Alexander V Barashev
email: a.barashev@liv.ac.uk.

Stanislav

S I Golubov

On the onset of void ordering in metals under neutron or heavy-ion irradiation

Keywords: kinetics, lattice defects, neutron irradiation, non-equilibrium phenomena, pattern-related phenomena, radiation effects user supplied): displacement cascades, interstitial clusters void swelling, void lattice, displacement cascades, interstitial clusters

Formation of void lattices is observed in a number of metals and alloys under high-energy particle bombardment. Here we derive the conditions for destabilisation of homogeneous void arrangement using an approach developed for the description of lane formation in pedestrian crowds. The model is based on the Foreman's mechanism of void alignment due to onedimensionally-migrating clusters of self-interstitial atoms. The results show that spatial correlations between voids should exist above some very small size, unless correlations with other defects prevail. It is shown that spatial correlations of voids with dislocations and secondphase precipitates should also evolve and provide a powerful driving force for further swelling.

Introduction

The formation of a void lattice was first reported in 1971 by Evans [1] in molybdenum under nitrogen ion irradiation, by Kulchinski, Brimhall and Kissinger [2] in nickel under selenium ion bombardment and by Wiffen [START_REF] Wiffen | Proc. Int. Conf. Radiation-Induced Voids in Metals[END_REF] in molybdenum, niobium and tantalum under neutron irradiation. Since then it was observed in bcc tungsten, the fcc Al, the hcp Mg, and some alloys. Jäger and Trinkaus [START_REF] Jäger | [END_REF] reviewed the characteristics of defect ordering, where many original references can be found, and analysed the theories proposed, including those based on the elastic interaction between voids, phase instability theory and low-dimensional atomic transport. They concluded that in cubic metals the void ordering is probably due to the one-dimensional (1-D) glide of self-interstitial atom (SIA) clusters along close-packed crystallographic directions. Two features of void ordering support this conclusion. First, that the symmetry and crystallographic orientation of a void lattice are always the same as those of the host lattice. And second, that the void lattices are formed under neutron and heavy-ion but not electron irradiation, i.e. when the primary damage is produced in displacement cascades. The only exception of void ordering during electron irradiation in a stainless steel containing a high concentration of nitrogen [5] was proposed to be most probably due to SIA loop punching from small nitrogen bubbles preferentially absorbing SIAs. Voids in hcp Mg (and bubbles in hcp Ti, Co, Zr and some Zr alloys) can order in planes parallel to the basal plane with variable void plane separation [6,7]. Jäger and Trinkaus concluded that for hcp metals a decision between 1-D or 2-D SIA transport proposed by Evans [8][9][10] was not possible at the moment due to the lack of a sufficient body of information. Now we know from molecular dynamics (MD) simulations using empirical potentials for the inter-atomic interaction that SIA clusters bigger than four SIAs in hcp Zr migrate one-dimensionally in the basal plane [11]. Hence, the observed planar void ordering in must also be accounted for using the same mechanism as for the void lattices in cubic metals.

It is worth noting that already in 1972 Foreman [START_REF] Foreman | UKAE Authority Harwell Report[END_REF] suggested that voids are aligned along close-packed directions by 1-D SIA fluxes. Dubinko et al. [START_REF] Dubinko | [END_REF] were first to study the effect of the glide of perfect interstitial loops due to interaction with voids on void lattice formation. Then, in 1992, Foreman et al. [14] MD studies of displacement cascades in copper. This finding resolved a serious problem of the theory called 'Singh-Foreman catastrophe' [15], that is, accumulation of a high density of immobile SIA clusters produced in displacement cascades that is not observed in experiments, and promoted formulation of the 'Production Bias Model' of radiation damage in its modern form. The model succeeds in explaining several striking observations, for example, enhanced swelling rates near grain boundaries [16] and in materials with small grain size [17], and under neutron compared to electron irradiation [18,19]. This is due to the recognition of two distinguishing features of defect production by high-energy recoils: first, the formation of SIA clusters directly in displacement cascades, as shown both experimentally [20] and in MD simulations [14,21,22] (see also review [23]); and, second, the 1-D mobility of SIA clusters [14,[24][25][26][27].

After the review by Jäger and Trinkaus [START_REF] Jäger | [END_REF], several papers were published which exploited the Foreman's idea for the description of the void lattice. Some of them used Monte Carlo method to simulate void lattice formation (e.g. [28][29][30]). In our view, these attempts have not added much to the very Foreman's idea. This is mainly because they suffer from an intrinsic deficiency of the small box size compared to the mean-free path of the SIA clusters, which introduces methodological errors in the simulations, as partly discussed at the end of this paper. Two attempts were made to describe analytically the void lattice formation, but both failed in our view. In the analysis by Walgraef and Ghoniem [31], reactions of the 1-D diffusing SIA clusters with voids were incorporated incorrectly into the basic equation system (compare, e.g. equation system (1) in [31] with equations ( 5) and ( 6) in [18]). Another attempt was made by Semenov and Woo [32,33]. Their analysis was based on the assumption of the existence of the stationary size-distribution function of voids. This led to a conclusion that the formation of a void lattice is only possible if the fraction of the SIAs in the form of 1-D mobile clusters is less than 1% and thus less than the dislocation bias. However, as shown in [34], the stationary state does not exist for such a small fraction. In addition, the value of 1% is at least an order of magnitude smaller than that obtained in MD simulations [22].

In this paper we consider the void ordering in the framework of a model developed for the lane formation in pedestrian crowds [35]. It suits naturally to the description of the phenomenon, since, according to experiments, voids are first nucleated in a random arrangement and only then form a lattice, the perfection of which increases with increasing irradiation dose [START_REF] Jäger | [END_REF]. The main aim of the present work is to derive general conditions for the onset of void ordering, after which the conventional rate equations should not apply.

Analysis

Basic equations

We assume that the displacement cascades produce 3-D mobile single vacancies and SIAs and 1-D mobile SIA clusters. The fraction of the clustered SIAs is denoted by ε i g . We also assume that the void nucleation stage is over and the mobile defects interact only with already existing voids of the number density N and dislocations of the density ρ . The swelling rate is then [17] dS dφ where S = 4π r 3 N / 3 , r is the mean void radius, φ is the irradiation dose in displacements per atom, which accounts for intra-cascade recombination, p j is the probability that a j-type defect is absorbed by a void, Z v and Z i are the capture efficiencies of dislocations for vacancies and single SIAs, respectively, and r d is the dislocation capture radius for SIA clusters. The main purpose of writing equation ( 1) is to describe the reaction scheme and introduce probabilities p j .

= p v -(1 -ε i g )p i -ε i g p cl = = 4π rN 4π rN + Z v ρ -(1 -ε i g ) 4π rN 4π rN + Z i ρ -ε i g πr 2 N πr 2 N + πρr d / 2 , (1)
We use a mathematical model similar to that developed for the description of lane formation in pedestrian crowds [35]. With the total void density to be constant, the local density changes due to voids moving in and out of the local area and hence obeys a continuity equation

∂N ∂φ = -div(NV -D∇N ) , ( 2
)
where V is the mean velocity and D is the diffusion coefficient of voids, defined locally:

V = 〈 ν j ∆ j j ∑ 〉 , (3)
D = 1 6 〈 ν j ∆ j 2 j ∑ 〉 . (4)
Here, j = v,i,cl for the components corresponding to void collisions with vacancies, single SIAs and SIA clusters, respectively, ∆ j are the vectors of void displacements after collisions and the averaging is performed over voids in the local region. The collision frequencies are

ν v = p v / ΩN , ν i = 1 -ε i g ( ) p i / ΩN and ν cl = ε i g p cl / mΩN ,
where Ω is the atomic volume and m is the mean number of SIAs in a cluster. In the following two sections, we derive the void velocity and diffusion coefficient as functions of microstructure parameters.

