Luca Di Stasio
email: luca.di.stasio@ltu.se.

Janis Varna

Zoubir Ayadi

matrix interface crack in UD composites under transverse loading: Effect of the fiber volume fraction and of the distance to the free surface and to non-adjacent debonds

Keywords: Polymer-matrix Composites (PMCs), Thin-ply, Energy Release Rate, Debonding, Finite Element Analysis (FEA)

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

Introduction

Stimulated by the ever more stringent requirements in terms of weight and mechanical performances of the aerospace industry, in recent years the com-posite community has returned its attention to the mechanisms of intralaminar crack initiation with a focus on thin-ply laminates. Alternative design approaches are now considered based on this non-conventional laminate in applications ranging from cryogenic pressure vessels [START_REF] Mccarville | 3.5 Design, Manufacture and Test of Cryotank Components[END_REF], to airplanes' wings [START_REF] Kim | Effects of shallow biangle, thin-ply laminates on structural performance of composite wings[END_REF], and even reusable space launchers [START_REF] Kopp | The aurora space launcher concept[END_REF].

Thin-ply laminates are the result of a technological innovation, the spread tow technology, which consists in opening or spreading the tows, in which fibers (carbon, glass, aramid, basalt among others) are usually shipped in, into very thin tapes used for laminate production. Ply thicknesses of less than 50 µm can nowadays be mass-produced, and record thicknesses of around 20 -25 µm, or ∼ 4 -5 times the average fiber's diameter, have been achieved. In its current form the technique, sometimes referred to as "FUKUI method", was firstly proposed towards the end of the 1990s [START_REF] Kawabe | A pneumatic process for spreading reinforcing fiber tow[END_REF] and perfected in the subsequent decade [START_REF] Kawabe | New spreading technology for carbon fiber tow and its application to composite materials[END_REF][START_REF] Kawabe | New carbon fiber tow-spread technology and applications to advanced composite materials[END_REF].

Several experimental investigations on thin ply laminates have highlighted their main properties [START_REF] Sasayama | Effect of lamina thickness on first ply failure in multidirectionally lami-nated composites[END_REF][START_REF] Sihn | Experimental studies of thin-ply laminated composites[END_REF][START_REF] Yokozeki | Experimental characterization of strength and damage resistance properties of thin-ply carbon fiber/toughened epoxy laminates[END_REF][START_REF] Yokozeki | Damage characterization in thin-ply composite laminates under out-of-plane transverse loadings[END_REF][START_REF] Arteiro | Large damage capability of non-crimp fabric thin-ply laminates[END_REF][START_REF] Amacher | Thin ply composites: Experimental characterization and modeling of sizeeffects[END_REF][START_REF] Cugnoni | Towards aerospace grade thin-ply composites: Effect of ply thickness, fibre, matrix and interlayer toughening on strength and damage tolerance[END_REF]: increased fiber content; more uniform packing of fibers; delay and even suppression of intralaminar cracking (called also transverse-, matrix-or micro-cracking) and delamination. A very insightful work documenting how these phenomena are affected by the morphology of thin-ply laminates is the microscopic study of Saito & al. [START_REF] Saito | Experimental evaluation of the damage growth restraining in 90 layer of thin-ply cfrp cross-ply laminates[END_REF], which focuses on the effect of ply thickness on the onset and propagation of intralaminar cracking. In their investigation, tensile tests were performed on carbon fiber/epoxy [0 2 , 90 n , 0 2 ] thin-ply laminates for n = 1, 2, 4 and the crack density was measured with a digital microscope at several levels of applied tensile strain in the range between 0% and 1.5%. Furthermore, they performed microscopic observations on the specimen's edge at each level of strain. They observed the onset of fiber/matrix interface cracks (referred to as debonds in the following) at lower levels of strain in thinner plies, while at the same time coalescence of debonds and through-the-thickness propagation of transverse cracks in thin plies were delayed and even suppressed as ply thickness decreased. In particular, they reported the first onset of debonds at 0.4% for n = 1, 2 and 0.7% for n = 4. For n = 1, however, at ε = 1.5% coalescence of debonds had started to take place but the crack had not completely propagated through the thickness, while for n = 2 and n = 4 the latter alreay happened at a value of strain respectively of 1.3% and 1%. Our inability to explain these observations with the currently accumulated knowledge demonstrates the necessity of further investigation of interactions between debonds and studies of the constraining (or accelerating) effect of presence of bonded fibers, free and constrained boundaries in the vicinity of a partially debonded fiber.

Early studies on the effect of ply thickness on the onset and propagation of transverse cracks were conducted on glass fiber/epoxy cross-ply laminates by Bailey, Parvizi and collaborators [START_REF] Garrett | Multiple transverse fracture in 90 • cross-ply laminates of a glass fibre-reinforced polyester[END_REF][START_REF] Parvizi | On multiple transverse cracking in glass fibre epoxy cross-ply laminates[END_REF][START_REF] Parvizi | Constrained cracking in glass fibrereinforced epoxy cross-ply laminates[END_REF], who firstly observed the beneficial effect of thickness reduction on the delay of transverse cracking. They furthermore pointed the attention to the appearance of debonds at the fiber/matrix interface and their subsequent coalescence as the mechanism at the origin of transverse cracks [START_REF] Bailey | On fibre debonding effects and the mechanism of transverse-ply failure in cross-ply laminates of glass fibre/thermoset composites[END_REF]. Moreover, they identified the main mechanical driver of the damage process in the mismatch of elastic properties, and particularly of Poisson's ratios, between fibers and matrix [START_REF] Bailey | On the transverse cracking and longitudinal splitting behaviour of glass and carbon fibre reinforced epoxy cross ply laminates and the effect of poisson and thermally generated strain[END_REF]. A full understanding of damage onset and propagation in thin-ply laminates thus requires comprehension of the mechanisms governing its very first stage, i.e. the fiber/matrix interface crack. First results were obtained through analytical models in the case of a single fiber with an arc crack (debond) in an infinite matrix under transverse tension by England [START_REF] England | An arc crack around a circular elastic inclusion[END_REF] and Perlman & Sih [START_REF] Perlman | Elastostatic problems of curvilinear cracks in bonded dissimilar materials[END_REF], who obtained the stresses at the interface and calculated the stress intensity factors at the crack tip, and by Toya [22], who evaluated the Energy Release Rate (ERR). Drawing upon the results for the straight bi-material interface crack by Comninou [START_REF] Comninou | The interface crack[END_REF], the effect of crack face contact in fiber-matrix debonding was investigated in [START_REF] París | The fiber-matrix interface crack -a numerical analysis using boundary elements[END_REF][START_REF] Varna | The effect of crack-face contact on fiber/matrix debonding in transverse tensile loading[END_REF].

In [START_REF] García | Debonding at the fibre-matrix interface under remote transverse tension. one debond or two symmetric debonds?[END_REF], it was showed in terms of ERR why the case of a single asymmetric debond is more likely to be observed under remote transverse tension than two symmetric debonds on the same fiber. The effect of different types and combinations of loads on debonding have been studied for the single fiber model: compression [START_REF] Correa | Effects of the presence of compression in transverse cyclic loading on fibre-matrix debonding in unidirectional composite plies[END_REF], residual thermal stresses [START_REF] Correa | Effect of thermal residual stresses on matrix failure under transverse tension at micromechanical level: A numerical and experimental analysis[END_REF], and biaxial configurations with different combinations of tension and compression [START_REF] Correa | Effect of the presence of a secondary transverse load on the inter-fibre failure under tension[END_REF][START_REF] Correa | Effect of a secondary transverse load on the inter-fibre failure under compression[END_REF]. The effect of the presence of nearby bonded fibers on the debonding of a fiber embedded in an infinite matrix has been studied under uniaxial transverse tension [START_REF] Sandino | Numerical analysis of the influence of a nearby fibre on the interface crack growth in composites under transverse tensile load[END_REF], biaxial tension [START_REF] Sandino | Composite materials under transverse biaxial loads: Study of the influence of a nearby fibre on the interface crack growth under tension[END_REF] and uniaxial transverse compression [START_REF] Sandino | Interface crack growth under transverse compression: nearby fibre effect[END_REF]. The effect of inter-fiber distance on debond growth has been studied for a partially debonded fiber at the center of a hexagonal cluster inside a homogenized UD composite in the case of fully bonded neighbouring fibers [START_REF] Zhuang | Effects of inter-fiber spacing on fiber-matrix debond crack growth in unidirectional composites under transverse loading[END_REF] and of two partially debonded fibers out of the surrounding six [START_REF] Varna | Growth and interaction of debonds in local clusters of fibers in unidirectional composites during transverse loading[END_REF]. An understanding of crack shielding and finite ply thickness effects on debond growth in non-homogenized microstructural models of UDs seems thus to be lacking: this is the problem that we want to address in the present work. Mode I and Mode II energy release rates will be analyzed using stress fields calculated with the FEM for a variety of Repeating Unit Cell (RUC) of the composite with square packing of fibers under transverse tensile loading. The choice of a square packing configuration for the fibers is motivated by its simplicity, as it allows to easily separate the effect of fibers (fully bonded and/or partially debonded) placed along the loading direction from that of fibers placed in the through-the-ply-thickness direction. These RUCs represent composites with different distances between partially debonded fibers and a varying number of bonded fibers between them, which allows to study the effect of crack shielding on the ERR. In the ply thickness direction, the varying number of perfectly bonded fiber rows exposes the effect of the proximity of the free boundary of the composite on debond growth. Finally, using coupling of thickness direction displacements on horizontal boundaries of the RUC, the accelerating effect of the interaction between debonds of fibers located on the same vertical line is studied.

