Cedric Peyrot

Dr Blanka Didak

Dr Laure Guillotin

Ludovic Landemarre

Dr Pierre Lafite

Loic Lemiegre

Prof. Dr Richard Daniellou
email: richard.daniellou@univ-orleans.fr

Cédric Peyrot

Dr Loïc Lemiègre

Dr C Peyrot

Enzymatic synthesis of a series of thioglycosides: Analogs of arbutin with efficient antipigmentation properties

Keywords: Arbutin analogs, Carbohydrates, Enzyme Catalysts, Thioglycoligase, Whitening Agent

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Introduction

Due to the global warming, the world population is more and more exposed to the sun, thus leading to a continuous rise in the appearance of skin pigmentary disorders. One of them, namely hyperpigmentation, is generally due to a local overproduction of melanin correlated to a significant increase of melanosome number in the epidermis. However, sun exposure is not the only external parameter, and hormonal or contraceptive treatments can also trigger this kind of disorders. To limit these unsightly aspects, there are common physical or chemical skin treatments [1] , but other approaches have been also developed, including melanin transfer inhibition from melanosomes to keratinocytes, melanosomes alteration or inhibition of the DNA transcription coding for tyrosinase [2] . Some chemical treatments, notably hydroquinone [3] , have effects on these targets (Figure 1). It has been part of the reference molecules as whitening agent for more than 50 years. However, its significant side effects like irritation or itch phenomena, have led to its disappearance in cosmetic products [4] . Indeed, its oxidation leads to quinones formation and generation of free radicals which can cause cell membrane alterations and induce the effects mentioned above [5] . To restrict these side-effects, manufacturers have turned their attention to hydroquinone analogs, with one protected phenolic moiety, such as mequinol or monomethyl ether of hydroquinone (MMEH) [4a] . Nevertheless, the current best alternative is the use of its bglucosylated substituent, namely arbutin (Figure 1). This natural compound is extracted from the leaves and barks of many plants such as bearberry [6] . The presence of the glucosyl moiety increases its water solubility while retaining a good antipigmentation activity [7] . In addition, this molecule has the advantage of being less toxic to cells. Moreover, glycoconjugates are described as being capable to interact with the membrane receptors present on the cell surfaces (lectins) thanks to their sugar moiety so as to inhibit the melanin transfer from melanocytes to keratinocytes [8] (Figure 2)

Accepted Manuscript

European Journal of Organic Chemistry In an attempt to obtain improved tyrosinase inhibitors, various arbutin analogs have been proposed, i.e. α-arbutin, deoxy-arbutin or analogs in the form of disaccharides [9] . All these glycosides present interesting properties but, in all the cases mentioned above, their chemical and/or enzymatic hydrolysis leads to hydroquinone release, which might cause serious skin disorders. The use of a thioglycoside, more resistant against chemical or enzymatical hydrolysis, would limit the hydroquinone release while retaining biological activities [10] . Thioglycoconjugates are already commonly used as stable ligands for the crystallographic analysis of protein structures or as glycosidase inhibitors [11] . They can even serve for the development of carbohydrate-based vaccine antigen [12] .Until recently, their only access was through very tedious chemical synthesis steps [13] , or through metallic catalysis reactions [14] , in particular because of numerous protection and deprotection steps [15] , thus limiting drastically the commercial applications. One solution to overcome this limitation relies in the use of enzymatic synthesis, which in addition will owe the advantage of decreasing environmental impact [16] . In 2003, Withers et al described a major advance in terms of thioglycoconjugates synthesis by proposing the first enzymes capable to promote the formation of this type of linkage [17] , a biocatalyst obtained thanks to judicious mutations on the acid/base residue of the catalytic site starting from a native glycosidase [18] .With a similar approach based on site-directed mutagenesis, our group has been able to access to a novel thioglycoligase (DtGlyE159Q) from Dictyoglomus thermophilum [19] . In particular, this mutant, whose hydrolytic activity has been greatly reduced by the replacement of the catalytic E159 by a glutamine, has the capacity to efficiently catalyze the binding of a saccharidic unit on thiophenolic derivatives, which pKa (around 7 (https://doi.org/10.1021/ja058344i)) is of high importance to lead to very good reaction yields.

Herein, we thus wish to report the efficient biocatalyzed synthesis of numerous S-arbutin analogs and their biological properties as promising whitening agents. We therefore focused on cosmetic alternatives that rely on tyrosinase inhibition at the melanosome level and therefore prevent the synthesis of melanin [20] .

Results and discussion

1) Biocatalyzed synthesis

In order to obtain arbutin analogs, we particularly focused on thiophenolic compounds, which could mimic hydroquinone. First, to keep a good analogy, our choice fell on p-hydroxythiophenol, as well as p-aminothiophenol to rule on the phenolic moiety influence on the biological activities (Scheme 1). Second, the acid/base catalytic residue being mutated, it was essential to use activated sugars to carry out the enzymatic glycosylation step. For this, we chose p-nitrophenyl-ß-D-glucopyranoside (pNPGlc) as sugar donor. In order to favor the nucleophilic attack of sulfur moiety on the glycosyl-enzyme complex, we kept the pH of the reaction buffer fixed at 9 (Tris-HCl, 20mM), so that thiophenol acceptors were mostly present in their thiophenolate forms. The reactions were thus carried out and followed by TLC. p-Hydroxythiophenol did not react, as only starting material could be detected after enzymatic incubation. This absence of reactivity was thought to be related to the formation of unreactive quinone. Regarding p-aminothiophenol acceptor, a new glucosylated compound S-1 was detected and, after purification, characterized by NMR. We therefore sought to optimize the enzymatic thioglycosylation reaction based on the same conditions as those used previously. The examples present in the literature highlight the necessity to use a large excess of acceptor equivalents [21] . Thus, we initially decided to carry out the reaction in the presence of 40, 30 and 20 eq of acceptors (Table 1 entries 1-3) and we did not observe any degradation of the enzyme due to the acceptor under the thiol or disulfide forms. The yields obtained after purification demonstrated that an excess of acceptor caused a negative effect, with 20 eq. leading to the best yield (Table 1, entries 1-3). In parallel, the reaction time, initially set at 48 hours, was reduced to 24 hours. In addition, to limit the potential negative impact of this reduction time, the quantity of enzyme was multiplied by 5. The reactions were carried out under these conditions in the presence of 20, 10 and 5 equivalents of acceptor (Table 1, Entries 4-6). Indeed, when the number of acceptor equivalents decreased, so did the yield. However, the conditions of entry 4 allowed an optimized yield of 99% in just 24 hours. The increase of the quantity of enzyme had a very positive effect on the reaction, while remaining in catalytic quantity. In addition, when the acceptor equivalents decreased, we still kept excellent yields (Table 1, Entries 5-6: 92% and82%, respectively). Two stages of gradual scale up were carried out, starting from 100 mg and then 600 mg of sugar donor (Table 1, Entries 7-8). The yields being good in 24 hours on small quantities, we tried to reduce by 1.7 the enzyme equivalents for the scaling up reaction. The yields obtained for both conditions were very good with respective values of 91% and 96%. Under these conditions, they were even quite higher than those obtained previously while the amount of enzyme has been reduced. At this stage, we were therefore able to significantly optimize the reaction by reducing the number of acceptor equivalents required while retaining excellent yields. The reaction time has also been reduced and the enzyme quantity fine-tuned.

We then sought to obtain analogues of molecule S-1 carrying different saccharidic moieties. In the literature, numerous glycosidases are described as being versatile with respect to their substrate [22] . We therefore carried out the same reaction according to the best conditions (Table 1, Entry 4) starting from different activated sugar donors. The results obtained are presented in Table 2 andFigure 3A.

