Theocharis Baxevanis

F R Éd

AND Éric Dufour

Gilles Pijaudier-Cabot

INTERFACE CRACK PROPAGATION IN POROUS AND TIME-DEPENDENT MATERIALS ANALYZED WITH DISCRETE MODELS

Keywords: Zip model, FPZ size, size effects, creep, ageing, fracture, viscoelasticity, time effect, concrete failure, discrete approach. 1

. The creep -fracture interaction is analyzed in the second part of the paper. It is found based on a Maxwell model that the size of the process zone depends on the fracture propagating velocity and on the distribution of forces in the interface due to the interaction between the interface and the rest of the specimen. The observed decrease of the size of the process zone, in creep experiments, compared to the size of the process zone in a static process, is justified by the proposed model for an interface that is less viscous than the rest of the material.

Introduction

Progressive failure of quasi-brittle heterogeneous materials is a succession of micro-cracks nucleation, propagation and arrest. First, the material response is elastic, then microcracking appears and eventually these microcracks coalesce in order to form a macro-crack which propagates suddenly. The above rupture events are controlled by the randomness of the distribution of the material properties and also by internal correlation lengths that are in general unknown. Such correlation lengths have a strong influence on the failure of quasi-brittle materials. Instead of a perfect crack, with a very small process zone of nonlinear response at its tip, a rather large fracture process zone (FPZ) develops in the material ahead of the tip of the macrocrack [START_REF] Bažant | Fracture and size Effect in Concrete and Other Quasi-Brittle Materials[END_REF]. This process zone induces a size effect, which is typical of quasi-brittle heterogeneous structures. Among several possible explanations for such a size effect on the structural strength, a simple one is that it is a purely deterministic effect, resulting from the energetic interaction between the elastic part of the structure and the FPZ. From the point of view of design of structures, e.g. reinforced concrete structures, this size effect is important. According to the size effect law proposed by Bažant, the nominal strength of a structural component can be decreased by as much as 50 percent if the real size of the structure is five times larger than the laboratory specimen size. This size effect law incorporates also the definition of a fracture energy, seen as the energy dissipation in the asymptotic with polystyrene content in model materials.

limit of a structure of infinite size with a very large initial notch which would fail at crack initiation.

A relevant continuum model has to take into account this size effect, and at the same time to capture strain softening due to progressive cracking in a way that is physically and mathematically sound. This is usually performed by the introduction of an internal length in the continuum model (see e.g. the review in [START_REF] Bažant | Nonlocal integral formulations for plasticity and damage: Survey of progress[END_REF]). The internal length serves two purposes at the same time: [START_REF] Andersen | Tricritical behavior in rupture induced by disorder[END_REF] it is a way of constraining the energy dissipation to occur in a region of finite size upon strain localization due to softening and to keep the governing equations well posed [START_REF] Pijaudier-Cabot | Strain localization and bifurcation in a nonlocal continuum[END_REF], [START_REF] Batrouni | Heterogeneous interfacial failure between two elastic blocks[END_REF] the occurrence of the FPZ upon cracking and the inherited size effect are captured at the same time. As a matter of fact, size effect, e.g. of geometrically similar notched three point bending beams, is an indirect technique for the calibration of the internal length and other model parameters for isotropic non local damage models for concrete [START_REF] Bellégo | Calibration of nonlocal damage model from size effect tests[END_REF]. Theoretical strain localization analyses show that the width of the fracture process zone ought to be proportional to the internal length. Such a result was already observed numerically by Bažant et al [START_REF] Bažant | Nonlocal continuum damage, localization instability and convergence[END_REF] and used in order to design an approximate method for the determination of the internal length [START_REF] Bažant | Measurement of the characteristic length of nonlocal continuum[END_REF]. Experiments and finite element analyses on several model materials made with mix of mortar and polystyrene beads [START_REF] Haidar | Correlation between internal length, fracture process zone and size effect in mortar and model materials[END_REF] have shown that the width of the FPZ and Irwin's length (which can be seen as the length of the FPZ ahead of the macrocrack tip) are also correlated to the variation of the internal length in the non local continuum. Figure [START_REF] Andersen | Tricritical behavior in rupture induced by disorder[END_REF] shows that the FPZ measured experimentally from acoustic events analysis during fracture, the internal length obtained after calibration of the non local damage model on the various model materials (using size effect test data) and Irwin's length correlate quite well. Irwin's length is defined as

l cl = EG f σ 2 (1)
in this equation, E is the Young's modulus, G f is the fracture energy and σ is the tensile strength. These experiments on model materials, with various amount of polystyrene beads, were designed in order to mimic the ageing process in cementitious materials due to calcium leaching, i.e. a progressive dissolution of the material due to contact with water. In this case, the width and the length of the FPZ increase with the amount of material porosity and can be considered as proportional to the internal length, at least in the three point bending beam tests on notched specimen considered in [START_REF] Haidar | Correlation between internal length, fracture process zone and size effect in mortar and model materials[END_REF]. Furthermore, the same experimental study has shown that large variations of the fracture energy occur with the amount of polystyrene content. Such variations were also observed in fracture tests performed after accelerated leaching on mortar beams [START_REF] Bellégo | Coupled chemical and mechanical damage in calcium leached cementitious structures[END_REF]. Without variations of the internal length, it was not possible to model the consequences of the leaching process on the fracture properties of aged materials, and in particular the variations of the size effect parameters observed by Le Bellégo et al [START_REF] Bellégo | Coupled chemical and mechanical damage in calcium leached cementitious structures[END_REF] which indicate variations of the size of the FPZ.

There is another situation of structural response where similar variations of the size effect properties have been observed. It is the case of creep -fracture interaction studied in [3]. The authors conducted size effect tests on geometrically similar specimens loaded at different rates and observed that the slower the loading rate, the more brittle the response. Pijaudier-Cabot et al [START_REF] Pijaudier-Cabot | Creep-damage interaction in concrete structures[END_REF] performed a series of three point bending beam tests (notched specimens) with and without applying a sustained load for 3 months up to 85 percent of the maximum load prior to the fracture test. They arrived to a similar conclusion, namely that creep influences the residual capacity of the beams, but also that upon size effect tests, the specimens which were subjected to creep exhibited a more brittle response. This property yields a shift to the right of the data set on the size effect plot as shown in Figure [START_REF] Batrouni | Heterogeneous interfacial failure between two elastic blocks[END_REF]. The same phenomena has been observed by Bažant and Gettu [3] and Bažant and Li [START_REF] Bažant | Cohesive crack with rate-dependent opening and viscoelasticity: I. mathematical model and scaling[END_REF] for linear creep.

In the present paper, we use a discrete random modeling based on a fiber bundle model in order to investigate the size of the FPZ. It is the so-called ZIP model developed in [START_REF] Delaplace | Avalanche statistics of interface crack propagation in fiber bundle model: Characterization of cohesive crack[END_REF]. We consider the two examples depicted in the above paragraphs, namely the case of a material whose porosity increases and the case of creepfracture interaction. As we will see in section 2, it is quite straightforward to demonstrate that, according to the discrete model, when the microstructural size increases (in order to capture the heterogeneities in a representative volume) the FPZ length ought to increase too. The case of creep -fracture interaction needs additional mathematical considerations, but upon some simplifying assumptions, it will be shown in section 3 that the FPZ length decreases as creep develops.

Zip model -Case of interface with variation of porosity

During the past decades, fiber bundle models have received considerable attention and have been studied extensively. Originally introduced to explain ruptures in heterogeneous materials under tension [START_REF] Daniels | The statistical theory of the strength of bundles of threads[END_REF], fiber bundle models have been applied to cracks and fractures, earthquakes, and other related breakdown phenomena [START_REF] Andersen | Tricritical behavior in rupture induced by disorder[END_REF] and [START_REF] Zapperi | First order transition in the breakdown of disordered media[END_REF]. They are directed towards the study of the material heterogeneities, i.e. at a scale much smaller than the representative volume of the material. These models consist of a set of parallel fibers having statistical distributed strength. The sample is loaded parallel to the fiber direction, and a fiber breaks if its elongation exceeds a threshold value. When a fiber breaks, its load is transferred to other surviving fibers in the bundle, according to a specific transfer rule. Among the possible options of load transfer are the assumption of equal load sharing (global-sharing rule) [START_REF] Daniels | The statistical theory of the strength of bundles of threads[END_REF] and the much studied variants -local load-sharing rule-where the load on the failing fiber is distributed equally among the nearest surviving fibers [START_REF] Harlow | Approximations for the strength distribution and size effect in an idealized lattice model of material breakdown[END_REF]. There are also a number of studies that may be placed among the two extremes that global and local load-sharing rules constitute. Among them is the study by Delaplace et al [START_REF] Delaplace | Avalanche statistics of interface crack propagation in fiber bundle model: Characterization of cohesive crack[END_REF], who constructed an hierarchically connected fiber bundle model. They have worked out analytically and numerically the statistics of avalanches in a system whose geometry mimics the propagation of crack front at the interface between a rigid support and an elastic beam. Other work on hierarchical fiber bundle models can be found in [START_REF] Newman | Time-dependent fiber bundles with local load sharing[END_REF], [START_REF] Delaplace | Analyse Statistique de la Localisation dans les Matériaux Hétérogènes Quasi-Fragiles[END_REF] and [START_REF] Zhang | Complex fiber bundle model for optimization of heterogeneous materials[END_REF]. For a review of the literature in the subject, one may refer for instance to the work of Batrouni et al [START_REF] Batrouni | Heterogeneous interfacial failure between two elastic blocks[END_REF].