RVE models & FE discretization

Introduction & Nomenclature

In this paper, we analyze debond development in unidirectional (UD) composites subjected to in-plane transverse tensile loading. The interaction between debonds in UD composites is studied developing models of different Repeating Unit Cells (RUC) of laminates (see Fig. 1 to Fig. 3) where only the central fiber in the cell has a damage in the form of a fiber/matrix interface crack (debond). The composite RUC may be repeating in the in-plane transverse direction only (representing an ultra-thin composite) or repeating also in the composite thickness direction, representing an infinite composite in a limiting case. Thus, the conditions at the UD composite's upper and lower boundaries are one of the parameters for the investigation. The used RUCs allow for considering the composite with debonds as a sequence of stacked damaged and undamaged fiber rows, each row with only one fiber in the thickness direction. Since all of these RUCs feature regular microstructures with fibers placed according to a square-packing configuration, they are Representative Volume Elements (RVE) of composites with a certain distribution of debonds. Introducing in-plane coordinates x and y, where x is in the transverse direction of the UD composite under consideration, the strain in the y-direction due to a load in the x-direction is small, caused in turn by the very small minor Poisson's ratio of the UD composite. Additionally, debonds are considered to be significantly longer in the fiber direction than in the arc direction. Therefore, we use 2D models under the assumption of plane strain, defined in the xz section of the composite. Thus, the analysis presented applies to long debonds, with a focus on understanding the mechanisms of growth along their arc direction. The composites are subjected to transverse tensile strain, applied as a constant displacement in the x-direction along the vertical boundary of the RUC as shown in Figures 1 to 4. As the models are differentiated by the number of rows of fibers and by the spacing between debonds along the vertical and horizontal directions, the corresponding RUCs can be distinguished from each other based on the number n of fibers in the horizontal direction and k in the vertical direction. Furthermore, the horizontal surfaces can be either free or vertical displacement coupling can be applied. We thus introduce a common notation n × kf ree and n × kcoupling to denote a RUC with n × k fibers and, respectively, a free upper surface or with kinematic coupling applied to it.

The specific combinations of particular choices of n, k, and boundary conditions are detailed in Section 2.2, together with the description of the corresponding models of damaged composite they are representing.

Models of Representative Volume Element (RVE)

The first two models feature, as shown in Fig. 1, an ultra-thin UD lami-130 nate with only one row of fibers across its thickness, k = 1. This is quite an extreme model from the microstructural point of view; however, it allows to focus the analysis on the interaction between debonded fibers placed along the x-direction. Furthermore, as the horizontal surfaces are considered free, the interaction is stronger in this case than in any other, making the trends very clear 135 and the predictions of this model rather conservative. In retrospective, if only 20 years ago such a model would have been considered too abstracted from the physical reality, the recent advancements in the spread tow technology make this approach appealing also as a limiting case for practical considerations. In the sub-model of Fig. 1a, every n th fiber in the composite is partially debonded on alternating sides of the fiber. The symmetries of the model allow the use of the upper part of the RUC, as highlighted in Fig. 1 to 3. Following the notation introduced in Section 2.1, we will refer to this model as n × 1f ree.

In the sub-model n = 1, Fig. 1b, a debond appears on each fiber on alternating sides and the corresponding RUC contains only one fiber. We will refer to this model as 1 × 1f ree.

The second set of models in Fig. 2 and Fig. 3 considers laminates with multiple rows of fibers across the thickness: a finite number of rows in the first two sub-models in Fig. 2; an infinite number in the model of Fig. 3. In Fig. 2a, the RUC contains n = 1 fiber in the x-direction, k fibers across the thickness and the central fiber is debonded. This model will be referred to in the following as 1 × kf ree. Thinking in terms of rows, in this model we have a central row where each fiber is debonded. This row is surrounded from each side by (k-1) /2 rows with perfectly bonded fibers. In the sub-model in Fig. 2b, each n th fiber in the central row is debonded and this row is surrounded by (k-1) /2 rows of undamaged fibers from each side. We will refer to this model as n × kf ree (because the horizontal boundary of the RUC is free of any constraint).

Finally, the model in Fig. 3 represents an UD composite with an infinite number of rows; all of them with partially debonded fibers. As all fibers have debonds, the corresponding RUC is made of a single partially debonded fiber with kinematic coupling conditions applied to the upper boundary to assure periodicity. This model is referred to as 1 × 1coupling.

Finite Element (FE) discretization

Each RUC is discretized using the Finite Element Method (FEM) within the Abaqus environment, a commercial FEM package [START_REF] Simulia | ABAQUS/Standard User's Manual[END_REF]. The length l and height h of the model are determined by the number of fibers n in the horizontal direction and k across the thickness (see 2.2) according to Eq. 1: where 2L is the length of a one-fiber unit, see Fig. 4, defined as a function of the fiber volume fraction V f and the fiber radius according to

l = 2nL h = kL; (1)
L = R f 2 π V f . (2)
x z The fiber radius R f is assumed to be the same for each fiber in the model and equal to 1 µm. The latter value is not physical and it has been chosen for simplicity. It is worth to note at this point that, in a linear elastic solution as the one presented here, the ERR is proportional to the geometrical dimensions and recalculation of the ERR for fibers of any size, thus, requires a simple multiplication. Furthermore, notice that the relationships in Eqs. 1 and 2 ensure that the local and global V f are everywhere equal.

n = 1 k = 1
The debond is placed symmetrically with respect to the x axis (see Fig. 4)

and we characterize it with an angular size of ∆θ (the full debond size is thus 2∆θ). For large debond sizes (≥ 60 • -80 • ), a region of variable size ∆Φ appears at the crack tip in which the crack faces are in contact and slide on each other.

Due to its appearance, frictionless contact is considered between the two crack faces to allow free sliding and avoid interpenetration. The presence of friction at the interface is considered in [START_REF] Varna | The effect of crack-face contact on fiber/matrix debonding in transverse tensile loading[END_REF], where the authors model the contact interaction between crack faces using Coulomb's friction with a coefficient µ = 0.25 and show that, for a debond with ∆θ = 80 • , the crack sliding displacement is always different from zero in every point of the crack and only slightly lower than that of the frictionless case. This in turn means that the estimation of G II in the case of frictionless contact provide an upper bound and thus the results presented here represent a conservative estimation (a higher ERR corresponds to higher likelihood of crack propagation). Symmetry with respect to the x axis is applied on the lower boundary. The upper boundary is in general free, except for the model 1 × 1coupling (Fig. 3) which requires kinematic coupling of vertical displacements also on the upper side. Kinematic coupling on the x-displacement is applied along the left and right sides of the model in the form of a constant x-displacement ±ε x l, corresponding to transverse strain εx equal to 1%. and G T OT ) represent the main output of the FEM analysis; they are evaluated using the VCCT [START_REF] Krueger | Virtual crack closure technique: History, approach, and applications[END_REF] implemented in a custom Python routine and, for the total ERR, the J-integral [START_REF] Rice | A path independent integral and the approximate analysis of strain concentration by notches and cracks[END_REF] is obtained by application of the Abaqus built-in functionality. A glass fiber-epoxy system is considered throughout this paper, and it is assumed that their response lies always in the linear elastic domain.

x z ∆θ L L L ūx = εx l ūx = -ε x l Ω f Ω m R f O ∆Φ
The latter assumption lies on the work of Asp and co-workers, who show that epoxy subjected to a tri-axial stress state as the one observable in the inter-fibers region fails at very low strains (∼ 0.5% -0.8%) [START_REF] Asp | Effects of a composite-like stress state on the fracture of epoxies[END_REF] and in a brittle manner [START_REF] Asp | A criterion for crack initiation in glassy polymers subjected to a composite-like stress state[END_REF],

and that the magnitude of deviatoric stresses (evaluated in terms of equivalent Von Mises stress) in the fiber/matrix interface neighborhood does not justify the occurrence of plastic deformations in the case of debond propagation [START_REF] Asp | A criterion for crack initiation in glassy polymers subjected to a composite-like stress state[END_REF].

The properties used are listed in Table 1.

Validation of the model

The model is validated in Fig. 5 against the results reported in [START_REF] París | Kinking of transversal interface cracks between fiber and matrix[END_REF][START_REF] Sandino | Numerical analysis of the influence of a nearby fibre on the interface crack growth in composites under transverse tensile load[END_REF],

230
obtained with the Boundary Element Method (BEM) for a single fiber with a symmetric debond placed in an infinite matrix. This situation is modeled using To allow for a comparison, the results are normalized following [START_REF] Sandino | Numerical analysis of the influence of a nearby fibre on the interface crack growth in composites under transverse tensile load[END_REF] with 235 respect to a reference Energy Release Rate G 0 defined as

the 1 × 1 -f ree RVE with V f = 0.0079%,
G I , BEM G II , BEM G T OT , BEM G I , FEM G II , FEM G T OT , FEM
G 0 = 1 + k m 8µ m σ 2 0 πR f ( 3
)
where µ is the shear modulus, k is the Kolosov's constant defined as 3 -4ν for plane strain conditions, R f is the fiber radius and the index m refers to the properties of the matrix. σ 0 is the stress at the boundary, computed as the average of the stress extracted at each boundary node along the right side (arithmetic average as nodes are equispaced by design along both the left and right sides). The agreement is good: the difference between the BEM solution, which is considered more accurate, and the FEM solution does not exceed 5%.

