Kevin Schmidmayer
email: kevin.schmidmayer@inria.fr

Luc Biasiori-Poulanges

On bubble cloud growth in shock-droplet interaction

Keywords: compressible, cavitation, droplet, multiphase flow, acoustics

Investigations of shock-induced cavitation is highly challenged by the multiphase nature of the mechanisms involved. Thermodynamically well-posed multiphase numerical models accouting for phase compression and expansion however allow to elucidate the underlying physics. A description of the bubble cloud growth and its effects on the early droplet dynamics is proposed, as well as a critical discussion on the analytical predictions of the cavitation event previously reported.

Introduction

Shock-induced droplet aerobreakup occurs in a rich variety of technical applications such as rain erosion damage, bulk dissemination of liquid agents or liquid jet atomization. Over the last decades, the aerobreakup phenomenology has been extensively addressed assuming that the incident shock does not affect the fragmentation processes. However, research efforts have recently shown that internal shock reflections inside the droplet might result in cavitation bubbles. Previous attempts to explicit the conditions for the bubble to grow, and possible consequences on the droplet dynamics, failed because of inappropriate methods. Recently, Kyriazis et al. (2018) demonstrated that thermodynamically wellposed models incorporating phase change (although not primary) are suitable for simulating the growth of bubbles, thus standing out as a promising method to elucidate the physics of shock-induced cavitation inside a water droplet. Based on numerical simulations and ray theory, we herein address the bubble cloud growth and its effect on the early stage of the droplet dynamics.

Numerical modelling and wave analysis

A thermodynamically well-posed, pressure-non-equilibrium, multi-component flow model [START_REF] Saurel | Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixtures[END_REF]) that conserves mass, momentum and total energy is used. The effects of viscosity and surface tension are insignificant on the observation time scale. The model requires pressure relaxation to recover a unique equilibrium pressure, achieved by an infinite-relaxation procedure. A second-order-accurate scheme (MUSCL) is used to solve the model and is combined with an adaptive mesh refinement technique to resolve the wide range of spatial and temporal scales of shock fronts and interfaces. This modelling is implemented in ECO-GEN [START_REF] Schmidmayer | ECOGEN: An open-source tool for multiphase, compressible, multiphysics flows[END_REF], which has been validated for several gas bubble dynamics problems, including free-space and near-wall bubble collapses, and other multicomponent problems such as liquid-gas shock tubes and water column and droplet breakup due to high-speed flow (see [START_REF] Dorschner | On the formation and recurrent shedding of ligaments in droplet aerobreakup[END_REF] and references herein). The computational setup is shown in Fig. 1(a). The shock wave travels at Mach 2.4 in air at atmospheric conditions, and interacts with a two-dimensional water column. The initial air volume fraction in water is 10 -6 . This corresponds to the preexisting nuclei in non-purified water. Considering the difference in the acoustic impedance between both phases, the modelling enables to simulate each phase response, within the mixture, to compression and expansion effects, i.e. cavitation without phase change.

Numerical simulations are complemented with the analytical analysis of the wavefronts propagating within the droplet. These fronts are shaped by the internal reflections, which can be interpreted by applying the classical ray-tracing method to geometrical acoustics. The incident shock is modelled as an incoming parallel bundle of rays, which is transmitted to the droplet before experiencing internal reflections that obeys the fundamental law of refraction. Within the context of shockdroplet interaction and in the absence of bubble growth, the parametric equations for the transmitted wavefront and its n-reflections have been derived by Biasiori-Poulanges & El-Rabii (2020), and thoroughly validated against ECO-GEN. Mainly based on direct comparisons of the numerical schlieren and the theoretical wavefronts, the present work illuminates the effect of bubble growth on the early stages of the shock-droplet interaction. very good agreement between theory and numerics up to the growth of the bubble cloud that annihilates the downstream segment of F 2 , as clearly seen from t = 9.58 µs.

Results and Discussion

Not considering phase change or wave interplay, previous works on shock-induced cavitation inside confined fluid volume suggests that the location of strongest shock-induced cavitation is at C [START_REF] Obreschkow | Confined shocks inside isolated liquid volumes: A new path of erosion?[END_REF]. In this context, it is instructive to compare the numerical results with experimental observations, under the same initial conditions [Fig. 1(c)]. The upper-half image plots the numerical time history of the bubble cloud collapse, which occurs at the yellow cross. The lower-half shows the shadowgraph of the bubble cloud from [START_REF] Sembian | Plane shock wave interaction with a cylindrical water column[END_REF] experiments. The origin of the cloud and the collapse location on the x-axis is estimated to be at the pink cross. The solid lines are in good agreement with the numerics, while the dashed lines do not agree with the numerical schlieren.

Figure 1(c) illuminates the significant error between the experiments and the predicted cloud location at C, while a significantly better match is reported with the numerics. In addition, it demonstrates that not accounting for the wave in-terplay and the bubble cloud cavitation leads to unfortunate descriptions of the wave system and its role in the shockinduced cavitation event.

Conclusion

This work illuminates the effect of the bubble cloud growth on the early stages of the shock-droplet interaction. Our numerical results reveals that the cloud modifies, and ultimately annihilates, the wave development as predicted by geometrical acoustics. It also shows that the analytical prediction of the cavitation location is incorrect, while our simulations are in good agreement with the experimental observations. However, these results are limited by the infinite relaxation rate, i.e. instantaneous phase equilibrium, which results in the sudden growth of the cloud. Consequently, its size is overestimated and much larger than the experimental observations. Future research will address this limit by assuming a finite relaxation rate, calibrated against experiments.

Figure 1 :

 1 Figure 1: (a) Computational setup. Domains 1 and 2 refer to post-shock and pre-shock states, respectively. (b) Comparison of (top) numerical results with (bottom) theory and (c) experiments of Sembian et al. (2016). Times on the wavefronts are indicated in µs. Time steps are (b) 1.5 µs and (c) 6.0 µs. Note that only the transmitted and the once-reflected wavefronts are plotted.

Figure 1

 1 Figure1(b) shows the comparison between numerical schlieren (top) and the wavefronts predicted by the ray tracing approach (bottom). The light gray lines corresponds to the transmitted wavefront F 1 . When meeting the droplet boundary, F 1 experiences a first internal reflection resulting in the development of the singly-reflected wavefront F 2 (dark lines). F 2 consists of two segments and exhibits a singular point S. The reflection of F 1 is completed when it reaches the downstream droplet surface [see dark lines on Fig.1(c)]. As time proceeds, S traces out a surface, the so-called caustic, shown in red on Figs.1(b)-(c). The caustic has a cusp singularity at C, where S vanishes. Figure1(b) reports a very good agreement between theory and numerics up to the growth of the bubble cloud that annihilates the downstream segment of F 2 , as clearly seen from t = 9.58 µs.Not considering phase change or wave interplay, previous works on shock-induced cavitation inside confined fluid volume suggests that the location of strongest shock-induced cavitation is at C[START_REF] Obreschkow | Confined shocks inside isolated liquid volumes: A new path of erosion?[END_REF]. In this context, it is instructive to compare the numerical results with experimental observations, under the same initial conditions [Fig.1(c)]. The upper-half image plots the numerical time history of the bubble cloud collapse, which occurs at the yellow cross. The lower-half shows the shadowgraph of the bubble cloud from[START_REF] Sembian | Plane shock wave interaction with a cylindrical water column[END_REF] experiments. The origin of the cloud and the collapse location on the x-axis is estimated to be at the pink cross. The solid lines are in good agreement with the numerics, while the dashed lines do not agree with the numerical schlieren.Figure1(c) illuminates the significant error between the experiments and the predicted cloud location at C, while a significantly better match is reported with the numerics. In addition, it demonstrates that not accounting for the wave in-

Acknowledgments

Authors acknowledge the financial support from the ETH Zurich Postdoctoral Fellowship program.


Petr Havlásek
email: petr.havlasek@cvut.cz

Václav Nežerka
email: vaclav.nezerka@fsv.cvut.cz

Petr Bittnar
email: petr.bittnar@fsv.cvut.cz

Pavel Padevět
email: pavel.padevet@fsv.cvut.cz

Milan Jirásek
email: milan.jirasek@fsv.cvut.cz

Zdeněk Bittnar
email: bittnar@fsv.cvut.cz

Efficient Approach to Measuring Strength and Deformation of Passively Confined Concrete

Keywords: Confined concrete, Experiment, Passive confinement, Digital image correlation (DIC)

This paper presents a novel experimental method developed to determine the increase in concrete strength and the evolution of both axial and lateral deformation under confined compression. In contrast to expensive and time-consuming triaxial experiments, the present method requires only a uniaxial testing machine with displacement control. In the proposed setup, a concrete cylinder is surrounded by metal rings, which are activated by the lateral expansion of concrete and provide passive confinement. The magnitude of confinement can be easily controlled by the dimensions and spacing of the rings. In contrast to triaxial tests, the stress history is much closer to that encountered in real structural members. The most suitable measuring setup is a combination of conventional and non-contact measurements, namely of the vertical extensometers for the axial strain and digital image correlation (DIC) for the hoop strain. The results indicate that the lateral deformation needs to be averaged over a representative volume as its distribution over the height of the specimen becomes highly nonuniform in the postpeak. The proposed setup was successfully verified on 20 concrete cylinders with 4 different ring layouts. The measured strength increase Strength and Deformation of Passively Confined Concrete due to confinement is consistent with models from the literature and thus with triaxial tests that were used for their development and calibration. The experimental scatter decreases with increasing magnitude of confinement and decreasing spacing of the rings.

