Cédat Denis

Maximilien Libert

Marion Le Flem

Olivier Fandeur

Colette Rey

Michel Clavel

Jean-Hubert Schmitt

Denis Cedat
email: denis.cedat@ecp.fr

Experimental characterization and mechanical behaviour modelling of Molybdenum -Titanium

Keywords: Molybdenum, Composite, Thermally activated behaviour

Introduction

In the scope of refractory materials development for structural applications in the core of the future nuclear reactors such as Gas Fast Reactors, materials combining carbide and refractory metals are of great interest to avoid the major drawbacks of monolithic ceramics (poorly damage tolerant) and metals (neutronic incompatibility). This work focuses on the study of the molybdenum-titanium carbide composite processed by hot isostatic pressing. The investigated material (MoTiC 30vol.% ) was identified as one of the most promising structural materials in term of mechanical strength to operate at temperatures greater than 1000 K (standard using) and up to 1900 K (accidental scenario) under a fast neutron flux.

Mechanical studies of Mo-TiC composite have highlighted the need for a knowledge of the properties of the individual components. Thus, this paper consists of relating the microstructure, the behaviour of each phase and the mechanical interactions between the different relevant components to the global behaviour. In this aim, the plasticity mechanisms of molybdenum polycrystal are investigated, carbide being assumed as elastic.

The following work will be concentrated on determining the role of the low/high temperature behaviour in molybdenum, on the composite mechanical behaviour. The concept of thermal activation energy as well as the dislocations interaction mechanisms is used in a modelling taking into consideration the effect of the dislocation density evolution on the flow stress of the deformed material. The understanding of these mechanisms will be investigated through simulation of compressive tests by using a crystalline approach implemented in ABAQUS finite element code. Such modellings have shown their efficiency to determine the nondirectly measurable physical mechanisms [START_REF] Peirce | Material rate dependence and localized deformation in crystalline solids[END_REF][START_REF] Tabourot | Use of mesoscopic simulations to model the dislocations densities evolution law[END_REF][START_REF] Bhattacharyya | Evolution of grain scale microstructure during large strain simple compression of polycrystalline aluminium with quasi-columnar grains: OIM measurements and numerical simulations[END_REF][START_REF] Hoc | Experimental and numerical analysis of localization during sequentials tests for IF-Ti steel[END_REF][START_REF] Barbe | Intergranular an intragranular behavior of polycrystalline aggregates. part 2 : Results[END_REF][START_REF] Barbe | Intergranular an intragranular behavior of polycrystalline aggregates. part 1[END_REF][START_REF] Stainier | A micromechanical model of hardening, rate sensitivity and thermal softening in bcc single crystals[END_REF][START_REF] Erieau | Modelling of deformation and rotation bands and of deformation induced grain boundaries in IF steel aggregate during large plane strain compression[END_REF]. In this modelling, the determination of material parameters, based on experimental characterization, will be described in this paper. The composite powder is obtained by milling of TiC and Mo in alcohol during two hours, with zirconia balls. After alcohol evaporation, the mixing powder is introduced in a latex can and shaped by cold isostatic pressing at 2500 bar during 1 minute. After this step, the green sample is placed in a titanium can and an out gassing in high-vacuum at 600°C during 12 hours was carried out. The sintering step is produced by hot isostatic pressing at 1600°C during 2 hours under a pressure of 1610 MPa. The density obtained is d = 8,52 with a total porosity of 1,24 % (isolated porosity = 1,13%).

Materials and experimental

The microstructure of the sintered Mo/TiC composite (Mo-TiC 30%vol ) is referred to a core/shell or core/rim structure with molybdenum as binder phase (fig. 1). It can be seen that there is titanium carbide core, surrounded by (Mo,Ti)C rim in a molybdenum matrix. The chemical and physical characterizations of this composite [START_REF] Cédat | Microstructural characterization of a composite Mo reinforced by 25% at. TiC[END_REF] have shown that the (Mo,Ti)C phase exhibits a composition close to the solid solubility of molybdenum in the titanium carbide: TiC-Mo 10-15at.% . It was assumed that (Mo,Ti)C phase, originating from the Mo and TiC powders, is caused by the diffused phases recrystallisation as the shell.

b) Mechanical characterization

As was mentioned in several studies, plastic deformation is often assumed temperature insensitive. In actual practice, however, the flow stress of ductile bcc metals does show some temperature sensitivity in various ranges of temperature. However this behaviour change couldn't be attributed to brittle/ductile transition which occurs in molybdenum close to 298K. The molybdenum, in its ductile domain, presents low/high temperature behaviour due to different mechanisms of plasticity.

Strain rates change tests were carried out from 5.10 -2 s -1 and 5.10 -4 s -1 at different temperatures on cylinder specimens which were of 12 mm width and 18 mm length. The variation of the flow stress with the strain rate is shown in figure 2.a, which shows the stressstrain curved measured at several temperatures added to the strain rate change tests. The figure 2.b shows the stress "jumps" at different strain levels and presents a decrease of stress with the increase of temperature. As expected from the results of strain rate change tests, the sensitivity of the flow stress on the deformation temperature decreases. As a matter of fact, an increase in strain rate is equivalent to a decrease in temperature in a thermally activated process. In fact, the temperature change tests (Fig. 3) were realized on same samples shape and give results similar to the strain rate change tests. Such strain rate sensitivity or, alternatively, temperature sensitivity of the flow stress indicates that the plastic deformation process is thermally activated. By considering the titanium carbide elastic, this thermally activated behaviour of the composite comes from molybdenum properties what it has been investigated on a previous work [10].

Constitutive laws of thermally activated model

The numerical simulation, undertaken at different scales, predicts the local mechanical fields in the different phases. The used crystalline plasticity model (cristalECP) is developed in the framework of large transformations [START_REF] Peirce | Material rate dependence and localized deformation in crystalline solids[END_REF][START_REF] Teodosiu | Finite implementation of the large elastoplastic deformation of multicrystals. Large Plastic Deformation[END_REF] (with small elastic distortion but large lattice rotation) and has been implemented in ABAQUS finite element code, using a User Subroutine UMAT.

Within the framework of the classical theory of thermally activated dislocation motion, a unified constitutive description of the mechanical behaviour of molybdenum is developed. In this paper, a constitutive model is proposed to describe the stress-strain response of commercially pure molybdenum over the temperature range 298-993K.

a) Constitutive Law

The model accounts for crystallographic glide and lattice rotation. The mechanisms responsible for viscoplastic flow of molybdenum are essentially the crystallographic slip of two main slip systems {110} <111> and {112} <111> related to the body centered crystal structure (bcc). By neglecting the slip systems {123}<111> in this work, 24 slip systems are considered.

In bcc transition metals, the low-temperature and high-strain-rate plastic deformation properties differ drastically from those of the fcc metals. In particular, there is a rapid increase of flow stress with decreasing temperature and breakdown of Schmid law of critical resolved shear stress. These mechanical behaviour changes are characterized by the temperature of transition (T t ) which has been identified around 500K in the case of molybdenum.