Diffusion coefficient of void

A void of radius r contains Ω = 3 / 4 3 r M π vacancies and arrival of an additional vacancy to it should displace its centre of mass by a vector, which is approximately equal to

∆ v = rn M + 1 ≈ 3Ω 4π r 2 n , ( 5
)
where n is a unit vector normal to the void surface at the point of defect absorption. Here we use a conventional assumption that a defect is absorbed by a void at the point they touch each other and that the local surface diffusion is sufficient to keep void spherical (see, e.g. kinetic Monte Carlo simulations [28,30] and analytical treatment [36]). This issue was generally discussed by Balluffi [START_REF] Balluffi | Proc. Int. Conf. On the Fundamental Aspects of Radiation Damage in Metals[END_REF] and the assumption seems reasonable for a small void at low temperature, when the void surface represents a highly distorted region and the surface diffusion is limited to distances of the order of a facet size. Arrival of an SIA to the same point should shift the void by the same distance but in the opposite direction

∆ i = -∆ v ,
D = 3 2N Ω 4π r 2 ( ) 2 mε i g p cl + (2 -ε i g )p v     . (6)

Void velocity

The void velocity is determined by anisotropy of void collisions with mobile defects, hence by a nonzero ∇ν j . Here, we assume that this anisotropy is due to existence of a gradient of void concentration ∇ν j ∝ ∇N . An additional driving force due to gradient of void radius is ignored here for simplicity, to preserve the visual transparency of the original model of lane formation in pedestrian crowds (see also discussion in section 2.4 after equation ( 19)). Then, a component of the void velocity due to interaction with j-type defects can be calculated as an average over defects arrived to different points of the void surface

V j = r σ dσ ∂ν j ∂N n∇N ( )∆ j ∫ L j . ( 7
)
Here, σ = 4π r 2 for 3-D and σ = π r 2 for 1-D migrating defects (separately for each cluster Burgers vector k b , k = 1 to n, n=4 for the bcc and n=6 for the fcc crystal lattice), dσ is the corresponding surface element and L j is a characteristic size of the volume for averaging, which has replaced the averaging over different voids in the mean-field approach. For 3-D migrating vacancies and SIAs, this size is of the order of the mean-free path

L v,i ≈ L 3D ≈ 4πrN + Z v ρ ( ) -1/2 . ( 8
)
For 1-D migrating clusters, for each Burgers vector, it is of the order of the projection of the mean-free path L 1D on the direction of ∇N :

L cl k = L 1D k ≈ L 1D cos b k ,∇N ( ) , (9)
L 1D = πr 2 N + πρr d / 2 ( ) -1 . ( 10
)
By performing the integration indicated in equation ( 7), one can obtain the following equations for the partial velocities:

NV i = (1 -ε i g )A i ∇N L 3D , ( 11
)
NV v = -A v ∇N L 3D , ( 12
)
NV cl = ε i g A cl 3 4n ∇N + e k ,∇N ( ) e k L 1D k k =1 n ∑ , ( 13
)
where hand side of equation ( 13) describes movement of voids in the directions of the cluster Burgers vectors.

A j = p j

Conditions for the onset of void ordering: a linear stability analysis

Let us substitute equations ( 6) and ( 11)-( 13) into equation ( 2) and analyse its stability to small variations of the void number density. Obviously, a homogeneous solution, N hom , satisfies equation (2). To derive the basic conditions for its destabilisation, we apply the linear stability analysis and insert the ansatz N = N hom + % N exp(iqR + aφ) , where R is the radius vector, q is a wave vector and a is the amplification factor. This results in the following equation

a q 2 e iqR = ε i g A cl 3 4n 1 + cos 2 b k ,q ( )     〈e iqR 〉 L 1D k k =1 n ∑ + + (1 -ε i g )A i 〈e iqR 〉 L 3D -A v 〈e iqR 〉 L 3D -De iqR . ( 14
)
We note that 3 cos 2 (b k , q) k ∑ = n in cubic structures, hence, for long waves, such that q -1 >> L 1D , equation ( 14) is reduced to (taking p i = p v for simplicity)

a q 2 = ε i g 4π rN p cl 2 -p v 2 ( ) -D < 0 , (15)
which is always negative since p cl ≤ p v (voids do not shrink). For shorter wavelengths, we notice that the amplification factor a can be positive provided that two conditions are satisfied at the same time: (i) the third term in the right-hand side of equation ( 14) is zero and (ii) some of the terms of the sum are nonzero. The condition (i) requires q -1 << L 3D . The condition (ii) introduces limitations for both the magnitude and the direction of the wave vector. First, one needs r q >> -1

, otherwise all terms in the sum vanish.

And, second, it requires

L 1D cos b k ,∇N (
) << r for at least one k, i.e. the wave vector should be perpendicular to b k . The maximum a is achieved for a wave vector perpendicular to a plane containing any two Burgers vectors. In this case, two terms of the sum are nonzero, while the terms with other Burgers vectors are equal to zero, because their averaging is performed over distances

L 1D cos(b k ,∇N ) ≥ L 3D .
Summarising the analysis presented above, we conclude that the most favourable conditions for destabilisation of the homogeneous solution are realised when

r << q -1 << L 3D , ( 16
) q [b k , b k ' ] , (17)
where

a = q 2 3ε i g A cl / 2n -D ( ) > 0 . (18)
Or, in a more explicit form, as

4π r 3 3Ω > nm 3 1 + ρr d 2r 2 N       1 + 2 -ε i g ε i g m p v p cl       . ( 19
)
Figure 1 shows the dependence of a critical void radius, above which voids start to order, on the void number density, given by equation (19). The calculations were performed for the bcc lattice (n=4), m=10, ε i g =0.5, Z i = Z v = 1 and two values of r d and dislocation density shown in the figure . As can be seen from the figure, the critical void radius is very small, much smaller than the void saturation radius ~2π r d / Z v , predicted by equation ( 1) [16]. (The latter can readily be derived in the limiting case when the production bias is much higher than the dislocation bias:

ε i g >> B = Z i / Z v -1.
For a comprehensive analysis, see an accompanying paper [START_REF] Barashev | [END_REF].)

[Insert figure 1 about here] It is worth mentioning that the analysis presented here is similar to that used by Hähner and Frank [36]. The main difference is in consideration of a realistic mechanism for the lattice formation, which is supported by MD and first principal calculations, and is a requirement for explanation of some experimental data. Some better mathematical transparency of the present analysis is achieved by using several simplifying assumptions, for example that void velocity, equation (7), is proportional to the void concentration, while the contribution of the gradient of void radius is neglected. There are several reasons why gradients of void radius are not as important, but we have not found any concise explanation of this. Note, however, that according to the results obtained by Hähner and Frank, the fluctuations of void radii do not initiate instability, which supports our assumption. As far as we are concerned, Hähner and Frank were also the first to conclude that the void spacing in void lattices is likely to be determined by that in the disordered state, rather than be an 'intrinsic material parameter'. Although they used unrealistic mechanism for void ordering based on radiation-produced SIAs in metastable crowdion configuration, the main conclusion obtained in the present paper that void lattice should start for very small voids can be derived from their results in the limiting case, when the mean time before conversion of a crowdion to a dumbbell configuration is infinitely long: τ C → ∞ . So, assuming that the sink strength of dislocations is negligible as compared to that of voids, which agrees with the experimental observations of void lattices, it can be obtained that void ordering should start at any void size and such wave vectors that perpendicular to the closepacked planes (e.g. {110} in the bcc crystal lattice) and which modulus satisfies the condition: q -1 < L 3D . This is similar, but less stronger condition than the right-hand part of equation ( 16). The restriction corresponding to equation (19), which originates in diffusion of voids, is absent in the analysis by Hähner and Frank. Finally, the main important conclusion of the present work that the void ordering due to clusters of SIAs should start at very small voids is supported by their work, too.

Thus, the conditions for the onset of void ordering show that spatial correlations between voids should always be present, except for very early stages of irradiation, when voids are too small. Essentially, the real question is not why voids sometimes form super-lattices, but why they do not form them. We believe that we know the answer to the first question, although not all aspects of the process are yet understood. We suppose the answer to the latter question to be as follows. Spatial correlations always develop under cascade damage conditions. The void lattices represent only one type of such correlations, which is realised when the void number density is high enough. At lower void density, spatial correlations with other defects, e.g. second-phase precipitates and dislocations, are formed. Which correlations should prevail depends on the densities of different defects. In the analysis presented above we assumed that voids are correlated with each other rather than with other elements of microstructure. This is taken into account by assuming that the void velocity is caused by the gradient of void number density.

Let us derive applicability of this assumption for a system containing voids and edge dislocations. In the framework of the model considered, the probability that an SIA cluster has voids at both ends of its path is p cl 2 and the probability that it has a void at one of the ends and a dislocation at the other end is 2 p cl (1p cl ) . A void lattice should evolve if the former is much greater than the latter, i.e. when

r 2 N >> ρr d . (20)
Note that the dependence of the amplification factor on the defect densities in equation ( 18) is rather weak, hence it is most probable that this requirement determines the observed conditions for the void lattice formation, that is, high void and low dislocation density. The very absence of a void lattice, i.e. spatial correlations between voids, must be an indication that correlations of voids with other defects prevail.

On spatial correlations between voids and other lattice defects

It follows from the analysis that other microstructural features, e.g. dislocations and secondphase precipitates, must also be involved in shadowing of voids from the SIA clusters and establish spatial correlations with voids. So, for example, an existence of a gradient of dislocation density, should cause a void movement with the velocity given by equation (17),

where N is replaced by ρ . The following analysis would be similar to that for void ordering and, if the inhomogeneity of the dislocation density is taken in the form ρ = ρ hom + % ρ exp(iq d R) , the equation (18) for the amplification factor becomes

a = % ρ % N q d 2 A d 2 exp i q d -q ( )R     -q 2 D , (21)
where A d = 3ε i g r d A cl / 4nr 2 . As can be seen, the first term in the right-hand side of equation ( 21) is real and positive if q = q d , i.e. when spatial positions of voids correlate with dislocations.

