Laurent Molez

Yves Berthaud

Benoît Bissonnette

Denis Beaupré

Modelling drying and loading effect in structural concrete repair

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.

INTRODUCTION

Second half of twentieth century is marked by an intense activity of construction of concrete infrastructures. These structures have a limited lifespan, and several of them require repairs today. Thus the field of concrete repairs knows, since ten years, a significant rise in the sector of construction. For owners, achieving durable repairs is of primary importance because the costs involved are generally very high. Data presented at table 1 give an idea of the extent of work in the field of repairs. Experience shows, however, that design of durable concrete repairs can be as complex as the design of new structures because each damaged structure imposes its own set of conditions [START_REF] Emmons | Factor affecting the durability of concrete repair: the contractor's viewpoint[END_REF] In this study, concrete repair is defined as a thin concrete layer ( 50 to 150 mm) cast on an existing structure to replace the deteriorated part close to the surface. These two concrete layers have differ- [START_REF] Saucier | The challenge of numerical modeling of strains and stresses in concrete repairs[END_REF] resume the effect of hygral, thermal and mechanical change. Shrinkage is identified to be the most severe parameter [START_REF] Saucier | Air-void stability, part v: temperature, general analysisand performance index[END_REF][START_REF] Bissonnette | Le fluage en traction: un aspect important de la problématique des réparations minces en béton[END_REF][START_REF] Laurence | La fissuration due au retrait restreint dans les réparations minces en béton: apports combinés de l'expérimentation et de la modélisation[END_REF]). In the other hand, tensile creep can partly relax the induced strains [START_REF] Bissonnette | Le fluage en traction: un aspect important de la problématique des réparations minces en béton[END_REF]. The cracking of repair materials can cause the most serious deterioration processes leading to repair failures, since the aggressive agents can then penetrate through these cracks.

We first present the numerical model developed in this research study. Hydration process is not modelled, but its effects are taken into account in the evolution of physical and mechanical characteristics.

Endogenous shrinkage is considered. The drying is modelled by a non-linear diffusion equation, and drying shrinkage is assumed to be proportional to relative humidity changes. Mechanical behaviour is introduced with respect of basic and drying creeps. A nonlinear continuum damage model is applied to concrete rupture. In second time, numerical results are compared to experimental ones from previous study [START_REF] Molez | Structural concrete repair behavior: coupled effect of drying and loading[END_REF][START_REF] Molez | [END_REF].

NUMERICAL MODEL 2.1 Hydration effects

Hydration mechanisms are not modelled, but their effects have to be taken into account. Degree of hydration ξ can be evaluated from the experimental evolution of compressive strength f c (t) (Ulm 1999):

ξ(t) = ξ 0 + (1 -ξ 0 ) f c (t) f c (∞) (1)
where ξ 0 is the percolation threshold as defined by [START_REF] Acker | Caractérisation des pâtes de ciment et des bétons -Méthodes, analyse, interprétations[END_REF] and f c (∞) is the compressive strength at an infinite time.

Thereafter, the evolution of strength will be expressed according to the following equation:

R(t) = R(∞) t m 1 m 2 + t m 1 (2)
where R(∞) is the strength at an infinite time, and m 1 , m 2 parameters determined from experimental results (figure 2).

Thermal effects

We assumed that ambient temperature is constant. The thickness of new concrete layer is small then temperature growth due to chemical reactions is low. Consequently, in all this study, thermal effects are neglected.

Drying model

Mass conservation equation, expressed according to the mass water content w, is:

∂w ∂t = -div(J v + J l ) + ∂w ad ∂t (3)
where J v is the flux of water in vapour form and J l the flux of water in liquid form. The derivative term of w ad represent the variation of water mass content due to self-desiccation.

Using Fick law and ideal gas law, vapour flux can expressed according to the relative humidity h r . Similarly, using Darcy law and Kevin law, the liquid flux can be written in terms of h r . In additions, experimental results (Baroghel-Bouny 1994;[START_REF] Therrien | Étude de l'évolution des propriétés de transfert hygrométrique d'un mortier au jeune âge[END_REF]) show that desorption isotherms are linear (i.e. the slope of the curve w = w(h r ) is constant) for 0.4 < h r < 0.95. Then we obtain the classical diffusion equation:

∂h r ∂t = div(D(h r ) grad h r ) + ∂h ad ∂t (4)
Thereafter, we will neglect the variation of relative humidity ∂h ad /∂t due to hydration reactions. However associated shrinkage (part of endogenous shrinkage) is introduced further.

The macroscopic diffusion coefficient depend on h r . When h r is important, humidity transfers takes place in water of capillary spaces. In opposition, when relative humidity is low, transfer is of vapour form and surface diffusion occurs in areas of hindered adsorption. According to theses observations, we choose the following expression of D(h r ) (Bažant 1988) :

D(h r ) = D 0 (t)   a + 1 -a 1 + 1-hr 1-hc n   (5)
where a, n and h c are adjustable parameters specific to material. h c would be the humidity threshold where transfer change from first phenomenon to the second. Those parameters can be identified with experimental data for weight loss (figure 3).

Structure of the porous network evolves during hydration. Some results [START_REF] Therrien | Étude de l'évolution des propriétés de transfert hygrométrique d'un mortier au jeune âge[END_REF] show that diffusion coefficient is dependent on maturity of concrete. However, it is difficult to obtain an experimental relation D(ξ). So, we fit the evolution of D to the inverse of the evolution of strength.

D 0 (t) = D 0 (t ∞ ) m 2 + t m 1 t m 1 (6)
Boundary conditions of convective type are imposed at the surface Γ in contact with air at relative humidity h a r . Convective flux is then written:

J Γ = β(h Γ r -h a r ) (7)

Shrinkage

Shrinkage of concrete is due to chemical reactions and drying of porous media. Thus, an endogenous shrinkage and a drying shrinkage can be identified.

In numerical simulations, experimental results of endogenous shrinkage are directly used. At each step, a homogeneous deformation is imposed in repair concrete layer.

Drying shrinkage is obtained from variation of relative humidity through a coefficient of hydrous dilatation:

εds = α h (t) ḣr 1 (8)
This relation does not reflect micro-mechanisms of shrinkage (capillary depression, Gibbs-Bangham theory and disjoining pressure theory), but a theoretical study (Lassabatere 1994) has shown that it gives a not so bad approximation.

According to concrete structuration, α h (t) should depend on the maturity of material. Some experimental results [START_REF] Therrien | Étude de l'évolution des propriétés de transfert hygrométrique d'un mortier au jeune âge[END_REF] show this dependence. However, it is difficult to obtain an experimental evolution law. Thus, α h is considered constant and determined for each concrete (figure 4).

Creep

Creep of concrete can be separated in two parts: basic creep and drying creep. The origins of compressive or tensile creep are discussed in Bažant (1988), [START_REF] Bissonnette | Le fluage en traction: un aspect important de la problématique des réparations minces en béton[END_REF], or [START_REF] Molez | [END_REF].

