N Bourenane

Y Hamlaoui
email: hamlaoui_youcef@yahoo.fr

F Pedraza

Electrochemical investigation of ceria nanoparticles as corrosion inhibitor for steel in alkaline medium

Keywords: ceria nanoparticles, alkaline medium, corrosion inhibition, surface morphology, bath temperature

This work investigates the synthesis and the corrosion inhibitor efficiency ceria nanoparticles may exert on steel reinforcement in alkaline solution of saturated Ca(OH) 2 + 0.5M CaCl 2 up to 50 °C. The corrosion mechanisms are thoroughly investigated by electrochemical and various characterization techniques. The electrochemical results show that ceria nanoparticles act as anodic inhibitor and provide maximum inhibition efficiency (80 %) at a concentration of 800 ppm. The resulting surface is smooth and free from corrosion products due to the formation of a physically adsorbed film according to the thermodynamic calculations.

1.-Introduction

Corrosion of reinforcing steel, which occurs when chloride ions migrate through the concrete pores is considered as the most important cause that determines the durability of reinforced concrete structures. This type of corrosion often occurs in buildings and most highway bridges which are usually constructed with rebar reinforcing steel. The cement paste through which the contaminants initiate the process determines the corrosion resistance of steel reinforcing bars [1]. During hydration, Portland cement reacts chemically with water where the anhydrous compounds, which are slightly soluble, dissolve in solution to give different ionic species. The main hydration products are calcium-silicate-hydrate "C-S-H", and Portlandite or Ca(OH) 2 "C-H" where the amount of C-S-H or Ca(OH) 2 depend mainly on the water/cement ratio and the hydration time [2][3].

A cement paste hardened in ordinary conditions contains between 50 and 70 % of C-S-H and 25 to 27 % of Ca(OH) 2 . The C-S-H is the most important hydrated phase in the hardened cement that the main mechanical properties of the material will depend on. However, Portlandite is the compound that determines the concrete alkalinity (pH of concrete is between 12.4 and 13.5) and may lower the physical properties of the cement paste since it is more soluble than the C-S-H gels. However, Portlandite adjusts the interstitial solution pH value to approximately 12.5. When this phase is completely solubilized, the C-S-H fixes the solution pH between 10.5 < pH < 12.5. It is therefore the most soluble phase that imposes the pH until its total dissolution [4]. It is noted that the interstitial solution contains cations such as Ca 2+ , Na + and K + and anions such as SO 4 2-and OH -. Calcium ions come from gypsum and portlandite whereas sodium and potassium ions are derived from alkaline oxides (Na 2 O and K 2 O) present in cement. In addition, such cations can come from contamination agents (chlorides or sulphates) and/or from water [5]. Nevertheless, the presence of portlandite fixes the pH at 12.5 and the calcium concentration at 21 mmol/kg according to the equilibrium reaction R. 1:

Ca(OH) 2  Ca 2+ + 2 OH - R. 1
The most employed solutions to reproduce the chemical characteristics of concrete are alkaline solutions with Ca(OH) 2 at saturation with and without NaOH, KOH and carbonatebased solution (NaHCO 3 , Na 2 CO 3 ) for uncontaminated or for carbonated concrete, respectively [6][7][8]. Indeed, alkaline solutions in the presence of chloride ions in the form of NaCl or CaCl 2 were used to study the electrochemical behaviour and the depassivation mechanism [9] whereas buffered carbonate solutions are rarely used to study the corrosion behaviour of steel reinforcement in concrete [10].

It is well-known that the corrosion mechanism of steel in concrete involves two steps. The The exposure of reinforced concrete to chloride ions is the primary cause of premature corrosion of steel reinforcement. The intrusion of chloride ions from seawater or present in water or aggregate used in concrete, into reinforced concrete can cause steel corrosion if oxygen and moisture are also available to sustain the reaction. Chlorides dissolved in water can permeate through sound concrete or reach the steel through cracks. Thus, when the chloride content at the surface of the steel exceeds a certain limit known as the chloride threshold value, corrosion will occur if water and oxygen are also available. Although chlorides are directly responsible for the initiation of corrosion, they appear to play only an indirect role in the rate of corrosion after initiation. The primary rate-controlling factors are the availability of oxygen, the electrical resistivity and saturation degree of the concrete as well as pH and temperature.

Several studies have been conducted to develop methods and products to reduce or even to arrest the corrosion of steel reinforced concrete [12][13][14] including among others, cathodic protection [3,[START_REF] Manna | Effect of steel substrate for phosphate treatment[END_REF] (usually used for chloride contaminated concrete structures and for which it is necessary to keep the original surface [START_REF] Manna | Effect of steel substrate for phosphate treatment[END_REF]), surface oxidation [16], epoxy coating [17][18], electrochemical re-alkalization and electrochemical chloride extraction (ECE) [19]. Yet, the use of organic and inorganic corrosion inhibitors is considered to be the simplest and the least expensive [20].

In this view, cerium salts were successfully applied as corrosion inhibitors for metals in NaCl solution mainly for Al [21], Mg [22], Zn [23], carbon steel [24][25][26] and for Zn and rebar steel in concrete [27,28]. Ceria deposited films act as a physical barrier which prevent dissolved O 2 and or aggressive anions such as Cl -and SO 4 2-to penetrate through defects or cracks to the surface of the substrate. Moreover, it was shown that ceria based coatings repair defects after being damaged through Ce 3+ /Ce +4 redox process. The exchange process 2 CeO 2 ↔ Ce 2 O 3 + 1⁄2 O 2 is favoured since the redox reaction Ce 4+ ↔ Ce 3+ occurs easily [29][30][31]. Thus, following the success of cerium salts used as corrosion inhibitor, ceria particles have also been widely studied as pigments in organic protective coatings [32][33][34], as corrosion inhibitor of Zn [35], of steel in Na 2 SO 4 solution [36], in HCl and H 2 SO 4 [37]. To the best of our knowledge, only very limited work was conducted on the effectiveness of ceria nanoparticles as corrosion inhibitor in concrete pore solution [START_REF] Thiruvoth | Materials Today: Proceedings[END_REF]. Therefore, in the current study, the corrosion behaviour of steel is examined during immersion in alkaline solution of saturated lime Ca(OH) 2 contaminated by CaCl 2 chloride ions containing ceria nanoparticles as corrosion inhibitor. Also, the effect of the temperature on the inhibition efficiency of the inhibitor was evaluated between 25 and 55°C. For all such purposes, electrochemical techniques and surface morphological characterization were conducted.

2.-Experimental set-up

2.1.-Materials

Reinforcing steel bar of 10 mm in diameter according to E24 specification (NF EN 10025)

with chemical composition C-0.17, Mn-1.4, P-0.045, S-0.045 N-0.009 and balance Fe, was used. Coupon samples of 10-mm of diameter were cut from the steel bar. The active surface of the electrode was polished using a series of SiC papers ranging from 180 to 2500 grit, degreased by sonication for 30 min in acetone and then with ethanol, rinsed several times with distilled water and finally dried with hot air. The samples were kept in desiccators until immersion.

2.2.-Synthesis of ceria nanoparticles.

Cerium nitrate (Ce(NO 3 ) 3 .6H 2 0, Strem, 99,9 %) and acetic acid as organic ligand were used for nanoceria synthesis in colloidal dispersion. More details can be found in recent research involving the use of dispersion colloids for ceria preparation in the references [START_REF] Jima | [END_REF][40][41]. In a double-envelope stirred reactor, 0.5 M cerium nitrate and 0.5 M acetic acid were dissolved in 100 ml distilled water at 60 °C. 24.5 % ammonia was slowly added drop by drop to the above solution until reaching a pH solution equal to 9 then the bath temperature was increased to 6 90 °C. Right after, the reactor was hermetically closed in order to reduce the amount of the dissolved oxygen almost to 50 %. After 24 hours of stirring, the precipitate was recovered by centrifugation at 4000 rpm during 50 minutes in the presence of excess ethanol. The resulting precipitate was washed and centrifuged with distilled water and ethanol alternately three times in order to eliminate residual ions such as acetate. Finally, the collected ceria nanoparticles were redispersed in distilled water to constitute the inhibitor with a content of 10 wt% of ceria nanoparticles.

It was noted that during the addition of ammonia to alkalinize the medium, changes of the solution colours occur depending on the pH of the solution (Figure 1): for pH < 3.4, the solution is transparent (Fig. 1.a), a light brown coloration characteristic of Ce(OH) 3 precipitate subsequently appeared (Figures1.b and 1.c). For solution pH maintained between 9 and 9.5 (Fig. 1.d and 1.e), the colour of the medium becomes violet characterizing an oxidation reaction of Ce(OH) 3 to Ce(OH) 4 with a very significant increase of the medium viscosity. After 24 h of reaction time, the solution turns into a pale yellow colour (Fig. 1.f).

2.3.-Electrochemical Tests

A conventional three-electrode cell was used with a saturated calomel electrode as reference electrode, a platinum wire as counter electrode and the steel rebars as working electrode.

Electrochemical tests were carried out with a potentiostat Biological SP 150 Instrument at room temperature (21±2 °C). Ca(OH) 2 saturated solution was prepared with 3 g L -1 of Ca(OH) 2 in distilled water. The pH was 12.6. The concentration of the chloride ions in the solution was 0.5 M. Polarization resistance, R p , was obtained from the slope of potentiodynamic scan carried out in the anodic direction from -20 to +20 mV vs E oc (open circuit potential) at scan rate of 0.5 mV/s.

The potentiodynamic polarization curves were obtained by using a sweep rate of 0.5 mV/s between E oc ±250 mV. The impedance data were obtained at the E oc between 100 kHz and 10 mHz, with 4 mV as the applied sinusoidal perturbation. The EIS response from the samples was fitted using ZView software (Scribners Associates, USA). The potentiodynamic polarization and EIS techniques were performed after 30 min of immersion time.

2.4.-Characterization

Non-contact white-light optical profilometry was performed using a WLI FRT. Imaging of the surface of the samples was conducted using vertical scanning interferometry (VSI) mode with full resolution. Analysis of the surface roughness, thickness, and grain size was done using FRT Mark III software (FormFactor Inc). Surface analysis was also performed using a

Zeiss Axio Imager M1 Trinocular Frame optical microscope. Moreover, the surface morphology was analyzed more thoroughly using a scanning electron microscope (SEM) of FEI Quanta 250 equipped with X-ray energy-dispersive spectroscopy (EDS) for chemical analysis of films.

