Pierre Morenon
email: morenon@insa-toulouse.fr

Stéphane Multon

Alain Sellier

Etienne Grimal

François Hamon

Philippe Kolmayer

Flexural performance of reinforced concrete beams damaged by Alkali-Silica Reaction

Keywords: Alkali-Silica Reaction (ASR), flexural mechanical behaviour, reinforced concrete, modelling, diffuse cracking, localized macrocracks

Alkali-Silica Reaction (ASR) affects concrete and decreases its mechanical characteristics. However, for some reinforced concrete structures, the global mechanical behaviour can appear to be improved by ASR: for similar reinforcement, the first flexural cracking of reactive beams usually occurs for higher loading than in non-reactive beams. The flexural failures of two reinforced beams, one reactive and one non-reactive, are numerically simulated here in order to discuss the origins of the delay in cracking observed for the reactive beam. The poromechanical model used considers the swelling anisotropy, and is able to differentiate ASR diffuse cracking from structural macrocracks and the coupling of both crack

Introduction

Alkali-Silica Reaction (ASR) affects the concrete of structures such as dams, bridges or basements [START_REF] Léger | Finite element analysis of concrete swelling due to alkaliaggregate reactions in dams[END_REF][START_REF] Malla | Analysis of an arch-gravity dam with a horizontal crack[END_REF][START_REF] Godart | Guide to Diagnosis and Appraisal of AAR Damage to Concrete in Structures[END_REF]. This internal swelling reaction generates damage in the material, which can lead to a degradation of the concrete characteristics and the structure stability. However, the behaviour of some reinforced structures seems to be improved by ASR [START_REF] Ohno | The mechanical behaviour of reinforced of beams coated after Alkali-Silica Reaction damage[END_REF][START_REF] Habibi | Alkali aggregate reaction in nuclear concrete structures: part 3: structural shear wall elements[END_REF][START_REF] Multon | Évaluation expérimentale et théorique des effets mécaniques de l'alcaliréaction sur des structures modèles[END_REF][START_REF] Multon | Effets structuraux de l'alcali-réaction: Apports d'une expérimentation sur éléments de structures à la validation de modèles[END_REF]. In flexion, the first cracking occurs for higher loading if the beam has undergone AAR. To consider this phenomenon in structural analysis, a numerical approach is proposed and confronted with the behaviour of real beams.

In this context, a faithful cracking model is essential. Usually, two kinds of tensile cracking are observed in concrete structures affected by ASR. The first is ASR diffuse cracking, which leads to decreases in the material characteristics (tensile and compressive strengths, elastic modulus) [START_REF] Swamy | Structural implications of alkali silica reaction[END_REF][START_REF] Koyanagi | Mechanical properties of concrete deteriorated by alkali-aggregate reaction under various reinforcement ratios[END_REF][START_REF] Larive | Combined contribution of experiments and modeling to the understanding of alkali-aggregate reaction and its mechanical consequences[END_REF]. The second kind of cracking is localized structural cracks. This can come from external loads as in all structures. It is also encountered in ASR-affected structures because of swelling gradients [START_REF] Fournier | Alkali-aggregate reaction in concrete: a review of basic concepts and engineering implications[END_REF]. The hydric state and the alkali release rate influence the swelling kinetics and amplitude through the structure. This can induce significant swelling gradients and thus tensile stresses and localized cracks. According to their opening, such cracks can lead to stress concentrations in steel bars (reinforcement), and to water leaks or durability problems. Cracks are an open door to aggressive agents such as CO2 or chlorides.

To obtain accurate assessments of damaged structures and, in particular, to determine the failure mode of a structure and its durability, mechanical modelling should be able to differentiate between diffuse cracking and localized cracks. Therefore, two distinct criteria need to be defined: one for diffuse cracking of the material and one for structural localized cracking. Moreover, as the calculations have to be performed over a long time scale, all the delayed deformations should be evaluated. In particular, concrete creep has to be taken into account to obtain reliable stress evaluation for structures damaged by ASR over several decades.

The first part of this paper presents the main features of the model. This type of modelling, clarified and implemented in several finite element software products since 1997 [START_REF] Sellier | Modélisation physico-chimique de la réaction alcali-granulat: apport au calcul des structures dégradées[END_REF], has been continuously improved and is now able to clearly differentiate diffuse and localized damage and to combine these two kinds of damage with creep. The second part of the paper is the application of the model to a flexural test to failure on laboratory reinforced beams (with and without ASR) in two phases: the service life (ageing, also modelled by [START_REF] Grimal | Concrete modelling for expertise of structures affected by alkali aggregate reaction[END_REF][START_REF] Comi | Two-phase damage modeling of concrete affected by alkali-silica reaction under variable temperature and humidity conditions[END_REF][START_REF] Alnaggar | Modeling time-dependent behavior of concrete affected by alkali silica reaction in variable environmental conditions[END_REF]) and the residual strength (failure test, never modelled). The comparison of forcedeflection curves leads to a discussion on the impact of ASR, on the flexural performance of these structures and on their durability.

Modelling

Rheological model

The concrete behaviour law is established in the poromechanical framework. The rheological model is built to accept the two kinds of cracking (Figure 1) that are encountered in structures affected by ASR. The first one is ASR diffuse cracking, which is a direct consequence of the ASR gel pressure inside the material (Figure 1) at the microscopic scale.

The structural localized cracks are evaluated separately (Figure 1). The creep is also taken into account by Sellier's model, which contains a Kelvin-Voigt (visco-elastic) and an anisotropic non-linear Maxwell level in order to reproduce multi-axial delayed strains due to loading [START_REF] Sellier | Concrete creep modelling for structural applications: non-linearity, multi-axiality, hydration, temperature and drying effects[END_REF]. The model takes into account creep reversibility and irreversibility, nonlinearity with stress, anisotropy, dependence to moisture and temperature to be as realistic as possible and usable in structural calculations [START_REF] Sellier | Concrete creep modelling for structural applications: non-linearity, multi-axiality, hydration, temperature and drying effects[END_REF]. The rheological model uses a poromechanical scheme to take ASR gel pressure, water pressure (shrinkage), and external stress into account in the creep calculation. This global scheme directly couples the concrete swelling and the tensile creep produced around the reactive sites [START_REF] Morenon | Impact of stresses and restraints on ASR expansion[END_REF].

Figure 1: Rheological model

In Figure 1, the total stress 𝜎 𝑖𝑗 is calculated from the damage 𝐷 𝐴𝑆𝑅 (due to ASR) and 𝐷 𝑠𝑡𝑟𝑢𝑐𝑡 (due to structural cracks), and from the effective stress 𝜎 ̃𝑖𝑗 (Equation ( 1)). In the context of damage theory, the effective stress 𝜎 ̃𝑖𝑗 corresponds to the stress in the undamaged zone of the material, which itself can be defined in the framework of poromechanics theory [START_REF] Coussy | Mécanique des milieux poreux[END_REF] by Equation (2) [START_REF] Sellier | Concrete creep modelling for structural applications: non-linearity, multi-axiality, hydration, temperature and drying effects[END_REF], with 𝜎 ̃𝑖𝑗 ′ the stress part in the solid and (-𝑏 𝑔 𝑃 𝑔 -𝑏 𝑤 𝑃 𝑤 ) the contribution of the interstitial phase pressures (𝑃 𝑔 for the pressure due to ASR and 𝑃 𝑤 for the pressure due to shrinkage induced by water). Each interstitial pressure is affected by the corresponding Biot coefficient (respectively 𝑏 𝑔 and 𝑏 𝑤 [START_REF] Biot | General Theory of Three-Dimensional Consolidation[END_REF]). 𝛿 𝑖𝑗 is the Kronecker symbol, equal to 1 only if 𝑖 = 𝑗. The effective poromechanical stress increment 𝜎 ̃𝑖𝑗 ̇′ is calculated from the stiffness matrix 𝑆 0 and the elastic strain. The elastic strains can be obtained by subtracting the non-elastic strain increments (𝜀ṗ 𝑙 𝑘𝑙 for the plastic strain, 𝜀ċ 𝑟 𝑘𝑙 for the creep strain and 𝜀ṫ ℎ 𝑘𝑙 for the thermal strain) from the total strain increment 𝜀k 𝑙 according to Equation [START_REF] Godart | Guide to Diagnosis and Appraisal of AAR Damage to Concrete in Structures[END_REF].

𝜎 𝑖𝑗 = (1 -𝐷 𝐴𝑆𝑅 ) (1 -𝐷 𝑠𝑡𝑟𝑢𝑐𝑡 ) 𝜎 ̃𝑖𝑗 (1)

ASR diffuse cracking

ASR-gel pressure 𝑃 𝑔 creates an orthoradial tensile stress around each reaction site (Figure 2)

When the tensile strength, 𝑅 𝑡 𝑚𝑖𝑐𝑟𝑜 , is reached at this microscale, diffuse cracking begins and leads to the decrease of macro material characteristics such as the elastic modulus [START_REF] Pan | Meso-scale particle modeling of concrete deterioration caused by alkali-aggregate reaction[END_REF][START_REF] Alnaggar | Lattice Discrete Particle Modeling (LDPM) of Alkali Silica Reaction (ASR) deterioration of concrete structures[END_REF].