Basic creep:

Basic creep can be written according to viscoelasticity theory. A compliance function J is defined by:

ε(t) = J(t 0 , t)σ 0 (9)
where t 0 is the age at loading σ 0 , and t -t 0 the load duration.

For a stresses history, creep strain can be written:

ε(t) = ε(t 0 ) + t t 0 BJ(τ, t 0 ) σ(τ )dτ ( 10
)
where B is the fourth order tensor of Poisson's terms. Adopting a Dirichlet series expression for compliance function (Bažant 1988):

J(t 0 , t) = 1 C 0 (t 0 ) + n µ=1 1 C µ (t 0 ) 1 -exp(- t -t 0 τ µ ) (11)
makes it possible to free from storage of stresses history, and strain can be calculated from the preceding step. Previous expression can be explained in terms Kelvin chain (figure 5) with ageing modulus E µ = C µ -τ µ Ċµ and constant dashpot viscosity η µ = τ µ C µ . Strain evolution is calculated step by step. For step [t n , t n+1 ], one can write: If C µ and σ = ∆σ/δt are constant during this step, integral can be easily calculated (see [START_REF] Molez | [END_REF] for detailed calculus):

∆ε = ε(t n+1 ) -ε(t n ) E η¡ E¡ η¢ E¢ η E η£ E£ µ µ σ σ
∆ε = 1 C 0 (t n + ∆t / 2 ) + n µ=1 1 -1 -exp(-∆t τµ ) τ µ / ∆t C µ (t n + ∆t / 2 )   B ∆σ+ n µ=1 1 -exp(- ∆t τ µ ) ∆ε * µ (t n ) (12
) with the recursive expression:

∆ε * µ (t n ) = 1 -exp(-∆t τµ ) τ µ / ∆t C µ (t n -∆t / 2 ) B ∆σ+ ∆ε * µ (t n-1 ) exp(- ∆t τ µ ) (13)
Thus the increment of deformation is calculable according to the increment of load and the history of loading defined in previous step. The increment of basic creep is obtained by subtracting the increment of instantaneous strain from previous equation. We obtain:

∆ε bc =   n µ=1 1 -1 -exp(-∆t τµ ) τ µ / ∆t C µ (t n + ∆t / 2 )   B ∆σ+ ∆ε hist
(14) Characteristic relaxation times τ µ are fixed a priori according to a logarithmic series (Bažant 1988;[START_REF] Granger | Calcul des ouvrages de construction[END_REF]:

τ µ = τ 1 10 µ-1 (15)
and C µ parameters are obtained by the method of least squares. An example of results is given in figure 6.

Drying creep:

Creep experimental tests show that an extra creep can be measured when test specimen is allowed to dry. This phenomenon is called drying creep. Different interpretations are proposed. [START_REF] Bažant | Concrete creep at variable humidity: constitutive law and mechanism[END_REF] suggest a mechanism of stress-induced shrinkage. On the contrary, drying creep can be interpreted as a drying-induced creep (Bažant 1988). These different mechanisms can be written mathematically as

¤ ¥ ¤ ¦ ¤ ¤ ¦ ¥ ¤ ¤ §¦ ¤ ©¤ ¤ © ¤ ¥ ¤ ¤ ¤ ! " # $ " ¦ # $ " % & # &$ " ' ( ) 0 1 2 1 0 0 3 ) ) ( 4 5 6 7 6 7 8 9 @ A B C D FE HG I P ¥ ¤ Q SR T U T
εdc = κ f s B σ | ḣr | (16)
Coefficient κ f s is determined from basic and drying creep test. We can notice that the sign of ( εdc ) ij depends only on the sign of (σ) ij . So in uniaxial traction, we notice an increase of drying creep whether it is in moistening or drying conditions. Then, total creep can be obtained by adding basic and drying creep (figure 7).

V W V X V V X W V V §X V ©Y V V ©a V W V b V c V c d e f g h i c d e p &q d i X d e p &q d i Y r d e p q d i s t u v w x w v v y u u t H W V S X V V S

Mechanical damage

Total strains ε total are supposed to be the sum of instantaneous strains ε i , shrinkage strains ε s and creep strains ε c :

ε total = ε i + ε s + ε c (17)
The stress-strain relation of damaged material is written:

σ = C d (ε total -ε s -ε c ) = C d ε i (18)
In the present analysis, we use an isotropic scalar damage model [START_REF] Mazars | Application de la mécanique de l'endommagement au comportement non linéaire et à la rupture de béton de structure[END_REF]. In this model, the mechanical effect of progressive micro-cracking due to external loads is described by a single internal variable which degrades the Young's modulus of the material. The constitutive relations are:

σ ij = (1 -d)Λ ijkl (ε i ) kl (19)
where σ ij and (ε i ) kl are the components of the stress and strain tensors respectively (i, j, k, l ∈ [1, 3]), Λ ijkl are the initial stiffness moduli, and d is the damage variable. The material is initially isotropic, with E and ν the initial Young's modulus and Poisson's ratio respectively. Damage is a function of the positive strains which means that it is mainly due to micro cracks opening in tension mode. In order to avoid ill-posedness due to strain softening, the mechanical model has to be enriched with an internal length (Pijaudier-Cabot and Bažant 1987).

NUMERICAL EXAMPLES

In this section, we present some results of numerical simulations of concrete repairs. Test configuration is the same one as that used in previous experimental study [START_REF] Molez | Structural concrete repair behavior: coupled effect of drying and loading[END_REF][START_REF] Molez | [END_REF]. Figure 8 shows the geometry of the investigated concrete repair. A 2D analysis of one half of the beam, using four-noded plane stress elements, was performed (figure 9).

Examples of calculated humidity distribution, corresponding deformation and induced damage are shown in figures 12, 13 and 14.

Numerical results can be compared to experimental ones. During experimental tests, crack width and deflection has been measured for different configurations: drying only, drying and loading at 33% of ultimate load, and flexural testing. Those results are compare in figures 10 and 11.

These numerical results show a good agreement with experimental ones.

9: FE mesh ¡ ¢ ¢ ¡ ¡ ¢ ¢ ¡ £ ¤ ¥ ¦ § ¨© ¨ ¦ ¤ ¥ ¦ § ¨© ! " # $ % & ' ( ) 0 1 2 3 4 µ 5 6
Figure 10: Comparison of experimental (dots) and calculated (lines) average crack width for a repaired beam submitted to drying and loading.

CONCLUSIONS

To be able to make a complex analysis (to complete experimental study) of the behaviour of structural concrete repairs, the main phenomena acting in those structures have been integrated in a finite elements code. So, we can take into account the evolution of mechanical properties during hydration process and induced volume change (endogenous shrinkage). The drying of the material is modelled by a non linear law. Furthermore, the diffusion law is coupled with the evolution of hydration process by a diffusion coefficient which depends on the maturity of the material. Drying shrinkage is connected with the variation of humidity by a proportionality law that is experimentally verified for relative humidity included between 0.5 and 0.95. Viscoelastic behaviour is modelled by a Kelvin chain. The effects of hydration are taken into account with variable coefficients of the Kelvin chain. The influence of drying on creep is introduced by means of an extra term proportional to variation of relative humidity and imposed stress. A damage model is used to reproduce the mechanical effect of progressive micro-cracking of concrete.