Experimental observations suggest that screw dislocations in bcc transition metals are much less mobile than edge dislocations and consequently, screw dislocation slip rate mobility is the limiting factor which controls both low-temperature and high-strain rate plasticity. It is also conceded that a double-kink Peierls barrier mechanism is responsible for the actual mobility of screw dislocations.

Thus the slip rate is controlled by the glide of screw dislocations in the low temperature range (below the temperature T t ). The motion of dislocations is opposed by: (i) short-range barriers due to the lattice friction which can be overcome by thermally assisted formation of double kinks ( s eff τ ), (ii) short-range forces from dislocation forest which can be overcome by athermal shear stress term ( s int τ ), and (iii) long-range forces due to precipitates ( 0  ). These interactions lead to the following relation:

s int s eff 0 s τ τ τ τ    (1)
Where

s eff τ
is the thermal component of the applied shear stress s  , associated to the first kind of obstacles, resolved in a slip system (s). Thus, the chosen viscoplastic law is able to describe the competition between lattice friction and forest hardening. According to Louchet [START_REF] Kubin | Description of low-temperature interstitial hardening of the b.c.c. lattice from in situ experiments[END_REF], the slip rate on a slip system (s) depends of the mean dislocation velocity, which follows an Arrhenius-type law:

    s eff ss 0 B Δ G τ γ γ exp signe τ kT       (2)
Where s  , s eff  are respectively the shear stress, the effective shear stress on (s), k B is the Boltzmann constant and the absolute temperature. Indeed, the flow law used is able to account for the overcoming of Peierls valley by double kink mechanism (Fig. 5), and specificities of dislocation during double kink mechanism are used as input into the slip rate: The activation energy is directly connected with the effective shear stress s eff τ on the system (s). The effective shear stress is equal to the applied resolved shear stress s  deduced from the temperatureindependent internal stress s int τ due to obstacles. The phenomenological description, proposed by Kocks and al. [START_REF] Kocks | Thermodynamics and kinetics of slip-Progress in Materials Science[END_REF], described most of the shortrange barriers encountered by dislocations as the following profile:

  q p R s eff 0 s eff τ τ 1 ΔG τ ΔG                     (4)
Where 0 G D is the activation energy when 0 s eff τ  , p and q were parameters describing the shape of the energy profile associated to the Peierls barriers, whereas R  is the effective stress to supply in order to create a double kink at T=0K, when the contribution of the thermal activation is strictly useless. The glide of dislocation on a system (s) being active when

s eff τ is greater than R  .

b) Hardening law

The hardening evolution bases on the competition between the lattice friction, which generates a effective stress s eff τ , and the forest hardening coupled with the crossing of obstacle, for which it is respectively necessary to supply s int τ and 0  .

Thus the yield stress required to activate the plastic slip is then equal to the sum of these three contributions (equ. 1).

By considering the dislocation-dislocation interactions on a same system plan (Selfhardening) and the interaction of the system (s) with each of 23 systems (u) potentially latent (Latent-hardening), the internal stress s int τ is directly dependent on the local dislocation densities of each system (s):

  0 s u u su 2 s int τ τ ρ a b μ τ    ( 5
)
Where  is the shear modulus and su a the crystallographic part of the interaction matrix as described by Franciosi [START_REF] Franciosi | Glide mechanisms in bcc crystals: an investigation of the case of α-iron through multislip and latent hardening tests[END_REF].

The hardening law, based on the Rauch model [START_REF] Rauch | Etude de l'écrouissage des métaux, aspects microstructuraux et lois de comportement[END_REF], translates the competition between lattice and forest hardening:

  u u su 2 2 s eff s eff s 0 s ρ a μb 4 τ 2 1 2 τ τ τ      (6)
In the restricted strain rate change (5.10 -2 s -1 -5.10 -4 s -1 ), thermal atomic vibrations assist dislocations to overcome the short-range obstacles at lower values of applied shear stress than that required at 0K. Under such conditions, an increase in temperature reduces the effective resolved shear stress s eff τ .

Thus, at temperature below T t , the forest hardening's contribution appeared as very small compared to effective stress, what leads to: ² 0 ()

su u ss u eff s eff ba          (7)
At temperature above T t , there will be sufficient thermal energy for the barriers to be overcome by thermal activation. Then, by considering ( ) 0 eff t T   , the hardening law is thus classically linked to forest hardening in agreement with the well known relation:

0 s su u u ba       (8)

c) Dislocation evolution law

The strain-hardening model takes into account the creation and the annihilation of dislocations through characteristic parameters K and g c . This law is a generalisation of the relation proposed by Mecking and Estrin [START_REF] Estrin | A unified phenomenological description of work hardening and creep based on one-parameter models[END_REF]. It describes the evolution of the total dislocation densities rates with strain for each slip system:

Where K is a material parameter. This equation is derived from the balance between accumulation, based on Orowan's relationship, and annihilation rates.

The annihilation of dislocations is controlled by the main distance g c , which the temperature dependence was expressed by an Arrhenius law. Thus two dislocations can annihilate together if they interact at a distance smaller than g c .

4_Determination model constants and comparison with experimental results

The model, used to describe the mechanical behaviour in molybdenum, brings in many materials constants.

The activation energy values suggest that the thermally activated mechanical behaviour of molybdenum take a real importance in the deformation of the composite, which presents a transition temperature close to 423-573K. These results will be used in the simulation part to fix the parameter corresponding to the free activation energy DG 0 .

By considering the carbide as elastic, the thermally activated mechanical behaviour in molybdenum is the main phenomenon which occurs during the deformation of the composite. It turns out essential to implement behaviour laws able to report this phenomenon evolving versus temperature.

To understand the underlying mechanisms in detail, it is, however, necessary to carry out identification of the key parameters at large length scale experiments. For example, the activation energy for double kink motion of screw dislocation has been identified from literature with accurate atomistic simulations on dislocation interaction [START_REF] Xu | Accurate atomistic simulations of the Peierls barrier and kinkpair formation energy for {111} screw dislocations in bcc Mo[END_REF], as well as macroscopic experimental procedures with temperature change tests. Even if most of these parameters could be identified thanks to experimental procedures and bibliography, some of behaviour law coefficients can't be determined by experimental tests. Thus, an inverse method was chosen, using SiDoLo [START_REF] Pilvin | SiDoLo version 2.4, Notice d'utilisation[END_REF][START_REF] Cailletaud | Identification and inverse problems related to material behaviour[END_REF], to fix these parameters. The resolution of the identification problem consists in minimizing a functional which measures, for a set of given parameters, the distance between the forecasts of the model and the physical reality represented by the experimental observations. The model's parameter evaluation is reduced to the resolution of a problem of non linear optimization. The molybdenum parameters identified are presented in the table 2:

0  a ss a su 0  g c0 E gc D grain R  0  0 G D p q
70 MPa 0,01 10 -11 m -2 14 nm 2,17.10 -2 eV 3 µm 498 MPa 10 -1 1,1 eV 0,2 1,5

Finite element simulation of compressive test

To constitute the Finite Element mesh, the grid defined by Electron Back Scatter Diffraction measurement is used. Scanning Electron Microscope (SEM) coupled with Electron Back Scatter Diffraction (EBSD) was carried out in order to characterise the microstructure and the orientation of each grain. This aggregate is constituted by several layers chosen to obtain a composition of MoTiC 30%vol. and the texture of the as-received material. However, the layers are not obtained from successive polishing on the same area. Thus, the numerical aggregate built doesn't take into account the real shape of the grain in depth (fig. 4). By this method, geometrical characteristics of the grains and the crystallographic orientations were taken into account in the numerical aggregate which size is about 30 x 30 x 10 µm 3 . The mesh size is 0,3 µm and the elements used are C3D8R.