The ERRs' maxima are in the same positions and the size of the contact zone is the same. Nevertheless, an analysis of phenomena leading to less than 5% differences in ERR would not be reliable and, therefore, it is not recommended.

Results & Discussion

Effect of Fiber Volume Fraction

As shown in Figs. 6 and7, respectively for Mode I and Mode II, the fiber content has a drastic effect on the Energy Release Rate at the tip of the fiber/matrix interface crack. The effect of four levels of fiber volume fraction are compared, 30%, 50%, 60% and 65%, on two microstructural models: a 11×11-f ree (every and 65% the values of Mode I ERR are rather similar when measured at 10 • and at 20 • , approximately creating a plateau. Furthermore, increasing the fiber volume fraction delays the onset of the contact zone, which corresponds in Fig. 6 to the first value of ∆θ for which G I is equal to zero. For V f = 30%, the contact zone first appears for a debond of 60 fiber in infinite matrix model (Fig. 5). For higher fiber contents, the contact zone's onset is delayed to a debond's size approximately equal to 70 • . For Mode II, Fig. 7, there is a distinct maximum in the curve and its shape does not depend on the fiber content. The maximum value of the ERR increases by ∼ 2.1 times when V f changes from 30% to 65%. The effect is thus similar to Mode I, but with a significantly lower magnitude. Similar to Mode I, the debond's size for which the peak value of Mode II occurs remains unchanged, at 60 • . It is worthwhile to notice that the ratio of Mode II to Mode I peak values is max(G II ) max(G I ) ∼ 2.2 0.9 ∼ 2.4 for V f = 30%, while it is ∼ 4.7 4.7 ∼ 1 for V f = 65%. The general increasing trends observed in Figs. 6 and7 are related to the fact that, given that the global and local V f are everywhere identical in the models presented, an increase in fiber content corresponds to a decrease in the average distance between fibers. Thus, the distances for the decay of the local stress and strain fields in the matrix domain become shorter, leading to higher stresses in general and causing higher values at the crack tip. The difference in relative magnitude between Mode I and Mode II and the delay in the contact zone's onset are instead due to the interplay between two different mechanisms, both caused by the ordered microstructural arrangement of the model. In the models considered, a fully bonded fiber is always placed along the horizontal direction, aligned with the partially debonded fiber and exactly in front of the debond.

11
G I   J/m 2   V f = 30% V f = 50% V f = 60% V f = 65% (
G I   J/m 2   V f = 30% V f = 50% V f = 60% V f = 65% (
G II   J/m 2   V f = 30% V f = 50% V f = 60% V f = 65% (
G II   J/m 2   V f = 30% V f = 50% V f = 60% V f = 65% (
By increasing V f , the former moves closer to the latter and for small debonds this causes a magnification of the x-strain at the crack tip. For small debonds (≤ 20 • -30 • ) in fact, the crack tip is approximately normal to the x-direction and thus an increase in ε x causes an increase in G I . On the other hand, for large debonds (≥ 70 • -80 • ) the crack growth direction is almost aligned with the x-axis, thus a magnification in the x-strain translates into an increase of Mode II ERR. However, this increasing effect on G II is partially counteracted by the presence of a fully bonded fiber on top of the debonded fiber and aligned with it.

As fibers are more rigid than the surrounding matrix, the presence of the former will restrain horizontal displacements, thus hampering strong increases in G II for large debonds. Furthermore, due to the mismatch in the Poisson's ratios, the fully bonded fiber placed above generates an upward-directed component of the vertical displacement field in the matrix, which tends to open the debond and causes the delay in the contact zone's onset. The interplay between these mechanisms is governed by the average inter-fiber distance and, in turn, by the fiber volume fraction.

These observations are in agreement with the results reported in [START_REF] Sandino | Numerical analysis of the influence of a nearby fibre on the interface crack growth in composites under transverse tensile load[END_REF], where the effect on the ERR of a partially debonded fiber of two fully bonded nearby fibers, placed symmetrically with respect to the loading direction, is studied for different angular positions (denoted as θ 2 ) and radial distances in a model with an effectively infinite matrix (V f ∼ 0.09%). The effect of the former is studied for a constant value of the radial distance between the debonded and bonded fibers, which corresponds to a local V local f of ∼ 62% assuming hexagonal packing. They report an increase in both Mode I and Mode II ERR with respect to the single fiber case when the two fibers are placed at an angle of respectively Comparing the results from [START_REF] Sandino | Numerical analysis of the influence of a nearby fibre on the interface crack growth in composites under transverse tensile load[END_REF] with those presented in this paper, an hypothesis can be furthermore formulated about the robustness of the results of the present article with respect to deviations in fiber position: it seems reasonable to assume a tolerance to deviations of max. ±30 • with respect to the loading direction and of max. ±20 • with respect to the through-the-thickness direction.

±25 • , ±30 • , ±140 • , ±150 • , ±155 • , i.
The effect of the local fiber content is also investigated in [START_REF] Sandino | Numerical analysis of the influence of a nearby fibre on the interface crack growth in composites under transverse tensile load[END_REF], by changing the radial distance between the partially debonded fiber and the fully bonded ones. They observe that the further the fully bonded fibers are placed from the central one, i.e. the lower the local V f , the lower is their effect on the ERR.

The magnitude of the effect is however small: the maximum increase of the total ERR is of ∼ 1.15 times for θ 2 = 30 • and 150 • when increasing V local f from 28% to 62%; the total ERR decreases by a factor of ∼ 0.62 for θ 2 = 60 • , ∼ 0.74 for θ 2 = 90 • and ∼ 0.5 for θ 2 = 120 • when increasing V local f from 28% to 62%. Analogous results can be found in [START_REF] Zhuang | Effects of inter-fiber spacing on fiber-matrix debond crack growth in unidirectional composites under transverse loading[END_REF], where the authors consider a centrally-placed partially debonded fiber sorrounded by an hexagonal cluster inside an homogenized UD composite. They observe a reduction in the ERR when the local fiber volume fraction is increased, i.e. when the spacing between fibers is reduced. The strongest change is reported for Mode II, which decreases by a factor of ∼ 0.73 when the local fiber volume fraction is decreased from 66% to 78%. Thus, the trends presented in [START_REF] Sandino | Numerical analysis of the influence of a nearby fibre on the interface crack growth in composites under transverse tensile load[END_REF][START_REF] Zhuang | Effects of inter-fiber spacing on fiber-matrix debond crack growth in unidirectional composites under transverse loading[END_REF] are in agreement with our results on the effect of V f and support the considerations made so far. The stark difference in magnitude however highlights the contrast between the effect of the local fiber volume fraction of a cluster of fibers inside an infinite medium and of the global V f of long-range microstructural arrangements, such the ones considered in this article. The similarity in trends with the concurrent difference in magnitudes can be explained in relation to the characteristics of the elastic solution computed. In the first case the local fiber volume fraction controls the distance, with respect to the central partially debonded fiber, at which a localized perturbation zone appears in the far-field elastic solution; in the second case the global V f determines the characteristic lengths of a global periodic solution.

Interaction between debonds in UD composites with a single row of fibers

The interaction of debonds appearing at regular intervals in an ultra-thin UD composite with a single row of fibers is studied for Mode I (Fig. 8) and Mode II (Fig. 9) and fiber content equal to 30% (Figs. 8a and9a) and 60% (Figs. 8b and9b). The models treated are 3 × 1f ree, 5 × 1f ree, 7 × 1f ree, 11 × 1f ree, 21 × 1f ree, 101 × 1f ree and 201 × 1f ree, corresponding respectively to a debond every 3 rd , 5 th , 7 th , 11 th , 21 st , 101 st and 201 st fiber (Fig. 1a). Given that the upper surface of the UD row is left free, the interaction with the debonded fiber in the next RUC is stronger than in any other case and the results of this section are thus the most conservative in terms of debond's growth: the ERRs should be the largest. The effect is enhanced in composites with high V f and especially for G II : at V f = 60% the highest G II value for the 201 × 1f ree composite in Fig. 9b is more than 3 times higher than the G II value value for the 21 × 21f ree composite in Fig. 7b. Even the maximum is shifted to larger angles. The G I value is for some cases only 30% higher.

From both Fig. 8 and Fig. 9, it can be seen that the presence of a debond close to the analyzed debond decreases the strain magnification effect discussed in Sec. 3.1 and thus reduces the value of the ERR. This phenomenon is called "crack shielding" [START_REF] García | Debonding at the fibre-matrix interface under remote transverse tension. one debond or two symmetric debonds?[END_REF]. For Mode I, the presence of a free surface, and inversely the absence of a fully bonded fiber along the vertical direction, implies the absence of the counteracting upward-oriented vertical component of the displacement field due to the mismatch in Poisson's ratios. This in turn translates into the constancy of the value of ∆θ corresponding to contact zone's onset, always equal to 60 • .