This cracking (ASR cracking) is diffuse because it can develop at the level of each reactive swelling site [START_REF] Sanchez | Reliable quantification of AAR damage through assessment of the Damage Rating Index (DRI)[END_REF]. It can be oriented according to the multi-axial stress state [START_REF] Morenon | Impact of stresses and restraints on ASR expansion[END_REF], which depends on the external stress applied, 𝜎 ̃𝐼. The ASR plastic criteria are described below. Details of the numerical implementation algorithm used to combine the poromechanics, the creep, the plastic flow and the orthotropic damage occurrences are available in [START_REF] Sellier | Anisotropic Damage and Visco-Elasto-Plasticity Applied to Multiphasic Materials, LMDC -Laboratoire Matériaux et Durabilité des Constructions de Toulouse[END_REF]. Only the main features of this implementation are presented here.

𝜎 ̃𝑖𝑗 = 𝜎 ̃𝑖𝑗 ′ -𝛿 𝑖𝑗 𝑏 𝑔 𝑃 𝑔 -𝛿 𝑖𝑗 𝑏 𝑤 𝑃 𝑤 (2)
𝜎 ̃𝑖𝑗 ′ ̇ = 𝑆 0 ( 𝜀k 𝑙 -𝜀ṗ 𝑙 𝑘𝑙 -𝜀ċ 𝑟 𝑘𝑙 -𝜀ṫ ℎ 𝑘𝑙 )

(3)

Chemical advancement

The evolution of the gel pressure depends on the ASR chemical advancement, 𝐴 𝑎𝑠𝑟 . In this work, the advancement is evaluated from an empirical equation ( 4) already used by [START_REF] Grimal | Creep, Shrinkage, and Anisotropic Damage in Alkali-Aggregate Reaction Swelling Mechanism-Part I. A Constitutive Model[END_REF]. It depends on:

a characteristic time 𝜏 𝑟𝑒𝑓 𝑎𝑠𝑟 , calibrated using a free swelling test,

two coefficients to take account of environmental impacts:

o one for the temperature effect, 𝐶 𝑇,𝑎𝑠𝑟 , managed by an Arrhenius law (5) (𝐸 𝑎𝑠𝑟 is the thermal activation energy (≈ 40,000 J.Mol -1 [START_REF] Larive | Apports combinés de l'expérimentation et de la modélisation à la compréhension de l'alcali-réaction et de ses effets mécaniques[END_REF]), 𝑅 is the perfect gas constant (8.3145 J.mol -1 K -1 ), and 𝑇 𝑟𝑒𝑓 is the temperature corresponding to the 𝜏 𝑟𝑒𝑓 𝑎𝑠𝑟 calibration), o one for the humidity effect, 𝐶 𝑊,𝑎𝑠𝑟 (6), proposed in [START_REF] Morenon | Impact of stresses and restraints on ASR expansion[END_REF] based on Poyet's law [START_REF] Poyet | Influence of Water on Alkali-Silica Reaction: Experimental Study and Numerical Simulations[END_REF]. It uses a minimum threshold to initiate the reaction 𝑆 𝑟 𝑡ℎ,𝑎𝑠𝑟 (here 0.1) and evolves non-linearly in order to strongly accelerate the reaction kinetics with the water saturation of the material.

a kinetics and amplitude term < 𝑆𝑟 -𝐴 𝑎𝑠𝑟 > + , which limits the advancement at the saturation degree 𝑆𝑟, following the conclusion of [START_REF] Poyet | Influence of Water on Alkali-Silica Reaction: Experimental Study and Numerical Simulations[END_REF].

Water has two main roles in ASR gel formation: it can be absorbed by ASR products and it allows the diffusion of ionic species necessary for aggregate attack and products precipitation.

That is the reason why the lack of water in the porosity stops the chemical reactions; hydroxyl ions do not come into contact with the silica in the aggregates. The effect of water on the chemical reaction is taken into account in equation ( 4) by means of the porosity saturation ratio (Sr). For low saturation degrees, aggregate attack cannot occur and the chemical advancement is stopped. Therefore, the gel volume does not reach its total amplitude.

Gel pressure

Then, the balance of solid and liquid volumes changes due to ASR is noted 𝜙 𝑔 [START_REF] Multon | Effets structuraux de l'alcali-réaction: Apports d'une expérimentation sur éléments de structures à la validation de modèles[END_REF]. It is obtained from the ASR advancement, 𝐴 𝑎𝑠𝑟 , and from the balance of volume change 𝜙 𝑔 ∞ corresponding to the total reaction. The relatively simple equations set (4-8) are usually sufficient to obtain relevant representation of water effects on samples and for structural modelling [START_REF] Grimal | Concrete modelling for expertise of structures affected by alkali aggregate reaction[END_REF][START_REF] Poyet | Influence of Water on Alkali-Silica Reaction: Experimental Study and Numerical Simulations[END_REF].

𝜙 𝑔 = 𝜙 𝑔 ∞ . 𝐴 𝑎𝑠𝑟 (7)
The production of gels by ASR leads to increasing the pressure around the reaction site. It can induce cracking in the aggregate and / or in the cement paste depending on aggregate nature [START_REF] Ponce | Different manifestations of the alkali-silica reaction in concrete according to the reaction kinetics of the reactive aggregate[END_REF]. The intra-porous pressure induced by the ASR gel, 𝑃 𝑔 , (Figure 2) is calculated with a matrix-gel interaction modulus 𝑀 𝑔 (taken equal to 27 700 MPa according to [START_REF] Morenon | Impact of stresses and restraints on ASR expansion[END_REF]) and from the volume of ASR gel, 𝜙 𝑔 :

𝑃 𝑔 = 𝑀 𝑔 < 𝜙 𝑔 -(𝜙 𝑔 𝑣 ( 𝑃 𝑔 𝑅 ̃𝐼 𝑡 ) + 𝑏 𝑔 𝑡𝑟(𝜀 𝑒 + 𝜀 𝑐𝑟 ) + 𝑡𝑟(𝜀 𝑝,𝑔 )) > + (8) ⏟ 𝜙1 ⏟ 𝜙2 ⏟ 𝜙3
In this equation, the volumes 𝜙1, 𝜙2 and 𝜙3 can impact the evolution of the pressure for mechanical and / or chemical reasons:

𝛿𝐴 𝑎𝑠𝑟 𝛿𝑡 = 1 𝜏 𝑟𝑒𝑓 𝑎𝑠𝑟 𝐶 𝑇,𝑎𝑠𝑟 𝐶 𝑊,𝑎𝑠𝑟 < 𝑆𝑟 -𝐴 𝑎𝑠𝑟 > + (4)
1) A part of the gel produced by ASR, 𝜙1, migrates into the porosity inside the aggregate and/or the concrete (according to the aggregate nature) [START_REF] Struble | Swelling properties of synthetic alkali silica gels[END_REF][START_REF] Garcia-Diaz | Mechanism of damage for the alkali-silica reaction[END_REF][START_REF] Vayghan | Composition-rheology relationships in alkali-silica reaction gels and the impact on the gel's deleterious behavior[END_REF]. In this work, it is taken to depend on a characteristic void volume, 𝜙 𝑔 𝑣 (calibrated in a free swelling test), on the gel pressure applied, 𝑃 𝑔 , (the higher the pressure is, the greater is the reachable volume of connected porosity, as in a Hg intrusion test [START_REF] Sellier | Une modélisation de la réaction alcalis-granulat intégrant une description des phénomènes aléatoires locaux[END_REF]), and on the effective tensile strength, 𝑅 ̃𝐼 𝑡 . The increase of the proportion of gel that permeates into the porosity connected to the reaction site with increasing pressure can thus be evaluated without supplementary equations. It is especially necessary to model the expansion under multi-axial compression due to external loadings [START_REF] Liaudat | ASR expansions in concrete under triaxial confinement[END_REF].

2) The volume of porosity is modified by the strains induced by external loading. Positive strains lead to a decrease of the pressure while negative strains lead to a pressure increase.

It is evaluated from 𝜙2. 𝑏 𝑔 is the Biot coefficient, which comes from the poromechanics, and 𝑡𝑟(𝜀 𝑒 + 𝜀 𝑐𝑟 ) is the volume induced by the elastic and the creep strains ("𝑡𝑟( )"

represents the matrix trace). Creep and swelling are thus strongly coupled.

3) In this work, the volume of diffuse cracking is evaluated from the plastic strains 𝑡𝑟(𝜀 𝑝,𝑔 ).

The behaviour law of ASR products [START_REF] Swamy | Structural implications of alkali silica reaction[END_REF], considers the effects of these microcracking as permanent strains of the matrix. Even if the term 𝜙3 in equation ( 8) assumes that the microcracks are completely filled by the ASR products, microscopic observations show that in reality they are only partially filled [START_REF] Sanchez | Reliable quantification of AAR damage through assessment of the Damage Rating Index (DRI)[END_REF]; this simplification assumption is balanced by the fitting of 𝜙 𝑔 ∞ which is chosen to minimize model deviation toward real swelling, the consequence is that 𝜙 𝑔 ∞ certainly includes also a volume fraction of empty microcracks induced by ASR. This pressure model configuration seems sufficient to model mechanical behaviour of ASR samples and beams without resorting to more complex models of ASR diffusion or aging, what limits the parameters number.