The identification phase shows that the numerical tool well reproduces the behaviour of small lab specimens, and also bigger structures.

¡ ¢ £ ¤ £ ¢ ¥ ¤ ¥ ¢ ¦ ¤ ¦ ¢ ¤ ¢ ¢ ¦ ¢ ¥ ¢ £ ¢ § ¡ ¢ ¤ ¢ © ¢ ¢ ! " # $ % & ' ( 0 ) 1 2 ) 4 3 5 6 7 9 8 A @ B A B D C

Figure 1 :

 1 Figure 1: Schematical illustration of the stress build-up in repairs

Figure 2 :

 2 Figure 2: Experimental (dots) and calculated (lines) evolution of compressive strength for an ordinary concrete (BO) and a selflevelling concrete (BAP).

Figure 3 :

 3 Figure 3: Comparison of experimental and calculated weight loss.

Figure 4 :

 4 Figure 4: Evolution of total shrinkage. Comparison of experiments and numerical simulation.

Figure 5 :

 5 Figure 5: Kelvin chain equivalent to the proposed model

Figure 6 :

 6 Figure 6: Experimental (dots) and calculated (lines) basic creep of an ordinary concrete

Figure 7 :

 7 Figure 7: Experimental (dots) and calculated (lines) total creep (basic + drying creep) of an ordinary concrete

  Figure 8: Test specimens

Figure

  Figure 9: FE mesh

Figure 11 :Figure 14 :

 1114 Figure 11: Flexural behaviour: experimental (dots) and calcu-lated (lines) results.

Table 1 :

 1 Value of construction work in some countriesFrance 1 UK 2 Canada 3 USA 4

	billion of		£	$ CAD	$ US
	Building	new	47,3		551
		repairs	44		294
	Nonbuilding new	17		103
		repairs	9,5		67
	All	repairs	14,9	16
	1 Ministère de l'Équipement (2001)
	2 Cusson (2001)

3 Emberson and Mays (1996)

4 US department of commerce

(2000)


Joël Sarout

Laurent Molez

Yves Guéguen

Nasser Hoteit

Shale Dynamic Properties & Anisotropy under Triaxial Loading

This paper is concerned with the experimental identification of the whole dynamic elastic stiffness tensor of a transversely isotropic clayrock from a single cylindrical sample under loading. Measurement of elastic wave velocities (pulse at 1 MHz), obtained under macroscopically undrained triaxial loading conditions are provided. Further interpretation of the velocity measurements is performed in terms of (i) dynamic elastic parameters ; and (ii) elastic anisotropy. A discussion concerning microcracks and fluid content (saturation) issues follows. Experiments were performed on a Callovo-Oxfordian shale, Jurassic in age, recovered from a depth of 613 m in the eastern part of Paris basin in France.

INTRODUCTION

Callovo-Oxfordian Shale

Clayrocks, and shales in particular, represent approximately two-third of all sedimentary rocks. In oil and gas drilling operations, shales constitute 80 % of all the drilled sections, mainly because they overlie most hydrocarbon bearing reservoirs. Furthermore, several countries are considering clayrocks as possible host lithologies for radioactive waste confinement, and therefore carrying out research programs to estimate feasibility of such solution. In this trend, the french agency for radioactive waste management, ANDRA, is evaluating the reliability of the Callovo-Oxfordian layer, Jurassic in age, located in the eastern part of France (Bure), at a depth ranging from 400 m to 700 m. The static mechanical properties of this clayrock have been largely studied for the past few years (complete bibliography in [START_REF] Escoffier | Caractérisation Expérimentale du Comportement Hydromécanique des Argilites de Meuse/Haute-Marne[END_REF]). However, very few velocity measurements, all at atmospheric pressure, were performed, and no velocity data at all under mechanical loading are available in the literature for this rock. The main outcome of interest here from these research programs is the transversely isotropic nature of Bure clayrock demonstrated experimentally in several labs for several locations and depths in the layer [START_REF] David | Personal Communication[END_REF]). The main plane of symmetry is quasi-horizontal and corresponds to a very fine layering (bedding). This clayrock is then called a shale in the following.

Shale Lithologies

In general, the monitoring of elastic velocities in shales under mechanical loading are relatively uncommon in view of the abundance of this lithology in the shallow earth crustal rocks [START_REF] Stanley | Attenuation anisotropy in shale at elevated confining pressures[END_REF]. This relative rareness is partly due to: (i) the inherent difficulty of transmitting transducer signals through the waterproof pressure chamber in a loading apparatus ; and, on the other hand, (ii) the specificities in preparing and handling shale samples for mechanical testing (chemical sensitivity to water, extremely low permeability...). Most of the dynamic experimental studies reported in the literature on shale samples were performed under hydrostatic loading conditions [START_REF] Johnston | Ultrasonic P and S wave attenuation in dry and saturated rocks under pressure[END_REF][START_REF] Jones | Ultrasonic velocities in cretaceous shales from the williston basin[END_REF][START_REF] Lo | Experimental determination of elastic anisotropy of berea sandstone, chicopee shale, and chelmsford granite[END_REF][START_REF] Johnston | Seismic anisotropy of shales[END_REF][START_REF] Hornby | Experimental laboratory determination of the dynamic elastic properties of wet, drained shales[END_REF][START_REF] Stanley | Attenuation anisotropy in shale at elevated confining pressures[END_REF]. Only [START_REF] Yin | Acoustic Velocity and Attenuation of Rocks: Isotropy, Intrinsic Anisotropy, and Stress Induced Anisotropy[END_REF] carried out triaxial tests.

Main Goals

The specific experimental setup available in our lab allows for the simultaneous measurement of five different velocities and two directions of strains on the same sample, under triaxial and pore pressurecontrolled conditions of loading. This procedure reduces the number of experiments on differently oriented samples usually needed to identify the dynamic properties of a rock. It also minimizes the errors due to the particular difference between two samples of the supposedly same lithology or in the same physical state (stress and saturation history since recovery from 613 m deep).

The main outcomes of this experiment are : (i) identification of the apparent dynamic stiffness tensor of the Callovo-Oxfordian shale from elastic velocity measurements ; (ii) assessment of velocity anisotropy, and its evolution under triaxial loading. This last step allows for the quantification of the intrinsic and stressinduced anisotropies, leading eventually to an estimation of the microcracks density and distribution evolutions in the shale sample under loading [START_REF] Vernik | Hydrocarbon-generaion-induced microcracking of source rocks[END_REF].

In the specific experiment described here, there are no strain data available during the hydrostatic loading because of experimental difficulties in strain gages gluing. However, an external LVDT allows for the estimation of the sample global axial strain during the deviatoric loading.