The aggregate is loaded on compression in direction u 2 . The applied displacement u 2 leads to a strain rate of about 5.10 -4 s -1 . The other boundary conditions are u 2 =0 on the bottom of the aggregate and u3=0 on the face normal to 3  . The set of parameters were defined to model the mechanical behaviour in molybdenum and the crystalline approach applied on the numerical aggregate gives the following results (fig. 5) on compressive tests performed at different temperatures. The m ises s values on the map (fig. 7) show that a stress concentration takes place in the carbide phase whereas molybdenum accommodates the deformations and experiences large strain level.

Discussion

The results of the simulation investigated on the composite don't give satisfactory response. Nevertheless, the misfit between computed and experimental compressive curves is assumed to come from to the carbide behaviour. Although, according to the spheres theory, it is intuitively clear that percolation of particles in structure will happen once its concentration reaches a critical value in the system. This critical value is close 33% in volume what is about the concentration of carbide particles in the composite.

To get a 3D representation of the material, high energy X-ray tomography at the European Synchrotron Radiation Facility (ESRF, Grenoble, France) has been used. This technique basically consists in recording several radiographs of a sample at different angular positions and in reconstructing a 3D map of the X-ray attenuation coefficient within the material using adequate algorithm and software. Fig. 8 illustrates a 3D image of the microstructure of the Mo-TiC material obtained with the highest resolution (0.28 µm) available on the ID19 beamline (X-ray energy: 65 keV) that has been used at ESRF. Fig. 8 represents the set of all points in the carbide phase that can be linked together to the bottom and top faces of the parallelepiped by paths entirely contained in the carbide phase. Consequently, the high 3D connectivity of the carbide phase would be a possible explanation of the observed mechanical behaviour of the composite. In fact, the microstructure of the composite could be seen as a foam of carbide in a matrix of molybdenum.

Thus, the compressive test curve of MoTiC 30%vol . at room temperature could be divided in three phases. The first stage corresponds to an elastic deformation of the carbide structure. The second stage starts with the fracture of many titanium carbides' particles which leads to the loss of percolation. At this stage, the microstructure is supposed to be a composite of molybdenum reinforced by TiC particles. The damage of TiC occurs all along this stage. The stage III corresponds to the strain-hardening of molybdenum.

As a matter of fact, the aggregate used in simulation doesn't take into account the percolation of the carbide phase what could explain the discrepancy between experimental and simulated results of stress during the transition from the stage I to the stage II. Moreover, the difference of strain-hardening mode would be due to the damage of titanium carbide which is not taken into account in the model. Although, the map on fig. 10 shows that carbide particles are subjected to large tensile stress. The tensile strength of ceramic leading to damage is much less important than compressive strength:

15 com pressive tensile s s  Thus, it is assumed that stress component in TiC is sufficiently high to promote premature damage.

Conclusion

A crystalline plasticity model has been developed in the framework of large transformations to describe the elastic-viscoplastic behaviour of heterogeneous materials. It has been applied to a polycrystalline ceramic/metal composite, and take simultaneously account the plasticity mechanisms through the low/high temperature transition for the viscous part and the elastic behaviour for ceramic. The macroscopic behaviour during compressive tests at various temperatures is successfully simulated for molybdenum. The identification of the model parameters is based on physical and microstructural considerations such as the grain size, the glide of dislocations. Nevertheless, discrepancies between experiments and model exist for the composite. They are attributed to the elastic formulation used which is too restricted to represent the ceramic behaviour.

The proposed assumptions recommend the consideration of the damage and of the percolation in carbide phase. The implemented damage could be reduced by including a plastic yield criterion associated with a critical shear stress.

Fig. 1 :

 1 Fig.1: SEM picture showed microstructure in Mo-TiC 30%vol.

Fig. 2 :

 2 Fig. 2: a) Strain rate change Tests on MoTiC 30%vol. b) Stress jumps vs temperature at different strain levels

Fig. 3 :

 3 Fig. 3: Temperature change tests on MoTiC 30%vol.

Fig 5 :

 5 Fig 5: Crossing of the Peierls valley by double kink mechanism

Fig 4 :

 4 Fig 4: The five EBSD maps used in the numerical aggregate

Fig. 5 :

 5 Fig. 5: Experimental and simulated compressive behaviour in molybdenum.

Figure 6 Fig. 6 :

 66 Figure 6 presents the results of simulation on the numerical aggregate of figure4. A large difference can be observed between simulation and experimental test.

Fig. 7 :

 7 Fig.7: a) EBSD map showing carbide phase on the first layer of the aggregate. b) m ises s

Fig. 8 :

 8 Fig. 8: 3D representation of the composite (30µm x 28µm x 14µm) interconnected parts of the molybdenum (red) and carbide (yellow) phase.

Fig. 9 :

 9 Fig. 9: Stainstress curve of MoTiC 30%vol. at room temperature.

Fig. 10 :

 10 Fig.10: a) EBSD map showing carbide phase on the first layer of the aggregate. b) . m ax p rin c s (MPa) map on the first layer of the numerical aggregate in the composite.

procedure a) Material Description

		Both titanium carbide (d 50 =3µm) and molybdenum (d 50 =5µm) were supplied by HC
	Starck. TiC and Mo chemical analysis is given in table 1:
				TiC Composition (wt. %)
	C total	0	N	Ca	Co	W	Ni	Al	Fe	S
	19.23 0.6126 0.0279 0.002	0.032	0.39	<4E-4 0.0014 0.0061 0.0019
				Mo Composition (wt. %)
		Mo		O			Fe		K
		99,8		0.0620		9 ppm		29 ppm

Table 1 :

 1 Table of composition of commercial TiC and Mo composition

Table 2 :

 2 Set of model parameters identified for molybdenum International Journal of Refractory Metals & Hard Materials, 2009, 27 (2), pp267-273


V Pinel

F Albino

Consequences of volcano sector collapse on magmatic storage zones: insights from numerical modeling

Keywords:

Major volcano flank collapses strongly affect the underlying magmatic plumbing system. Here, we consider the magma storage zone as a liquid pocket embedded in an elastic medium, and we perform numerical simulations in two-dimensional axisymmetric geometry as well as in three dimensions in order to evaluate the consequences of a major collapse event. We quantify the pressure decrease induced within and around a magma reservoir by a volcano flank collapse. This pressure reduction is expected to favour replenishment with less evolved magma from deeper sources. We also estimate the impact of the magma pressure decrease, together with the stress field variations around the reservoir, on the eruptive event associated with the edifice failure. We show that, for a given magma reservoir geometry, the collapse of a large strato-volcano tends to reduce the volume of the simultaneous eruption; destabilization of large edifices may even suppress magma emission, resulting in phreatic eruptions instead. This effect is greater for shallow reservoirs, and is more pronounced for spherical reservoirs than for