ASR plastic criteria

For a free swelling concrete, when the gel pressure 𝑃 𝑔 causes an orthoradial tensile stress equal to the strength 𝑅 𝑡 𝑚𝑖𝑐𝑟𝑜 in the vicinity of the reactive spots, the diffuse cracking due to ASR begins. The criterion (9) written for each main tensile stress direction, depends on the external loading, 𝜎 ̃𝐼, through the effective stress, 𝜎 ̃𝐼′ [START_REF] Larive | Combined contribution of experiments and modeling to the understanding of alkali-aggregate reaction and its mechanical consequences[END_REF]. If the concrete is free to swell, the criterion is activated when 𝑃 𝑔 reaches 𝑅 𝑡 𝑚𝑖𝑐𝑟𝑜 . If the concrete is under an external compressive stress, in direction 𝐼 for instance, the criterion will be reached first for a pressure equal to 𝑅 𝑡 𝑚𝑖𝑐𝑟𝑜 in directions 𝐼𝐼 and 𝐼𝐼𝐼. This implies that diffuse cracking propagates in the planes perpendicular to these two directions while cracking in the plane perpendicular to the direction I (Figure 2) is delayed or impeded. In direction 𝐼, the pressure has to rise to 𝑅 𝑡 𝑚𝑖𝑐𝑟𝑜 plus the intensity of the compressive stress 𝜎 ̃𝐼 to cause diffuse cracking (Equation ( 10) in ( 9)).

𝑓 𝐼 𝑡 𝐴𝑆𝑅 = 𝜎 ̃𝐼′ -𝑅 𝑡 𝑚𝑖𝑐𝑟𝑜 𝐼 ∈ [𝐼, 𝐼𝐼, 𝐼𝐼𝐼] (9)
𝜎 ̃𝐼′ = 𝑃 𝑔 + min (𝜎 ̃𝐼 ; 0 ) 𝐼 ∈ [𝐼, 𝐼𝐼, 𝐼𝐼𝐼]

Once the criterion is reached in one direction, a plastic strain 𝜀 𝐼 𝑝,𝑔 is induced. This plastic strain represents the diffuse cracking due to ASR.

ASR damage

Anisotropic tensile damage, 𝐷 𝐼 𝑡,𝑔 , due to gel pressure in the three principal directions, is deduced from the corresponding ASR plastic strains 𝜀 𝐼 𝑝,𝑔 in each main cracking direction (( 11) from [START_REF] Capra | Orthotropic modelling of alkali-aggregate reaction in concrete structures: numerical simulations[END_REF]) and from a characteristic strain 𝜀 𝑘,𝑔 . The characteristic strain is taken as constant and equal to 0.3 % [START_REF] Capra | Anisotropic modelling of alkali-aggregate reaction in concrete[END_REF].

𝐷 𝐼 𝑡,𝑔 = 𝜀 𝐼 𝑝,𝑔 𝜀 𝐼 𝑝,𝑔 + 𝜀 𝑘,𝑔 (11)

Localized cracks

Localized cracks can be caused by external loads or ASR swelling gradients, due to humidity or alkali transfers for instance. They can lead to leakage, reinforcement stress concentration and decreased durability. This part describes the model used to reproduce the concrete behaviour in tension and particularly for the determination of the localized cracks induced by the structural effects. This cracking is always oriented perpendicular to the principal tensile directions.

Rankine plastic criterion

To consider the behaviour of concrete in tension, the model [START_REF] Sellier | Anisotropic Damage and Visco-Elasto-Plasticity Applied to Multiphasic Materials, LMDC -Laboratoire Matériaux et Durabilité des Constructions de Toulouse[END_REF] associates plasticity and damage theory [START_REF] Carol | Damage and plasticity in microplane theory[END_REF][START_REF] Jefferson | Craft--a plastic-damage-contact model for concrete. I. Model theory and thermodynamic considerations[END_REF]. The initialization of localized cracks is managed by an anisotropic Rankine plastic criterion (Equation ( 12) from [START_REF] Sellier | Orthotropic damage coupled with localized crack reclosure processing[END_REF]). Numerically, the hypothesis is that up to three orthogonal localized cracks can exist for each finite element. During a test in direct tension, the apparent stress 𝜎 ̃𝐼 increases until the tensile strength 𝑅 𝐼 𝑡 is reached (Figure 3, point ①) for the peak strain 𝜀 𝑝𝑒𝑎𝑘 𝑡 . This path could be linear (perfect elasticity with 𝐸 the elasticity modulus) or nonlinear (diffuse damage 𝐷 𝑝𝑟𝑖𝑜𝑟𝑝𝑒𝑎𝑘 before the peak). In the latter case, the effective stress at the peak, 𝑅 ̃𝐼 𝑡 , as defined in damage theory, is greater than the tensile strength 𝑅 𝐼 𝑡 that can be measured on the concrete [START_REF] Grimal | Concrete modelling for expertise of structures affected by alkali aggregate reaction[END_REF]. The difference between the effective stress at the peak, 𝑅 ̃𝐼 𝑡 , and the measured tensile strength, 𝑅 𝐼 𝑡 , is due to the damage occurring before the peak, which characterizes the nonlinearity of the tensile behaviour of concrete before the peak. Details about this part of the model are available in [START_REF] Rahal | Finite element modelling of permeability in brittle materials cracked in tension[END_REF]. The prior peak damage 𝐷 𝑝𝑟𝑖𝑜𝑟𝑝𝑒𝑎𝑘 is maximum at that point and is equal to 𝐷 𝑝𝑒𝑎𝑘 [START_REF] Comi | Two-phase damage modeling of concrete affected by alkali-silica reaction under variable temperature and humidity conditions[END_REF].

𝑅 ̃𝐼 𝑡 = 𝑅 𝐼 𝑡 (1 -𝐷 𝑝𝑟𝑖𝑜𝑟𝑝𝑒𝑎𝑘 ) (13)
𝐷 𝑝𝑒𝑎𝑘 = 𝑅 𝑡 𝐸. 𝜀 𝑝𝑒𝑎𝑘 𝑡 (14)

Plastic strain and damage

After the peak (Figure 3), the concrete behaviour is managed by a theory of plastic flow coupled with anisotropic damage. When the strain is higher than the peak tensile strain, 𝜀 𝑝𝑒𝑎𝑘 𝑡 , a localized crack is assumed to be initiated in the finite element. For any strain (Figure 3, point ②), the total stress is calculated from the peak damage 𝐷 𝑝𝑒𝑎𝑘 and the localized damage in the corresponding main direction 𝐷 𝐼 𝑠𝑡𝑟𝑢𝑐𝑡 (( 15) and ( 16)). 𝑤 𝐼 𝑘 is a characteristic crack opening (calculated to dissipate the fracture energy Gft). 𝑤 𝐼 𝑝𝑙,𝑚𝑎𝑥 is the maximal value of the crack opening in the principal direction 𝐼, obtained from the maximal plastic strain 𝜀 𝐼 𝑝𝑙,𝑚𝑎𝑥 multiplied by the finite element size in this direction.

𝜎 𝐼 = (1 -𝐷 𝑝𝑒𝑎𝑘 )(1 -𝐷 𝐼 𝑠𝑡𝑟𝑢𝑐𝑡 ) 𝜎 ̃𝐼 (15) 𝐷 𝐼 𝑠𝑡𝑟𝑢𝑐𝑡 = 1 -( 𝑤 𝐼 𝑘 𝑤 𝐼 𝑘 + 𝑤 𝐼 𝑝𝑙,𝑚𝑎𝑥 ) 2 (16)
An energy regularization is necessary for the post-peak behaviour law where the localized crack occurs. Here, the continuous Hillerborg method [START_REF] Hillerborg | Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements[END_REF] is used to adapt the volumetric dissipated energy and obtain a constant surface energy.

Combining diffuse cracking and localized cracks

Diffuse cracking and localized cracks are combined through the weakest link method.

The effective stress 𝜎 ̃𝐼 (12) is affected by the ASR diffuse cracking 𝐷 𝐼 𝑡,𝑔 in this direction [START_REF] Morenon | Impact of stresses and restraints on ASR expansion[END_REF] (for the sake of simplicity no prior peak damage is considered here).

𝜎 ̃𝐼 = (1 -𝐷 𝐼 𝑡,𝑔 ) (1 -𝐷 𝐼 𝑠𝑡𝑟𝑢𝑐𝑡 )𝜎 𝐼 (17)
For a direct tensile test, the tensile stress is lower on a sample affected by ASR than on a sound one (Figure 4). In this model, the ASR damage has the same impact on the effective stress and on the effective modulus.

Ageing simulation of laboratory structures

Based on the validation of the coupling between diffuse cracking and creep presented in [START_REF] Morenon | Impact of stresses and restraints on ASR expansion[END_REF], the model is applied to a flexural failure test carried out on reinforced concrete beams (with and without ASR) in two phases: the service life (ageing) and the residual strength (failure test) [START_REF] Multon | Évaluation expérimentale et théorique des effets mécaniques de l'alcaliréaction sur des structures modèles[END_REF][START_REF] Multon | Effets structuraux de l'alcali-réaction: Apports d'une expérimentation sur éléments de structures à la validation de modèles[END_REF][START_REF] Multon | Effect of moisture conditions and transfers on alkali silica reaction damaged structures[END_REF][START_REF] Multon | Water distribution in concrete beams[END_REF]. The model responses and the experimental results are compared for the displacements and the cracking patterns of the beams. The effects of reinforcement on swelling and on stress induced in the reinforced beams are analysed in comparison with a beam without reinforcement that is also damaged by ASR in the same conditions.