EXPERIMENTAL PROCEDURE 2.1 Experimental Setup

The rock physics group in the geology laboratory of the Ecole Normale Supérieure recently acquired a triaxial cell designed for the exploration of the hydromechanical behavior of shallow earth crustal rocks. This apparatus allows for pore pressure, hydrostatic and deviatoric stress to be applied independently on a cylindrical porous rock sample (up to ∅ 40 mm × L 80 mm). The pore, confining and axial pressures may reach 100, 300 and 800 MPa (on the largest samples), respectively. The originality of this loading cell is to allow for a maximum of 32 waterproof signal wires through the wall of the pressure chamber. The control and data acquisition are performed by means of a dedicated software designed in Labview TM environment. The lab is temperature-controlled with an accuracy of ± 0.5 • C.

Sample Description and Preparation

The tested shale was provided by the french agency for radioactive waste management, ANDRA. The received large cores (∅ 90 mm × L 280 mm) were recovered in 1995 from a depth of 613 m in the eastern part of Paris basin, near Bure, in France. These cores were since preserved in a so-called T1 cell, isolated from gas exchange with the storing environment, and maintained under small confining stress ( 1 MPa). The samples are dry-cored perpendicularly to the horizontal bedding, ground for parallel end faces, and preserved in waterproof membranes. In this specific experiment, the sample is 60 mm long with a 30 mm diameter. Few days before testing, during the phase of transducers gluing and curing on the sample, this one is maintained in a 100% relative humidity (RH) environment in order to avoid drying. Usually, after putting it in 100%RH atmosphere, the mass of the sample increases then stabilizes. However, it is assumed that the sample may still be superficially unsaturated. It is tested under macroscopically undrained conditions within the loading cell.

Transducers Arrangement

For the experiment reported here, 22 signal wires were used. Both axial and circumferential strain gages and nine piezoelectric transducers were glued on the cylindrical sample. However, the strain gages signals are rapidly lost at the beginning of the test. The specific arrangement of piezoelectric transducers allows for the measurement of five different elastic velocities as shown in figure 1. Indeed, let us define the reference frame (x 1 , x 2 , x 3 ), with (x 1 , x 2 ) being the horizontal bedding plane of symmetry of the transversely isotropic cylindrical shale sample. Then, measurements of three compressional velocities and two shear velocities, in three different directions are provided. These velocities are referenced with respect to the bedding plane, i.e., V P (0 • ) for the bedding-parallel compressional velocity, V P (45 • ) for the 45 • -to-bedding compressional velocity, V P (90 • ) for the bedding-perpendicular compressional velocity, V SH (0 • ) for the horizontally polarized, beddingparallel shear velocity, and V SV (0 • ) for the vertically polarized, bedding-parallel shear velocity. In the case of transverse anisotropy, this latter is equal to the shear wave velocity V S (90 • ) perpendicular to the bedding. Note that the specifically designed compressional piezoelectric transducers are virtually load insensitive. On the other hand, the shear piezoceramics are simply glued to a fitting metal piece, itself glued on the sample, and are therefore load sensitive.

Measurement Technique

For velocity measurement, the classical Ultrasonic Pulse Transmission technique is used [START_REF] Birch | The velocity of compressional waves in rocks to 10 kilobars[END_REF][START_REF] Tosaya | Acoustical Properties of Clay-Bearing Rocks[END_REF][START_REF] Yin | Acoustic Velocity and Attenuation of Rocks: Isotropy, Intrinsic Anisotropy, and Stress Induced Anisotropy[END_REF]). This method consists in measuring the travel time of a solitary elastic pulse through the rock sample of known travelling wave path length. The frequency of the solitary pulse in this experiment is 1 MHz.

Test Conditions & Loading Path

The chemical sensitivity of this shale to external water and its very low permeability precluded any pore pressure control of the test. Therefore, the tests are performed in macroscopically undrained conditions within the loading cell. The in-situ stress is estimated from depth and rock density. Indeed, we assume hydrostatic state of stress in this large sedimentary basin, neglecting any deviatoric stress possibly due to tectonic stress build-up. This in-situ stress is estimated around 15 MPa.

The loading starts with confining pressure cycle between 0 and 55 MPa, i.e., (i) loading from 0 to 20 MPa ; (ii) unloading from 20 to 3 MPa ; (iii) reloading from 3 to 55 MPa ; then (iv) unloading from 55 to 15 MPa. The loading rate is 0.013 MPa/s during the pressure-controlled hydrostatic loading. After the fourth stage, a deviatoric stress is applied and increased until sample rupture. The deviatoric loading is displacement-controlled in order to avoid sudden sample failure. The velocity measurements are made after stabilization of all sample parameters at a given loading stage.

EXPERIMENTAL RESULTS

Error Analysis

For this particular experiment, corrections due to the rock deformation in the estimation of the velocity change during the hydrostatic loading are not feasible with accuracy because of a lack of strain measurement. However, a previous experiment on a similar sample, under the same conditions, allowed for strain data acquisition. Therefore, in the hydrostatic part of the present experiment, these strain data are used to correct velocity data. Both samples where cored exactly at the same depth, one centimeter far from each other, and no apparent heterogeneity was noted.

Furthermore, a complete error analysis of the Ultrasonic Pulse Transmission technique has been carried out [START_REF] Yin | Acoustic Velocity and Attenuation of Rocks: Isotropy, Intrinsic Anisotropy, and Stress Induced Anisotropy[END_REF] and [START_REF] Hornby | Experimental laboratory determination of the dynamic elastic properties of wet, drained shales[END_REF]. This analysis includes the errors in the travel time picks for the measured and reference travel times, and the error in the sample size measurements. In our experimental case, using Hornby (1998) equation ( 6), the relative error reaches a maximum of 1.1 % in the velocity measurements, depending on the wave type and propagation direction. Velocity (m/s) Axial deformation (%)

V sv (0°) V sh (0°) V p (90°) V p (45°) V p (0°)
Velocity (m/s) V sv (0°) V sh (0°) V p (90°) V p (45°) V p (0°)
Figure 4: Elastic wave velocities evolution under deviatoric loading, at 15 MPa confining pressure.

Experimental Data

Figure 2 shows the evolution of the five measured velocities V P (0 • ), V P (45 • ), V P (90 • ), V SH (0 • ) and V SV (0 • ), under confining pressure varying between 0 and 55 MPa. All velocities increase with increasing pressure, and a hysteresis effect exists upon pressure release at 20 and 55 MPa. Note that in the 0 -55 MPa pressure range, ∆ p V P (0

• ) < ∆ p V P (45 • ) < ∆ p V P (90 • ).
Even the amplitudes of the hysteresis velocity lags are ordered in the same way, i.e.,

∆ h V P (0 • ) < ∆ h V P (45 • ) < ∆ h V P (90 • ).
Figure 3 shows the axial stress-axial strain curve during the deviatoric part of the loading path. The sample failure happened probably around the peak axial stress (51 MPa), and for a 1.9 % axial strain.