1 Introduction

Motivation

Cement production is responsible for approximately 8% of global anthropogenic CO 2 emissions [START_REF]Editorial Board: Concrete needs to lose its colossal carbon footprint[END_REF]. The urge to limit these emissions led to the adoption of several mitigation strategies [START_REF] Scrivener | Eco-efficient cements: Potential economically viable solutions for a low-CO2 cement-based materials industry[END_REF]: i) partial replacement of clinker in Portland cement, ii) utilization of recycled concrete, and iii) adoption of more advanced methods in the design of concrete structures. All of these methods require a thorough description of the material's mechanical properties. Moreover, the increasing volume of concrete production demands a higher consumption of quality natural aggregates, which are becoming scarce. Incorporating lowergrade or recycled aggregates as well as Portland cement substitutes directly results in greater variability of the mechanical properties of concrete.

To avoid inefficiently oversized structures designed to meet structural safety, and in the spirit of performance-based design, it is essential to determine the governing mechanical characteristics of concrete more accurately using readily available experiments. The resulting experimental data can provide a basis for the calibration of advanced constitutive models for concrete to fully utilize its structural potential. Such an approach can lead to more efficient utilization of the material without sacrificing structural safety and, in turn, to substantial savings in the consumption of raw materials and reduction in CO 2 emissions.

Advanced constitutive models for concrete, such as continuum damageplasticity models [START_REF] Grassl | Damage-plastic model for concrete failure[END_REF][START_REF] Grassl | CDPM2: A damage-plasticity approach to modelling the failure of concrete[END_REF], microplane models [5], or discrete lattice models [START_REF] Cusatis | Lattice discrete particle model (LDPM) for failure behavior of concrete[END_REF], can accurately reproduce the response of compressed concrete under lateral confinement, which can be provided by properly designed transverse reinforcement. Such confinement enhances both the strength and the ductility of concrete [START_REF] Erwin Richart | Failure of plain and spirally reinforced concrete in compression[END_REF][START_REF] Imran | Experimental study of plain concrete under triaxial stress[END_REF][START_REF] Harries | Experimental investigation of the behavior of variably confined concrete[END_REF][START_REF] Berthet | Compressive behavior of concrete externally confined by composite jackets. Part A: experimental study[END_REF][START_REF] Li | Stress-strain behavior of actively and passively confined concrete under cyclic axial load[END_REF][START_REF] Wei | Compression behavior of concrete columns confined by high strength steel wire[END_REF][START_REF] Yin | Interlocking spiral confinement for rectangular columns[END_REF], thereby substantially increasing the load bearing capacity without increasing concrete consumption. To provide essential data for calibration, a novel experimental setup which allows conducting compressive tests under confinement has been recently developed at CTU in Prague. As opposed to conventional tests relying on results from triaxial testing, the new setup should be more accessible in engineering practice.

State of the art in confined concrete experiments

The experimental setups for testing the response of confined concrete can be divided into two basic groups. Active confinement refers to the case where the magnitude of lateral confinement can be prescribed and controlled independently of the deformation of the tested specimen. A prerequisite for this type of test is the triaxial machine (Hoek cell [START_REF] Hoek | A simple triaxial cell for field and laboratory testing of rock[END_REF]), primarily developed for testing rocks. The target magnitude of confinement is typically imposed already before dominant loading and is maintained throughout the experiment at a constant level [START_REF] Li | Stress-strain behavior of actively and passively confined concrete under cyclic axial load[END_REF]. There are some exceptions to this rule, e.g., in [START_REF] Imran | Experimental study of plain concrete under triaxial stress[END_REF], where conventional triaxial tests are supplemented with a multistep and cyclic history of confinement. The range of prescribed confinement is very wide and depends on the testing apparatus; it covers the magnitudes common for concrete columns reinforced with spirals (several MPa) as well as for concrete filled steel tubes (possibly tens of MPa).

The need for special experimental equipment is not the only drawback of the active confinement approach. The cell size predetermines the size of concrete cylinders (typically 54 × 108 mm), which might not be sufficient to represent concrete with larger aggregates. Moreover, the testing procedure is laborious and time-consuming, making the experiment expensive and also hindering the testing of a large number of specimens at a specific age. In addition, specimen instrumentation is not trivial and inexpensive. The axial strain can be measured with resistance strain gauges oriented vertically at midheight and regularly spaced along the circumference to determine the average strain. An externally mounted LVDT inherently adds spurious compliance of the supports and leads to undesired averaging of strain over the entire height of the specimen. Both axial and lateral strain gauges often need to withstand strains exceeding 30 × 10 -3 in magnitude. Furthermore, the length of the hoop strain gauge should encompass the entire circumference of the cylinder because a shorter measurement distance can provide local and misleading information. As the view of the specimen is blocked during loading in the triaxial machine, the formation and propagation of cracks remain obscured, which hinders the classification of failure modes. One must not forget that concrete is a path-sensitive material whose response depends on the entire loading history. Therefore, the response determined under active confinement may differ from the behavior of a real structural member even when the magnitude of confinement is similar [START_REF] Havlásek | Numerical modeling of axially compressed circular concrete columns[END_REF].

The other setup, passive confinement, is based on restraining the lateral expansion of loaded specimens, which results from elastic (Poisson effect) and inelastic (plastic deformation and microcracking) processes. The confinement is never as uniform over the specimen height as in the preceding case. Friction in the supports hinders lateral deformation, which in turn leads to a nonuniform distribution of confining-induced stress over the specimen height. The stiffness of the restraint determines the evolution of the confinement and the material strength determines its maximum attainable magnitude. The shape of the specimen cross-section has a strong impact on the distribution of the confinement about the circumference.

In this study, attention is limited to cylinders, for which the lateral confinement (about their circumference) can be idealized as uniform and which can provide a direct insight into the material behavior rather than an understanding of the confinement efficiency, such as in the case with rectangular cross-sections and different corner radii [START_REF] Ceccato | Simulation of concrete failure and fiber reinforced polymer fracture in confined columns with different cross sectional shape[END_REF][START_REF] Wang | Effect of corner radius on the performance of cfrp-confined square concrete columns: Test[END_REF]. In contrast to active confinement, the requirements on specimen dimensions are not so strict; the maximum cross-section area is limited by the confined strength and the capacity of the testing machine.

Metallic or composite materials are commonly used to produce a restraint to free lateral expansion of concrete and thereby to generate confinement. The advantages of metallic materials are the consistency of their properties and the elasto-plastic behavior. Additionally, metallic tubes can serve as a mold for concrete, which simplifies the specimen preparation. However, the stress state that develops in the tube wall upon loading is chiefly biaxial. In the axial direction of the specimen, the tube is subjected to compression, which can substantially contribute to the overall measured axial force, while in the circumferential direction the tube is in tension. The ratio between these stresses with similar magnitudes gradually evolves during loading. This biaxial stress state also implies earlier yielding, especially since it combines tension with compression. As a result, the value of the hoop stress and the corresponding confining stress are not easy to determine. Since the objective is not to measure the performance of a concrete-steel composite, but rather to provide data for the identification of concrete behavior, continuous metallic tubes are not suitable for the given purpose.

The problem associated with undesired vertical restraint can be eliminated by using spiral reinforcement [START_REF] Kim | The behaviour of concrete columns with interlocking spirals[END_REF][START_REF] Wei | Behaviour of concrete confined by both steel spirals and fiber-reinforced polymer under axial load[END_REF], which does not require longitudinal reinforcement or spacers as opposed to circular (welded) rings. However, increasing spiral pitch results in a non-uniform distribution of confinement over the height of the specimen and it is more difficult to attain the target magnitude of confinement as the rebars are produced at a discrete scale and changing the spiral pitch disrupts the confinement efficiency factor.

The behavior of composite materials (carbon/glass fiber reinforced polymers, CFRP/GFRP [START_REF] Harries | Experimental investigation of the behavior of variably confined concrete[END_REF][START_REF] Berthet | Compressive behavior of concrete externally confined by composite jackets. Part A: experimental study[END_REF][START_REF] Cao | Study on the active confinement of gfrp-confined expansive concrete under axial compression[END_REF]) in tension is elastic-brittle, which implies that the lateral confinement grows linearly with lateral expansion until the surrounding shell ruptures, typically at maximum load. The history of stress in concrete is different from typical reinforced concrete structures, but it is essential to understand the behavior of concrete in retrofitted structures. In contrast to metallic reinforcement, the properties of FRP are highly variable [START_REF] Bisby | Strain localisations in FRP-confined concrete: new insights[END_REF] and can significantly differ from the manufacturer's specifications [START_REF] Harries | Experimental investigation of the behavior of variably confined concrete[END_REF]. The specimens can be cast in standard size molds [START_REF] Choi | Behavior and modeling of confined concrete cylinders in axial compression using FRP rings[END_REF][START_REF] Wei | Behaviour of concrete confined by both steel spirals and fiber-reinforced polymer under axial load[END_REF][START_REF] Guo | Compressive behavior of FRP ring-confined concrete in circular columns: Effects of specimen size and a new design-oriented stress-strain model[END_REF], which simplifies the comparison with the reference uniaxial strength. Furthermore, the FRP layer(s) can be applied shortly before testing, which eliminates possible shrinkage strains that might delay the initiation of the confining pressure. However, this is not a rule and FRP tubes prepared beforehand can also be used as a mold [START_REF] Mirmiran | Dilation characteristics of confined concrete[END_REF]. Provided that the FRP is uniaxially woven and oriented in the hoop direction, its stiffness in the axial direction can be neglected. Stiffness and strength can be controlled by material selection, by the diameter of the cylinder [START_REF] Guo | Compressive behavior of FRP ring-confined concrete in circular columns: Effects of specimen size and a new design-oriented stress-strain model[END_REF], and by the number of plies of the composite [START_REF] Harries | Experimental investigation of the behavior of variably confined concrete[END_REF].