Methodology and test conditions

All the experimental results analysed in this section are taken from Multon's work [START_REF] Multon | Évaluation expérimentale et théorique des effets mécaniques de l'alcaliréaction sur des structures modèles[END_REF][START_REF] Multon | Water distribution in concrete beams[END_REF][START_REF] Multon | Effets structuraux de l'alcali-réaction: Apports d'une expérimentation sur éléments de structures à la validation de modèles[END_REF][START_REF] Multon | Effect of moisture conditions and transfers on alkali silica reaction damaged structures[END_REF]. The concrete is the same as the one used for the multi-axial confinement samples that were studied with this model in [START_REF] Morenon | Impact of stresses and restraints on ASR expansion[END_REF]. Thus, the material parameters are the ones obtained in [START_REF] Morenon | Impact of stresses and restraints on ASR expansion[END_REF]. They are given in metres between supports. The lower 70 mm of the beams was immersed in water while their upper face was subjected to air-drying at 30% relative humidity (RH) for 428 days. No moisture transfer could occur on the lateral faces as they were covered with watertight aluminium. The upper faces were re-saturated with liquid water after 428 days (Figure 5) and saturation conditions were maintained for up to 700 days, while the boundary conditions in the other parts were not modified. The temperature was kept at 38 °C during the entire test.

These beams were simply supported on steel bars 0.1 metre from each end and halfway up (Figure 5). Because of the double symmetry, only a quarter of the beam is modelled, to reduce the calculation time (Figure 5). The water saturation degree has a significant impact on the amplitude and the kinetics of ASR swelling. As the ASR-affected beams were subjected to hydric gradients here, a hydric calculation is necessary to reproduce the different swelling in the structures [START_REF] Bangert | Chemo-hygro-mechanical modelling and numerical simulation of concrete deterioration caused by alkali-silica reaction[END_REF]. The calculations are chained: a hydric calculation is performed first and is followed by the poromechanics calculation. This chaining is possible because, on the one hand, the water pressure is mainly controlled by the desorption curve and not by liquid water compressibility and, on the other hand, the moisture transfer is driven by capillary forces in undamaged zones.

The advantage of this chaining method is to separate the resolution of the different physical mechanisms (hydric and mechanical) and the drawback is that there is no feedback from the mechanical to the hydric state. In particular, the transfer is little modified by the cracking. A combination between mechanics and transfer based on Rahal's work [START_REF] Rahal | Modelling of change in permeability induced by dilatancy for brittle geomaterials[END_REF][START_REF] Rahal | Finite element modelling of permeability in brittle materials cracked in tension[END_REF] is a perspective of this work, which would improve the precision of the modelling response.

Beam meshes

The meshes used in the calculations are presented in Figure 6. The mesh of the nonreinforced beams is refined in the lower part to obtain accurate calculations in this area. For the reinforced beam, a finer mesh is used in the zones around the reinforcements to improve the modelling of the steel -concrete interface.

Determination of the water saturation degree in the structures

Alkali-silica reaction is very sensitive to the degree of saturation of the concrete [START_REF] Multon | Évaluation expérimentale et théorique des effets mécaniques de l'alcaliréaction sur des structures modèles[END_REF][START_REF] Larive | Apports combinés de l'expérimentation et de la modélisation à la compréhension de l'alcali-réaction et de ses effets mécaniques[END_REF][START_REF] Poyet | Influence of Water on Alkali-Silica Reaction: Experimental Study and Numerical Simulations[END_REF][START_REF] Poyet | Etude de la degradation des ouvrages en beton atteints par la reaction alcalisilice -approche experimentale et modelisation numerique des degradations dans un environnement hydro-chemo-mecanique variable[END_REF].

The water facilitates the diffusion of the reactive species and contributes to the final volume of ASR-gels, which are hydrophilic. Accurate quantification is thus necessary to obtain relevant analysis.

To simulate drying (upper part) and capillary rise (lower part), a model of water diffusion is used [START_REF] Coussy | Mécanique des milieux poreux[END_REF]. The water diffusion coefficient 𝐷 (19) takes account of the permeation transfers and the dependence of diffusion on the water saturation degree, 𝑊, according to Mensi's model [START_REF] Mensi | Séchage du béton: analyse et modélisation[END_REF]. This model is simple but it gives realistic representation of moisture gradients in the beams which is an important point for their structural analysis.

The calibration is based on the data provided by the experimental programme [START_REF] Multon | Évaluation expérimentale et théorique des effets mécaniques de l'alcaliréaction sur des structures modèles[END_REF][START_REF] Multon | Effets structuraux de l'alcali-réaction: Apports d'une expérimentation sur éléments de structures à la validation de modèles[END_REF]. For the coefficient 𝑎 (equation ( 19)), the values vary between 1.2 10 -13 m².s -1 and 5.8 10 -12 m².s -1 ).

The coefficient 𝑏 is taken as 0.051 for the zone in imbibition and 0.06 for the zone under drying [START_REF] Grimal | Concrete modelling for expertise of structures affected by alkali aggregate reaction[END_REF][START_REF] Morenon | Modélisation des réactions de gonflement interne des bétons avec prise en compte des couplages poro-mécaniques et chimiques[END_REF]. The profiles presented in Figure 7 for these two parameters have been evaluated to obtain a correct prediction of moisture transfers in the beams (Figure 8 and Figure 9). Variations between the top and the bottom of the beams can be due to differences in concrete porosity observed during the study [START_REF] Multon | Évaluation expérimentale et théorique des effets mécaniques de l'alcaliréaction sur des structures modèles[END_REF]. Differences for parameter b between drying and rewetting are due to usual hysteresis behaviour between concrete sorption and desorption [START_REF] Aligizaki | Pore structure of cement-based materials: testing, interpretation and requirements[END_REF] and due to cracking appeared during the drying period. The initial saturation degree was determined on companion specimens made under the same conditions (degree of saturation 0.85 after 28 days of curing in endogenous conditions). The difficulty to use more precise modelling lies in the lack of experimental data, as sorption curves, to calibrate their parameters for the studied concrete. Thus, the total mass variation and water content profiles in the height of the structures are given in Figure 8 and in Figure 9. The evaluation of moisture in the depth of the beams is correct (Figure 9). This result gives confidence for the structural analysis. The results obtained by the modelling in terms of deflection at mid-span and mean longitudinal strain are compared to the experimental results for the three reactive beams in Figure 10. Deflection analysis is divided into two parts: before and after the wetting of the upper face (428 days) (Figure 10).

Imbibition-drying phase (before 428 days):

The deflections move towards negative values during the first 50 days because shrinkage in the upper part of the beams is the predominant phenomenon. Then, swelling is initiated in the lower part, which was immersed (Figure 10). In the reactive beam without reinforcement (WR red curve), the evolution of the deflection accelerates because the lower part of the beam is subjected to high positive strains while the upper part shortens slightly under the effect of shrinkage when drying. In the two reinforced beams, the longitudinal steel bars quickly restrain the ASR swelling in this direction. A compressive stress is then created in the concrete in the direction of the main reinforcements. The middle part (mid-height of the beam) continues to swell with a delay due to the influence of the saturation degree on the ASR kinetics. It is the expansion in the middle part combined with the restraint of expansion by the steel bars that causes the inversion of the deflection evolution between 150 and 428 days.

For the amplitude, all the calculated deflections agree quite well with the experimental deflections obtained at the end of drying (428 days) as in [START_REF] Grimal | Concrete modelling for expertise of structures affected by alkali aggregate reaction[END_REF][START_REF] Comi | Two-phase damage modeling of concrete affected by alkali-silica reaction under variable temperature and humidity conditions[END_REF][START_REF] Alnaggar | Modeling time-dependent behavior of concrete affected by alkali silica reaction in variable environmental conditions[END_REF].

Figure 10: Comparison between deflections found in experimental data [43] and by modelling

Upper face wetting phase (after 428 days):

In this period, all beams moved upwards (positive deflection) as the upper face was then in contact with water. The upper part of the beams was saturated, implying a rapid positive deflection because the kinetics and the ASR swelling potential depend directly on the presence of water. In the case of late water supply, ASR can appear very quickly, as already discussed in [START_REF] Multon | Effect of moisture conditions and transfers on alkali silica reaction damaged structures[END_REF]. The displacements are well reproduced by the model. The increase obtained during the upper face wetting seems to be managed in terms of amplitude. From 480 days, the deflections of the reinforced and strongly reinforced beams seem to evolve more slowly in the model than in reality. This can be due to the impact of transverse cracks caused by restrained shrinkage obtained during the first 428 days in the upper part of the beam. The number of these cracks is faithfully reproduced by the model for both beams but it is difficult to evaluate the cracks depths and their impacts on hydric transfer during the rewetting. The differences between calculation and experiment can be due to the absence of coupling between transfer and mechanics. Indeed, the saturation degree modelling predict a fast and uniform rewetting in the beams to fit the global mass balance while important rewetting could be localized around cracks in the reality. It should not have the same structural consequences. The result is a too fast stabilization of the deflection. This modelling could be improved in the future. All these conclusions are confirmed by the comparison of longitudinal strains.

To prepare the failure bending tests on the normally reinforced beam, it is now interesting to analyse its stress state at the end of the ageing phase.