Figure 4 shows the evolution of the five measured velocities V P (0 • ), V P (45 • ), V P (90 • ), V SH (0 • ) and V SV (0 • ), under this deviatoric loading. All velocities decrease with increasing axial stress except V P (0 • ) which first increases until 1.3 % axial strain, then reduces slightly.

THEORETICAL CONSIDERATIONS

Dynamic Stiffness Matrix

Definition

Transversely isotropic materials (hexagonal symmetry) may be adequately described in the elastic regime by five independent stiffness constants relating stress to strain components. They form a fourth order tensor which may be written in the Voigt notation as a twelve components matrix in the reference frame (x 1 , x 2 , x 3 ), i.e.,

       C 11 C 12 C 13 0 0 0 C 12 C 11 C 13 0 0 0 C 13 C 13 C 33 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 66        , ( 1
)
where

C 12 = C 11 -2C 66 .
When evaluated from elastic wave velocity measurements, this tensor characterizes the dynamic apparent behavior of the material. These five independent C ij constants may be related uniquely to the five classical Young's, Poisson's and shear dynamic moduli, i.e. ,E d 1 , E d 3 , ν d 12 , ν d 13 and µ d 13 for instance. These five independent C ij constants are related to the five elastic wave velocities measured experimentally as follows :

C 11 = ρV 2 P (0 • ), C 33 = ρV 2 P (90 • ), C 44 = ρV 2 SV (0 • ), C 66 = ρV 2 SH (0 • ), ( 2
)
C 13 = -C 44 + [(C 11 + C 44 -2ρV 2 P (45 • )) (C 33 + C 44 -2ρV 2 P (45 • ))] 1/2 .

Evolution under Loading

Figure 5 and 6 illustrate the evolution of the dynamic stiffness parameters C ij with increasing confining pressure and axial stress, respectively. Note that figure 5 allows for the estimation of the crack-free dynamic moduli at low pressure.

Anisotropy Parameters

Definition

In the specific context of transverse isotropy analysis, it is convenient to express three of the five dynamic elastic constants in (1) in terms of three anisotropy parameters, initially introduced by Thomsen (1986), i.e.,

ε = C 11 -C 33 2 C 33 , γ = C 66 -C 44 2 C 44 , δ = (C 13 + C 44 ) 2 -(C 33 -C 44 ) 2 2 C 33 (C 33 -C 44 ) . ( 3
)
where ε measures the P-wave anisotropy, γ measures the S-wave anisotropy and δ is totally independent of horizontal velocity and may be either positive or negative for geomaterials [START_REF] Thomsen | Weak elastic anisotropy[END_REF][START_REF] Hornby | Experimental laboratory determination of the dynamic elastic properties of wet, drained shales[END_REF].

The main advantage of these parameters is their dimensionless nature, therefore facilitating qualitative insight into the material behavior.

Evolution under Loading

Figure 7 and 8 show the evolution of velocity anisotropy parameters ε, γ and δ under hydrostatic and deviatoric loadings, respectively. Note the constant value of δ under hydrostatic loading and its drastic decrease until 2.25 % axial strain and then stabilization at a negative value under deviatoric load.

PHYSICAL INTERPRETATION: DISCUSSION

Anisotropy Issue

The main factors contributing to velocity anisotropy of rocks are: (i) preferred orientation of void space (pores and cracks) ; (ii) textural-structural features (bedding or foliation) ; and (iii) anisotropic constituent minerals. Crack-induced anisotropy reduces with increasing hydrostatic load, leading eventually to the estimation of the intrinsic anisotropy at high confining pressure, due to factors (ii) and (iii) only. If we assume that crack-induced (subscript c) and intrinsic (subscript i) anisotropies are additive [START_REF] Vernik | Velocity anisotropy in shales: A petrophysical study[END_REF], we may extract the intrinsic anisotropies ε i and γ i from Fig. 7 at 55 MPa, i.e., ε i = 0.23,

γ i = 0.16. (4)
This allows for the estimation of the crack-induced anisotropy variation under hydrostatic loading, i.e., ∆ p ε c = 0.19 and ∆ p γ c = 0.05. The δ parameter seems constant under hydrostatic loading, i.e., δ 0.08. However, it is very sensitive to the deviatoric load, and decreases drastically with the increase of this latter to a value of δ c = -0.28 (Fig. 8). Furthermore, under deviatoric loading, both ε and γ anisotropy parameters decrease until 1.3 % axial strain, then stabilize at ε d = 0.11 and γ d = 0.14. After, 1.9 % axial strain, γ increases back. We therefore deduce that after closing the bedding-parallel microcracks under confining pressure, the deviatoric loading induces a damage to which the horizontal S-velocities are sensitive through the γ parameter. This damage may further be interpreted in terms of bedding-perpendicular microcracks. Note that for ε parameter, the crackinduced and intrinsic anisotropies are of the same order of magnitude, which confirms the importance of charaterizing the microcracked state of shales through anisotropy analysis.

The global analysis of the anisotropy data leads to a conceptual modeling of the microcracking history of a shale after coring and recovery. In general, (i) stress relief, (ii) microhydraulic fracturing, (iii) bottomhole stressing, (iv) dessication, and (v) in-situ mi-crocracks, contribute to the final state of microcracking in a rock sample in the lab. For this shale, microcracks close under confining pressure, preferentially in the direction perpendicular to the bedding (beddingparallel microcracks). The deviatoric loading induces damage (bedding-perpendicular microcracks).

Fluid Content and Saturation Issue

As pointed out by [START_REF] Jones | Ultrasonic velocities in cretaceous shales from the williston basin[END_REF], the partial saturation of the sample manifests itself by the velocity hysteresis under cyclic confining pressure. They noticed that this hysteresis is drastically reduced when the shale sample is rehumidified, and even cancels at very high confining pressure (400 MPa). Indeed, at ambient pressure, the void space is partially gas-saturated, in particular at the boundaries of the sample where gas exchange with atmosphere may occur (surface drying). When submitted to a hydrostatic load, the microcracks closure reduces the void space, which is eventually totally occupied by the pore water at very high confining pressure. At this stage, hysteresis due to partial fluid saturation is observable. In our experiment, it seems that even at 55 MPa confining pressure, hysteresis still exists, which allows us to assume that even at this pressure value, the pore space is still not fully saturated by water.

This brings out the question of effective pressure in the sample under loading. We will assume that during the hydrostatic loading, the confining pressure is equal to the effective pressure, since the pore space is not water-saturated and the incompressibility of gas is negligible compared to that of liquid. The sample acts as under drained conditions.

Two other phenomena related to solid/fluid interactions are likely to occur in shales: (i) chemical softening as a result of fluid-clay interaction and swelling (smectites in particular) ; and (ii) physicochemical softening due to lubrication of grain interfaces. These issues are responsible for the inherent difficulty of shale experimental testing.