Introduction

Large flank collapses have been recognized as common phenomena in the long-lived evolution of volcanic edifices. A large number of studies focus on the causes of, and/or triggers for, these destabilization events. They show that the origin of the destabilization can be related to exogenous processes such as weathering, but in most cases volcanic activity itself is involved [START_REF] Mc Guire | Volcano instability: A review of contemporary themes[END_REF]. In particular, the ability of magmatic intrusions to favour large flank collapse either during vertical dyke emplacement or during sill formation has been observed in the field [START_REF] Famin | Volcanic destabilization by magma injections in a detachment[END_REF], and investigated through modeling [START_REF] Paul | A numerical model for simulating deformation of Mount St. Helens volcano[END_REF][START_REF] Iverson | Can magma-injection and groundwater forces cause massive landslides on Hawaiian volcanoes[END_REF]. [START_REF] Siebert | Threats from debris avalanches[END_REF] emphasized the potential hazards represented by sector collapses, which fully justify studies investigating volcano stability [START_REF] Voight | Failure of volcano slopes[END_REF][START_REF] Borselli | Flank collapse scenarios at volcan de Colima, Mexico: A relative instability analysis[END_REF]. From a risk assessment perspective, the direct impact of a sudden and drastic sector collapse is also investigated through studies or modelling related to the volume and extension of the associated debris avalanche deposits [START_REF] Borselli | Flank collapse scenarios at volcan de Colima, Mexico: A relative instability analysis[END_REF].

Another field of study encompasses describing and quantifying the consequences of such an event on the magmatic plumbing system evolution. The long-term history of volcanic edifices reveals that partial destruction of an edifice is usually followed by a change in eruption rate and/or magma composition [START_REF] Presley | Postshield volcanism and catastrophic mass wasting of the Waianae Volcano, Oahu, Hawaii[END_REF][START_REF] Hildenbrand | Volcano-tectonic and geochemical evolution of an oceanic intra-plate volcano: Tahiti-Nui (french Polynesia)[END_REF][START_REF] Hora | Volcano evolution and eruptive flux on the thick crust of the Andean Central Volcanic Zone: 40 Ar/ 39 Ar constraints from Volcán Parinacota, Chile[END_REF][START_REF] Longpré | Volcanic and geochemical evolution of the Teno massif, Tenerife, Canary Islands: Some repercussions of giant landslides on ocean island magmatism[END_REF][START_REF] Boulesteix | Eruptive response of oceanic islands to giant landslides: New insigths from the geomorphologic evolution of the Teide-Pico Viejo volcanic complex (Tenerife, Canary)[END_REF]. For oceanic volcanoes, this observation has usually been related to an increase in decompression melting subsequent to collapse [START_REF] Presley | Postshield volcanism and catastrophic mass wasting of the Waianae Volcano, Oahu, Hawaii[END_REF][START_REF] Hildenbrand | Volcano-tectonic and geochemical evolution of an oceanic intra-plate volcano: Tahiti-Nui (french Polynesia)[END_REF], although [START_REF] Manconi | The effects of flank collapses on volcano plumbing systems[END_REF] also evoked the depressurisation of a magmatic storage zone. For continental volcanoes, [START_REF] Pinel | Some consequences of volcanic edifice destruction for eruption conditions[END_REF], using an analytical elastic model for the two-dimensional plane strain approximation, quantified the pressure decrease induced within a magmatic reservoir by the partial destruction of an edifice. They also detailed the influence of such an event on the volume of magma erupted during the failure event.

Meanwhile, other surface load variations, occurring over a larger time scale, have been proven to have a significant impact on eruptive behaviour. In particular, a temporal correlation is observed between ice retreat induced by climate warming and volume of magma erupted, with an increase of eruption rates during postglacial periods [START_REF] Jellinek | Did melting glaciers cause volcanic eruptions in eastern California? probing the mechanics of dike formation[END_REF][START_REF] Sinton | Postglacial eruptive history of the western volcanic zone, Iceland[END_REF]. The effect of ice retreat on both magma melting and storage has been investigated (see [START_REF] Sigmundsson | Climate effects on volcanism: Influence on magmatic systems of loading and unloading from ice mass variations with examples from Iceland[END_REF] for a review). More recently, new modeling has shown that magma propagation within the upper crust is also affected by ice unloading, with an increased likelihood of magma storage within the crust during transport towards the surface. This is in good agreement with some geodetic observations performed around Vatnajökull ice cap in Iceland [START_REF] Hooper | Increased crustal capture of magma at volcanoes with retreating ice cap[END_REF].

In this study, we calculate the pressure reduction induced by a sudden flank collapse event within and around a magma storage zone located beneath a volcanic edifice. We only consider one-shot catastrophic flank collapses, rather than the effects of large, progressive landslides. We then quantify how the flank collapse affects the volume of erupted magma during the associated eruption, resulting from the storage zone withdrawal. Results are derived from numerical simulations incorporating the equation of elasticity, performed with the commercial software COMSOL both in axisymmetric geometry and in three dimensions. We also discuss the potential impact of large flank collapses on the long-term eruptive history, based on petrological observations.

Pressure decrease induced by a volcano flank collapse

Broadly speaking, a major sector collapse is equivalent to a surface unloading event. In reality, the edifice portion which fails is not removed from the Earth's surface, but is redistributed over a larger area. As previously shown by [START_REF] Pinel | Some consequences of volcanic edifice destruction for eruption conditions[END_REF], using analytical solutions, and by [START_REF] Albino | Influence of surface load variations on eruption likelihood: Application to two Icelandic subglacial volcanoes, Grimsvötn and Katla[END_REF], using numerical models in axisymmetric geometry, an unloading event always induces a pressure decrease within the underlying crust. This pressure reduction is of the same order of magnitude as the load removed from the surface.

A conical load removed over an elastic half-space

If we consider the crust to be an elastic, homogeneous medium characterized by its Young's modulus, E, and Poisson's ratio, ν, the stress changes induced at depth by a conical load can be derived by integration of the point load solution. At the axis, the vertical stress due to a cone of radius, R e , and maximum height, H e , as a function of depth below the surface, z, is given by:

σ zz = P e [1 - z R 2 e + z 2 ], (1)
with P e = ρ c gH e , where ρ c is the load density. The horizontal components are equal and given by [START_REF] Pinel | The effect of edifice load on magma ascent beneath a volcano[END_REF]:

σrr = σ θθ = Pe 2 [(1 + 2ν) -2(1 + ν) z Re ln( Re+ √ R 2 e +z 2 z ) + z √ R 2 e +z 2 ] for z > 0 σrr = σ θθ = Pe 2 (1 + 2ν) for z = 0, (2)
It follows that the pressure, P , defined as one third on the stress tensor trace, induced by a conical load, is equal to:

P = 2Pe 3 (1 + ν)[1 -z Re ln( Re+ √ R 2 e +z 2 z )] for z > 0 P = 2Pe 3 (1 + ν) for z = 0 (3)
Stress and pressure reduction induced by the removal of conical load are

shown in figure 1. The stress component most affected by the load is, as expected, the vertical one σ zz . The amplitude of the perturbation is greatest at the surface, and is directly related to the height of the load removed. The pressure reduction decreases with depth and becomes negligible at depths greater than three times the radius of the load.