The instrumentation of passively confined concrete specimens is not standardized. The measurement setup typically uses two [START_REF] Cao | Study on the active confinement of gfrp-confined expansive concrete under axial compression[END_REF][START_REF] Guo | Compressive behavior of FRP ring-confined concrete in circular columns: Effects of specimen size and a new design-oriented stress-strain model[END_REF], three [START_REF] Berthet | Compressive behavior of concrete externally confined by composite jackets. Part A: experimental study[END_REF][START_REF] Choi | Behavior and modeling of confined concrete cylinders in axial compression using FRP rings[END_REF][START_REF] Mirmiran | Dilation characteristics of confined concrete[END_REF] or even four [START_REF] Wei | Behaviour of concrete confined by both steel spirals and fiber-reinforced polymer under axial load[END_REF] externally mounted LVDTs or extensometers with a measurement distance equal to half or the entire specimen height. Occasionally, an independent technique is used to supplement these measurements, e.g., strain gauges [START_REF] Berthet | Compressive behavior of concrete externally confined by composite jackets. Part A: experimental study[END_REF][START_REF] Cao | Study on the active confinement of gfrp-confined expansive concrete under axial compression[END_REF][START_REF] Wei | Behaviour of concrete confined by both steel spirals and fiber-reinforced polymer under axial load[END_REF], laser sensors [START_REF] Wei | Behaviour of concrete confined by both steel spirals and fiber-reinforced polymer under axial load[END_REF] or non-contact image-based methods [START_REF] Guo | Compressive behavior of FRP ring-confined concrete in circular columns: Effects of specimen size and a new design-oriented stress-strain model[END_REF]. Strain gauges of various lengths are by far the most common device for measurements of the hoop strain, especially in the case of FRP whose behavior is linear until failure. The other techniques include chain extensometers [START_REF] Choi | Behavior and modeling of confined concrete cylinders in axial compression using FRP rings[END_REF], horizontal LVDTs [START_REF] Berthet | Compressive behavior of concrete externally confined by composite jackets. Part A: experimental study[END_REF][START_REF] Wang | Effect of corner radius on the performance of cfrp-confined square concrete columns: Test[END_REF], and non-contact measurements [START_REF] Bisby | Strain localisations in FRP-confined concrete: new insights[END_REF][START_REF] Guo | Compressive behavior of FRP ring-confined concrete in circular columns: Effects of specimen size and a new design-oriented stress-strain model[END_REF][START_REF] Bisby | Strengthening fire-damaged concrete by confinement with fibre-reinforced polymer wraps[END_REF]. The non-contact measurements have demonstrated good agreement with the strain gauges, but at the same time have revealed extreme variability in the distribution of the hoop strain, even in the elastic range. Conventional measurements rely on the hypothesis that the hoop strain measured at mid-height is representative of the investigated specimen. The measurements carried out within the present research indicate that this assumption is not always valid.

2 Materials and Methods

Experimental design

In the newly developed experimental setup, a cylindrical concrete specimen is equipped with evenly spaced metal rings that provide axisymmetric passive confinement that improves both strength and ductility.

The benefits of the proposed approach include:

(i) elimination of special or single-purpose testing equipment-the proposed setup requires only a commonly available uniaxial testing machine with displacement control, (ii) replacement of conventional strain gauges by non-contact measurements, (iii) simple and fast assembly with quick replacement and fitting of the specimens, (iv) variability that allows changing the target value of confinement.

A schematic representation of the experimental setup and photographs of specimens with 8 and 12 rings ready for testing are shown in Fig. 1. A similar concept has recently been used to evaluate the influence of ring width and spacing on increased concrete strength [START_REF] Safitri | Nuroji: Concrete strength enhancement due to external steel ring confinement[END_REF]. The proposed design adopts the recommended height-to-diameter ratio H : D = 2 : 1 [START_REF] Tc | Recommendations of TC 148-SSC: Strain softening of concrete -test methods for compressive softening test method for measurement of the strain-softening behaviour of concrete under uniaxial compression[END_REF]. In this particular case, the diameter of the concrete cylinder, D 0 , is slightly less than the recommended value of 100 mm, which facilitates the manufacturing process. Every experimental setup should properly represent the behavior of the investigated material; therefore, as recommended in EN 12390-1 [START_REF]Testing Hardened Concrete Part 1: Shape, Dimensions and Other Requirements for Specimens and Moulds[END_REF], the internal diameter D 0 should exceed 3.5× the maximum aggregate size. According to Van Mier [START_REF] Van Mier | Failure of concrete under uniaxial compression: An overview[END_REF], this factor is in the range 3-5, and the minimum dimension of the cross-section should be at least 100 mm [START_REF] Tc | Recommendations of TC 148-SSC: Strain softening of concrete -test methods for compressive softening test method for measurement of the strain-softening behaviour of concrete under uniaxial compression[END_REF]. It should be noted that every 1 MPa of lateral confinement increases the strength by ≈ 4 MPa [START_REF] Erwin Richart | Failure of plain and spirally reinforced concrete in compression[END_REF], which in combination with a large diameter D 0 (required by the aggregate size or by a requirement of a high value of the confinement effectiveness factor, k e (3)) could easily exceed the capacity of the testing machine.

The magnitude of the lateral confinement can be modified by adjusting the geometry of the metal rings and/or their material properties. The development of passive confinement depends on the lateral expansion of concrete restrained by the rings. Therefore, the confinement builds up from zero in the initial unloaded state and reaches its maximum when the rings reach their tensile strength. Fig. 2 schematically describes a symmetric half of the representative segment of the specimen with one ring characterized by its height h, thickness t, and internal diameter D 0 . The uniform spacing between the rings is denoted as g (gap), and the total height of the representative segment can be expressed as h + g. The average value of passive confinement, σ L , can be derived from the equilibrium condition

F L = 2F R (1)
of the stress resultants of lateral confinement in concrete, F L , and tension in the metal ring, F R . After substituting F L = σ L D 0 (g + h) and F R = σ R th, the average confinement is expressed as

σ L = 2 σ R th D 0 (g + h) (2)
As described in Section 2.2, highly uniform confinement can be achieved by minimizing the gap width, g. However, it needs to be ensured that the gaps can accommodate the high axial deformation and do not close during the experiment and, additionally, that the neighboring rings do not interfere with the vertical (axial) extensometers. The height of the rings should be sufficient to allow for the placement of a strain gauge, which might be necessary for the initial verification of non-contact measurements. An excessive height brings about a potential risk of spurious friction in the vertical direction. Before yielding, the behavior of rings is elastic and defined by Hooke's law σ R = E R ε R where E R is the Young modulus of the metal and ε R is the hoop strain in the rings, which is equal to the (average) lateral strain in concrete cylinder, ε L .

Special care should be taken to choose a suitable material for the metal rings. The transition between the elastic and plastic response should be as sharp as possible. Section 3 shows that the confined concrete strength is usually attained close to this transition, so the value of the yield stress, f Y , needs to be known with sufficient precision to properly evaluate the corresponding lateral confinement. Provided the hardening of the metal is not significant, a constant value of stress can be assumed after the onset of the yielding, σ R = f Y .

The last but definitely not least important requirement is the compatibility of the standard mold sizes for the reference unconfined specimens with the range of metal tubes available on the market.

Confinement effectiveness

In axially compressed concrete members reinforced, for example, with spirals or rings, the distribution of lateral confinement is not uniform and depends on the layout of both transverse and longitudinal reinforcement. Even more complex is the confinement in columns with multi-spiral reinforcement [START_REF] Yin | Development of multi-spiral confinements in rectangular columns for construction automation[END_REF][START_REF] Yin | Helical rebar structure[END_REF].

The distribution of confinement is variable in both the lateral and the axial directions. Just below the reinforcement, the concrete is under triaxial compression and the magnitude of confinement is very high; on the contrary, at the surface between the rebars, the lateral stress is negligible. Concrete cover is usually not considered in the analysis at all, as it tends to spall off at very low levels of axial deformation (≈ -2 × 10 -3 ). Toward the center of the member, the homogeneity of the lateral confinement rapidly increases. In practical engineering applications, concrete in the structural member is, for simplicity, divided into two regions-the effectively and ineffectively confined parts-whose behavior is idealized as uniform [START_REF] Mander | Theoretical stress-strain model for confined concrete[END_REF].

The overall structural strength is determined by the weakest section with the highest proportion of ineffectively confined (unconfined) concrete. In the case of circular rings, such a section is between the rings. In the present case, the strength of the unconfined part (red color in Fig. 2) is assumed to be equal to the uniaxial compressive strength, f c . The strength of the effectively confined concrete (shown in the weakest section in green), f c,c , is evaluated from the average confinement σ L (2). To reflect this idea in structural analysis and design, a simple multiplicative factor k e referred to as the confinement effectiveness is introduced. This factor, proposed in the pioneering work of Mander [START_REF] Mander | Theoretical stress-strain model for confined concrete[END_REF], has been adapted directly or in a modified way into almost every design code of practice [START_REF]European Committee for Standardization: EN 1992-1-1 Eurocode 2: Design of Concrete Structures -Part 1-1: General Rules and Rules for Buildings[END_REF][START_REF]Eurocode 2: Design of Concrete Structures -Part 1-1: General Rules, Rules for Buildings, Bridges[END_REF][START_REF]Fédération Internationale du Béton: Model Code[END_REF].