Table 2. Exemplification of the method starting from different sugar donors (pNP sugar 0.13 mmol, p-aminothiophenol 20 eq, DtGlyE159Q 1.10 -4 eq, 24h, pH9 (Tris-HCl, 20 mM), RT)

Entry

Abbreviations Donors Yield

1 pNP-Gal p-nitrophenyl-ß-D-galactopyranoside 79% a 2 pNP-Fuc p-nitrophenyl-ß-D -fucopyranoside 52% a 3 pNP-Xyl p-nitrophenyl-ß-D -xylopyranoside - 4 pNP-GlcA p-nitrophenyl-ß-D-glucuronic acid - [a] Isolated yields after column purification step
The enzyme accepted other sugar donors. Its versatility allowed us to obtain compound S-2 carrying a D-galactosyl unit, with a good yield of 79% (Table 2, Entry 1) as well as compound S-3 in D-fucose series with a lower yield of 52% (Table 2, Entry 2). Unfortunately, the enzymatic reaction did not occur in presence of pNP-Xyl or pNP-GlcA (Table 2, Entries 3-4).

Finally, we were also wondering whether the nature of the glycosidic linkage could also have an effect on anti-pigmentation activity. Using a simple and rapid methodology, already published by Peyrot et al, we were able to quickly access the p-aminophenyl ß-D-glycopyranosides from the corresponding pNP sugars, by reduction of the nitro moiety without any protection or deprotection steps [23] . The protocol was developed in presence of dithiothreitol (DTT) in methanol under microwave activation for 2 h. This quick methodology provided access to compounds O-1, O-2 and O-3 (Figure 3B) with very good respective yields of 97, 91 and 92% [23] .

This allowed us to obtain a library of 8 arbutin analogs in S-or Oglycosidic series carrying different substituents on the aromatic ring (Figure 3). We were then able to assess the anti-pigmentation activity of these 7 compounds on different targets. To compare with arbutin, S-arbutin was synthesized using the methodology developed by Brachet et al through nickel catalysis allowing to access to the perfect analog with a thioglycosides linkage [14a] .

Figure 3. Target molecules for biological evaluations.A) this study. B) Peyrot et al. [23] 2) Tyrosinase inhibition.

According to the targeted mechanism intended to decrease pigmentation, various types of approaches exist to evaluate the anti-pigmentation activity. The simplest and most common is to assess tyrosinase inhibition, the enzyme producing melanin in melanosomes. Based on a protocol described by Masamoto et al., we were able to perform in-vitro inhibition studies for each molecule [24] .The enzymatic activity was monitored by spectrophotometry and the IC50 values were recovered with GraphPad. The IC50 correspond to the concentrations needed to observe 50% of activity loss. All the data have been grouped in Table 3, and arbutin was used as the positive control.

Table 3. Tyrosinase inhibition IC50 for each arbutin analog according to the Masamoto protocol [24] Compounds IC50 mM The IC50 values mentioned in the literature are somehow variable depending on the natural source of the tyrosinase but remain in the order range of mM [9a, 9b] . Our experimental value for arbutin was therefore in agreement with the literature, which thus validates the experimental protocol. The glycosidic linkage nature, the sugar moiety as well as the substitution on the aromatic ring may have an influence on the inhibition activity of tyrosinase.

When we focused on the influence of nature of the aromatic cycle, it appeared that the phenolic group led overall to better tyrosinase inhibition (Table 3, Arbutin vs O-1). The nature of the sugar also demonstrated an important influence. Indeed, the presence of a fucose was unfavorable (Table 3, Compound S-3) whereas the presence of a galactose unit made it possible to gain in efficiency (Table 3, Compound S-2). In the case of galactose, the presence of the sulfur atom even made it possible to obtain a more effective molecule (Compounds S-2 and O-2). At this stage, the compound S-2 appeared as the best candidate, both regarding its IC50 value but also its stability against hydrolysis. Considering the first results, and the low activity differences regarding tyrosinase inhibition between S-arbutin and compound S-2, we have chosen to focus on the latter, obtained by a biocatalysed procedure.

3) Lectin recognition.

We then investigated the potency of inhibition of melanin transfer from melanocytes to the keratinocytes by the compounds. This process takes place through the activation of lectin-type receptors on the cell surface [8a] . These lectins are capable of specifically recognizing glycan motifs, then triggering the migration process of melanin. The aim of this study was to consider the influence, or not, of the sulfur atom at the level of the sugar linkage. Given the promising in vitro results obtained for the compound S-2, we carried out this recognition test with this compound as well as its analogue in O-series O-2. To assess whether these new molecules can potentially be recognized by lectin-like receptors involved in melanin transfer (expressed at the surface of melanosomes and/or keratinocytes), we first set up in vitro screening on different lectins based on the GLYcoPROFILE® technology described by Landemarre et al [25] . The specificity and the nature of each lectin are reported in the supporting information. Lectin recognition tests were carried out in a 96-well plate (LEctPROFILE® plates). Compounds S-2&O-2 were assessed for their ability to competitively displace the reference labelled ligand for each lectin. In that case, absorbance or fluorescence reads made it possible to determine the percentage of resulting bound reference ligands, bearing either a chromophore or a fluorophore. The results obtained for compound S-2 and O-2 are respectively presented in Figure 4. The other important element was the preservation of a dose effect. In fact, when the inhibitor molecule was introduced in higher quantity, the binding percentage to receptors was greater, regardless the glycosidic linkage nature. Thioglycosides therefore seemed to be an excellent alternative for inhibiting the melanin transfer.

We further moved to in vitro tests on cell culture. In order to study and compare the potential interaction of S-2 and O-2 compounds with lectin-like receptors expressed at the surface of keratinocytes, we use the method of "neoprofile". A set of fluorescent neoglycoproteins (galactosylated, glucosylated and rhamnosylated) well known to interact in a "glycan-specifically" manner with the surface of normal human epidermal keratinocytes (NHEK) were incubated with the cells in the presence or not of the compounds. After washing, the resulting inhibition of neoglycoprotein shows the specificity of interaction of the corresponding compound. The binding percentage was indirectly measured through the ratio between the studied mixture fluorescence and the reference binding fluorescence. According to cell toxicity up to 1 mg/mL (determined by MTT test on same keratinocytes, data not shown), we chose to fix this concentration as the cell maximum concentration for the following studies. Several lower concentrations were then tested depending on the compounds. Thus, the higher inhibition percentage results in better compound recognition (Figure 5).

Accepted Manuscript

European Journal of Organic Chemistry Regarding the galactose lectins, the S-2 compound showed better inhibition than O-2, and therefore, the binding intensity (inhibition percentage) was markedly greater in the case of compounds carrying a thioglycosidic linkage, suggesting a nature bond positive effect on the cellular interaction. Looking at the other lectins (glucose, rhamnose), once again the S-2 interacted stronger with the other receptors than the O-2. Overall, the presence of a thioglycosidic bond was favorable. However, it induced a slight loss of sugar specificity. This is not necessarily negative in our case, because it would allow better saturation of the various cell receptors to limit the transfer of melanin. In addition, the dose effect was retained significantly. In view of the literature which indicates that the melanin transfer from the melanosome to the keratinocyte can be inhibited by galactosylated or approximate structures [8a] , the compound S-2 could be used at a concentration much lower than that of the compound O-2.

Conclusion

We have developed an original approach through enzymatic catalysis allowing access to arbutin analogs with very good yields. One of the best tyrosinase inhibitor was found to be compound S-2, a thiogalactoside. The molecular and cellular in vitro studies demonstrated a conservation of the osidic specificity of thioglycosides recognized by lectin-like as cell membrane receptors. In addition, the presence of the sulfur atom increased this recognition, while retaining the dose effect. The stability of the compound also gave it a longer lifespan and therefore a longer interaction with the receptors. This made it a double action compound, obtained by an eco-responsible process with high application potential in cosmetics.