A conical load removed from above a magmatic reservoir

Most tomographic studies performed on volcanoes [START_REF] Monteiller | An efficient algorithm for double-difference tomography and location in heterogeneous media, with an application to the Kilauea volcano[END_REF][START_REF] Prôno | Pwave velocity structure of Piton de la Fournaise volcano deduced from seismic data recorded between 1996 and 1999[END_REF] reveal that the crust is far from being homogeneous around a magmatic system. In particular, shallow magma storage zones have been detected in many locations by either petrologic, seismic or geodetic studies [START_REF] Gardner | Experimental constraints on pre-eruptive water contents and changing magma storage prior to explosive eruptions of Mount St Helens volcano[END_REF][START_REF] Sturkell | Volcano geodesy and magma dynamics in Iceland[END_REF][START_REF] Peltier | Cyclic magma storages and transfers at Piton de la Fournaise volcano (La Réunion hotspot) inferred from deformation and geochemical data[END_REF]. The pressure reduction induced by an unloading event within these magma pockets will depend both on the crustal deformation and on the equation of state of the melt embedded in the crust. Here we consider an ellipsoidal magmatic reservoir filled with fluid, embedded in an homogeneous elastic crust. Initially the liquid has the same density as the surrounding crust and is characterized by its bulk modulus, K. We only deal with the perturbation induced by a conical load removed from the Earth's surface, on which the initial stress field has no influence.

Within the magma reservoir the pressure change, ∆P , is related to the reservoir volume change, ∆V , through the bulk modulus definition:

∆P = -K ∆V V , (4)
with V being the initial volume of the reservoir.

The change in reservoir volume is also a function of the chamber wall displacement, which depends on both the conical load and the magma pressure change. This volume change is calculated numerically, using the equations of elasticity with the commercial software COMSOL. The domain of calculation is a 100*100 km square box with a mesh of about 100 000 triangular units that is refined around the volcanic edifice and magma reservoir. No displacement perpendicular to the boundary is allowed at the basal and lateral boundaries; the upper boundary is considered as being a free surface.

The load is modeled with a normal stress applied at the upper surface, and a normal stress equal to the magma overpressure is applied at the reservoir walls. Numerical solutions have been validated using well-known analytical solutions as detailed in [START_REF] Albino | Influence of surface load variations on eruption likelihood: Application to two Icelandic subglacial volcanoes, Grimsvötn and Katla[END_REF]. Pressure reduction within and around the magma reservoir induced by a conical load of 2 km radius, 1 km height, and density 2800 kg/m 3 , is shown in figure 2 for two different chamber geometries: a spherical chamber, and a vertically-elongated chamber (prolate shape). Calculations are performed for a chamber top at 1 km depth and with a maximum chamber radius of 1 km. The pressure variation within the crust departs strongly from the homogeneous case (figure 1) in the vicinity of the reservoir, this difference being more pronounced for the spherical reservoir than for the prolate one. An increase in pressure is even observed (negative values of the pressure reduction) at the chamber margins, the effect being most extreme at the chamber top. This is due to the deformation of the reservoir wall induced by the unloading event and partially counterbalanced by pressure variations within the magma reservoir. Within the magma reservoir, pressure always decreases as a consequence of the unloading event, the effect being, once again, larger in the case of the spherical reservoir than in the case of the prolate one. The amplitude of the magma pressure reduction increases with the value of the bulk modulus. This can be explained by the fact that for incompressible magmas (larger value of K) no reservoir volume change occurs, thus all the volume reduction induced by the unloading event has to be compensated by a pressure reduction within the chamber.

The effect of compressibility is shown in figure 3. In a compressible magma, buoyancy forces appear due to magma density variation; however since the model used here is valid in most of natural cases, as discussed by [START_REF] Pinel | Some consequences of volcanic edifice destruction for eruption conditions[END_REF], these forces have not been included here.

Figure 5 shows the pressure reduction within a spherical reservoir with a top at 1 km depth, induced by the removal of the upper 20 % of the edifice, which corresponds to a mean value based on field observations (figure 4a). This pressure reduction is more marked when considering larger edifices and smaller magma reservoirs. Calculations performed also show that the amplitude of the pressure reduction decreases for deeper magma reservoirs and is less marked in the case of a prolate reservoir than in a spherical one.

Effect of volcano flank collapse on an ongoing eruption

As described above, partial destruction of an edifice always induces a pressure decrease in the underlying storage zone. Simultaneously, this drastic change in surface load strongly modifies the stress field at the reservoir walls, and thus presumably dyke initiation and closure in this zone. While longterm conduit systems develop at the surface of silicic strato-volcanoes, this shallow level conduits are connected to deeper reservoirs via dykes as shown by deformation data [START_REF] Mattioli | GPS measurement of surface deformation around Soufriere Hills volcano, Montserrat from October 1995 to[END_REF], seismicity [START_REF] Roman | Assessing the likelihood of volcanic eruption through analysis of volcanotectonic earthquake fault-plane solutions[END_REF], and magma flow studies [START_REF] Costa | Control of magma flow in dykes on cyclic lava dome extrusion[END_REF]. Thus it is important to consider dyke opening and closure at the reservoir walls when dealing with magma transport to the surface.

Principle

For any given state of stress, one may define a threshold pressure, P r , required for dyke initiation at the chamber wall. Here, we consider that once this pressure threshold is reached within the magma reservoir, a dyke initiates and magma leaves the storage zone to reach the surface and feed an eruption, such that pressure within the magma chamber cannot exceed this threshold value. Following [START_REF] Albino | Influence of surface load variations on eruption likelihood: Application to two Icelandic subglacial volcanoes, Grimsvötn and Katla[END_REF], we consider that tensile rupture of the reservoir wall occurs when the deviatoric part of the hoop stress exceeds the tensile strength, T s , of host rocks. This criterion is consistent with the rupture criterion used in dyke propagation studies [START_REF] Lister | Buoyancy-driven fluid fracture: the effects of material toughness and low-viscosity precursors[END_REF].

Once a dyke has initiated, magma pressure within the reservoir decreases until it reaches a second threshold value, P cl , at which point the dyke closes [START_REF] Pinel | On the relationship between cycles of eruptive activity and volcanic edifice growth[END_REF]. It follows that the volume of erupted magma is directly proportional to the pressure difference: ∆P e = P r -P cl .

