles limitations du droit de propriété ? 19 conception reprend l'article 609 du projet de réforme du droit des biens établissant le droit réel de jouissance « de la même manière que l'usufruit », en soulignant son caractère spécial, contrairement à l'usufruit qui se veut général. La création de ce type de jouissance spéciale apparaît particulièrement opportune dans la gestion familiale du patrimoine, lorsque le démembrement classiquement prévu par le Code civil ne correspondra pas aux attentes des parties. Il pourra alors être prévu en guise de nue-propriété, en lui adjoignant une obligation de gestion de la société ou de l'exploitation familiale.

Ainsi, le contentieux de la copropriété illustre cette appréhension du droit réel de jouissance spéciale. En effet, le caractère perpétuel concédé à ce droit réel se justifiait, dans l'arrêt précité, du fait qu'il soit accessoire au fonds, « attaché à un lot de copropriété » 28 , similaire à une servitude 29 . Il s'agissait d'une piscine dont les propriétaires avaient autorisé, par convention, l'accès à l'ensemble des copropriétaires. Cette élaboration impose donc aux personnes bénéficiaires de ce droit d'en supporter aussi les charges, sans pouvoir s'en libérer comme c'est le cas pour les servitudes. Ce droit n'est donc pas fondé sur un caractère personnel, mais se rattache à l'ensemble de la copropriété.

Cette distinction constituerait une justification de la solution rendue par la Cour de cassation en l'espèce. Tant l'absence de caractère spécial du droit de jouissance du vendeur que de la copropriété au moment de la vente, ne permettaient pas la requalification du droit réel établi par les parties. En l'espèce, le silence des juges sur le droit réel de jouissance spéciale permet toutefois d'envisager une certaine clarification sur sa durée. D'une part, un droit réel de jouissance spéciale peut être perpétuel lorsque sa spécialité est accessoire au fonds. La jurisprudence traduit cette possibilité par la reconnaissance d'un droit réel perpétuel de jouissance de crû et à croître 30 , mais aussi d'un droit de passage pour piéton réservé sur un fonds privé aux habitants d'une ville 31 , ou par le droit de « seconde herbe » au profit des habitants d'une entité territoriale 32 . D'autre part, lorsque le droit réel de jouissance spéciale s'exprime à travers un caractère intuitu personae, le rattachement aux conditions de l'usufruit semble nécessaire et le caractère perpétuel impossible 33 . L'évolution de la 28 Dans ce sens V. J. FRANÇOIS, « Qu'est-ce qu'un droit réel de jouissance spéciale ? », chron. D. 2019.1660, spéc. p. 1666. 29 Par absence de fonds dominant, la charge constituée ne pouvait être une servitude. La Cour de cassation la présente comme « une charge imposée à certains lots, pour l'usage et l'utilité des autres lots appartenant à d'autres propriétaires ». 30 Cass. 3 e civ., 23 33 Cette distinction semble avoir été exploitée par la Cour de cassation pour rejeter la qualification d'une servitude. V. Cass. 3 e civ., 22 juin 1976 : « L'existence d'une servitude exige un fonds dominant et un fonds servant. Ainsi, la concession d'un droit de chasse ne saurait avoir le caractère d'une servitude réelle, le fonds en faveur duquel il est accordé ne recueillant aucune utilité et le profit ou l'agrément que ce droit peut amener ne concernant que la personne du propriétaire du fonds et non le fonds lui-même ». La Cour de cassation qualifie ce droit de chasse de droit réel

  Conclusions de B. STURLESE, op. cit., pp. 26 et 27 : ce constat permet à l'auteur d'estimer la demande de requalification « vaine et inopérante ». Il ajoute plus loin « l'assujettissement total et perpétuel des utilités d'usage de la propriété d'un fonds doit demeurer proscrit, quelle que soit la qualification susceptible d'être donnée au droit en cause ». 20 Cass. 3 e civ., 28 janvier 2015, n°14-10.013, Bull. civ. III, n°13 ; D. 2015. 599, note B. MALLET-BRICOUT, 988, chron. A-L. MEANO, et 1863, obs. N. REBOUL-MAUPIN ; Justice & cassation, 2015.270, rapp. M-T. FEYDEAU et 277, avis B. STURLESE ; AJDI, 2015. 304, obs. N. LE RUDULIER ; RDI 2015. 175, obs. J-L. BERGEL ; RTD Civ. 2015.413, obs. W. DROSS et 619, obs. H. BARBIER. 21 CA Paris, 22 janvier 2015, n°13-649.