In this section, we focus on a system of elastic-perfectly brittle fibers loaded in parallel between a rigid substratum and a semi-infinite elastic beam. The geometry is suited to an inhomogeneous, but steady state of loading and it can be considered as a schematic model for mode I crack propagation (the rigid side of the interface representing an axis of symmetry). The nonlinear interface serves to concentrate cracking onto a line, the same as in the cohesive crack model. A normal displacement is imposed at one point of the elastic beam that can move along the interface as if a wedge was pushed in a double cantilever geometry (Figure ( 3)). The fiber strength is randomly distributed with a uniform probability of critical fiber extension between 0 and 1. As the wedge is moving along the x-axis, the interface decouples into three different parts due to the randomness of fiber strength. Namely, under the wedge and in a small area ahead all fibers are broken. This region constitutes the crack itself. Further ahead there is an active area. Moving along this zone, broken fibers become more and more scarce and their influence can be neglected.

The exponential decay of fiber rupture ahead of the crack tip provides the means to define the size of the fracture process zone.

A simplified model where the displacement profile along the upper beam is imposed is used. The interface opening has an exponential shape: for any abscissa x, the profile y is given by

y(x) = exp U (t) -x ξ (2)
where the length scale

ξ = √ 2 (EI/k) 1/4 , (3)
is fixed (E is the Young modulus of the beam, I its transverse geometrical inertia and k the stiffness of the fibers divided by the spacing in between the fibers), and U (t) is the time-dependent horizontal displacement of the wedge. As shown in [START_REF] Delaplace | Avalanche statistics of interface crack propagation in fiber bundle model: Characterization of cohesive crack[END_REF], this simplified model has the same statistical properties as the one where an elastic beam is connected with elastic-perfectly brittle fibers to the rigid substratum. We are not going to derive again the complete results obtained by the authors of [START_REF] Delaplace | Avalanche statistics of interface crack propagation in fiber bundle model: Characterization of cohesive crack[END_REF] who used analytical solutions and analyzed the distribution of avalanches. We will recall just the main useful results, and in particular the motivations for using a simplified approach, with a fixed beam profile (2) instead of a deformable one.

In the spirit of continuum modeling (i.e. for length scales much larger than the fiber separation) and using beam theory, one can write an equation for the mean deflection of the beam y(x) (also fiber elongation), based on the survival uniform force that fibers apply onto the beam. This force for elastic fibers would be simply ky(x).

EI d 4 y(x) dx 4 = -ky(x)(1 -Y (x)), (4)
where

Y (x) = max x ≥x y(x ) . (5)
This equation holds for a uniform distribution of critical fiber extension between 0 and 1, and for y < 1, whereas d 4 y(x)/dx 4 = 0 for larger y. The boundary conditions are y(∞) = dy(∞)/dx = 0, y(0) = 1 and d 2 y(0)/dx 2 = 0 (no bending moment applied at the loading point). The distinction between y and Y is made because damage is an irreversible process and thus one has to compute the maximum damage having been met by the corresponding section of fibers. (1 -Y (x)) actually represents the probability of survival of fibers. There is no analytical solution to this problem. However, since the quadratic nonlinear term becomes unimportant at a large distance from the origin, the asymptotic shape will have the form

y (x) = Ae -x/ξ cos x/ξ + φ , (6)
where ξ is defined in (3), φ is such that cos(φ) = 1/A > 0 in order for y to satisfy the boundary conditions, while A is chosen such that

∂y (x) ∂x x=0 = - 1 ξ 1 + A 1 -(1/A) 2 (7)
is equal to the corresponding value of the exact solution. The oscillatory component is the one that makes the deflection nonmonotonous and requires the distinction between y and Y . Omitting this oscillatory term and the nonlinearity, one retrieves (2) which is found to yield a correct assumption compared to full numerical calculations with a deformable beam in the zip model [START_REF] Delaplace | Avalanche statistics of interface crack propagation in fiber bundle model: Characterization of cohesive crack[END_REF].

A typical time variation of total force acting on the beam (see Fig . ..) shows a constant mean value through time with large fluctuation though. There exists various definitions of avalanches. The one used in [START_REF] Delaplace | Avalanche statistics of interface crack propagation in fiber bundle model: Characterization of cohesive crack[END_REF] is the most natural. It consists in fixing a level of force, and computing the distance ∆U over which the crack can propagate which is the retrival of the force level. The smallest avalanche is 1 (fiber spacing) and the largest one can be in theory infinite when computed for a level of force corresponding to the maximum ever reached but with an infinitely small probability. The avalanches are characterized by their statistical distribution, p(∆U, F ). In order to give a global characterization of the signal without considering a specific value of the force, one may consider the above avalanches for any crack length ∆U such that a fiber would break, and then average over all breaking events. These avalanches are the "forward" avalanches as described in Fig . ... In order to investigate the effect of ξ on the size of the FPZ, we shall just consider that the avalanche distribution follows two regimes with respect to a crossover value ∆ * of the avalanche size ∆ ≡ ∆U . In the first regime the statistical distribution of avalanches is a power law p(∆) ∝ ∆ τ1 with an exponent τ 1 . In this regime the force versus crack length displays correlations similar to a random walk (τ 1 -1.5). In the second regime, the distribution of avalanches is again a power law but with a different exponent τ 2 . In this regime, it can be shown that the forces are uncorrelated (τ 2 -2) (Figure ( 4)). The cross over value ∆ * scales the extent of the FPZ. Same as in Hillerborg's fictitious crack model [START_REF] Hillerborg | Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements[END_REF], it is the length of the FPZ, i.e. the size beyond which fiber breakage is not influenced by the crack tip.

In Figure ( 4) the scaled variable ∆/ξ and the scaled distribution ξ 1.5 p(∆) were used to show that curves corresponding to different values of ξ collapse onto one master curve. Hence, the size of the fracture process zone scales as ξ . More specifically the size of the FPZ increases as ξ increases (Figure [START_REF] Bažant | Cohesive crack with rate-dependent opening and viscoelasticity: I. mathematical model and scaling[END_REF]).

The Young modulus E of the beam is a positive, monotonically decreasing function F of porosity g; E = E 0 F(g), where E 0 is a reference value. Porosity g here is defined as the ratio of the voids area to the total area of the beam. Furthermore, the ratio k of the average stiffness of the interface is the same function:

k = k 0 F(g),
where k 0 is a reference value. In the fracture process zone, however, it is not the average porosity of the interface that is of importance. In the 2D material that the zip model aims at describing, fracture follows the weakest possible path. Consequently, the porosity of the interface should be less than the average interface porosity as the crack connects the pores in the material in such a way that the work of fracture is minimum. In the case of a regular array of voids for instance, the interface porosity g l along the dotted line in Figure ( 6) is g l = n2r/R ∝ √ g. One can generalize this remark and set that g l scales as g b , where b will be in principle a real number smaller than 1. One can argue also that in the porous material, the roughness of the crack will increase, thus increasing the fracture energy. In the zip model, it would be equivalent to change the fiber spacing which is not sensitive on the avalanche statistics. Hence, submitting stiffnesses accounting for porosity into (3), one gets

ξ = √ 2 E 0 F(g)I k 0 F(g b ) 1/4 . (8)
Since g < 1 and F is a positive, monotonically decreasing function of g, one can conclude that F(g)/F(g b ) increases with increasing g. It is then obvious from (8) that increasing porosity yields an increase in ξ and consequently an increase in the size of the FPZ. So the present theoretical model is consistent with experimental observations on the dependence of the size of FPZ on porosity, as well as with numerical computations with a non local damage model.

Zip model -Case of viscoelastic fibers loaded between a rigid substratum and a viscoelastic beam

The model described below is a generalization in viscoelasticity of the previous model developed in [START_REF] Delaplace | Avalanche statistics of interface crack propagation in fiber bundle model: Characterization of cohesive crack[END_REF], (Figure ( 3)). All fibers are viscoelastic, following a Maxwell model, and perfectly brittle (i.e. fail suddenly at a given elongation) with the same stiffness k and viscous coefficient µ. However, the strength of the fibers are random uncorrelated variables in order to incorporate inhomogeneity into the model, same as in the elastic case. The fibers are loaded in parallel between a rigid support and a semi-infinite viscoelastic beam of Young modulus E, transverse geometrical inertia I and viscous coefficient M . One could as well consider, instead of the rigid part, another viscoelastic beam even with different properties without any change in the formulation of the problem and in the qualitative results that will be drawn eventually.

In notation, the position of the fiber i is denoted as x i and its deformation as y i . The strength of each fiber denoted as y cr i is, as mentioned above, a random uncorrelated variable having a statistical distribution in the interval [0, 1]. The spacing between the fibers is set to 1 (i.e. x i+1 -x i = 1), thus defining a fixed microstructural size. A normal displacement of 1 is imposed at one point of the beam, which may move along the interface, as if a wedge is pushed in a double cantilever geometry (Figure (3)).

The fiber deformation y i is related to force F i as

∂F i (t) ∂t + k µ F i (t) = k ∂y i (t) ∂t (Maxwell model ), (9)
where t denotes the time and ∂ denotes partial differentiation with respect to the indicated variable. It can also be shown without altering the assumptions of beam theory that the deformation of the beam y(x, t) satisfies a similar equation, if again the Maxwell model is used,

∂F (x, t) ∂t + E M F (x, t) = EI ∂ 5 y(x, t) ∂t∂x 4 , (10)
where F (x, t) denotes the force acting on the beam and the other parameters are already defined.