We consider an initial state, corresponding to the situation just before the edifice partial destruction, with a pressurized storage zone. For this initial state, we can define the two threshold values, P r (i) and P cl (i) (see figure 6). As magma intrusions often acts as the triggers for edifice destabilization [START_REF] Mc Guire | Volcano instability: A review of contemporary themes[END_REF], we make the assumption that the system is about to erupt, such that the magma pressure, P (i), is equal to the threshold value required for dyke initiation, P r (i). A major collapse of the edifice then occurs, and our final state is a truncated edifice. The unloading event results in a pressure reduction (∆P ) within the magma reservoir, such that the final magma pressure within the reservoir is P (f ) < P (i). In the final state, the two threshold pressures, here denoted P r (f ) and P cl (f ) are different from the ones in the initial state (see figure 6). Based on the evolution of the threshold pressures, which can either increase or decrease, three different scenarios can result. Where the pressure difference, P -P cl , increases with the edifice partial destruction (case 1 of figure 6, where P (f )-P cl (f ) > P (i)-P cl (i)), the edifice collapse is followed by an eruption with a volume of erupted magma larger that it would have been in the absence of edifice destruction. Where the pressure difference, P -P cl , decreases with the edifice partial destruction (case 2 of figure 6, where P (f ) -P cl (f ) < P (i) -P cl (i)), the erupted volume associated with the edifice collapse is smaller that it would have been in absence of edifice destruction. In the event of the magma pressure within the reservoir dropping below the threshold pressure for dyke closure (case 3 of figure 6, where P (f ) < P cl (f )), the incipient eruption is aborted and no magma is erupted at the surface as a consequence of the edifice collapse.

Factors determining which scenario occurs are the initial edifice size and geometry, the amount of edifice destruction, the size, shape and depth of the magma reservoir, and magma gas content.

Numerical results for axisymmetric models

We performed numerical simulations for edifices of different initial sizes, with various reservoir depths and sizes, and two different reservoir shapes:

a spherical one and a vertically-elongated one (prolate shape). We considered the effect of partial destruction, corresponding to the removal of the upper 20 % of a strato-volcano with a conical shape and a slope of 30 degrees (figure 9 B). A slope of 30 degrees is an upper limit for the upper part of strato-volcanoes based on a compilation of Digital Elevation Models, and corresponds to the maximum frequency of major slope failure events on Quaternary volcanoes [START_REF] Voight | Failure of volcano slopes[END_REF]; collapse of 20 % of the initial volcano can be considered as a mean value based on a compilation of field observations (figure 4a). In all simulations we consider an incompressible magma; effects of compressibility and gas content are discussed at the end of this section, and in section 5.

For a spherical reservoir located at 1 km depth beneath the volcanic edifice, each of the three scenarios previously described (figure 6) can occur (see figure 7 b). When the collapse affects small edifices, the erupted volume is larger than, but still close to, that expected in the absence of edifice destruction. As the edifice size increases, so the volume of erupted magma after the edifice collapse tends to decrease. This volume reduces to zero when large strato-volcanoes are partially destroyed by flank collapse, resulting in the abortion of any incipient eruption. For a shallower magma reservoir, a smaller edifice size is required to reach the point of aborted eruption, (see figure 7 a), whereas all effects of edifice collapse on erupted magma volume are reduced with a deeper chamber (see figure 7 c).

Effects of an edifice collapse event on the subsequent eruption also depend on the magma reservoir shape. Figure 8 shows that, for a prolate reservoir, the influence of the collapse is smaller, and a larger edifice size is required in order to decrease the amount of erupted magma, than for a spherical reservoir at the same depth. Above a prolate reservoir with a top at one kilometer depth, magma eruption is only aborted when the edifice radius is greater than 6 km.

For storage zones located at a few kilometers depth beneath the volcanic edifice, it is possible to include compressibility effects (see section 5), which can be important when volatiles are present. The inclusion of compressibility effects mainly acts to reduce the magma pressure decrease within the storage zone following the unloading event, as shown in figure 3. It follows that the volume of erupted magma will thus be larger than in the case of an incompressible magma. Larger edifice size is required to counteract this, and reduce the amount of erupted magma.

3D effects: Influence of the shape of the load removed

To be more realistic, we also carry out 3D models in order to simulate asymmetric flank collapses and mimic the resulting horseshoe-shaped craters For comparison, elastic parameters are the same as in the previous axisymmetric models. We run models for the same reservoir geometry: a sphere with a 3 km radius, with a top situated at two different depths (1 and 3 km).

Three different values for the initial edifice radius (1, 3 and 6 km) are considered. Table 1 gives the comparison between symmetric and asymmetric collapses. Here the symmetric case is recalculated so that the removed part of the edifice is re-distributed as a deposit layer of constant thickness from the edifice to a distance equal to 5 times the edifice radius. This distance is chosen in order to obtain the same order of magnitude for the deposit thickness as for the asymmetric case.

Values of the ratio of erupted volume after collapse to erupted volume without collapse are almost the same for both collapse geometries. The erupted volume after collapse only remains the same as that without any edifice destruction for small edifices (case 1, for an edifice of 1 km radius).

However, this erupted volume decreases when the initial edifice size increases (case 2, for an edifice of 3 km radius and a chamber top at 1 km depth, or for an edifice of 6 km radius and a chamber top at 3 km depth). For large edifices and shallow storage zones, no magma is erupted, since eruption is aborted by the collapse event (case 3 for an edifice of 6 km radius and a chamber depth of 1 km).

The closure pressure tends to decrease for asymmetric failure compared to symmetric failure. This effect produces a small increase in erupted volume compared to the symmetrical model for case 2 (Table 1). Calculated differences are minimal between the two failure geometries, with a difference in total erupted volume of only a few percent. In the same way, our results for erupted volume do not change significantly when taking into account the mass load redistribution due to runout deposits at the periphery of the volcano.

From these calculations, it appears that the results are not significantly affected by the collapse geometry; thus the less time-consuming axisymmetric models, can be used to perform parametric studies.

Phreatic eruptions

Several large volcanic failure events have not resulted in emission of juvenile material, but instead lead to a phreatic eruption. Such phreatic events were defined by [START_REF] Siebert | Volcanic hazards from Bezymiannyand Bandai-type eruptions[END_REF] as Bandai-type eruptions after the name of the Bandai-san volcano in northeast Japan, which produced large phreatic explosions associated with a major debris avalanche in 1888 [START_REF] Yamamoto | Pyroclastic density current from the 1888 phreatic eruption of Bandai volcano, NE Japan[END_REF]. At Bandai-san, the volcanic failure was triggered by an earthquake and there is no direct evidence of magma involvement in this catastrophic event. As no juvenile products were erupted, the straightforward conclusion is that no magma was present at shallow level. However, pressurized fluids had been stored at a shallow depth, and a heat source is necessary to explain the vigorous hydrothermal system. Our results provide an alternative framework to interpret such phreatic events, in which magma might be trapped in reservoirs beneath the edifice. We show that for shallow reservoirs, even if magma was present, a large edifice collapse could potentially abort the incipient eruption.

Effect of edifice destabilization on the long-term eruptive history

The three main modifications documented with regard to the erupted magma after a major flank collapse are: an increase in eruption rate [START_REF] Beget | Cyclic formation of debris avalanches at Mount St Augustine volcano[END_REF][START_REF] Siebert | The 1883 and late-prehistoric eruptions of Augustine volcano, Alaska[END_REF][START_REF] Boulesteix | Eruptive response of oceanic islands to giant landslides: New insigths from the geomorphologic evolution of the Teide-Pico Viejo volcanic complex (Tenerife, Canary)[END_REF], a change in erupted magma composition towards less evolved and denser magmas [START_REF] Manconi | The effects of flank collapses on volcano plumbing systems[END_REF][START_REF] Longpré | Volcanic and geochemical evolution of the Teno massif, Tenerife, Canary Islands: Some repercussions of giant landslides on ocean island magmatism[END_REF][START_REF] Boulesteix | Eruptive response of oceanic islands to giant landslides: New insigths from the geomorphologic evolution of the Teide-Pico Viejo volcanic complex (Tenerife, Canary)[END_REF], and a change in magma storage pressure [START_REF] Rutherford | Magmatic conditions and processes in the storage zone of the 2004-2006 Mount St Helens dacite[END_REF].