  mai 2012, n°11-13.202. 31 Cass. 3 e civ., 6 mars 1996, n°93-17.424, RDI 1996.536, obs. J-L. BERGEL. 32 Cass. 3 e civ., 25 mars 1992, n°89-21.866, Bull. civ. III, n°106 ; D. 1993. 65, note E-S. DE LA MARNIERRE, et 306, obs. A. ROBERT ; RDI 1992.176, obs. J-L. BERGEL, et 1993.532, obs. PH. DELEBECQUE et P. SIMLER.

De manière subsidiaire, la septième branche, en corrélation avec la première branche, souligne la possibilité offerte à l'acquéreur de renoncer à son droit de jouissance, qu'en déclarant la clause comme privée d'objet, la cour d'appel serait contrevenu aux articles 544, 637, 686, 1134, 1108 et 1126 du Code civil dans leur rédaction antérieure au Code civil.

Les cinquièmes et neuvième branches soulignaient l'ambiguïté de la clause qui nécessitait alors une interprétation, alors que les troisième et septième branches défendaient qu'une convention ne devait être annulée du fait d'une qualification contraire à l'économie générale du contrat, mais interprétée en vue de recouvrir son objet.

La Cour de cassation, confrontée à un conflit de droits sur les herbes et les arbres des berges d'un cours d'eau, affirme que les articles

544, 546 et 552 du Code civil « sont déclaratifs du droit commun relativement à la nature et aux effets de la propriété, mais ne sont pas prohibitifs » et que « ni ces articles, ni aucune autre loi, n'excluent les diverses modifications et décompositions dont le droit ordinaire de propriété est susceptible ».

J. CARBONNIER, Droit civil, Les biens, 2000, n°44 ; F. ZENATI-CASTAING et TH. REVET, Les biens, 2008, n°296 et s.

C. LARROUMET et B. MALLET-BRICOUT, Les biens, droits réels principaux, Economica, 6 e édition, 2019, n°574-3, p. 344 : « cette clarification jurisprudentielle, après deux siècles de tergiversations doctrinales et jurisprudentielles, doit être approuvée ».

Cet article est invoqué par le demandeur au pourvoi au sein des deuxième, quatrième et huitième branches et fondé au regard de « l'économie du contrat ».

Ainsi, si les parties avaient établi un droit réel de jouissance spéciale, une requalification de la clause par les juges aurait été certainement effectuée afin de privilégier la qualification légale de la servitude.

Conclusions de B. STURLESE, Defrénois, n°27, 4 juillet 2019, p 27 : « le droit réel de jouissance spécial, pour être admissible, doit nécessairement être spécial au sens substantiel ».

V. sur la question R. LIBCHABER, RDC 2013. 584 : « il est bien clair qu'en l'occurrence, la SACD est à jamais empêchée de jouir du bien qu'elle a acquis ». V. également J. DUBARRY et M. JULIENNE, RLDC, févr. 2013, p. 7 : on « peut regretter, si les mots ont un sens et alors que rien ne l'y obligeait, que la troisième chambre civile ait repris la terminologie de droit de jouissance spéciale pour qualifier un droit aussi large que celui concédé dans la présente affaire » ; J-B. SEUBE, RDC. 2013.627.

Des servitudes de cour commune ont ainsi pu être reconnues sans présenter ses conditions d'existence. Les juges ont ainsi relevé que « l'existence d'un fonds dominant et d'un fonds servant n'est pas essentielle à l'existence d'une servitude créée par la loi ». V. Cass. 1 e civ., 30 avril 1963, Bull. civ. I, n°239 ; Cass. 1 e civ., 15 décembre 1999, n°97-22.161, JCP 2000. I. 265, n°22, obs. H. PERINET-MARQUET ; JCP N 2000. 682, obs. D. SIZAIRE, Defrénois 2000. 314, obs. C. ATIAS ; RDI 2000.147.