In the spirit of continuum modeling (i.e., for length scales much larger than the fiber separation) the system defining the mean deflection of the beam y(x, t) reads:

∂F (x, t)/∂t + E M F (x, t) = EI ∂ 5 y(x, t)/∂t∂x 4 ∂F (x, t)/∂t + k µ F (x, t) = -k ∂ y(x, t) 1 -Y (x, t) /∂t, (11)
where

Y (x, t) = max t ≤t y(x, t ) . ( 12
)
The above system holds for a uniform distribution of fiber extension between 0 and 1, and for y < 1, whereas ∂ 4 y/∂x 4 = 0 for larger y. The boundary conditions are such that ∀t, y(∞, t) = ∂y(∞, t)/∂x = 0, ∂ 2 y (U (t), t) /∂x 2 = 0, (13a) and for t ≥ t 0 , y (U (t), t) = 1, (13b)

where U (t) denotes the time-dependent horizontal displacement of the wedge along the x-axis, t 0 is the time at which the wedge starts moving, and the relation ∂ 2 y (U (t), t) /∂x 2 = 0 implies that no bending moment is being applied at the loading point, while the initial conditions are y(x, 0) = y 0 (x) and F (x, 0) = F 0 (x). ( 14)

3.1. Simplified version of the model: constant displacement profile. Let us derive a simplified model, where the displacement profile along the upper beam is imposed, that serves for analytical study. For this purpose we neglect oscillatory and non linear term due to Y .??? This simplifying assumption will allow for closedform expressions. In the same spirit as in [START_REF] Delaplace | Avalanche statistics of interface crack propagation in fiber bundle model: Characterization of cohesive crack[END_REF] (Eqns. ( 4)-( 7)), the profile of the upper beam is derived, but now for the case of a viscoelastic beam connected to a viscoelastic interface, although the interface still fails for a sufficient elongation. The governing equations read now

∂F (x, t)/∂t + E M F (x, t) = EI ∂ 5 y(x, t)/∂t∂x 4 ∂F (x, t)/∂t + k µ F (x, t) = -k ∂y(x, t)/∂t. (15)
In notation, we denote the system [START_REF] Bellégo | Calibration of nonlocal damage model from size effect tests[END_REF] with boundary conditions ( 13) and initial conditions ( 14) as (Σ * ). We eliminate F in ( 15) to obtain

∂ ∂t EI ∂ 5 y(x, t) ∂t∂x 4 + k ∂y(x, t) ∂t = - k µ EI ∂ 5 y(x, t) ∂t∂x 4 - E M k ∂y(x, t) ∂t . ( 16
)
By separation of variables, i.e. by assuming that ∂y(x, t)/∂t = u(t)v(x),

we take from ( 16)

∂u(t)/∂t u(t) = -k µ EI ∂ 4 v(x)/∂x 4 -E M kv(x) EI (∂ 4 v(x)/∂x 4 ) + kv(x) = α, ( 18
)
where α is a positive constant. Indeed, the displacement y(x, t) is a monotonically increasing function and hence u(t) and ∂u(t)/∂t are both positives. Solving the above equation for u and v, and then integrating [START_REF] Mazzotti | Experimental study of nonlinear creep of concrete at high stress levels[END_REF], we find a solution of system ( 15)

y(x, t) = 1 α e -x/ξ cos(x/ξ)e αt F (x, t) = k k µ +α e -x/ξ cos(x/ξ)e αt , (19)
where

ξ = √ 2 EI k k µ + α E M + α 1/4 . (20)
By multiplying in [START_REF] Pijaudier-Cabot | Strain localization and bifurcation in a nonlocal continuum[END_REF] with α, taking into account the linearity of (15) (i.e. any linear combination of solutions is a solution itself), and omitting the oscillatory term cos(x/ξ), we end up to another solution of ( 15)

y(x, t) = e (αξt-x)/ξ F (x, t) = αk k µ +α e (αξt-x)/ξ . (21)
x y (x=0, y=1)

Figure 7. At time t 0 , such that e t0 -1 = α, the vertical displacement of the loading point is equal to 1. The wedge now starts moving along the x-axis.

Further, for initial conditions

y(x, 0) = e -x/ξ , F (x, 0) = αk k µ + α e -x/ξ , (22) (21)
is a solution of system (Σ * ) for t 0 = 0. It represents the problem of the wedge moving along x-axis under constant velocity v = αξ, (

). Indeed, for the deformation y(x, t) at position x to become equal to 1,

y(x, t) = e (αξt-x)/ξ = 1, (23)
time t = x/ (αξ) is needed.
It should be noted here that α is a parameter that characterizes the deformation rate. Although the deformation rate is not constant (∂y(x, t)/∂t = e (αξt-x)/ξ ) the process can be viewed as quasi-static since for constant ξ the total force exerted on the wedge, which is the sum of all fiber contributions, remains fixed within the statistical randomness of the fiber strength.

Suppose now that the initial conditions are

(y 0 (x) = 0, F 0 (x) = 0) , (24)
then y(x, t) = 1 α e -x/ξ (e αt -1)

F (x, t) = k k µ +α e -x/ξ (e αt -1), (25)
is one solution of system [START_REF] Bellégo | Calibration of nonlocal damage model from size effect tests[END_REF]. For t 0 , such that e t0 -1 = α, the above equation (25) gives

y(x, t 0 ) = e -x/ξ , F (x, t 0 ) = αk k µ + α e -x/ξ . (26)
Hence, for t ≥ t 0 , solution (25) reads

y(x, t) = e (U (t)-x)/ξ F (x, t) = αk k µ +α e (U (t)-x)/ξ , (27)
where U (t) = ξ ln e αt -1 α . U (t) represents the time-horizontal displacement of the wedge for t ≥ t 0 (Figure [START_REF] Bažant | Measurement of the characteristic length of nonlocal continuum[END_REF]). In this form it is apparent that (25) satisfies the boundary conditions [START_REF] Harlow | Approximations for the strength distribution and size effect in an idealized lattice model of material breakdown[END_REF]. Thus (25) is a solution of (Σ * ).

Although the wedge does not move, in this case, under constant velocity, the process is still quasi-static since again the total force exerted on the wedge remains constant within the statistical randomness.

We showed above that, at least for two different initial conditions, namely for ( 22) and (24), the system (Σ * ) admits an asymptotic solution of the form y(x, t) = e (U (t)-x)/ξ F (x, t) = αk k µ +α e (U (t)-x)/ξ , (see [START_REF] Zapperi | First order transition in the breakdown of disordered media[END_REF] and ( 27)).

(28)

These solutions represent the motion of a wedge whose time-dependent horizontal displacement for t ≥ t 0 is U (t) while before is at rest. To derive the above solutions (27), we omitted the oscillatory component cos(x/ξ) and made the assumption that a fiber breaks when it is strained up to its critical extension y cr i which is statistically distributed in the interval [0, 1]. Although from experimental evidence we know that in viscoelasticity the threshold cannot be placed in general neither to the total deformation nor to the elastic deformation. Indeed for low stress levels although creep strain can be large -even larger than that corresponding to peak stress for short term loading-there is no significant damage [START_REF] Mazzotti | Experimental study of nonlinear creep of concrete at high stress levels[END_REF]). On the other hand, if one wishes to model the coupling between creep and damagen the threshold cannot be set only on the elastic part. Our assumption is justified by the fact that the loading path is identical for all fibers and thus they are expected to fail at the same total deformation. Indeed, the elastic deformation y el i of every fiber remains through out the process the same fraction of its total deformation y tot i

y el i (x, t) = αk k µ + α y tot i (x, t). (29)
This situation corresponds to experimental material tests.

3.1.1. Avalanche distribution. As it is already pointed out, the model described here is the "viscoelastic" analog of the model in [START_REF] Delaplace | Avalanche statistics of interface crack propagation in fiber bundle model: Characterization of cohesive crack[END_REF]. For matters of convenience, we refer to the model described here as "viscoelastic" model and to the one developed in [START_REF] Delaplace | Avalanche statistics of interface crack propagation in fiber bundle model: Characterization of cohesive crack[END_REF] as "elastic" model. Looking now at the parameters ξ (20) and ξ (3) that scale the size of the fracture process zone in the two models respectively, we see that in the "viscoelastic" model ξ depends not only on the stiffness fraction E/κ as it is for the "elastic" model (ξ ) but on the relaxation times κ/µ and E/M as well, and on the parameter α which is related to the rate of deformation. α may also be seen as a parameter that scales the fracture propagating velocity. Indeed, the time-horizontal displacement of the wedge U (t) depends on α (see [START_REF] Zapperi | First order transition in the breakdown of disordered media[END_REF] and ( 27)) and in turn sets the fracture propagating velocity. Indeed Figure [START_REF] Bažant | Fracture and size Effect in Concrete and Other Quasi-Brittle Materials[END_REF] shows that the average distance between the moving wedge and the crack tip remains constant, up to the randomness of the model, thoughout the total time of computations. Further, when both the interface and the rest of the material are elastic in the "limit" (µ, M → +∞), the "elastic" case can be reproduced, ξ → ξ . This is also the case (ξ → ξ ) when the rate of deformation is too large to allow for creep, i.e. when α → +∞ or when the interaction between the interface and the rest of the material does not depend on their relaxation times, i.e. when (k/µ = E/M ). ξ can be either greater or less than ξ , with the two end members of the spectrum to be the case of an elastic beam and a viscoelastic interface and the case of a viscoelastic beam and an elastic interface (Figure [START_REF] Daniels | The statistical theory of the strength of bundles of threads[END_REF]). The case of a beam with relaxation time E/M greater than that k/µ of the interface corresponds to the observed decrease of the size of the fracture process zone in creep experiments compared to the size of the FPZ in a rate independent process. This should be the case, i.e. the composite should be more viscous than the interface, in order for our model to be realistic; in order for the fibers to remain alive when the matrix material in between has failed. The fibers may represent for example, based on the work of Hillerborg et al [START_REF] Hillerborg | Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements[END_REF] who studied fracture in concrete with the help of a cohesive crack, the aggregates that bridge the crack and allow for stress transfer across the crack. The fiber bundle model developed in this paper mimics the propagation of the crack front at an interface which ahead of the crack tip collapses onto a line as in the cohesive crack model. The relaxation time of the aggregates is usually less than that of concrete.