As shown in section 2, an edifice flank collapse always causes a pressure decrease within underlying storage zones. Where the shallow reservoir is still connected to a deeper source of magma, this reservoir depressurisation should induce a rapid replenishment as observed after a reservoir withdrawal due to an eruptive event (Lu et al., 2010). The deeper source is expected to contain more primitive magmas, and the replenishment should thus increase the amount of less differentiated magmas within the shallow storage zone. A reduced edifice size also allows the eruption of denser products which would otherwise have stalled at shallow depth [START_REF] Pinel | The effect of edifice load on magma ascent beneath a volcano[END_REF].

The evolution of the threshold pressure required for dyke initiation, which corresponds to the maximum pressure within the reservoir, should have an impact over a longer time scale. At Mount St Helens, the pressure evolution of the storage zone can be seen through petrological studies of rocks spanning the last thousand years [START_REF] Gardner | Experimental constraints on pre-eruptive water contents and changing magma storage prior to explosive eruptions of Mount St Helens volcano[END_REF][START_REF] Rutherford | Magmatic conditions and processes in the storage zone of the 2004-2006 Mount St Helens dacite[END_REF]. The observed pressure range is between 130 MPa and 300 MPa, with several episodes of storage pressure reduction occurring over periods of a few years. The most recent episode of storage pressure decrease corresponds to the renewed activity in 2004-2006, with a pressure decrease in the order of 50 MPa compared to the magma erupted in 1980. Such episodes of pressure decrease have been interpreted as being due to a rise of the storage zone [START_REF] Gardner | Experimental constraints on pre-eruptive water contents and changing magma storage prior to explosive eruptions of Mount St Helens volcano[END_REF]. However, it has been shown that each episode of pressure decrease follows directly on from a large edifice destabilization, such as the one which occurred in 1980 [START_REF] Hopson | Compositional trends and eruptive cycles at Mount St[END_REF][START_REF] Hausback | Record of prehistoric debris avalanches on the north flank of Mount St. Helens volcano, Washington[END_REF]. [START_REF] Pinel | Magma chamber behavior beneath a volcanic edifice[END_REF] and [START_REF] Pinel | On the relationship between cycles of eruptive activity and volcanic edifice growth[END_REF] propose that this pressure variation could be explained by a magma pressure decrease within a fixed storage zone induced by the edifice partial destruction rather than by upward migration of the storage location. At Mount St Helens, the petrological data are thus consistent with a decrease of the threshold pressure required for dyke initiation following flank failure. Where the shallow reservoir is fed by a source of constant pressure at depth, such a decrease should result in an increase of the eruption rate [START_REF] Pinel | On the relationship between cycles of eruptive activity and volcanic edifice growth[END_REF], in good agreement with the observations.

Discussion

We investigate the influence of the reservoir shape by considering only vertically-elongated ellipsoids. Calculations with horizontally-elongated reservoirs (oblate shape) could have been performed, but the edifice collapse impact is only significant for shallow reservoirs, and it has been shown

previously that shallow, oblate reservoirs strongly favour caldera formation [START_REF] Roche | Les mécanismes de formation des calderas: étude expérimentale et modélisation[END_REF][START_REF] Geyer | Relationship between caldera collapse and magma chamber withdrawal: an experimental approach[END_REF]. Caldera collapse formation is a complex and specific phenomenon, already studied elsewhere [START_REF] Pinel | Influence of preexisting volcanic edifice geometry on caldera formation[END_REF], and beyond the scope of this paper in relation to major flank collapse. Thus we choose to ignore oblate-shaped chambers here.

In this study we assume that rocks encasing the reservoir behave elastically. Volcano flank collapses are often sudden events. For instance, detachment of the northern flank of Mount St Helens, USA, in May 1980, occurred in a few seconds, as testified by eye-witnesses. Besides, we show that an unloading event significantly affects the stress field only at depths of less than three times the radius of the removed load (figure 1). At the time scale of an eruptive cycle, geodetic measurements recorded during replenishment or eruptive events [START_REF] Sturkell | Volcano geodesy and magma dynamics in Iceland[END_REF]Lu et al., 2010;Lu and Dzurisin, 2010) prove that the elastic assumption is valid, at least for shallow reservoirs.

This geophysical observation justifies our elastic assumption when looking at the impact of a major flank collapse on the subsequent eruptive event for continental volcanoes, for which the lateral extension of the edifice remains small compared to the elastic crustal thickness. Estimations of the Maxwell relaxation time for upper crustal rocks in volcanic areas are around 30-80 ky [START_REF] Jellinek | Did melting glaciers cause volcanic eruptions in eastern California? probing the mechanics of dike formation[END_REF], and the usual duration for cone-building episodes is less than 100 ky for continental volcanoes [START_REF] Davidson | ) Encyclopedia of volcanoes[END_REF].

Based on this consideration, our elastic model can still be used to discuss the influence of a major flank collapse on the long-term eruptive history of a given continental volcanic system. However, the elastic assumption becomes less valid when looking at the effect of destabilization on large oceanic volcanoes. The largest debris avalanche deposits (reaching 5000 km 3 ) are observed around large ocean-island volcanoes [START_REF] Mc Guire | Volcano instability: A review of contemporary themes[END_REF]. The lateral extension of the removed load is then close to, or even larger than, the elastic crustal thickness. For instance, the El Golfo landslide, which affected El Hierro island in the Canary Islands, had an inferred lateral extension of more than 10 km, for an elastic crustal thickness in this area of around 20 km, based on seismic and gravity data [START_REF] Watts | A seismic study of lithospheric flexure in the vicinity of Tenerife, Canary Islands[END_REF]. To investigate the impact of these large-scale flank collapses on the magmatic plumbing system of oceanic volcanoes, it would be necessary to take the viscous response of the mantle into consideration, too, as proposed by [START_REF] Sigmundsson | Multiple effects of surface load changes and associated stress change on magmatic systems[END_REF] in their study of the influence of long-term ice retreat on magma storage zones.

The model developed in this paper deals with a magma storage zone beneath the volcanic edifice such that it cannot account for decompression of the magma emplaced at shallow depth within the volcanic edifice. This effect should be taken into account to fully describe what occurred during the May 1980 eruption of Mount St Helens, where the volcano collapse was followed by a 1 km 3 Plinian ash eruption [START_REF] Bradley | Mount St. Helensfrom the 1980 Eruption to 2000[END_REF]. However the effect of magma compressibility within the chamber can also be included when buoyancy forces induced by magma density variations remain small relative to magma pressure changes (see [START_REF] Pinel | Some consequences of volcanic edifice destruction for eruption conditions[END_REF] for a complete dis-cussion). This condition is verified for magma with a few percent of volatiles stored at depths greater than a few kilometers.

