Chen Liu

Ezequiel E Ferrero

Eduardo A Jagla

Kirsten Martens

Alberto Rosso

Laurent Talon

The Fate of Shear-Oscillated Amorphous Solids

The behavior of shear-oscillated amorphous materials is studied using a coarse-grained model. Samples are prepared at different degrees of annealing and then subject to athermal and quasistatic oscillatory deformations at various fixed amplitudes. The steady-state reached after several oscillations is fully determined by the initial preparation and the oscillation amplitude, as seen from stroboscopic stress and energy measurements. Under small oscillations, poorly annealed materials display shear-annealing, while ultra-stabilized materials are insensitive to them. Yet, beyond a critical oscillation amplitude, both kind of materials display a discontinuous transition to the same mixed state composed by a fluid shear-band embedded in a marginal solid. Quantitative relations between uniform shear and the steady-state reached with this protocol are established. The transient regime characterizing the growth and the motion of the shear band is also studied.

Amorphous solids are a vast class of materials, common in nature and ubiquitous for human applications. Their mechanical behaviour is strongly affected by the preparation protocol: poorly annealed materials, such as emulsions, foams or gels, are soft and ductile, while well annealed materials such as metallic glasses, ceramics and silica are hard and brittle [START_REF] Bonn | Yield stress materials in soft condensed matter[END_REF][START_REF] Nicolas | Deformation and flow of amorphous solids: Insights from elastoplastic models[END_REF]. Under a uniform deformation, the former tend to melt into a liquid, while the latter fail with a sharp stress-drop and the appearance of a thin liquid shear-band [START_REF] Ozawa | Random critical point separates brittle and ductile yielding transitions in amorphous materials[END_REF][START_REF] Popović | Elastoplastic description of sudden failure in athermal amorphous materials during quasistatic loading[END_REF][START_REF] Barlow | Ductile and brittle yielding in thermal and athermal amorphous materials[END_REF]. Although we tend to distinguish between these two kind of yielding, a debate is open: ductile materials could also display stress overshoots [START_REF] Barlow | Ductile and brittle yielding in thermal and athermal amorphous materials[END_REF][START_REF] Ozawa | Rare events and disorder control the brittle yielding of amorphous solids[END_REF][START_REF] David | Finite-size study of the athermal quasistatic yielding transition in structural glasses[END_REF][START_REF] Fielding | Yielding, shear banding and brittle failure of amorphous materials[END_REF]. Beyond finite-size issues [START_REF] Ozawa | Rare events and disorder control the brittle yielding of amorphous solids[END_REF][START_REF] David | Finite-size study of the athermal quasistatic yielding transition in structural glasses[END_REF][START_REF] Fielding | Yielding, shear banding and brittle failure of amorphous materials[END_REF], dealing only with transient states does not help to settle the discussion. This is where an oscillatory deformation protocol [START_REF] Fiocco | Encoding of memory in sheared amorphous solids[END_REF][START_REF] Leishangthem | The yielding transition in amorphous solids under oscillatory shear deformation[END_REF][START_REF] Das | Annealing glasses by cyclic shear deformation[END_REF][START_REF] Hexner | Periodic training of creeping solids[END_REF][START_REF] Sastry | Models for the yielding behaviour of amorphous solids[END_REF][START_REF] Khirallah | Yielding in an integer automaton model for amorphous solids under cyclic shear[END_REF][START_REF] Yeh | Glass stability changes the nature of yielding under oscillatory shear[END_REF][START_REF] Kawasaki | Macroscopic yielding in jammed solids is accompanied by a nonequilibrium first-order transition in particle trajectories[END_REF][START_REF] Regev | Onset of irreversibility and chaos in amorphous solids under periodic shear[END_REF][START_REF] Regev | Reversibility and criticality in amorphous solids[END_REF] becomes handy, since it opens the possibility of characterizing a transition in terms of stationary states. In molecular dynamics (MD) simulations, two different "phases" have been identified: For moderate strain amplitudes Γ, the material appears solid and is progressively annealed [START_REF] Leishangthem | The yielding transition in amorphous solids under oscillatory shear deformation[END_REF][START_REF] Das | Annealing glasses by cyclic shear deformation[END_REF][START_REF] Yeh | Glass stability changes the nature of yielding under oscillatory shear[END_REF][START_REF] Kawasaki | Macroscopic yielding in jammed solids is accompanied by a nonequilibrium first-order transition in particle trajectories[END_REF][START_REF] Regev | Onset of irreversibility and chaos in amorphous solids under periodic shear[END_REF] as oscillation cycles accumulate. Above a critical amplitude Γ c , further shear-annealing is prevented and a shear-band coexists with a solid [START_REF] Leishangthem | The yielding transition in amorphous solids under oscillatory shear deformation[END_REF][START_REF] Das | Annealing glasses by cyclic shear deformation[END_REF][START_REF] Kawasaki | Macroscopic yielding in jammed solids is accompanied by a nonequilibrium first-order transition in particle trajectories[END_REF][START_REF] Regev | Onset of irreversibility and chaos in amorphous solids under periodic shear[END_REF][START_REF] Fiocco | Oscillatory athermal quasistatic deformation of a model glass[END_REF]. Despite a rapidly growing literature [START_REF] Fiocco | Encoding of memory in sheared amorphous solids[END_REF][START_REF] Leishangthem | The yielding transition in amorphous solids under oscillatory shear deformation[END_REF][START_REF] Das | Annealing glasses by cyclic shear deformation[END_REF][START_REF] Hexner | Periodic training of creeping solids[END_REF][START_REF] Sastry | Models for the yielding behaviour of amorphous solids[END_REF][START_REF] Khirallah | Yielding in an integer automaton model for amorphous solids under cyclic shear[END_REF][START_REF] Yeh | Glass stability changes the nature of yielding under oscillatory shear[END_REF][START_REF] Kawasaki | Macroscopic yielding in jammed solids is accompanied by a nonequilibrium first-order transition in particle trajectories[END_REF][START_REF] Regev | Onset of irreversibility and chaos in amorphous solids under periodic shear[END_REF][START_REF] Regev | Reversibility and criticality in amorphous solids[END_REF][START_REF] Fiocco | Oscillatory athermal quasistatic deformation of a model glass[END_REF][START_REF] Priezjev | Reversible plastic events during oscillatory deformation of amorphous solids[END_REF][START_REF] Priezjev | Collective nonaffine displacements in amorphous materials during large-amplitude oscillatory shear[END_REF][START_REF] Regev | The irreversibility transition in amorphous solids under periodic shear[END_REF][START_REF] Regev | Critical diffusivity in the reversibility-irreversibility transition of amorphous solids under oscillatory shear[END_REF][START_REF] Mungan | Networks and hierarchies: How amorphous materials learn to remember[END_REF][START_REF] Bhaumik | The role of annealing in determining the yielding behavior of glasses under cyclic shear deformation[END_REF][START_REF] Parmar | Strain localization above the yielding point in cyclically deformed glasses[END_REF][START_REF] Das | Unified phase diagram of reversible-irreversible, jamming, and yielding transitions in cyclically sheared soft-sphere packings[END_REF], basic questions remain to be addressed: (i) the nature of the transition at Γ c , (ii) the relation between steady-states in the two protocols (oscillatory and uniform shear), and (iii) how those states are reached.