G I   J/m 2   1 × 1 -f ree 3 × 1 -f ree 5 × 1 -f ree 7 × 1 -f ree 11 × 1 -f ree 21 × 1 -f ree 101 × 1 -f ree 201 × 1 -f ree (a) V f = 30%.
G I   J/m 2   1 × 1 -f ree 3 × 1 -f ree 5 × 1 -f ree 7 × 1 -f ree 11 × 1 -f ree 21 × 1 -f ree 101 × 1 -f ree 201 × 1 -f ree (b) V f = 60%.
For V f = 30%, Mode I is reduced when the spacing between debonds (in terms of number of fully bonded fibers between them in our models) decreases, but the magnitude of change is significant only in the range when the spacing is reduced from a debond every 5 th fiber to one every 3 rd . For comparison, the difference of peak G I values for V f = 30% between 5 × 1f ree and 3 × 1f ree is ∼ 0.2 J m 2 (around 30% of the lower value), while between 201 × 1f ree and 5 × 1f ree is ∼ 0.05 J m 2 (around 7% of the lower value). A similar observation can be made for V f = 60%, but for larger spacings: no difference can be seen between the case of a debond placed every 101 st and every 201 st fiber. These observations suggest the existence of characteristic distance dependent on the fiber volume fraction which governs the interaction between debonds: in low V f composites (V f = 30%) the convergence to a non-interactive solution is faster (less interaction between debonded fibers in neighboring RUCs).

G II

  J/m 2   1 × 1 -f ree 3 × 1 -f ree 5 × 1 -f ree 7 × 1 -f ree 11 × 1 -f ree 21 × 1 -f ree 101 × 1 -f ree 201 × 1 -f ree (a) V f = 30%.
G II   J/m 2   1 × 1 -f ree 3 × 1 -f ree 5 × 1 -f ree 7 × 1 -f ree 11 × 1 -f ree 21 × 1 -f ree 101 × 1 -f ree 201 × 1 -f ree (b) V f = 60%.
Figure 9: Effect of the interaction between debonds appearing at regular intervals on Mode II ERR in an UD with a single row of fibers at different levels of fiber volume fraction V f , subject to an applied transverse strain εx of 1%.

Without costraint on the upper surface, the strain magnification effect creates a larger displacement gap in the x-direction, which increases Mode II for larger debonds. When debonds are far apart, the series of rigid elements in the ultra-thin composite row (constituted by fully bonded fibers and their surrounding matrix) creates higher x-strains than in average in the element with the debonded fiber, which in turn generates higher tangential displacements at the crack tip for larger debonds. Conversely, when debonds are closer (smaller number of rigid elements between them), the strain concentration in the debonded element is more similar to the applied strain (the magnification is reduced) and the tangential displacement component at the crack tip decreases for large ∆θ. This is the mechanism behind the change in the value of ∆θ for which the peak of G II occurs: from 70 • to 50 • at 30%, and from 80 • to 40 • at 60% going from the higher to the smaller spacing of debonds. Differently from Mode I, the presence of a characteristic distance is harder to establish. For V f = 30% (Fig. 9a), it seems reasonable to establish it at around 100 fully bonded fibers between each debond. For V f = 60% (Fig. 9b), the difference between models 101 × 1f ree and 201 × 1f ree is still sizable, thus preventing the establishment of such characteristic distance. It is possible to observe, however, that the change between 101 × 1f ree and 201 × 1f ree is significantly smaller than between 21 × 1f ree and 101 × 1f ree (2 J m 2 vs 11 J m 2 ), thus suggesting the existence of the characteristic distance outside the range studied.

Nevertheless, one should question whether the single row composite with free surface is an appropriate RUC for defining the upper bound for G II : G II may be more affected by the free surface than by the effect of the interaction between debonds in the row.

Influence of rows of fully bonded fibers on debond's ERR in the middle row

The effect of the presence of rows of fully bonded fibers on debond's growth in the central row with all fibers partially debonded is studied for Mode I (Fig. 10) and Mode II (Fig. 11) and fiber content equal to 30% (Figs. 10a and11a) and 60% (Figs. 10b and11b). The models treated are 1 × 3f ree, 1 × 5f ree, size, due to the introduction of an additional positive component of the vertical displacement which translates into an opening displacement at the debond's tip.

1 × 7 -f ree, 1 × 11 -f ree, 1 × 21 -f ree, 1 × 101 -f
G I   J/m 2   1 × 3 -f ree 1 × 5 -f ree 1 × 7 -f ree 1 × 11 -f ree 1 × 21 -f ree 1 × 101 -f ree 1 × 201 -f ree (a) V f = 30%.
G I   J/m 2   1 × 2 -f ree 1 × 3 -f ree 1 × 4 -f ree 1 × 6 -f ree 1 × 11 -f ree 1 × 51 -f ree 1 × 101 -f ree (b) V f = 60%.
Comparing Fig. 9b with Fig. 11b, we observe that the presence of bonded fiber rows significantly reduce the G II and its maximum is shifted back to 60 • , thus confirming the hypothesis in Section 3.2 that the absence of G II convergence with the increasing distance in a single-row composite is caused more by the free surface than by the interaction between debonds.

The results of both Mode I and Mode II show that the introduction of an increasing number of fully bonded fiber rows doesn't change the ERR calculated at the crack tip after adding more than one row (the convergence is very fast).

A small effect, mostly on Mode I, of the number of bonded fiber rows can be observed at low fiber content (Figs. 10a and11a), while for high fiber content the smaller model with only one fiber row above the partially debonded one is already representative.

Effect of multiple rows of bonded fibers on debonding in the central row of a UD composite with different distances between debonded fibers

The ERR of debonds appearing at regular intervals in the central row of fibers in UD composites is affected by the number of rows with bonded fibers. with debonded fibers reduces the interaction with the free surface and thus has a restraining effect on the ERR, that counteracts the magnification due to an increasing number of fully bonded fibers in the horizontal direction. The interplay is further modulated by the fiber content. Observing Fig. 12, it is possibe to note how the free surface interaction decays fast: the presence of 5 fiber rows across the thickness is already sufficient to prevent any significant effect of additional fiber rows on the ERR of a debond in the central row.

∆θ • G II   J/m 2   1 × 3 -f ree 1 × 5 -f ree 1 × 7 -f ree 1 × 11 -f ree 1 × 21 -f ree 1 × 101 -f ree 1 × 201 -f ree (a) V f = 30%.
∆θ • G II   J/m 2   1 × 2 -f ree 1 × 3 -f ree 1 × 4 -f ree 1 × 6 -f ree 1 × 11 -f ree 1 × 51 -f ree 1 × 101 -f ree (b) V f = 60%.
The results in Fig. 13 show instead the effect of increasing the distance The comparison of the 1 × 1f ree model with one row multi-fiber models n × 1f ree in Figure 8 and Figure 9 show that the former provides in general the lowest value of the ERR (the highest crack shielding case).

G II   J/m 2   × 5 -f ree × 5 -f ree × 5 -f ree × 5 -f ree × 5 -f ree (b) GII .
The 1 × 1coupling model is compared with 1 × 3f ree and 1 × 201f ree models in Fig. 14. In all three models the distance between debonds in the x-direction is the same and the difference is in the vertical direction. The In each RUC, the fiber volume fraction is spatially homogeneous (no fiber clustering is considered) and the fiber distribution is uniform by design, which establishes a direct relationship between fiber content and inter-fiber distance.

The main conclusions of this work are summarized here.

1. With a decreasing number of fully bonded fibers between two partially debonded fibers in the central row, the ERR decreases. It seems to exist a characteristic distance between debonds which defines the transition to a non-interactive solution. However, this distance depends on the number of perfectly bonded fiber rows surrounding the central row and on the fiber volume fraction. This distance can be estimated to be around 100 fully bonded fibers for a 1-fiber-row thick UD with V f = 30% and 200 for a 1-fiber-row thick UD with V f = 60%, while it is expected to be around 100 fibers for a 5-fibers-row thick UD with V f = 60%.

2. The presence of a free surface close to the debond leads to higher Mode I and Mode II ERRs and a shift of the peak G values to larger debonds. 4. The presence of a debond in the fiber above the central partially debonded one only delays the appearance of the contact zone, while no significant effect on the ERR has been observed.

5.

Increasing the fiber content (decreasing the inter-fiber distance), magnifies in general the effects described in the previous points.

6. The results and conclusions presented agree well with previous observations reported in the literature [START_REF] Sandino | Numerical analysis of the influence of a nearby fibre on the interface crack growth in composites under transverse tensile load[END_REF][START_REF] Zhuang | Effects of inter-fiber spacing on fiber-matrix debond crack growth in unidirectional composites under transverse loading[END_REF]. A mechanical explanation of the observed trends has been presented based on the mismatch in elastic properties, particularly Poisson's ratios, and the positions of fibers and debonds with respect to the loading direction.

1 (

 1 a) Single row of fibers with a debond appearing every n fibers: model n × 1 -f ree (n = 3 in the figure).