Concluding remarks

The conditions for the onset of void ordering have been analysed in the framework of a model developed for the lane formation in pedestrian crowds and Foreman's mechanism of void alignment due to 1-D migrating SIA clusters. The results show that spatial correlations between voids should almost always be present unless correlations with other lattice defects of a higher density prevail. The condition of high void / low dislocation density is thus necessary for void ordering, since it ensures the development of correlations between voids rather than between voids and other lattice defects, such as edge dislocations and second-phase precipitates. The very absence of a void lattice must be an indication of the existence of correlations of voids with other defects. An accompanying paper [START_REF] Barashev | [END_REF] is devoted to study the consequences of these correlations on void swelling and generally defect accumulation. Importantly, the critical void size for the onset of void ordering is much smaller than the void saturation radius predicted by the theory for voids distributed randomly. Hence, the saturation of swelling observed in void lattices cannot be a consequence of this, as was thought before [8,9]. This statement is also supported by a previous finding that the formation of the free channels between voids in void lattices provides the escape routes for the SIA clusters to dislocations and leads to a significant increase of the void saturation radius [39]. In addition it is shown in [35] that, in conditions when void ordering is observed, i.e. a high void density, the irradiation dose required for an ensemble of randomly distributed voids to reach the steady state is very large, about several hundred dpa, thus much higher than in experiments, which is ~10 dpa.

It is important that such spatial correlations provide a powerful mechanism for void growth and in an accompanying paper we argue that spatial correlations of voids with dislocations and second-phase precipitates should be a common feature under cascade-produced irradiation [START_REF] Barashev | [END_REF]. Association of large voids with various precipitates (G, η, Laves etc.) [START_REF] Cawthorne | Symp. The Nature of Small Defect Clusters[END_REF][START_REF] Mansur | Proc. Conf. Phase Stability under Irradiation[END_REF][START_REF] Rowcliffe | [END_REF][43][START_REF] Pedraza | Radiation Induced Changes in Microstructure: 13th Int. Symp. (Part 1)[END_REF][START_REF] Boothby | [END_REF] and growth of voids in the compression side of edge dislocations [46,[START_REF] Lanore | Proc. Int. Conf. On the Fundamental Aspects of Radiation Damage in Metals[END_REF] is well-known for a long time. Recently, an evidence for the correlations between voids and other defects has been observed by Kozlov, Portnykh et al. [START_REF] Kozlov | [END_REF]49] in 20% cold-worked 16Cr-15Ni-2Mo-2Mn austenitic steel irradiated up to ~100 dpa in a BN-600 fast reactor in the temperature range 410 to 600ºC. The transmission electron microscopy has revealed voids of three main types: a-type associated with dislocations, b-type associated with G-phase precipitates and c-type, which were formed homogeneously in the matrix. The c-type voids were the smallest and made the smallest contribution to swelling, while the a-type voids were the largest.

It is relevant here to give arguments against apparently wrong interpretation of experimental data and some Monte Carlo results on void ordering by Evans [50], which led him to conclude that 2-D rather than 1-D SIA migration should be responsible for the void lattice formation. First, it is evident that a correct modelling of void ordering must use simulation boxes bigger than the mean-free path of 1-D migrating SIA clusters. This can be estimated using equation (10) and is typically of the order of micrometers, thus an order of magnitude longer than for 2-D diffusing objects. Evans used much smaller box size, which was suitable for 2-D but not for 1-D migrating defects, and his observation of an existence of a barrier for the precise alignment of voids due to 1-D motion of SIAs likely originates from this methodological error. Second, according to equations ( 16) and (17) of the present analysis, during early stages of void ordering, voids form close-packed planes rather than close-packed lines, in exact correspondence with experiments. Hence, Evans' expectation of 'the local ordering to be either along the relevant close-packed planes of along close-packed directions, depending on whether the SIA transport is planar or linear' is invalid. We also emphasise that, generally, substantial modelling and experimental evidence collected to date support 1-D rather than 2-D SIA transport.

Finally, we note that the considered spatial correlations tend to decrease the void-SIA collision frequency. The condition of the minimum of interaction intensity in the evolving structures is common in self-organisation processes and is considered as a generalisation of the Onsager principle of the minimal dissipation of entropy [35].

( ) 2 /

 2 4π rN and e k = b k / b . As can be seen from equations(11) and (12), 3-D migrating SIAs and vacancies move voids towards regions of higher and lower void concentration, respectively, thus maximising their growth rate. The effect of SIA clusters is double fold. The first term in equation (13) is analogous to equation(11) for single SIAs and describes movement of voids along the void concentration gradient. The last term in the right-

  are any two of n different Burgers vectors of the SIA clusters. These conditions define the void movement leading to formation of close-packed planes b k , b k '    of voids, and this is consistent with the observed isomorphism of the finally evolved void lattices and the host lattice structure. Substituting conditions (16) and (17) into equation (14), one obtains the destabilisation condition as

Figure 1 .

 1 Figure 1. Dependence of the critical void radius for the onset of void ordering on the void number density.

  discovered 1-D mobility of small SIA clusters in

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http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters

Acknowledgements

The authors express their gratitude to Dr B.N. Singh (RisØ National Laboratory, Denmark), Dr H. Trinkaus (Forschungscentrum Jülich, Germany), and Drs S.J. Zinkle, R.E. Stoller and Yu.N. Osetsky (Oak Ridge National Laboratory, USA) for careful reading and useful discussions of the manuscript. The research was sponsored by a research grant from the UK Engineering and the Physical Sciences Research Council and by EU-funded project FP 7 GetMat (A.V.B) and by the Office of Fusion Energy Sciences U.S. Department of Energy, under contract DE-AC05-00OR22725 with UT-Battelle, LLC (S.I.G).


Hugues Raimbourg A-B
email: hugues.raimbourg@univ-orleans.fr

Kohtaro Ujiie

Achim Kopf

Toshio Hisamitsu

Yozo Hamano

Saneatsu Saito

Masataka Kinoshita

The role of compaction contrasts in sediments in décollement initiation in an accretionary prism

To understand how décollements develop into the pristine sedimentary succession entering subduction zones, we have performed mechanical tests on samples from the sediment column entering the Nankai accretionary prism, Japan (ODP site 1173). Both poroelastic compliance and plastic shrinkage upon application of a large effective pressure sharply decrease with depth in a -thick domain in the upper section of the Lower Shikoku Basin unit, i.e. in a domain stratigraphically close to the actual 2 location of the décollement near the toe of the prism. These property contrasts provide a potential explanation for the outward migration of the décollement into the incoming sediments.

When approaching the deformation front, a given material particle is affected by an increase in stress, which has a component of vertical loading due to the deposition of overburden trench sediment, but also a component of lateral compression transmitted from the accretionary wedge. Depending on its initial mechanical state, the amount of lateral shortening in the incoming Nankai sediment column varies with depth and causes horizontal velocity gradients that concentrate into the mechanical transition zone (upper section of the Lower Shikoku Basin at appx. 450-550m depth) into which the décollement eventually propagates. Future work has to assess the role of this plastic deformation relative to other governing factors such as friction coefficient and excess pore pressure, both at Nankai and along other active margins.

Introduction

Convergent margins can be either non-accreting, where all the incoming sediment sequence is underthrust by subduction, or accreting, where part of these sediments are incorporated into a growing accretionary prism [START_REF] Uyeda | Backarc opening and the mode of subduction[END_REF][START_REF] Von Huene | Observations at convergent margins concerning sediment subduction, subduction erosion, and the growth of continental crust[END_REF]. Along accreting margins, the fraction of incoming material accreted to the overriding plate is controlled by the position of the décollement, which is therefore a major factor in prism growth, dynamics and global mass balances.

The décollement can often be seismically imaged as a high amplitude, reverse polarity reflector (e.g. [START_REF] Moore | Structure of the Nankai Trough accretionary zone from multichannel seismic-reflection data[END_REF] in Nankai, [START_REF] Shipley | Underthrust sediments, fluid migration paths, and mud volcanoes associated with the accretionary wedge off Costa Rica: Middle America Trench[END_REF] in Costa Rica), interpreted as the interface separating high porosity/fluid pressure underthrust sediments from low porosity/fluid pressure accretionary prism material [START_REF] Tobin | Elevated fluid pressure and extreme mechanical weakness of a plate boundary thrust, Nankai Trough subduction zone[END_REF].

The coexistence of these two independent hydrologic systems implies that the interface, or décollement, has larger permeability than the material above and below [START_REF] Henry | Fluid flow along a decollement layer: A model applied to the 16°N section of the Barbados accretionary wedge[END_REF]. The décollement is therefore often considered as an highly fractured horizon, where the pore pressure is relatively high compared to surrounding material, as a result of channelized flow of deep fluid [START_REF] Saffer | Episodic fluid flow in the Nankai accretionary complex: Timescale, geochemistry, flow rates and fluid budget[END_REF]. In turn, high pore fluid pressure reduces the effective stress and promotes further fracturing or fault initiation.