To answer these questions, in this Letter we use a coarse-grained approach, which goes beyond natural limitations of MD simulations. We find that the transition from solid to flow at Γ c is discontinuous. We quantitatively relate properties of the emerging phases in oscillatory shear with those of the uniform-deformation protocol. Moreover, we describe transient stages previously unexplored. To do that, we prepare samples at differ-ent degrees of annealing, characterized by the initial energy E init per unit volume, and subject them to oscillatory deformations at various fixed amplitudes Γ until steady-states are reached. We find that once a driving condition (E init ,Γ) is chosen, it univocally defines the material's fate in the steady-state; allowing us to fill a sort of phase diagram. Very well-annealed samples (E init < E * , with E * a critical annealing level) are insensitive to small oscillations, while poorly-annealed samples (E init > E * ) exhibit shear-annealing for large enough sub-critical Γ. We therefore distinguish between stable solids, that keep memory of their initial condition, and marginal solids which result from poorly-annealed samples loosing memory of their initial conditions due to the shear-annealing by oscillations. Yet, when the oscillation amplitude reaches the critical threshold Γ c (E init ), every sample undergoes a discontinuous transition, comprising a jump in stress and energy, towards the same mixed solid-fluid state. Part of the system melt in a shear-band while the rest becomes a critical solid of energy E * , independently on the initial condition. The fluid in the band corresponds to the one of the stationary state at large uniform quasi-static shear and holds at most the uniform yield stress Σ y . As Γ is further increased, the width of the shear-band scales as a power of (Γ -Σ y /G), with G the effective shear-modulus, further relating uniform and oscillatory protocols. Also, a rich transient dynamics towards the steady-state above Γ c is unveiled. We show a band-width growth as a function of the number of oscillation cycles that is reminiscent of critical domain coarsening. Surprisingly, once the band reaches its stable width it ballistically sweeps out the deeply-annealed regions, if present, turning them in the critical solid of energy E * , and finally displays an (anomalous) diffusion. Overall, our Letter offers answers to fresh issues raised by MD simulations of glasses. Methods -Our modelling is based on the evolution of a two-dimensional scalar strain field in a disordered potential, previously used to study yielding under uniform deformation [START_REF] Jagla | Strain localization driven by structural relaxation in sheared amorphous solids[END_REF][START_REF] Aguirre | Critical exponents of the yielding transition of amorphous solids[END_REF][START_REF] Ferrero | Elastic interfaces on disordered substrates: From mean-field depinning to yielding[END_REF][START_REF] Jagla | Tensorial description of the plasticity of amorphous composites[END_REF][START_REF] Ferrero | Properties of the density of shear transformations in driven amorphous solids[END_REF]. Initial sample annealing is achieved by tuning the configuration of the disordered landscape (lower initial energies E init for better annealed systems). Within this picture, the strain field behaves as a manifold propagating in a disorder medium [START_REF] Aguirre | Critical exponents of the yielding transition of amorphous solids[END_REF][START_REF] Ferrero | Elastic interfaces on disordered substrates: From mean-field depinning to yielding[END_REF][START_REF] Lin | Scaling description of the yielding transition in soft amorphous solids at zero temperature[END_REF]. Two essential ingredients are incorporated: the quadruple Eshelby's elastic interaction [START_REF] Nicolas | Deformation and flow of amorphous solids: Insights from elastoplastic models[END_REF][START_REF] Eshelby | The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems[END_REF] and a disorder that rather than originating in quenched impurities mimics the random positions of particles. The local disorder changes irreversibly, even if the strain manifold revisits the same location. Details of our method are presented in the Supplemental Material (SM) [35].

Results -In view of their relevance in interpreting the shear oscillation results, we first reproduce some key features in uniform shear. In Fig. 1(a) we show stress-strain curves for individual samples prepared at different degrees of annealing E init (see also SM[35]). After an initial linear elastic response of slope G ≈ 0.91, all samples reach at large strain γ 3 a common plateau corresponding to the yield stress Σ y = 0.375 ± 0.003. The characteristic strain γ y where plasticity becomes important is estimated as γ y = Σ y /G 0.413. Poorly annealed materials display a monotonic crossover from an elastic solid to a liquid crossing γ y , while well annealed systems remain elastic above γ y up to a failure strain γ f where a sharp stress downfall occurs. The overshoot gets larger for better annealed samples [START_REF] Ozawa | Random critical point separates brittle and ductile yielding transitions in amorphous materials[END_REF][START_REF] Popović | Elastoplastic description of sudden failure in athermal amorphous materials during quasistatic loading[END_REF][START_REF] Barlow | Ductile and brittle yielding in thermal and athermal amorphous materials[END_REF][START_REF] Fielding | Yielding, shear banding and brittle failure of amorphous materials[END_REF]. In general, the total energy per volume of the material can be decomposed into two contributions:

E tot = E sf + Σ 2 /(4µ),
where the second contribution is due to macroscopic elastic deformation (with µ = 1/2 the shear modulus) and E sf , called 'stress-free energy' afterward, characterizes the state of the material with zero macroscopic stress. In the large uniform shear limit γ → ∞, where the memory of the initial state is completely erased, all samples converge to a homogeneous stationary liquid state with E stat sf = -0.122 ± 0.001 ≡ E liq (see SM [35] for details). Now, in our oscillatory protocol, the material is deformed by quasi-statically varying the strain from 0 to Γ, then to -Γ and subsequently back to 0 to complete one cycle of amplitude Γ. This cycle is repeated until a steady-state is reached. By varying Γ and E init , we obtain an assortment of stationary states. Figure 1(a) shows the steady-state maximum stress [START_REF] Leishangthem | The yielding transition in amorphous solids under oscillatory shear deformation[END_REF][START_REF] Yeh | Glass stability changes the nature of yielding under oscillatory shear[END_REF][START_REF] Fiocco | Oscillatory athermal quasistatic deformation of a model glass[END_REF]] Σ(Γ) as a function of Γ in comparison with uniform-shear load curves. Consistent with MD results, all samples exhibit a sharp jump of Σ(Γ) at a critical strain amplitude Γ c (E init ). The moderately annealed samples (E init = -0.095, -0.12 and -0.149) fail at the same Γ c ≈ 0.4725 ≡ Γ * , independently on the initial degree of annealing; while the more annealed samples (E init = -0.168 and -0.199) experience a later failure at Γ c (E init ) ≈ γ f (E init ) > Γ * , stronger as better annealed is the sample. Once above Γ c , the steady-state stress Σ(Γ > Γ c ) for any initial state, is well identified with Σ y of a stationary flowing state in the quasi-static uniform shear deformation. Figure 1(b) displays the stationary stress-free energy E stat sf as a function of Γ [START_REF]Esf characterizes the intrinsic state without external load, while the stroboscopic energy[END_REF]. For very small Γ, the stationary state E stat sf strongly depends on E init . Increasing Γ within the solid phase (Γ < Γ c ), two scenarios separated by a critical annealing E * ≈ -0.16 are observed. Samples with E init < E * keep a perfect memory of their initial states up to melting, evidenced by E stat sf (Γ) = E init for all Γ < Γ c (E init ) [START_REF] Yeh | Glass stability changes the nature of yielding under oscillatory shear[END_REF][START_REF] Bhaumik | The role of annealing in determining the yielding behavior of glasses under cyclic shear deformation[END_REF]. Yet, for E init > E * the phenomenology is richer: Increasing Γ, at some point, the stationary-state E stat sf starts to decrease with the amplitude, loosing memory of the initial condition. This is a manifestation of shear-annealing in the steady-state, as similarly observed in [START_REF] Leishangthem | The yielding transition in amorphous solids under oscillatory shear deformation[END_REF][START_REF] Yeh | Glass stability changes the nature of yielding under oscillatory shear[END_REF][START_REF] Bhaumik | The role of annealing in determining the yielding behavior of glasses under cyclic shear deformation[END_REF][START_REF] Parmar | Strain localization above the yielding point in cyclically deformed glasses[END_REF][START_REF] Das | Unified phase diagram of reversible-irreversible, jamming, and yielding transitions in cyclically sheared soft-sphere packings[END_REF]. Notice that, for initially weakly annealed systems, there exist also a transient process of shear-annealing (described later on), where samples reach a lower-energy steadystate by oscillating at a fixed Γ. E stat sf for E init > E * ends up collapsing to a common curve (purple dashed line in Fig. 1(b)), where the initial condition becomes irrelevant and E stat sf depends only on Γ. We call "marginal solids" the stationary states lying on the purple dashed line, which terminates at Γ * at the critical energy E * . Once Γ > Γ c (E init ), both Σ(Γ) and E stat sf (Γ) fall onto a unique curve, independent on E init . The system develops a permanent liquid shear-band embedded in a solid phase, and the steady-state energy E stat sf increases with Γ (Fig. 1(b)), see also [START_REF] Yeh | Glass stability changes the nature of yielding under oscillatory shear[END_REF][START_REF] Fiocco | Oscillatory athermal quasistatic deformation of a model glass[END_REF][START_REF] Parmar | Strain localization above the yielding point in cyclically deformed glasses[END_REF]. From the previous discussion, we can associate to each E init an amplitude Γ a (E init ) above which the steady-state E stat sf (Γ, E init ) is independent on the initial annealing E init .

As summarized in a diagram Fig. 1(c), Γ a (E init ) (dashed line) identifies with Γ c (E init ) (solid line) below E * , and bifurcates above it. The solid steady-states resulting from driving conditions below Γ a (E init ) are strongly reminiscent of the initial state and called "stable solids". In contrast, those driving conditions in between Γ c and Γ a correspond to solids that forget their initial condition and end up being "marginal solids" in the steady-state. Driving conditions in the region above Γ c (E init ) lead to a steady-state mixed phase composed by a fluid shear band surrounded by a critical marginal solid, which we detail in the following.