The simplest choice would be to consider the confinement effectiveness as the ratio k e = A ′ c /A c where A ′ c is the area of concrete effectively confined in the weakest section and A c is the total area of concrete bound by reinforcement. With the assumption that (in the elevation view) the boundary dividing the effectively and ineffectively confined concrete is a 2 nd -order parabola with a 45 • angle intercept at the surface, the maximum width of the ineffectively confined concrete zone is g/4 and the diameter of the effectively confined concrete is thus D 0 -g/2. The resulting confinement effectiveness factor is then expressed as

k e = A ′ c A c = π (D 0 /2 -g/4) 2 π (D 0 /2) 2 = 1 - g 2D 0 2 (3)
To obtain a uniform stress distribution in concrete, the gap width g should be considerably smaller than the internal diameter D 0 . For small values of the ratio g/D 0 , k e is close to 1 -g/D 0 . For instance, k e = 0.95 corresponds to g/D 0 = 0.051, k e = 0.9 to g/D 0 = 0.102, and k e = 0.85 to g/D 0 = 0.156. In practice, however, a modified definition of the confinement effectiveness factor

k e = σ ′ L /σ L (4)
is usually preferred. It is then used to evaluate the confined concrete strength on the entire concrete cross section A c from the reduced effective confinement, σ ′ L . Provided that the increase in concrete strength due to confinement, ∆f c,c , is a linear function of confinement, the two definitions coincide and the compressive strength of the specimen can be computed as

f spec c,c = f c A c + ∆f c,c (σ L )A ′ c A c = f c + k e ∆f c,c (σ L ) = f c + ∆f spec c,c (σ ′ L ) (5)

Experimental program

Table 1 summarizes the four different configurations of the experimental setup that were tested. The label that identifies a given configuration specifies the number of rings and their thickness. For instance, '8R t3' refers to 8 rings, each 3 mm thick. The approximate range of interest for the target confinement was between 2.5 and 5.0 MPa. This range is typical, e.g., in the design of multi-spiral columns developed in Taiwan [START_REF] Yin | Development of multi-spiral confinements in rectangular columns for construction automation[END_REF]. Two setups labeled '8R t3' and '12R t2' with the same total areas of reinforcement yield very similar average confinement, and so the results should reveal the differences caused by the confinement effectiveness factor. Dimensions common to all series: H = 200 mm, D = 100 mm, h = 9 mm.

Materials

Concrete

The composition of the concrete mixture with the 0.42 water-to-cement ratio is specified in Table 2. Basic material properties measured at the age of 28 or

Metal rings

The metal rings, which are the source of passive confinement, were cut from AW6060 T66 aluminum tubes with external diameter D = 100 mm and wall thickness t = 2 or 3 mm, which are commonly available on the market. The rings were first cut with a band saw and then finished with a lathe to achieve the desired height h = 9 mm. The tensile behavior of aluminum AW6060 T66 was verified experimentally on 100 mm long coupons, with a cross section 1.8 × 15 mm. The specimens were cut from a square hollow section in the direction perpendicular to its longitudinal axis. The measured stress-strain diagram is shown in Fig. 3. In contrast to high-ductility reinforcing steel (e.g. B500C or SD420W) used in seismic areas, the transition from elastic to plastic behavior is much smoother, but the advantage is that the subsequent hardening is very slow, and thus the confinement during the late stages of the test remains almost constant.

The recorded stress-strain diagram agrees nicely with the standard characteristics of this alloy (E = 69.5 GPa, minimum tensile strength 215 MPa). Deformation was evaluated from the elongation of a 25 mm long extensometer (solid lines with points) and from the cross-head displacement (dashed lines).

In the main experimental study on the behavior of concrete under passive confinement, the hoop strain in the metal rings at the peak axial load level is typically between 3 × 10 -3 and 4 × 10 -3 ; see Figs. 10 and13. For this reason, the value of the yield stress used in further analyses is set to f Y = 190 MPa, which is considerably higher than the initial yield stress (approx. 160 MPa) but lower than the stress at the yield plateau (approx. 200 MPa). On average, the tensile strength 220 MPa is attained at strain 80 × 10 -3 , which is in accordance with the standard specifications (f u = 215 MPa). On the basis of the two measurements, the strain at failure is approx. 140 × 10 -3 .

Specimen preparation

Compatibility of the outer diameter D of the rings with the inner diameter of the standard cylinder mold 100 × 200 mm allowed for a simple and accurate assembly of the aluminum and 3D-printed plastic rings with height g (or g/2). Before that, the inner surface of the aluminum rings was treated with floor wax to prevent a chemical reaction.

Approximately 4 hours after concrete casting (and properly vibrating), the molds were wrapped with foil to minimize evaporation. The next day, the specimens were demolded, and the spacer rings were removed. The specimens were then transferred to a closed container with relative humidity ≈ 95% and room temperature. Two days prior to testing, the horizontal load bearing surfaces were ground [START_REF]Testing Hardened Concrete Part 3: Compressive Strength of Test Specimens[END_REF].

Instrumentation and testing program

Conventional measurements

To measure the hoop strain of the passively confined specimens, the four middle rings (T2, T1, B1, B2 in Fig. 1) were equipped with strain gauges HBM-1-LY41-10/120 (measuring length 10 mm, width 8 mm) oriented in the circumferential direction. Only 3 of 5 specimens (Nos. 7, 8 and 9) in each series were instrumented in this way. Additionally, axial shortening was recorded by two 100 mm long extensometers placed at mid-height and attached to the surface of the concrete cylinder. In the case of plain concrete specimens, only axial deformation was measured.

DIC measurements

The secondary aim of the presented research was to demonstrate that simple and cheap 2D DIC measurements can extract valuable data from uniaxial tests on cylindrical specimens if carried out and evaluated properly. The evolution of deformation was recorded as a sequence of images taken in 5-second intervals using a high-definition camera Canon 70D equipped with a 200mm objective lens, minimizing the fish-eye distortions. The observed specimens, sprayed with black and white speckles to provide a random pattern needed for DIC, occupied the entire image frame when placing the camera approximately 10 m from the tested specimens. This setup yielded a pixel resolution of 16.95 px/mm.

A suite of open-source noncommercial software, Ncorr [START_REF] Blaber | Ncorr: Open-source 2D digital image correlation matlab software[END_REF] and Ncorr post [START_REF] Nežerka | An integrated experimental-numerical study of the performance of lime-based mortars in masonry piers under eccentric loading[END_REF], was used for the evaluation of the displacement and deformation fields and for data extraction. To quantify relative displacements, virtual extensometers were used to gather the data (Ncorr post), while the distributions of strains and displacements along a line were calculated by averaging the displacement/strain matrix values within a strip parallel to that line (Ncorr post CSTool); see Fig. 4.

Assuming a uniform distribution of hoop strain over the circumference of the aluminum rings, the in-plane projection can be used to evaluate hoop strain. The strains on the rings are calculated as ε = ∆L/L where L is the length of the extensometer and ∆L is its change, and both ∆L and L are reduced by the same factor to obtain the 2D projection. The same approach was adopted for hoop strain measurements on FRP wraps used as confinement for cylindrical specimens using particle image velocimetry (PIV) [START_REF] Bisby | Strain localisations in FRP-confined concrete: new insights[END_REF][START_REF] Guo | Compressive behavior of FRP ring-confined concrete in circular columns: Effects of specimen size and a new design-oriented stress-strain model[END_REF][START_REF] Bisby | Strengthening fire-damaged concrete by confinement with fibre-reinforced polymer wraps[END_REF]. A demonstration of this conjecture is provided in Fig. 5; the study was performed on the rings T1 and B4 of 12R t2-8 specimen. Although the results in Fig. 5 Fig. 4 Placement of virtual extensometers for extraction of displacements at discrete locations (left) and a 60 mm wide averaging strip for evaluation of displacement/strain distribution along a red line (right), on a map of vertical displacement (mm) at the peak load for the 12R t2-8 specimen. Extensometer couples 1a-1b and 2a-2b (located near edges ∼ EXT L/R length) were used for evaluation of vertical displacements, while extensometer couples 3a-3b to 14a-14b were placed on the rings 60 mm apart to evaluate their horizontal deformation. are almost independent of the length of the extensometer, the hoop strain is highly variable about the circumference, therefore, it is important to maximize the extensometer length.

Testing

The specimens were tested at the age of 28 days (12 rings) and 29 days (8 rings). Every specimen series with rings (5 specimens) was supplemented with a set of 4 reference plain concrete specimens tested on the same day; altogether the experimental program comprises 16 plain concrete specimens and 20 specimens with rings.

A loading frame with 2.5 MN maximum force was used for testing. The instrumented specimens were properly centered (max. 10% difference in reading of extensometers under 40 kN load) and loaded at a constant rate of 100 kN/min until the load of 200 kN was reached. The test was then switched to displacement control at a rate of 0.48 mm/min. (Note that this rate corresponds to the displacement of the loading cylinder, while the actual imposed displacement is about 2 times less.) The test was terminated once several rings had ruptured (typically at less than 25% of the maximum force). One run of the experiment (excluding centering) took approx. 15-30 minutes.

Results and Discussion

To achieve maximum clarity of the results presented, the following conventions are adopted. Positive values of stress and strain denote tension and extension, respectively. The axial stress is computed as the loading force divided by the net area of the concrete cross-section; this area differs for the setups with 2 and 3 mm thick rings. The rings are numbered from the center towards the bases according to Fig. 1. The axial strain corresponds to the average of two extensometers, physical (contact measurements) or virtual (DIC). The designation introduced in Table 1 characterizes the geometry of the experimental setup. Where applicable, an additional number refers to particular specimens: 1-4 to companion plain concrete specimens and 5-9 to specimens with rings. Specimens 5 and 6 were not equipped with strain gauges to measure hoop strain.

Stress-strain response

This section presents the axial stress-strain response of the specimens with metal rings. Here, the axial strain was measured using a pair of (contact) extensometers whose consistent readings indicated well-centered specimens and their symmetric behavior even in the post-peak. The rupture of the first ring, which typically occurs at axial strains between -20×10 -3 and -25×10 -3 , defines the measuring range of this experimental setup. This is fully sufficient and comparable to other experiments in the literature [START_REF] Imran | Experimental study of plain concrete under triaxial stress[END_REF][START_REF] Harries | Experimental investigation of the behavior of variably confined concrete[END_REF][START_REF] Li | Stress-strain behavior of actively and passively confined concrete under cyclic axial load[END_REF] at similar confinement magnitudes.