3.-Results and discussion

3.1.-Ceria nanoparticles

The SEM image of the synthesized ceria nanoparticles (Figure 2.a) shows a homogeneously distributed non-spherical shaped morphology with a diameter of agglomerated particles around of 70 nm, which is in good agreement with previous works [42,43]. Besides, a few clusters were also identified in the SEM image, which exhibits irregularly shaped and large sized particle aggregates. The FT-IR spectrum in Fig. 2.b shows the characteristic chemical bands of synthesized CeO 2 nanoparticles. The observed bands are in good agreement with the literature [44] where the intense absorption band appearing at 3227 cm -1 is characteristic for the OH bond of the carboxylic acid and water. The bands observed at 2824 and 2816 cm -1 are attributed to the C-H bond. The band at 2359 cm -1 is not well defined and is due to the absorption of atmospheric CO 2 . The bands appearing in the range between 594 and 668 cm -1 are related to asymmetric and symmetric vibrations of Ce-O. Other bands corresponding to Ce-O are observed at 843 cm -1 and between 917 and 1558 cm -1 .

3.2.-Corrosion inhibition at room temperature

Fig. 3 depicts the polarization curves for the substrate in the corrosive medium with and without inhibitor addition at room temperature. Table 1 summarizes the corrosion parameters including the corrosion potential (E corr ), the corrosion current densities (i corr ) and the anodic and cathodic Tafel slopes (β a and β c ) obtained from the polarization plots by extrapolating the linear portion of the curve. The efficiency of inhibition (EI (%)) was calculated using the following equation (Eq. 1) and the obtained values are listed in Table 1.

Eq. 1

Where i corr et i corr (inh) are the corrosion current density without and with inhibitor addition, respectively.

As shown in Fig. 3, the corrosion potential of the substrate shifted towards more cathodic values for inhibitor concentrations of 100 and 200 ppm. Yet, the Tafel slopes are similar to those of the blank material, i.e. with no inhibitor. This implies that low concentrations of inhibiting nanoparticles do not markedly develop any passive layer or ensure any improvement of the corrosion resistance as also described in previous works for Al alloys [32] and steel [36]. However, for inhibitor concentrations higher than 200 ppm, the anodic slopes shifted the E corr towards anodic values thereby suggesting that ceria nanoparticles act as an anodic inhibitor. This feature can be explained in terms of the nature of the formed layer which provides a "physical barrier" possibly promoted by oxidation of the substrate upon the reduction of Ce 4+ into Ce 3+ following reaction R. 4 as also proposed by Li et al. [45].

2CeO 2 ⇄ Ce 2 O 3 + 1⁄2 O 2 R. 4
It is well known that CeO 2 is suitable for the storage and the release of oxygen under varying conditions between reduced and oxidized state of cerium ions [45]. The passivation mechanism in this case can be explained by a prior adsorption of the particles followed by a redox process in which the reduction of the passivating anion takes place on the defects located on the surface of the oxide and hydroxide layer [43,46]. The electrochemical parameters calculated from polarization curves (Table 1) show an increase of the inhibition efficiency with increasing the inhibitor concentration reaching 78% for 800 ppm, which corresponds to the minimum effective concentration "MEC".

To confirm such MEC at 800 ppm, the more sensitive electrochemical impedance spectroscopy (EIS) diagrams were conducted at the OCP (Fig. 4). The Nyquist plot (Fig. 4.a)

shows that the substrate immersed in the blank solution displays two capacitive loops but the impedance arc is flattened at intermediate frequencies. This phenomenon is usually caused by the constant phase element (CPE) [47]. Moreover, the impedance spectra did not show a imperfections and roughness of surface. In contrast, the diagrams for the substrate in the corrosive solution with inhibitor addition were fitted using an equivalent circuit "EC" composed of three-time constants (Fig. 5b) where the not well defined straight line at low frequencies is attributed to an initiation of a diffusion process (Warburg resistance "Z w ") induced by oxygen diffusion as reported by Eichler et al. [48] and or by the evolution of a redox process attributed to Ce oxide (reaction R. 4). Except for the EIS diagrams performed in the solution with small amount of the inhibitor added (less than 200 ppm) where no inhibition efficiency was shown, the diameter of the semi-circle in the Nyquist plots (giving information about the global resistance) increases with increasing the inhibitor concentration to reach its maximum value at 800 ppm and decreases afterwards. The linear polarization resistance values (Table 1) evaluated at low frequencies give more significant information about the global resistance. Indeed, it appears from the EIS diagrams that the polarization resistance at low frequencies increased more than 3 times for the solution with the addition of 800 ppm than in the inhibitor free solution. Furthermore, the Bode plots show that the maximum ( max )

shifts slightly toward lower frequencies with increasing inhibitor concentration from 200 to 800 ppm. In summary, it is found that  max increases from 73° to 82° for, respectively the solution without and with 800 ppm of inhibitor, which is indicative of an increase of the capacitive behaviour and that the film becomes more protective.

The main EIS parameters are gathered in Table 2. It can be noted that the EI% inhibition efficiency of the EIS technique was calculated by comparing the values of R ct of the substrate in Ca(OH) 2 and 0.5 M CaCl 2 without and with inhibitor addition according to equation Eq. 2.

Eq. 2

Table 2 shows that the minimum effective concentration corresponds to 800 ppm, which is in good agreement with d.c polarization results. The increase of the layer resistance "R c " with the increase of the inhibitor concentration (Table 2) suggests that the inhibitor delays the degradation of the substrate through the formation of a protective layer on the surface, which decreases the active surface area. Besides, the decrease in the inhibitor film capacitance indicates an increase of the film thickness. Furthermore, the increase in charge transfer resistance (R ct ) and the decrease of the double layer capacitance values CPE dl with increasing the concentration of inhibitor indicate the absence of interfacial dispersion [48][49][50]. This is likely due to a precipitation of the cerium oxides in the defects of the film, which block the anodic active sites.

The surface morphology of the substrate after polarization tests in the aggressive medium with and without inhibitor addition was evaluated by optical microscopy, scanning electronic microscopy and white light interferometry. As shown in Fig. 6, the surface of the substrate in the blank solution undergoes severe degradation (black spots) that gradually diminishes with the inhibitor addition. The surface of the substrate after polarization in the aggressive medium with 800 ppm of inhibitor seems to be free from corrosion products, which is in good agreement with the electrochemical results. The same morphology is also observed with SEM images (Fig. 7). Except for the samples polarized in the presence of 600 and 800 ppm of the inhibitor, all other surfaces were not smooth and showed the presence of pores, which indicates that the samples undergo localized corrosion. This is in good agreement with another study showing that pitting corrosion starts on pores of the passive layer formed in chloride containing concrete solution [51].

The EDS analysis of the film formed on the surface of the substrate treated in the aggressive medium with 800 ppm shows that the film is composed of iron, oxygen and calcium. In addition, cerium was also detected. These results confirm that the protective formed film on the surface of the steel is composed of a mixture of the corrosion products and ceria nanoparticles. To confirm this behaviour, the morphology of the surface of the substrate was also examined by using the white light interferometer. The 3D images (Fig. 8) and the calculated parameters of the film are gathered in Table 3. It can be clearly seen that the overall surfaces of the substrates in the presence of the inhibitor are considerably smoother and free from dark spots (black areas is attributed to the surface of the substrate which means lower thickness) compared to the surface of the substrate in the blank solution. The calculated surface parameters (Table 3) showed that the higher average thickness and maximum grain size is observed for the substrate treated with 800 ppm of ceria nanoparticles. However, the average roughness values calculated with the FRT Mark III software, decreases continuously with increasing inhibitor concentration, which is attributed to the formation of homogenous and smooth ceria-iron oxides protective films. Iron oxides are the result of the substrate dissolution occurring before or simultaneously with ceria deposition leading to an increase of the grain size of the film particles (Table 3) as also proposed in other investigations [52,53].

3.2.-Effect of bath temperature

The temperature of the corrosive baths is an important parameter influencing the corrosion rate and the inhibition efficiency as well as modifying the mechanism of the adsorption of the inhibitor on the electrode surface [54]. In order to evaluate the effect of the temperature of the solution on the corrosion inhibition efficiency of ceria nanoparticles, the tests were carried out at bath temperatures ranging from 20 to 50 °C without and in the presence of the optimal concentration of ceria nanoparticles (800 ppm). The evaluation of the inhibition efficiency was firstly investigated by potentiodynamic polarization techniques as shown in Figure 9. In the non-inhibited solution, the evolution of corrosion currents densities i corr and anodic Tafel slopes with increasing bath temperature shows a continuous increase (Table 4) as a result of an increase of the metal dissolution reaction. In contrast, the cathodic slopes remain practically constant irrespective of temperature, which may indicate that the oxygen reduction reaction occurs with the same activation mechanism. On the other hand, the increase of i corr with increasing bath temperature of the inhibited solution with ceria nanoparticles seems to be significantly lower than the one observed in the blank solution. Moreover, the inhibitor remains effective in the studied temperature range (see Table 4). According to Boudellioua et al. [55], the difference in the decrease of i corr values in inhibited and blank solution is mainly due to the film formed on the surface of the substrate.

It should also be noted that the recovery rate (ɵ) calculated from Eq. 3 dropped by about 50 % when the bath temperature increased from 40 to 50 °C, thereby indicating a decrease in the inhibition efficiency.

Eq. 3

According to Mobin et al. [56], this decrease in the inhibition efficiency could be attributed to the intensification of thermal agitation and to the increase of the solution conductivity of the environment, hence the corrosivity of the medium. The drop on recovery rate would be in line with the works from Ngala et al. [57] and Rosenberg and coll. [START_REF] Rosenberg | Mechanisms of corrosion of steel in concrete[END_REF], who suggested that the inhibitor is physically adsorbed, hence the physisorption process predominates. The hypothesis was assessed by calculating the activation energy where the variation of the corrosion current densities with the temperature is expressed by the Arrhenius law (Eq. 4):

Eq. 4

Where E a is the activation energy, R is the ideal gas constant, A is the Arrhenius constant and T is the absolute temperature.