Stress state in the reinforced beam

The influence of reinforcement on swelling and stress in concrete has a great impact on the service and ultimate behaviours of damaged structures. This model faithfully reproduces the impact of stresses and restraint on ASR expansions [START_REF] Morenon | Impact of stresses and restraints on ASR expansion[END_REF]. In the reinforced beams [START_REF] Multon | Évaluation expérimentale et théorique des effets mécaniques de l'alcaliréaction sur des structures modèles[END_REF], the swelling is restrained by the longitudinal steel bars. This causes tension in the steel bar and compression in the concrete. The compressive stress obtained in concrete is comparable to a chemical prestress [START_REF] Koyanagi | Failure Behavior of Reinforced Concrete Beams Deteriorated by Alkali-Silica Reactions[END_REF][START_REF] Swamy | Structural implications of alkali silica reaction[END_REF][START_REF] Monette | Residual strength of reinforced concrete beams damaged by alkali-silica reaction-Examination of damage rating index method[END_REF]. With the isotropic gel pressure presented in the model, the irreversible strain due to ASR in the unrestrained direction controls the gel pressure and, thus, the prestress in the restrained direction. for a free ASR swelling of 0.3% [START_REF] Morenon | Impact of stresses and restraints on ASR expansion[END_REF]. Thus, flexural cracking can be delayed under applied loading due to the chemical prestress induced by ASR in reinforced structures.

Experimentally, a reinforced wall subject to ASR can effectively crack later than a sound wall, thanks to this prestress [START_REF] Habibi | Alkali aggregate reaction in nuclear concrete structures: part 3: structural shear wall elements[END_REF].

In the reinforcement, the maximum stress (140 MPa) obtained by calculation is lower than the elastic limit of the steel (500 MPa). This is consistent with the structural analysis presented in [START_REF] Multon | Chemomechanical Assessment of Beams Damaged by Alkali-Silica Reaction[END_REF]. The elastic behaviour of the steel bars during ageing is thus validated.

Effective mechanical characteristics of the normally reinforced beam

Figure 12 presents the damage due to ASR in the normally reinforced beam. The damage is variable in the beams and highly anisotropic due to the presence of the reinforcement. This is due to the swelling anisotropy induced by the restraint of the steel bars [START_REF] Morenon | Impact of stresses and restraints on ASR expansion[END_REF]. The upper part of the beam is slightly affected by the ASR damage (<0.2) while the lower part is strongly affected in the directions parallel to the longitudinal steel bars (0.4 in the Y and Z directions).

As a result, the concrete is little damaged perpendicular to the longitudinal direction. At the end of ageing, the elastic modulus and effective tensile strength are thus anisotropic and variable in the beam. This is the reason why the modulus measured on cores drilled from a structure may be not be representative of the mechanical behaviour of the structure as a whole. In the modelling, the mechanical properties are weighted by the ASR damage in each main direction in order to obtain realistic anisotropic values. The failure test of the reinforced beams (one reactive and one non-reactive) was performed at the end of ageing [START_REF] Multon | Évaluation expérimentale et théorique des effets mécaniques de l'alcaliréaction sur des structures modèles[END_REF]. The non-reactive beam was subjected to the same moisture conditions as the reactive beam. This enabled the relevance of the model for simulating the residual resistance of these structures, when sound or damaged by ASR, to be assessed. It also provided a complementary validation of the previous calculations because the stress and damage states, before these failure tests were performed, had a significant influence on the failure behaviour.

After 700 days, the beams were taken to failure with a 4-point flexural test using a slightly modified span (2.75 m vs. 2.8 m during the swelling phase) and supports on the underside (Figure 13). The steel behaviour was modelled using a bilinear law: a first elastic phase with a modulus of 200 GPa and an elastic limit at 500 MPa, and a second phase of hardening with a slope of 5

GPa (calibrated by inverse analysis of the non-reactive beam as no measurement of this data was performed during the experimental programme). In this phase, special care was taken concerning the anchoring of the steel bars: the mesh in the area of the supports was modified to simulate perfect anchoring. The calculation was carried out with a steel that was not under maximum strain (failure) and was thus stopped at the value of the force-deflection peak of the experimental curve (40 mm of deflection). Two main points were investigated in this work:

the capability of the model to predict the first flexural cracks, and the load for structure yielding according to the stress and damage states due to ASR.

Deflections and applied load

Failure of the plain beams

The paper focuses on the flexural performance of reinforced concrete beams. But, as the model captures the concrete behaviour considering the combination of ASR, creep, shrinkage and mechanical damage, it is important to evaluate the capacity of the model to reproduce the performance of the two plain beams. The results of the failure tests for the beams without reinforcement is presented in Figure 14. The modulus of elasticity of the beam without ASR is 46,400 MPa to take into account the cement hydration in agreement with the conclusions of [START_REF] Multon | Évaluation expérimentale et théorique des effets mécaniques de l'alcaliréaction sur des structures modèles[END_REF]. The behaviours of both beams are well-reproduced. The peak deflection of the nonreactive beam is a little too high (19%) but its rigidity is consistent. Without reinforcement, the ASR beam is weaker than the non-reactive one (experimentally 33%, numerically 45%).

Indeed, the ASR damage reduces the mechanical characteristics such as the tensile strength.

The rigidity of the ASR beam is a bit too low (25%) but the force peak is really consistent.

The consideration of hydration could be a way of improvement for the modelling of these tests, in accordance with the conclusions of [START_REF] Multon | Évaluation expérimentale et théorique des effets mécaniques de l'alcaliréaction sur des structures modèles[END_REF]. reproduced. The stiffness differences after the first bending crack may be due to the displacement of the sensor used to measure the deflection. The sensor was placed near a crack (following the observations given in [START_REF] Multon | Évaluation expérimentale et théorique des effets mécaniques de l'alcaliréaction sur des structures modèles[END_REF]).

Failure of the reactive reinforced beams

The reactive beam (dark blue curves: experiment in dashed line and modelling in continuous line) follows the same evolution but the first flexural crack appears for a higher load (120 kN versus 75 kN). The start of the steel yielding is similar to that for the non-reactive beam. The experimental behaviour of the reactive beam is well reproduced throughout the test. The load for the first flexural crack is slightly overestimated (140 kN versus 120 kN). In the longitudinal direction of the reactive beam, there is no decrease in Young's modulus, thanks to the anisotropic criteria. Thus, there is no loss of rigidity for the reactive beam, despite the amount of expansion in the free direction.

Discussion

Impact of ASR on the flexural performance of reinforced concrete beams ASR seems beneficial to the bending behaviour of a reinforced beam. For the same applied force, the deflection is smaller for the reactive beam and cracking is delayed. This behaviour is particularly apparent for deflections of less than 5 mm. In the service limit state (SLS), the admissible force for these beams is 124 kN [START_REF] Multon | Évaluation expérimentale et théorique des effets mécaniques de l'alcaliréaction sur des structures modèles[END_REF], corresponding to a deflection of 1.7 mm for the reactive beam and to a deflection to 3.9 mm for the non-reactive beam. This is an improvement of the flexural performance of the ASR beam due to the ASR-induced stress in the reinforced concrete (chemical prestress [START_REF] Fujii | The static and dynamic behavior of reinforced concrete beams with cracking due to alkali-silica reaction[END_REF][START_REF] Inoue | Structural behaviors of reinforced concrete beams affected by alkali-silica reaction[END_REF][START_REF] Clark | Critical review of the structural implications of the alkali silica reaction in concrete, Bridges Design[END_REF][START_REF] Ohno | The mechanical behaviour of reinforced of beams coated after Alkali-Silica Reaction damage[END_REF]) and to the anisotropic damage (no Young's modulus decrease in the longitudinal direction).

Impacts on durability

The structural cracking delay of the reinforced and reactive beam could imply a better protection from aggressive external agents (such as salts, sulfates or carbon dioxide) and water. However, diffuse cracking created parallel to the reinforcement due to ASR expansions facilitated their access. The durability of a beam, in particular of its reinforcement, which could become corroded faster, can therefore be affected. This aspect could largely balance the positive effect of the chemical pre-stress mentioned above. A computational coupling of the state of cracking with a numerical model of corrosion by carbonation or sulfate attacks which would consider the consequences of longitudinal diffuse cracks on transfer properties could be a perspective on this point.

Minimum model skills to reproduce the improvement of the flexural behaviour

In order to reproduce the cracking delay of the ASR beam numerically, it is necessary to generate a good amplitude of ASR chemical prestress parallel to the reinforcement. Based on the above faithful modelling of this phenomenon, the capacities of the models necessary to obtain relevant quantification are established.

-The model used should be able to reproduce the anisotropy of the swelling in reinforced concrete to avoid cracking perpendicular to the reinforcements. An isotropic model will not be able to reproduce the cracking delay.

-The consistent amplitude of the ASR chemical prestress depends on the coupling between the pressure induced by ASR, the diffuse cracking in the unrestrained direction and the creep during the ageing stage.

-The model should be able to differentiate between the diffuse cracking (during ageing)

and the structural cracking (during failure test).

Conclusion

The aim of this paper was to understand and model the impact of ASR on the flexural performance of reinforced concrete beams. A numerical approach was used to reproduce this behaviour and to discuss the real impact of ASR on reinforced concrete. The validation was performed on ageing structures in the laboratory. The simulation of the service life (ageing)

and the residual strength (failure test) have been presented and analysed. The good reproduction of the behaviour of the ASR reinforced beam allows some conclusions to be drawn:

1) First flexural cracks appear at higher load for ASR affected reinforced beam than for non-reactive, beam, due to the ASR chemical prestress generated by the restrained swelling in the reinforcement direction.

2) The rigidity of a reinforced beam is little impacted by ASR, due to anisotropic damage induced by the presence of the steel bars.

3) The cracking delay seems to be positive from the mechanical point of view. However, durability problems could appear because of the ASR diffuse cracking induced parallel to the reinforcement during the ageing phase. A corrosion model, coupled with a transfer model considering the cracking state could be a perspective on this point.