3.2.

Model for the complete problem. We now come back to the complete problem (Σ), i.e. system [START_REF] Delaplace | Avalanche statistics of interface crack propagation in fiber bundle model: Characterization of cohesive crack[END_REF] with boundary conditions (13) and initial conditions [START_REF] Hillerborg | Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements[END_REF]. Our aim is rather to compute the approximate average shape of the deformed beam. One can see that in system (Σ) for the damaged interface the quadratic nonlinear term ∂(yY )/∂t, which is additional compared to system (Σ * ) for the undamaged interface, becomes unimportant far away from the boundary x = 0. Thus the asymptotic shape can have a similar expression as the one used in the previous section.

   y (x, t) = A(t)e -x/ξ cos -x ξ + φ(t) e αt F (x, t) = αk k µ +α A(t)e -x/ξ cos -x ξ + φ(t) e αt , (30)
where φ is such that cos(φ) = 1/A > 0 in order for the solution y to satisfy the boundary condition y (0, t) = e αt , while A is chosen such that

∂y (x, t) ∂x x=0 = - 1 ξ 1 + A(t) 1 -1/A(t) 2 e αt (31)
expression (32) solves the problem of a wedge moving along x-axis with a time horizontal displacement of U * (t) = ξ αt+ln A(t) . This asymptotic solution has the same statistical properties with the simplified version of the model (28) derived in the previous section. Indeed, as it was pointed out in [START_REF] Delaplace | Avalanche statistics of interface crack propagation in fiber bundle model: Characterization of cohesive crack[END_REF], the avalanche distribution of (32) depends solely on ξ and it is independent of U * . One may also come to the same conclusions for the case of initial conditions (24).

Conclusions

A model describing the crack propagation at the interface between a rigid substratum and a beam has been considered. The interface is modeled using a fiber bundle model (i.e. using a discrete set elements having a random strength). The distribution of avalanches, defined as the distance over which the crack is propagated under a fixed force, has been studied in order to capture correlations of breaking events in the course of fracture. Fiber breakage kinetics is related to a correlation length, which sets the size of the fracture process zone that occurs ahead of the crack due to progressive failure.

Ageing is considered as a variation of porosity of the interface. It corresponds for instance to diffusion controlled dissolution processes in cementitious materials (e.g. calcium leaching). Results obtained in [START_REF] Delaplace | Avalanche statistics of interface crack propagation in fiber bundle model: Characterization of cohesive crack[END_REF] and described in the present paper show that the size of the fracture process zone is proportional to a length scale. This length scale increases (so does the size of the FPZ) upon increasing porosity and the discrete model is consistent with experiment data and numerical analysis.

The creep -fracture interaction has been analyzed in the second part of the paper. A simplified model is proposed, which is very similar to the static model used in the first part, but with a length scale that depends now on the time dependent characteristics of viscoelastic fibers and viscoelastic beam. It has been found that the size of the process zone depends on the fracture propagating velocity and on the distribution of forces in the interface due to the interaction between the interface and the rest of the specimen. The observed decrease of the size of the process zone, in creep experiments, compared to the size of the process zone in a static process, is justified by the proposed model for an interface that is less viscous than the rest of the material. This condition is realistic since the fibers should remain alive when the matrix material in between has failed, thus the proposed model is consistent with experimental evidence.

Figure 1 .

 1 Figure 1. Evolution of the width of the FPZ measured experimentally (l F P Z ), internal length (l c ), and the Irwin's length (l cl ) with polystyrene content in model materials.

Figure 2 .

 2 Figure 2. Evolution of size effect test data for 3 point bending tests on notched specimens subjected to creep prior fracture (after Pijaudier-Cabot et al. [20])

Figure 3 .

 3 Figure 3. Schematic representation of "zip" model. The wedge moves along the x-axis.

Figure 4 .

 4 Figure 4. Avalanche distribution for the simplified model of imposed beam deformation (eqn (2)) for two different values of ξ , namely ξ = 100 ( ) and ξ = 1000 (×). Scaled variable ∆/ξ and scaled distribution ξ 1.5 p(∆) are used. The dashed line is guideline with slope -1.5; continuous line is guideline with slope -2.05.

Figure 5 .

 5 Figure 5. Avalanche distributions for the simplified model of imposed beam deformation (eqn (2)) for two different values of ξ , namely ξ = 100 ( ) and ξ = 1000 (×). The size of the fracture process zone (crossover value) for ξ = 100 is less than the respective size for ξ = 1000. The dashed lines are guidelines with slope -2.05 and the dotted one with slope -1.5, while the continuous ones mark the crossover values.

Figure 6 .

 6 Figure 6. Schematic representation of a material as a square of edge R. The circles of radius r in within represent n 2 uniformly distributed voids.

Figure 9 .

 9 Figure 9. Avalanche distributions for the simplified model of imposed beam deformation (eqn (28)) for different mechanical properties of beam and fibers. Namely, the avalanche distributions for elastic beam EI = 12 × 10 10 and elastic fibers k = 1, ( ), for viscoelastic beam EI = 12 × 10 10 , E/M = 10 4 and elastic fibers k = 1, (×), and for elastic beam EI = 12 × 10 10 and viscoelastic fibers k = 1, k/µ = 10 4 , (♦). The dashed lines are guidelines with slope -2.05 and the dotted one with slope -1.5.

kµ

  +α A(t)e (αξt-x)/ξ (32) For initial conditions y(x, 0) = A(0)e -x/ξ , F (x, 0) = αk k µ + α A(0)e -x/ξ ,

  Figure 8. The evolution of the averages on the relative displacement of the crack tip, U (t), minus the relative displacement of the wedge U (t) in time intervals of 10000 broken fibers, is plotted, against time t * , which is equal to the total time (which corresponds to 10 8 broken fibers) divided by the length of the aforementioned time intervals. Since these averages remain through out the process very close to zero, the velocity of the crack tip coincides with the velocity of the wedge.

		0.004
	average (U'(t)-U(t))	-0.002 0.000 0.002
		-0.004	0	2000	4000	6000	8000	10000
					time t*

Acknowledgments. We thank A. Delaplace for providing us the numerical code used in [11]. This work was partially supported from the EU project 'Degradation and Instabilities in Geomaterials with Application to Hazard Mitigation' (DIGA-HPRN-CT-2002-00220) in the framework of the Human Potential Program, Research Training Networks.


A smart grid is an electrical network enabling a two-way flow of both data and electricity. During electrical power generation in the Smart Grid, a vast amount of electricity is lost. Mainly in transmission and distribution lines, where 17% of electricity is lost in transmission lines, and 50% is lost in distribution lines. This is called Transmission and Distribution Losses (T&D), and it describes when the amount of electricity generated is greater than the amount of electricity delivered to end-users. There are two types of T&D losses. First are Technical Losses (TL), which are ordinarily 22.5%, and directly depend on the network characteristics and mode of operation. Second is Non-technical Losses (NTL) at 16.6% and related to meter reading, defective meter and error in meter reading, billing of customer energy consumption, lack of administration, financial constraints, and estimating the unmetered supply of energy as well as energy thefts. Previously proposed solutions for detecting TL and NTL depend on monitoring user behavior, analysis, and feature extraction. In contrast, new proposed solutions for detecting TL and NTL made use of Machine Learning (ML) Algorithms and Synthesis data, differing from each other in applying various ML approaches (Supervised, Unsupervised, Semi-Supervised) and diverse Data processing techniques. As the rising need to detect TL and NTL in the Smart Grid, the industry is not equipped with an applicable solution to overcome the problem. The overall deliverables of this study were: 1. multi-layers system model, 2. Machine learning model, 3. new methodology. In principle, the model uses a new method to detect TL and NTL and Real-time data from Smart Grid.

Introduction

As the rising need to detect TL and NTL in the Smart Grid, the industry is not equipped with the applicable solution to overcome the problem. The deliverables of this study were: 1. multi-layers system model, 2. Machine learning model, 3. new methodology. In principle, the overall model uses Real-time data from Smart Grid, and a new method to calculate TL and NTL.

Conclusion

Abstract

Introducing A New Method To Detect Technical And Non-technical Losses In The Smart Grid Using Machine Learning Algorithms

Retaj Alsulami Retajaasulami@gmail.com Saad Aldoihi, PhD Saldoihi@kacst.edu.sa Motaz Alfarraj, PhD Motaz@kfupm.edu.sa

Results and Discussion

Discussion

Since most of the studies were conducted using synthesis data, the authors recommend electrical companies to release genuine data to encourage Researchers and Data Scientists to tackle the problem in Academic and Research environment.

A smart grid is an electrical network enabling a two-way flow of both data and electricity. In the process of generating electrical power electricity in Smart Grid, a huge amount of electricity is lost. Mainly in transmission and distribution lines, where 17% of electricity is lost in transmission lines and 50% is lost in distribution lines. This is called Transmission and Distribution Losses (T&D), and it describes when the amount of electricity generated is greater than the amount of electricity delivered to end-users.

There are two types of Transmission and Distribution (T&D) losses.

First are technical losses (TL), which are ordinarily 22.5%, and directly depend on the network characteristics and the mode of operation.

Second are non-technical losses (NTL) at 16.6%, and related to meter reading, defective meter and error in meter reading, billing of customer energy consumption, lack of administration, financial constraints, and estimating the unmetered supply of energy as well as energy thefts.

Methodology

Mixed research method was utilized to meet the aim and objective of this research.

The research methods used are qualitative and simulation.