The development of the Arrhenius equation (Eq. 5) in its useful form (Eq. 6) makes it possible to determine the enthalpy "ΔH a " and the entropy "ΔS a " of adsorption [START_REF] Noor | [END_REF].

Eq. 5

Eq. 6

where N is Avogadro's number and h is Planck's constant. S and H are respectively, the entropy and enthalpy of activation for the adsorption to occur.

Figure 10 shows the graphical representation of the Arrhenius law under both forms Eq. 5 and

Eq. 6 of the steel substrate immersed in saturated calcium hydroxide solution and 0.5 M of CaCl 2 with and without inhibitor addition. The obtained thermodynamic parameters are summarized in Table 5. The calculated values show that E a for the substrate immersed in the aggressive medium with inhibitor addition is greater than that in absence of inhibitor, which is defined as the additional energy necessary for the formation of the protective film [60]. On other hand, the weak positive values of ΔH a reveals that the ceria nanoparticles adsorb on the surface of mild steel through an endothermic physisorption process as it involves formation of electrostatic weak bonds generated at the surface of the substrate [61].

Furthermore, the changes in entropy (∆S a ) imply that the activated complex represented the rate-determining step for the association rather than the dissociation step [62]. According to Popova et al. [63] when the inhibition efficiency decreases as bath temperature increases, it

indicates the presence of a few active sites blocked by the adsorbed inhibitor and in this case the activation energy in the inhibited solution becomes higher compared to the free solution.

The surface of the metal immersed in saturated calcium hydroxide solution and 0.5 M of CaCl 2 without and with 800 ppm of ceria nanoparticles was characterized by SEM (Fig. 11).

The SEM images clearly show a very heterogeneous structure of the surface of the substrate immersed in the aggressive medium without inhibitor. The samples indeed seem to undergo a pitting attack well observed at bath temperature 40 °C (Fig. 11.a). However, in the inhibited solution, the micrographs presented in Figure 11.b reveal a uniform and thin protective film free from cracks or defects. With increasing solution temperature the surface appeared less smooth and more cracked indicating that the deposited film became less resistant against corrosion, although it still remains effective compared to the uninhibited solution. Based on the surface morphologies and the electrochemical results, it seems that the precipitation of the ceria nanoparticles inside the initiated pits diminishes their rate of propagation and blocks the initiation of cracks. However, when the bath temperature increases, the holes appear more frequently and seem to reach a critical size where cracks can initiate. According to Zand et al.

[64], the coating defects may serve as initiators for pit corrosion through reduced coating thickness and/or as the initiation sites of fatigue cracks. It should be noted that even by increasing the bath temperature the behaviour of the substrate in the inhibited solution remains better compared to that in the inhibited solution.

4.-Conclusions

In this study, the electrochemical behaviour of a E24 (NF EN 10025) rebar steel immersed in saturated Ca(OH) 2 with 0.5 M CaCl 2 with and without ceria nanoparticles as corrosion inhibitor was investigated. Thereafter, the inhibitor efficiency was evaluated and the effect of the bath temperature on it. A significant corrosion inhibition effect leading to a high polarization resistance compared to the uninhibited solution was observed. The maximum inhibition efficiency (about 80 %) was obtained when adding 800 ppm of ceria nanoparticles to the corrosive medium. The EIS diagrams showed the initiation of a new relaxation time attributed to a diffusion process. The surface morphology of the substrate immersed in the aggressive medium underwent localized corrosion, whereas with the addition of 800 ppm of ceria nanoparticles smooth surfaces, free from corrosion products were obtained. The surface parameters deduced in the studied bath temperature range showed that the inhibitor remains effective despite the increase of the corrosion current densities. The calculated thermodynamic parameters suggest that the inhibitor particles adsorb on the surface through an endothermic physisorption process.

5.-Data availability

The raw data required to reproduce these findings are available to download upon request.

The processed data required to reproduce these findings are available to download upon request.

6.-References

Table1: Polarization parameters for the substrate immersed in saturated Ca(OH) 2 with 0.5 M CaCl 2 at room temperature with and without addition of inhibiting ceria nanoparticles.

Conc

  first step consists in the oxidation of the steel where the released Fe 2+ ions combine with OH - resulting from the cathodic reaction to give rise to the ferrous hydroxide Fe(OH) 2 . This reaction is favoured by the high alkalinity (pH values > 12.5) of the pore solution near the surface of the steel which usually occurs in uncontaminated concrete. Ferrous hydroxide Fe(OH) 2 react with water and oxygen to form hydrated ferric oxide (Fe 2 O 3 ; H 2 O), hydrated magnetite (Fe 3 O 4 ; H 2 O), or evolve into goethite (α-FeOOH) or lepidocrocite (γ-FeOOH) according to the reactions R. 2 and R. 3 [11]:

  perfect semi-circle. Similarly, the Bode diagrams clearly show phase angle values far from -90° (Fig.4.b). Therefore, the equivalent circuit shown in Fig.5ato fit the EIS plots contains a solution resistance "R s " in series with two time constants. The capacitive loop at high frequencies is attributed to outer porous layer (R c -CPE c ) and the second one at low frequencies reflects the charge transfer reaction of the steel substrate (R ct -CPE dl ). Where R c , R ct , represent the layer and charge transfer resistance. CPE c (constant phase element) and CPE dl are the non-ideal layer capacitance and double layer capacitance respectively. The exponent n (0 < n < 1) is a non-linear coefficient which varies with the electrode surface roughness. The presence of the CPE instead of a capacitance indicates dispersion effects by

Figure 1 :

 1 Figure 1: Colour evolution during ceria nanoparticles preparation with increasing solution pH. (f) is the same pH 9.5 as (e) but has been maintained under magnetic agitation for 24 h.

Figure 2 :

 2 Figure 2: SEM image (a) and FTIR spectrum (b) of synthesized ceria nanoparticles.

Figure 3 :

 3 Figure 3: Polarization curves for the substrate immersed in saturated Ca(OH) 2 with 0.5MCaCl 2 at room temperature with increasing addition of inhibitor. N.B. "blank" had no inhibitor.

Figure 4 :

 4 Figure 4: EIS diagrams (a) Nyquist and (b) phase plots for the substrate in saturated Ca(OH) 2 with 0.5 M CaCl 2 at room temperature with inhibitor addition. N.B. "blank" had no inhibitor.

Figure 5 :

 5 Figure 5: Electrical equivalent circuit (EEC) model used for modelling EIS diagrams for the substrate in saturated Ca(OH) 2 with 0.5 M CaCl 2 at room temperature (a) without and (b) with addition of the inhibitor.

Figure 6 :

 6 Figure 6: Optical macrographs showing the evolution of the surface of the substrate after polarization tests in saturated Ca(OH) 2 with 0.5 M CaCl 2 (a) without and with (b) 100 (c) 200 (d) 400 (e) 600 (f) 800 and (g) 1000 ppm of ceria nanoparticles at room temperature.

Figure 7 :

 7 Figure 7: SEM images showing the evolution of the surface of the substrate after polarization tests in saturated Ca(OH) 2 with 0.5 M CaCl 2 at room temperature with and without (blank) inhibitor addition. EDS spectra of the surface of the substrate immersed in blank solution and of particles deposited on the surface of the substrate immersed in the solution with 800 ppm of ceria nanoparticles. N.B. "blank" had no inhibitor.

Figure 8 :

 8 Figure 8: WLI (white-light interferometer; three-dimensional) images of the surface morphologies for the substrate after polarization test in in saturated Ca(OH) 2 with 0.5 M CaCl 2 at room temperature with and with inhibitor addition. N.B. "blank" had no inhibitor.

Figure 9 :

 9 Figure 9: Polarization curves for the substrate immersed in saturated Ca(OH) 2 with 0.5 M CaCl 2 without (a) and with addition of 800 ppm of ceria nanoparticles (b) at different bath temperatures.

Figure 10 : 5 M

 105 Figure 10: Arrhenius plots of mild steel immersed for 30 min in saturated Ca(OH) 2 with 0.5 M CaCl 2 solution with and without ceria nanoparticles.

Figure 11 :

 11 Figure 11: SEM images showing the evolution of the surface of the substrate after polarization tests in saturated Ca(OH) 2 with 0.5 M CaCl 2 without (on the left) and in the presence of 800 ppm (on the right) at different bath temperatures.

Figure 1 :

 1 Figure 1:

Figure 6 :

 6 Figure 6:

Figure 9 :

 9 Figure 9:

Table 2 :

 2 Fitting results for the EIS data for the substrate immersed in saturated Ca(OH) 2 with 0.5M CaCl 2 at room temperature with and without addition of inhibiting ceria nanoparticles.

		-E corr	I corr	a	c	Rp	EI
	(ppm)	(mV/SCE)	(µA/cm²)	(mV/dec)	(mV/dec)	(.cm 2 )	(%)
	Blank	524	7.30	184.2	153.7	3162.00	-----
	100	545	11.41	327.5	190.8	2060.62	-----
	200	543	11.1	326.5	190.6	2218.19	------
	400	501	3.73	234.6	137.7	5610.47	48.94
	600	484	3.03	289.4	143.1	7655.96	58.52
	800	477	1.63	224.0	142.6	10519.61	77.68
	1000	409	4.45	257.4	143.4	4008.66	39.09

Table 3 :

 3 Surface parameters of the substrate after polarization tests realized in saturated Ca(OH) 2 with 0.5M CaCl 2 at room temperature with and without addition of inhibiting ceria nanoparticles.

	Conc	Thickness	Grain size	Roughness
	ppm	(µm)	(µm)	(µm)
	Blank	1.693	0.005	0.436
	100	1.996	0.009	0.406
	200	2.054	0.009	0.399
	400	2.090	0.010	0.382
	600	2.128	0.011	0.363
	800	2.405	0.012	0.350
	1000	2.340	0.012	0.353

Table 4 :

 4 Fitting results for the EIS data for the substrate in saturated Ca(OH) 2 with 0.5M CaCl 2 without and with the presence of 800 ppm of ceria nanoparticles at different temperatures.