6 CONCLUDING REMARKS Anisotropic elastic properties of a quasi-saturated Jurassic shale under macroscopically undrained conditions were determined by means of ultrasonic velocity measurements. These five velocity measurements were performed on the sample while under hydrostatic and deviatoric loadings in a triaxial cell. The experimental method is based on a single core test, minimizing the errors due to the particular difference between two samples of the supposedly same lithology or in the same physical state (stress and saturation history since recovery).

Analysis and quantification of anisotropy may lead to the estimation of critical parameters such as microcracks density and orientation, as well as crack aspect ratio, using Hudson's or Kachanov's models. This type of experiment will provide with input parameters for modeling wave propagation in porous, fluidinfiltrated, very low permeability shales. A natural way to model such medium is to use Biot's dynamic theory of poroelasticity, possibly extended to account for squirt-flow phenomenon due to the existence of an equant porosity connected to an anisotropic one (crack-shape).

Figure 1 :

 1 Figure 1: Piezoelectric transducers arrangement and velocity measurements on the shale sample.

Figure 2 :Figure 3 :

 23 Figure 2: Elastic wave velocities evolution under hydrostatic loading.

Figure 6 :

 6 Figure 5: Dynamic stiffness parameters evolution under hydrostatic loading.

Figure 7 :

 7 Figure 7: Anisotropy parameters evolution under hydrostatic loading.

Figure 8 :

 8 Figure 8: Anisotropy parameters evolution under deviatoric loading, at 15 MPa confining pressure.

7 ACKNOWLEDGEMENTS Support and shale samples for this research program have been provided by the french agency for radioactive waste management, ANDRA. This support is gratefully acknowledged. I would also like to thank Guy Marolleau for his crucial help in the setting of the experiment in the triaxial cell.


come

Tableau extensions for reasoning with link keys

Maroua Gmati, Manuel Atencia, Jérôme

Tableau extensions for reasoning with link keys

Maroua Gmati 2,1 , Manuel Atencia 1,2 , and Jérôme Euzenat 2,1 1 Univ. Grenoble Alpes and 2 INRIA, France gm.gmati.maroua@gmail.com,{Manuel.Atencia,Jerome.Euzenat}@inria.fr Abstract. Link keys allow for generating links across data sets expressed in different ontologies. But they can also be thought of as axioms in a description logic. As such, they can contribute to infer ABox axioms, such as links, or terminological axioms and other link keys. Yet, no reasoning support exists for link keys.

Here we extend the tableau method designed for ALC to take link keys into account. We show how this extension enables combining link keys with classical terminological reasoning with and without ABox and TBox and generate non trivial link keys.

Motivation

Part of the added value of linked data lies in the links between entities denoting the same individual in data sets issued by different sources as it allows for making inferences across data sets. For instance, links may identify the same books and articles in different bibliographical data sources. So finding the manifestation of the same entity across several data sets is an important task of linked data.

One way of identifying entities is to use link keys which generalise keys usually found in data bases to the case of different data sets. A link key [START_REF] Atencia | Data interlinking through robust linkkey extraction[END_REF] is a statement of the form:

{ auteur, creator , titre, title } linkkey Livre, Book stating that whenever an instance of the class Livre has the same values for properties auteur and titre as an instance of class Book has for properties creator and title, then they denote the same entity. Such keys are slightly more complex than those of databases because, in RDF, properties are not necessarily functional (they may have several values) and their values may be other objects. One further difference is that RDF data, together with ontologies expressed in the OWL or RDFS languages, are logic theories. In such a context, a link key is a statement as any other logical statement. As such, it may contribute deducing other statements. Indeed, the above link key entails: Hence, it is possible to reason on link keys in different ways:

{ auteur,
-deducing link keys from OWL statements, -deducing link keys from link keys, -deducing OWL statements from link keys.

Our goal is to study reasoning procedures for link keys. For that purpose, we define a preliminary extension of the tableau method for ALC dealing with link keys and we provide examples for each of the inference types above.

In the following, we first discuss related work ( §2) and define more precisely the problem ( §3). Then we present a tableau extensions allowing for ABox reasoning with link keys ( §4) and for reducing link key inference to that ABox reasoning ( §5).

Related work

Data interlinking is a very active area [START_REF] Ferrara | Data linking for the semantic web[END_REF]. Two main approaches are used for coping with this problem: numerical methods and logical methods. The numerical methods usually compute a similarity between resources based on their property values to establish links between those which are highly similar [11; 13]. Logical methods for data interlinking use an axiomatic characterisation of what makes two resources the same to find the links between different data sets [12; 1; 3]. This work belongs to the logic-based approach. It uses a generalisation of keys in relational databases, called link keys, for expressing the condition for identifying resources across different ontologies. Keys in databases indicate that a set of properties uniquely identifies individuals. Relational properties are functional (have only one value) and concrete (the value comes from a data type).

RDF data differ from relational data in their properties, which are not functional, and their values, which may be resources. Hence, keys have been generalised to cope with this problem [START_REF] Atencia | Defining key semantics for the RDF datasets: experiments and evaluations[END_REF]. RDF property values are considered the same if they are the same concrete value or are interpreted as the same individual. Coping with non functionality lead to define two different types of keys: in-keys and eq-keys. Eq-keys require that the properties of two objects have exactly the same values for them to be equal, while in-keys only require that each property shares at least one common value. In this work, we focus on in-keys.

Keys may be introduced in description logics either as global constraints in a specific KBox [7; 10], or as a new concept constructor [START_REF] Borgida | Adding uniqueness constraints to description logics (preliminary report)[END_REF]. [START_REF] Calvanese | Viewbased query processing for regular path queries with inverse[END_REF] discusses the introduction of keys in the DLR logic but does not provide any reasoning method. Keys based on features (functional roles whose value is from a concrete domain) have been introduced within the ALCOK(D) and SHROIC(D) logics [START_REF] Lutz | Keys, nominals, and concrete domains[END_REF] and an extension of the tableau method has been provided to deal with these logics.

Keys identify objects within a single data source with a single schema. Link keys have been designed for coping with heterogeneous data sources [START_REF] Euzenat | Ontology matching[END_REF]. They can be seen either as a generalisation of keys across two data sets or as a merge between keys and alignments. They express conditions by which two individuals, from two different classes, must be considered the same by comparing values of properties.

Link keys raise two distinct problems: the first one is to extract link keys from data sets [START_REF] Atencia | Data interlinking through robust linkkey extraction[END_REF]; the second one is to take advantage of link keys to generate links. These two problems may be thought of as two steps of a link generation procedure: first extract link keys, then generate links from them.

Here we tackle a third problem (not unrelated to the second one): reasoning with link keys, i.e., infering links, ontological and assertional statements as well as other link keys. We define this problem more precisely below.

Preliminaries

Data interlinking is the process of generating links across data sets that can help finding equivalent resources representing the same entity on the web for linked data. These links are usually owl:sameAs statements between two resources across different RDF data sets. We will consider that these data sets are description logic knowledge bases (KB = T, A ) made of a TBox T and an ABox A. Description logics [START_REF]The description logic handbook: theory, implementations and applications[END_REF] are at the basis of OWL, so this is quite natural.