Although there is an enhancement in deformation and fluid flow in a mature décollement, it is unclear how this discontinuity in physical properties of the seaward-migrating frontal thrust extends into material outboard of the deep-sea trench.

In Barbados prism, the presence of a high-porosity, radiolarian-rich horizon (Moore et al., 1998;[START_REF] Moore | Synthesis of Results: Logging While Drilling, Northern Barbados Accretionary Prism[END_REF] may explain décollement localization. On the other hand, such lithological heterogeneity is not relevant in Nankai area (Fig. 1), where both Upper Shikoku Basin (U.S.B.) and Lower Shikoku Basin (L.S.B.) units consist dominantly of hemipelagic silty mudstones (Shipboard Scientific Party, 2001b).

As detailed in section 2, alternative candidates for the controlling properties are permeability (2.1), shear strength (2.2) or yield strength (2.3). The two latter factors refer to the material mechanical state, which is usually assessed by geotechnical tests, i.e. rather by the stress necessary to trigger plastic deformation/slip than by the amount of strain for a given stress increment. In this manuscript we propose to reverse this point of view and to consider the finite strain affecting incoming material upon arrival into the prism. The amount of compaction versus depth constitutes a new possible controlling factor for décollement localization when considering the horizontal compression in the vicinity of prism toe. For the new model, we first consider the state of stress in the vicinity of the prism toe and describe, based on the analogy with cutting tool processes, how horizontal compression affects incoming material seaward of décollement tip and deformation front (section 3). We then assess the material response to horizontal compression using mechanical testing under isotropic loading conditions up to maximum of twice in-situ effective stress (section 4). We finally define a model combining the horizontal compression with the mechanical profile to propose a new mechanism of décollement localization based on the differential horizontal compaction of a sedimentary column approaching the Nankai prism (section 5).

Décollement localization: state of the art

Permeability/pore fluid pressure hypothesis

As the décollement is the surface where the shear localizes, its propagation into incoming material is favoured in horizons where the shear strength is reduced. The presence of high pore fluid pressure, which lower the shear strength, is believed to account for the formation of the protodécollement seaward of the Nankai prism toe (Le Pichon and [START_REF] Le Pichon | Erosion and accretion along subduction zones: a model of evolution[END_REF][START_REF] Le Pichon | Accretion and erosion in subduction zones: The role of fluids[END_REF]. In their model, the rapid loading of low-permability hemipelagites by the deposition of high-permeability turbidites near the trench results in the development of a minimum in effective pressure below the turbidite-hemipelagite transition, which attracts décollement propagation (See also Discussion 6.3).

Shear strength hypothesis

In Barbados, the décollement corresponds to a relatively broad zone with high smectite content, while its lower limit coincide with a minimum in smectite [START_REF] Deng | Abundance of smectite and the location of a plate-boundary fault, Barbados accretionary prism[END_REF]. Smectite is characterized by a low friction coefficient, as evidenced by shear experiments performed on clay as well as natural Nankai samples (Ashizuri transect) [START_REF] Brown | Compositional and fluid pressure controls on the state of stress on the Nankai subduction thrust: A weak plate boundary[END_REF].

For western Nankai (Muroto transect), [START_REF] Brown | Compositional and fluid pressure controls on the state of stress on the Nankai subduction thrust: A weak plate boundary[END_REF] linked smectite content and low intrinsic friction with the location of the outward-growing frontal thrust. There is indeed a maximum in smectite content in Site 1173 in the upper section of L.S.B., at depths compatible with the décollement [START_REF] Steurer | Clay mineralogy of mudstones from the Nankai Trough references sites 1173 and 1177 and frontal accretionary prism site 1174[END_REF]. Additional ring shear experiments on samples from the two sites of this study (1173,1174) reveal a wide range of friction coefficients from µ = 0.32 -0.47 in U.S.B. facies to values as low as 0.14 -0.16 in the décollement zone at site 1174 and 0.16 in the projected décollement depth at site 1173 (Kopf, unpublished data). The extension along strike of this minimum is nevertheless not completely clear, as in site 808 of the Muroto transect, the smectite content [START_REF] Underwood | Sediment geochemistry, clay mineralogy, and diagenesis: A synthesis of data from Leg 131, Nankai Trough[END_REF] as well as experimentally-determined friction profiles [START_REF] Kopf | Friction experiments on saturated sediments and their implications for the stress state of the Nankai and Barbados subduction thrusts[END_REF] are relatively smooth across the décollement.

Cementation hypothesis

Using site 1173 sediments as pristine material not yet affected by prism influence, one-dimensional consolidation tests revealed large variations in mechanical strength, possibly related to the presence of cement [START_REF] Morgan | Consolidation state and strength of underthrust sediments and evolution of the décollement at the Nankai accretionary margin: Results of uniaxial reconsolidation experiments[END_REF][START_REF] Spinelli | Diagenesis, sediment strength, and pore collapse in sediment approaching the Nankai Trough subduction zone[END_REF].

Within the shallower U.S.B, consolidation tests and XRD analysis illuminated the presence of silica cement [START_REF] Karig | Reconsolidation tests and sonic velocity measurements of clay-rich sediments from the Nankai Trough[END_REF][START_REF] Spinelli | Diagenesis, sediment strength, and pore collapse in sediment approaching the Nankai Trough subduction zone[END_REF], which is also indirectly inferred from the unusually large porosity of these sediments for their depth. Due to the increase in temperature with depth, this silica cement disappears at larger depths and samples from the uppermost section of the L.S.B. (391 and 425 mbsf), were shown to behave like uncemented sediment [START_REF] Spinelli | Diagenesis, sediment strength, and pore collapse in sediment approaching the Nankai Trough subduction zone[END_REF]. Uniaxial (vertical) consolidation tests carried out on L.S.B. sediments in Site 1173 (476 mbsf), as well as in sites 1174 [START_REF] Morgan | Consolidation state and strength of underthrust sediments and evolution of the décollement at the Nankai accretionary margin: Results of uniaxial reconsolidation experiments[END_REF] and 808 [START_REF] Karig | Reconsolidation tests and sonic velocity measurements of clay-rich sediments from the Nankai Trough[END_REF] below the décollement, yielded peak strengths much larger than vertical effective stress for hydrostatic conditions, interpreted as evidences of the presence of rock-strengthening cement of authigenic illite [START_REF] Morgan | Consolidation state and strength of underthrust sediments and evolution of the décollement at the Nankai accretionary margin: Results of uniaxial reconsolidation experiments[END_REF][START_REF] Sunderland | Microstructural variations in sediments from the toe of the Nankai accretionary prism: Results of scanning electron microscope analysis[END_REF]. This authigenic clay cement appears to disappear downward within the lower section of the L.S.B. [START_REF] Ujiie | Deformation and fluid pressure variation during initiation and evolution of the plate boundary decollement zone in the Nankai accretionary prism[END_REF][START_REF] Sunderland | Microstructural variations in sediments from the toe of the Nankai accretionary prism: Results of scanning electron microscope analysis[END_REF].

The strength profile of incoming sediment appears therefore as complex, as there are two cementing agents, silica and clays, distributed over distinct depth ranges, possibly not overlapping. Furthermore, S.E.M. analysis show evidence of clay cementation within material of the décollement in site 1174, which is absent in the deeper, underthrust sediments [START_REF] Ujiie | Deformation and fluid pressure variation during initiation and evolution of the plate boundary decollement zone in the Nankai accretionary prism[END_REF]. The connection between décollement and material strength is therefore not straightforward.

State of compaction of incoming sediments hypothesis

While the mechanical state of incoming material is a priori a major control on décollement localization, one important process is the large porosity reduction affecting the sediment as it approaches the trench (e.g. Fig. 1) (Shipboard Scientific Party, 2001a;[START_REF] Screaton | Porosity loss within the underthrust sediments of the Nankai accretionary complex: Implications for overpressures[END_REF]. The principal factor governing compaction is the progressive increase in overburden stress resulting from the deposition of a thick turbiditic trench wedge sequence. In addition, compressive horizontal stress active in convergent margins is likely to contribute also to compaction, although its amplitude and spatial distribution is poorly constrained. The lateral shortening has nevertheless large consequences on horizontal velocity and differential displacement of particles if heterogeneously distributed with depth in the incoming sediment column. Horizontal compaction constitutes therefore a potential new factor in décollement localization, whose validation requires to assess how (i) horizontal compression affects incoming material (section 3), (ii) the shortening resulting from this compression varies with depth (section 4) and (iii) velocity variations relate with horizontal compaction amplitude (section 5).