Experimental Section

2.1. Chemical synthesis p-Nitrophenyl-ß-D-glucopyranoside, p-nitrophenyl-ß-D-galactopyranoside, p-nitrophenyl-ß-D-fucopyranoside,p-nitrophenyl-ß-D -xylopyranoside, pnitrophenyl-ß-D-glucuronicacidwere purchased from Carbosynth. p-Aminothiophenol, p-hydroxythiophenol and arbutin were purchased from Sigma Aldrich. All chemicals were used directly without purification. Chromatographic purifications of products were accomplished using a silica column in dichloromethane/methanol eluent. 1 H NMR spectra were recorded on Avance III HD NanoBay Bruker at 400 MHz ( 13 C: 100 MHz) and were calibrated with residual D2O or CD3OD protons signals at δ4.79 or 3.31 ppm respectively. Data are reported as follows: chemical shift (δ ppm), multiplicity (s = singlet, d = doublet, t = triplet, dd = doublet of doublet, m = multiplet), coupling constant (Hz), integration and assignment. 13 C NMR spectra were calibrated with CD3OD signal at δ49.2 ppm. Data are reported as follows: chemical shift (δ ppm) and attribution. All NMR assignments were made using COSY, HMBC and HSQC spectra. HRMS were performed on a Maxis Bruker 4G system.

Chemical thioglycosylation

p-Hydroxyphenyl-ß-D-glucopyranoside (S-arbutin) was synthesized as colorless oil (90% yield) from ß -thioglucose and p-hydroxyanisole according to an already published protocol [14a] . Characterization data were identical with those already described.

Enzymatic thioglycosylation: General procedure 1: p-nitrophenyl-ß-Dglycopyranoside (1.0 equiv., 40 mg) and 20 equiv. of aminothiophenol were dissolved in Tris-Base/Tris HCl buffer (20 mM, pH 9, 2 mL). Thioglycoligase mutant DtGly E159Q (80 nmol, 1.10-4 equiv.) was successively added and the mixture was stirred at room temperature overnight. Reaction was concentrated under reduced pressure and then purified by flash chromatography.

p-aminophenyl-ß-D-thioglucopyranoside (S-1) was obtained using pnitrophenyl-ß-D-glucopyranoside (0.13 mmol, 1.0 equiv.) according to the general procedure 1. The desired compound was obtained after flash chromatography (DCM/MeOH 9/1) as a white solid (38 mg, 99 %). 1 H NMR (D2O, 400 MHz) δ: 7.44 (2H, d, J = 8.4 Hz, H-8), 6.85 (2H, d, J = 8.4 Hz, H-9), 4.57 (1H, d, J = 10.0 Hz, H-1), 3.89 (1H, dd, J = 1.6, 12.4 Hz, H-6), 3.72 (1H, dd, J = 5.6, 12.4 Hz, H-6'), 3.51 (1H, t, J = 8.8 Hz, H-3), 3.42 (2H, m, H-4, H-5), 3.27 (1H, t, J = 9.6 Hz, H-2). 13 p-aminophenyl-ß-D-thiogalactopyranoside (S-2) was obtained using pnitrophenyl-ß-D-galactopyranoside (0.13 mmol, 1.0 equiv.) according to the general procedure 1. The desired compound was obtained after flash chromatography (DCM/MeOH, 9:1) as a white solid (30 mg, 79 %). 1 H NMR (CD3OD, 400 MHz) δ: 7.34 (2H, d, J = 7.2 Hz, H-8), 6.63 (2H, d, J = 7.6 Hz, H-9), 4.31 (1H, d, J = 9.2 Hz, H-1), 3.86 (1H, s, H-4), 3.71 (2H, m, H-6), 3.51 (1H, t, J = 9.6 Hz, H-2), 3.46 (2H, m, H-3, H-5). 13 C NMR (CD3OD, 100 MHz) δ: 149.48 (C10), 136.30 (C8), 120.89 (C7), 116.44 (C9), 91.62 (C1), 80.46 (C5), 76.39 (C3), 70.98 (C2), 70.43 (C4), 62.54 (C6). HRMS (m/z) [M+Na]+ calculated for C12H17NNaO5S 310.07196; found: 310.07159 p-aminophenyl-ß-D-thiofucopyranoside (S-3) was obtained using pnitrophenyl-ß-D-fucopyranoside (0.14 mmol, 1.0 equiv.) according to the general procedure 1. The desired compound was obtained after flash chromatography (DCM/MeOH, 9:1) as a white solid (20 mg, 52 %). 1 H NMR (CD3OD, 400 MHz)δ: 7.31 (2H, d, J = 8.0 Hz, H-8), 6.64 (2H, d, J = 8.0 Hz, H-9), 4.28 (1H, d, J = 8.4 Hz, H-1), 3.56 (1H, s, H-4), 3.56 (1H, q, J = 6.4 Hz, H-5), 3.46 (2H, m, H-2, H-3), 1. 24 (3H, d, J = 6.4 Hz, H-6). 13 glucopyranoside(O-1) was synthesized as a white solid (97% yield) from p-nitrophenyl-ß-D-glucopyranoside according to an already published protocol [23] . Characterization data were identical with those already described.

p-aminophenyl-ß-D-galactopyranoside (O-2) was synthesized as a white solid (91% yield) from p-nitrophenyl-ß-D-galactopyranoside according to an already published protocol [23] . Characterization data were identical with those already described.

p-aminophenyl-ß-D-fucopyranoside(O-3) was synthesized as a white solid (97% yield) from p-nitrophenyl-ß-D-fucoopyranoside according to an already published protocol [23] . Characterization data were identical with those already described.

Biological procedures

Protein expression and purification

Production and purification of DtGlyE159Q was performed as described previously by Guillotin et al [19] . Escherishia coli Rosetta (DE3) transformed with expression plasmids were grown in LB medium supplemented with chloramphenicol (34 μg/mL) and kanamycin (30 μg/mL) at 37 °C until OD600 reached 0.6. Induction was then done by addition of 1 mM IPTG and incubated overnight a 25 °C. Cells were harvested, lyzed by freezethaw cycles and sonication. Then, supernatant was clarified by heat treatment for 15 min at 70 °C before centrifugation. Finally, supernatant was loaded on a Nickel column (HisPure, Thermo Scientific) and purified by elution with lysis buffer containing 500 mM imidazole. The concentration of the various fractions collected was obtained by Bradford assay.

Lectin Array assays

Lectin Array assays were performed according GlycoDiag's protocol already described [25] . Briefly, GLYcoPROFILEs were performed on LEctPROFILE® plates from GLYcoDiag (Orléans, France). The interaction profiles of each compound were determined through an indirect method based on the inhibition by the compound of the interaction between a specific couple lectin-glycan (neoglycoproteins or glycoproteins). Briefly, a mix of neoglycoproteins or glycoproteins (fixed concentration) and the corresponding compounds (range of concentrations) prepared in PBS supplemented with 1 mM CaCl2 and 0.5 mM MgCl2 is deposed in each well (50 µL each) in triplicates and incubated two hours at room temperature. After washing with PBS buffer, the conjugate streptavidine-DTAF for fluorescence plate or extravidineperoxydase for absorbance palte is added (50 µL) and incubated 30 min more. The plate was washed again with PBS. Finally, 100 µL of PBS was added for the readout of fluorescent plate performed with a fluorescence reader (λex = 485 nm, λem = 530 nm, Fluostar OPTIMA, BMG LABTECH,France) For the absorbance plate, the plate was washed with PBS buffer, and a solution of OPD (SIGMAFAST™ OPD (o-Phenylenediamine dihydrochloride), 100 μl) for the detection of the peroxidase activity. The plate was incubated 15 min protected from light. The coloration was stopped by adding HCl (100 μl, 1 mM) and the readout performed with an absorbance reader. The signal intensity is inversely correlated with the capacity of the compound to be recognized by the lectin and expressed as inhibition percentage with comparison with the corresponding tracer alone (Neoglycoproteins or glycoproteins). NeoPROFILE@ Assays NeoPROFILE@ Assays were performed according GlycoDiag's protocol already described [26] . Compounds interactions with carbohydrate recognition receptors expressed at the surface of NHEK cells were measured and achieved with fluoresceinylated neoglycoproteins according to GLYcoDiag technology. (α-Galactose-BSA, ß-Glucose-BSA, α-Rhamnose-BSA). Briefly, cells were first grown to confluence (80-90%) in 96-well plates. Once the confluence reaches, cells were washed several times with PBS and incubated with fluorescent neoglycoproteins (fixed concentrations) and compounds (range of concentrations) and incubated 4h at 4 °C. Then, the plates were carefully rinsed with PBS. 200 uL of fresh PBS was added for the fluorescence readout (λex = 485 nm, λem = 530 nm). The amount of neoglycoproteins stayed in interaction with cells was compared with the amount of neoglycoprotein in absence of product.