Fig. 2 shows the averaged stress-free energy profile across a section orthogonal to the shear-band (evidenced in the strain field in the left-inset). The energy profile characterizes the band by a bell-shape. The interior of the band (top of the bell) has the same energy as the stationary liquid in uniform shear deformation E liq ≈ -0.122. The width w s of the band increases with Γ and is well fitted by w s (Γ) = a(Γ -Γ 0 ) α . A good collapse is found when rescaling the profiles with α = 1/2 (right-inset of Fig. 2), reminiscent of the transient band dynamics in uniform shear [START_REF] Jagla | Shear band dynamics from a mesoscopic modeling of plasticity[END_REF]. We further notice two peculiarities: First, from the fit, Γ 0 identifies with γ y instead of Γ * , implying that at the transition, the shear-band has a finite width w s (Γ * ) > 0 (because γ y < Γ * ). Second, the energy profile is independent on E init and the energy at the bottom of the bell-shape coincides with the critical solid energy E * . This allows to conclude that the mixture phase is completely independent on the initial degree of annealing and uniquely characterized by the width w s : the stationary regime is composed of a fraction of critical solid at energy E * and a shear-band made of the liquid of energy E liq . Since E liq > E * , the monotonic growth of E stat sf (Fig. 1(b)) can be rationalized in terms of the shear-band widening with Γ:

E stat sf E * + (E liq -E * ) a(Γ-γy) α L
. This is tested in Fig. 1(b). It also implies that a discontinuity of E stat sf at the transition Γ c = Γ * , even though small, is expected [START_REF] Leishangthem | The yielding transition in amorphous solids under oscillatory shear deformation[END_REF][START_REF] Yeh | Glass stability changes the nature of yielding under oscillatory shear[END_REF][START_REF] Bhaumik | The role of annealing in determining the yielding behavior of glasses under cyclic shear deformation[END_REF][START_REF] Parmar | Strain localization above the yielding point in cyclically deformed glasses[END_REF][START_REF] Das | Unified phase diagram of reversible-irreversible, jamming, and yielding transitions in cyclically sheared soft-sphere packings[END_REF]. In addition, we observed that w s is proportional to the linear system size, as also seen in the fact that E stat sf shows no size dependence [35]. We now discuss the transient oscillatory dynamics. Figs. 3(a-b) show the evolution of E sf (n), as the oscillation cycles n accumulate, for different driving conditions (E init , Γ). Below Γ c (Fig. 3(a)), ultra-stable systems (E init < E * ) are unperturbed by the oscillations, while systems with E init > E * shear-anneals if the amplitude is large enough. At amplitudes above Γ a (E init ) all initial conditions are shear-annealed to the same stationary state, as observed for our three softest samples in Fig. 3(a). Above Γ c (Fig. 3(b)), all samples go to the same stationary state at large n. Our observations for the number n T of cycles to reach steady-states are compatible with a divergence when approaching Γ c reported in previous works [START_REF] Leishangthem | The yielding transition in amorphous solids under oscillatory shear deformation[END_REF][START_REF] Sastry | Models for the yielding behaviour of amorphous solids[END_REF][START_REF] Khirallah | Yielding in an integer automaton model for amorphous solids under cyclic shear[END_REF][START_REF] Kawasaki | Macroscopic yielding in jammed solids is accompanied by a nonequilibrium first-order transition in particle trajectories[END_REF][START_REF] Fiocco | Oscillatory athermal quasistatic deformation of a model glass[END_REF]. Data for (Γ < Γ c ) is shown in SM [35]. We switch now to the observation of different dynamical regimes for the mixed phase. For poorly annealed samples, the onset of the shear-band and its stabilization in a stationary width w s (Γ) occurs gradually and rapidly as the solid region shear-anneals to the critical state in a small number of cycles. Then the shear band diffuses in the material (see Fig. 4(a) and Video 1 [35]) with a mobility that increases with Γ. In well annealed systems instead, the transient is richer and we identify three dynamical regimes (Fig. 4(b), see also Video 2[35]): (i) Shear band formation and growth. The initial band growth is observed at the top of Fig. 4(b) (see [35] for details). This initial coarsening of the shearband is studied in Fig. 3(c) at different Γ. In all cases, the band width grows as ∼ n 1/3 until reaching the stationary width w s ∼ (Γ -γ y ) 1/2 . (ii) Melting of the solid. The now fully-formed shear band displays a ballistic motion preferentially invading the deeply annealed solid and leaving behind the critical solid of energy E * . (iii) Shear band diffusion. When the shear-band has visited the entire system, all the regions outside the band have been modified into the critical marginal solid. The stationary state is reached and the band diffuses forever, maintaining its characteristic Γ-dependent width.

Discussion -Our mesoscopic model allows to recover the phenomenology of the oscillatory shear of amorphous materials reported by MD simulations [START_REF] Ozawa | Random critical point separates brittle and ductile yielding transitions in amorphous materials[END_REF][START_REF] Leishangthem | The yielding transition in amorphous solids under oscillatory shear deformation[END_REF][START_REF] Yeh | Glass stability changes the nature of yielding under oscillatory shear[END_REF][START_REF] Bhaumik | The role of annealing in determining the yielding behavior of glasses under cyclic shear deformation[END_REF][START_REF] Das | Unified phase diagram of reversible-irreversible, jamming, and yielding transitions in cyclically sheared soft-sphere packings[END_REF] and go beyond. While some emerging features can be already captured by even further simplified toy models as seen recently [START_REF] Sastry | Models for the yielding behaviour of amorphous solids[END_REF], keeping an spatial extent for the system and its geometry gives access to the full picture. We are able to univocally classify the oscillatory stationary states according to a "phase diagram" in the space of oscillation amplitude Γ and initial annealing level E init , and relate them with the steady-states of the uniform-deformation protocol. Interestingly, the oscillatory protocol always shows a discontinuous transition in both stress and energy at a critical strain amplitude Γ c , even in the case of ductile materials that melt homogeneously under uniform deformation. Notably, ductile materials mechanically anneal and harden at sub-critical amplitudes under oscillatory shear. Independently on the initial state, every system sheared at large amplitudes ends up in a steady-state displaying a permanent shear-band. This band contains a stationary fluid identical to the one obtained at large uniform deformations and is embedded in a solid matrix which is not at all arbitrary. The solid surrounding the shear-band in this mixed phase is the critical marginal solid that has the energy of the critical annealing level E * . A line of amnesia Γ a (E init ) discriminate driving conditions (E init , Γ) between stable solids and solids that shear-anneal and evolve towards a marginal line in the steady-state. Our approach gives also transparent access to the transient regimes of shear-band formation, growth and motion. For deeply-annealed samples, a small but finite width band first spreads the system, then grows with a power-law ∝ n 1/3 of the number of oscillation cycles. Ulteriorly, the band invades ballistically the deeplyannealed solid, leaving behind the critical solid of energy E * . Finally, is diffuses anomalously over all the system. These results shed light to clarify some previous observations in the literature and open interesting directions of new research. One could ask, for example, whether the emergence of a transient state of multiple shear-bands is possible in the large system limit, and how do they merge in a single band in the steady-state (or not). Also, to which extent the incorporation of "reversible" plastic events observed in atomistic simulations [16-20, 23, 24] enriches the overall phenomenology. Those are absent in our present model by choice, but easy to include by locally quenching the disordered potential [START_REF] Khirallah | Yielding in an integer automaton model for amorphous solids under cyclic shear[END_REF]. Further, the critical annealing E * detected in the oscillatory protocol could play also a relevant role in uniform shear, where the ductile-brittle yielding transition is a matter of vivid discussion [START_REF] Ozawa | Random critical point separates brittle and ductile yielding transitions in amorphous materials[END_REF][START_REF] Barlow | Ductile and brittle yielding in thermal and athermal amorphous materials[END_REF][START_REF] Fielding | Yielding, shear banding and brittle failure of amorphous materials[END_REF]. It is indeed suggestive that the load curve of our closest to E * sample sits near to the apparent transition between ductile/brittle yielding probed by the uniform shear curves. More importantly, it will be interesting to test experimentally these predictions, with new measurements closer to the quasi-static limit, since in the standard setups the oscillations are relatively fast and the material softens instead of annealing and hardening. In general, we hope that our results will motivate new research directions in the study of low temperature amorphous materials, from realistic large-scale atomistic simulations to experiments.

Figure 1 .

 1 Figure 1. Stationary properties-(a) Symbols: Maximum stress Σ(Γ) versus the oscillatory strain amplitude Γ. Curves: Stressstrain produced by quasi-static uniform shear. Different colors code for different degrees of initial annealing, signaled in (b). System size is 128 2 . (b) Stress-free steady energy E stat sf versus the Γ. The purple dashed-line corresponds to "marginal solids". The orange solid line displays a fit of E stat sf (Γ) discussed in the text. (c) "Phase diagram" classifying driving conditions (Einit,Γ) according to the steady-state they will reach: stable solids (strongly reminiscent of the initial state), marginal solids (shearannealed solids independent of the initial condition), and solid-liquid mixtures characterized by an erratic shear-band. Color circles indicate Γc for different initial degrees of annealing Einit.

Figure 2 .

 2 Figure 2. Energy profiles averaged along the y-axis for different Γ. x = 0 is set to the maximum location. Left inset: map of strain field during a half-cycle in the steady-state. Right inset: profiles collapse using ws(Γ) = a(Γ -Γ0) α with Γ0 = γy = 0.413 and α = 1/2.

Figure 3 .