3 Change in stress conditions near the prism toe

Force balance near prism toe

In accretive margins, the material approaching the trench is eventually either accreted into the accretionary prism or underthrust beneath it (Fig. 2A). The décollement, i.e. the frontier between accreted and underthrust domains, is a mechanically weak zone, possibly because of high pore fluid pressure, so transmitted shear stresses are of low magnitude and the stress field is decoupled between accretionary prism and underthrust sequence [START_REF] Westbrook | Extensive underthrusting of undeformed sediment beneath the accretionary complex of the Lesser Antilles subduction zone[END_REF][START_REF] Taira | Sediment deformation and hydrogeology of the Nankai Trough accretionary prism: Synthesis of shipboard results of ODP Leg 131[END_REF][START_REF] Le Pichon | Accretion and erosion in subduction zones: The role of fluids[END_REF]. The state of stress within the accretionary prism is at first order described by the critical Coulomb wedge theory [START_REF] Davis | Mechanics of fold-and-thrust belts and accretionary wedges[END_REF][START_REF] Dahlen | Noncohesive Critical Coulomb Wedges -An Exact Solution[END_REF][START_REF] Dahlen | Critical taper model of fold-and-thrust belts and accretionary wedges[END_REF]. The wealth of low angle thrust faults visible on the seismic cross-sections of the outer wedge of Nankai accretionary prism [START_REF] Moore | New insights into deformation and fluid flow processes in the Nankai Trough accretionary prism: Results of Ocean Drilling Program Leg 190[END_REF][START_REF] Moore | Structural and seismic stratigraphic framework of the NanTroSEIZE Stage 1 transect[END_REF] supports the idea that the prism is, in the long-term, in a critical compressional state with the maximum compressive stress close to horizontal (volume "In" of Fig. 2A; see also results of mechanical testing of samples from ODP site 808 [START_REF] Feeser | Stress-regime-controlled yield and strength behavior of sediment from the frontal part of the Nankai accretionary prism[END_REF][START_REF] Moran | In-situ stress conditions at Nankai Trough, site 808[END_REF]), which sharply contrasts with the underthrust sequence, either free of contraction [START_REF] Taira | Sediment deformation and hydrogeology of the Nankai Trough accretionary prism: Synthesis of shipboard results of ODP Leg 131[END_REF] or even stretched in the horizontal direction [START_REF] Byrne | Evidence for a weak and overpressured decollement beneath sediment-dominated accretionary prisms[END_REF].

If the classical critical wedge solution accounts for many of prism features, it is not clear how lateral compression in the toe extends seaward (volume "Close" of Fig. 2A).

In the vicinity of the prism toe, the principal forces at stake are: (1) Dragging by the subducting lithosphere, whose velocity is constant all over its surface, (2) shear stresses on the décollement and (3) compressive forces in the wedge. Because shear stresses on the weak décollement are of limited magnitude, the forces in the wedge (in position "In", Fig. 2) must be transmitted seaward to be balanced by basal shear stresses, which implies that horizontal compression affects material seaward of the décollement tip. The second crucial point is that such compression must affect the whole sequence of incoming material (volume "Close") and not only its upper section, as it is balanced by shear stresses on its base. Lateral compression vanishes far from the prism tip at the same pace as basal shear stresses diminish, until a far-off state (volume "Far", corresponding to site 1173), where the influence of the prism is negligible.

2 Analogy with metal-cutting process

Some insights on the stress state can be gained by regarding the analogy of classical metal cutting [START_REF] Drucker | An analysis of the mechanics of metal cutting[END_REF] (Fig. 2C). Although timescale and lengthscale, as well as the mechanisms of deformation are completely different from granular materials entering a subduction zone, the geometry and the kinematic boundary conditions of the metal cutting process and frontal accretion are very similar. When considering the tool+chip=accretionary prism and metal layer=incoming sediment sequence being analogues, we find: (1) The velocity with respect to the tool is fixed as constant in the metal layer over its lateral and basal boundaries [START_REF] Woon | Investigations of tool edge radius effect in micromachining: A FEM simulation approach[END_REF]. This translates as constant convergence velocity all over the sediment sequence far off the trench and all over its base, as a result of dragging by underlying lithosphere. (2) Upon arrival on the tip (seaward end of the décollement) of the cutting tool (accretionary prism), an incoming column is split into two sections. While the velocity of the lower section is hardly affected, the upper section's velocity changes sharply as the material is incorporated into the chip (prism). In the sketches (Fig. 2B/C), the only difference lies in the fact that chip and tools remain distinct entities in metal cutting but are a single volume in prism accretion process.

From this analogy, a few essential insights on accretionary prism stress can be drawn from the abundant literature about material processing:

-Shearing and splitting of incoming material column results in the formation of a zone of high strain and stress, typically extending ahead of the tool (i.e. seaward of the décollement tip) over a distance of the order of a few times the tool edge radius. When taken into the subduction zone setting, this distance is equivalent to the thickness of accreted layer [START_REF] Drucker | An analysis of the mechanics of metal cutting[END_REF][START_REF] Kim | A finite-element analysis of machining with the tool edge considered[END_REF][START_REF] Liu | Finite element analysis of the influence of tool edge radius on size effect in orthogonal micro-cutting process[END_REF][START_REF] Wu | Modeling of flow stress in orthogonal micro-cutting process based on 56 strain gradient plasticity theory[END_REF].

-The stress field within this region includes, in addition to shear parallel to the convergence direction, a component of horizontal compression, which may reach a magnitude similar to the order of the shear stresses (cf. the amplitude of the quantity labelled as "overpressures" in [START_REF] Woon | Investigations of tool edge radius effect in micromachining: A FEM simulation approach[END_REF]).

Mechanical experiments on Nankai Trough samples

Experimental Procedure

We sampled 12 small cylinders (diameter x height = 38 x 11~17 mm) in Hole 1173A, orientated so that the cylinder base is vertical (i.e. parallel to the core axis) and the height is horizontal (Fig. 1D). These samples span an interval of 250-660 mbsf (meters below sea-floor), which includes the lower part of U.S.B. and most of the underlying L.S.B. and comprises the stratigraphic equivalent of the décollement (390-420 mbsf) (Fig. 1).

The mechanical tests consist of isotropic loading of the sample under controlled conditions of confining ( : oil) and pore pressure ( : de-ionized water) and monitoring the evolution of height using linear variable displacement transducers (LVDTs) (precision~1 micron) (Fig. 3).

The isotropic loading conditions chosen for our tests do not enable us to derive all mechanical (poroelastic and plastic) properties of the tested sample, hence we cannot estimate precisely the sample response to different, more complex loading conditions, such as the uniaxial loading (either horizontal or vertical) discussed hereafter. On the other hand, our experimental results can be considered as coarse indicators of the magnitude of the deformation to be expected, whatever the precise loading conditions, and we use them in this sense in the following. This assumption is further supported by our experimental measurements (Raimbourg et al., in press), showing that, at least in the poroelastic field, the mechanical anisotropy is very limited, so that measurements done in an arbitrary direction are to some extent representative of sample response in every direction.

As we do not monitor the volume of fluid flow in and out of the sample, we cannot precisely estimate pore volume changes and, hence, true compressibility. Nevertheless, for isotropic behavior, total volume/porosity variations can be assessed from uniaxial measurements of height variations as .

Each experiment can be divided into a first, poroelastic phase, where the effective pressure ( ) applied was kept low, and a second phase, where large effective pressures were applied to assess the plastic response of the material (Fig. 4).

For the sake of comparison between samples, every experiment lasted for two days, including the re-saturation phase. Tests were conducted at room temperature using de-ionized water. Changes in pore water composition may induce swelling of smectitic clays [START_REF] Deng | Abundance of smectite and the location of a plate-boundary fault, Barbados accretionary prism[END_REF], but as stressed by these authors, such swelling is only significantly large if smectite is the major constituent of the rock. In Hole 1173A, smectite relative abundance is generally below 30% except in the upper section of L.S.B.

where it is in the range 30-40% [START_REF] Brown | Compositional and fluid pressure controls on the state of stress on the Nankai subduction thrust: A weak plate boundary[END_REF]. We thus consider this effect negligible. Furthermore, for swelling of clay to be significant, a large portion of in-situ pore water must be exchanged with external, de-ionized fluid -something we consider unlikely given the experimental setup: In our experiments where poroelastic deformation involves only a very small amount of porosity variations and fluid exchange, the largest fluid flow corresponds to sample plastic shrinkage, but in this case only expulsion of in-situ fluids rather than fluid exchange occurs (see also [START_REF] Morgan | Consolidation state and strength of underthrust sediments and evolution of the décollement at the Nankai accretionary margin: Results of uniaxial reconsolidation experiments[END_REF]. On this basis, we suppose that the use of de-ionized water does not induce large perturbations in the mechanical behavior of the tested samples, except maybe for desorption at some clay mineral surfaces at the sample edges.

Thorough coupling between the entire sample face and the tubing is ensured by channels cut within the holder end-caps and the insertion, on both sides of the sample, of a perforated steel disk and a flexible steel mesh (noted as "spacers" on Fig. 3). Two piezometers on both sides of sample holder monitor throughout the test the evolution in P-wave velocity of the sample (Fig. 3).