  Single row of fibers with debonds appearing on each fiber: model 1 × 1 -f ree .

Figure 1 :

 1 Figure 1: Models of ultra-thin UD composites with a single "row" of fibers and debonds repeating at different distances.

3 ( 3 (

 33 a) Multiple rows of fibers with debonds appearing on each fiber beloging to the central row: model 1 × k -f ree (k = 3 in the figure). b) Mutiple rows of fibers with a debond appearing every n fibers within the central row: model n × k -f ree (n = 3 and k = 3 in the figure).

Figure 2 :

 2 Figure 2: Models of UD composites with different "rows" of fibers and debonds repeating at different distances.

Figure 3 :

 3 Figure 3: Model of UD composites with an infinite number of "rows" of fibers and debonds appearing on each fiber: model 1 × 1 -coupling .

Figure 4 :

 4 Figure 4: Schematic of the model with its main parameters.

  Material E [GP a] µ [GP a] ν [-] meshed using second order, 2D, plane strain triangular (CPE6) and rectangular (CPE8) elements, which have respectively 6 and 8 nodes per element. Each node has 2 degrees of freedom, i.e. the horizontal displacement u x and the vertical displacement u z . A regular mesh of quadrilateral elements with an almost unitary aspect ratio is required at the crack tip. The angular size δ of an element in the crack tip region is always equal to 0.05 • . The overall number of elements needed to discretize the model depends on the debond size ∆θ (larger debonds have larger contact zones which require more elements for their correct resolution), the fiber volume fraction (which determines the size of the RVE) and the number of fully bonded fibers present in the model. As an example, the discretization of the 1×1-f ree model at V f = 60% requires a total of 132507 elements for ∆θ = 10 • and of 296606 elements for ∆θ = 140 • , which corresponds to a minimum required RAM respectively of 445 M B and 1014 M B and to a minimum RAM needed to minimize I/O operations respectively of 1.63 GB and 3.87 GB. To put it into perspective, the wallclock time required for their solution is respectively 1.3 [min] and 14.95 [min] on a laptop with a 2.5 GHz Intel Core i5 processor and 6 GB of installed RAM. The crack faces are modeled as element-based surfaces and a small-sliding contact pair interaction with no friction is established between them. The Mode I, Mode II and total Energy Release Rates (ERRs) (respectively referred to as G I , G II

  which corresponds to a RUC's length and height of respectively ∼ 200R f and ∼ 100R f .

Figure 5 :

 5 Figure 5: Validation of the single fiber model for the infinite matrix case with respect to the BEM solution in [31].

  th fiber in the central fiber row is partially debonded and, on the top of this row, we have 5 undamaged fiber rows), Figs. 6a and 7a, and a 21 × 21f ree (every 21 th fiber in the central fiber row is partially debonded and, on the top of this row, we have 10 undamaged fiber rows), Figs.6b and 7b.Comparing Fig.6awith 6b, and Fig.7awith 7b, we can observe that the ERRs' values are very similar for RUCs with 11 × 11 and 21 × 21 fibers, though they are slightly higher for the larger RUC where the next debonded fiber and the free surface are further away from the debonded fiber. From these results we conclude that both RUCs are large enough to represent a single debonded fiber in an infinite array of bonded fibers. Obviously, there exists a specific effect of the fiber content. For Mode I, Fig.6, the maximum value of the ERR increases by ∼ 5.2 times when V f changes from 30% to 65%. The debond's angular size for which the peak value occurs remains unchanged at 20 • , but for V f = 60%

  a) Model 11 × 11 -f ree.

  b) Model 21 × 21 -f ree.

Figure 6 :

 6 Figure 6: A view of the effect of fiber volume fraction on Mode I ERR in two exemplificative models, subject to an applied transverse strain εx of 1%.

  a) Model 11 × 11 -f ree.

  b) Model 21 × 21 -f ree.

Figure 7 :

 7 Figure 7: A view of the effect of fiber volume fraction on Mode II ERR in two exemplificative models, subject to an applied transverse strain εx of 1%.

  e. closest to the loading direction. Notice that for ±25 • and ±155 • the two fully bonded fibers are almost in contact, with an inter-fiber distance of ∼ 0.04 times their radius. This result confirms the considerations made in the previous paragraph about the x-strain magnification caused by the presence of fully bonded fibers along the loading direction.The effect is further analyzed and discussed in Sec. 3.2 and Sec. 3.4. In the range ±40 • -±130 • instead, the presence of the other fibers causes a reduction of the ERR and, particularly in the range 80 • -120 • , results are very close and almost insensitive to variations in θ 2 , which supports the previous conclusion about the effect of a fully bonded fiber on top the partially debonded one. This effect is treated in more detail in Sec.3.3.

Figure 8 :

 8 Figure8: Effect of the interaction between debonds appearing at regular intervals on Mode I ERR in an UD with a single row of fibers at different levels of fiber volume fraction V f , subject to an applied transverse strain εx of 1%.

ree and 1 ×

 1 201f ree, corresponding to a UD composite with respectively 3, 5, 7, 11, 21, 101 and 201 rows of fibers (Fig. 2a). The results shown strengthen the arguments made in Sec. 3.1 and Sec. 3.2.It can, in fact, be seen in Fig.10that an increasing number of bonded fiber rows across the thickness delays the onset of the contact zone to a debond of 70 • in

Figure 10 :

 10 Figure 10: Influence of rows of fully bonded fibers on debond's growth in Mode I ERR in a centrally located row of debonded fibers at different levels of fiber volume fraction V f , subject to an applied transverse strain εx of 1%.

Figure 11 :

 11 Figure 11: Influence of rows of fully bonded fibers on debond's growth in Mode II ERR in a centrally located row of debonded fibers at different levels of fiber volume fraction V f , subject to an applied transverse strain εx of 1%.

21 × 3 × 3

 2133 f ree 21 × 5f ree 21 × 7f ree 21 × 11f ree 21 × 21f ree (a) GI . f ree × 5f ree × 7f ree × 11f ree × 21f ree (b) GII .

Figure 12 : 5 × 5

 1255 Figure 12: Effect on Mode I and Mode II ERR of the presence of an increasing number of rows of fully bonded fibers in UD composites with debonds appearing every 10 th fiber (model 21 × k -f ree). V f = 60% and εx = 1%.

Figure 13 : 3 . 5 .

 1335 Figure 13: Effect on Mode I and Mode II ERR of increasing the spacing between debonds appearing in the central row of fibers in a UD composite with a fixed number of rows across the thickness. V f = 60%, k = 5 and εx = 1%.

1 × 1 - 24 0 1 × 1 -coupling 1 × 3 -f ree 1 × 1 × 1 -coupling 1 × 3 -f ree 1 ×

 1241113111131 coupling model describes the interaction between debonds in different rows of debonded fibers whereas the 1×k-f ree models describe the effect of the proximity of the composite's free surface. The Mode I ERR in the 1 × 3f ree and 1 × 201f ree model is very similar to the 1 × 1coupling model, which leads to a rather surprising conclusion. In both models we have, on the top of the central one, a large amount of fibers (bonded in two cases and debonded in the third case). It appears that the effect of bonded and debonded fibers on the central debond is the same. This implies that, for Mode I ERR, the interaction between debonds in elements placed on top of each other is small. The same comparison for Mode II shows a sizeable difference in the range 50 • -90 • , while the results almost coincide for smaller values of ∆θ. The lower values of G II of the 1 × 1coupling model in the range 50 • -90 • are due to the shielding effect of a debond of the same size in the fiber just above the central one (modeled by the coupling boundary condition), which leaves the strip of matrix between the two fibers free to deform away from both of them due to 201f ree (a) GI . 201f ree (b) GII .

Figure 14 :Fig. 14a. 4 .

 144 Figure 14: Comparison of the ERR between the single fiber model with coupling conditions along the upper boundary and the 1 × k -f ree model. V f = 60% and εx = 1%.

3 .

 3 The presence of fibers (fully or partially bonded) in the composite thickness direction, along the same vertical line as the analyzed central fiber, appears to have a restraining effect on both G I and G II . The free composite surface effect on the ERR decays very fast: adding more than 2 fully bonded fibers below and above the central row leads to stable constant values of ERR.

Table 1 :

 1 Summary of the mechanical properties of fiber and matrix. E stands for Young's modulus, µ for shear modulus and ν for Poisson's ratio.

Acknowledgements

Luca Di Stasio gratefully acknowledges the support of the European School of Materials (EUSMAT) through the DocMASE Doctoral Programme and the European Commission through the Erasmus Mundus Programme.


H Bouhani

A Endichi

H Zaari

A Benyoussef

M Hamedoun

M Balli

El Kenz

O Mounkachi

On the origin of the giant magneto-caloric effect in Ho single crystals: First principles study and Monte Carlo simulations

Keywords: Magneto-caloric effect, anisotropy, XMCD, magnetism, First principles calculations

Large rotating magneto-caloric effect (MCE) has been recently observed in HoMn2O5 single crystal due to the strong magneto-crystalline anisotropy making a significant step towards efficient magnetic refrigeration systems. We tried to explain, theoretically, the reasons behind the magnificent performance in terms of magneto-caloric effect based on DFT calculations and Monte Carlo simulations. X-ray magnetic circular dichroism (XMCD) was performed in order to disentangle the magnetic contribution responsible for the MCE, as well as, the investigation of electronic and magnetic properties. The obtained results would enable the understanding of electronic and magnetic behaviors experimentally observed in HoMn2O5 single crystal and also the optimization the MCE for magnetic refrigeration.