 3 Figure 3. Transient properties-(a-b) Evolution of stress-free energy Esf as the number of cycles n increases, for different amplitudes. Colors coding for different Einit are the same as in Fig.1. System size is 128 2 . (a) Solid phase (Γ < Γc). (b) Mixed phase (Γ > Γc). Inset in (b) shows a zoom-in to appreciate the difference in E stat sf for different Γ.(c) Shear-band width w as a function of n for different Γ. The initial condition is a well annealed solid (Einit = -0.168). The inset shows raw-data and the main panel the scaling w(Γ -γy) -1/2 vs. n(Γ -γy) 1/2 . The red dashed line is a power-law ∼ n 1/3 . System size is 256 2 .

Figure 4 .

 4 Figure 4. Energy profiles Eprofile(x) evolving as function of the number n of oscillation cycles for Γ = 0.7. Light colors represent higher energies and evidence the position of the shear-band. (a) A poorly annealed sample with Einit ≈ -0.12.(b) Well annealed sample with Einit ≈ -0.199. The red dashed-line in (b) indicates the stabilization of the band width at n ≈ 220. When the blue dashed-line is reached, the band has visited the entire system.

ACKNOWLEDGMENTS

We are indebted to M. Ozawa for illuminating discussions on the subject. We thank S. Sastry for kind feed-back on an earlier version of this manuscript. EEF acknowledges support from PICT-2017-1202. Authors acknowledge support from the collaboration project ECOS Sud-MINCyT A16E01. CL, AR and LT acknowledges support by "Investissement d'Avenir" LabEx PALM (Grant No. ANR-10-LABX-0039-PALM)


Emilie Bucourt

Virginie Martaillé

Philippe Goupille

Isabelle Joncker-Vannier

Brigitte Huttenberger

Christian Reveillere

Denis Mulleman

Robert Courtois
email: robert.courtois@univ-tours.fr

A comparative study of fibromyalgia, rheumatoid arthritis, spondyloarthritis and Sjogren's syndrome; impact of the disease on quality of life, psychological adjustment and use of coping strategies

Keywords: Fibromyalgia, rheumatic diseases, impact of the disease, psychological adjustment, coping, rheumatic diseases

Background: Fibromyalgia, rheumatoid arthritis, spondyloarthritis and Sjögren's syndrome are chronic rheumatic diseases with very different clinical characteristics, but which share symptoms such as pain and fatigue. The aim of the study was to examine the impact of the disease on psychological adaptation in fibromyalgia compared to other rheumatic diseases (rheumatoid arthritis, spondyloarthritis and Sjögren's syndrome).

Method:

In a multicenter study, 165 women with rheumatic diseases (48 with fibromyalgia, 47 with rheumatoid arthritis, 47 with spondyloarthritis, 23 with Sjögren's syndrome) completed the GHQ-28 (emotional distress), FSS (fatigue), FIQ (impact of the disease), CSQ (coping), and MINI (comorbidity with DSM IV axis-I disorders). We used the Kruskal-Wallis test, Mann-Whitney U-test and the Chi² test to compare comorbid anxiety and depressive disorders, and to compare the impact of the disease on patients' mental well-being and daily life and adjustment (coping strategies).

Results: Anxiety and depressive disorders were more common in fibromyalgia patients; they had higher scores on impact of the disease, physical symptoms, pain and fatigue than rheumatoid arthritis patients, and reported more fatigue than patients with spondyloarthritis.

Overall, they used more maladaptive coping strategies (less use of Distancing from pain than patients with rheumatoid arthritis and spondyloarthritis, less use of Ignoring pain sensations, and more use of Catastrophizing than those with rheumatoid arthritis). No differences were found between fibromyalgia and Sjögren's syndrome on impact and adjustment.

Conclusions: Compared to other rheumatic diseases, fibromyalgia has a greater impact on daily life; patients have more difficulty adjusting to the disease and generally use poorer strategies to cope with pain.

Introduction

Fibromyalgia, rheumatoid arthritis, spondyloarthritis and Sjögren's syndrome are chronic diseases with different clinical characteristics, but which share symptoms such as pain and fatigue. Unlike the other three rheumatic diseases, which are clearly recognizable by their clinical, imaging and biological features, fibromyalgia is a diagnosis of exclusion, which can only be established once other conditions have been ruled out and could affect the way that patients cope and adjust to the disease. All these diseases have a major impact on patients' daily life, requiring considerable effort to cope with the illness. As most existing studies comparing fibromyalgia with other rheumatic diseases only consider it when it is associated with another rheumatic disease, they do not allow a clear distinction to be made between how patients adjust psychologically to each disease.

Studies have examined symptoms of fibromyalgia in inflammatory or autoimmune rheumatic diseases and the relationship with disease activity [START_REF] El-Rabbat | Clinical significance of fibromyalgia syndrome in different rheumatic diseases: Relation to disease activity and quality of life[END_REF][START_REF] Haliloglu | Fibromyalgia in patients with other rheumatic diseases: prevalence and relationship with disease activity[END_REF]. Other studies found that fibromyalgia patients had higher pain scores [START_REF] Laursen | Health related quality of life and quantitative pain measurement in females with chronic non-malignant pain[END_REF][START_REF] Tander | A comparative evaluation of health related quality of life and depression in patients with fibromyalgia syndrome and rheumatoid arthritis[END_REF] and greater fatigue variability (5) than patients with rheumatoid arthritis. Pain in rheumatoid arthritis is associated with depression, while in fibromyalgia, it is linked more to stress levels [START_REF] Drummond | Painful effects of auditory startle, forehead cooling and psychological stress in patients with fibromyalgia or rheumatoid arthritis[END_REF]. Hence, pain and fatigue levels seem to be predictors of depression in rheumatoid arthritis, whereas distress markers are predictors of fibromyalgia [START_REF] Wolfe | Predicting depression in rheumatoid arthritis: The signal importance of pain extent and fatigue, and comorbidity[END_REF]. Patients with fibromyalgia reported more depressive episodes and anxiety disorders (panic disorder, post-traumatic stress disorder) than patients with rheumatoid arthritis [START_REF] El-Hage | Fibromyalgie : une maladie du traumatisme psychique?[END_REF][START_REF] Patten | Mental disorders in a population sample with musculoskeletal disorders[END_REF], and had higher scores on emotional distress, depression, anxiety and stress [START_REF] Arnold | Comorbidity of fibromyalgia and psychiatric disorders[END_REF][START_REF] Sayar | Alexithymia and anger in patients with fibromyalgia[END_REF].

Patients with spondyloarthritis perceived a change in their daily life due to physical disability, while patients with fibromyalgia experienced these changes due to pain, anxiety, fatigue and disability [START_REF] Ovayolu | Health-related quality of life in ankylosing spondylitis, fibromyalgia syndrome, and rheumatoid arthritis: a comparison with a selected sample of healthy ındividuals[END_REF]. Finally, patients with fibromyalgia express lower general well-being than those with spondyloarthritis. They also report a poorer quality of life than patients with Sjögren's syndrome (particularly social and physical dimensions) [START_REF] Hincapie-Zapata | Quality of life in multiple sclerosis and other chronic autoimmune and non-autoimmune diseases[END_REF]. So far, no study has compared these diseases in terms of coping strategies. The aim of this study was to compare fibromyalgia, rheumatoid arthritis, spondyloarthritis and Sjögren's syndrome regarding (i) their impact on daily life, and (ii) psychological adjustment (coping strategies).

Material and method

Participants

The inclusion criteria were: a) women aged over 18 and under 70 years; b) duration of disease between 6 months and 10 years; c) social security cover; d) no difficulty communicating in French, and e) to have been informed about the study by her doctor. A rheumatologist systematically confirmed the diagnosis using international disease classification criteria: ACR [START_REF] Wolfe | The american college of rheumatology 1990 criteria for the classification of fibromyalgia[END_REF] and FIRST criteria [START_REF] Perrot | Development and validation of the Fibromyalgia Rapid Screening Tool (FiRST)[END_REF] for fibromyalgia; ACR criteria for rheumatoid arthritis [START_REF] Arnett | The american rheumatism association 1987 revised criteria for the classification of rheumatoid arthritis[END_REF]; ASAS criteria for spondyloarthritis [START_REF] Rudwaleit | The development of Assessment of SpondyloArthritis international Society classification criteria for axial spondyloarthritis (part I): classification of paper patients by expert opinion including uncertainty appraisal[END_REF]; and Vitali et al.'s criteria [START_REF] Vitali | Classification criteria for Sjögren's syndrome: a revised version of the European criteria proposed by the American-European Consensus Group[END_REF] for Sjögren's syndrome. He also ensured that patients met the inclusion and exclusion criteria for the study. The exclusion criteria were: association with another rheumatic diseases, HIV infection, history of cancer or blood malignancy, thyroid dysfunction and other endocrinopathies, viral hepatitis and all other comorbidities that the investigator considered to be severe (chronic heart failure, renal failure, respiratory failure, etc.).