Initial sample re-saturation

In order to fully re-hydrate the samples in the oil-filled high-pressure vessel prior to testing, the confining pressure and pore fluid pressure This re-saturation procedure is similar to that of [START_REF] Morgan | Consolidation state and strength of underthrust sediments and evolution of the décollement at the Nankai accretionary margin: Results of uniaxial reconsolidation experiments[END_REF], although there is a large time lapse (~10 years) between core recovery and laboratory testing in our case. Doubts concerning sample saturation and potential evaporation of pore water can be almost excluded in spite of their long stay in the IODP core repository as:

-Height variations during this saturation phase are limited to a few tens of microns, i.e.

a few per mil -Porosity measured using oven-drying to assess water content [START_REF] Blum | Physical properties handbook: a guide to the shipboard measurement of physical properties of deep-sea cores[END_REF] and grain density determined on-board by pycnometer for nearby samples is, to a few percent, the same as on-board measurements carried out after coring (Fig. 5).

1 st experimental stage

After the re-saturation phase, several steps of varying were applied within a range below . We repeated these operations for several values of confining pressure, both to check the reversibility of the deformation and to assess a potential effect of the confining pressure on poroelastic moduli. In the range from 0.5 to 1.5 -3 MPa The time necessary to reach "perfect" equilibrium is much larger than in a typical experimental time-scale, because of slow plastic deformation. Upon application of a large , we observed nevertheless a strong decrease in deformation rates, so that, over a few hours, one can consider that some equilibrium, at least with respect to initial, rapid deformation mechanisms, is reached. Similarly, for an initially large applied while the volume of fluid in the external circuit connected to the sample is kept constant, decreases exponentially towards an asymptotic value. We considered equilibrium was reached based on such asymptotic evolution, resulting in durations of at most a few hours for the largest increments in the slowest experiments. Based on the above, we defined the duration of each test to be a full two days.

Upon application of large , we also checked that the time necessary for equilibrium is much larger than the time necessary for the pressure to be transmitted through the sample. The latter was estimated by a test similar to the pulse-decay method to determine permeability [START_REF] Brace | Permeability of granite under high pressure[END_REF], i.e. by applying a differential fluid pressure between upper and lower sample surface and measuring its decrease with time.

Results and interpretation of the tests

Experimental mechanical state profile

The poroelastic compliance , determined in the horizontal direction, displays a non-linear trend of decrease with depth (Fig. 6B). A large decrease in compliance by a factor of >2, is concentrated in the depth interval 450~550 mbsf, while both 250~450 mbsf and 550~650 mbsf intervals have roughly constant properties.

The sample shrinkage for a large applied, i.e. , shows a similar trend of rapid decrease in the 450~550 mbsf interval despite some scatter (Fig. 6A): While samples from depths shallower than 450 mbsf are shortened by 2% to almost 10%, the shortening of samples deeper than 550 mbsf is less than 1.5%. This shortening includes both poroelastic and plastic deformation as is increased from to . However, due to the overall relatively low values of both compliance and , most of the sample shortening corresponds to plastic deformation. In summary, both poroelastic compliance and plastic deformation amplitude expresses a large variation in material state around 450~550 mbsf, which we hence define as the "transitional domain" (see below, and Fig. 6). (1993) and [START_REF] Morgan | Consolidation state and strength of underthrust sediments and evolution of the décollement at the Nankai accretionary margin: Results of uniaxial reconsolidation experiments[END_REF] All these experiments concern relatively deep samples with respect to the domains we consider here, as the shallowest, core 1173-51 at 476 mbsf, is already in what we defined as the transitional domain of depth, where plastic shrinkage gets limited (Fig. 6).

Comparison with previous results

Karig

Some of their samples were recovered far off the prism or below the décollement at the toe, so that the material has been variously affected by prism-related deformation and consolidation and is difficult to be compared to our data. For four samples one can nevertheless apply an approach similar to ours and consider the amount of strain for an experimentally applied vertical effective stress twice as large as in-situ value (estimated for hydrostatic pressure; see section 4.2.1 above). As a result, U.S.B. samples (DSDP cores 582-68, 582-73) with high porosity (~50%) are affected by vertical strains of 6-8% [START_REF] Karig | Reconsolidation tests and sonic velocity measurements of clay-rich sediments from the Nankai Trough[END_REF], while samples with a lower porosity, 1173-51 (~45%) and 1174-78 (~38%) by 4.7% and 4%, respectively [START_REF] Morgan | Consolidation state and strength of underthrust sediments and evolution of the décollement at the Nankai accretionary margin: Results of uniaxial reconsolidation experiments[END_REF]. The contrast in vertical strain is probably even larger, as actual (hence ) for sample 1174-78 is much lower than the value calculated assuming hydrostatic pore pressure [START_REF] Morgan | Consolidation state and strength of underthrust sediments and evolution of the décollement at the Nankai accretionary margin: Results of uniaxial reconsolidation experiments[END_REF], so that the strain (4%) is probably overestimated.

In summary, similarly to our study, the amplitude of the plastic shrinkage strongly decreases with depth and is, in deeper and less porous samples, half of its value in shallower and more porous material.

Model of differential horizontal compaction

Velocity reduction from horizontal shortening

Both the schematic stress analysis of the prism toe area as well as the analogy with metal-cutting processes show that the sedimentary column approaching the prism toe is affected by lateral compression ahead of the décollement tip. The lateral shortening enhances porosity reduction in the sedimentary column, which is already affected by the rapid loading of trench turbidites. The effect of lateral compression is nevertheless not the same as vertical loading, as in the former case the shortening is parallel to rock velocity (See Appendix). For mass conservation sake, in steady state regime, as the flux of solid material through a given surface must be constant laterally (i.e. in the direction parallel to convergence), horizontally shortened (hence denser) material at the trench has to move more slowly than in its initial position far off the trench. In other words, the change in porosity associated with lateral shortening has to be compensated by a velocity decrease.

The porosity decrease associated with compaction as the sedimentary column approaches prism toe can be defined as where is the porosity decrease corresponding to the (virtual) case where only vertical loading occurs and the additional decrease in porosity resulting from lateral compression.

The progressive reduction in velocity associated with incipient lateral compression for material moving from a position off the prism ( , , ) to the trench ( , , ) can be estimated as Where is the (negative) porosity reduction from lateral shortening, assessed at trench (i.e. for ) and in the limit .

Application of differential compaction to Nankai Trough

Despite the fact that the loading conditions in experimental and actual settings are much different, we postulate that the large differences obtained by isotropic loading are somehow significant of some "general" material behavior and are to some extent indicative of the material response to any loading path, in particular the complex, 3-dimensional loading active near prism toe.

Based on our results, the sedimentary column entering the trench can be modeled as composed of a shallow domain (above 450 mbsf), easily compactable, a deep domain This bi-layered material, when affected by vertical loading and horizontal compression, undergoes some lateral shortening scaled by its resistance to compaction: in the lower layer can be neglected with respect to the upper one. Accordingly, velocity is constant in the lower layer while it is reduced in the upper layer, resulting in differential displacement and strain distribution in the transition layer (Fig. 7).

In order to estimate this strain, we make the following assumptions:

-The thickness of the transition layer between these 2 layers is of the order of 100m Then the shear deformation accommodated by the transition layer can be estimated as

(See Appendix)
This large value shows that even a small component of "horizontal compaction", of the order of 1%, may result in large shear strain within the transition zone in the upper section of L.S.B., which in turn may attract the décollement propagation.

Discussion

Rock properties: cementation and critical porosity

Cement distribution

The complex pattern of cement distribution in Nankai incoming sediments, as highlighted by geomechanical testing and other data (see 2. This apparent paradox, where our shallow deformable domain is actually the one where cement is present, is compatible with the anisotropy of magnetic susceptibility (AMS) profile (Fig. 8) that was related to bulk strain [START_REF] Borradaile | Tectonic applications of magnetic susceptibility and its anisotropy[END_REF]. The vertical compaction fabrics, expressed by the magnetic parameter F [START_REF] Hrouda | Magnetic anisotropy of rocks and its application in geology and geophysics[END_REF], low throughout the U.S.B., is higher in the uppermost L.S.B., decreases down to 400 mbsf then starts increasing again. This reversal in compaction trends around 400 mbsf, may somehow correspond to the lower limit of the cement zone within the L.S.B., with deeper sediments much more affected by vertical compaction than shallower, more resistant ones. Accordingly, the capacity to shrink in response to horizontal compression when approaching the prism toe is much reduced in the uncemented, lower section of the L.S.B. compared to shallower sediments. This inevitably results in the differential strain patterns observed in the mechanical transition zone at the top of the L.S.B. -a fact that very likely contributes to décollement localization in the incoming sequence.

Critical porosity and time-scale of compaction

The porosity corresponding to the transition in mechanical behavior spans a porosity range of 42-46% (See Fig. 1 and Raimbourg et al. (in press)), i.e. close to the "critical porosity" where physical properties strongly vary and close-packing of grains prevails [START_REF] Nur | Critical porosity: A key to relating physical properties to porosity in rocks[END_REF], although this value of the critical porosity is defined for sandstones and may not apply directly to the clay-rich material considered here.