I. Introduction

Multiferroic materials have been intensively studied over the past decade for their high interest and their potential applications in the field of spintronic [START_REF] Cheong | Multiferroics: a magnetic twist for ferroelectricity[END_REF]. Indeed, these multifunctional systems are usually characterized by a fundamental state involving a strong interplay between magnetism and ferroelectricity [START_REF] Aizu | Possible Species of Ferromagnetic, Ferroelectric, and Ferroelastic Crystals[END_REF][START_REF] Balli | Anisotropy-enhanced giant reversible rotating magnetocaloric effect in HoMn2O5 single crystals[END_REF], opening the way for several future applications.

One of the fields of great interest is the magneto-caloric effect. Recently, a large rotating MCE was experimentally observed in multiferroics RMn2O5 (R=Ho,Tb) [START_REF] Balli | Giant rotating magnetocaloric effect at low magnetic fields in multiferroic TbMn2O5 single crystals[END_REF][START_REF] Noda | Magnetic and ferroelectric properties of multiferroic RMn2O5[END_REF][START_REF] Fournier | Giant rotating magnetocaloric effect at low magnetic fields in multiferroic TbMn2O5 single crystals[END_REF]. Following this discovery, an innovative concept for the liquefaction of helium and hydrogen using the rotating MCE was proposed [START_REF] Balli | Anisotropy-enhanced giant reversible rotating magnetocaloric effect in HoMn2O5 single crystals[END_REF] which is an important step towards the development of magnetic refrigeration. It is worth noting that the majority of existing prototypes and patents are based on the rotation of magneto-caloric materials in a magnetic field since there is a much lower contribution of the magneto-crystalline anisotropy to the magnetoc-caloric effect at the magnetic phrase transition when compared to that generated by the change of magnetic transition [START_REF] Nikitin | Giant Rotating Magnetocaloric Effect in the Region of Spin-Reorientation Transition in the NdCo5 Single Crystal[END_REF].

These multiferroics materials are usually frustrated magnets whose spin-spin and spin-lattice interactions exclude the simple magnetic arrangement of spins. The consequences of their frustrated magnetic structures are spectacular, it manifests mainly as the strong correlation between magnetic and electrical properties [START_REF] Hur | Electric polarization reversal and memory in a multiferroic material induced by magnetic fields[END_REF]. The RMn2O5 compounds (R = rare earth) have an orthorhombic structure [START_REF] Blake | Spin structure and magnetic frustration in multiferroic RMn2O5[END_REF] consisting of Mn 4+ O6 octahedral and Mn 3+ O5 bipyramids connected by their edges and corners. This structure is illustrated in FIG. 1. Perfect knowledge of magnetic orders involving Mn 3+ and Mn 4+ ions is essential for a better understanding of multiferroic features in RMn2O5 compounds. Recent studies of RMn2O5 compounds show rich magnetic properties [START_REF] Balli | Anisotropy-enhanced giant reversible rotating magnetocaloric effect in HoMn2O5 single crystals[END_REF][START_REF] Balli | Giant rotating magnetocaloric effect at low magnetic fields in multiferroic TbMn2O5 single crystals[END_REF]. The majority of these materials present at least four magnetic transitions: a first transition occurs at T1 = 42 -45 K in which the spins of Mn ions adopt a sinusoidal incommensurate (ICM) order. Then, another transition takes place at T2 = 38-41 K where the spins are configured in a commensurate (CM) antiferromagnetic order then a return to sinusoidal incommensurate order at T3 = 18 K. Below 10 K, the magnetic order of the rare earth ions appears and adopts an incommensurate antiferromagnetic order, becoming commensurate on further cooling [START_REF] Blake | Spin structure and magnetic frustration in multiferroic RMn2O5[END_REF]. A dielectric constant step is observed at the CM-ICM magnetic phase transition [START_REF] Cao | First-Principles Modeling of Multiferroic RMn2O5[END_REF] being the key to understand the strong magnetoelectric correlation which is indicated by the modification of dielectric function due to the ordering of Mn spins [START_REF] Radulov | Colossal magnetostriction effect in HoMn2O5[END_REF]. In this work, we mainly focus on the electronic, magnetic and magnetocaloric properties of HoMn2O5 single crystal and therefore, explain the reasons behind the large rotating magneto caloric effect recently observed in HoMn2O5 crystals [START_REF] Balli | Anisotropy-enhanced giant reversible rotating magnetocaloric effect in HoMn2O5 single crystals[END_REF].

II. Computational details

All calculations were done using the Full Potential Augmented Plane Wave (FP-LAPW) method implemented in WIEN2K code [START_REF] Blaha | WIEN2k_7.3, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties[END_REF] based on density functional theory [START_REF] Hohenberg | Inhomogeneous Electron Gas[END_REF] with the exchange correlation functional treated in the Generalized Gradient Approximations [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] taking into account the spin orbit interactions. To investigate the magnetic properties, we have used 7*6*9 Monkhorst-Pack scheme [START_REF] Monkhorst | Special points for Brillouin-zone integrations[END_REF], the total energy was converged to less than 10 Ryd and the cut-off energy is set to -8.0

Ryd. The radii of the muffin-tin spheres were: RMT (Ho) = 2.53, RMT (Mn1) = 1.86, RMT (Mn2) = 1.97 and RMT (O) = 1.67 a.u (atomic units). Moreover, magnetic anisotropy is calculated using DIPAN program [16,[START_REF] Blaha | Marks: WIEN2k An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties revised edition WIEN2k 18[END_REF] that calculate the dipolar magneto-crystalline anisotropy and the magnetic dipolar hyperfine field by a direct lattice summation over the magnetic moment of all the sites.

In order to disentangle the magnetic contribution responsible for the magneto-caloric effect in HoMn2O5, we have studied the differentiated role of Ho and Mn ions using XMCD and XAS spectrum.

Therefore, we calculated, separately, the values of both spin and orbital magnetic moment.

XMCD represents the absorption difference between the right circular polarization ( + ) and the left circular polarization ( -). The left and right directions are referenced to the propagation direction of the incident radiation while respecting the axis of the magnetic field. In particular, the photons are polarized left (right). When the direction of propagation is antiparallel, their helicity isћ (+-ћ) [START_REF] Thole | X-ray circular dichroism as a probe of orbital magnetization[END_REF].

The objective of XMCD spectrum is to determine the orbital and spin moment, knowing that there is a relationship between the integral of the XMCD spectrum and the mean value of the projection of the orbital angular momentum on the magnetization axis. This relationship was suggested by TholeCarra and Van Der Laan [START_REF] Carra | X-ray circular dichroism and local magnetic fields[END_REF], later, the same authors proposed a second sum rules for spin moment [START_REF] Wu | Limitation of the Magnetic-Circular-Dichroism Spin Sum Rule for Transition Metals and Importance of the Magnetic Dipole Term[END_REF] ( ) -<Tz>= expectation value of magnetic dipole operator The sum rules allow to obtain separately the orbital magnetic moment <Lz> and the spin magnetic moment <Sz> [START_REF] Thole | X-ray circular dichroism as a probe of orbital magnetization[END_REF] from the integrated XMCD.

( ) ( ) ( ) ( )( ) > Lz < . n + l + l l + l l + c c = μ + μ + μ ω d μ μ ω d + ± j + + j - - - - - - ∫ ∫ 2 4 1 2 2 1 1 1 0 (I.1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) > - - - - - - - - - - ∫ ∫ ∫ Sz < n + l c + c c + l l = μ + μ + μ ω d μ μ ω d + c c μ μ ω d + ± j + j + + j
For M4, 5 the transition from 3d (l=2) to 4f (l=3):

( )( ) 1 4 5 4 4 5 ( / ) ((7 ( ) 6 ( )) / ( ( )) * 14 1 10( / ) b f M M M M ms Sz d d d n Tz Sz µ ω µ µ ω µ µ ω µ µ - + - + - + - + = = - - - + - + ∫ ∫ ∫ h ( ) 4 4 5 4 5 ( / ) 2(( ( )) / ( )) * 14 L b f M M M M m Lz d d n µ ω µ µ ω µ µ + - + - + + = = - - + - ∫ ∫ h (I.3)
For L23, the transition from 2p (l=1) to 3d (l=2):

( )

3 2 3 2 3 ( ) 4 10 3 ( ) L L b L d L L d m L z n d ω µ µ µ ω µ µ + - + + - + - = = - - + ∫ ∫ h (I.4) ( ) 3 3 2 3 2 1 3 6 ( ) 4 ( ) 7 10 1 2 ( ) L L L b d L L d d T z m s S z n S z d ω µ µ ω µ µ µ ω µ µ + - + - - + + - + - - -   = = - - +     +   ∫ ∫ ∫ h (I.5)
A strong magneto-crystalline anisotropy is experimentally observed in HoMn2O5 single crystals where the magnetization tends to orient preferentially along the b-axis [START_REF] Balli | Anisotropy-enhanced giant reversible rotating magnetocaloric effect in HoMn2O5 single crystals[END_REF][START_REF] Balli | Comment on ``Giant anisotropy of magnetocaloric effect in TbMnO3 single crystals[END_REF]. In order to get more insight on the gigantic anisotropy shown by this compound, the energy of the system following each axis has been calculated using DIPAN program implemented in WIEN2k package.