We decided to extend the tableau method used for checking entailment in the ALC family of description logics for several reasons: -ALC is a subset of OWL; -The tableau method is extensible, so it is possible to add rules for dealing with more expressive logics. We could have started with procedure specific to less expressive logics (EL, DL-Lite, OWL-RL), but we could barely extend them.

An ALC TBox is a set of general concept inclusion axioms of the form C C . Concepts are defined by:

C = A|⊥| |C C |C C |¬C|∀R.C|∃R.C
and roles are simply atomic roles (R = r).

The ABox is made of assertions of the form C(a) and r(a, b). We will use two specific statements a = b and = b which are interpreted as usual. These two predicates are the transcription of owl:sameAs and owl:differentFrom.

The semantics of such logics is defined by interpretations I = ∆ I , • I such that ∆ I is a non empty set and • I is a function such that: a I ∈ ∆ I , C I ⊆ ∆ I , and r I ⊆ ∆ I × ∆ I with:

(¬C) I = ∆ I \ C I ⊥ I = ∅ I = ∆ I (C C ) I = C I ∩ C I (∀r.C) I = {δ ∈ ∆ I |∀δ ; δ, δ ∈ r I ⇒ δ ∈ C I } (C C ) I = C I ∪ C I (∃r.C) I = {δ ∈ ∆ I |∃δ ∈ C I ; δ, δ ∈ r I }
An interpretation satisfies an axiom (denoted by I |= α) in the following conditions:

I |= C(a) iff a I ∈ C I I |= r(a, b) iff a I , b I ∈ r I I |= a = b iff a I = b I I |= a = b iff a I = b I I |= C C iff C I ⊆ C I
A model of a knowledge base KB is an interpretation satisfying all its axioms and an assertion α is entailed by a knowledge base (denoted by KB |= α) if it is satisfied by all the models of KB.

We extend description logics with a KBox K which contains link keys instead of simple keys. The KBox is a set of link keys: { p i , q i } i∈I linkkey w in C, D with C and D two classes coming from different data sets and p i and q i roles, from the data sets of C and D respectively, indexed by a finite set of indices I. Since we concentrate specifically on weak in-link keys, we use the keyword linkkey w in . The semantics of description logics is extended to cover link keys: An interpretation I satisfies ({ p i , q i } i∈I linkkey w in C, D ) iff, for any δ ∈ C I and η ∈ D I , i∈I

(∃z i ∈ ∆ I ; δ, z i ∈ p I i ∧ η, z i ∈ q I i ) ⇒ δ = η
Any key {p i } i∈I keyF or C is equivalent to the link key { p i , p i } i∈I linkkey C, C . In this paper, we only consider hierarchical KBoxes, i.e., KBoxes in which there cannot be circular dependencies between link keys. It is possible, to establish entailment rules for link keys considered as assertions:

{ p i , q i } i∈I linkkey w in C, D |= { p i , q i } i∈I∪J linkkey w in C, D { p i , q i } i∈I linkkey w in C, D , C C |= { p i , q i } i∈I linkkey w in C , D { p i , q i } i∈I linkkey w in C C , D |= { p i , q i } i∈I linkkey w in C, D D
Proving all such rules one by one is tedious, so an inference procedure for doing this would be useful.

Links and Abox entailments with link keys

The basic way of applying link keys is to start with two datasets A and A described by two ontologies T and T and a set K of link keys across these ontologies and to generate links, i.e., statements of the form a = b with a and b from each data set.

We consider this problem more widely as that of reasoning in a knowledge base1 KB = T ∪ T , K, A ∪ A . We will consider more precisely the decision problem of checking the entailment of any ABox axiom α from such a knowledge base.

Problem: ABOX AXIOM ENTAILMENT INSTANCE:

-A knowledge base KB = T, K, A -An ABox assertion α. QUESTION: Does KB |= α?

Tableau rule for applying link keys

The tableau method is the classical technique to reason with ALC. Explaining the method is out of the scope of this paper (see [4; 5]). To summarise, this method attempts to find a model of a knowledge base KB = T, A in negation normal form.

For that purpose, it starts with a representation of the ABox A and applies rules (see Appendix) guided by T until no rule is applicable [START_REF] Baader | Description logics[END_REF]. In such a case, there exists a model of KB. However, there are special constraints, called clashes, which express the impossibility to build a model: if such a clash is satisfied, then the current representation cannot be turned into a model and the algorithm must explore eventual alternative representations. Finally, for guaranteeing the termination of the process due to infinitely expanding rules, provisions are taken for detecting this and blocking some parts of the representation to be expanded. We rely here on the classical tableau method for ALC and use a graphical representation of partial models in which nodes (x) represent individuals labeled (L(x)) by sets of class descriptions and edges ( x, y ) represent relations labeled (L( x, y ) by role descriptions. The tableau method may be used for finding a model or for proving that there exist no model of a knowledge base.

In order to tackle the ABox Axiom entailment problem within the tableau method we introduce the Linkkey-rule:

Linkkey-rule

Condition: { p i , q i } i∈I linkkey w in C, D ∈ K, ∃x, y, not blocked, such that C ∈ L(x), D ∈ L(y), and ∀i ∈ I, ∃z i , such that p i ∈ L( x, z i ) and q i ∈ L( y, z i ) Action: L(x) := L(x) ∪ L(y)

Replace y by x in all edges starting from or ending at y Suppress node y This rule is sound, i.e., any model has to satisfy it, as it strictly follows the semantics of link keys. It generalises rule T14 in [START_REF] Lutz | Keys, nominals, and concrete domains[END_REF] to link keys.

The use of this rule for checking a link a = b can be illustrated on the straightforward Example 1: For proving the entailment of a = b, we proceed by refutation, i.e., we prove that it is not possible to create a model satisfying the antecedents and the negation of the consequence (a = b). A representation of such a model is created and the rules are applied on it. The Linkkey-rule merges the two nodes satisfying the link key condition which makes them fall under the =-clash.

T = {} K = { p, q linkkey w in C, D } A = {C(a), D(b), p(a, v), q(b, v), a = b}

Combining link key reasoning and ABox reasoning

Example 2 shows the use of these rules for chaining the use of two link keys. However, it may be used in any ABox reasoning development. C(a),p(a,c),E(c),q(c,v),D(b),r(b,d),F (d),s(d,v) |= a = b? Knowledge base:

T = {} K = { p, r linkkey w in C, D , q, s linkkey w in E, F } A = {C(a), p(a, c), E(c), q(c, v), D(b), r(b, d), F (d), s(d, v), a = b}
Solving the ABox entailment problem may not be the most efficient way to generate links from RDF especially if the size of the considered ABox is very large. A more interesting use of such reasoning is for checking link key entailment.

Link key entailment

The link key entailment problem aims at checking if a link key is entailed by a knowledge base. Because this resorts to the terminological level, i.e., without regard to a particular ABox, it is defined only on a knowledge base made of a TBox and a KBox. Indeed, some link keys may be entailed from terminological axioms, some others from other link keys of a mix of this.