Measures

To estimate the prevalence of anxiety and depressive disorders, we used the Mini International Neuropsychiatric Interview (MINI) [START_REF] Sheehan | The Mini-International Neuropsychiatric Interview (M.I.N.I.): The development and validation of a structured diagnostic psychiatric interview for DSM-IV and ICD-10[END_REF][START_REF] Lecrubier | Mini International Neuropsychiatric Interview French Version 5[END_REF]. This is a structured diagnostic interview exploring the presence or absence of the main DSM-IV axis-I disorders. It indicates comorbid anxiety and depressive disorders.

First, to assess the emotional distress caused by the disease, we used the General Health Questionnaire (GHQ) [START_REF] Goldberg | A user's guide to the General Health Questionnaire[END_REF][START_REF] Pariente | The GHQ-28 questionnaire in French: A validation survey in a panel of 158 general psychiatric patients[END_REF]. This is a self-administered questionnaire evaluating the level of distress experienced by the individual using a 4-point Likert scale. Its 28-item version investigates four areas of subjective emotional distress: Somatic symptoms, Anxiety-insomnia, Social dysfunction, Severe depression.

To assess the impact of the illness on daily life, first we used the Fibromyalgia Impact Questionnaire (FIQ) [START_REF] Burckhardt | The fibromyalgia impact questionnaire: development and validation[END_REF][START_REF] Perrot | Quality of life in women with fibromyalgia syndrome: validation of the QIF, the French version of the fibromyalgia impact questionnaire[END_REF], which is a self-administered questionnaire with 10 items assessing the physical, psychological and social impact of the illness on the quality of the patient's daily life using an 11-point Likert scale. Although this questionnaire is designed specifically for fibromyalgia, we assumed that it would be relevant for comparison of fibromyalgia with other diseases in these aspects. Secondly, to assess the common somatic symptoms of the disease in daily life, we used the Visual Analogic Scale (VAS), which assesses the level of pain on a scale of 0 to 10, and the Fatigue Severity Scale (FSS) [START_REF] Krupp | The fatigue severity scale: application to patients with multiple sclerosis and systemic lupus erythematosus[END_REF], a selfadministered 9-item questionnaire assessing the level of fatigue and its functional consequences using a 7-point Likert scale.

Finally, to assess how patients adjust to living with a rheumatic disease, we used the Coping Strategies Questionnaire (CSQ) [START_REF] Rosenstiel | The use of coping strategies in chronic low back pain patients: relationship to patient characteristics and current adjustment[END_REF][START_REF] Irachabal | Stratégies de coping des patients douloureux : adaptation française du coping strategies questionnaire (CSQ-F)[END_REF], a 21-item self-administered questionnaire assessing five specific cognitive strategies of coping with pain: Distraction (thinking of things that can divert attention from pain), Catastrophizing (worrying and focusing only on the negative aspects of pain), Praying (saying to yourself that you have to hope and pray that the pain will one day go away), Distancing from pain (trying to perceive sensations other than pain), and Ignoring pain sensations (denying that pain makes us suffer and affects us). Usually, problem-focused coping strategies (Distancing from pain, Ignoring pain sensations) are better for health outcomes, while emotion-focused strategies (avoidant coping: Distraction, Catastrophizing, Praying) are less favorable.

Procedure

The data were collected in three French hospitals (Tours, Orleans and Paris) by a rheumatologist and a psychologist. Informed written consent was obtained from all the participants and an ethics committee approved the study.

Data analysis

Internal consistency of all the questionnaires was good, with a Cronbach's alpha coefficient >.70. As the variables were not normally distributed in each group, we used nonparametric analysis techniques. First, we compared comorbid anxiety and depressive disorders in fibromyalgia and the other rheumatic diseases (rheumatoid arthritis, spondyloarthritis and Sjögren's syndrome). To this end, we used the Chi² test for categorical variables or Fisher's exact test when there were fewer than five subjects in a sample. Next, we conducted a Kruskal-Wallis test (one-way ANOVA on ranks) to compare the impact of the disease on daily life and psychological adjustment in the different rheumatic diseases. After that, we performed a Mann-Whitney U test to compare fibromyalgia with the other diseases for continuous variables. We also conducted a hierarchical cluster analysis using Ward's linkage method with squared Euclidian distance, followed by a k-means cluster analysis, which allocates each case to the cluster with the nearest center point. To compare clusters, we used the Kruskal-Wallis test (including a pairwise post-hoc test). Statistical analyses were performed with Statistica ® version 12.

Results

Participants' characteristics

The sample comprised 165 French women with rheumatic diseases, mean age 47. [START_REF] Perrot | Quality of life in women with fibromyalgia syndrome: validation of the QIF, the French version of the fibromyalgia impact questionnaire[END_REF] Average duration of disease was 4.13 years (± 2.83, range .5 to 10) with significant differences between groups (H (3, 165) = 9.40, p<.05). However, the effects of age and duration of disease were small and limited and indicated only a tendency towards a slight decrease of the functional repercussions with the evolution of the disease and the age of the patient (data not shown). Forty percent of the women were menopausal (35% with fibromyalgia, 49% with rheumatoid arthritis, 26% with spondyloarthritis, and 61% with Sjögren's syndrome), with significant differences between groups (Chi² = 10.26, p<.05). For patients with fibromyalgia, the FIRST criterion mean was 5.92 (±.28, range 5 to 6) and tender points mean was 16.79 (±2.30, range 11 to 18).

Internal consistency

The internal consistency of all scales was evaluated for each rheumatic disease.

Cronbach's alpha coefficients were acceptable (between .69 and .94), as shown in Table 1.

Descriptive data are provided in Table 2.

- -----------------------------------------------------------------------------------------------------Please Insert Tables 1 and2 here - -----------------------------------------------------------------------------------------------------

Comparison of the prevalence of anxiety and depressive disorders in patients with fibromyalgia and other rheumatic diseases

Psychiatric disorders were assessed with the MINI in order to establish the prevalence of comorbidities. Tables 3 and4 show only significant results in the comparison of comorbid anxiety and depressive disorders in fibromyalgia and the other rheumatic diseases.

Fibromyalgia patients had experienced more recurrent (past) major depressive episodes than those with other diseases (rheumatoid arthritis, spondyloarthritis or Sjögren's syndrome).

Anxiety disorders (agoraphobia and generalized anxiety disorder) were more common in fibromyalgia than in spondyloarthritis or Sjögren's syndrome; post-traumatic stress disorder was more common in fibromyalgia than in rheumatoid arthritis.

All analyses were then processed with menopausal and premenopausal women separately. Menopausal women with fibromyalgia presented with significantly more depressive disorders than those with other disorders, with the exception of menopausal women with spondyloarthritis who presented with more major depressive episodes with melancholic features. For premenopausal women, only the results for recurrent major depressive episodes were significant. There was a greater prevalence of current generalized anxiety disorder among premenopausal women with fibromyalgia than among those with other rheumatic diseases. This was the same for menopausal women, except in comparison with Sjögren's syndrome. There was a higher prevalence of agoraphobia among premenopausal women with fibromyalgia than among those with rheumatoid arthritis or spondyloarthritis. There was a higher prevalence of agoraphobia among, menopausal women with fibromyalgia than among those with Sjögren's syndrome.

- -----------------------------------------------------------------------------------------------------Please Insert Tables 3 and4 here - ---------------------------------------------------------------------------------------------------- 5 shows the results of comparative analyses for emotional distress, impact of the disease, pain, fatigue, and coping strategies. Patients with fibromyalgia had higher scores than those with rheumatoid arthritis for emotional distress (GHQ-28) (Somatic symptoms and Severe depression), for perceived fatigue, and impact of the disease. Patients with fibromyalgia were more likely to use a Catastrophizing strategy and less likely to use Distancing from pain and Ignoring pain sensation than patients with rheumatoid arthritis. They experienced greater pain, fatigue and more somatic symptoms and used the Ignoring pain sensations strategy less than patients with spondyloarthritis. There was no difference in the perception of the impact of the disease and in adjustment between fibromyalgia and Sjögren's syndrome patients.

- -----------------------------------------------------------------------------------------------------Please Insert Table 5 here

------------------------------------------------------------------------------------------------------
Menopausal women with fibromyalgia reported a significantly greater impact of disease, pain and fatigue than those with rheumatoid arthritis, greater fatigue and less Distancing from pain than spondyloarthritis patients, and greater Distraction and Ignoring pain sensations than Sjögren's syndrome patients. Premenopausal women with fibromyalgia reported more Somatic symptoms and fatigue than those with rheumatoid arthritis and more fatigue than those with spondyloarthritis. Menopausal women seemed to be more affected by the disease, pain and fatigue than premenopausal women, but comparisons between the two groups were not significant (available on supplementary data).

Major depressive episodes and emotional distress

In order to examine the relationship between major depressive episodes (MDE) diagnosed with the MINI, emotional suffering evaluated by the General Health Questionnaire (GHQ), and pain or the strategies for coping with pain, we first investigated the functional impact of the presence or absence of current or recurrent MDE (Table 6). Next, we conducted a k-means cluster analysis. A three-cluster solution seems to reflect the particularities of each group while remaining easy to interpret (Figure 1). Cluster 1 was characterized by an absence of mood disorder (current or past MDE) and low or moderate scores on Severe depression, Anxiety/insomnia, Somatic symptoms and Social dysfunction. Cluster 2 included patients presenting with mood disorders (current and past MDE) and high scores on GHQ subscales.