Fig

  Fig 1, New method

Fig 3 ,

 3 Fig 3, Machine Learning Model

Fig 2 ,

 2 Fig 2, System Overview Model


Nada Ibrahim

Ingrid Allart-Simon

Gina R De Nicola

Renato Iori

Jean-Hugues Renault

Patrick Rollin

Jean-Marc Nuzillard

Advanced NMR-Based Structural Investigation of Glucosinolates and Desulfoglucosinolates

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.

ABSTRACT:

Glucosinolates (GLs) constitute a class of plant secondary metabolites that are characteristic of the order Brassicales. They contain a common hydrophilic moiety connected to a mostly hydrophobic sidechain whose constitution is the most frequent structural variant. Their transformation by myrosinases leads to intensively studied and highly reactive compounds of biological relevancy.

In other respects, the enzymatic desulfation of GLs produces derivatives (DS-GLs) that are useful for their analysis. A collection of 31 compounds representing 17 different sidechains either in GL or DS-GL form was established in order to report accurate descriptions of their 1 H, 13 C, and 15 N NMR parameters. The description of the 1 H NMR spectra was achieved using the PERCH software, which accurately analyzes the complex coupling patterns that arise from strongly coupled nuclei. The chemical shift assignments were supported by 2D COSY, HSQC, and HMBC spectra. The impact of desulfation and the influence of the sidechain nature on chemical shift values are discussed. The result of the spectroscopic analysis and the 3D chemical structure model of each studied molecule were grouped in Structure and Data Format (SDF) files. The NMR parameters were also collected in a simple text file, a spreadsheet file, and a relational database.

Glucosinolates invariably comprise a hydrophilic β-D-glucopyranosyl unit, an O-sulfated anomeric (Z)-thiohydroximate function (Chart 1a), connected to a rather hydrophobic sidechain whose constitution, depending on plant species, is the most frequent structural variant and derives from α-amino acid biosynthetic precursors (Chart 1b). [START_REF] Blažević | Glucosinolates: Novel Sources and Biological Potential[END_REF][START_REF] Agerbirk | [END_REF] A small number of GLs have their carbohydrate unit esterified by aromatic acids such as benzoic, p-coumaric, or isoferulic acid. [START_REF] Agerbirk | [END_REF] Several methionine derived GLs, which constitute the largest group of GLs, bear in their sidechain an additional sulfur atom at different oxidation states (sulfide, sulfoxide, or sulfone).

One of the most studied GLs is glucoraphanin, a compound found in the Brassicaceae, notably in broccoli and Tuscan black kale. 3 To date, about 130 different GLs have been identified, [START_REF] Agerbirk | [END_REF] and these compounds are associated in plants to myrosinase (E.C.3.2.1.147), an atypical glucohydrolase able to convert GLs into the corresponding isothiocyanates (ITCs). Thus far, ITCs are the most intensively studied GL breakdown products and have recently attracted intense research interest regarding their protective effect against cancer and neurodegenerative disorders.

A recent study estimated a possible neuroprotective effect of glucomoringin, 4-(α-Lrhamnopyranosyloxy)benzyl glucosinolate, bioactivated with the enzyme myrosinase to form the corresponding ITC moringin, 4-(α-L-rhamnopyranosyloxy)benzyl ITC, in the treatment or prevention of Parkinson's disease. 4 The possible pest control by ITCs from GLs for agricultural crop protection has been also reported. 5,6 The relationships of the GL-myrosinase system with insects have therefore been thoroughly studied. 7,8 The analysis of GLs by HPLC is a difficult task that is often carried out on desulfated GLs (DS-GLs) which result from the reaction of GLs with sulfatases. 9 DS-GLs being standards for GL identification, this study covered both native and desulfated GLs. Chart 1 shows the invariant part of GLs and the variable parts of the compounds involved in the present study. Each of the GLs studied here has a trivial name, most often given according to the plant from which it has been isolated for the first time, and a "code", which is an abbreviated form of the name. The codes are by no means officially assigned to the compounds but are of common use among GL specialists.

The compounds studied were related to 17 sidechains (Table 1) and in 14 cases both GL and DS-GL forms were available. Only GRE, DS-GCL, and DS-GSY had no counterpart in our collection, thus leading to the analysis of 31 compounds, designated hereafter as the (DS)GL collection.

The present work was first intended to provide reference NMR data on selected GLs and DS-GLs for identification purposes. The description of the 1 H NMR spectra in terms of NMR parameters, considered here as chemical shifts and coupling constants, was hampered by the presence of strongly coupled spin systems, even in spectra recorded at 600 MHz. The PERCH software was consequently used for accurate spin network analysis. 10 In brief, the PERCH software determines a starting set of NMR parameters from a molecular structure and refines them so that the corresponding reconstructed 1 H NMR spectrum fits best with the experimental one. The necessity of reporting analysis results in a way that could fit with their dissemination via computerized databases has led to the design of a human-and machine-readable data format. 11 The influence of desulfation and of the sidechain constitution on chemical shifts is also discussed, in relation with molecular 3D structures.

RESULTS AND DISCUSSION

Compounds, Public Databases, and Study Outcome. An attempt to find Chemical Abstracts Service 12 (CAS) Registry Numbers of the compounds of the (DS)GL collection failed for DS-GSY and DS-GCL. The DS compounds unknown to CAS (as well as the others) were prepared as analytical standards by action of sulfatase on intact GLs. All GLs are known to include a Zconfigured imino double bond, as documented in section "Lossen and Beckmann Rearrangements" of a review article by Walter and Schaumann. 13 Since the configuration of the imino double bond does not always appear explicitly in the publication from which CAS collected the registered structures, this configuration is often wrong or left "unspecified" in the CAS database. Other databases such as ChemSpider or PubChem contain 2D and 3D GL structures that are incorrect. The side chains of GIB, GRA, and GRE contain a methylsulfinyl group, whose R absolute configuration of the sulfur atom is rarely specified. 14 The first sample of DS-GRA that we analyzed was indeed a mixture of diastereoisomers obtained by nonstereoselective oxidation of DS-GER. 15 A sample of DS-GRA from desulfation of GRA was also analyzed (vide infra).

The publicly available outcome of the present study consists of, i) a collection of SDF 16 files which comprise 3D structures, atom naming information, 1 H, 13 C, and 15 N chemical shifts, and [START_REF] Blažević | Glucosinolates: Novel Sources and Biological Potential[END_REF] H-1 H scalar coupling constants determined by PERCH; ii) a collection of NMR files that include 1D and 2D time domain data, 1 H, 13 C, 1 H-13 C HSQC, 1 H-13 C HMBC, and 1 H- 15 N HMBC spectra, and 1 H spectra reconstructed by PERCH; iii) a spreadsheet file that summarizes all the NMR-related information and a simple text file version of it; and iv) a relational database file that includes the content of the spreadsheet file as well as structural information that would enable the rebuild of the 3D molecular structures from scratch.

The 3D models of the (DS)GL collection members were built initially for the 1 H NMR parameter determination assisted by the PERCH software. The structure files in MMS PERCH format were exported in the MOL 16 format in order to prepare the assembly of the SDF files. The assembly process also involved a re-indexing and a naming of the atoms. Each (DS)GL is associated to a human-readable SDF file that contains, i) atom data, with one line per atom including 3D coordinates, the first line of atom data being related to atom indexed 1; ii) bond data that refer to atom indexes as previously defined; iii) atom naming data correlating numerical atom indices and textual atom names; and iv) property value pairs, including NMR parameter names derived from atom names and their associated numerical values. It must be emphasized that not only chemical shift values are associated to atom indexes (as usual for resonance assignment) but that, less commonly, coupling constant values are systematically related to index pairs of coupling atoms (for coupling assignment).

Atom re-indexing was carried out according to Chart 1a. Heavy (i. e. non-hydrogen) atoms in the DS-GL substructure that is common to all structures are numbered from 1 to 16, the sidechain heavy atoms are then numbered according to Chart 1b, always starting at index 17. In the sulfate group of an intact GL, the sulfur atom receives the next available index and the three bound oxygen atoms receive the next three available indexes. Hydrogen atoms are then indexed in the order of the heavy atoms they are attached to. Heavy atom naming is straightforward: atom 1 is named C1, the name being built from the atomic symbol followed by the atom index. The rhamnose carbon atoms in the GMG sidechain, indexed from 24 to 29, were exceptionally named from C1' to C6' instead of C24 to C29 and their attached hydrogen atoms named from H1' to H6'. Diastereotopic methylene hydrogen atoms receive a suffix in their names, so that H6R and H6S designate the pro-R and pro-S hydrogen atoms bound to C-6. Identically, suffixes Z and E in terminal methylene groups are appended to the names of pro-Z and pro-E hydrogen atoms. Suffixes A, B, and C are appended to the names of the equivalent hydrogen atoms in methyl groups.

NMR parameter names are used as property names in the data section of the SDF files. For example D_C1 is the name associated to the value of the chemical shift of C-1, D denoting delta, the usual symbol of chemical shift. In the same way, J_H1_H2 and J_H2_H1 are the names of the coupling constant value shared by H-1 and H-2, so that a textual search for "J_H2" in an SDF file easily highlights all the couplings of H-2 without missing its coupling with H-1.

Once all information pieces about each (DS)GL have been integrated in an SDF file, the atom 3D coordinates from PERCH structure exportation were re-calculated by conformational analysis. Conformers were generated and optimized using a classical force-field molecular mechanics approach. The coordinates of lowest energy conformer were reported in the final, publicly released SDF files (Supporting Information).

At this stage, it was possible to sequentially read all the SDF files in order to produce a single textual data table, for which Table 2 presents the sections for sinigrin and DS-sinigrin. A simple importation into a spreadsheet of the whole text, using the tabulation character as separator, provides an easy way to read it and to save it as an Excel file, for example.