The compaction processes for a porosity exceeding close-packing structure require only the displacement and rearrangement of individual grains, i.e. very little time. In contrast, compaction and further porosity decrease for a material already closely packed is not achieved by grain rearrangement but requires processes such as pressure dissolution and precipitation. The time-scale of compaction is, therefore, much longer for chemical processes to operate and the porosity loss for a given stress increment is much lower.

Distribution of compaction near prism toe

Evidences for horizontal compaction

On the basis of the correlation of logging-while-drilling (LWD) signals between adjacent sites, [START_REF] Hunze | Compaction in the Nankai and Barbados accretionary prisms: New insights from logging-while-drilling data[END_REF] were able to assess the porosity variations, for a given sedimentary sequence, from a reference site far from the trench to positions within the prism. As a result, the depth-averaged amplitude of the total (vertical and lateral) porosity decrease is in the range of 10% in the Barbados prism (comparison of Sites 1044 and 1045) and 25% and 15% for two distinct depth sections in Nankai prism (comparison of Sites 1173 and 808). Moreover, discriminating within this reduction the relative contribution of vertical/horizontal compaction, [START_REF] Hunze | Compaction in the Nankai and Barbados accretionary prisms: New insights from logging-while-drilling data[END_REF]'s results show that porosity loss due to lateral shortening is as large as 5-10%, which represents a major contribution to total loss and is found in agreement with [START_REF] Saito | Compaction and dewatering processes of the oceanic sediments in the Costa Rica and Barbados subduction zones: estimates from in situ physical property measurements[END_REF].

Similarly, quantitative rock fabric analysis, either based on anisotropy of magnetic susceptibility (AMS) analysis [START_REF] Owens | Magnetic fabric studies of samples from hole 808C, Nankai Trough[END_REF][START_REF] Housen | Strain decoupling across the decollement of the Barbados accretionary prism[END_REF][START_REF] Ujiie | Deformation and fluid pressure variation during initiation and evolution of the plate boundary decollement zone in the Nankai accretionary prism[END_REF], X-ray goniometry [START_REF] Behrmann | Textures and microfabrics in fine-grained muds and mudstones from site 808, Nankai accretionary prism[END_REF][START_REF] Morgan | Decollement processes at the Nankai accretionary margin, southeast Japan: Propagation, deformation and dewatering[END_REF] or anisotropy of P-wave velocity analysis [START_REF] Brückmann | Acoustic anisotropy and microfabric development in accreted sediments from the Nankai Trough[END_REF], attest the presence of a component of lateral shortening parallel to the convergence velocity within the accretionary prism (i.e. in the material located above the décollement).

Although such results demonstrate the presence of horizontal shortening within the prism, they do not provide information as to how the lateral compressional field may extend seaward of the décollement tip. Few studies have attempted to examine this question. Along the Ashizuri transect in Nankai, seaward of the deformation front, there is a large porosity decrease (from ~60% to ~50%) within the U.S.B. sediments between ODP hole 1177 (Shipboard Scientific Party, 2001a), ~15 km from deformation front and DSDP site 582 [START_REF] Bray | Physical properties of sediments from the Nankai Trough, Deep Sea Drilling Project, Leg 87A, Sites 582 and 583[END_REF], ~3 km from deformation front. Although the larger part of this porosity decrease is probably the result of loading by overlying trench turbidites (~100m-thick in 1177 and ~550m-thick in site 582), there may be a contribution of tectonic shortening, but it has not yet been estimated. Unfortunately, in their kinematic model, [START_REF] Morgan | The estimation of diffuse strains in the toe of the western Nankai accretionary prism: A kinematic solution[END_REF] also assume that in the domain seaward of the deformation front only vertical compaction operates, preventing the quantification of horizontal shortening there.

The other example where two adjacent sites were drilled outboard of the décollement tip is Barbados [START_REF] Moore | Synthesis of Results: Logging While Drilling, Northern Barbados Accretionary Prism[END_REF]: Here, the porosity profiles inferred from logging data (Shipboard Scientific Party, 1998) show that site 1048, located ~3 km from the frontal thrust (assumed to roughly correspond to décollement tip), has not yet been affected by compaction with respect to the reference site 1044, located ~6.5 km from tip. From this single example, it should be assumed that the horizontal compression related to the accretionary prism does not extend as far as 3 km from décollement tip, which is in agreement with the comparison with metal-cutting processes, where the width of the high-strain zone is shown to be of the order of a few times the off-scraped thickness, roughly 200m in Barbados case.

The above examples do not provide unambiguous evidence with regard the assumption of the model. An additional difficulty is the amplitude of the porosity variations: The heterogeneous depth distribution, starting 1 km seaward of the décollement tip, of only 1% porosity decrease as a result of lateral shortening triggers large differential displacement and strain (See 4.2.2). However, this small amount of compaction is well within the range of scatter of porosity profiles determined either through core sample measurement or from P-wave velocity-porosity relationships and is therefore hardly detectable.

Origin of horizontal compaction in the wedge

As a corollary to its central assumption, our model provides an alternative explanation defined as easily deformable is unable to sustain a large increase in without compacting, so that an increase in the bulk load (i.e. in ) is instantaneously entirely converted into an increase in . On the other hand, the deep, hardly deformable material of the lower section of L.S.B. can stand a large increase in without compacting, which implies that an increase in the bulk load is only partially transmitted to the pore fluid, resulting in a much smaller instantaneous pore pressure build-up. This contrasted compaction behavior has presumably a large impact on the evolution of the pore pressure profile during progressive loading, either vertical or horizontal. We consider in the following only vertical, sedimentary loading, but the conclusions would be qualitatively the same when taking horizontal compaction also into account.

Le Pichon and [START_REF] Le Pichon | Erosion and accretion along subduction zones: a model of evolution[END_REF] and Le [START_REF] Le Pichon | Accretion and erosion in subduction zones: The role of fluids[END_REF] interpreted the décollement localization within incoming hemipelagic sediments as a consequence of their rapid loading by more permeable trench turbidites, which results in the development of a high pore pressure zone/low effective stress zone in a horizon below the limit between trench and hemipelagic material.

Permeability profile is an essential feature of such model, as turbidites are required to be much more permeable than underlying hemipelagites, so that the pore pressure gradient is lower (Fig. 9). Another parameter of prime importance is how the increase in the sedimentary load (i.e.

) is transmitted to the pore fluid and converted into increase. The simplest approach, adopted by Le Pichon and [START_REF] Le Pichon | Erosion and accretion along subduction zones: a model of evolution[END_REF] and Le [START_REF] Le Pichon | Accretion and erosion in subduction zones: The role of fluids[END_REF] is to consider that the timescale of compaction is much smaller than timescales of (i) loading by sedimentation and (ii) fluid flow. In such a case, adjustment of the material porosity to the variations in the load as well as in the pore fluid pressure is instantaneous. Throughout its evolution towards the accretionary prism, all the points of the sedimentary column plot on a compaction curve depending on the material. When assuming a rapid loading with respect to fluid outflow, the increase in affecting a given rock volume in the hemipelagite section is first entirely transmitted to the pore fluid (constant and , increase in ) before outflow of fluid enables to increase and and to decrease (Fig. 9B).

Building from their model on, let's assume that the behavior of the low-permeability hemipelagite section with respect to compaction is not homogeneous and that it is constituted of an easily deformable upper section and a hardly deformable lower section.

When this column is submitted to a load increase resulting from rapid turbidite deposition, within its shallower portion all the load increase is transmitted to the pore fluid (hence of the order of ). In contrast, in the deeper portion of the succession, a significant fraction of is supported by the framework without being transmitted to the pore fluid (Fig. 9C).

As a result, within the low permeability hemipelagite, the compaction behavior governs the distribution of the pore pressure and the depth portion where is lowest is localized where the material is more easily compactable. This effect is even amplified within the U.S.B. part of the deformable section, as the anomalously large original porosities observed at site 1173 ((Shipboard Scientific Party, 2001a), Fig. 1C), probably related to silica cementation [START_REF] Spinelli | Diagenesis, sediment strength, and pore collapse in sediment approaching the Nankai Trough subduction zone[END_REF], provide a long-lasting source of pore fluid pressure before on the compaction curve once the cement got destructed by loading.

In conclusion, the difference in mechanical response to loading within the incoming

Conclusion

As Nankai accretionary prism is not a perfect wedge, closed by an acute angle, the horizontal compressive stresses active there must to some extent be transmitted seaward of the decollement tip and affect diffusely the whole incoming sedimentary sequence.

As a consequence, the compaction affecting this material incorporates a part of horizontal shortening.

The response of samples collected at different depths within the incoming sequence to experimentally-imposed compaction show very contrasted behavior between shallower, easily compactable material and deeper, hardly compactable material. As a result, the amplitude of the horizontal shortening, as the sequence approaches the prism toe and is increasingly affected by horizontal compression, varies with depth. This differential horizontal shortening leads to differential displacement, hence strain, which localizes in the transitionary domain between easily and hardly deformable domains. This strain concentration eventually guides the propagation of the decollement within incoming material.