Patients in Cluster 3 had a past MDE, no current MDE, and moderate scores on GHQ subscales.

- -----------------------------------------------------------------------------------------------------Please Insert Table 6, Figure 1 and2 here ----------------------------------------------------------------------------------------------------- (32.4%), 3 spondyloarthritis patients (8.8%) and 3 Sjögren's syndrome patients (8.8%). The clinical characteristics of patients in Cluster 1 were also better than those in Cluster 2, while those of Cluster 3 were between the two (See Figure 2). Significant differences on Kruskal-Wallis tests are indicated in the figure, and we also performed pairwise post-hoc tests to assess the differences between groups (group interactions).

Most fibromyalgia patients were included in the second (47.9%) or third (35.4%) cluster, with only 16.7% in the first. The majority of rheumatoid arthritis patients were in the first cluster (58.7%) (vs 17.4% and 23.9% in Clusters 2 and 3 respectively). The situation was similar for both spondyloarthritis patients (66.0% in Cluster 1 vs 27.7% in Cluster 2 and 6.4% in Cluster 3) and Sjögren's syndrome patients (56.5%, 30.4%, and 13.0% in Clusters 1, 2 and 3 respectively).

Discussion

Consistent with previous studies (2-5), patients with fibromyalgia showed higher levels of pain, fatigue, perceived emotional distress, anxiety and depressive comorbidity (with a significant suicidal risk), and they made less use of the strategies of pain distancing and ignoring pain than patients with other rheumatic diseases.

In fibromyalgia, levels of pain and fatigue are markers of distress [START_REF] Wolfe | Predicting depression in rheumatoid arthritis: The signal importance of pain extent and fatigue, and comorbidity[END_REF], and fatigue is associated with a reduction in positive affectivity, which plays a role in reducing the impact of pain [START_REF] Zautra | Depression and Reactivity to Stress in Older Women With Rheumatoid Arthritis and Osteoarthritis[END_REF]. Symptoms of anxiety and depression in fibromyalgia are linked to physical symptomatology and functional disability and to an increase in pain and tender points [START_REF] Homann | Percepção de estresse e sintomas depressivos: funcionalidade e impacto na qualidade de vida em mulheres com fibromialgia[END_REF][START_REF] Thieme | Comorbid depression and anxiety in fibromyalgia syndrome: relationship to somatic and psychosocial variables[END_REF][START_REF] Aguglia | Fibromyalgia syndrome and depressive symptoms: Comorbidity and clinical correlates[END_REF].

Depression appears to be fairly reactive to fibromyalgia [START_REF] White | Chronic widespread musculoskeletal pain with or without fibromyalgia: psychological distress in a representative community adult sample[END_REF] and the pain experienced by patients increases the risk of a major depressive episode [START_REF] Raphael | Psychiatric comorbidities in a community sample of women with fibromyalgia[END_REF]. There is a significant bilateral causal link between depression and pain: pain depresses the patient and depression promotes pain [START_REF] Radat | Pain and depression: cognitive and behavioural mediators of a frequent association[END_REF][START_REF] Wessely | Taking the pain out of depression: Dual action antidepressants in the relief of pain in depression, fibromyalgia and other chronic pain syndromes[END_REF][START_REF] Williams | Depression and pain: an overview[END_REF]. In this study, the emotional distress of fibromyalgia patients was associated with social dysfunction and somatic symptoms, and patients experienced more comorbid anxiety and depressive disorders, especially recurrent major depressive episodes and agoraphobia, than patients with other rheumatic diseases. The greater anxiety (agoraphobia, post-traumatic stress disorder and generalized anxiety disorder) and higher incidence of depressive disorders (major depressive episodes) could partly explain why fibromyalgia patients feel more affected by their illness in daily life than patients with other rheumatic diseases and could play a role in the way they adjust to the condition. Maladaptive coping strategies also contribute to the strengthening of mood disorders (e.g. [START_REF] Raphael | Psychiatric comorbidities in a community sample of women with fibromyalgia[END_REF][START_REF] Garnefski | Cognitive coping strategies and symptoms of depression and anxiety: a comparison between adolescents and adults[END_REF][START_REF] Hudson | Comorbidity of fibromyalgia with medical and psychiatric disorders[END_REF]. In this study, the comparative table with and without MDE (current or past MDE) and the nonhierarchical cluster analysis highlight the importance of mood disorders in fibromyalgia patients and their association with levels of pain, fatigue and maladaptive coping strategies. Our results also show the importance of checking for past MDEs due to their association with emotional distress. They can be indicators of the vulnerability of patients, but it is difficult to establish whether the functional repercussions are the result of mood disorders or represent aspects of behavior that are closely related to fibromyalgia.

In the present study, fibromyalgia patients used Catastrophizing more than patients with rheumatoid arthritis, Distancing from pain less than patients with rheumatoid arthritis, and Ignoring pain sensations less than patients with rheumatoid arthritis and spondyloarthritis.

Patients with fibromyalgia rarely use active coping strategies to improve their quality of life [START_REF] Theadom | Exploring the role of sleep and coping in quality of life in fibromyalgia[END_REF] and are more likely to resort to maladaptive coping strategies than patients with other rheumatic diseases. The use of maladaptive strategies is an indication of how patients give up trying to manage pain or assign this responsibility to an outside source [START_REF] Radat | Pain and depression: cognitive and behavioural mediators of a frequent association[END_REF]. Catastrophizing is linked to the intensity of pain and the severity of depression, which are markers of maladjustment [START_REF] Radat | Pain and depression: cognitive and behavioural mediators of a frequent association[END_REF][START_REF] Edwards | Pain, catastrophizing, and depression in the rheumatic diseases[END_REF].

The comparison between fibromyalgia and rheumatoid arthritis shows that fibromyalgia patients feel the effects of the disease more strongly (higher level of pain, fatigue and impact of illness on daily life) and present with more anxious and depressive disorders, fewer adaptive coping strategies, and more catastrophizing than patients with rheumatoid arthritis. Rheumatoid arthritis is considered as the archetype of rheumatic diseases and is easier to diagnose than fibromyalgia. The lack of objective diagnostic criteria for fibromyalgia can often lead patients to consult multiple practitioners before the diagnosis is considered. This doctor shopping (or "medical nomadism") results in multiple and often conflicting opinions, which may give patients a sense of being misunderstood and of injustice, that they are given no support or consideration for their suffering, and fear about their future [START_REF] Ferrari | Perceived injustice in fibromyalgia and rheumatoid arthritis[END_REF][START_REF] Laroche | La fibromyalgie en France : vécu quotidien, fardeau professionnel et prise en charge. Enquête nationale auprès de 4516 patients[END_REF][START_REF] Sullivan | The Role of Perceived Injustice in the Experience of Chronic Pain and Disability: Scale Development and Validation[END_REF]. This would in turn lead to an increase in emotional insecurity [START_REF] Van Houdenhove | Prevalence and Psychodynamic Interpretation of Premorbid Hyperactivity in Patients with Chronic Pain[END_REF] and a lack of understanding of their physical state, resulting in an inability to cope with the disease on a daily basis and heightened anxiety and depressive disorders. This acute perception by patients of the effects of fibromyalgia on their daily life and their poor use of coping strategies could also be seen in connection with personality, which plays an essential role in how we interpret our environment (health beliefs and underlying dysfunctional thoughts), notably Neuroticism, which is the tendency to perceive reality as threatening and painful and is connected with psychological distress and mood disorders.

Neuroticism has been shown to be higher in fibromyalgia patients than in those with other rheumatic diseases [START_REF] Bucourt | Comparison of the Big Five personality traits in fibromyalgia and other rheumatic diseases[END_REF] and could explain the greater presence of depression in fibromyalgia.

There was no difference between fibromyalgia and Sjögren's syndrome patients in the way they perceived the impact of their illness on their daily life, and in the use of coping strategies. This could be explained by the difficulty of establishing a diagnosis in both these diseases, often delayed for several years because of the variability of the clinical features and the insidious progress of Sjögren's syndrome and the lack of objective diagnostic criteria for Fibromyalgia, which often leads to loss of confidence, discouragement and feelings of uncertainty about the future. Although these diseases differ in severity, prognosis and treatment, it appears that the difficulty of making a diagnosis could have a negative impact on how patients cope and adjust to the disease.