In a more database-oriented perspective, we also provide the (DS)GL NMR data as an SQLite file as well. SQLite is a lightweight, public-domain, zero-configuration relational database system that is embedded in many software offerings, which one uses every day without being aware of it. 17 The GLs.db database file that was created from the SDF files contains seven tables.

The NAMES table associates compound codes (like DS-SIN) to compound names (DS-sinigrin).

The database tables named HNMR, CNMR, and NNMR associate a compound code, a position name (1 for C-1, 6 for H-6R and H-6S), an atom label, and the corresponding chemical shift value. The JHH table associates a compound code, a position name, and an atom label for two hydrogen atoms, and the corresponding coupling constant value. Two tables were dedicated to structural information: ATOMS associates a compound name, an atom label, a chemical symbol, an electric charge, and a chirality flag (R/S for stereogenic centers, including those at sulfur in sulfoxides) and BONDS that associates a compound code, two atom labels, the multiplicity of the bond between the two atoms, and a configuration flag (Z/E). The ATOMS and BONDS tables content is sufficient in order to build a 3D model of the molecules. The GLs database was created in order to be able to extract data from the tables in any possible way using an SQL query like "select Code, J from JHH where Atomname1 = "H1" AND Code NOT LIKE "DS%" for the collection of the J(H-1, H-2) coupling constants among all intact GLs. Determination, Assignment, and Collection of NMR Parameters. The determination and assignment of 1 H, 13 C, and 15 N chemical shifts of each compound in the (DS)GL set started with the recording of 1D and 2D spectra. The basic rules of 1 H and 13 C chemical shift interpretation [START_REF] Silverstein | Spectrometric Identification of Organic Compounds[END_REF] and the analysis of 2D 1 H-1 H COSY, 1 H-13 C HSQC, and 1 H-13 C HMBC [START_REF] Bross-Walch | [END_REF] spectra easily produced 1 H and 13 C NMR chemical shift assignments due to the structural simplicity of the studied compounds. The assignment process is exemplified hereafter by means of the spectra of glucolimnanthin (GLI) in Figure 1.

Like in all (DS)GLs, C-7 plays a central role, as it correlates in the HMBC spectrum with H-1 in the sugar unit and with H-8 in the sidechain. The signal of H-1 is well separated and generally close to the HOD solvent signal; it correlates in the COSY spectrum to H-2. The HSQC spectrum, when recorded with J-modulation (CH2 signal in opposite phase with those of CH and CH3 groups), readily gives the position of both the C-6 protons. In the COSY spectrum, the latter correlates with H-5, H-5 with H-4, and H-4 with H-3. In the present case, the near complete All NMR spectra of the GLs were recorded in D2O, a solvent that mimics at best the natural environment of these compounds. NMR in D2O precludes observation of the exchangeable protons, mainly those from the hydroxy groups of the glucose unit, a loss that is not detrimental for the structural analysis of (DS)GLs. Sodium 3-(trimethylsilyl)propionate-2,2,3,3-d4 (TSP-d4) was used as internal reference compound for 1 H and 13 C NMR spectra. This gave the opportunity to evaluate the possibility of referencing 13 C chemical shifts from the [START_REF] Blažević | Glucosinolates: Novel Sources and Biological Potential[END_REF] H shifts. 20 The study was carried out on a set of 28 samples for which 1 H and 13 C NMR spectra were recorded. The resonance frequency of the methyl groups of TSP-d4 was calculated by adding their offset frequency to the spectrometer base frequency (BF1) and setting the Spectrum Reference (SR) calibration parameter to 0. The average apparent chemical shift of TMS in 1 H NMR spectra (νTMS -BF1) was -0.093 ppm (-55.8 Hz), thus meaning that if SR equals 0 then 0.093 ppm must be added to the chemical shift values read on spectra. The standard deviation of this calibration parameter was 6.5 ppb (3.9 Hz), ensuring that calibration without any added reference compound may be safely undertaken. The average apparent chemical shift of TMS in 13 C NMR spectra (SR( 13 C) = 0) was -2.92 ppm (-440.5 Hz) with a 28 ppb (4.2 Hz) standard deviation. A 13 C NMR spectrum calibration with a null SR and without TSP-d4 internal reference results in a systematic error of about 3 ppm on chemical shifts. The calculation of SR( 13 C) from SR( 1 H) using TSP-d4 as reference in D2O at 298K can be achieved using Ξ = νTMS( 13 C)/ νTMS( 1 H). The average 10 9 .Ξ value is 251449491 with a standard deviation of 9. The application of this mean value to the set of νTMS( 1 H) leads to an average error of -0.01 ppm on 13 C chemical shifts with a 0.03 ppm standard deviation, which is compatible which the order of magnitude of chemical shift accuracy that is required by chemistry journals. Using the Ξ value from reference for TSP at 295K would result in a 0.2 ppm glitch of the 13 C chemical shift scale. 21 The choice of TSP-d4 as a water-soluble NMR reference compound is common albeit questionable. As salt of a strong base and a strong acid, DSS (or 4,4-dimethyl-4-silapentane-1sulfonic acid sodium salt) would in theory be a better choice since the protonation state of the sulfonate group is not expected to be modified by one of the analyzed compounds. Conversely, the carboxylate anion in TSP may undergo protonation at low (<5) pH. GL samples were available as potassium salts, i. e. salts of a strong base and a strong acid. The pH value of their solution is close to 7 so that any possible change in the protonation state of TSP should not induce a significant variation of the methyl group chemical shifts. Another water-soluble standard, DSA (4,4-dimethyl-4-silapentane-1-ammonium trifluoroacetate), may also show some chemical shift dependence on pH variation due to the presence of an ammonium group. 22 The 1 H- 15 N HMBC spectra were used to indirectly determine the 15 Handling 1 H NMR Data. The 1 H NMR spectrum of sinalbin (SNB) was among the first recorded ones; the first order analysis of its glucose signals was found to be inappropriate, as immediately visible by the inspection of the anomeric signal: while a doublet was expected, a set of 10 lines was observed, as reported in Fig. 2a. This high complexity of a signal is rationalized by the presence of a virtual coupling between H-1 and H-3 caused by the nearly equal resonance frequencies of H-2 and H-3. In our experience, this situation is likely to happen with benzylic sidechains (DS)GLs and was observed for glucolimnanthin (GLI), glucomoringin (GMG), or glucotropaeolin (GTL), as well as their desulfated homologues. The complex aspect of the H-1 signal in benzylic (DS)GLs has most probably been observed but never reported, except for a recent publication by us. 23 The AA'XX' (Δν = 217 Hz at 600 MHz) spin system of aromatic 1 H nuclei in SNB also showed two patterns, each featuring six peaks (Figures 2b and2c) that cannot be analyzed at first-order. Therefore, use of iterative NMR parameter refinement software was needed to provide spectra interpretation. The goal of using such software is to extract the coupling constant and chemical shift values for which the corresponding simulated spectrum fits best with the experimental one. It is worth noting that a coupling constant value is associated to the pair of nuclei it originates from, information that is rarely reported. The PERCH software makes use of a hypothesized structure for the molecule under investigation in order to propose a realistic guess of the NMR parameters that are subsequently refined using the integral transform based method. 10 The matching between the refined and the initially predicted parameters, as well as the matching between the predicted and the experimental spectra contribute to the rating of the supposed structure as being the one of the analyzed compound. PERCH can therefore be recognized as a structure validation tool, even though we used it without doubting that the structure of the analyzed compounds were correct (with one exception, see below). We only used PERCH for its ability to extract NMR parameters even from the spectra of highly complex strongly coupled spin systems. Such an approach has been put into practice in the field of natural product chemistry. 24 The advantage of our approach, offering free accessibility to raw, Fourier transformed and interpreted NMR data, lies in the possibility for any interested reader to check and re-analyze data according to personal current interest. The analysis of the spectrum of glucolimnanthin (GLI) by PERCH is exemplified in Figure 3 and shows how complex signals that would be reported as multiplets can be fully interpreted.

One of the problems inherent in the assignment of 1 H NMR chemical shifts to hydrogen atoms has been the handling of diastereotopic hydrogen atoms. In the first place, we paid attention to chemical shift assignments of the pro-R and pro-S C-6 methylene protons and to their coupling constants with H-5 in the glucose units because it is a common part of all (DS)GL compounds.

Taking into account the detailed analysis of the 1 H NMR spectrum of methyl β-glucopyranoside in D2O, 25 it appeared that the most deshielded H-6 should be the one with the smallest coupling constant with H-5, regardless of the diastereotopic assignment. Considering (DS)GL [START_REF] Blažević | Glucosinolates: Novel Sources and Biological Potential[END_REF] H NMR spectra, it also appeared that the most deshielded H-6 has the small coupling constant in all cases, except for compounds GLI and GMG. For GLI, PERCH converged towards two identical values of δ[H-6(pro-R)] and δ[H-6(pro-S)], thus making the result of chemical shift versus coupling constant consistency test undetermined. For GMG, the chemical shift difference was 0.01 ppm and the coupling constant difference was lower than 0.01 Hz, thus making the consistency test unreliable. Following the conclusions of Nishida et al. we imposed the assignment that makes H-6(pro-S) more deshielded than H-6(pro-R). NMR by itself cannot provide directly any argument to decide which assignment is the correct one. In PERCH, NMR parameters are predicted on the basis of a set of conformers that is generated during a short molecular dynamics run. 26,27 Considering 31 compounds for which prediction was carried out, the inequality δ[H-6(pro-S)] > δ[H-6(pro-R)] was wrong in nine cases and J(H6S/H5) < J(H6R/H5) was wrong in 14 cases. The consistency between predicted orders of chemical shifts and coupling constants was wrong in seven cases. Parameter optimization thus reduced the number of inconsistencies from seven to two (among which an undecided and a borderline case for the latter). From these observations, it appeared that PERCH predictions are not necessarily accurate regarding its ability to provide correct assignments of diastereotopic protons in methylene groups, at least in our standard usage conditions. Moreover, the accuracy of the initial NMR parameter prediction determines the relevance of the adjusted parameters because an inversion in the initial order of H-6(pro-R) and H-6(pro-S) chemical shifts may persist during parameter adjustment. The assignment inversion of pro-R and pro-S H-6 that occurred in nine cases for the adjusted parameters was corrected by swapping δ[H-6(pro-R)] and δ[H-6(pro-S)] values as well as the corresponding J(H-6(pro-R), H-5) and J(H-6(pro-S), H-5)values.