	TT	-E corr	I corr	a	c	Rp
	(°C)	(mV/ECS)	(µA/cm²)	(mV/dec)	(mV/dec)	(.cm 2 )
	20	-524	7.30	184.2	153.7	3162	.......
	30	-501	7.62	151.6	224.4	1584	.......
	Blank
	40	-551	13.55	139.9	163.9	946	.......
	50	-587	23.03	88.6	160.6	656	.......
	20	477	1.63	224.0	142.6	10519	0.77
	30	480	3.55	214.1	199.1	6426	0.53
	inhibitor
	40	472	6.48	181.2	188.2	3054	0.52
	50	523	16.28	221.0	155.5	857	0.29

Table 5 :

 5 Thermodynamic parameters for the substrate in saturated Ca(OH) 2 with 0.5M CaCl 2 with and without the presence of 800 ppm of ceria nanoparticles

		ΔH a (kJ.mol -1 )	ΔS a (J.mol -1 .K -1 )	Ea (kJ.mol -1 )
	Blank	28.87	-246.68	31.40
	With ceria
	nanoparticles	56.44	-163.27	59.00


Keywords: atomic configurations, fermions, generating functions, supershell, recurrence relations, statistical modeling, Gram-Charlier expansion series

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  and numerical expressions for the number of atomic configurations contained in a supershell (2020 J. Phys. B: At. Mol. Opt. Phys. 53 115002) To cite this article: Jean-Christophe Pain and Michel Poirier 2020 J. Phys. B: At. Mol. Opt. Phys. 53 199601 View the article online for updates and enhancements. The sentence before equation (

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P Cañadas

C Wagner-Kocher

P Royer

Fiber orientation and crimp level might control the auxetic effect of biological tissues

Keywords: Auxetic behavior, Biocomposite, Micromechanics, Constitutive relation, Crimped fibers, Negative Poisson's ratio, Collagen fiber, Fibrous soft tissues, Annulus fibrosus, Arterial tissue

We propose an analytical micromechanical model for studying the lamellarcomposite-like structure of fibrous soft tissue. The tissue under consideration is made up of several lamellae, and is designed to resemble the annulus fibrosus (AF) tissue or media layer of arterial tissue, for example. The collagen fibers are arranged in parallel in each lamella and the fiber orientation differs from one lamella to its neighbors. The parallel fibers in each lamella of AF tissue, for example, have been observed to have a crimped microstructure. The proposed model incorporates this quality, considering fiber waviness as a sinusoidal shape and taking into account the fiber dispersion in different layers, where both fiber and matrix are considered as solid phases. We find that collagen-fiber waviness and layer orientation have a significant influence on Poisson's ratio. The effective Poisson's ratio predicted by the proposed model demonstrates that the crimped collagen fiber microstructure might weaken the auxetic effect of fibrous soft tissue, which might explain why, as the literature suggests, the auxetic behavior is more difficult to observe than large Poisson's ratios. As opposed to the many studies that use the well-known hyperelastic fiber-based constitutive model, in which out-of-plane expansion is often observed, the present work explains the auxetic response found in modeling and in experimental data from the perspective of collagen fiber microstructure.

Introduction

Fibrous soft tissue composed of several families of collagen fibers is found, for example, in annulus fibrosus (AF) and the media layer of arterial tissues. AF tissue is one of the components that make up the intervertebral discs, which support the movements of the spine. The arteries are vessels that transport blood from the heart to the other tissues and organs of the body. Arteries can be distinguished by their layers: intima, media, and adventitia.

AF and arterial tissues are usually considered as orthotropic materials. Although the linear and nonlinear elastic properties of these tissues are well-established in the literature, the link between their macroscopic behavior and microscopic structure remains only partially understood. The Poisson's ratios of these soft tissues have motivated studies of their internal microstructures; more specifically the collagen fiber structure and arrangement. In recent years, the auxetic behavior of AF and arterial tissues have received increased attention [START_REF] Yuan | Fabrication of multi-scale and tunable auxetic scaffolds for tissue engineering[END_REF] [START_REF] Fereidoonnezhad | A new anisotropic soft tis[END_REF]. Some materials or structures exhibit negative Poisson's ratios, and can therefore be described as having an auxetic property, showing unusual and counterintuitive mechanical behavior; when stretched (respectively, compressed), these materials become thicker (respectively, thinner) in the direction perpendicular to that of the load.

Most of the tensile experiments reported in the literature for AF are uniaxial tests [3][4][5] [START_REF] Grace | Theoretical and uniaxial experimental evaluation of human annulus fibrosus degeneration[END_REF] [START_REF] Cherblanc | Mechanical behaviour of annulus fibrosus: the role of the fluid phase[END_REF][8] [START_REF] Dusfour | Heterogeneous mechanical hyperelastic behavior in the porcine annulus fibrosus explained by fiber orientation: An experimental and numerical approach[END_REF], although biaxial tensile tests have also been performed [START_REF] Bass | Biaxial testing of human annulus fibrosus and its implications for a constitutive formulation[END_REF] [START_REF] Grace | Human annulus fibrosus material properties from biaxial testing and constitutive modeling are altered with degeneration[END_REF]. The measurements of the Poisson's ratio for AF reveal a large range of values, from -0.57 [START_REF] Derrouiche | A chemo-mechanical model for osmoinelastic effects in the annulus fibrosus[END_REF] to 2.32 [START_REF] Wagner | Theoretical model and experimental results for the nonlinear elastic behavior of human annulus fibrosus[END_REF]. The difference in experimental results could be due to many factors, such as differences in: the types of AF tissue analyzed, the subsections of AF tissue analyzed (inner or outer AF) [3], extension conditions, test specimen status (degenerated or non-degenerated AF), strain-measurement techniques, specimen-storage-solution osmolarity, and so on. Guerin and Elliott [4] and O'Connell et al. [START_REF] Grace | Theoretical and uniaxial experimental evaluation of human annulus fibrosus degeneration[END_REF] found that AF degeneration influences the results of Poisson's ratio measurements. Moreover, Wagner and Lotz [START_REF] Wagner | Theoretical model and experimental results for the nonlinear elastic behavior of human annulus fibrosus[END_REF] showed that the loading type, that is, tension or compression, has an effect on Poisson's ratio results. Furthermore, Derrouiche et al. [START_REF] Derrouiche | A chemo-mechanical model for osmoinelastic effects in the annulus fibrosus[END_REF] presented results showing that the different saline concentrations and strain rates used in extension tests also affect such measurements. Although the range of reported Poisson's ratio results is considerable, the orthotropic features of AF along the longitudinal, circumferential, and radial directions are consistently observed. Comparison of the experimental results in the literature reveals that, when loading AF tissue in the the circumferential direction, the Poisson's ratio that characterizes the strain in the radial direction is usually much lower than the Poisson's ratio that characterizes the strain in the longitudinal direction [3] [START_REF] Wagner | Theoretical model and experimental results for the nonlinear elastic behavior of human annulus fibrosus[END_REF][8] [START_REF] Derrouiche | A chemo-mechanical model for osmoinelastic effects in the annulus fibrosus[END_REF]. Even negative ν θr values (auxetic behavior) were recently reported by Baldit et al. [START_REF] Baldit | Experimental analysis of the transverse mechanical behaviour of annulus fibrosus tissue[END_REF], Derrouiche et al. [START_REF] Derrouiche | A chemo-mechanical model for osmoinelastic effects in the annulus fibrosus[END_REF], and Dusfour et al. [START_REF] Dusfour | Heterogeneous mechanical hyperelastic behavior in the porcine annulus fibrosus explained by fiber orientation: An experimental and numerical approach[END_REF]. In other words, with extension of AF tissue in the circumferential direction, it is observed to expand in the radial direction in some cases.

In order to predict the mechanical behavior of AF using mechanical modeling, Elliott and Setton [3] implemented a linear anisotropic material model to determine a complete set of model properties under idealized kinematic states. According to the model predictions presented by these authors, interactions between fiber populations in the multilamella AF contribute significantly to the mechanical behavior of the material, suggesting that a model for AF made up of physically isolated concentric lamellae may not be appropriate. Derrouiche et al. [START_REF] Derrouiche | A chemo-mechanical model for osmoinelastic effects in the annulus fibrosus[END_REF] presented a chemo-mechanical approach to studying the intrinsic osmo-inelastic response of AF tissue in relation to the microstructure of the layered and reinforced soft tissue, the biochemical environment, and the mechanical loading conditions. This chemo-mechanical approach successfully captured the variations in osmolarity, strain rate, and auxeticity. Kandil et al. [START_REF] Kandil | Interlamellar-induced time-dependent response of intervertebral disc annulus: A microstructure-based chemo-viscoelastic model[END_REF] proposed a chemo-viscoelastic model as part of a microstructure-based approach to predicting the regional dependency of the annulus response, in which the auxetic behavior is identified in the plane of the lamellae. Dusfour et al. [START_REF] Dusfour | Heterogeneous mechanical hyperelastic behavior in the porcine annulus fibrosus explained by fiber orientation: An experimental and numerical approach[END_REF] found an auxetic response by applying the HGO model ( [START_REF] Holzapfel | A new constitutive framework for arterial wall mechanics and a comparative study of material models[END_REF]), but found poor agreement between model and experimental results. Furthermore, AF is widely modeled as a fiber-induced anisotropic hyperelastic material [6][16][17] [START_REF] Peng | An anisotropic hyperelastic constitutive model with fiber-matrix shear interaction for the human annulus fibrosus[END_REF]. These models describe the fibers and the matrix using the principle invariants of the Green deformation tensor and structural tensors representing the collagen fiber populations [START_REF] Grace | Human annulus fibrosus material properties from biaxial testing and constitutive modeling are altered with degeneration[END_REF].