Fig. 6 illustrates the statistically processed results, different colors denote the individual experimental setups. Solid lines correspond to the the mean response of specimens with satisfactory failure mode. The scatter (standard deviation, shaded area) increases approximately linearly with axial strain in the post-peak. The minimum and maximum responses (dashed lines) are very close to the bounds of this shaded region.

Except for the discarded specimens exhibiting shear failure, the results are very consistent in terms of strength, maximum strain (ductility), strain at peak stress, and the softening rate in the post-peak. The noisy character of the lines in the late post-peak range is due to the removal of the failed specimens from the statistical evaluation.

Only a portion of the experimental data is presented. The data was typically trimmed after the rupture of the first ring, resulting in a sudden drop in confinement and rapid propagation of damage in unrestrained concrete. In addition to this, the uniformity of readings between the extensometers was lost and the measured data could become misleading.

Consistent with other studies, e.g. [START_REF] Imran | Experimental study of plain concrete under triaxial stress[END_REF][START_REF] Harries | Experimental investigation of the behavior of variably confined concrete[END_REF][START_REF] Li | Stress-strain behavior of actively and passively confined concrete under cyclic axial load[END_REF], the compressive strength and the corresponding strain (Table 4) increase with confinement. Series 8R t3 and 12R t2, which have a very similar reinforcement ratio (the same cross-sectional area of the rings but different internal diameter D 0 ), reach similar strength. Owing to better confinement efficiency, the strength is higher in the case with Fig. 6 Statistically processed stress-strain diagrams for all specimen groups. Solid line is the average, dashed lines are the minimum and maximum responses, shaded area is the average ± standard deviation.

closely spaced rings, whereas the setup with loosely spaced rings results in higher strain at peak stress. Additionally, the standard deviation of strength is rapidly decreasing with higher confinement and closer ring spacing (Table 4).

The summary presented in Table 4 reveals that the number of discarded specimens that failed in shear decreases with an increasing number of rings and/or confinement. Furthermore, from the available data it appears that the inappropriate failure mode does not affect the strength and corresponding strain, but significantly influences post-peak behavior, which is characterized by higher softening rate. Therefore, any deviation from the prevailing stressstrain behavior might serve to detect an inappropriate failure mode, which, in contrast to plain concrete specimens, is more difficult to detect, even from a sequence of (uncorrelated) digital images.

Distribution and evolution of axial and lateral strain

This section presents the results obtained using a noncontact DIC with a 60 mm wide averaging strip (depicted in Fig. 4b), which allowed efficient processing of the hoop strain and vertical (axial) displacement over the entire specimen height. As demonstrated in Table 4, the behavior of specimens with 12 rings is more regular than with 8 rings. Furthermore, with ring thickness t = 2 mm the confinement (see Table 6) becomes more representative of the present experimental program. For this reason, specimen 12R t2 8 was selected as an example to illustrate the typical behavior both visually and by means of the DIC measurements. The photos shown in Fig. 7, taken at the beginning of loading and then in the post-peak at 80% and 60% of the peak load, illustrate typical behavior. Until the peak load, no damage is visible except for local and very limited concrete spalling between the rings. Lateral deformation starts to be visually noticeable in the post-peak (Fig. 7b). This deformation is uniformly distributed in the middle part between the springs (≈ 100 mm). Shortly before the failure of the metal rings (Fig. 7c), concrete becomes severely damaged in the central part, while the outer (top and bottom) regions remain undamaged. Damage is diffuse and uniform; no localized deformation is apparent. Towards the end of the experiment, the rings start to bend, the lateral strain is no longer uniform, and concentrates at the mid-height. The results of the DIC analysis provide detailed insight into the evolution and distribution of deformation, which is shown at several time snaps in Fig. 8. The response at the peak load is drawn by a solid black line, the numbers in the legend indicate the fraction of the peak load. To facilitate orientation, the horizontal gray bands represent individual rings. The values were obtained by averaging over a 60 mm wide band; the drops to zero around H = 50 and 150 mm are caused by external springs blocking the view of the specimen.

The evaluated vertical (axial) strain was very noisy. Therefore, the left part of Fig. 8 shows the vertical displacement. The graph reflects the loading setup in which the top surface is supported and an upward displacement is imposed at the bottom. The right part of Fig. 8 presents the distribution of the hoop strain over the height. Local peaks between the rings signal concrete damage and associated spalling in that region. To improve the clarity of the DIC results and to eliminate the noise due to concrete damage, both quantities-the axial (vertical) displacement and the hoop strain-are evaluated in the middle of the ring height. Additionally, the axial displacement is zeroed with respect to the top ring. The persistent irregularities can probably be attributed to the sliding of the rings on the concrete cylinder. Horizontal dashed lines indicate the 100 mm long central part of the specimen monitored by a couple of external extensometers.

The evolution and distribution of vertical displacement is, for the experimental setup 12R t2, presented in Fig. 9. Different line types are used to distinguish the individual specimens while the colors mark the specific load levels (normalized with respect to the corresponding strength). Except for large magnitudes of axial strain, the distribution in the central part can be idealized as linear and at the same load levels as parallel and without strain localization. Therefore, it can be concluded that the measurements with 100 mm long external extensometers can deliver representative data of the central part of the specimen which is sufficient to characterize the material behavior. For the experimental setup 12R t2, the distribution of hoop strain over the specimen height is shown in Fig. 10. The left part shows the response up to the peak and the right part up to 70% of the specimen strength in the postpeak, which typically corresponds to an axial strain exceeding in magnitude 20 × 10 -3 . Similarly to the previous graphs, colors correspond to individual load levels and line types serve to distinguish individual specimens. As shown in the figure, the response is almost perfectly regular and the hoop strain is nicely symmetric about the horizontal axis of the specimen. The maximum is typically attained in one of the middle rings (B1 or T1). In the pre-peak, the distribution of the hoop strain is almost uniform in the central part of the specimen. When approaching the maximum load level, this distribution gradually evolves to parabolic and finally, in the post-peak, the overall distribution over the height is highly nonlinear and resembles a bell shape. To properly capture the behavior of the representative concrete volume, the hoop strain needs to be averaged. The responses of the other experimental setups which show similar trends are presented in the supplementary material [START_REF] Havlásek | Data to: Efficient Approach to Measuring Strength and Deformation of Passively Confined Concrete[END_REF]. Hoop strain [×10 -3 ] 0.9 pre 1.0 peak 0.9 post 0.8 post 0.7 post Fig. 10 Distribution of hoop strains over the height of specimens 12R t2.

Conventional measurements vs. DIC

This section compares the results obtained with conventional measurements (i.e., external axial extensometers and lateral resistive strain gauges) and the non-contact DIC method. The aim is to validate the DIC measurements and to select a suitable and reliable combination of experimental techniques to accurately describe the evolution of axial and lateral deformation of the representative concrete volume.

In the literature [START_REF] Imran | Experimental study of plain concrete under triaxial stress[END_REF][START_REF] Harries | Experimental investigation of the behavior of variably confined concrete[END_REF][START_REF] Li | Stress-strain behavior of actively and passively confined concrete under cyclic axial load[END_REF], the response of confined concrete is usually presented in terms of an extended stress-strain diagram. In this diagram, both the axial and the hoop strains are linked to the axial stress. An example of such a diagram is presented in Fig. 11a for specimen 12R t2 8 (shown previously in Figs. 7 and8). This overall view demonstrates that a very similar evolution of the average axial strain was measured with both physical (100 mm long, red lines) and virtual DIC (84 mm long, purple points) extensometers. In this particular case, the DIC method offers a longer range (45 × 10 -3 instead of 25 × 10 -3 ) but on the other hand the extensometers attached to the concrete provide more uniform behavior (dashed red lines vs. tiny purple points).

A detailed view of the stress vs. axial strain relationship shown in Fig. 11b reveals that the present DIC measurements suffer from irregular oscillations, which are common to both extensometers. These oscillations are most profound in the first half of the loading branch and are responsible for a spurious offset in this diagram.

The measurements of the hoop deformation are drawn in green and blue in Fig. 11a andc. Only the results for the central 4 rings (top to bottom: T2, T1, B1, B2) are shown, because these were examined by both methods. A detailed view of the loading branch indicates that the magnitudes of the oscillations are similar for both axial and hoop deformations. In the latter case, they become more apparent because the magnitude of the lateral deformation is initially scaled by the Poisson ratio. In contrast to a good agreement of the axial deformation, the hoop deformation exhibits large differences between the two methods. This inconsistency is attributed to variations in the hoop strains along their perimeter, as reported by Bisby and Take [START_REF] Bisby | Strain localisations in FRP-confined concrete: new insights[END_REF]. The strain averaging procedure over a 60-mm projection using DIC provides more reliable data than strain gauges of very limited length (10 mm in this case).

The evolution of axial and hoop strain was evaluated for all specimens with metal rings. The axial strain measured by the two methods exhibits a symmetric scatter of approx. 25%. The measuring range of both methods is comparable but the reusable external extensometers can deliver more consistent and smoother data not polluted by the noise of the DIC. For this reason, the physical extensometers should be preferred.

On the other hand, in the case of hoop strain the DIC technique is found superior to conventional measurements with strain gauges. First, the conventional measurements are not representative in the post-peak because the non-uniform strain distribution over the circumference becomes amplified by the material behavior of the rings. Owing to their very little hardening, the rings start yielding in some areas whereas remain elastic elsewhere. Second, DIC delivers more consistent results among specimens with the same setup. The accuracy can be considerably improved if the readings from the central rings are averaged. Finally, the strain gauges are not reusable, their price rapidly increases with measuring length and their usage is overall cumbersome (laborious gluing and soldering, slow transition between specimens during testing).