Proton NMR parameter evaluation of a compound by PERCH is possible only if its 3D structure has been established by means of the integrated MMS molecular modelling system.

During interactive molecule building, each atom receives a label that does not correspond to the standardized ones in Chart 1. Even though MMS is able to assign pro-R and pro-S status to diastereotopic hydrogen atoms in methylene groups and to assign absolute configurations to stereogenic centers (except the sulfur-centered ones), atom labels do not include pro-R/pro-S or pro-E/pro-Z qualifiers when MMS files are converted to the MOL files from which SDF files were produced. Therefore, an equivalence table between MMS A/B and pro-R/pro-S or pro-Z/pro-E qualifiers was built by hand and exploited at the time the MMS-generated MOL files were converted into SDF files. For each compound, a so-called "master file" was created, containing all different pieces of information needed to build the SDF files, such as i) the method to replace MMS labels by standardized labels in the variable part of the GL or DS-GL structures;

ii) the interpretation of A/B MMS qualifiers; iii) the location of PERCH-generated NMR parameter files; iv) the location of the 13 C NMR peak list files; v) the 1 H and 13 C calibration parameters (original 1D NMR files were not calibrated, meaning that the spectrum referencing parameter was left to zero); vi) the list of MMS labels of the carbon atoms in the decreasing order of their chemical shifts; and vii) the uncorrected 15 N NMR chemical shift value of the imino nitrogen atom.

The automation of the 1 H NMR data handling was made possible by the processing of the ACA files (Automated Consistency Analysis) that contained the result of the NMR parameter determinations carried out by PERCH. An ACA file is indeed an XML file (eXtensible Markup Language). XML is a data description language with a structured and human-readable syntax similar to the one of HTML, for which efficient data look-up software libraries are available. 28 The search for the relationship between MMS atom labels and chemical shift values in an ACA file is therefore straightforward, as well as the search for the relationship between pairs of MMS atom labels and coupling constant values; this automated search step avoided the manual extraction of data from the PERCH graphical user interface. For 13 C chemical shifts, a peak list in XML format was created by the TopSpin software. This list was easily read and the chemical shift values were corrected by addition of the calibration parameter.

Relationships between Structures and NMR Parameters.

A superficial observation of the 2D structure of GLs might lead to conclude that the D-glucopyranose unit just ignores the remote sidechain. A convincing proof of the opposite was shown by the 1 H and 13 C NMR spectra of semisynthetic (RS)-DS-GRA, 15 a mixture of epimers that differ by the absolute configuration of the sulfoxide group. PERCH analysis of the 1 H NMR spectrum of DS-GRA recorded at 600 MHz was initially performed at a time the sample was thought to be a single diastereoisomeric form. PERCH showed a poor fit for the H-6(pro-R) resonance, all other protons being correctly processed. The expected signal should be a doublet of doublet (dd), corresponding to the coupling of H-6(pro-R) with H-5 and H-6(pro-S). However, the observed signal resembled a ddd that did not fit well with a dd (top 1D trace in Figure 4b). A 1 H, non-tilted, 2D J-resolved spectrum was recorded in order to obtain a clue to the origin of the extra splitting (Figure 4). It showed that the ddd was indeed a pair of dds, the narrowest splitting, of about 3 ppb or 1.8 Hz, corresponding to the difference of resonance frequencies of the H-6(pro-R) in what was revealed to be a mixture of two epimeric sulfoxides. Figure 4a displays the 1D 1 H NMR spectrum of (R)-DS-GRA obtained by desulfation of natural GRA and shows that H-6(pro-R) is more deshielded in (R)-DS-GRA than in (S)-DS-GRA. Inspection of the 13 C NMR spectrum of (RS)-DS-GRA permitted measurement of peak splittings for C-7 (Δδ = 15 ppb), C-1 (Δδ = 14 ppb), C-4 (Δδ = 11 ppb), C-19 (Δδ = 20 ppb), and C-18 (Δδ = 42 ppb) thus revealing the influence of the sulfoxide configuration on the sidechain and on the sugar part of DS-GRA. Such a long distance influence between the end-of-chain sulfoxide and the sugar moiety would not have been easy to anticipate if not experimentally proven. A key point about 13 C NMR chemical shift values in the D-glucopyranose unit is the invariability of the chemical shift order: δ(C-1) > δ(C-5) > δ(C-3) > δ(C-2) > δ(C-4) > δ(C-6). The order of 1 H NMR shift values for H-2, H-3, H-4 and H-5 is more subject to changes, depending on the nature of the sidechain, even though δ(H-1) > δ(H-6), the latter being greater than all other ones. Assuming the given order for 13 C, the determination of 1 H chemical shifts can be secured by the 1 H-13 C HSQC spectrum, the particular case of H-6(pro-R) and H-6(pro-S) having been considered in the previous section. When the sidechain is of the benzylic type, the order δ(H-4) > δ(H-3) ≈ δ(H-2) > δ(H-5) prevails, the similarity of δ(H-3) and δ(H-2) resulting in the virtual coupling of H-1 and H-3 as shown in Figure 2a. In the case of nonbenzylic sidechains, the observed order is mostly δ(H-3) > δ(H-5) > δ(H-4) > δ(H-2) with δ(H-3) close to δ(H-5) and δ(H-4) close to δ(H-2). The difference in 1 H chemical shifts in the D-glucopyranose moiety between non-benzylic and benzylic (DS)GL is illustrated in Figure 5 by portions of the HSQC spectra of DS-GCL and GLI.

The search for low energy conformers of the (DS)GLs was partly motivated by an attempt to compare the structures of GLs with those of their corresponding DS-GLs and by the search for a possible source of interaction between the sugar moiety and the sidechain. The main result, exemplified by SNB and GRA in Figures 6a and6c, is the presence of a hydrogen bond between an oxygen atom of the sulfate group and the hydrogen of HO-2 in all the GLs. The desulfated molecules do not show such a structural feature as demonstrated in Figures 6b and6d in the case of DS-SNB and DS-GRA. Desulfation, therefore, induces a clear conformational change, even though no particular evidence of it is evident in the NOESY spectra. The hydrogen bond in intact GLs keeps C-8 close to the sugar moiety, thus setting the molecule in a folded conformation, with S-11 located at the turning point.

The main effect of desulfation on chemical shift values is most probably due to the electron withdrawing effect of the sulfate group: the 15 N nucleus N-10 is shielded by 8.0 ppm on average by desulfation and the adjacent imino C-7 position is shielded by 8.1 ppm on average. Nearly no effect can be detected for H-1. The example given in Table 2 illustrates the low impact of desulfation on the chemical shifts of sinigrin (SIN). Figure 7 also shows how desulfation has a negligible influence on 1 H chemical shifts of gluconasturtiin (GST) and only affects noticeably C-7 of this compound.

Beyond Glucosinolates, NMR Parameter Reporting. Reporting correct structures and NMR parameter assignments is an important part of the activity of organic chemists as the corresponding data constitute a basis for the identification of known molecules during dereplication 29 and of new molecules during structure elucidation. 30 At the beginning of the application of NMR to organic molecules, the attention was focused on the reporting of 1D NMR parameters, which were the only ones available back then. The advent of 2D NMR made the 1D spectra interpretation more reliable when dealing with compounds of constantly increasing molecular complexity. In the context of a digital world, there is presently no incentive to publish NMR parameters in a format that would be readily understandable by computers with a high level of reliability. 11 The traditional way of reporting NMR parameters, i. e., the one used by scientific journals in general and by the Journal of Natural Products (JNP) in particular is hardly usable by computers. The "line notation" or the "table notation" as reported on page 4 of the "Author Guidelines" of JNP constitute, therefore, dead-ends for significant findings. 31 One of the reasons is that line headers in tables are position numbers in molecular structures whose drawing is separated from NMR data across article pages and in which the numbering is often incompletely or implicitly defined. The back-conversion of a printed structure drawing into a MOL or SDF file is a complex problem for which the best solution is to avoid having to solve it.

The SDF files we created for each analyzed (DS)GL gather at the same place the molecular constitution, the atom numbering scheme, and the NMR parameters, all reported in a humanreadable/writable and a computer-readable/writable format. The NMR time-domain (raw NMR data) and frequency-domain data (the NMR spectra) are grouped together in a way that is understandable by most, if not all, NMR software and in a way specified by the presently most representative NMR instrumentation actor worldwide. The link between NMR data files and SDF files is ensured in the present case by the compound code; this simple option might have been replaced by a more general, but probably less readable, naming scheme.

The idea of aggregating NMR data, NMR parameters, and molecular structures has already been put into practice in the still developing NMR Mark-up Language project. 32 As indicated by its name, this project relies on an XML-based approach and provides data files that are computer-friendly but poorly adapted to the reading and writing by humans. In an even more general approach, the Allotrope Foundation attempts to develop a technology aiming to rule in a general way the production, exchange, exploration and storage of chemical and biological laboratory data. [33] Besides these heavyweight actors of analytical data handling, discussions are underway for the definition of an NMR parameter exchange format based on SDF files, whose details will soon be released and follow the guidelines proposed in references 11 and 34.