Tyrosinase inhibition tests

Tyrosinase inhibitor activity was measured by spectrophotometry based one the method presented by Masamoto et al [24] . 10 μL of inhibitor solution at different concentrations in DMSO were placed into 96-wells microplate mixed with ammonium formate buffer (60 μL, 50 mmol/L, pH 6.4) and 20 μL (0.8 mg/mL) of tyrosine was added. Just before the absorbance read at 450nm, 10 μL of mushroom tyrosinase (5000 U/mL) was added. The mixture was incubated at 27 °C for 10min. The absorbance value was recovered every 20 sec during 10 min. Arbutine were used as a positive control. Each measurement was performed at least in triplicate. The IC50, corresponding to the needed concentration to observe 50% loss of the enzyme activity, were calculated with GraphPad.

Figure 1 :

 1 Figure 1: Hydroquinone and Arbutin analogues

Figure 2 :

 2 Figure 2: Tyrosinase and melanine transfer inhibitions.

Figure 4 .

 4 Figure 4. GLYcoPROFILE of compounds S-2 and O-2 at different concentrations

Figure 5 .

 5 Figure 5. Dose-dependent inhibition by S-2 and O-2 of carbohydrate-specific lectins on NHEK cells.

  , 100 MHz) δ: 149.93 (C10), 136.81 (C8), 121.44 (C7), 116.88 (C9), 91.95 (C1), 76.93 (C2), 76.38 (C5), 73.60 (C4), 71.28 (C3), 17.42 (C6). HRMS (m/z) [M+Na]+ calculated for C12H17NNaO4S 294.07705; found 294.07693 Chemical synthesis of the O-glycosidic series p-aminophenyl-ß-D-

Scheme 1: Enzymatic thioglycosylation scheme Table 1. Optimization of enzymatic thioglycosylation conditions

		pNPGlc	Acceptor	DtGly E159Q	Time	Yield
	Entry
		mg	mmol	Eq	Eq	(h)	(%)
	1	40	1.3 10 -1	40	2.9 10 -5	48	75 a
	2	40	1.3 10 -1	30	2.9 10 -5	48	69 a
	3	40	1.3 10 -1	20	2.9 10 -5	48	82 a
	4	40	1.3 10 -1	20	1.4 10 -4	24	99 a
	5	40	1.3 10 -1	10	1.4 10 -4	24	92 a
	6	40	1.3 10 -1	5	1.4 10 -4	24	82 a
	7	100	3.3 10 -1	5	8.4 10 -5	24	91 a
	8	600	2.0	5	8.4 10 -5	24	96 a
	[a] Isolated yields after column purification step

Acknowledgements

The authors thank the regions Centre Val de Loire and Bretagne for their financial support

FULL PAPER

8

Entry for the Table of Contents

A biocatalyzed synthesis led to the preparation of a series of thioglycoside analogs of arbutin. These original molecules demonstrated potent tyrosinase inhibition properties, as well as the capacity to interact with lectins at the surfaces of cells. The presence of the sulfur atom prove therefore to increase the biological potency of these glycosides.

Institute and/or researcher Twitter usernames: daniellou_r


Adomas Baliuka

Elsa Dupraz

Rengaraj Govindaraj

Michael Auer

Peter Freiwang

Lukas Knips

Harald Weinfurter

Open-Source LDPC Error Correction for QKD

est destinée au dépôt

LDPC codes are the subject of active research with many applications, such as for Wi-Fi and digital television. They have been used for QKD error correction for decades, together with methods such as Cascade [2].

A single LDPC code operates on a fixed number of symbols and is optimized for a specific noise level of the quantum channel. In practice, the quality of the quantum channel fluctuates over time and across applications of a single QKD system. To achieve efficient error correction would thus require a large set of LDPC codes. However, storing, selecting and using hundreds or even thousands of different codes is not feasible in practice. Rate adaption solves this issue by modifying a single LDPC code, thus adjusting it to the current channel.

Error correction in QKD is a special case of Slepian-Wolf coding [3]. So far, the QKD community has been using rate adaption methods tailored to forward error correction, such as puncturing and shortening, which are sub-optimal for Slepian-Wolf coding [4]. Recently, more efficient rate adaption methods specialized for Slepian-Wolf coding have been developed [1,3,4]. Error correction in QKD can benefit from these insights to achieve better efficiency.

Due to the flexible nature of the proposed rate adaption, our coding scheme can be used not only for one-way error correction, but also for interactive protocols, which can achieve higher efficiency, i.e., less leaked information, at the cost of more classical communication. Furthermore, while our current codes are optimized for DV-QKD, our construction and rate adaption methods can be applied more generally. We invite contributions from the research community and plan to add support for more protocols, such as CV-QKD, in the future, incorporating further developments in QKD and channel coding. We hope that the proposed methods will enable simpler, efficient and practical implementations of error correction for QKD post-processing.

Figure 1 :

 1 Figure 1: Error correction efficiency f = r/h 2 (QBER), where r is the rate of information leakage, and h 2 is binary entropy.


Student Member, IEEE Yi Wu
email: yi.wu@insa-rennes.fr

Member, IEEE Erwan Fourn
email: erwan.fourn@insa-rennes.fr

Senior Member, IEEE Philippe Besnier
email: philippe.besnier@insa-rennes.fr

Senior Member, IEEE Cédric Quendo
email: cedric.quendo@univ-brest.fr

Direct Synthesis of Multiband Bandpass Filters With Generalized Frequency Transformation Methods

Keywords: Frequency transformation methods, microstrip technology, multiband bandpass filter (MBPF), quad-band filter, quint-band filter, triple-band filter

This article presents a new generalized direct synthesis method to design multiband bandpass filters (MBPFs).

Based on the proposed frequency transformation technique, it can be used to synthesize an arbitrary number of bandpass filters starting from a low-pass prototype filter. This synthesis allows determining analytically all the resonant angular frequencies and slope parameters of all the bandpass filters from the low and high cut-off angular frequencies used as initial specifications whatever the number of bands. To validate our method, triple-, quad-, and quint-band third-order Chebyshev bandpass filters are designed and implemented in microstrip technologies. Very good agreements were achieved between simulation responses and measurements.

I. INTRODUCTION

F ILTERS are key components in radio frequency trans- ceivers in handing multiband operations. Their integration is challenging with regard to requirements. In new terminal generations providing several communication standards simultaneously, and especially in mobile terminals, embedded systems, or satellites, the footprint of the filtering parts needs to be reduced in order to meet mass and volume requirements of the whole system. Multiband bandpass filters (MBPFs) indeed then appear as an attractive solution to select multiple bands with a relatively compact circuit size. Thus, [START_REF] Holme | Multiple passband filters for satellite applications[END_REF], [START_REF] Uhm | A triple-passband waveguide filter with dual-mode resonators for ka band satellite applications[END_REF] present some examples of MBPFs to simplify the design of communication system for satellite applications. Many researchers pay great attention on MBPFs leading to a large amount of circuit topologies implemented in different technologies such as microstrip [START_REF] Xu | Compact microstrip dual-/tri-/quad-band bandpass filter using open stubs loaded shorted stepped-impedance resonator[END_REF]- [START_REF] Gomez-Garcia | A frequency transformation for co-designed multi-passband/multi-embedded-notch RF filters[END_REF], waveguide [START_REF] Zhu | Triple-band cavity bandpass filters[END_REF], or substrate integrated waveguide (SIW) [START_REF] Guo | Design method for multiband filters with compact configuration in substrate integrated waveguide[END_REF]- [START_REF] Zhou | Resonance characteristics of substrate-integrated rectangular cavity and their applications to dualband and wide-stopband bandpass filters design[END_REF].