Several studies have reported measurements of the Poisson's ratio of arteries [START_REF] Doli | Static anisotropic elastic properties of the aorta in living dogs[END_REF] [START_REF] Skacel | Poisson's ratio and compressibility of arterial wall-improved experimental data reject auxetic behaviour[END_REF]. In most studies, arteries are stretched in an in vitro environment, except for those of Patel et al. [START_REF] Doli | Static anisotropic elastic properties of the aorta in living dogs[END_REF] and Hasegawa et al. [START_REF] Hasegawa | Non-invasive evaluation of poisson's ratio of arterial using ultrasound[END_REF], who evaluated the Poisson's ratio of dog and human arteries, respectively, in vivo. However, the Poisson's ratios of arteries reported so far are inconsistent, and both large (ν > 1.0) and negative (ν < 0.0) values have been measured [START_REF] Liu | A linearized and incompressible constitutive model for arteries[END_REF], while arteries have been widely observed to be orthotropic along the longitudinal, circumferential, and radial directions. The causes for these significant discrepancies in experimental results could be due to many factors, such as the different types of artery tissue studied [START_REF] Hasegawa | Non-invasive evaluation of poisson's ratio of arterial using ultrasound[END_REF], the different layers (media or intact vessel) studied, or differences in extension conditions, strain measurement techniques, test specimen conditions (fresh, or frozen and thawed), and so on. As arterial tissue can be divided into three main types (intima, media, and adventitia), rather than studying the intact arterial wall, more recently the media and adventitia layers were isolated and studied separately [START_REF] Lucas H Timmins | Structural inhomogeneity and fiber orientation in the inner arterial media[END_REF][27] [START_REF] Skacel | Poisson s ratio of arterial wall-inconsistency of constitutive models with experimental data[END_REF][30] [START_REF] Skacel | Poisson's ratio and compressibility of arterial wall-improved experimental data reject auxetic behaviour[END_REF]. Comparing the experimental extension conditions in the literature, the strain velocity, loading pressure, and the maximum stretch strain might also affect the measured values [19][22][20][28] [START_REF] Víctor A Acosta | Characterization of chemoelastic effects in arteries using digital volume correlation and optical coherence tomography[END_REF]. Santamaría et al. [START_REF] Víctor A Acosta | Characterization of chemoelastic effects in arteries using digital volume correlation and optical coherence tomography[END_REF] and Skacel and Bursa [START_REF] Skacel | Poisson's ratio and compressibility of arterial wall-improved experimental data reject auxetic behaviour[END_REF] noted that using frozen and thawed artery specimens leads to different experimental results compared to using fresh tissue. Indeed, Santamaría et al. [START_REF] Víctor A Acosta | Characterization of chemoelastic effects in arteries using digital volume correlation and optical coherence tomography[END_REF] note that frozen and thawed tissues may have nonphysiological hydraulic permeability properties, and that freezing and thawing may cause the destruction of cell membranes. Moreover, several individual negative out-of-plane Poisson's ratios were measured with frozen and thawed test specimens among the 12 specimens used in experiments [START_REF] Skacel | Poisson s ratio of arterial wall-inconsistency of constitutive models with experimental data[END_REF][31] but no negative Poisson's ratio was found in tests with fresh specimens [START_REF] Skacel | Poisson's ratio and compressibility of arterial wall-improved experimental data reject auxetic behaviour[END_REF]. Negative Poisson's ratios were measured by Timmins et al. [START_REF] Lucas H Timmins | Structural inhomogeneity and fiber orientation in the inner arterial media[END_REF], Lillie et al. [START_REF] Ma Lillie | Mechanical anisotropy of inflated elastic tissue from the pig aorta[END_REF], Skacel and Bursa [START_REF] Skacel | Poisson s ratio of arterial wall-inconsistency of constitutive models with experimental data[END_REF], and Santamaría et al. [START_REF] Víctor A Acosta | Characterization of chemoelastic effects in arteries using digital volume correlation and optical coherence tomography[END_REF] in arterial tissues, and Skacel and Bursa [START_REF] Skacel | Poisson's ratio and compressibility of arterial wall-improved experimental data reject auxetic behaviour[END_REF] reject their previous experimental observations. In addition, auxetic behaviors have been reported for tendon tissues [START_REF] Chavaunne T Thorpe | Effect of fatigue loading on structure and functional behaviour of fascicles from energy-storing tendons[END_REF][33] [START_REF] Vella | 3d printed clamps to study the mechanical properties of tendons at low strains[END_REF], skin tissue [START_REF] Veronda | Mechanical characterization of skin-finite deformations[END_REF][36], tibia bone [START_REF] Williams | Properties and an anisotropic model of cancellous bone from the proximal tibial epiphysis[END_REF], axoloti embryonic epithelia [START_REF] Wiebe | Tensile properties of embryonic epithelia measured using a novel instrument[END_REF], and for the bovine cornea [START_REF] Patten | Suprafibrillar structures of collagen, evidence for local organization and auxetic behaviour in architectures[END_REF].

[20][21][22] [23][24][25][26] [27][28][29][30]
Timmins et al. [START_REF] Lucas H Timmins | Structural inhomogeneity and fiber orientation in the inner arterial media[END_REF] proposed that the auxetic response could be due to the variable and inhomogeneous alignment of elastin and collagen fibers in the arterial wall. Such lateral expansion was predicted based on a nonlinear hyperelastic anisotropic model of arterial wall with two families of perfectly aligned collagen fibers [START_REF] Holzapfel | A new constitutive framework for arterial wall mechanics and a comparative study of material models[END_REF], (known as the HGO model), as presented by Gasser et al. [START_REF] Gasser | Hyperelastic modelling of arterial layers with distributed collagen fibre orientations[END_REF], who further proposed a GOH model with dispersed fiber orientations. Skacel and Bursa [START_REF] Skacel | Poisson s ratio of arterial wall-inconsistency of constitutive models with experimental data[END_REF] reported a negative Poisson's ratio exhibited by a GOH model and provided a comprehensive analysis of the auxetic response with distributed fiber orientations. Nolan et al. [START_REF] David R Nolan | A robust anisotropic hyperelastic formulation for the modelling of soft tissue[END_REF] also identified auxetic behavior with a compressible form of the HGO model called HGO-C, analyzing the predicted lateral stresses induced during uniaxial stretching, and proposed a modified anisotropic (MA) model to avoid them. Similarly, Latorre et al. [START_REF] Latorre | The relevance of transverse deformation effects in modeling soft biological tissues[END_REF] noticed an unrealistic transversal deformation response predicted by the HGO and GOH models, and proposed their what-you-prescribe-is-what-you-get (WYPIWYG) model as a solution. Volokh [START_REF] Volokh | On arterial fiber dispersion and auxetic effect[END_REF] applied the HGO model to study auxetic behavior based on the angular integration (AI) approach. Fereidoonnezhad et al. [START_REF] Fereidoonnezhad | A new anisotropic soft tis[END_REF] consider that auxetic behavior is primarily influenced by the ratio of fiber-to-matrix stiffness and is accentuated by strain-stiffening fibers in a constant stiffness matrix. These authors propose a bilinear strain-stiffening fiber and matrix model (BLFM), which allows close control of the fiber-matrix stiffness ratio to eliminate auxetic behavior.

The AF and media layer of arterial tissue consist of several families of collagen fibers, in which the collagen fibers are observed to have a crimped microstructure. With our study, we aim to investigate the microstructure function mechanism in fibrous soft tissue such as AF and arterial media so as to understand how material microstructure parameter characteristics might regulate the macroscopic effective Poisson's ratio. We adopt the method of Xiao et al. [START_REF] Xiao | A micromechanical model of tendon and ligament with crimped fibers[END_REF] to find the effective stiffness matrix of composite reinforced with fibers of a particular sinusoidal waviness. Then, our proposed model takes into account the fiber dispersion in different families of collagen fibers and considers fiber waviness as a sinusoidal shape. The layout of the present paper is as follows: section 2 presents the micromechanical model we use to predict the constitutive relations of a tissue-like biocomposite and section 3 summarizes the mechanical parameters of the tissue found in the literature and provides the material parameters used to study its mechanical behavior. In Section 4, we present our theoretical calculation results and compare them with different studied cases of physiological ranges found in real tissues. This is followed by a discussion of the implications and limitations of our micromechanical model and our conclusions in section 5.

In this study, we propose to model the AF or media of arterial tissue as a fiber-reinforced composite of which the fibers have a crimped microstructure. The AF and media of arterial tissues consist of several concentric lamellae, with the collagen fibers of each individual lamina being arranged in parallel. This model takes into account the dispersion of the fibers in different families and considers fiber waviness as a sinusoidal shape. Furthermore, in our assumption, the collagen fibers are only corrugated within the plane of the lamella. The proposed analytical solution is based on the effective stiffness matrix of composite reinforced with fibers of a particular sinusoidal waviness, as proposed by Xiao et al. [START_REF] Xiao | A micromechanical model of tendon and ligament with crimped fibers[END_REF] for the unique direction of fibers, and three-dimensional effective moduli of laminate composite proposed by Sun and Li [START_REF] Sun | Three-dimensional effective elastic constants for thick laminates[END_REF] which helps us to take in account the different families of fibers.

Composite reinforced with straight fibers

In order to develop our model, we first present an analytical solution based on the Eshelby equivalent inclusion method [START_REF] Mori | Average stress in matrix and average elastic energy of materials with misfitting inclusions[END_REF] for a stiffness matrix of composite reinforced with straight fibers.

The mechanical strain interaction between the fibers and matrix that make up a composite reinforced with straight fibers, based on the Eshelby equivalent inclusion method, can be expressed as εf = B : εm ,

where εf and εm are the average strain tensors in the fibers and matrix, respectively, and B is the concentration tensor whose details can be found in Parnell's monograph [START_REF] William | The eshelby, hill, moment and concentration tensors for ellipsoidal inhomogeneities in the newtonian potential problem and linear elastostatics[END_REF], which can be expressed as

B = [I + (S E : S m ) : (C f -C m )] -1 , ( 2
)
where I is the identity tensor, C f and C m are the elastic stiffness tensors of isotropic fiber and matrix, respectively, S m is the elastic compliance of the isotropic matrix (S m = [C m ] -1 ), and S E is the Eshelby tensor.

The average stress of fibers σf is σf = C f : εf = C f : B : εm .