Figure 2 :

 2 Figure 2: Scheme of the ASR plastic criterion for free swelling (a) and swelling under uniaxial compressive stress (b)

Figure 3 :

 3 Figure 3: Model of concrete tensile behaviour under external loading or displacement gradients 𝑓 𝐼 𝑡 𝑠𝑡𝑟𝑢𝑐𝑡 = 𝜎 ̃𝐼 -𝑅 ̃𝐼 𝑡(12)

Figure 4 :

 4 Figure 4: ASR cracking and localized crack combinations

Figure 5 :

 5 Figure 5: Geometry, limit conditions, environmental conditions and reinforcement of the beams [7,43]

Figure 6 :

 6 Figure 6: Meshes used

Figure 7 :Figure 8 :Figure 9 :

 789 Figure 7 : Boundary conditions for the hydric simulation and definitions of Mensi's law coefficients a and b

Figure 11

 11 Figure 11 represents a profile of the longitudinal stress in the height of the beam, obtained at

Figure 11 :

 11 Figure 11: Computed stresses along elevation in the concrete in the longitudinal direction of the beam at mid span at 700 days In the literature, this maximum chemical prestress obtained in contact with steels is estimated

Figure 12 : 6 ] 4 . Simulation of behaviour up to failure of aged structures 4 . 1 .

 126441 Figure 12: Comparison of cracking patterns obtained with the model and by experiment for the reinforced beam [6]

Figure 13 :

 13 Figure 13: Mechanical limit conditions for the two stages: ageing and breaking

Figure 14 :

 14 Figure 14: Force-Deflection curves of the failure test (model, theoretical results [6] and experiment [6] of the beams without reinforcement)

Figure 15 :

 15 Figure 15: Force-Deflection curves of the failure test (model and experiment [6] of the normally reinforced beams)The reactive beam cracking is delayed compared to the first cracking of the non-reactive

Figure 16 :

 16 Figure 16: Force-Deflection curves of the failure test (model and experiment [6] of the strongly reinforced beam)

4 )

 4 To reproduce the behaviour of reinforced concrete damaged by ASR numerically, models have to produce a realistic ASR chemical prestress in the reinforcement direction (coupling between ASR expansion, diffuse cracking and concrete creep) to evaluate the anisotropy of the mechanical properties, and to differentiate ASR diffuse cracking and structural macrocracks.

Table 1

 1 𝜙 𝑔 𝑣 =0.13% in Table1). Elastic, creep and ASR plastic strains used to obtain a relevant representation of ASR action on concrete in Equation (8) lead to the differences between the imposed volumetric ASR gel potential 𝜙 𝑔 ∞ minus 𝜙 𝑔 𝑣 and the strains measured on concrete samples.

and enable the deformations of the cylindrical specimens studied by Multon to be retrieved: shrinkage in non-saturated conditions, free swelling, and creep of sound and reactive specimens in both uniaxial and triaxial tests. The volumetric ASR gel potential 𝜙 𝑔 ∞ (calibrated on a free swelling test) is about 0.54%. A part of this potential is supposed ineffective in Equation (8) (volume

Table 1 : Material parameters calibrated on specimens in [17]

 1 Three reactive beams are modelled in this part: one without reinforcement, one normally reinforced and one strongly reinforced. Their dimensions are 3.0 x 0.5 x 0.25 metres for 2.8

	Parameter	Symbol	Value
		General parameters
	Elastic modulus	E	37.2 MPa
	Tensile strength	𝑅 𝑡	3.7 MPa
	Fracture energy in tension	𝐺 𝑓𝑡 𝑡	1.0 10 -4 MJ/m²

Acknowledgments

Financial support was provided by the Centre d'Ingénierie Hydraulique (CIH) of Electricité de France (EDF) in the framework of a PhD thesis approved by the French ANRT (Association Nationale de la Recherche et de la Technologie). This support is gratefully acknowledged.


Ali Bentrad

Satyanad Kichenassamy

HYPERGEOMETRIC FUNCTIONS AND SINGULAR SOLUTIONS OF WAVE EQUATIONS WITH LORENTZ-INVARIANT POTENTIAL

Keywords: 2020 Mathematics Subject Classification. 35L15, 33C05 Singular solutions, hypergeometric equation. 1

New solutions of equation u tt -∆u+Au/(r 2 -t 2 ) = 0, in series of hypergeometric functions (HGF), are constructed. A sharpened estimate on hypergeometric functions is derived to prove the convergence of the series.

1. Introduction 1.1. Main results and motivation. Hypergeometric solutions of linear PDEs have been studied almost since the beginning of the analytic study of PDEs, because they have built-in singularities and therefore permit a global representation of solutions near regular and singular points, unlike the representation in power series. Among the first examples are the Euler-Poisson-Darboux and Tricomi equations, see [10, book iv, chap. iii] and [START_REF] Delache | Calcul de la solution élémentaire de l'opérateur d'Euler-Poisson-Darboux et de l'opérateur de Tricomi-Clairaut, hyperbolique, d'ordre 2[END_REF]. These special equations have been studied from different perspectives in [START_REF] Begehr | Transformations,Transmutations, and Kernel Fuctions[END_REF][START_REF] Gilbert | Function Theoretic Methods in Partial Differential Equations[END_REF]. However, recent results show that many other PDEs also possess hypergeometric solutions. In particular, degenerate equations which fall beyond the scope of general existence results have been found to have solutions expressible in terms of HGF or series of HGF, see e.g. [START_REF] Urabe | On the theorem of Hamada for a linear second-order equation with variable multiplicities[END_REF][START_REF] Urabe | Hamada's theorem for a certain type of the operators with double characteristics[END_REF][START_REF] Urabe | Singular Cauchy problem for a certain linear and 2nd order equation[END_REF][START_REF] Bentrad | A linear Fuchsian equation with variable indices[END_REF][START_REF] Bentrad | Représentation hypergéométrique de la solution pour une classe d'opérateurs d'ordre deux[END_REF][START_REF] Bentrad | Singularities of solutions to some second-order partial differential equations[END_REF]. The resulting information on singularities of solutions does not seem to be accessible via other methods.

These results open up the possibility that hypergeometric solutions are representative of the singular behavior of solutions of PDEs of rather general form; in a special case, this has been demonstrated in [START_REF] Bentrad | A linear Fuchsian equation with variable indices[END_REF].

The present work gives another example of this phenomenon, for the equation: [START_REF] Anco | Exact solutions of semilinear radial wave equation in n dimensions[END_REF] Lu := u tt -∆u + Au r 2 -t 2 = 0, for 0 < t < r < R, where u = u(x, t), r = |x|, R is a positive constant, x ∈ R N , and N ≥ 1. Note that the expression r 2 -t 2 is Lorentz-invariant. The constant A is assumed to be real and less than 1 4 (N -1) 2 , to make the various parameters defined below real; the expressions we obtain admit of complex extension. The solutions are considered outside the light-cone only for the same reason. The constant 1 4 (N -1) 2 is the Hardy constant in dimension N + 1.

The technical difficulties are twofold: (a) recognizing that the problem can be reduced to a hypergeometric equation (HGE) and (b) estimating the resulting solution.

Theorems 5 and 6 perform the reduction, and give solutions of the form:

(2) u(x, t) = r l φ( x r )f ( t 2 r 2 ),
where l is an integer and φ is a spherical harmonic: -∆ S N-1 φ = λφ, where ∆ S N-1 is the Laplace-Beltrami operator on the (N -1)-sphere.

When N is odd and φ is constant, f may be expressed in terms of elementary functions. The estimate is proved in Lemmas 2 and 3, and gives the convergence of a series of HGFs representing the solution of the Cauchy problem with general analytic data proportional to φ(x/r) (Theorem 8). The solutions are valid for 0 < t < r < R -t, which is the domain of dependence predicted by finite speed of propagation. The usual estimates on the HGF, see Lemma 4, are not sufficient to obtain this result: they would only give convergence for r ≤ R/2 (Theorem 9). Remark 1. Solutions of inhomogeneous HGE with right-hand side z m-1 are also known [START_REF] Babister | Transcendental functions satisfying nonhomogeneous linear differential equations[END_REF]; they would enable one to obtain exact solutions to equation [START_REF] Anco | Exact solutions of semilinear radial wave equation in n dimensions[END_REF] with suitable right-hand sides. Recall that hypergeometric solutions of the inhomogeneous wave equation with a constant righthand side arise as a step in the construction of interacting waves for nonlinear equations with piecewise linear nonlinearities [START_REF] Keller | Singularities of semilinear waves[END_REF].

Many of the HGF in this paper may be expressed in terms of Legendre functions P µ ν [2, formula 15.4.25]. However, general results on Legendre functions do not seem to give the estimates required for the convergence proof.

Replacing t by it in our solutions gives solutions of a Schrödinger equation with inverse-square, or Hardy potential, away from the origin in R N +1 :

(3) -u tt -∆u + Au r 2 + t 2 = 0, This equation arises by linearization about the "very singular solution" of -∆u = gu p [START_REF] Brezis | Singular solutions for some semilinear elliptic equations[END_REF][START_REF] Véron | Singular solutions of some nonlinear elliptic equations, Nonlinear Anal[END_REF]. Eigenfunctions of the Schrödinger equation with inverse-square potential on a bounded domain require Bessel functions [START_REF] Brezis | Blow-up solutions of some nonlinear elliptic problems[END_REF], as in the case of zero potential. We do not consider the wave equation with potential A/r 2 , which is not Lorentz-invariant.