The shifting from a paper-based to a computer-based dissemination of chemical knowledge will have to proceed through a change in data publication habits. This change is motivated by a search for more efficiency in the rapid identification (dereplication) of known natural products, on elaboration of better NMR parameter prediction tools based on error-free databases, and therefore on availability of better tools for unknown structure identification.

In conclusion, a collection of 16 glucosinolates and 15 desulfated glucosinolates, including 14 pairs of compounds with identical sidechains, has been studied by NMR, using D2O consistently as solvent. The accurate interpretation of the 1 H NMR spectra initially required the assignment of the 1 H and 13 C resonances using standard 2D HSQC, COSY, and HMBC techniques. The extraction of 1 H-1 H coupling constants from the 1 H NMR spectra was achieved by computerassisted iterative refinement of chemical shifts and coupling constants using the PERCH software. This approach to spectrum interpretation goes beyond the commonly undertaken firstorder analysis and also assigns to each coupling constant value the pair of atoms it originates from. The question of the assignment of chemical shifts of diastereotopic proton pairs remains a challenge, which has been discussed for the hydrogen atoms in the hydroxymethyl group of glucose units. The 15 N NMR data of the studied molecules are reported for the first time. All the NMR parameters were collected as text in a computer file and organized in a structured database for a possible exploitation by an automated process. The most visible effect of desulfation on chemical shifts is a shielding of the carbon and nitrogen atoms of the imino functional group, thus resulting in only small modifications of the 1 H NMR spectra. The presence of a benzylic sidechain, both in native and desulfated compounds, induces a strong similarity of chemical shift values of the hydrogens at positions C-2 and C-3 in the glucose unit and creates therefore strong coupling patterns in the 1 H NMR spectra of these compounds, clearly visible on the signal of the anomeric proton. A study of the 3D structure of the compounds in the collection was undertaken by force-field molecular modelling in order to attempt to relate the nature of sidechain, the sulfation state, the 3D structure, and NMR parameters. This study showed the presence of an intramolecular hydrogen bond between the sulfate group of native compounds and the sugar unit, a bond that induces a spatial proximity between the sidechain and the face of the sugar ring that contains axial hydrogens at C-2 and C-4. Even though it was not possible to rationalize the particularity of benzylic compounds by considering their geometry, the calculated 3D structures were aggregated with their corresponding NMR parameters in Structure Data Formatted files, in a way that is currently considered as a future standard for NMR data dissemination.

EXPERIMENTAL SECTION. All samples originate from the Consiglio per la Ricerca in

Agricoltura e l'Analisi dell'Economia Agraria, Agricoltura e Ambiente (CREA-AA), Via di Corticella 133, 40128 Bologna, Italy (previously Industrial Crop Research Centre, Agricultural Research Council). They were provided in air-free sealed vials due to their possible hygroscopic nature. Intact GLs were isolated as their potassium salts.

MHz Bruker Avance III spectrometer equipped with a 5 mm BBFO+ probe; and ii) a 600 MHz Bruker Avance III spectrometer equipped with a 5 mm TCI cryoprobe, both driven by the TopSpin NMR software (v2.1), which was also used for data processing. Spectra were recorded by means of standard Bruker pulse sequences, namely cosygppr for COSY, hsqcedetgpsisp2.2 for HSQC, hmbcgplpndqf for 1 H-13 C HMBC and hmbcf3gpndqf for 1 H- 15 N HMBC Conformational analysis was carried out by means of the Macromodel software, with the MM3 force-field and water as implicit solvent, using default parameters for starting geometry generation and for atom coordinate optimization. 35 The handling of MOL and SD files, atom labelling, NMR parameter storage, and the determination of atom and bond configuration were carried out using the OEChem ToolKit. 36

  superimposition of H-2 and H-3 resonances complicates their differenciation by means of the COSY spectrum. The assignments of H-1 to H-6 and C-1 to C-6 may be done by the observation of their HMBC correlations and by the observance of the chemical shift order of the carbon atoms (see in section "Relationships between Structures and NMR Parameters"). The assignment of the resonances from the sidechain clearly depends on the nature of this sidechain. Because H-8 is identifiable by means of the HMBC spectrum and C-8 by the HSQC spectrum from H-8, the resonances arising from the aromatic and the methyl positions remain to be assigned. The intense singlet resonance of the Me-24 atoms correlates with the signal of the aromatic C-21, thus giving their respective assignments. C-17 is readily identified because it is the only unassigned quaternary carbon. The sharp signal of H-22 is readily assigned because H-22 is only subject to long-range couplings. The H-19 signal is readily located because it shows two aromatic vicinal couplings and appears as a broad pseudo-triplet. The HMBC correlation between C-8 and H-18, as well as the one between H-8 and C-18, provide the chemical shifts of H-18 and C-18 and, by elimination, those of H-20 and C-20, thus completing the resonance assignment task. Interestingly, the 1 H NMR spectra of GLI and DS-GLI are nearly superimposable: the most important variation is at one of the C-8 methylene protons, shielded by 0.11 ppm in DS-GLI. It is therefore almost impossible to distinguish GLI and DS-GLI by sole inspection of their 1 H NMR spectra. The 13 C NMR spectra of GLI and DS-GLI are highly similar, the most noticeable difference being the shielding of C-7 by 8.2 ppm in DS-GLI.

  N chemical shift of the nitrogen atoms, exploiting the correlation of the H-8 signals with those of N-10. For each studied compound, a spectrum was first recorded with a 100 ppm window in F1 and a small number (16) of t1 increments in order to obtain a quick (3 min recording time) estimation of the 15 N chemical shift. A 20 ppm spectral width and 128 increments (11 min recording time) were then used to obtain a better resolved spectrum with a better signal-to-noise ratio. The collected 15 N chemical shifts were stored uncorrected (SR = 0, BF = 60.813685 MHz) in the NMR parameter files. This study constitutes the first determination of 15 N NMR resonances in glucosinolates.

Figure 1 .

 1 Figure 1. a) 1 H (top: overview, bottom: details), b) 13 C, c) 1 H-13 C HSQC, d) 1 H-1 H COSY, and e) 1 H-13 C HMBC NMR spectra of glucolimnanthin (GLI) recorded at 600 MHz for 1 H and 150 MHz for 13 C. Atoms are numbered according to Chart 1. The * symbols in the HSQC spectrum (c)

Figure 2 .

 2 Figure 2. Expansions of the 1 H NMR spectrum of sinalbin, (SNB) recorded at 600 MHz, showing

Figure 3 .

 3 Figure 3. Top: 1 H NMR signals of glucolimnanthin (GLI), recorded at 600 MHz, and their

Figure 4 .

 4 Figure 4. a) 1 H NMR signal of H-6(pro-R) in desulfo-glucoraphanin (DS-GRA). b) 2D J-resolved

Figure 5 .

 5 Figure 5. Parts of the HSQC spectrum of a) desulfoglucocleomin (DS-GCL) and b) of glucolimnanthin (GLI) that show typical patterns of 1 H NMR chemical shift distributions for H-2,

Figure 6 .

 6 Figure 6. Lowest-energy conformers of a) sinalbin (SNB) and b) desulfosinalbin (DS-SNB)

Figure 7 .

 7 Figure 7. 1D NMR spectra of gluconasturtiin (GST) and its desulfated analogue (DS-GST) in D2O. a) and b) comparison of the 1 H NMR spectra of GST and DS-GST (residual HOD signals marked with X), c) and d) comparison of the 13 C NMR spectra of GST and DS-GST.

Table 1 .

 1 List of the Studied GLs and DS-GLs. Each Sidechain (Chart 1b) Refers to a Code and to a GL Usage Name. The "No" Entries Correspond to Compounds Unavailable to Us.

	Code Name	GL	DS-GL	Botanical origin
	EPRO epi-progoitrin	Yes	Yes	Crambe abyssinica
	GBB glucobarbarin	Yes	Yes	Barbarea vulgaris
	GCL	glucocleomin	No	Yes	Cleome hassleriana
	GER	glucoerucin	Yes	Yes	Eruca sativa
	GIB	glucoiberin	Yes	Yes	Iberis amara
	GLI	glucolimnanthin Yes	Yes	Limnanthes alba
	GMG glucomoringin	Yes	Yes	Moringa oleifera
	GNA gluconapin	Yes	Yes	Brassica rapa
	GRA glucoraphanin	Yes	Yes	Brassica oleracea var. acephala
	GRE	glucoraphenin	Yes	No	Raphanus sativus
	GRH glucoraphasatin	Yes	Yes	Raphanus sativus var. longipinnatus
	GST	gluconasturtiin	Yes	Yes	Barbarea verna
	GSY	glucosisymbrin	Yes	No	Sisymbrium loeselii
	GTL	glucotropaeolin	Yes	Yes	Lepidium sativum
	PRO	progoitrin	Yes	Yes	Brassica napus
	SIN	sinigrin	Yes	Yes	Brassica juncea
	SNB	sinalbin	Yes	Yes	Sinapis alba

ACKNOWLEDGMENT

Financial support by CNRS, Conseil Regional Champagne Ardenne, Conseil General de la Marne, Ministry of Higher Education and Research (MESR) and EU-programme FEDER to the PlAneT CPER project is gratefully acknowledged. We thank Dr. Matthias Niemitz, PERCH Solutions Ltd., for his advices on the use of the PERCH software.

AUTHOR INFORMATION

Corresponding Author *jm.nuzillard@univ-reims.fr ORCID: 0000-0002-5120-2556

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

Notes

The authors declare no competing financial interest.