III. Results and Discussions : 1. Electronic properties

We have displayed the density of state using GGA approximation to describe the magnetic and electronic properties shown in FIG. 2. The main contribution to the magnetism comes from 4f states of Holmium which is responsible for the magnetic moment formation on Holmium atoms. The oxygenmanganese molecular bond is formed by p electrons of oxygen and s electrons of manganese. Each oxygen atom is twice ionized, involving two electrons in its p orbitals. The ionization of the manganese atom depends on the chemical formula of the compound: Mn is 4 times ionized, Mn 3+ and Mn 4+ are deficient in electrons, respectively involving 4 and 3 electrons in its d layers. In the absence of perturbation, the manganese d orbitals are degenerate but in a crystal field of given symmetry. The crystal matrix favors an arrangement of some orbitals by partially lifting their degeneration. For HoMn2O5 compound, Mn occupies two sites of different symmetry: Mn 4+ at the center of an octahedron and Mn 3+ at the center of a pyramid. Each symmetry acts differently on the distribution of the electronic levels of Mn. The d orbitals are divided into two sub-groups for octahedral and tetragonal symmetries: at t2g level of low triple-degenerate and another level doubly degenerate Eg. As expected, we can see that the degeneracy of its d orbital is lifted especially in Dz 2 lodge. This anomaly is known as the crystal field deformation or Jahn Teller effect [START_REF] Chang | Charge-orbital ordering and ferroelectric polarization in multiferroic TbMn${}_{2}$O${}_{5}$ from first principles[END_REF].

One can also notice that 4f Holmium orbitals favors the spin orbit coupling effect while the 3d orbitals of manganese favors the crystalline field. T2g levels have dxy, dyz and dxz orbital character lying on the same level while there is separation in energy between the e2g pics which indicate a deformation field or splitting in Mn1. In the contrary, the partial density of states for Mn2 shows a t2g splitting with a 2/3 filled band while there is no e2g splitting as shown in FIG. 3.

Magnetic properties

As mentioned in the introduction, HoMn2O5 single crystals shows an interesting magnetic properties following a specific direction. Indeed, the calculation of the energy of each axis demonstrate that system has its lowest energy following the b-axis which has been reported in experimental data [START_REF] Balli | Anisotropy-enhanced giant reversible rotating magnetocaloric effect in HoMn2O5 single crystals[END_REF].

The high magnetic moment of Holmium atoms makes the compound an excellent candidate for MCE materials. Therefore, we calculate that spin and orbital magnetic moment, separately, using XMCD spectrum to disentangle the magnetic contribution responsible for the MCE in HoMn2O5 single crystals.

A. XMCD at the Ho M4,5 edge

The absorption edge that provides the most direct information for the rare earth 3d-4f transitions is the M4,5 edge. The theoretical treatment of that edge is generally done using a model where the interactions with the environment such as the crystal field are considered to be of weak perturbation except for systems where the rare earth exhibits valence instability or mixed valence [START_REF] Band | Khomskii Atomic mechanisms for the appearance of mixed valence states in lanthanides and actinides[END_REF]. This approach is valid due to localized Ho 4f electrons and there is therefore no overlap between the 4f levels of two neighboring atoms.

In FIG. 4, the Ho 3d-4f absorption spectrum is characterized by M4 and M5 that are separated in energy by the strong spin-orbit effect. At M5 edge, the intensity is very high compared to M4 edge indicating a large contribution of the orbital moment. We calculated, therefore, the spin and orbital magnetic moment. The values are as follows:

Tab. 1. Spin, orbital and total magnetic moment for Ho, Mn1 and Mn2 The expected value of a free Ho 3+ (4 10 f ) magnetic moment with respect to hund's rule ground state (J=8, S=2 and L=6) is 10.3 taking into consideration that the atom is considered as isolated whereas, it is necessary to take into account the atomic environment.

The negative (positive) signal indicates a parallel alignment of Ho sublattice to the applied magnetic field and may refer to a double transition: 3d to 4f and 4f to 5s known as the quadrupole transition.

FIG. 4 (b) represents the integral of XAS (XMCD) that allow the calculation of the spin and orbital magnetic moment.

B.XMCD at the Mn L2,3 edge

We tried now to acquire in depth information about the magnetic contribution from the Mn ions. The results shown in Tab.1, indicates that only a small magnetic moment is induced within the antiferromagnetic cycloidal configuration confirming that along the easy axis b, the contribution of the Mn lattice to the full magnetization is negligible and mainly originates from the Ho 3+ sublattice. XMCD spectra representing M4.5 (wine line) and the integral of the spectra (black line) within GGA approximation

Magnetocaloric properties

To investigate the magneto-caloric effect in HoMn2O5 single crystal, a classical model that combines both Ising and XY models was proposed to deal with the anisotropy since the manganese spin's components can be oriented in any direction in a two-dimensional plane while the holmium's spin components direction are either parallel or antiparallel. Indeed, the spins are represented by twocomponent vectors that satisfy the = + constraint.

The proposed hamiltonian is: Where:

1314 1332 1350 1368 0.00 6.90x10 -5 1.38x10 -4 2.07x10 -4 Energy (eV) 0.00 2.50x10 -4 5.00x10
S %& : Holmium spin, : /01201343 /1 #5 4671 and σ : /01201343 /1 "5 spin J : Interaction coupling between first /1 #5 nearest neighbors J : Interaction coupling between first /1 "5 nearest neighbors

J " : Interaction coupling between first /1 #5 -/1 "5 nearest neighbors J # : Interaction coupling between second /1 #5 -/1 "5 nearest neighbors J $ : Interaction coupling between first <= "5 -/1 "5 nearest neighbors J : Interaction coupling between first <= "5 -/1 #5 nearest neighbors ). The equilibrium properties are reached when averaging the generated configuration starting from different initial conditions.

The data generated with 10 steps per spin discarding the first 10 # generated configurations, our calculations were performed for a system with N= 60.

The program used for calculations predicts the following thermodynamic quantities: the magnetization, specific heat and the susceptibility which are given respectively by:

/ O = P ∑ S %& P %Q and M S& = P . T / U 2 + / W 2 (I.7) X = Y Z [ \ [< / _=_02 > -< / bcbde > ] (I.8) gh = 1 Z i \ [< j > -< j > ] (I.9)
Where E is the internal energy of the system, Kb is the Boltzmann constant and N is the number of spin in each sublattice. The isothermal magnetic entropy change was calculated using an integral version of Maxwell's relation: Another important parameter that is usually used to compare the MCE is the relative cooling power (RCP) which is defined as the product of the maximum isothermal magnetic entropy and the full width at half maximum and it is given in Fig. 6 as the RCP reaches 353 J/Kg for an applied magnetic field of 7 T. These results are comparable with those reported in literature [START_REF] Balli | Anisotropy-enhanced giant reversible rotating magnetocaloric effect in HoMn2O5 single crystals[END_REF] FIG. 6. Relative cooling power (RCP) as a function of the applied magnetic field

Conclusion

First principles calculations reveal that the saturation magnetization reaches it maximum following the easy axis b. The same calculations shows that the main contribution to the MCE comes from Ho ions.

The MCE effect in HoMn2O5 single crystal is studied based on Monte Carlo simulations giving similar results compared to experimental study. This work confirms that the use of the intrinsic anisotropic properties is a promising way to optimize the MCE for magnetic refrigeration application at low temperature regime.

Compliance with ethical standards

Conflit of interest The authors declare there is no conflict of interest.

FIG. 1 .

 1 FIG. 1. (a) Side view and (b) top view along b axis of the crystallographic structure of HoMn2O5 single crystals. The green spheres are holmium atoms, the orange stands for the manganese atoms and the red corresponds to oxygen atoms.

  orbital quantum number of the valence state -c = orbital quantum number of the core state -( + ) ( -) = absorption spectrum for left (right) circularly polarized light.

FIG. 2 . 7 FIG. 3 .

 273 FIG. 2. The total and partial density of states obtained within (GGA+ Spin Orbit)

FIG. 4 .

 4 FIG. 4. (a) XAS spectra representing M4.5 (wine line) and the integral of the spectra (black line) within GGA approximation and (b)

  J ) : Interaction coupling between first <= "5 nearest neighbors J * : Interaction coupling between second <= "5 nearest neighbors With > ? = @ A, > ? = @ A and h: External magnetic field. The MCE in HoMn2O5 single crystal is studied based on Monte Carlo simulation using the hamiltonian given by Eq. (I.7) where random spins were chosen and then flipped with Boltzmann based probability using Metropolis algorithm [23] B CDE = exp. (-∆K L M N

FIG. 5 .

 5 FIG. 5. (a) Temperature dependence of magnetization following y axis for HoMn2O5 elements (b) Isothermal magnetic entropy change for different applied magnetic field

Fig. 5 .