Similarly, the potential A/(r 2 -t 2 ) arises naturally by linearizing the homogeneous or "self-similar" solution [START_REF] Grundland | Symmetry breaking and bifurcating solutions in the classical complex φ 6 theory[END_REF][START_REF] Anco | Exact solutions of semilinear radial wave equation in n dimensions[END_REF] (4)

U = (r 2 -t 2 ) -1/(p-1) for r > t,
of the nonlinear wave equation

(5) u tt -∆u = gu p , p = 1, in N space dimensions, where p = (N + 1)/(N -1), and

g = 2 p -1 (N -1 - 2 p -1
).

This solution is real for r > t. Linearization leads to [START_REF] Anco | Exact solutions of semilinear radial wave equation in n dimensions[END_REF] with

A = -gp = 2p p -1 ( 2 p -1 -N + 1).
The constant A is positive if and only if 0 < p < (N + 1)/(N -1). For p < 1, the self-similar solution U remains bounded on the light-cone; it is a polynomial if p = 1 -1/k, with k = 2, 3,. . . Homogeneous solutions such as U have weak singularities for all time; they differ from solutions with blowup, for which Cauchy data in the remote past are not themselves singular-such solutions may be constructed systematically by Fuchsian Reduction [START_REF] Kichenassamy | Fuchsian Reduction: Applications to Geometry, Cosmology, and Mathematical Physics[END_REF][START_REF] Kichenassamy | Nonlinear Wave equations[END_REF][START_REF] Kichenassamy | Stability of blow-up patterns for nonlinear Wave equations[END_REF]. The present paper shows that certain problems with characteristic singularities are also amenable to a reduction to the HGE, which is the prototype of a Fuchsian equation. 1.2. Organization of the paper. Background information on the hypergeometric equation is collected in Section 2, to make the paper self-contained. For further details, see e. g. [START_REF]Handbook of Mathematical Functions[END_REF][START_REF] Kampé De Fériet | La Fonction Hypergéométrique[END_REF][START_REF] Ponnusamy | Asymptotic expansions and inequalities for hypergeometric functions[END_REF][START_REF] Slater | Generalized Hypergeometric functions[END_REF]. The comparison Lemmas 2 and 3 are perhaps new. Lemma 4 is a very simple form of estimates in [19, p. 283]. For details on the recent results on estimates for the HGF motivated by results found in Ramanujan's notebooks, see [START_REF] Ponnusamy | Asymptotic expansions and inequalities for hypergeometric functions[END_REF].

Section 3 constructs exact solutions for the Cauchy problem with initial data of special form (Theorem 6). In Section 4, these solutions are superposed to obtain more general solutions with controlled behavior (Theorems 8 and 9).

(6) z 1-c F (a -c + 1, b -c + 1 ; 2 -c ; z).
The only singularities of the HGE are 0, 1 and infinity, and they are regular.

2.2.

A simple reduction. Any linear Fuchsian equation with three regular singularities is reducible to the HGE by a change of unknown of the form w(z) = z α (1 -z) β w(z). We record a special case of this result, for later use.

Lemma 1. Consider a solution f (z) of D(D + c -1)f -z(D + a)(D + b)f + Bzf 1 -z = 0,
where B is a constant. If

(7) β(β + a + b -c) + B = 0,
and

f (z) = (1 -z) β g(z), then D(D + c -1)f -z(D + a + β)(D + b + β)f = 0.
Proof. The result may be proved either by appealing to general properties on the hypergeometric equation or in a self-contained manner. We give both arguments. First, let β be one of the roots of equation [START_REF] Bentrad | A linear Fuchsian equation with variable indices[END_REF]. The equation solved by f has three regular singular points, at 0, 1, and infinity. The indices at 0, 1 and infinity are respectively (0, 1 -c), (β, β ′ ), and (a, b), where β and β ′ are the two roots of equation [START_REF] Bentrad | A linear Fuchsian equation with variable indices[END_REF]. It follows that g solves an equation with the same singular points, which are regular, and for which the indices at 0, 1 and infinity are respectively (0, 1 -c), (0, β ′β), and (a + β, b + β). Since zero is an index both at 0 and at 1, this equation must be equivalent to the hypergeometric equation for the function F (a + β, b + β; c; z). This proves the result.

The self-contained proof runs as follows. Let y = z/(1 -z), so that Dy = y + y 2 . We find

Df = (1 -z) β (D -βy)g; D 2 f = (1 -z) β (D 2 -βy(2D + 1) + β(β -1)y 2 )g.
Substituting into the equation for f , and dividing through by (1 -z) β , we find, after some algebra,

D(D + c -1)g -z[(D + a)(D + b) + β(2D + 1)]g + Qg = 0, where Q = β(β -1)y 2 (1 -z) -β(c -1)y + βyz(a + b) + By = yzβ(β + a + b -1) + [B -β(c -1)]y = β(β + a + b -1)y(z -1) + [B + β(β + a + b -c)]y.
Since y(z -1) = -z, we find

D(D + c -1)g -z(D + a + β)(D + b + β)g + [B + β(β + a + b -c)]yg = 0,
from which the desired result follows.

a n z n ≪ n≥0 b n z n if |a n | ≤ b n for every n ≥ 0. Lemma 2. If c ≥ 1 2 and 0 ≤ a ≤ b -1 2 , then (8)
F (a, b ; c ; z 2 ) ≪ 1 2 (1 + z) 1-2b + (1 -z) 1-2b . If c ≥ 1 2 and 0 ≤ a ≤ b + 1 2 , (9) F (a, b ; c ; z 2 ) ≪ 1 2 (1 + z) -2b + (1 -z) -2b .
Proof. Consider first the case a ≤ b -1 2 . The assumptions on the parameters ensure that F (a, b ; c ; z 2 ) ≪ F (b -1 2 , b ; 1 2 ; z 2 ). Now

F (b - 1 2 , b ; 1 2 ; z 2 ) = n≥0 (b -1 2 )b • • • (b + n -1) 1 2 • 1 • • • (n -1 2 )n z 2n = n≥0 (2b -1) 2n (1) 2n z 2n .
This expression is equal to the right-hand side of ( 8), thanks to the identity

(10) (1 -z) -d = n≥0 (d) n (1) n z n . If a ≤ b + 1 2 , we observe that F (a, b ; c ; z 2 ) ≪ F (b, b + 1 2 ; 1 2 ; z 2 ), to which the previous considerations apply, with b replaced by b + 1 2 . Lemma 3. If c ≥ 3 2 and 0 ≤ a ≤ b -1 2 , then (11)
F (a, b ; c ; z 2 ) ≪ 1 4z(1 -b) (1 + z) 2-2b -(1 -z) 2-2b . If b > 0, c ≥ 3 2 , and 0 ≤ a ≤ b + 1 2 , (12)
F (a, b ; c ; z 2 ) ≪ 1 2z(1 -2b) (1 + z) 1-2b -(1 -z) 1-2b . Proof. Assume first 0 ≤ a ≤ b -1 2 . We now have F (a, b ; c ; z 2 ) ≪ F (b -1 2 , b ; 3 2 ; z 2 ). Since ( 3 2 ) n = 2( 1 2 ) n ( 3 2 + n -1) = ( 1 2 ) n (2n + 1), zF (b - 1 2 , b ; 3 2 ; z 2 ) = n≥0 (b -1 2 ) n (b) n ( 1 2 ) n (1) n z 2n+1 2n + 1 .
This series represents the anti-derivative of F (b -1 2 , b ; 1 2 ; z 2 ) that vanishes for z = 0. The first statement follows; the second is reduced to the first as before.

We conclude with an estimate in terms of powers of (1 -z).

c ; z) ≪ Γ(c)Γ(d) Γ(a)Γ(b) (1 -z) -d . Proof. Since a + b = c + d, we have, for n ≥ 1, (a + n)(b + n) (c + n)(d + n) = 1 + ab -cd (c + n)(d + n) . Also, ab -cd = (c -a)(c -b) > 0. Therefore, (a) n (b) n (c) n (d) n increases with n. Let Γ 0 = Γ(c)Γ(d) Γ(a)Γ(b) . Since, for any x > 0, Γ(x) = lim n→∞ n x-1 (1) n (x) n , (a)n(b)n (c)n(d)n tends to Γ 0 as n → ∞. Therefore, (a) n (b) n (c) n (1) n ≤ Γ 0 (d) n (1) n .
The result follows, thanks to (10).

Hypergeometric solutions

We now seek solutions of equation ( 1) of the form (2). We first obtain solutions of Lu = 0, and then select those with Cauchy data (r l φ λ (x/r), 0), or (0, r l φ λ (x/r)), where φ λ is an eigenfunction of the Laplace-Beltrami operator on the sphere: -∆ S N-1 φ λ = λφ λ , with λ constant and nonnegative.

3.1.

A family of solutions of Lu = 0. Let u = φ λ (x/r)v(r, t). We obtain

u tt -∆u = v tt -v rr - N -1 r v r + λ r 2 v φ λ .
Next, let z = t 2 /r 2 and v(r, t) = r l (1 -z) β g(z). We show that one may choose β so that g solves a hypergeometric equation:

Theorem 5. Assume that (14) β l := 1 2 l + N -1 2 1 + 1 - A (l + N -1 2 ) 2 . Let (15) a ± = - 1 2 (l -1 + N 2 ) ± 1 2 ( N 2 -1) 2 + λ.
Then, equation Lu = 0, with

u = r l (1 -z) β g(z)φ λ (x/r),
is equivalent to the HGE for g, with parameters

(a l , b l , c) := (a + + β l , a -+ β l , 1/2).
Remark 2. The parameters a l and b l depend on l, N, λ, A, but we single out the dependence in l since we will let it tend to infinity. They are real if and only if

(l + N -1 2 ) 2 ≥ A.
In that case, a l ≥ b l .