More specifically, the synthesis of MBPFs, whatever the implementation technology, can be done through numerous methods. One of the most popular ones consists in deriving coupling matrix to achieve multiband responses [START_REF] Cameron | Advanced coupling matrix synthesis techniques for microwave filters[END_REF]- [START_REF] Zhang | Analytical synthesis of generalized multi-band microwave filters[END_REF]. Actually, this method is based on the optimization of the whole coupling matrix. High quality of initial values and good optimization algorithms (such as genetic algorithm or sequential quadratic programming) are needed to achieve satisfying results. Such methods may be considered inefficient in some cases because the convergence is not always guaranteed and the final topology is difficult to control [START_REF] Zhang | Analytical synthesis of generalized multi-band microwave filters[END_REF], [START_REF] Crnojević-Bengin | Advances in Multi-Band Microstrip Filters[END_REF]. A second popular method to design MBPF is based on stepped-impedance resonators (SIRs) with two main concepts. The first one makes use of multisection SIRs with different resonant frequencies [START_REF] Kuo | Design of microstrip bandpass filters with a dual-passband response[END_REF]- [START_REF] Tu | Design of sext-band bandpass filter and sextaplexer using semilumped resonators for system in a package[END_REF]. These resonators can have different shapes such as ring or stub-loaded shapes or consist of an assembling of quarter-or half-wavelength lines. For such filters, each resonator can resonate at the center frequencies of at least two bands. By coupling these resonators together, one can realize multibands responses. However, MBPFs' design, especially beyond triple-bands, is still challenging because it is really difficult to simultaneously satisfy all the desired requirements for all bandpass, including resonant frequencies, bandwidths, and so on. A third method consists in using dual behavior resonator (DBR) filter [START_REF] Quendo | Narrow bandpass filters using dualbehavior resonators[END_REF], [START_REF] Quendo | Planar tri-band filter based on dual behavior resonator (DBR)[END_REF]. A DBR is based on the parallel association of N band-stop resonators, which can create N transmission zeros to form N -1 bands. Based on this method, dual-and triple-band bandpass filters have been successfully synthesized and implemented. However, for this type of filter, it is hardly possible to achieve more bandpass, because of the harmonic frequencies, and high-order filters because slope parameters are not taken into account into the synthesis.

Another MBPFs' synthesis technique is based on frequency transformation methods. The first article was presented in [START_REF] Macchiarella | Design techniques for dualpassband filters[END_REF] in the case of a dual-band filter. The main idea of this article is to map the lower and upper cut-off angular frequencies of each bandpass to the low-pass prototype ones (positive and negative). The dual-band resonator obtained using this transformation then consists in the association of a bandpass resonator and a band-stop one, the latter creating a transmission zero in the bandpass of the first to obtain the two bands. Similar techniques were used to design dual-and triple-band bandpass filters in [START_REF] Guan | Synthesis of dual-band bandpass filters using successive frequency transformations and circuit conversions[END_REF] and [START_REF] Di | Synthesis of cross-coupled triple-passband filters based on frequency transformation[END_REF] with different implementation topologies and technologies. In [START_REF] Ma | A design method of multimode multiband bandpass filters[END_REF], the authors also applied the frequency transformation approach to design triple-band bandpass filter. Additional transmission zeros were realized before each bandpass using optimized multimode resonators. However, the optimization-based multimode resonators are difficult to be implemented for filters more than triple bandpass. A generalized synthesis method was proposed in [START_REF] García-Lampérez | Single-band to multiband frequency transformation for multiband filters[END_REF]. In this article, dual-and triple-band cases were presented to validate the synthesis theory. This approach is based on coupling matrices and frequency-invariant susceptances. However, practical implementation of multiband filters with more than three bands seems rather difficult.

In this article, a general analytical method to design MBPFs with an arbitrary number of bands based on frequency transformation technique is presented. It is a direct generalization of the solution presented in [START_REF] Macchiarella | Design techniques for dualpassband filters[END_REF] and differs from [START_REF] García-Lampérez | Single-band to multiband frequency transformation for multiband filters[END_REF] by the use of the slope parameters instead of coupling matrices, leading to a relative greater simplicity and freedom in terms of design and implementation. The design of high-order MBPFs with an important number of bands is then relatively easy unlike most of the solutions discussed above. To demonstrate the effectiveness of this approach, several synthesis examples are also shown. The article is organized as follows: Section II presents the generalized frequency transformation technique in its mathematical aspects. Section III shows the ideal implementation in LC element structure and Section IV presents three examples of MBPFs implemented in microstrip technology: A triple-band filter, a quad-band filter, and a quint-band filter. The last section concludes this article.

II. GENERALIZED FREQUENCY TRANSFORMATION TECHNIQUE FOR MULTIBAND BANDPASS FILTERS

The transformation from a classical low-pass prototype to an N-band bandpass filter is schematically presented in Fig. 1. The proposed frequency transformation function is a generalization of the one proposed in [START_REF] Macchiarella | Design techniques for dualpassband filters[END_REF] in the dual-band case and can be written as follows:

= T (N ) (ω) = b 0 ω ω 0 - ω 0 ω - N -1 k=1 1 b k ω ω k - ω k ω (1)
where is the normalized angular frequency (associated with the normalized low-pass filter). First, let us denote

T (N ) (ω)
U (N ) (ω) = T (N ) (ω) -1. ( 2
)
Substituting equation (1) for T (N ) (ω) in (2) leads to a new expression of U (N ) (ω) consisting in a ratio of two polynomials

U (N ) (ω) = N(ω) D(ω) = ω 2N + 2N -1 p=0 n p ω p 2N -1 q=0 d q ω q (3)
where d q is equal to 0 when q is even and -n q when q is odd.

A first expression of the 2N n p ( p ∈ [0; 2N -1]) coefficients is then obtained as functions of the output parameters as detailed below.

We now assume that the lower cut-off angular frequencies ω L i and the upper cut-off angular frequencies ω H i (i ∈ [1; N]) are mapped to -1 and +1 in the normalized domain, respectively. As T (N ) (ω) is an odd function, we then have

T (N ) (-ω L i ) = T (N ) (ω H i ) = 1 i ∈ [1; N]. (4)
Therefore, the N high cut-off angular frequencies and the opposite of the N lower cut-off ones are the 2N roots of U (N ) (ω). A second expression of the n p coefficients is thus obtained as functions of the input parameters this time.

So, 2N equations (one for each n p coefficient) link the N resonant angular frequencies, the ω k , and the N slope parameters, b k ((k ∈ [0; N -1])), to the N low cut-off angular frequencies, ω L i , and the N high cut-off ones, ω H i (i ∈ [1; N]). As presented below, an analytical expression can then be obtained for each output parameters.

Section II-A-II-C present the general expressions of the n p coefficients ( p ∈ [0; 2N -1]) according to the cut-off angular frequencies (see Section II-B), on the one hand, and according to the resonant angular frequencies and slope parameters (see Section II-C), on the other hand. To simplify the writing of these equations and make the understanding easier, we first introduce specific mathematical expressions in Section II-A.