The final data, which can be used for further analysis and calibration of constitutive models for confined concrete, are presented in the form of combined graphs in Fig. 12. The axial strain is the average of the physical 100 mm long extensometers. The hoop strain is the average of the DIC measurements from the middle region equal to the length of the physical extensometers. In this global view, the noise of the DIC completely vanishes. In contrast to typical graphs from the literature, the lateral strain is here plotted as a function of the axial strain, not of the axial stress. In this way, the rate of lateral dilation can be directly assessed. The volumetric deformation, which is often presented, combines numbers with opposite signs and similar magnitudes, which could lead to large errors. However, even in this respect the results are consistent with data from literature [START_REF] Imran | Experimental study of plain concrete under triaxial stress[END_REF][START_REF] Li | Stress-strain behavior of actively and passively confined concrete under cyclic axial load[END_REF][START_REF] Chen | Plasticity in Reinforced Concrete[END_REF], the material strength is reached approximately at zero volumetric strain [START_REF] Havlásek | Data to: Efficient Approach to Measuring Strength and Deformation of Passively Confined Concrete[END_REF]. Fig. 12 Relations between the stress, conventionally measured axial strain and the average hoop strain measured using DIC. Dashed lines correspond to elasticity with ν = 0.2.

Increase in strength due to confinement

This section compares the present experimental data with selected models from the literature and design codes or recommendations, to support the credibility of the proposed experimental setup. Since most of the selected models were developed for steel reinforced concrete and the present experiment uses aluminum with a distinctly different stress-strain relationship, the only quantity that can be objectively compared is the increase in strength due to confinement. The experimental setup presented here does not produce a perfectly uniform distribution of the lateral confinement. Therefore, to establish a meaningful comparison, this structural effect which causes a strength reduction needs to be properly considered by all models. Table 5 presents slightly modified formulae of the selected models. In particular, the confinement effectiveness factor k e has been added to Richart's model and, in the code expressions, the characteristic or design strength has been replaced by the mean compressive strength.

Table 5 Increase in concrete strength due to lateral confinement according to the selected models.

Recommendation

Strength increase ∆f spec c,c [MPa] Richart [START_REF] Erwin Richart | Failure of plain and spirally reinforced concrete in compression[END_REF] 4.1keσ L

Mander [START_REF] Mander | Theoretical stress-strain model for confined concrete[END_REF]

fc 2.254 1 + 7.94keσ L fc -2 keσ L fc -2.254 (7)
MC 1990/99 [START_REF]Comité Euro-International du Béton (CEB): CEB-FIP Model Code 1990[END_REF][START_REF]Fédération Internationale du Béton: Structural Concrete: Textbook on Behaviour, Design and Performance: Updated Knowledge of the CEB/-FIP Model Code 1990[END_REF], EC2 [START_REF]European Committee for Standardization: EN 1992-1-1 Eurocode 2: Design of Concrete Structures -Part 1-1: General Rules and Rules for Buildings[END_REF] 5keσ L for keσ L < 0.05fc

Draft of EC2 [START_REF]Eurocode 2: Design of Concrete Structures -Part 1-1: General Rules, Rules for Buildings, Bridges[END_REF] 4keσ L for σ L < 0.6fc [START_REF] Berthet | Compressive behavior of concrete externally confined by composite jackets. Part A: experimental study[END_REF] 3.5keσ

3/4 L f 1/4 c
otherwise All models presume that, at compressive strength, the reinforcing steel is yielding at constant stress. In contrast with steel reinforcement, for aluminum, the transition between elastic and plastic behavior is very smooth, and the compressive strength in all four setups is attained this transition. Therefore, the value of lateral confinement σ L needs to be determined carefully, and its compatibility with the hoop strain at peak stress must be established. As shown in Fig. 13, a combination of the stress vs. axial and lateral strain diagrams (Fig. 12) and the stress-strain curve for aluminum (Fig. 3) has been used to correctly estimate the corresponding average value of the hoop stress in the metal rings, σ R .

The metal rings start yielding (proof stress, dashed arrows) shortly prior to reaching the peak load. At compressive strength (solid arrows), the stress in the rings is approximately 190 MPa. In the post-peak, the insignificant material hardening guarantees an almost constant magnitude of the confinement.

The lateral confinement was subsequently calculated using (2) and the reduction confinement effectiveness factor k e , defined in (3), was evaluated from Mander's formula [START_REF] Erwin Richart | Failure of plain and spirally reinforced concrete in compression[END_REF] for all models compared. The values of lateral confinement and effectiveness factor for the individual setups are summarized in Table 6. A graphical comparison of the selected models with the experimental Axial strain [×10 -3 ] 5 6 7 9

Fig. 13 Relations between the vertical stress in concrete, axial and average hoop strain, and tangential stress in rings. Solid arrows correspond to the peak load and dashed arrows to the proof stress. data is shown in Fig. 14. The reduced effective confinement σ ′ L from (4) and Table 6 is used on the horizontal axis to emphasize the structural effect of the ring layout. Formula (7), derived by Mander from the failure surface proposed by Willam and Warnke [START_REF] Willam | Constitutive models for the triaxial behavior of concrete[END_REF], is in best agreement with the present experimental data. This is a good message because it promotes the soundness of the experimental setup developed. Except for the final draft of the Model Code 2010, which nicely matches the three setups with the smallest confinement, the other purely empirical models slightly underestimate the strength increase due to confinement.

Conclusions

The paper describes an experimental setup recently developed to test concrete cylinders under passive confinement. In contrast to a triaxial test, the proposed setup requires only an ordinary uniaxial testing machine with displacement control, a pair of extensometers, a (single!) high-quality camera, and a computer for data processing. In addition to this, the stress history becomes much closer to that encountered in real structural members, and the specimen size can be almost arbitrary, provided that it remains large enough to properly represent the material and small enough to comply with the capacity of the testing machine. Comparison of the main features of the proposed method with the triaxial testing machine is summarized in Table 7. Despite the numerous differences outlined in this Table, the strength increase due to confinement determined from the proposed tests is in accordance with the widely accepted models from the literature (see Fig. 14) and therefore also with the triaxial tests that were used for their development and calibration. The confinement is provided by evenly spaced aluminum rings prepared from a tube commonly available on the market. The magnitude of confinement can be adjusted by balancing the tube wall thickness, the spacing, and the yield stress. The uniformity of the spacing was achieved with a set of dummy 3D printed plastic rings removed after casting. Metal rings did not cause spurious restraint in the vertical direction, but provided a very uniform lateral confinement. Additionally, the rings were suitable for applying a random pattern needed for DIC measurements that yielded consistent results with sufficient accuracy and measurement range.

The robustness of the proposed experimental setup was successfully validated using 20 concrete cylindrical specimens with 4 different layouts of the rings. The experimental scatter decreased with increasing magnitude of confinement and decreasing spacing of the rings.

The results indicate that the most suitable measuring setup is the combination of vertical (physical) extensometers for the axial strain with the DIC tracking the evolution of the hoop strain of the rings. The axial strain is almost constant in the middle half of the specimen height throughout the experiment. In contrast to this, the initially uniform distribution of hoop strain over the specimen height becomes highly nonlinear close to the peak and in the postpeak. Very consistent data can be obtained by averaging the hoop strain of the rings in the region given by the extensometers.

The original research data published in this paper are available at Mendeley Data [START_REF] Havlásek | Data to: Efficient Approach to Measuring Strength and Deformation of Passively Confined Concrete[END_REF]. This dataset contains also responses of the other experimental setups and presents examples of displacement contours obtained with DIC to illustrate the satisfactory and shear failure modes.

Fig. 1

 1 Fig. 1 Outline of the experimental setup and photos of specimens with 12 and 8 metal rings. Red color corresponds to conventional techniques while purple color refers to the image-based measurements.

Fig. 2

 2 Fig. 2 Symmetric half of the representative segment: assumed distribution of stress and equilibrium of resultant forces in the horizontal direction.

Fig. 3

 3 Fig. 3 Detailed (left) and overall (right) view of the experimental stress-strain diagram of AW6060 T66 aluminum alloy coupons subjected to uniaxial tension.

Fig. 5

 5 Fig.5Hoop strain measured on the 12R t2 8 specimen rings T1 and B4 (Fig.1) demonstrating the independence of measured values on the length of virtual extensometers.

Fig. 7 3

 7 Fig. 7 3 snapshots from the experiment on a specimen with 12 rings with thickness 2 mm (12R t2 8): (a) prior to loading; (b) postpeak, 80% strength, visually noticeable lateral deformation; (c) postpeak, 60% strength, shortly before rings rupture.

Fig. 8

 8 Fig. 8 Distribution of vertical displacement (left) and hoop strain (right) over height of specimen 12R t2 8.

Fig. 9

 9 Fig. 9 Distribution of vertical displacement over the height of specimens 12R t2.

Fig. 11

 11 Fig. 11 Evolution of axial and hoop strain on specimen 12R t2 8; overall view (a), detailed view (b) and (c). Smooth lines correspond to conventional measurements, noisy lines with points to DIC.

Fig. 14

 14 Fig. 14 Dependence of the specimen strength on the effective lateral confinement: comparison of experimental data (error bars indicate standard deviation) and prediction models.

Table 1

 1 Geometry of concrete specimens with metal rings.

	Series	No. rings, N	t [mm]	D 0 [mm]	g [mm]
	8R t2	8	2	96	16.0
	8R t3	8	3	94	16.0
	12R t2	12	2	96	7.7
	12R t3	12	3	94	7.7

Table 2

 2 Concrete composition.

	Ingredients	Quantity [kg/m 3 ]
	CEM I 42.5R (Mokrá, Czech Republic [37])	410
	Water	170
	Aggregates 0-4 mm, 4-8 mm, 8-16 mm	796.4 + 450.0 + 524.2
	Superplasticizer Stachement F	4.1

29 days on standard cylinders 100 × 200 mm stored under sealed conditions are summarized in Table

3

. In this table, the basic numbers are the mean values while the corresponding standard deviations are in parentheses. The reported properties of plain concrete include density, ρ, uniaxial compressive strength, f c , the strain (magnitude) at peak stress, ε f c , and the secant modulus of elasticity evaluated at 40% of the peak stress, E 40 . The axial strain was measured by two 100 mm long extensometers installed on the opposite sides of the specimen at its mid-height. The present concrete can be classified as C40/50, as the characteristic strength f ck = f cm -1.64 × stdev = 46.576 -1.64 × 2.406 = 42.629

[MPa]

.