Problem: LINK KEY ENTAILMENT INSTANCE:

-A knowledge base KB = T, K -A link key λ. QUESTION: Does KB |= λ?

Reducing link key entailment to knowledge base satisfiability

The tableau method cannot be directly used for refuting a link key axiom because there is no negation for link keys: a link key is an axiom of our logic, the negation of a link key is not.

Other authors have considered expressing keys as simple concept constructors [START_REF] Borgida | Adding uniqueness constraints to description logics (preliminary report)[END_REF]:

C key({p i } i∈I )
This could be transposed for link keys as:

C, D linkkey w in ({ p i , q i } i∈I )
such statements would solve half of the problem as it is possible to negate the subsumption statements, but this would lead to strange statements as they concern pairs of classes. They would also be stronger than, and not equivalent to, our actual link key statements.

Adding the negation of a link key to the logic is another solution to this problem. However, since its only use would be for the decision procedure, we preferred to avoid this solution.

We choose a simpler method given that our goal is simply to have negated link keys as the statement to refute: we use a set of ABox statements as witness of the unsatisfiability of a link key. This set is given by the function ρ:

ρ({ p i , q i } i∈I linkkey w in C, D ) = {C(x), D(y), x = y} ∪ {p i (x, v i ), q i (y, v i )} i∈I
Checking the entailment of a link key λ by a knowledge base ∅, T, K can be reduced to checking the satisfiability of the knowledge base KB = T, K, ρ(λ) . Any model in which the link key λ is not valid satisfies ρ(λ). Hence, if KB is satisfiable, then λ is not entailed.

Example 3 (Link key inference from other link keys and TBox). Problem: p, q linkkey w in C, D , C C |= p, q linkkey w in C , D ? Knowledge base:

T = {C C} K = { p, q linkkey w in C, D } A = {C (a)

=-clash

For instance, one of the example given in Section 3 is a link key entailed from another link key and terminological axioms. Example 3 shows how this is performed without introducing any new rule or clash in the tableau procedure.

This shows the importance of being able to reason with the ABox, since the refutation of the KB is mostly carried out by reasoning in the ABox even if the problem does not have an ABox. It also shows that link key rules can be adequately interleaved with ALC rules. This suggests that extensions can properly work in the same way.

Link key entailed from terminological axioms

Some other link keys may only be entailed by terminological axioms. We illustrate this by the counter-intuitive Example 4. This inference is of little use, but it shows that the method indeed proves this valid link key. The use of the tableau method allows both to check inference rules and to determine minimal logics in which they hold. Example 4 shows that the given entailment holds in any description logic, with ALC-style models, which accepts subsumption axioms ( ), universal quantification (∀) and the empty concept (⊥).

Conclusion and future work

Link keys are very useful for generating links from data sources, but they can be studied independently from data sources as axioms. In order to prove when a particular knowledge base, eventually with link keys, entails a particular link key, we proposed extensions of the tableau method for ALC enabling the interpretation of link keys. We showed that these extensions also allow for checking link key entailment.

We considered the tableau method because it is well-adapted to ALC and thus to OWL as a whole. Weaker fragments of OWL (EL, DL-Lite, OWL-RL) are supported efficiently by other reasoning methods. It would be interesting to investigate the opportunity to reason with and about link keys in this context.

This work is preliminary and many developments may be undertaken from here. We discuss a few of them.

First, we need to determine the properties of the proposed extension. We have yet no formal proof to offer, but basic arguments for these. Although correctness of rules and clash independently seems to be straightforward, proving the completeness of the designed procedure with various logics must be considered. Termination can be guaranteed with a blocking mechanisms and because no rule erases any other rule condition (the Linkkey-rule merges nodes, but preserves the constraints on these nodes). Finally, the current link key rule should not increase the complexity of existing tableau methods since the rule does not introduce branches. The Linkkey-rule may offer new development opportunities by merging nodes but (i) this process is bounded, and (ii) new tableau developments should not go beyond current complexity.

Then, we want to implement these extensions. This would allow us to check automatically the link key inference rules that we designed. It would also be interesting, in a further step, to develop techniques to generate (specific) entailed assertions in a forward deduction style.

Finally, it would be worth considering the other type of link key conditions (eqlink keys). However, this may not be easy to integrate with the open world aspect of description logic semantics.

A ALC+Linkkey rules

We provide the full set of rules for helping the reader to read the examples.

Example 1 (

 1 Simple link generation). linkkey w in C, D , C(a), D(b), p(a, v), q(b, v) |= a = b? Knowledge base:

Example 2 (

 2 Chaining link generation). linkkey w in C, D , q, s linkkey w in E, F ,

  , D(b), p(a, v), q(b, v), a = b}

Example 4 (

 4 Link key inference from TBox alone). Problem: C ∀p.⊥ |= p, q linkkey w in C, D ? Knowledge base:T = {C ∀p.⊥} K = {} A = {C(a), p(a, v), D(b), q(b, v), a = b}It is noteworthy that Example 4 does not use the Linkkey-rule; it only relies on the encoding of the problem and classical ALC reasoning.

A. 1

 1 Completion rules -rule Condition: C D ∈ L(x), x is not blocked; {C, D} ⊆ L(x) Action: L(x) := L(x) ∪ {C, D} -rule Condition: C D ∈ L(x), x is not blocked; C ∈ L(x), D ∈ L(x) Action: L(x) := L(x) ∪ {C}, or L(x) := L(x) ∪ {D} ∃-rule Condition: ∃r.C ∈ L(x), x is not blocked; ∃y; r(x, y) ∧ C ∈ L(y) Action: create a new node y with L( x, y ) = {r} and L(y) = {C} ∀-rule Condition: ∀r.C ∈ L(x), x is not blocked; ∃y; r(x, y) ∧ C ∈ L(y) Action: L(y) := L(y) ∪ {C} -rule Condition: C D ∈ T , x is not blocked, ¬C D / ∈ L(x) Action: L(x) := L(x) ∪ {¬C D} Linkkey-rule Condition: { p i , q i } i∈I linkkey w in C, D ∈ K, ∃x, y, not blocked, such that C ∈ L(x), D ∈ L(y), and ∀i ∈ I, ∃z i , such that p i ∈ L( x, z i ) and q i ∈ L( y, z i ) Action: L(x) := L(x) ∪ L(y) Replace y by x in all edges starting from or ending at y Suppress node y A.2 Clash conditions ¬-clash : ∃x; {C, ¬C} ⊆ L(x) ⊥-clash : ∃x; ⊥ ∈ L(x) =-clash : ∃ x, x ; =∈ L( x, x )

  creator , titre, title , éditeur, publisher } linkkey Livre, Book or { auteur, creator , titre, title } linkkey Livre, Novel whenever Novel is subsumed by Book.

We assume no unwanted name conflicts, i.e., the same name or URI in both data sets must have the same interpretation.