L'arrêt Caquelard permet de concorder différentes jouissances spéciales sur un même fonds : l'un des copropriétaire en exploitait les arbres, l'autre les herbes.

propriété invite à apprécier toute la pertinence de ce droit réel de jouissance spéciale, néanmoins, il ne faut plus l'aborder comme le couteau suisse de l'organisation de la propriété. Il est alors temps


Romain Azaïs
email: romain.azais@inria.fr

Salaheddine Habibeche

Farah Ben-Naoum

Christophe Godin
email: christophe.godin@inria.fr

Exploring symmetries in plant architectures

Keywords: symmetry, plant architecture, tree edit distance, algorithmic clustering

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Symmetries and invariance

Symmetries are ubiquitous in living forms and often reflect repetition in the mechanisms of morphogenesis. They are closely related to the mathematical concept of invariance. An object x is said to be invariant under a certain operator T when it remains unchanged under the action of the operator, i.e. Tx = x. Observing symmetries means identifying operators under the action of which an object remains invariant. However, in living forms, because of intrinsic biological noise and environmental effects, symmetries are often imperfect and correspond to Tx ≈ x. Here, we investigate how these concepts can be applied to identify symmetries in plant architectures.

Symmetries in plant architectures: the swap operator

A plant architecture can be modeled as a mathematical rooted tree [START_REF] Godin | A multiscale model of plant topological structures[END_REF], i.e. a connected graph with no cycle in which a node, called the root, has been selected. In addition, each node in the tree may carry some additional pieces of information such as the type and the geometry of the elementary component of the plant, or even its physiological state. A subtree is a subpart of a tree made of a node and all its descendants in the tree. Macroscopic parts that are susceptible to be similar in plant architectures are often subtrees (Arber, 1950;[START_REF] Ferraro | Toward a quantification of self-similarity in plants[END_REF][START_REF] Godin | Quantifying the degree of self-nestedness of trees: application to the structural analysis of plants[END_REF]. In order to identify the symmetries of plant architectures, we therefore introduce the swap operator that swaps two subtrees (see Fig. 1).

However, not every pair of subtrees can be swapped: if a subtree appears as a descendant of another, their swap does not make sense. A set of subtrees that can be swapped pairwise forms an antichain for the inclusion partial order: these subtrees are not related to each other by parent-child relationships. Now, we are going to see how antichains can be used to formalize the idea of symmetric (or close to symmetric) structures in trees.

Maximal symmetry antichain

A symmetry antichain is an antichain of subtrees such that their swap preserves the architecture of the plant, i.e. for any pair of subtrees t and s in the antichain, the tree T' obtained after their swap is equal (or at least very similar) to the original tree T. Such a symmetry antichain is said maximal when there is no other subtree within T that can be swapped with one of the antichain while preserving the general architecture. Detecting the symmetries actually means identifying all the maximal symmetry antichains.

In a real plant architecture, one does not expect to observe exact symmetries. One must therefore be able to quantify the similarity between subtrees. For this, we use the tree edit distance developed in [START_REF] Zhang | A constrained edit distance between unordered labeled trees[END_REF] and already used for the comparison of plant architectures in [START_REF] Godin | Quantifying the degree of self-nestedness of trees: application to the structural analysis of plants[END_REF]. We have thus developed an algorithm to build all maximal symmetry antichains of the plant architecture that proceeds through a clustering of all its subtrees based on the above tree edit distance, instead of an exact match.

Visualization of symmetries

Within an antichain, the subtrees are not related, but from one maximal symmetry antichain to another, this may be the case. We can therefore consider the graph of maximal symmetry antichains: two maximal symmetry antichains are linked by an edge when subtrees of one appear as descendants of subtrees of the other, and the weight of the edge corresponds to the proportion of such links. Using this graph, we can therefore not only observe all the symmetries of a plant architecture but also their nesting, which represents a major difficulty when dealing with tree data (see Fig. 2). This symmetry graph actually represents the symmetry signature of the tree architecture.

Towards reverse engineering?

By observing the symmetries of a plant, we can better characterize its structure but also study the production and growth mechanisms that generated it. The maximal symmetry antichains we have identified in this work were probably generated by identical biological, chemical and mechanical processes. This can be verified on simulated plants for which the construction rules are known. This also provides a new conceptual tool for the analysis of the functional origin (molecular, biophysical) of structure repetitions in plant architectures.

Figure 1 :

 1 Figure 1: Illustration of the swap operator: red and blue subtrees in the original data (left) are swapped both in the mathematical tree and in the 3D plant architecture it models (right).