Table 3

 3 Average concrete properties at the age of 1 month measured on cylinders 100 × 200 mm.

	Property	Unit	Mean value (stdev)
	ρ	[kg/m 3 ]	2375.2 (8.0)
	E 40	[GPa]	37.68 (1.62)
	fc	[MPa]	46.58 (2.41)
	ε f c	[10 -3 ]	1.92 (0.24)

Table 4

 4 Compressive strength and corresponding strain of concrete cylinders with metal rings. The specified values represent the average and the standard deviation (in parentheses) evaluated from the specimens with the correct failure mode. Strength and Deformation of Passively Confined Concrete

	Series	passed/total	f spec c,c	ε f c,c
			[MPa]	[10 -3 ]
	8R t2	3/5	63.77 (2.32) 3.98 (0.33)
	8R t3	5/5	67.06 (1.61) 5.20 (0.35)
	12R t2	4/5	69.94 (0.99) 4.88 (0.13)
	12R t3	5/5	76.93 (0.93) 7.42 (0.44)

Table 6

 6 Increase in compressive strength due to lateral confinement.

	Designation ∆f spec c,c	σ L (2) ke (3)	σ ′ L (4)
		[MPa]	[MPa]	[-]	[MPa]
	8R t2	17.20	2.85	0.840	2.40
	8R t3	20.49	4.37	0.837	3.65
	12R t2	23.37	4.28	0.922	3.94
	12R t3	30.35	6.55	0.920	6.03

Average confinement computed assuming

σ R = f Y = 190 MPa.

Table 7

 7 Comparison of the proposed experimental setup with triaxial testing.

			Proposed method	Triaxial machine
	Testing machine	Versatile testing machine with dis-	One-purpose and expensive Hoek
			placement control	cell
	Specimen size		Limited only by the capacity of	Determined using the Hoek cell
			the testing machine
	Visual control during	Direct	Hindered
	testing, onset of localisa-
	tion
	Confinement magnitude	Sufficient for typical RC members	Almost arbitrary
	Confinement evolution	Typical for RC members, trig-	Almost arbitrary, typically con-
			gered by lateral expansion of con-	stant throughout the experiment
			crete
	Confinement	distribu-	Non-uniform over the specimen	Uniform
	tion		height (friction at supports and
			hoop spacing)
	Vertical strain measure-	Reusable extensometers	Foil strain gauges
	ment
	Hoop strain measure-	DIC analysis for all metal rings	Foil strain gauges at mid-height
	ment
	Needed time for prepa-	Short (only centering)	Long (instrumentation, assembly,
	ration			pressure medium, and cleanup)

Strength and Deformation of Passively Confined Concrete

Funding

Financial support for this work was provided by the Czech Science Foundation (GA ČR), project number 19-25312S (Compression tests with confinement for advanced modeling of concrete columns).

Declarations


Simela Chatzikonstantinou

Aude Jegou

Sacha Brohée

Rinki Singh

Christian Bénar

Dr Stanislas Lagarde
email: stanislas.lagarde@ap-hm.fr

Fabrice Bartolomei
email: fabrice.bartolomei@ap-hm.fr

How May a Brief Seizure Lead to Prolonged Epileptic Amnesia?

Keywords: drug-resistant epilepsy, partial epilepsy, SEEG, memory, stimulation

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TEXT

Introduction

Pure amnestic seizures [START_REF] Palmini | Pure amnestic seizures in temporal lobe epilepsy[END_REF]) are defined as self-limited episodes with isolated, anterograde memory loss and have been attributed to bilateral dysfunction of mesial temporal structures. This seizure can occur in patients with different forms of temporal lobe epilepsy and has been more recently associated with a late-onset epileptic syndrome, called transient epileptic amnesia (TEA) [START_REF] Kapur | Transient epileptic amnesia a clinical update and a reformulation[END_REF][START_REF] Butler | The syndrome of transient epileptic amnesia[END_REF][START_REF] Mosbah | Clinical, neuropsychological, and metabolic characteristics of transient epileptic amnesia syndrome[END_REF]. It usually begins in later life, most commonly in the seventh decade [START_REF] Zeman | Transient epileptic amnesia: a description of the clinical and neuropsychological features in 10 cases and a review of the literature[END_REF]) and is generally responsive to antiepileptic therapy [START_REF] Gallassi | Epileptic Amnesic Syndrome: An Update and Further Considerations[END_REF][START_REF] Butler | The syndrome of transient epileptic amnesia[END_REF]. In this form of epilepsy, the seizures are typically purely amnestic and often associated with interictal memory impairment. In most cases, the episodes are brief, lasting less than 10 minutes, but cases with longer duration, ranging from 30 minutes to several hours can occur [START_REF] Tassinari | Transient Global Amnesia as a Postictal State from Recurrent Partial Seizures[END_REF][START_REF] Mosbah | Clinical, neuropsychological, and metabolic characteristics of transient epileptic amnesia syndrome[END_REF]. The mechanisms of such prolonged manifestations are not well known. Few cases have been documented using video-EEG [START_REF] Palmini | Pure amnestic seizures in temporal lobe epilepsy[END_REF][START_REF] Mosbah | Clinical, neuropsychological, and metabolic characteristics of transient epileptic amnesia syndrome[END_REF][START_REF] Jensen | Transient epileptic amnesia diagnosed using long-term electroencephalography[END_REF]) and the ictal or post ictal origin remains poorly understood. Cases of amnestic seizures recorded during intracerebral EEG exploration are rare [START_REF] Palmini | Pure amnestic seizures in temporal lobe epilepsy[END_REF].

In this work we report a case of prolonged anterograde amnesia (lasting several hours) following a brief seizure induced by stimulation of the left entorhinal cortex, recorded during stereo-EEG (SEEG). To gain insight into the mechanisms of the prolonged amnesia, we applied an entropy-based method of signal complexity analysis. It has been argued that a certain degree of complexity in brain signals is associated with healthy cognition, whereas low complexity may be a sign of pathological states [START_REF] Mateos | Measures of entropy and complexity in altered states of consciousness[END_REF].

Case Study

A 40-year-old, left-handed woman, with an history of focal pharmaco-resistant epilepsy, posthead trauma, was referred for presurgical evaluation.

Seizures were characterised by behavioural arrest followed by head version to the left, dystonic right-hand posturing, chewing automatic movements and loss of consciousness.

Occasionally, the seizures were preceded by an ascending warm epigastric sensation or "déjàvu" feeling. Some of the seizures were followed by prolonged periods (hours) during which her memory "was not functioning properly", she had difficulty remembering events and names and she would usually fully recover the next day after the seizure. Video-scalp-EEG found left temporal ictal discharges. Magnetic resonance imaging (MRI) revealed gliotic scar within temporal edge of the left insula. 18-FDG-PET showed moderate hypometabolism in the left insulo-opercular region extending to the left inferior parietal region. Interictal magnetoencephalography (MEG) showed predominant localisation of interictal spikes in the left temporal anterior areas. The neuropsychological assessment showed: 1) normal global intellectual efficiency without visuo-verbal dissociation (WAIS-IV), 2) normal memory performance (immediate and differed, without dissociation, MEM III), 3) normal denomination (DO80) .

Given the possible dissociation between anatomical lesion and putative epileptogenic zone (EZ), a stereo-EEG (SEEG) was undertaken. A signed consent was obtained from the patient and her relatives, who were informed that her data could be used for research purposes. 17 electrodes were implanted (Figure 1). The decision on electrode placement was guided by the assumptions about the location of the EZ. High-frequency stimulation at 50 Hz (pulse duration 1 ms, intensity 0.5-2mA) was applied in a bipolar fashion to each of the contact pairs in the grey matter for a period of 3-5 s (see Supplementary Material). Direct electrical stimulation of the left entorhinal cortex triggered a habitual seizure [START_REF] Trébuchon | Electrical Stimulation for Seizure Induction and Functional Mapping in Stereoelectroencephalography[END_REF]. Presurgical evaluation concluded to a left mesio-temporal EZ and a left anterior temporal lobectomy made the patient seizure free for 6 months until now.

During stimulation of the right rhinal cortex (50 Hz, 1.5 mA, 5 s), the patient experienced a sensation of "déjà vu" together with nausea, which she did not recognize as a typical seizure.

After this first stimulation associated with a SEEG unilateral mesio-temporal discharge (duration 55 seconds), there was no obvious memory impairment (normal episodic retrograde memory, no formal testing of anterograde memory). However, there was some persistent slowing of SEEG background activity in the right rhinal cortex. Five minutes later, during stimulation of the left rhinal cortex (50 Hz, 1.5 mA, 5 s ), there was a clinically asymptomatic electrical discharge in the left mesio-temporal, left insular, and left posterior cingulate regions (sparing the right rhinal cortex) during 32 sec. At the end of this discharge, the patient presented a mixed retrograde-anterograde amnestic state that gradually subsided over the next 3 hours. Specifically, she was unaware of the reason for her hospitalization and could not recall the discussions that had taken place between her and the medical staff in the preceding minutes. In addition, she was unable to encode new information, as tested repeatedly by medical staff using a three-word recall test. Older memories and other aspects of memory remained intact throughout. On SEEG traces, there was a slowing of background activity in the bilateral mesiotemporal regions with progressive recovery (Figure 1).

In this study, we chose permutation entropy (PE) to characterize the level of disorganization of the signals and to correlate with the amnestic state (see Supplementary Material). We compute PE in 6 regions of interest (left amygdala, left hippocampus, left entorhinal cortex, left para-hippocampal gyrus, left temporal pole and right entorhinal cortex). These regions were chosen because of the implication of mesial temporal regions in memory processing.