Appendix

The increase in stress results in compaction and porosity decrease in the material affected. To assess the effect of vertical and horizontal compaction, let's consider alternatively the simple cases where only vertical load or horizontal tectonic stresses increase.

Vertical load increase:

The vertical load increase triggers some porosity decrease that depends on the mechanical state of the material. As the material is laterally confined, for the sake of solid mass conservation, the porosity decrease is completely compensated by a decrease in thickness of the column without effect on the horizontal velocity (Fig. 10A).

Horizontal stress increase

Horizontal stress increase in the convergence direction also triggers a porosity decrease that depends on the mechanical state of the material (Fig. 10B). On the contrary to vertical compaction, the material is not confined in the vertical direction, so that horizontal compression is likely to induce some vertical expansion. Nevertheless, its extent is unknown, so as a minimum hypothesis we consider that the height of the sedimentary column is constant (vertical thickening of the column would result in an even larger velocity decrease than calculated hereafter).

In such case, the conservation of solid material requires that:

where is the horizontal velocity.

For the sake of simplicity, we suppose that the porosity decrease (negative) is linear over a length ( ):

If the variations in porosity are very low ( ), then mass conservation equations simplifies as:

yielding, for the horizontal velocity decrease:

which can be approximated by for .

In summary, the decrease in porosity associated with horizontal compaction is accommodated by a decrease in horizontal velocity.

Double-layer horizontal compaction

Now consider a sedimentary column constituted of two superposed layers, the upper one subjected to large plastic shrinkage when compressed, the lower one hardly deformable, and a transition layer in between (Fig. 10C). If this bi-material is subjected to horizontal compression, the upper material shrinks but the lower one does not, resulting in difference in horizontal velocity: while the lower one keeps the same velocity over , the upper material is progressively slowed down according to the law above.

The time necessary for the material to reach is In this case, the material compaction is not instantaneous (as supposed in B): Upon rapid loading by the turbidites at , although, like in (B), porosity adjustement is strongly inhibited by the low permeability, the transmission of the load from the framework to the pore fluid is variable with depth. In the shallow and easily deformable domain of the hemipelagite section, all the load increase is instantaneously transmitted to the fluid (long grey arrows) and the pore fluid pressure increases strongly, while in the deep, hardly deformable domain the solid framework is able to sustain some increase in the load without instantaneously compacting, which results in a much smaller transmission of the load increase to the pore fluid (small grey arrows) and lower increase in pore fluid pressure. The time-evolution of the pore pressure profile is in this case the result of the competition between water flow (controlled by permeability profile) and time-dependent compaction (controlled by the mechanical properties of the solid framework), but for slow compaction in the deeper domain of the hemipelagites, the lowest effective pressure is localized in the more deformable upper domain.

  set to a value well below the in-situ effective pressure, estimated as , where is the vertical stress of the overlying rock column (calculated using the bulk density profiles from shipboard moisture and density measurements (Shipboard Scientific Party, 2001b)), z its thickness and the fluid density. The samples are exposed to these conditions for 12h prior to the mechanical test.

(

  the upper limit of the range is dictated by in situ conditions for shallow samples), sample height variations are almost linear, enabling us to derive the poroelastic compliance: 4.1.3 2 nd experimental stage During the second phase, was fixed to in situ conditions ( ) and we applied cycles of first increasing in amplitude to a value , then of decreasing amplitude back to initial conditions. This procedure of changing through variations differs from typical experiments (e.g. Morgan and Ask (2004)) where changes in are controlled through variations for a fixed . Its justification lies in the concomitant estimation of P-wave velocity using the piezometers shown in Fig. 3, for conditions as close as possible to in-situ ones (See Raimbourg et al. (in press).In some of the samples, plastic yield, characterized by a large and rapid shrinkage, was observed for in the range . To estimate this potential plastic deformation, we assessed the extent of the height reduction as for each sample, where is the initial height and is the height variation between and

  .

  carefully carried out uniaxial consolidation tests of core samples from prism toe below the décollement (samples from ODP cores 808-74, 808-85, 1174-78), off the deformation front below the turbiditic deposits (DSDP cores 582-68, 582-73) and further off the prism (ODP core 1173-51). These tests enabled to measure and plot the evolution of differential stress with increase in mean effective stress (Morgan and Ask, 2004), highlighting the complexity of the compaction, i.e. post-yield behavior, decomposed into several distinct phases.

(

  below 550 mbsf) more resistant to compaction, and a relatively thin transitional domain around 450~550 mbsf, in the upper section of the L.S.B. unit, where mechanical properties change sharply.

  3, Spinelli et al. (2007) and Raimbourg et al. (in press)), appears at first sight at variance with our results: the porous material of U.S.B. where we observed the most intense compaction, show evidences for silica cementation. Such contrasting behavior may be seen in the light of the exceeding we applied, which may have triggered breakage of silica intergranular bonds and, subsequently, favored porosity collapse. Broken bonds and relative low porosity have already been described in décollement rocks of Site 1174 (Ujiie et al., 2003). In that respect, rather than showing variations in strength, the plastic deformation we measured for high may indicate that the sample initial condition is far from normal consolidation state, most likely owing to the presence of cement that maintains the porosity at anomalously large value. Based on this hypothesis, shallow samples with large plastic shrinkage have large initial porosities that result from the presence of cement. The heterogeneous distribution of cement may also explain the large scatter in experimental compaction of these shallow, deformable samples. On the contrary, samples recovered at larger depth, where cement is absent or rare, have a normal porosity with respect to their depth, so a much lower capacity to shrink once large effective stresses are applied.

  sediment sequence has large consequences on the evolution in the profile as turbidites rapidly deposit on top of it. In the framework of the model developed by Le Pichon and Henry (1992) and Le Pichon et al. (1993), we propose that the minimum in , developed in response to rapid overlying load increase, should lie within the highly porous and compactable U.S.B. material, therefore slightly offset with respect to actual décollement localization, which would imply that this phenomenon is not the prime control on outward migration of the frontal thrust.

FiguresFigure 1 :

 1 Figures

  Figure 1: (A) Map (Moore et al., 2005) and (B) interpreted cross-section of the Nankai

Figure 2 :

 2 Figure 2: (A) Sketch of the forces applied on the mechanical discontinuities of the

Figure 3 :

 3 Figure 3: Experimental device of isotropic loading, where we measure variations in

Figure 4 :

 4 Figure 4: Typical evolutions of the sample height during one experiment, from to

Figure 5 :

 5 Figure 5: Comparison of on-board porosity determined just after sample recovery and

Figure 6 :

 6 Figure 6: Mechanical properties of sediments from site 1173 determined in the

Figure 7 :

 7 Figure 7: Model of décollement initiation based on the compaction of pristine,

Figure 8 :

 8 Figure 8: Evolution with depth of magnetic foliation parameter F in Site 1173. F,

Figure 9 :

 9 Figure 9: Evolution of (hence ) within a sedimentary column of hemipelagic

Figure 10 :

 10 Figure 10:

to the strong contrast in fabrics between material from the prism and the underthrust section (e.g. [START_REF] Westbrook | Extensive underthrusting of undeformed sediment beneath the accretionary complex of the Lesser Antilles subduction zone[END_REF], [START_REF] Taira | Sediment deformation and hydrogeology of the Nankai Trough accretionary prism: Synthesis of shipboard results of ODP Leg 131[END_REF] and Le [START_REF] Le Pichon | Accretion and erosion in subduction zones: The role of fluids[END_REF]):

Such fabrics are usually interpreted in terms of stress decoupling by the décollement, i.e. horizontal strain is only recorded above the décollement because only in this domain the horizontal tectonic compression is large. In this interpretation the strain patterns are the consequence of the décollement. As an alternative explanation we propose that these strain patterns may be partially inherited from the compaction of the sediments off the trench, where the décollement is not yet developed. In such a scenario, the décollement is the consequence rather than the cause of such differences in deformation patterns with depth.

Pore pressure variations related to differential compaction and décollement localization

Excess pore pressure, i.e. pore pressure in excess of fluid hydrostatic pressure, is the driving force for compaction. Its amplitude is the result of the competition between sedimentary/tectonic loading (source of ) and outflow of fluid (sink of ), so that large excess pore pressure are favored by rapid loading and hindered flow of fluids (Shi andWang, 1985, 1988). Pore pressure build-up in response to an increase in loading is controlled by the deformation of the skeleton of the rock/sediment: The material that we horizontal compaction, the sediment column height is increased, then the velocity decrease is even larger than in the "minimum" hypothesis chosen here of a column of constant height. (C) If the incoming sediment column is composed of a compactable upper layer, non-compactable lower layer and a transition domain, then the horizontal compaction results in differences in horizontal velocity between lower and upper layers, which lead to shear strain in the transition layer. For (a), (b) and (c), the yellow rectangle corresponds to the initial shape of the column as it enters the zone of compaction, which gets transformed into the red surface after undergoing compaction.