A. Specific Mathematical Operators

Let Z (r→s) be a set of real numbers z i with i ∈ [r ; s] (r ∈ N and s ∈ N )

Z (r→s) = {z i } s i=r . ( 5
)
We then denote Z (r→s)

{ j }
the same set of real numbers z i with i = j ((i ; j ) ∈ [r ; s])

Z (r→s) { j } = {z i } s i=r \ {z j } = {z r , z r+1 , . . . , z j -1 , z j +1 , . . . , z s-1 , z s }. (6)
Let P a (Z (r→s) ) be the sum of all the different products of a elements of Z (r→s) . P a (Z (r→s) ) is thus a sum of sr + 1 a terms, each of them being a product of a elements of Z (r→s) . For instance

P 1 Z (1→3) = z 1 + z 2 + z 3 P 2 Z (1→3) = z 1 z 2 + z 1 z 3 + z 2 z 3 P 3 Z (1→3) = z 1 z 2 z 3 and P 2 Z (1→4) {3} = z 1 z 2 + z 1 z 4 + z 2 z 4 .
Finally, we denote

P 0 Z (r→s) = 1. (7)

B. Expression of the n p C Coefficients as Functions of the Cut-Off Angular Frequencies

In this section, an expression of the 2N n p coefficients used in ( 3) is obtained as functions of the frequency transformation input parameters, i.e., the cut-off angular frequencies of the N bands. We use here the fact that the N high cut-off angular frequencies ω H i and the opposite of the low cut-off ones ω L i , with i ∈ [1; N], are the zeros of U (N ) (ω). We thus obtain a first system of 2N equations (one for each cut-off angular frequency) with 2N unknowns (n p , p ∈ [0; 2N -1]).

According to the expressions presented in Section II-A, we denote

L (1→N ) = {ω L i } and H (1→N ) = {ω H i }, i ∈ [1; N].

Solving this first system leads to

n p = p r=0 (-1) N -r P N -r H (1→N ) P N +r-p L (1→N ) for p ∈ [0; N -1] = 2N r= p (-1) 2N -r P 2N -r H (1→N ) P r-p L (1→N ) for p ∈ [N; 2N -1]. (8)

C. Expression of the n p Coefficients as Functions of the Resonant Angular Frequencies and the Susceptance Slope Parameters

We now express the 2N n p coefficients as functions of the frequency transformation output parameters, i.e., the resonant angular frequency, ω 0 , and the slope parameter, b 0 , of the wide bandpass resonator, and the parameters ω k and b k (k ∈ [1; N -1]). We recall that these parameters are the resonant angular frequencies and the slope parameters of the stopband resonators, respectively. This is based on the introduction of the expression of T (N ) (ω) from ( 1) in [START_REF] Uhm | A triple-passband waveguide filter with dual-mode resonators for ka band satellite applications[END_REF], which allowed us to obtain the expression of U (N ) (ω) given in [START_REF] Xu | Compact microstrip dual-/tri-/quad-band bandpass filter using open stubs loaded shorted stepped-impedance resonator[END_REF]. We thus obtain a second system of 2N equations giving each n p coefficient as a function of the output parameters.

Here, we denote W (0→N -1) = {ω 2 k } the set of all the resonant angular frequency squares and W (1→N -1) = {ω 2 k } the set of the resonant angular frequency squares of the stopband resonators only. We distinguish three cases for p = 0, p odd, and p even, p ∈ [0; 2N -1].

1) For p = 0, we have

n 0 = (-1) N P N W (0→N -1) = (-1) N N -1 r=0 ω 2 r . (9)
2) For p odd, we denote p = 2q -1, q ∈ [1; N]. We then have

n p = n 2q-1 = (-1) N -q+1 ω 0 b 0 P N -q W (1→N -1) . ( 10
)
3) For p even, we denote p = 2q, q ∈ [1; N -1], and then we obtain

n p = n 2q = (-1) N -q P N -q W (0→N -1) + ω 0 b 0 N -1 k=1 ω k b k P N -q-1 W (1→N -1) {k} . ( 11
)

D. Expression of the Resonance Angular Frequencies and Susceptance Slope Parameters

Equations ( 9) to [START_REF] Zhou | Resonance characteristics of substrate-integrated rectangular cavity and their applications to dualband and wide-stopband bandpass filters design[END_REF] give an expression of the 2N n p as functions of the output parameters of the frequency transformation technique (i.e., ω k and b k , k ∈ [0; N -1]). In a practical way, we have to reverse this equation system in order to express the output parameters as functions of the 2N n p coefficients, these latter being calculated from the input parameters (ω L i and ω H i (i ∈ [1; N])) using [START_REF] Zhu | Triple-band cavity bandpass filters[END_REF].

First, using the expressions of n 0 , n 1 , and n 2N -1 and whatever the number of bands N, we can easily demonstrate that the resonant frequency ω 0 and the slope parameter b 0 of the wide bandpass filter can be written as

ω 0 = - n 0 n 2N -1 n 1 (12) b 0 = - n 0 n 1 n 2N -1 . ( 13
)
Using [START_REF] Chen | Dual-band and triple-band substrate integrated waveguide filters with chebyshev and quasi-elliptic responses[END_REF] for all the odd value of p, we can also demonstrate (see Appendix A) that the resonant angular frequencies of the band-stop resonators (ω k , k ∈ [1; N -1])) are the N -1 positive solutions of ( 14)

N -1 r=0 n 2r+1 ω 2r k = 0. ( 14
)
Once the resonant angular frequencies of the band-stop resonators are determined using [START_REF] Mokhtaari | Couplingmatrix design of dual and triple passband filters[END_REF], the remaining unknowns are the slope parameter coefficients b k , k ∈ [1; N -1] in [START_REF] Zhou | Resonance characteristics of substrate-integrated rectangular cavity and their applications to dualband and wide-stopband bandpass filters design[END_REF]. This equation can also be rewritten as follows:

N -1 k=1 ω k b k P N -q-1 W (1→N -1) {k} = P N -q W (0→N -1) -(-1) N -q n 2q n 2N -1 ( 15
)
for q ∈ [1; N -1]. Using [START_REF] Amari | Synthesis of cross-coupled resonator filters using an analytical gradient-based optimization technique[END_REF] for each value of q, we obtain an N -1 linear equation system with (N -1) unknowns (i.e., each b k or more precisely 1/b k ). This system can be written as follows:

X N -1,N -1 B N -1 = A N -1 (16)
where X N -1,N -1 is the (N -1 × N -1) matrix of x q,k coefficients with

x q,k = ω k P N -q-1 W (1→N -1) {k} . ( 17
)
B N -1 is the vector of the N -1 unknowns

B N -1 = 1 b 1 , 1 b 2 , . . . , 1 b N -1 t (18)
and A N -1 is the vector of already known coefficients a (N )

q A N -1 = a 1 , a 2 , . . . , a N -1 t (19)
with

a q = P N -q W (0→N -1) -(-1) N -q n 2q n 2N -1 . ( 20
)
The slope parameters are then determined by inverting the matrix

X N -1,N -1 B N -1 = X -1 N -1,N -1 A N -1 . ( 21
)
By solving the above matrices, the slope parameters can be written as

b 1 = n 1 n 2 3 ω 1 n 1 n 2 n 3 -n 0 n 2 3 -n 2 1 for N = 2 (22) b k = ω k N -1 j =1, j =k ω 2 k -ω 2 j N -1 j =1 Y j ω 2 j -2 k for N 3 ( 23
)
where k ∈ [1; N -1] and

Y j = n 2 j - n 0 n 2 j +1 n 1 - n 2 j -1 n 2N -1 n 2N -1 . ( 24
)
One should note that this synthesis technique leads to a unique solution for a given specifications. So, there is theoretically only one transmission zero between two consecutive bands whatever the order of the designed filter.