Particularly, the hippocampus is thought to be critical for long-term detailed episodic memories [START_REF] Vargha-Khadem | Differential Effects of Early Hippocampal Pathology on Episodic and Semantic Memory[END_REF], and the perirhinal region has been implicated in recognition memory and in context-free memory in general [START_REF] Mishkin | Hierarchical organization of cognitive memory[END_REF][START_REF] Barbeau | Les modèles de la mémoire : approches anatomo-fonctionnelle et représentationnelle-hiérarchique[END_REF]. We analysed five distinct periods of interest: We compared these periods of interest with a 10 mn baseline taken on the same afternoon but preceding electrical seizure stimulation (see Supplementary Material).

During the seizure induced by the second stimulation of left rhinal cortex we observed a significant and marked decrease of entropy within the mesial temporal region (amygdala, hippocampus, entorhinal, temporal pole, and para-hippocampal) whereas the right entorhinal cortex (involved in the previous stimulus-induced seizure) showed non-significant increase of entropy. The immediate post-seizure period and POI1, when the patient had marked global amnesia, was both characterised by significant increased entropy (compared to baseline) within left amygdala, hippocampus and temporal pole, and also right entorhinal cortex (with rather unchanged entropy within left entorhinal and para-hippocampal cortices). Furthermore, we observed a higher entropy during immediate post seizure than during POI 1 (probably reflecting the beginning of the PE decrease process) POI 2, clinically characterised by a chronic state of anterograde amnesia, was characterised by a decreased of entropy within left amygdala, temporal pole, entorhinal, and para-hippocampal cortices, also affecting right entorhinal cortex (with unchanged entropy within left hippocampus). Finally, POI 3, clinically characterised by complete recovery of the memory functioning, showed a normalisation of the entropy value across areas, at the exception of the left temporal pole (Figure 2 and Supplementary Material). Noteworthy, the right entorhinal stimulation induced first a decrease of entropy during the seizure followed by an increase of entropy during postseizure period, both limited to the right entorhinal cortex (Supplementary Material).

Discussion

We studied a case of transient amnesia induced by direct electrical stimulation of the rhinal cortices in a patient with refractory focal structural epilepsy explored by SEEG. This report shows that a stimulated seizure, even of short duration, involving the mesial temporal structures, particularly the hippocampi and rhinal cortices, is able to induce a transient but prolonged post-ictal amnesia associated with prolonged changes in the SEEG signal and entropy. The dynamics of these modifications are correlated with the clinical evolution, the recovery of a normal entropy corresponding to a return to a normal memory state. In the original paper by Palmini et al, which introduced the entity of pure amnestic seizures, amnesia was attributed to epileptic discharge limited to uni-or bilateral mesio-temporal structures also depending on the underlying structural damage [START_REF] Palmini | Pure amnestic seizures in temporal lobe epilepsy[END_REF]). Interestingly, our results are in agreement with these observations, although in our case the amnesia was essentially a post-ictal manifestation. Moreover, the left insular scar of our patient may have weaken her memory system and favour her post-ictal memory deficit, as left insular infarction demonstrated poorer verbal memory performance [START_REF] Manes | Verbal memory impairment after left insular cortex infarction[END_REF].

Τhe question of whether amnestic seizures reflect ictal activity, or a postictal state has been posed since the early years of their description of the TEA syndrome or more generally in temporal lobe epilepsies [START_REF] Tassinari | Transient Global Amnesia as a Postictal State from Recurrent Partial Seizures[END_REF]. It has been hypothesized by various clinical descriptions that both scenarios could possibly occur, depending on the patient and the underlying pathology [START_REF] Zeman | Transient epileptic amnesia: a description of the clinical and neuropsychological features in 10 cases and a review of the literature[END_REF]. Interestingly, in a study of 30 patients with episodic memory dysfunction, interictal memory impairment, and a diagnosis of epilepsy (defined as epileptic amnesia syndrome, by the authors), 76.7% of them exhibited transient anterograde amnesia after the seizures (sometimes clinically subtle: short loss of contact and automatisms). Therefore, postictal amnesia was the most notable symptom leading patients to seek medical attention [START_REF] Gallassi | Epileptic Amnesic Syndrome: An Update and Further Considerations[END_REF]. Long term amnestic seizures with anterograde amnesia lasting more than 1 hour have been reported in this syndrome [START_REF] Butler | The syndrome of transient epileptic amnesia[END_REF][START_REF] Mosbah | Clinical, neuropsychological, and metabolic characteristics of transient epileptic amnesia syndrome[END_REF]. We show here that these probably relate to prolonged post-ictal alteration of entropy within mesio-temporal structures. An underlying structural pathology may favour this type of manifestation (cf. post-traumatic lesions in our patient). In TEA syndrome, chronic alterations of hippocampal regions have been suggested by morphometric studies [START_REF] Butler | Transient epileptic amnesia: regional brain atrophy and its relationship to memory deficits[END_REF]) and PET metabolic imaging [START_REF] Mosbah | Clinical, neuropsychological, and metabolic characteristics of transient epileptic amnesia syndrome[END_REF].

The interpretation of the entropy changes in post ictal phases remains hypothetical. The seizure is accompanied by a marked drop in entropy which probably corresponds to a massive decrease in signal complexity [START_REF] Yang | Epileptic Seizure Prediction Based on Permutation Entropy[END_REF]). To our knowledge, no specific study has investigated the correlation between entropy change and post-ictal recovery. The immediate post-ictal phase ("Post seizure") and the "POI1" period (persistence of a fairly global amnesia) are mainly characterized by an increase in entropy within the SEEG signal in the bilateral mesio-temporal structures. This apparent paradoxical entropy increase could be interpreted either as a compensatory and recuperative phenomenon or as an abnormal entropy too close to a stochastic random signal. It is likely that the information contained in the EEG signal to be informative requires an equilibrium in the entropy values. The post-ictal phase with pure anterograde amnesia (POI2,) corresponds to a decrease in entropy observed in the entorhinal cortex on both sides. This decrease in entropy corresponds classically to a decrease in signal complexification and information depletion [START_REF] Bandt | Permutation entropy: a natural complexity measure for time series[END_REF].

In conclusion, we report a case with prolonged post ictal amnesia associated with alteration of signal complexity within bilateral mesio-temporal structures. It could be a process involved in the semiology of prolonged amnestic seizures particularly observed in the transient epileptic amnesia syndrome.

FIGURES LEGENDS

Figure 1 A. Axial and coronal slice views of the electrodes of interest (post-operative CT and preoperative MRI fusion). The contacts are numbered in ascending order from the most mesial part (i.e. contact number 1 is the most medial contact of the electrode). 3D representations of the electrodes of the patients. A postoperative computerized tomography (CT) scan was performed to verify the absence of bleeding and the position of each recording lead. Then, a fusion of CT/MRI data was performed to identify and locate precisely each contact along the electrode trajectory using a specific inhouse software (Gardel, available at: http://meg.univ-amu.fr/wiki/GARDEL:presentation) (Medina Villalon et al. 2018). The signals were recorded on a 256-channel Deltamed™ system and were sampled at 1024 Hz. Signal analysis was performed with Anywave software (available at: https://meg.univ-amu.fr/wiki/AnyWave) (Colombet et al. 2015). Electrical stimulation was produced by a regulated neurostimulator designed for safe diagnostic stimulation of the human brain (Inomed®).

The specific entropy method chosen was the "Permutation Entropy" (PE) [START_REF] Bandt | Permutation entropy: a natural complexity measure for time series[END_REF] which considers the complexity of the electrophysiological signals. PE is based on a measure of Shannon entropy of the relationships between neighbouring values of the time series. The method is rendered more robust than standard information-based method by calculating the entropy for n "embedding dimensions" in the range of 3 to 7, (i.e., different number of neighbours or samples). We used a fixed delay t which indicates the distance between neighbours (t = 1 for consecutive samples). With π describing the dynamics between the neighbour values (e.g.,/ ' < / '(& ), )(*) probability of this order and n embedding dimensions. hn gives a value between 0 and 1, 0 meaning that the signal is perfectly organized (for example in the case of a pure sine wave) and 1 completely disorganized (completely random noise). The algorithm used to apply permutation entropy on our data is the one developed by Unakafova (Unakafova and Keller 2013) due to its algorithmic performance. The efficiency of this method stems from the use of pre-filled tables to describe the order relation between neighbouring values. Before applying permutation entropy on a given seizure signal, we performed careful selection of channels by keeping those in the grey matter and removing the white matter channels. A bipolar montage was used (i.e., difference of consecutive channels). Permutation entropy was computed on each channel, on a sliding window of 10s with an overlap of 5s, an order of 3 and a delay of 1 sample.

Statistically we evaluated the z-score of the PE value by period and region and the p-value determined by using Wilcoxon test was corrected by applying a Bonferroni correction to account for multiple comparisons.

  1. Seizure: period with ictal discharge 2. Post-seizure: the 225 seconds just after this ictal discharge 3. Period of Interest (POI) 1 = early amnesia: the first 500 seconds of amnesia, characterised by a marked amnesia with anterograde encoding and with major retrograde autobiographical amnesia 4. POI2 = persistent anterograde amnesic state: 1 hour after the onset of amnesia characterised by the recovery of retrograde memory but persistence of an anterograde amnesia 5. POI3 = complete recovery: the day after the episode when the patient has completely recovered her usual memory (in the morning, first period with certainty of memory recovery because of clinical testing)

B.Figure 2 A)

 2 Figure 2

we thank Dr Romain Carron for the implantation of SEEG electrodes and Prof Didier The research leading to these results received funding from Agence Nationale de la Recherche, under Grant/ Award Number: ANR-11-IDEX-0001-02 and ANR-13-PRTS-0011-

DATA AVAILABILITY STATEMENT

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Supplementary Table