 5 Fig.5. (a) presents the temperature dependence of magnetization following b axis for Ho 3+ , Mn 3and Mn 4+ . |ΔS| values are shown in Fig.5(b) for several magnetic field. The magnetic field variation from 0 to 7 T yields to a maximum of entropy change of 12.20 J/Kg.K close to the value reported by Balli et al[START_REF] Balli | Anisotropy-enhanced giant reversible rotating magnetocaloric effect in HoMn2O5 single crystals[END_REF] which is about 13.1 J/Kg.K. The maximum of |ΔS| increases with increasing the external magnetic field since the magnetic moments are governed by the interactions of exchange between ions in a magnetic structure passing to a disordered magnetic configuration (paramagnetic region).

Acknowledgments

This work was supported by the PHC Toubkal/17/49 project.


B La Scola

Manon Boxberger

Hussein Anani

Bernard La Scola
email: bernard.la-scola@univ-amu.fr

new bacterium isolated from human ileum of a patient with Crohn disease

Keywords: Culturomics, taxono-genomics, Alterileibacterium massiliense Alterileibacterium massiliense gen. nov. sp. nov. strain Marseille

published or not. The documents may come

Introduction

Alterileibacterium massiliense was isolated using culturomics approach that is based on the use of a large culture conditions panel in order to describe the microbial composition of a sample by high-throughput culture [START_REF] Lagier | Microbial culturomics: paradigm shift in the human gut microbiome study[END_REF][START_REF] Lagier | The rebirth of culture in microbiology through the example of culturomics to study human gut microbiota[END_REF][START_REF] Lagier | Culture of previously uncultured members of the human gut microbiota by culturomics[END_REF][START_REF] Lagier | Current and past strategies for bacterial culture in clinical microbiology[END_REF]. A taxono-genomics approach including matrixassisted laser desorption-ionization time-of-flight mass spectrometry (MALDI-TOF MS), phylogenetic analysis, main phenotypic description and genome sequencing, was used to describe it [START_REF] Fournier | From culturomics to taxonomogenomics: A need to change the taxonomy of prokaryotes in clinical microbiology[END_REF][START_REF] Ramasamy | A polyphasic strategy incorporating genomic data for the taxonomic description of novel bacterial species[END_REF].

Isolation and growth conditions

In April 2016, an unidentified bacterial strain was isolated from the human ileum of a patient with Crohn disease and provisionally named Ileibacterium massiliense [START_REF] Mailhe | Ileibacterium massiliense' gen. nov., sp. nov., a new bacterial species isolated from human ileum of a patient with Crohn disease[END_REF]. Tentative identification was done by MALDI-TOF MS on a Microflex LT spectrometer (Bruker Daltonics, Bremen, Germany) as previously described [START_REF] Seng | Ongoing revolution in bacteriology: routine identification of bacteria by matrix-assisted laser desorption ionization time-of-flight mass spectrometry[END_REF]. The obtained spectra (Figure 1) was imported into MALDI Biotyper 3.0 software (Bruker Daltonics) and analyzed against the main spectra of the bacteria included in two databases (Bruker Daltonics and constantly updated MEPHI databases (http://www.mediterraneeinfection.com/article.php?larub=280&titre=urms-database). This strain was cultured routinely on columbia sheep blood agar (Biomerieux, Marcy l'Etoile, France) at 37°C under anaerobic conditions.

Strain identification

To identify this bacterium, the 16S rRNA gene was amplified using the primer pair fD1 and rP2 (Eurogentec, Angers, France) and sequenced using the Big Dye® Terminator v1.1 Cycle Sequencing Kit and 3500xLGenetic Analyzer capillary sequencer (Thermofisher, Saint-Aubin, France) as previously described [START_REF] Drancourt | 16S ribosomal DNA sequence analysis of a large collection of environmental and clinical unidentifiable bacterial isolates[END_REF]. The 16S rRNA nucleotide sequence was assembled and corrected using CodonCode Aligner software (http://www.codoncode.com).

The 16s rDNA gene sequence strain Marseille-P3115 exhibited a 90.7 % sequence similarity with Mogibacterium neglectum ATCC700924 T (GenBank accession no. AB037875) the phylogenetically closest species with standing in nomenclature (Figure 3). We consequently classify this strain as a member of a new species within the new genus Alterileibacterium, family Clostridiales XIII. Incertae Sedis, phylum Firmicutes.

Phenotypic characteristics (Table 1)

Microcolonies are white and circular with a mean diameter of 0.08 mm. Bacterial cells of this gram-positive bacterium are easily discolored and appear rather as Gram-negative bacilli that differs with their neighbor's genus Eubacterium and Mogibacterium. They present a mean diameter of 0.3 μm and a mean length of 1 μm. (Figure 2) Strain Marseille-P3115 showed catalase-negative and oxidase-negative activities.

Genomics

Genomic DNA of the bacterium was sequenced using the MiSeq Technology (Illumina Inc, San Diego, CA, USA) with the mate pair strategy as previously described [START_REF] Diop | Microbial culturomics unravels the halophilic microbiota repertoire of table salt: description of Gracilibacillus massiliensis sp[END_REF].

Total information of 5.1 Gb was obtained from a 544K/mm 2 cluster density with a cluster passing quality control filters of 96.8 % (10.139.000 passing filter paired reads). Within this run, the index representation for A. massiliense was determined to 4.43%. The 449.618 paired reads were trimmed using the Trimmomatic software [START_REF] Bolger | Trimmomatic: a flexible trimmer for Illumina sequence data[END_REF], GapCloser [START_REF] Llorens | The Gypsy Database (GyDB) of mobile genetic elements: release 2.0[END_REF] was used to reduce gaps, then assembled with the Spades software [START_REF] Bankevich | SPAdes: a new genome assembly algorithm and its applications to single-cell sequencing[END_REF] in 2 scaffolds.

The genome of strain Marseille-P3115 is 1.450.823 bp long with a 35.9 mol% G+C content. The degree of genomic similarity of A. massiliense strain Marseille-P3115 with closely related species was estimated using the OrthoANI software [START_REF] Lee | OrthoANI: An improved algorithm and software for calculating average nucleotide identity[END_REF]. Values among closely related species (Figure 4) ranged from 63.39% between Eubacterium brachy ATCC (GenBank FNBF01000001.1) to 83.43% between Mogibacterium diversum ATCC700923 T (RefSeq NZ_CP027228.1) and Mogibacterium pumilium ATCC700696 T (RefSeq NZ_CP016199.1). When the isolate was compared to these closely species, values ranged from 64.02% with Eubacterium pyruvativorans ATCCBAA574 T (GenBank FNBF01000001.1) to 68.04% with Mogibacterium timidum ATCC33093 T (GenBank JALU01000001.1).

Conclusion

Strain Marseille-P3115 T exhibiting a 16S rRNA sequence divergence > 5 % with its phylogenetically closest species with standing in nomenclature, is consequently proposed as the type strain of the new genus Alterileibacterium massiliense gen. nov., sp. nov.

(Alter.il.ei.bac.te ' ri.um, Gr. adj. Alter 'other', Gr. n. ilei, 'ileum'; Gr. n. bakterion, 'bacterium'; N.L. neut. n. Ileibacterium, 'bacterium isolated from the human ileum sample 'mas.si.li. en'se, L. neut. adj., massiliense for Massilia, the Latin name of Marseille, where the strain was first isolated). This strain was provisionally named Ileibacterium massiliense [START_REF] Mailhe | Ileibacterium massiliense' gen. nov., sp. nov., a new bacterial species isolated from human ileum of a patient with Crohn disease[END_REF]. However, in the same time the genus name Ileibacterium gen. nov. was used to name an isolate of the family Erysipelotrichaceae [START_REF] Cox | Description of two novel members of the family Erysipelotrichaceae: Ileibacterium valens gen. nov., sp. nov. and Dubosiella newyorkensis, gen. nov., sp. nov., from the murine intestine, and emendation to the description of Faecalibaculum rodentium[END_REF]. This is the reason we modified the name of our isolate from Ileibacterium massiliense gen. nov. sp. nov. to Alterileibacterium massiliense gen. nov. sp. nov.

Nucleotide sequence accession number. The 16S rRNA gene and genome sequences were deposited in GenBank under accession number LT598557 and FNWE00000000, respectively.

Deposit in culture collections. Strain Marseille-P3115 T was deposited in two different strain collections under number CSURP3115 and DSM103486.

Figure 1 :Figure 2 .

 12 Figure 1: MALDI-TOF MS Reference mass spectrum for Alterileibacterium massiliense

Figure 3 :

 3 Figure 3: Phylogenetic tree showing the position of Alterileibacterium massiliense strain

Figure 4 :

 4 Figure 4: Heatmap generated with OrthoANI values calculated using the OAT software

FNBF01000001.1)

Acknowledgements

The authors are indebted to Catherine Robert for sequencing the genome, Aurelia Caputo for submitting the genome sequence to GenBank and the platform of electron microscopy of IHU for the electron micrographs.

Funding

This work was supported by the French Government under the "Investissements d'avenir" program managed by the Agence Nationale de la Recherche (ANR), [reference: Méditerranée-Infection 10-IAHU-03], by Région Provence-Alpes-Côte d'Azur and European funding FEDER PRIMI

Conflict of interest:

None to declare.