Proof. Let v(r, t) = r l f (z). Recall that D = z d dz . We shall first express Lu in terms of f , and then apply Lemma 1.

The derivatives of v with respect to t and x are as follows.

rv t = 2tr l-1 f ′ , r 2 v tt = 4 z r l D(D - 1 2 )f, rv r = r l (l -2D)f, r 2 v rr = r l (2D -l)(2D -l + 1)f, r 2 n -1 r v r = -(N -1)r l (2D -l)f.
It follows that

r 2 (u tt -∆u) = φ λ 4r l z D(D - 1 2 ) -z(D - l 2 )(D - l 2 + 1 2 ) + z N -1 2 (D - l 2 ) + 1 4 λz f = 4r l z D(D - 1 2 ) -z[(D - l 2 )(D - l + N 2 + 1) - λ 4 ] f.
Thanks to the choice of a ± in equation ( 15), we have

(D - l 2 )(D - l + N 2 + 1) - λ 4 = D 2 -(l -1 + N 2 )D + l 2 ( l + N 2 -1) - λ 4 = (D + a + )(D + a -).
On the other hand,

r 2 Av r 2 -t 2 = r l z Azf 1 -z .
Adding this to the above expression for r 2 (u tt -∆u), we obtain

z 4 r 2-l Lu = φ λ D(D - 1 2 ) -z(D + a + )(D + a -) + Az 4(1 -z) f.
Using Lemma 1 with (a, b, c) = (a + , a -, 1 2 ), we conlude that, if β l is given by ( 14), and

f (z) = (1 -z) β l g(z), then z 4 r 2-l Lu = (1 -z) β l φ λ D(D - 1 2 ) -z(D + a + + β l )(D + a -+ β l ) g.
This completes the proof.

3.2. Solutions with special Cauchy data. Theorem 6. Let

F l,λ := r l (1 - t 2 r 2 ) β l F a l , b l ; 1 2 ; t 2 r 2 and G l,λ := tr l (1 - t 2 r 2 ) β l+1 F a l+1 + 1 2 , b l+1 + 1 2 ; 3 2 ; t 2 r 2 . The functions F l,λ φ λ ( x r ) and G l,λ φ λ ( x r ) both solve Lu = 0. For t = 0, F l,λ = ∂ t G l,λ = r l and ∂ t F l,λ = G l,λ = 0.
Proof. We now know that, if F is a HGF with parameters (a l , b l , 1 2 ), then

r l φ λ ( x r )(1 - t 2 r 2 ) β l F solves Lu = 0. Taking F = F a l , b l ; 1 2 ; t 2 r 2
, the solution F l,λ results. Taking for F the second solution (6) of the HGE and replacing l by l + 1 leads, after rearranging, to the solution G l,λ . The behavior for t = 0 is checked by inspection.

Remark 3. If λ = 0 and N is odd, a l -b l is a half-integer, and the HGF may be expressed in terms of elementary functions. When a l -b l = 1 2 , we use [START_REF] Darboux | Leçons sur la Théorie Générale des Surfaces[END_REF]. For a l -b l > 1 2 , we use the identity

(c -a) n z c-a-1 (1 -z) a+b-c-n F (a -n, b ; c ; z) = d n dz n z c-a+n-1 (1 -z) a+b-c F (a, b ; c ; z) .

3.3.

Remarks on inhomogeneous problems. Because the hypergeometric equation with polynomial right-hand side may also be solved in closed form [START_REF] Babister | Transcendental functions satisfying nonhomogeneous linear differential equations[END_REF], the method of §3.1 readily yields solutions with righthand sides that are products of (1 -z) β l by polynomials in r and z.

We record next a different method that produces simple solutions of inhomogeneous problems of special form. As before, all computations are performed for r > t.

Theorem 7. Assume that A, A ′ and γ are given constants. If h(x, t) is homogeneous of degree γ and satisfies

h tt -∆h + A ′ h r 2 -t 2 = 0, then H := c(r 2 -t 2 ) α h solves LH = (r 2 -t 2 ) α-1 h provided that c[A -A ′ -2α(2γ + 2α + N -1)] = 1.
Proof. Let denote the wave operator. The assumptions imply that Lh

= (A -A ′ )h/(r 2 -t 2 ). Since (r 2 -t 2 ) α = -2α(2α + N -1)(r 2 - t 2 ) α-1 , LH = c(r 2 -t 2 ) α Lh -4cα(r 2 -t 2 ) α-1 (th t + rh r ) -2α(2α + N -1)(r 2 -t 2 ) α-1 ch. Since h is homogeneous of degree γ, th t + rh r = γh. Therefore, LH = (r 2 -t 2 ) α-1 h if and only if c[A -A ′ -2α(2γ + 2α + N -1)] = 1.
This completes the proof 4. Series solution of the Cauchy problem

Let v 0 (r) = l≥0 a l r l ; v 1 (r) = l≥0 b l r l
be two series with radius of convergence R > 0. Equation ( 1) has a unique solution such that u λ = φ λ v 0 , ∂ t u λ = φ λ v 1 for t = 0; it is equal to

(16) φ λ l≥0 a l F l,λ (r, t) + b l G l,λ (r, t).
Summation over λ yields the general solution. We focus here on estimations of u λ . The subscript λ will be suppressed in the following.

Theorem 8. If A < 1 4 (N -1) 2 , the series ( 16) is real and converges for t < r < R -t.

Remark 4. Since we are dealing with a wave equation with speed one, the solution of the Cauchy problem with data for 0 < r < R is uniquely determined on the domain of influence t < r < R -t. The domain of convergence in the theorem is therefore optimal.

Proof. The assumption on A ensures that β, hence a l and b l , are real. Also, a l and b l are positive if l is large. Since λ is an eigenvalue of the Laplacian on the sphere, is either 0 or ≥ 1; therefore, a l -b l ≥ 1 2 unless N = 2 and λ = 0, in which case a l = b l . In all cases, we may use Lemmas 2 and 3 if l is large.

If a l -b l ≥ 1 2 , we appeal to (8) to estimate F l,λ by r l r 2 -t 2 r 2

β l × 1 2 1 + t r 1-2b l + 1 - t r 1-2b l .
Now, β l = l + η 1 , b l = 1 2 l + η 2 , where η 1 and η 2 remain bounded as l → ∞. Rearranging, and using the inequality (r -t) l (r + t) -l ≤ 1, we obtain 2|F l,λ (r, t)| ≤ r l r 2 -t 2 r 2 l+η 1 r + t r -l+1-2η 2 + r -t r -l+1-2η 2 ≤ (r + t) l r 2η 2 -2η 1 -1 (r 2 -t 2 ) η 1 (r + t) -2η 2 + (r -t) -2η 2 .

Therefore, lim sup l→∞ |F l,λ | 1/l ≤ r + t. It follows that l≥0 a l F l,λ converges for r + t < R. The other series which makes up the solution is handled similarly. If a l ≤ b l + 1 2 , we appeal to (12) instead of (8). We conclude with a different estimate, which gives better results near the light-cone, but is weaker than the previous one for r > R/2. Theorem 9. As l → +∞, F l,λ = O(2 l (1 -t 2 /r 2 ) β l +η 3 ), where η 3 = O(1/l).

Proof. We estimate F l,λ using Lemma 4. Since a l + b l -

2 .

 2 Estimates of hypergeometric functions 2.1. Definitions. The hypergeometric function (HGF) F (a, b ; c ; z) is defined for c ∈ -N by analytic continuation of the sum of the hypergeometric series ∞ n=0 (a) n (b) n (c) n (1) n z n , where (a) n = a(a + 1) • • • (a + n -1) for n ≥ 1, and (a) 0 = 1. It is symmetric in a and b. It solves the hypergeometric equation (HGE) with parameters (a, b, c): z(1 -z)w ′′ + [c -(a + b + 1)z]w ′ -abw = 0. After multiplication by z, and letting D = z d/dz, the HGE may be written D(D + c -1)w -z(D + a)(D + b)w = 0. A second solution of the HGE is

2. 3 .

 3 Majorants for hypergeometric functions. If the parameters a, b and c are real and positive, the coefficients of F (a, b ; c ; z) increase as a and b are increased, and c is decreased. This makes it possible to compare different hypergeometric functions. Three results of this type are given next. Term-by-term domination of power series is expressed by the symbol ≪; thus, n≥0

Lemma 4 .

 4 If a ≥ b > c > 0, and d = a + b -c, (13) F (a, b ;

1 2 =

 2 O(1/l), we conclude that |F l,λ (r, t)| ≤ C ′ (1 -t 2 /r 2 ) β l +η 3 ,whereC ′ = Γ( 1 2 )Γ(a l + b l -1 2 )/Γ(a l )Γ(b l ). Stirling's formula gives C ′ = O(2 l). The result follows.

Laboratoire de Mathématiques, CNRS FRE 3111, Université de Reims Champagne-Ardenne, Moulin de la Housse, B.P. 1039, F-51687 Reims Cedex 2, France Email address: ali.bentrad@univ-reims.fr Email address: satyanad.kichenassamy@univ-reims.fr