Conclusion

Numerical simulations using the elasticity equations help to constrain the potential impact of a major volcano flank collapse on the ongoing eruption, as well as the longer term eruptive history, of continental volcanoes. Development of models taking into account the viscous response of the mantle would be necessary in order to model more precisely the potential consequences of the larger flank collapses affecting oceanic volcanoes. Calculation is for an elastic half-space with Poisson's ratio equal to 0.25.

The dotted line is for the vertical component of the stress tensor, σ zz , the dashed line is for the horizontal components, σ rr = σ θθ , and the solid line is for the pressure reduction, P = (1/3)(σ rr + σ θθ + σ zz ). which is removed from the surface. The chamber resides in an otherwise elastic homogeneous half-space with Young's modulus E = 30 GPa and Poisson's ratio ν = 0.25. In the absence of a magma chamber (black dashed curve), the pressure profile obtained is equivalent to the one given by equation 3 (solid line in figure 1). Other lines are for pressure profiles where there is a magma reservoir, with varying values of the magma bulk modulus (blue line, incompressible magma; purple line, K = 20 GPa; red line, K = 10 GPa; orange line, K = 1 GPa; yellow line, K=0 GPa). a) The chamber is a sphere (radius 1 km, depth to chamber top 1 km). b) The chamber is a prolate ellipsoid (half-height 1 km, half-width 0.25 km, depth to chamber top 1 km). (1996), except for Parinacota volcano [START_REF] Hora | Volcano evolution and eruptive flux on the thick crust of the Andean Central Volcanic Zone: 40 Ar/ 39 Ar constraints from Volcán Parinacota, Chile[END_REF]. Edifice volumes before collapse are estimated using the present topography of the volcano. When no collapse scar is visible in the topography, corresponding to old events, the volume before collapse is taken to be equal to the present volume; when a large scar is visible, the volume before collapse is obtained by adding the present volume to the failure volume.

The total current volume of each volcano, at present time, has been calculated by the numerical integration of SRTM (3" arc) elevation. For volume calculation, we assume that edifice extension stops when the slope becomes small (< 10 • ), and we subtract the mean elevation of the basement to deduce volcano height. The ratio between the volcano collapse volume and the total edifice volume is between 10 % and 30 % (dashed lines). For all collapse models, we use the mean value of 20 % (solid line) for this ratio. b) Runout distances of debris avalanches versus volcano radii. Runout distances are taken from Table 1 in Mc Guire (1996). Most volcanoes produce avalanches which travel to distances of around 6 times the radius of the edifice (lower dashed line). However, in certain conditions, such as Colima volcano, the deposits can travel up to 12 times the radius of the edifice (upper dashed line). A value of 10 is used in the 3D asymmetric modelling (solid line). where P is the magma pressure when the eruption starts and P cl is the threshold pressure value under which dykes close at the chamber wall, ending the eruptive event. Before edifice destabilization, the system is about to erupt, such that this difference is given by ∆P e (i) = P r (i) -P cl (i), with P r the threshold pressure for dyke initiation at the chamber wall. After the major flank collapse, magma pressure within the chamber decreases by an amount ∆P to the value P (f ), and the pressure difference becomes ∆P e (f ) = P (f )-P cl (f ). Three different cases can occur: case 1, when ∆P e (f ) is greater than ∆P e (i), where the volume of magma erupted is larger than it would have been with no edifice collapse; case 2, when ∆P e (f ) is less than ∆P e (i), such that the volume of magma erupted is smaller than it would have been with

no edifice collapse and, case 3, when ∆P e (f ) becomes negative, which means that the incipient eruption is aborted and there is no magma erupted.

  of the final edifice geometry that are observed at many strato-volcanoes, such as Mount St Helens (Cascades, USA),Bezymianni and Shiveluch (Kamtchatka, Russia) or Galunggung (Indonesia). The volume of the landslide remains fixed at 20 % of the initial volcanic edifice, but the collapse now occurs only at one side of the volcano (figure9 C). The unloading associated with the edifice collapse is thus asymmetric, with potential consequences for failure conditions at the reservoir wall. The removed part of the edifice is re-distributed as a thin deposit layer around the volcano. This layer has a constant thickness from the base of the edifice to a distance of ten times the edifice radius, and is only emplaced in a sector of ±30 • from the collapse flank. The runout and sector angle are based on field observations, as shown in figure4b.

Table 1 :

 1 Comparison between results obtained with a truncated edifice (2D axisymmetric calculation, figure9 B) and with a sector collapse (3D calculation, figure9 C). Mass is redistributed at the volcano base: all around the volcano for the axisymmetric case, and in a wedge-shaped sector for the asymmetric flank collapse. Calculations are performed for a spherical reservoir with a 3 km radius, and a top at 1 km or 3 km depth, for three different initial edifice radii: 1, 3 and 6 km. The same experiments are represented by crosses in figure7.

Figure 1 :

 1 Figure 1: Stress reduction under the centre of a conical load (2 km radius, 1 km height, density of 2800 kg/m 3 ) which is removed from the surface.

Figure 2 :

 2 Figure 2: Pressure decrease within and around a magmatic reservoir, under the centre of a conical load (2 km radius, 1 km height, density of 2800 kg/m 3 )

Figure 3 :

 3 Figure 3: Pressure reduction within the magma chamber (∆P (K) ) induced by the removal of a surface conical load (radius 2km, height 1km, density 2800 kg/m 3 ) as a function of the bulk modulus, K, of the magma. Crustal Young's modulus and Poisson's ratio are equal to 30 GPa and 0.25, respectively.The pressure change is normalized by the pressure change in incompressible magma (∆P ∞ ). The shaded area shows the range of values characteristic of dry magmas (K between 1 and 20 GPa;[START_REF] Tait | Pressure, gaz content and eruption periodicity of a shallow, crystallising magma chamber[END_REF]). The solid curve is for the spherical reservoir and the dashed curve for the prolate one.

Figure 4 :

 4 Figure 4: a) Volume of major collapse versus edifice volume. Failure volumes are taken fromTable 1 in Mc Guire (1996), except for Parinacota volcano

Figure 5 :

 5 Figure 5: Pressure reduction within the magma reservoir induced by the removal of the upper 20% of a conical edifice with a slope of 30 degrees. Results are presented as a function of the reservoir and edifice radius. Calculations are performed for a spherical reservoir with a top at 1km depth, and filled with incompressible magma. Crustal Poisson's ratio is equal to 0.25.

Figure 6 :

 6 Figure 6: a) Evolution of magma pressure, P , and threshold pressure for failure, P r , induced by volcanic collapse. (b) Evolution of the erupted volume of magma following edifice destabilization. For a given reservoir geometry, the erupted volume is proportional to the pressure difference, ∆P e = P -P cl ,

  Figure1Click here to download Figure: Figure1.pdf

Table 1 in

 1 Mc Guire

Acknowledgments

The authors thank the Editor Malcolm J. Rutherford and two anonymous reviewers for providing helpful comments. We also thank Francesca van Wyk de Vries for English corrections.