III. IMPLEMENTATION WITH LC ELEMENTS

Starting from a classical ladder structure composed of series inductances and shunt capacitances, a low-pass prototype filter can be modified to use only parallel capacitances separated by J admittance inverters [START_REF] Matthaei | Microwave Filters, Impedance-Matching Networks, and Coupling Structures[END_REF]. Using the generalized frequency transformation technique presented in Section II, the i th capacitance, C L P i , of such a low-pass prototype can be transformed into a multiband bandpass resonator as shown in Fig. 2. The multiband bandpass resonator then obtained consists in the parallel association of one bandpass resonator (L (i) 0 ; C (i) 0 ) and N -1 band-stop ones (L (i) k ;

C (i) k ), k ∈ [1; N -1]
whose values are given by the following equations:

L (i) 0 = 1 b 0 C L P i ω 0 (25)
C (i) 0 = 1
L (i) 0 ω 2 0 ( 26
)
L (i) k = b k C L P i ω k ( 27
)
C (i) k = 1
L (i) k ω 2 k . ( 28
)
A M-order MBPF is then composed of M MBPRs separated by J admittance inverters as shown in Fig. 3. The J i,i+1 inverters are here defined in a classical way taking into account only the bandpass part of each MBPR [START_REF] Matthaei | Microwave Filters, Impedance-Matching Networks, and Coupling Structures[END_REF] J 01 = G S β (1)

0 b 0 g 0 g 1 ( 29
)
J i,i+1 = 1 b 0 β (i) 0 β (i+1) 0 g i g i+1 ( 30
)
J M,M+1 = G L β (M) 0 b 0 g M g M+1 ( 31
)
where the g i coefficients are the low-pass prototype parameters, G S and G L are the source impedance, and

β (i) 0 = ω 0 C (i) 0 .
To use only bandpass resonators, all the stopband LC-series elements can be transformed in LC-parallel ones using another set of J-inverters. A MBPR then consists only in LC-parallel resonators (see Fig. 4), the main one (L (i) 0 , C (i) 0 ) being separated from the others (L

(i) k , C (i) k ), k ∈ [1; N -1]
, and i ∈ [1; M], by new J-inverters defined as follows:

J (i) k = β (i) 0 β (i) k b 0 b k ( 32
)
with

β (i) k = ω k C (i)
k . We should mention here that the values of capacitances both for bandpass and band-stop resonators can be chosen arbitrarily. However, in this article, considering the technological constraints, we have chosen the optimal capacitances values for band-stop and bandpass resonators with a view to the implementation in microstrip technology.

IV. SYNTHESIS AND EXPERIMENTAL VALIDATION OF THREE TEST MBPFS

For validating the frequency transformation method and the implementation proposed in the previous sections, three examples of MBPFs have been synthesized and experimentally validated with 3, 4, and 5 bands, respectively. In the first case (tri-band filter), all the output parameter expressions obtained from [START_REF] Cameron | Advanced coupling matrix synthesis techniques for microwave filters[END_REF] to [START_REF] Chen | Design of a compact microstrip quint-band filter based on the tri-mode stub-loaded stepped-impedance resonators[END_REF] are presented directly here as an example. In the other two cases, for the sake of clarity, these expressions are presented in Appendix (see Sections B and C).

The three test filters are of order 3 with a Chebyshev approximation and 20-dB bandpass return loss (RL). There are implemented in a low-cost microstrip technology using a RO4003C Rogers substrate (dielectric constant: ε r = 3.55, height: h = 0.508 mm, dissipation factor: tan δ = 0.0027) with copper metallization (metal thickness: t = 17.5 μm, conductivity: σ = 5.8 × 10 7 S.m -1 ). All MBPRs use a star-like structure according to Fig. 4 topology connecting N stubs at the same point allowing the implementation of such multiband filters without a significant increase of the circuit size compared to a single-band one. All prototypes were simulated and optimized using ADS software from Keysight Technologies © . The fabrication of the circuits was made by laser engraving using a LPKF Protolaser U4.

A. Triple-Band Bandpass Filters

To synthesize a triple-band bandpass filter using the frequency transformation method proposed here, one needs to take N = 3 in [START_REF] Cameron | Advanced coupling matrix synthesis techniques for microwave filters[END_REF] to [START_REF] Chen | Design of a compact microstrip quint-band filter based on the tri-mode stub-loaded stepped-impedance resonators[END_REF]. The resonance angular frequencies and slope parameters are then given by [START_REF] Lu | Five-pole parallelcoupled microstrip cascade quadruplet filters for high selectivity and flat group delay[END_REF] and

ω 0 = - n 0 n 5 n 1 ( 33
)
ω 1 = -n 3 -n 2 3 -4n 5 n 1 2n 5 ( 34
)
b 0 = -n 0 n 1 n 5 ( 36
) b 1 = ω 1 (ω 2 1 -ω 2 2 ) Y 1 + Y 2 ω 2 1 (37) b 2 = ω 2 (ω 2 2 -ω 2 1 ) Y 1 + Y 2 ω 2 2 (38)
with

Y 1 = n 2 - n 0 n 3 n 1 - n 1 n 5 n 5 (39) Y 2 = n 4 - n 0 n 5 n 1 - n 3 n 5 n 5 ( 40
)
where the n p ( p ∈ [0; 5]) are determined using (8) according to the specifications of the designed triple-band bandpass filter given in terms of low and high cut-off angular frequencies. The latter are then transposed in terms of resonant angular frequencies and slope parameters. As an example, the arbitrary following specifications are proposed. 1) Bandpass 1: 2.00-2.10 GHz (BW: 100 MHz).

2) Bandpass 2: 2.45-2.65 GHz (BW: 200 MHz).

3) Bandpass 3: 2.95-3.20 GHz (BW: 250 MHz). The resulting resonant frequencies and slope parameters are

f 0 = ω 0 /2π = 2.604 GHz, f 1 = ω 1 /2π = 2.191 GHz, f 2 = ω 2 /2π = 2.812 GHz, b 0 = 4.735, b 1 = 4.
277, and b 2 = 5.818. A tri-band third-order bandpass filter is first implemented in an ideal LC structure (see Fig. 3), each resonator following the model of Fig. 4. All the resonator components and J-inverters, calculated from (25) to [START_REF] Matthaei | Microwave Filters, Impedance-Matching Networks, and Coupling Structures[END_REF], are summarized in Table I. Note that the three tri-band resonators are identical in order to simplify the microstrip implementation in the next step. This is possible by fixing the three C L P i components at the same value using an admittance scale factor when introducing the J-inverters in the low-pass prototype [START_REF] Matthaei | Microwave Filters, Impedance-Matching Networks, and Coupling Structures[END_REF]. Fig. 5 shows the frequency response of this filter. The three simulated bands are in perfect agreement with the specifications.

The filter was then implemented in microstrip technology with the substrate characteristics given in the introduction of this section. Fig. 6 presents the layout of the fabricated IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES Fig. 5. Ideal frequency response of the specified triple-band bandpass filter. Each star is made of a tri-band resonator together with its previous and next inverters. A tri-band resonator consists in one bandpass element, i.e., a short-circuit stub of length λ/4 at ω 0 , and two stopband ones consisting of a short-circuit stub of length λ/4 at ω 1 and ω 2 , respectively, and separated from the connection point by another quarter-wavelength line at the same respective frequency having the J-inverter role. Compared with the method described in [START_REF] Ma | A design method of multimode multiband bandpass filters[END_REF], our proposal allows a direct determination of the values of circuit elements from the transformation formulas. For implementation, as mentioned in Section III, we have chosen suitable values to avoid too narrow (less than 0.1 mm) or too wide (more than 4 mm) linewidth. All lengths and widths of the resonators and the inverters are summarized in Table II. The tri-band filter is 72.1 × 68.5 mm 2 without taking account for the two 50 access lines.

Fig. 8 presents the EM-simulated and measured frequency responses of the fabricated tri-band bandpass filter. The correlation between both responses is very good over the entire frequency band. The measured RL are 12.3, 14.9, and 14.1 dB from the first to the last bandpass and the measured insertion loss (IL) are 1.98, 2.17, and 2.01 dB, respectively. The isolation between the three bandpass are better than 58.1 and 57.6 dB.

B. Quad-Band Bandpass Filters

In the same way as for the tri-band case, the synthesis of a quad-band bandpass filter using this frequency transformation .