(3)

The average stress and strain tensors in the composite reinforced with straight fibers can be written in terms of matrix average stress σm as

σ = V m σm + V f σf = V m σm + V f C f : B : εm = V m σm + V f C f : B : (S m : σm ), (4) ε
= V m εm + V f εf = V m (S m : σm ) + V f B : (S m : σm ), (5)
where V f and V m are the volume fractions of the fibers and the matrix, respectively. From Equations 4 and 5, the compliance matrix S of the composite reinforced with straight fibers can be obtained as

S = (V m I + V f B) : (V m C m + V f C f : B) -1 . (6)
For the composite reinforced with straight fibers, which are parallel to the x-axis, details can be found in the Mura's monograph [START_REF] Mura | Micromechanics of defects in solids[END_REF]. By 6, we get the equivalent compliance matrix for straight-fiber-reinforced composite S with the following formula, where the details of each nonzero component can be found in Xiao et al. [START_REF] Xiao | A micromechanical model of tendon and ligament with crimped fibers[END_REF]

Sij =         S11 S12 S12 0 0 0 S12 S22 S23 0 0 0 S12 S23 S22 0 0 0 0 0 0 2( S22 -S23 ) 0 0 0 0 0 0 S66 0 0 0 0 0 0 S66         . ( 7
)

Composite reinforced with crimped fibers

Here, we present the analytical solution to finding the effective stiffness matrix Ĉsin ij (i, j = 1, 2, 3, 4, 5, 6) of composite reinforced with fibers of a particular sinusoidal shape [START_REF] Xiao | A micromechanical model of tendon and ligament with crimped fibers[END_REF], as shown in Figure 2(a). The fiber corrugation is assumed to be planar sinusoidal in the x-z plane and the corrugated fiber is shown in Figure 2 The crimped fiber shape is defined as

z = Asin( 2π L x), (8)
where A is the amplitude and L represents the wavelength of the wavy fiber as shown in Figure 2(b).

The average transformed inverse compliance matrix Ŝsin ij is obtained by integrating the compliance matrix S along one wavelength of the sinusoidal shape in the x-direction as (see the details in Hsiao et al. [START_REF] Hsiao | Elastic properties of composites with fiber waviness[END_REF])

Ŝsin ij (β(A, L)) = L 0 Ŝij (β(A, L))dx, (9)
where

Ŝij = R ij T sin ij -1 R ij -1 Sij T sin ij , (10)
and

β = arctan( 2πA L cos( 2π L x)) and x ∈ [0, L]. T sin
ij is the general transformation matrix expressed as

T sin ij =        cos 2 (β(x)) 0 sin 2 (β(x)) 0 2cos(β(x))sin(β(x)) 0 0 1 0 0 0 0 sin 2 (β(x)) 0 cos 2 (β(x)) 0 -2cos(β(x))sin(β(x)) 0 0 0 0 cos(β(x)) 0 -sin(β(x)) -sin(β(x))cos(β(x)) 0 sin(β(x))cos(β(x)) 0 cos 2 (β(x)) -sin 2 (β(x)) 0 0 0 0 sin(β(x)) 0 cos(β(x))        , (11)
and R ij can be expressed as

R ij =        
1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2

        . ( 12
)
We then have

Ĉsin ij = Ŝsin ij -1 . (13)
The nonzero element of matrix Ŝsin ij can be expressed as

Ŝsin ij =         
         . ( 14
)

Composite reinforced with crimped fibers with alternatively oriented layers

We propose an analytical solution that takes into account the multiple layers and the relative orientation of each layer to find the effective elastic stiffness matrix. To do so, single-layer solutions are also presented.

Single layer of the multilayered composite

Slayer1 ij = R ij T st ij -1 R ij -1 Ŝsin ij T st ij Clayer1 ij = Slayer1 ij -1 , (15)
where the general transformation matrix T st ij is

T st ij =         cos 2 (θ) 0 sin 2 (θ) 0 2cos(θ)sin(θ) 0 0 1 0 0 0 0 sin 2 (θ) 0 cos 2 (θ) 0 -2cos(θ)sin(θ) 0 0 0 0 cos(θ) 0 -sin(θ) -sin(θ)cos(θ) 0 sin(θ)cos(θ) 0 cos 2 (θ) -sin 2 (θ) 0 0 0 0 sin(θ) 0 cos(θ)         . ( 16
)
The nonzero element of matrix Slayer1

         . ( 17
)

Multiple layers

Here, we introduce an analytical approach to finding the effective elastic stiffness tensor of a composite composed of several layers by applying formulations given in [START_REF] Sun | Three-dimensional effective elastic constants for thick laminates[END_REF]. Each considered layer, or lamella, is a composite reinforced with fibers of uniform sinusoidal waviness. The coordinate system is set so that the x-and z-axes lie in the plane of the lamella and the y-axis is perpendicular to that plane. The laminate composite contains r orthotropic fiber-composite layers and each layer is rotated by a corresponding angle θ k (k = 1, 2, 3 • • • ) around the y-axis as shown in Figure 4(a). The effective macro-stresses σ total and macro-strains ε total are defined as

σ total ij = 1 V V σ ij dV, (18)
and

ε total ij = 1 V V ε ij dV, ( 19
)
where V is the volume that contains the total thickness of the lamellae. The in-plane dimensions are kept infinitesimal so that the stresses and strains in each layer are uniform in the planar directions. As the stresses and strains in each lamella are constant, Equations 18 and 19 can be integrated as

σ total ij = r k=1 V k (σ ij ) k , (20)
and

ε total ij = r k=1 V k (ε ij ) k , (21)
where (σ ij ) k and (ε ij ) k are the stresses and strains in the kth (k = 1, 2, 3 • • • ) layer, and

V k = e k h , (22)
where e k is the thickness of the kth lamella, and h is the total thickness of the laminate as shown in Figure 4(b).

We assume that the strain of each layer along the x-axis and z-axis is equal to the global strain of the multi-layer laminate along the x-axis and z-axis, respectively. Additionally, we assume that the stress on each layer along the y-axis is the same as the global stress along the y-axis. Thus, the terms of the stress and strain conditions are assumed as follows:

                       (ε xx ) k = ε total xx (ε zz ) k = ε total zz (γ xz ) k = γ total xz (σ yy ) k = σ total yy (σ xy ) k = σ total xy (σ yz ) k = σ total yz . (23)
The effective elastic stiffness matrix of the multilayer laminate Ctotal

         . ( 24
)
The nonzero elements of Ctotal ij can be obtained from Equations 20, 21, and 23 (see the details of the developments limited to straight fibers in Sun et al. [START_REF] Sun | Three-dimensional effective elastic constants for thick laminates[END_REF]).

The effective elastic compliance matrix of a multilayered composite Stotal

ij is [ Stotal ij ] = [ Ctotal ij ] -1 . ( 25
)
3 Material parameter study

We consider the lamellar structure in AF or arterial tissue as a composite reinforced with corrugated fibers and study its mechanical behavior by applying the proposed micromechanical model. Our parameter study is based on data reported in the literature. The elastic modulus of collagen has been reported at different scales, from molecular to whole tissues, revealing that the mechanical stiffness decreases as the hierarchy scale increases; in other words, E monomer > E f ibril > E f iber > E tissue [START_REF] Zhang | Effect of the structural water on the mechanical properties of collagen-like microfibrils: a molecular dynamics study[END_REF]. Although the collagen elastic modulus is most often reported at the lamella scale [START_REF] Skaggs | Regional variation in tensile properties and biochemical composition of the human lumbar anulus fibrosus[END_REF][52], there are a few examples in the literature of the elastic modulus studied at the single collagen fiber scale. Ambard and Cherblanc [START_REF] Ambard | Mechanical behavior of annulus fibrosus: a microstructural model of fibers reorientation[END_REF] showed the collagen fiber elastic modulus to range from 6.6M P a to 12.3M P a using a tensile test on lamb and pig annulus fibrosus tissues as part of their development of a rheological model. Also, the collagen Type I single-fiber elastic modulus was measured between 100 M P a and 360 M P a in rat tail tendon using an approach that combines optical tweezers, atomic force microscopy, and exploitation of Euler-Bernoulli elasticity theory [START_REF] Dutov | Measurement of elastic modulus of collagen type i single fiber[END_REF]. Type I collagen fibers are the most prevalent fibers [START_REF] Sharabi | The mechanical role of collagen fibers in the intervertebral disc[END_REF] in AF, and the type III collagen fiber is the most abundant collagen type in arterial tissue, although type I collagen is also present here in significant amounts [START_REF] Boris V Shekhonin | Distribution of type i, iii, iv and v collagen in normal and atherosclerotic human arterial wall: immunomorphological characteristics[END_REF]. We consider that the measured collagen fiber modulus for type I collagen reported by Dutov et al. [START_REF] Dutov | Measurement of elastic modulus of collagen type i single fiber[END_REF] can also be applied in our study. The mechanical properties of the ground matrix of bovine AF were measured by Cortes et al. using tensile and confined compression tests, revealing an aggregate modulus of 10.18 ± 3.32KP a [START_REF] Daniel | Extra-fibrillar matrix mechanics of annulus fibrosus in tension and compression[END_REF]. As the arterial tissue contains amounts of smooth muscle cells (SMCs) in the ground substance, we consider the elastic modulus of SMCs as the elastic modulus of the matrix, which is reported to be about 10KP a by Nagayama et al. from rat thoracic aortas [START_REF] Nagayama | Effect of actin filament distribution on tensile properties of smooth muscle cells obtained from rat thoracic aortas[END_REF]. The fiber volume fractions have been found to range from 0.05 to 0.245 based on the description that AF comprises 65%-90% wet weight (water) and 50%-70% dry weight (collagen) [START_REF] Ducheyne | Comprehensive biomaterials[END_REF]. Collagen recruitment was reported to be 10 ± 1% in rat abdominal aorta by O'Connell et al. [START_REF] Mary | The three-dimensional micro-and nanostructure of the aortic medial lamellar unit measured using 3d confocal and electron microscopy imaging[END_REF]. The Poisson's ratios of the fiber (0.3) and matrix (0.4) used in our study are based on a previous modeling study, which is summarized by Sharabi et al. [START_REF] Sharabi | The mechanical role of collagen fibers in the intervertebral disc[END_REF]. The fiber orientation in the circumferential direction varies between lamellae by 25 • to 45 • reported by Baldit et al. [START_REF] Baldit | Micromechanics of the intervertebral disk[END_REF]in porcine AF and ±10 • in the aortic medial [START_REF] Mary | The three-dimensional micro-and nanostructure of the aortic medial lamellar unit measured using 3d confocal and electron microscopy imaging[END_REF]. The individual medial lamellar unit measured 13.9 ± 1.2µm for the aortic medial [START_REF] Mary | The three-dimensional micro-and nanostructure of the aortic medial lamellar unit measured using 3d confocal and electron microscopy imaging[END_REF] measured 0.14 -0.52 mm in the lateral portion and inner layers of the AF [START_REF] Robert | Encyclopedia of the neurological sciences[END_REF]. The crimp level of collagen fibers, represented by A L , is assumed to vary between 0.08 and 0.25. This variation is based on the description provided by Sharabi [START_REF] Sharabi | Structural mechanisms in soft fibrous tissues: A review[END_REF], where it is stated that the crimp angle increases from 20°to 45°. The crimp angle is determined by taking the arctan of the sinusoidal amplitude divided by a quarter sinusoidal period. Furthermore, the porcine AF consists of 7 -25 lamellae [START_REF] Robert | Encyclopedia of the neurological sciences[END_REF][61] and rat abdominal aorta contains about 8 lamellae [START_REF] Mary | The three-dimensional micro-and nanostructure of the aortic medial lamellar unit measured using 3d confocal and electron microscopy imaging[END_REF]. The parameters used in our study are summarized in Table 1.