Menopausal women accounted for 40% of our sample. Overall, they presented with more depressive disorders than premenopausal women, but the differences were not statistically significant. It is widely accepted that menopausal transition may affect women and increase the risk of mood disorders (e.g. [START_REF] Cohen | Risk for new onset of depression during the menopausal transition: The harvard study of moods and cycles[END_REF], but other factors must be taken into account, such as age, duration of the disease, somatic symptoms, type of rheumatic disease, etc. While the menopause can interact with anxiety or pain, it can also accentuate the phenomena of dryness of the mouth and vagina, which cause the patient real discomfort, particularly in Sjögren's syndrome; as these symptoms can be difficult to talk about, they may not be adequately treated. Menopause should be systematically included in the medical monitoring of rheumatic diseases.

The limitations of the study are mainly related to the self-assessment of the impact of the illness on daily life and coping strategies and the sample size. The subgroup of Sjögren's syndrome patients was small (it is a much rarer disease) and the inclusion and exclusion criteria restricted the number of participants.

Conclusion

This study shows that patients with fibromyalgia have higher levels of pain, greater incidence of anxiety, somatic symptoms and depressive disorders, and a worse quality of life than patients with other rheumatic diseases. It also shows that patients with fibromyalgia have greater difficulty adjusting to the illness, with significant consequences on outcome. Helping patients develop appropriate coping strategies should be part of the management of rheumatic diseases, particularly fibromyalgia and Sjögren's syndrome. It would be desirable for rheumatology departments to have a psychologist on their staff to systematically check for any history of mood disorders (due to their association with levels of pain, fatigue and maladaptive coping strategies) and more generally, to assess how patients adapt to their disease.

-

  Comparison of the impact of the disease and the coping strategies used by patients with fibromyalgia and other rheumatic diseasesFirst, we observed overall effects. The ANOVA revealed significant differences between groups in the impact of the disease on daily life (H (3, 165) = 13.42, p<.05), Somatic symptoms (H (3, 165) = 11.64, p<.01), fatigue (H (3, 165) = 36.00, p<.001), and on the coping strategies of Distancing from pain (H (3, 165) = 10.75, p<.05) and Ignoring pain sensation (H (3, 165) = 12.76, p<.01). Next, we performed a Mann-Whitney U-test to compare fibromyalgia with each of the other rheumatic diseases. Table

Figure 1 :

 1 Figure 1: Nonhierarchical cluster analysis of the whole sample

Figure 2 :

 2 Figure 2: Pain, Fatigue, and Coping strategies of patients in the three clusters

-

  Cluster 1 was composed of 79 women (mean age 46.30 ± 11.24 years, range 23-65) with the highest rate of professional activity (86%; Chi² test, p < .01), and the lowest invalidity rate (8%, ns). It included 8 fibromyalgia patients (10.1%), 27 rheumatoid arthritis patients (34.2%), 31 spondyloarthritis patients (39.2%), and 13 Sjögren's syndrome patients (16.5%). Cluster 2 was composed of 51 women (mean age 48.43 ± 9.26 years, range 28-65 years) with the lowest rate of professional activity (63%), and the highest invalidity rate (22%). It included 23 fibromyalgia patients (45.1%), 8 rheumatoid arthritis patients (15.7%), 13 spondyloarthritis patients (25.5%) and 7 Sjögren's syndrome patients (13.7%). Cluster 3 (n = 34) was intermediate between Clusters 1 and 2, comprising patients with a mean age of 47.88 ± 11.95 years (range 21-65 years), 79% in professional activity, and an invalidity rate of 15%. It comprised 17 fibromyalgia patients (50.0% of the group), 11 rheumatoid arthritis patients

Table 1 : Internal Consistency

 1

	AR	FM	RA	SpA	SS

Table 2 : Descriptive analysis

 2 10.77 47.98  9.69 50.89  11.31 42.30  9.30 48.3  11.48  2.55 12.02  2.69 11.98  2.63 12.68  2.34 11.70  2.51

		All rheumatic diseases	FM	RA	SpA	SS
	Number	165	48	47	47	23
	Age, years 47.24  Mean disease duration (years) 4.13  2.83	3.54  2.76	4.85  2.96	4.50  2.81	3.13  2.30
	Years of education 12.15 Professional activity (%) .77  .42	.69  .47	.89  .31	.72  .45	.78  .42
	Professional invalidity (%)	.13  .34	.25  .44	.11  .31	.06  .25	.09  .29
	Menopause (%)	.40  .49	.35  .48	.49  .51	.26  .44	.61  .50
	FIRST	NA	5.92  .28	NA	NA	NA
	Tender points	NA	16.79  2.30	NA	NA	NA
	Anti-TNF treatment	NA	NA	.68 .47	.74 .44	NA
	Somatic symptoms (GHQ)	9.38  4.48 10.83  4.10 7.72  3.96	9.21  4.91 10.04  4.43
	Anxiety/insomnia (GHQ)	8.47  4.72	9.10 4.63	7.52  5.06	8.19  4.12	9.61  5.19
	Social dysfunction (GHQ)	9.70  3.49 10.17  3.55 8.98  3.17	9.70  3.46 10.17  4.02
	Severe depression (GHQ)	5.16  5.21	6.13  5.61	3.74  4.47	5.13  5.08	6.09  5.65
	Impact of disease (FIQ)	51.44  17.45 56.35  13.61 43.09 18.95 52.91  17.00 55.43  17.15
	Pain (VAS)	6.18  2.69	6.96  1.95	5.26  3.22	6.19  2.73	6.43  2.31
	Fatigue (FSS)	5.33  1.51	6.13  1.07	4.47  1.57	5.17  1.44	5.76  1.35
	Distraction (CSQ)	11.35  4.32 11.52  4.40 11.96  4.47 11.41  3.84 9.65  4.57
	Praying(CSQ)	4.50  3.47	5.15  3.30	4.62  3.64	4.04  3.29	3.83  3.75
	Distancing from pain (CSQ)	6.02  4.15	5.00  3.40	7.47  4.62	6.48  4.17	4.26  3.48
	Ignoring pain sensations (CSQ) 10.19  4.60 9.15  4.08 11.43  5.09 11.13  3.82 7.96  4.97
	Catastrophizing (CSQ)	7.19  3.84	8.10  4.06	6.34  3.55	7.30  4.10	6.78  3.10
	GHQ: General Health questionnaire; FIQ: Fibromyalgia Impact Questionnaire; FSS: Fatigue
	Severity Scale; CSQ: Coping Strategies Questionnaire; AR: all rheumatic diseases ; FM:
	fibromyalgia; RA: Rheumatoid Arthritis; SpA: Spondyloarthritis; SS: Sjögren's Syndrome.
	NA: not applicable

Table 3 : Significant comparison of comorbidity with anxiety disorders

 3

			FM	RA	Chi²	p Chi² (or Fisher's
			(n, %)	(n, %)	value	exact when n<5)
	PD lifetime	Abs. 41 (85.4%)	47 (100%)
		Pres.	7 (14.6%)	0 (0%)	7.40	.012
	Agoraphobia	Abs. 29 (60.4%) 38 (80.9%)
		Pres. 19 (39.6%)	9 (19.1%)	4.77	.029
	PD with agoraphobia Abs. 40 (83.3%) 46 (97.9%)
		Pres.	8 (16.7%)	1 (2.1%)	5.85	.031
	PTSD	Abs. 44 (91.7%)	47 (100%)
		Pres.	4 (8.3%)	0 (0%)	4.09	.043
	Current GAD	Abs. 29 (60.4%) 44 (93.6%)
		Pres. 19 (39.6%)	3 (6.4%)	14.71	.001
			FM	SpA	chi²	p Chi² (or Fisher's
			(n, %)	(n, %)	value	exact when n<5)
	Agoraphobia	Abs. 29 (60.4%) 40 (85.1%)
		Pres. 19 (39.6%)	7 (14.9%)	7.28	.011
	Current GAD	Abs. 29 (60.4%) 45 (95.7%)
		Pres. 19 (39.6%)	2 (4.3%)	17.21	.001
			FM	SS	Chi²	p Chi² (or Fisher's
			(n, %)	(n, %)	value	exact when n<5)
	Agoraphobia	Abs. 29 (60.4%) 21 (91.3%)
		Pres. 19 (39.6%)	2 (8.7%)	7.12	.011
	Current GAD	Abs. 29 (60.4%) 21 (91.3%)
		Pres. 21 (39.6%)	2 (8.7%)	7.12	.011
	Abs.: absence; Pres.: presence; PD: Panic Disorder; GAD: Generalized Anxiety Disorder;
	PTSD: Post-Traumatic Stress Disorder; FM: Fibromyalgia; RA: Rheumatoid Arthritis; SpA:
	Spondyloarthritis; SS: Sjögren's Syndrome.