Parameter

Results

In this section, the effects of fiber crimp level, layer orientation angle, and number of layers on the Poisson's ratio of the fibrous-tissue-like composite are quantified using a theoretical analysis.

The effective Poisson's ratio for a laminate of one or multiple layers is obtained from Sij (here, Sij represents Slayer1 ij or Stotal ij according to the particular application) as:

ν xy = - S21 S11 ν xz = - S31 S11 , ( 26
)
where the term ν ij is the effective Poisson's ratio, which characterizes the strain in the j direction produced by the loading in the i direction.

Single and double layer

In this section, we compare the difference in Poisson's ratio between a single layer (i.e., a single family of fibers) and a double layer (i.e., two families of fibers). The values of the model parameters are shown in Table 1. For the double-layer analysis, the two layers are set to have the same thickness of 0.5 mm (e 1 = e 2 ), and the orientation of the first and second layers is symmetrical on either side of the x-axis, which means θ 1 = -θ 2 . With these parameters, the model is able to predict the effective Poisson's ratio as functions of fiber crimp level (0.01 < A L < 0.3) and layer orientation angle (0 L and layer orientation θ. See the value settings for the parameters in Table 1. in this study. Note that the angle θ at which both ν xy and ν xz reach their limit values is influenced by the ratio of stiffness between the fiber and the in the physiological range of AF, the values of ν xz are usually larger than those of ν xz , which is in good agreement with the observations in the literature [START_REF] Derrouiche | A chemo-mechanical model for osmoinelastic effects in the annulus fibrosus[END_REF]; when subjected to loading in the circumferential direction, the Poisson's ratio characterizing the strain in the radial direction is typically much lower than the Poisson's ratio characterizing the strain in the longitudinal direction. Additionally, with the extension of AF tissue in the circumferential direction, an expansion in the radial direction is observed in some cases [START_REF] Dusfour | Heterogeneous mechanical hyperelastic behavior in the porcine annulus fibrosus explained by fiber orientation: An experimental and numerical approach[END_REF].

• < θ < 90 • ).

Ten-layer laminate

In this analysis, we study the effective mechanical properties of a composite model with ten layers (r = 10) as a function of the relative orientation of the successive layers and fiber crimp level; a laminate schematic diagram and the coordinate system are shown in Figure 4, where the x-and z-axes lie in the plane of the lamella and the y-axis is perpendicular to that plane, and the layer orientation angle settings are shown in Table 2. We note that the properties of the ten-layer symmetrically arranged laminate with the constant angle between successive layers are exactly the same as those of the two-layer laminate. In cases 1 and 2, the average orientation angle of ten layers is |40 • |, and in case 1, a wider range of orientation angles is tested. In case 3, the orientation angle is set at a constant 40

• between successive layers |θ k | =40 • , (k = 1, 2, 3 • • • ).
In case 4, the orientation angles increase from one layer to the next with a constant step of 18 • degrees. The thickness of each layer is set to 0.5 mm. These parameters predict the effective Poisson's ratio as a function of the orientation of the successive layers as well as fiber crimp level in the range 0.01 < A L < 0.3, as shown in Figure 6. 1.

Layer number θ 1 θ 2 θ 3 θ 4 θ 5 θ 6 θ 7 θ 8 θ 9 θ 10 Case 1 20 • -20 • 30 • -30 • 40 • -40 • 50 • -50 • 60 • -60 • Case 2 30 • -30 • 35 • -35 • 40 • -40 • 45 • -45 • 50 • -50 • Case 3 40 • -40 • 40 • -40 • 40 • -40 • 40 • -40 • 40 • -40 • Case 4 0 • 18 • 36 • 54 • 78 • 90 • 108 • 126 • 144 • 162 •

matrix. A decrease in the ratio

E f
Em results in an increase in the angle θ at which the limit values occur. For fiber crimp levels (0.08 < < 0.25) and orientation angles (25 • < θ < 45 • ) Figure 6 shows how the effective Poisson's ratios ν ij of a ten-layer laminate model vary with changes to fiber crimp level A L in the laminate as a whole and changes to orientation angle between each of its ten successive layers (or no change, as in case 3). The behavior of the Poisson's ratios in cases 1 and 2 is the same as that of case 3 where the orientation angle between successive layers is constant. Each effective ν ij value in case 2 is always between the values of cases 1 and 3 at each increment in A L , which indicates that the trend seen for the Poisson's ratios is that the larger the range of layer orientation angles in the laminate, the larger the difference between the global effective Poisson's ratio and the constant average case Poisson's ratio. Moreover, we observe a negative Poisson's ratio ν xy in case 3, but not in case 1, which shows that variation in the orientation between successive layers could also weaken the auxetic response. In case 4, where the ten fiber families are regularly oriented and collectively span 180 • , the Poisson's ratios ν xy and ν xz are between 0.3 and 0.5, canceling all the effects found for two fiber families oriented at 40 • . We conclude that only tissues with just two distinct orientations of fibers can have negative apparent Poison's ratios. We note that, in our parameter studies, not shown here, we also found that the minimum value of the Poisson's ratio ν xy and the maximum value of the Poisson's ratio ν xz vary as the Poisson's ratio of the matrix changes. Modification of the Poisson's ratio of the matrix from 0.4 to 0.1 induces a change in the effective Poisson's ratio ν xy from -3.0 to -0.85. Moreover, the orientation angle θ for which ν xy has the minimum value changes with modification of the elastic modulus ratio E f /E m .

Discussion and conclusions

In this paper, we use an analytical micromechanical model to study the effective Poisson's ratio of the lamellar-composite-like structure of fibrous tissue, with a focus on the roles of the orientation angle between the parallel fibers of successive lamellae and the corrugation (or crimp level) of those fibers. Xiao et al. [START_REF] Xiao | A micromechanical model of tendon and ligament with crimped fibers[END_REF] proposed an analytical model to study the tendon tissue that is based on an Eshelby tensor and takes into account the waviness of fibers. The originality of the method used by these latter authors is that it can be used to predict -without any strong assumptions, for example as to the equality of the deformation between phases -the relationship between the deformation of the fibers and the deformation of the matrix. However, the model used by Xiao et al. [START_REF] Xiao | A micromechanical model of tendon and ligament with crimped fibers[END_REF] is limited to unidirectional fibers. Hsiao et al. [START_REF] Hsiao | Elastic properties of composites with fiber waviness[END_REF] proposed an interesting 2D approach similar to that of Xiao et al., to take into account the waviness of fibers in different directions. However, a significant assumption is required as part of this 2D approach, namely that fibers and matrix have the same deformation, which limits the versatility of the model and the conclusions that can be drawn from it. Based on the method presented by Sun and Li [START_REF] Sun | Three-dimensional effective elastic constants for thick laminates[END_REF], we propose an original 3D model that, without any strong assumptions as to the affinity between the fibers and the matrix (no equality between their deformation), merges the 2D multidirectional model of Hsiao et al. [START_REF] Hsiao | Elastic properties of composites with fiber waviness[END_REF] with the 3D unidirectional wavy model of Xiao et al [START_REF] Xiao | A micromechanical model of tendon and ligament with crimped fibers[END_REF].

Our proposed micromechanical model can be used to quickly estimate the small strain mechanical properties of a composite laminate embedded with crimped fibers of alternate orientation between layers using theoretical calculations only. Our model can easily be used to analyze the influence of crimp level and relative fibre orientation on the effective mechanical properties of the modeled material. The modulus of fiber E f and matrix E m , the Poisson's ratios of fiber ν f and matrix ν m , the volume fraction of the fibers V f and matrix V m , fiber crimp level A L , the relative fibre orientation angle of successive layers θ k , layer thickness e k , and total number of layers r are taken into account as parameters potentially modifying the mechanical properties of the laminate. The model described is designed to mimic, and therefore allows us to study, the morphological aspects of AF and arterial wall, but the approaches and techniques employed are also applicable to other fiber-reinforced biological tissues and biocomposites.

For a single-layer composite, ν xy and ν xz are affected by both fiber crimp level A L and layer orientation θ. For a two-layer laminate, the range of ν ij is significantly expanded: the minimum value is even negative and the maximum value is 5.7. It is known that, for isotropic solids, the Poisson's ratio is in general smaller than 0.5, but the Poisson's ratio in certain microstructures of the composites under study can be significantly larger than 0.5, as shown in Figure 5. This is because of the effects of corrugated fibers and alternately oriented layers. As fibers are considered slightly corrugated, when the dispersed fibers are stretched, the fiber-matrix stiffness ratio causes the matrix between the fibers to be compressed, which can lead to expansion in the x-y plane. The proposed model could explain the negative Poisson's ratios found for AF or media of arterial tissue in the plane perpendicular to that of the fibers. Moreover, our analytical model confirms that just two fiber directions (not one, not three, etc) may confer an auxetic behavior in the plane perpendicular to the direction of fibers (ν xy < 0), while multiplying the number of different fiber directions reduces this auxetic behavior. The same phenomenon is demonstrated for the Poisson's ratio ν xz along the mean orientation of the fibres, with very high values possible when there are only two families of fibers, which fades when multiplying the fiber orientations. We note that our analytical model predicts that there must be an inter-fibrilar medium with finite compressibility in order to to see such transverse behaviors. Moreover, as A L increases, the negative Poisson's ratios gradually disappear, that is to say that one of the mechanical effects of the corrugated fibers in the microstructure of the laminate model studied here is to reduce its auxetic behavior. In conclusion, because of the fiber-matrix stiffness ratio and fiber dispersion, the laminate composite shows an auxetic behavior in the planes perpendicular to the plane of the fibers. However, this behavior is not easily measured because of the joint influence of fiber crimp level and successive layer orientation. Therefore, the present work provides evidence of the link between corrugated fiber structure and auxetic behavior, which answers speculation in the literature. Furthermore, compared to the well-known HGO constitutive model [START_REF] Holzapfel | A new constitutive framework for arterial wall mechanics and a comparative study of material models[END_REF] or HGObased models in the literature, which often observe out-of-plane expansion, this work could be a solution to avoid the auxetic response in modeling from the perspective of fiber microstructure.