Table 4 : Significant comparison of comorbidity with depressive disorders

 4

			FM	RA	Chi²	p Chi² (or Fisher's
			(n, %)	(n, %)	value	exact when n<5)
	Current MDE	Abs. 25 (52.1%) 37 (78.7%)
		Pres. 23 (47.9%) 10 (21.3%)	7.43	.001
	Recurrent MDE	Abs. 14 (29.2%) 29 (61.7%)
		Pres. 34 (70.8%) 18 (38.3%)	10.15	.001
	MDE with	Abs. 39 (81.3%)	47 (100%)
	melancholic features	Pres.	9 (18.8%)	0 (0%)	9.74	.003
	Suicide risk	Abs. 35 (72.9%) 44 (93.6%)
		Pres. 13 (27.1%)	3 (6.4%)	7.27	.012
			FM (n, %)	SpA (n, %)	Chi²value	p Chi² (or Fisher's exact when n<5)
	Recurrent MDE	Abs. 14 (29.2%) 36 (76.6%)
		Pres. 34 (70.8%) 11 (23.4%)	21.43	.001
			FM (n, %)	SS (n, %)	Chi²value	p Chi² (or Fisher's exact when n<5)
	Recurrent MDE	Abs. 14 (29.2%) 18 (78.3%)
		Pres. 34 (70.8%)	5 (21.7%)	15.14	.001
	MDE with	Abs. 39 (81.3%)	23 (100%)
	melancholic features	Pres.	9 (18.8%)	0 (0%)	7.12	.011
	Abs.: absence; Pres.: presence; MDE: Major Depressive Episode; Recurrent MDE (MINI-V
	for past MDE); FM: Fibromyalgia; RA: Rheumatoid Arthritis; SpA: Spondyloarthritis; SS:
	Sjögren's Syndrome.

Table 5 : Results of significant comparative analyses of fibromyalgia and other rheumatic diseases

 5

		Means FM Means RA	Z	p
	Somatic symptoms (GHQ)	10.83	7.72	3.30	.001
	Severe depression (GHQ)	6.13	3.74	2.22	.026
	Impact of disease (FIQ)	56.35	43.00	3.39	.001
	Pain (VAS)	6.96	5.26	2.49	.001
	Fatigue (FSS)	6.13	4.47	5.58	.001
	Distancing from pain (CSQ)	5.00	7.47	-2.52	.012
	Ignoring pain sensations (CSQ)	9.15	11.43	-2.44	.015
	Catastrophizing (CSQ)	8.10	6.34	2.40	.016
		Means FM Means SpA	Z	p
	Somatic symptoms (GHQ)	10.83	9.21	1.96	.050
	Fatigue (FSS)	6.13	5.17	3.58	.001
	Ignoring pain sensations	9.15	11.13	-2.18	.030

FM: Fibromyalgia; RA: Rheumatoid Arthritis; SpA: Spondyloarthritis; GHQ: General Health questionnaire; FIQ: Fibromyalgia Impact Questionnaire; VAS: Visual Analogic Scale; FSS: Fatigue Severity Scale; CSQ: Coping Strategies Questionnaire

Table 6 : Impact of the presence or absence of depression on coping strategies and on daily health status

 6

		Absence of	Presence of
		current MDE	current MDE
		(means)	(means)	Z	p
	Psychic suffering:
	-Somatic symptoms	8.72	11.85	-5.04 .001
	-Anxiety/insomnia	7.15	11.17	-4.99 .001
	-Social dysfunction	8.78	11.57	-4.80 .001
	-Severe depression	3.15	9.26	-6.74 .001
	Impact of disease	46.24	62.13	-5.69 .001
	Pain	5.52	7.54	-4.46 .001
	Fatigue	4.92	6.18	-5.18 .001
	Coping strategy:
	-Distraction	11.79	10.46	2.03	.05
	-Distancing from pain	6.61	4.81	2.21	.05
	-Ignoring pain sensations	10.94	8.67	3.20	.01
	-Catastrophizing	6.48	8.63	-2.73	.01
		Absence of	Presence of
		recurrent	recurrent
		MDE (means)	MDE (means)	Z	p
	Psychic suffering:
	-Somatic symptoms	8.42	10.78	-3.63 .001
	-Anxiety/insomnia	7.78	9.46	-2.35	.05
	-Social dysfunction	9.26	10.40	-2.26	.05
	-Severe depression	4.13	6.66	-3.47 .001
	Impact of disease:	48.00	56.35	-2.98	.01
	Fatigue:	4.94	5.89	-3.70 .001
	Coping strategy:
	-Catastrophizing	6.41	8.31	-2.79	.01
	MDE: Major Depressive Episode; Recurrent MDE (MINI-V for past MDE).

Acknowledgments:

We would like to thank Dr Odile Plaisant, Professor Pierre Bourgeois and Professor Bruno Fautrel (Assistance Publique -Hôpitaux de Paris, University Hospital of Pitié-Salpêtrière) and Dr Stéphanie Rist and Dr Carine Salliot (Regional Hospital of Orléans) for their involvement in selecting the patients and the quality of data collection.

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.


V Ostuni

J Morales

C Bourdelle

J-F Artaud

P Manas

R Dumont

N Fedorczak

M Goniche

P Maget

Y Sarazin

Confinement properties in the large aspect ratio, full tungsten environment of the WEST tokamak

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The parametric dependence of the confinement time with respect to the aspect ratio

Shot 55539

The performed studies take into account statistics calculated on plateaus of total power intersecting plasma current plateaus (quasi-steady states). Different diagnostic measurements are time averaged on each plateaus and added to the database.

The plasma energy content computed from polarimetry constrained equilibrium reconstructions is chosen for the analysis.

The WEST database contains more than 1000 entries:

• L-mode pulses (𝑃 𝑎𝑢𝑥 >0.5𝑀𝑊);

• deuterium only pulses;

• heated by lower hybrid and ion; cyclotron resonance heating;

• Lower single null configuration.

WEST is a metallic

  tokamak operating in L mode since December 2016. Its specificities are an aspect ratio value of 5-6 and the inner wall covered with tungsten tiles plus ITER like PFU on lower divertor. Its plasmas are dominantly electron heated (ICRH and LHCD) and torque free (no NBI). [1] In this work, we analyze extensively the database of WEST plasmas to characterize the operational domain, based on plateaus of quasi-steady phases of power and current. In WEST database, 20% of the detected plateaus are affected by a rapid collapse of the central electron temperature or do not reach more than 1.5 keV at the core. Since reaching plasmas with enhanced stability is fundamental to obtain high tokamak performances, the core radiative collapse is modelled using RAPTOR to understand the causality leading to colder plasma core. [2][3] • WEST L mode energy content agrees with ITER96L scaling predictions; • The scaling coefficients do not change significantly adding WEST shots directly to ITER96L confirming the weak aspect ratio dependence reported in the ITER96L; • Two different confinement states coexist in the WEST operation: one with Te(0)>3keV and one with Te(0) <1.5 keV at the same power/<n>; • The core T e rapid collapse from 3 to 1.5 keV cannot be explain solely by an increase of the density. An increase of the core W peaking as well as an off-axis shift of the LHCD power deposition are necessary to reproduce the observed T e (0) collpase.

  𝑰𝒑 𝜶 𝒑 𝑩 𝜶 𝑩 𝑷 𝒕𝒐𝒕 𝜶 𝑷 𝒏 𝒆 𝜶 𝒏𝒆 𝑴 𝜶 𝑴 𝑹 𝜶 𝑹 𝜺 𝜶 𝜺 𝒌 𝜶 𝒌 is applied on WEST database accounting only plasma current, the line averaged electron density and the total power. does not play an important role in the scaling. Even if WEST has an aspect ratio larger than the other tokamaks in ITER96L database, the regression coefficient still close to zero. 𝛀 𝒊 𝝉 = 𝝆 * 𝒙 𝝆 * 𝜷 𝒙 𝜷 𝝂 * 𝒙 𝝂 * 𝒒 𝒙 𝒒 𝑴 𝒙 𝑴 𝜺 𝒙 𝜺 𝑩 𝒙 𝑩 𝒌 𝒙 𝒌Two different confinement states coexist in the WEST L-mode operation. The hot branch, is characterized by a high central electron temperature, internal inductance and neutron flux and a low tungsten peaking. Thanks to the bolometry tomography it is possible to compute the power emission density profile. Above 1keV all the radiated power is due to tungsten emission. The 1D transport code RAPTOR is used to model the collapse. unstable The density rises leading to slight T e (0) drop. The acceleration of the T e (0) drop is concomitant with an increase of the core W peaking. At first a flat W profile is found to reproduce well the measured P rad , then, a peaked core W profile is assumed to reproduce the core bolometer chords increase. In WEST L-mode database 20% of the detected plateaus are affected by a rapid collapse of the central electron temperature or do not reach more than 1.5 keV . The increase of the tungsten in the center of the plasma is not sufficient to reproduce the collapse. The central value of the LHCD power deposition is reduced in an ad-hoc manner. With both conditions the speed of the collapse is reproduced. 𝑷 𝒓𝒂𝒅,𝑾 [𝑾/𝒎 𝟑 ] 𝒏 𝒆 𝑳 𝑾 (𝑻 𝒆 ) 𝑾 𝒑𝒆𝒂𝒌 = 𝒏 𝑾 𝟎.𝟎 𝒏 𝑾 𝟎.𝟑𝟓