Our study focuses solely on the linear behavior of corrugated fiber composites. Our model may be completed to describe non-linear composite behaviors. However, provided the microstructure is stable (e.g. no micro-buckling or micro-fractures), our linear model may predict the behavior of such composite for finite strains. Classically admitted and shown for the arterial tissue, the crimp level of the fibers is higher when the tissue is not stretched than when it's stretched. Our model would therefore predict a positive Poisson's ratio for low deformation when the fibers are undulated (i.e. for large A/L), and either a negative Poisson's ratio (theta-r plan) or a large Poisson's ratio (theta-z plan) for higher deformations when the fibers are straight due to stretching (i.e. low A/L). For AF tissue, this point seems to be in agreement with experimental measurements made by Dusfour et al. [START_REF] Dusfour | Heterogeneous mechanical hyperelastic behavior in the porcine annulus fibrosus explained by fiber orientation: An experimental and numerical approach[END_REF] in the laminate plan, but not out of the laminate plan. In this study, for the laminate plan, the Poisson's ratio increases with the circumferential stretch, reaching elevated values, as our linear model could predict. However, in the plan out of the laminate, their study shows that the samples are not auxetic beyond 4% circumferential strain which is not in agreement with our model. For arterial tissues, no measures of non-linear Poisson's ratios have been made to our knowledge. Thus, to link our model to experimental observations, we need to increase our knowledge on the evolution of the microstructure (and especially the crimp level) with macroscopic finite strains. Our model should also be completed to link the crimp level with the macroscopic deformation, which is not straightforward.

The present study is limited to testing the effects of small strains on the laminate material, which is considered as a linear elastic behavior. It is also noted that the [START_REF] Mori | Average stress in matrix and average elastic energy of materials with misfitting inclusions[END_REF] Eshelby equivalent inclusion method provides the best estimation when the fiber volume fractions are relatively low (below 60%) [START_REF] Kwon | Effective moduli of high volume fraction particulate composites[END_REF][66] [START_REF] Saadat | Effective elastic properties of a composite containing multiple types of anisotropic ellipsoidal inclusions, with application to the attachment of tendon to bone[END_REF]. The fiber volume fraction of AF or arterial tissue is estimated to be much less than 60%. Therefore, we consider the Eshelby method to be suitable for use sue model for elimination of unphysical auxetic behaviour. Journal of Biomechanics, 111:110006, 2020. in conjunction with the proposed model, but for other applications, the limits of fiber volume fraction should be reconsidered and an alternative method for estimation of fiber volume fraction may be needed. Moreover, in our model, the strain of each layer along the x-axis and z-axis is assumed to be equal to the global strain of the multi-layer laminate along the x-axis and z-axis, respectively, and the stress on each layer along the y-axis is assumed to be the same as the global stress along the y-axis, as shown in Equation 23. However, no accepted method exists at present to verify these assumptions for AF and arterial tissues. Indeed, our model is not designed to predict how the Poisson's Ratio varies throughout the thickness of these structures. Furthermore, we have considered the fibers to be corrugated only in the plane of the lamella.

However, recent evidence [START_REF] Kelsey D Hamilton | Reflected cross-polarized light microscopy as a method for measuring collagen fiber crimp in musculoskeletal tissues[END_REF] suggests that collagen fibers in the AF may also exhibit crimped properties in the plane perpendicular to the lamella plane, which has not been taken into account in this study. Although the macroscopic properties of AF and arterial tissues are well established in the literature, the link between their macroscopic behavior and microscopic structure still needs to be elucidated. Our study provides a method for predicting the macroscopic properties of fibrous soft tissues based on their fiber microstructure. It is noted that, here, the AF and arterial tissues are considered to be composite-like materials composed of fibers and solid matrix.

Although Derrouiche et al.'s [START_REF] Derrouiche | A chemo-mechanical model for osmoinelastic effects in the annulus fibrosus[END_REF] research shows that the auxetic behavior of AF is related to the osmolarity of the liquid phase, Dusfour et al.'s [START_REF] Dusfour | Heterogeneous mechanical hyperelastic behavior in the porcine annulus fibrosus explained by fiber orientation: An experimental and numerical approach[END_REF] research suggests that the auxetic behavior may be primarily influenced by the solid structure. This is because they conducted their experiments under steady-state stretching conditions, thus excluding the influence of the viscous and dynamic effects of the liquid phase. In our study, the fiber and matrix are considered as compressible solid phases; however, the structure and composition of real arterial tissue and AF are much more complex. By nature, our homogenization method requires precise knowledge of the mechanical behaviors of each component of the tissue under study. In order to make better use of our model, further characterization of the different components of the microstructure of the studied tissue is also necessary.

Figure 1 :

 1 Figure 1: Scheme of AF or arterial tissue components and microstructure.

  (b) with Cartesian coordinates. Here, β represents the tangent angle of the sinusoidal fiber.

Figure 2 :

 2 Figure 2: (a) Composite reinforced with fibers of uniform sinusoidal waviness. (b) Corrugated fibers and Cartesian coordinates.

Figure 3 :

 3 Figure 3: Composite orientation in the Cartesian coordinate system.

Figure 4 :

 4 Figure 4: Organization of the synthetic laminate (a) Multilayer laminate with coordinate system. Two lamellae are shown: the first and second lamellae are orientated by θ 1 and θ 2 around the y-axis with respect to the x-axis, respectively. (b) Schema of laminate total thickness h, and thickness of each layer e k (k = 1, 2, 3 • • • ).

Figure 5 :

 5 Figure 5: Poisson's ratio under the loading along the x-direction for single-(a)(b) and double-layer composite (c)(d) as a function of fiber crimp level AL and layer orientation θ. See the value settings for the parameters in Table1.

Figure 5

 5 Figure 5 shows how the effective Poisson's ratios ν xy and ν xz vary with changes to fiber crimp level A L and layer orientation θ for single-and double-layer composites. For the singlelayer composite, ν xy and ν xz are affected by both A L and θ. Interestingly, ν xy and ν xz show opposite trends with changes to A L and θ as shown in Figure 5 (a) and (b). When A L is small and θ is large, the effective Poisson's ratio ν xz has a minimum value of 0.004 and the effective Poisson's ratio ν xy has a maximum value of 0.63. For the fiber crimp level (0.08 < A L < 0.25) and orientation angle (25 • < θ < 45 • ) in the physiological range of AF, the values of ν xy and ν xz are neither negative nor larger than 0.5. For quasi-straight fibers ( A L < 0.05), we find a Poisson's ratio ν xz of over 0.6. For a double-layer composite with the two layers arranged symmetrically on the x-axis, ν xy and ν xz are also affected by both A L and θ. Moreover, ν xy and ν xz also show opposite trends with changes to A L and θ, as shown in Figure 5 (c) and (d). As opposed to the single-layer composite, a negative Poisson's ratio value for ν xy and a Poisson's ratio of greater than 0.5 for ν xz are observed for double-layer composite. When the fiber is almost straight ( A L < 0.05) the effective Poisson's ratio ν xy has a minimum value of -3.0 and ν xz has a maximum value of 5.7 when θ 1 = -θ 2 = 17 • in this study. Note that the angle θ at which both ν xy and ν xz reach their limit values is influenced by the ratio of stiffness between the fiber and the

Table 2 :

 2 Layer orientation angle settings for each of the ten lamellae of the model. Four cases are studied.

Figure 6 :

 6 Figure 6: Effective Poisson's ratios (a) ν xy , (b) ν xz for a ten-layer laminate model as a function of fiber crimp level and relative orientation of the successive layers of the laminate model. See the value settings for the parameters inTable1.

[ 3 ]

 3 Dawn M Elliott and Lori A Setton. Anisotropic and inhomogeneous tensile behavior of the human anulus fibrosus: experimental measurement and material model predictions. Journal of biomechanical engineering, 123(3):256-263, 2001. [4] Heather Anne L Guerin and Dawn M Elliott. Degeneration affects the fiber reorientation of human annulus fibrosus under tensile load. Journal of biomechanics, 39(8):1410-1418, 2006.

  reformulating the axis, the nonzero components of the Eshelby tensor S E are S E

2222 , S E 3333 , S E 2233 , S E 3322 , S E 2211 , S E 3311 , S E 2323 , and S E 1212 . Inserting the Eshelby tensor into Equation

Table 1 :

 1 Parameters used in our study. Please note the wide ranges for layer orientation angle, fiber crimp level, and number of layers (i.e., families of fibers).

		Meaning	Value
	E f	Collagen fiber modulus	100 MPa
	Em	Extracellular matrix modulus	10 KPa
	V f	Collagen fiber volume fraction	0.2
	Vm	Matrix volume fraction	0.8
	ν f	Collagen fiber Poisson's ratio	0.3
	νm θ A L e k	Extracellular matrix Poisson's ratio Layer orientation angle Collagen fiber crimp level Thickness of each layer	0.4 0 • -90 • 0.01 -0.3 0.5 mm
	r	Number of layers	1, 2, 10

Acknowledgements

We thank Pr. Loïc Daridon, Pr. Yann Monerie and Dr. Cristina Cavinato for the scientific discussions. This work was supported by CNRS (AAP "Osez l'Interdisciplinarité" and MoTiV Project).