TABLE III COMPONENT VALUES OF THE IDEAL LC QUAD-BAND

THIRD-ORDER BANDPASS FILTER method starts with the determination of the expression of the resonant angular frequencies and the associated slope parameters as functions of the low and high cut-off frequencies. To do so, one takes N = 4 in ( 12) to [START_REF] Chen | Design of a compact microstrip quint-band filter based on the tri-mode stub-loaded stepped-impedance resonators[END_REF]. The resulting equations are given in Appendix B for a sake of clarity. Arbitrary specifications are also proposed here as an example to design, fabricate, and measure a third-order quad-band bandpass filter. Note that a first example of quad-band bandpass filter was already presented in [START_REF] Wu | Direct synthesis of quadband band-pass filter by frequency transformation methods[END_REF] with four equal absolute bandwidths. The equations presented in [START_REF] Wu | Direct synthesis of quadband band-pass filter by frequency transformation methods[END_REF] and here are slightly different in their writing (especially the n p coefficients) but completely equivalent. This is only because the generalization to any number of bands was not established at the time [START_REF] Wu | Direct synthesis of quadband band-pass filter by frequency transformation methods[END_REF] was written. The specifications proposed in this article show different bandwidths for each band are as follows:

1) Bandpass 1: 2.00-2.08 GHz (BW: 80 MHz).

2) Bandpass 2: 2.30-2.40 GHz (BW: 100 MHz).

3) Bandpass 3: 2.65-2.80 GHz (BW: 150 MHz). 4) Bandpass 4: 3.04-3.20 GHz (BW: 160 MHz).

The resulting resonant frequencies and slope parameters are III. As shown in Fig. 9, the four obtained bands are, as previously, in perfect agreement with the specifications. Fig. 10 presents the layout of the fabricated quad-band bandpass filter and a picture is also shown in Fig. 7. Again, three stars separated by J-inverters are obtained, each star including six branches this time: one bandpass element, three stopband ones, and the previous and next J-inverters. All lengths and widths of the resonators and the inverters are summarized in Table IV. The quad-band filter is 97.7 × 68.9 mm 2 without taking into account the two 50 access lines.

f 0 = ω 0 /2π = 2.573 GHz, f 1 = ω 1 /2π = 2.146 GHz, f 2 = ω 2 /2π = 2.501 GHz, f 3 = ω 3 /2π = 2.
Fig. 11 presents the EM-simulated and measured frequency responses of the fabricated quad-band bandpass filter. As in the tri-band case, both responses are in near perfect agreement. The measured RL are 12.1, 14.2, 14.2, and 14.3 dB from the first to the fourth bandpass and the measured IL are 2.90, 2.94, 2.95, and 2.65 dB, respectively. Very good isolation are here again achieved between all the bandpass.

C. Quint-Band Bandpass Filters

In the same way as for the two previous cases, one can derive the equations of the resonant angular frequencies and the slope parameters for the quint-band bandpass filters as functions of the low and high cut-off angular frequencies by taking N = 5 in [START_REF] Cameron | Advanced coupling matrix synthesis techniques for microwave filters[END_REF] to [START_REF] Chen | Design of a compact microstrip quint-band filter based on the tri-mode stub-loaded stepped-impedance resonators[END_REF]. The resulting equations are given in Appendix C for the sake of clarity.

The arbitrary specifications proposed here to design an example of quint-band bandpass filter are as follows: IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES Fig. 11. EM-simulated (dotted lines) and measured (solid lines) S-parameters of the quad-band bandpass filter.

TABLE V

COMPONENT VALUES OF THE IDEAL LC QUINT-BAND THIRD-ORDER BANDPASS FILTER 1) Bandpass 1: 2.00-2.10 GHz (BW: 100 MHz).

2) Bandpass 2: 2.30-2.38 GHz (BW: 80 MHz).

3) Bandpass 3: 2.60-2.69 GHz (BW: 90 MHz). 4) Bandpass 4: 2.88-2.98 GHz (BW: 100 MHz). 5) Bandpass 5: 3.20-3.30 GHz (BW: 100 MHz).

They leads to the following resonant frequencies and slope parameters: Fig. 14 presents the simulated and measured frequency responses of the fabricated quint-band bandpass filter. Here again, simulations and measurements are in a good correlation, except a slight frequency shift in the higher transmission zeros. This can be explained by the low-cost technology process, which implies in particular that the vias were made by hand. The measured RL are 24.8, 17.5, 13.9, 10.9, and 10.8 dB from the first to the fifth band and the measured IL are 3.01, 4.95, 4.98, 4.75, and 3.45 dB, respectively. Very good isolation are again achieved between all the bandpass.

f 0 = ω 0 /2π = 2.548 GHz, f 1 = ω 1 /2π = 2.178 GHz, f 2 = ω 2 /2π = 2.480 GHz, f 3 = ω 3 /2π = 2.797 GHz, f 4 = ω 4 /2π = 3.

V. DISCUSSION

Using our proposed transform function, we always assume that lower cut-off angular frequencies ω L i and the upper cut-off angular frequencies ω H i (i ∈ [1; N]) are mapped to -1 and +1 in the normalized symmetrical domain, respectively. This features that only transfer functions that are symmetric in the low-pass domain can be used in this method. Chebyshev, Butterworth, and elliptic filters have all symmetrical structures where the symmetry is defined as g 0 = g n+1 , g 1 = g n , g 2 = g n-1 , etc., in the low-pass domain. There are all therefore compatible with our proposed synthesis method. For Gaussian low-pass prototype (flat-group-delay), filters with order n = 1 are identical to the first-order Butterworth low-pass prototype and can also be synthesized with this method. However, Gaussian filters with order n ≥ 2 are structurally asymmetrical as discussed in [START_REF] Hong | Microstrip Filters for RF/Microwave Applications[END_REF]. It is therefore impossible to synthesize multiband filter with this method. Nevertheless, flat group-delay can also be obtained with symmetrical structures using cross-coupling as presented in [START_REF] Manchec | Cross-coupled microstrip dual behavior resonator (DBR) filter[END_REF] and [START_REF] Lu | Five-pole parallelcoupled microstrip cascade quadruplet filters for high selectivity and flat group delay[END_REF].

As an example, we present a multiband elliptic filter with quint-band response. We consider a fourth-order filter with an elliptic approximation (23-dB RL with normalized TZs at ±2.2 j ). The specifications for this case are as follows:

1) Bandpass 1: 3.00-3.10 GHz (BW: 100 MHz).

2) Bandpass 2: 3.30-3.40 GHz (BW: 100 MHz).

3) Bandpass 3: 3.60-3.70 GHz (BW: 100 MHz). 4) Bandpass 4: 3.90-4.00 GHz (BW: 100 MHz). 5) Bandpass 5: 4.20-4.30 GHz (BW: 100 MHz). The resonating frequencies and slope parameters can be easily obtained by the generalized formulas in Section II. Each sub-band is a bandwidth-scaled and frequency-translated copy of the original low-pass prototype (i.e., each bandpass reaches 23-dB in-band RL and owns two TZs on both sides, as shown in Fig. 15). This example also shows that it is possible to introduce several TZs between each bandpass with this method using elliptic filters.

VI. CONCLUSION

In this article, a new generalized synthesis method for designing MBPF has been proposed. This method is relatively simple and leads to a completely analytical equation system. It uses slope parameters offering flexibility to implement the designed filters with noncoupled structures, using them directly, or with coupled ones calculating the coupling coefficients from these parameters [START_REF] Cameron | Microwave Filters for Communication Systems: Fundamentals, Design, and Applications[END_REF]- [START_REF] Cameron | General coupling matrix synthesis methods for chebyshev filtering functions[END_REF]. Note that from six bands and beyond a numerical resolution of (14) will be necessary because of the degree of this equation.

Three third-order MBPFs with a different number of bands (3, 4, and 5, respectively) were implemented in microstrip technologies to illustrate the efficiency of the synthesis method. A star-like structure has been chosen to realize each multiband bandpass resonator of the filters. This enabled to keep the filters compact and the implementation simple. Very good agreement has been achieved between theoretical responses and measurements. This implementation solution is in fact mainly limited by the maximum bandwidth that can be obtained with the bandpass elements. Increasing the number of bands would lead to narrow bandwidth for each band and sharp rejection between them. Such a rejection could be difficult to realize because it implies that the impedance of J-inverters separating the bandpass part to the stopband ones in each multiband resonator will increase potentially reaching technological limits. Nevertheless, such an implementation is just an example, which has the interests to be low cost and easy to do, and any other topology or technology allowing wider bandwidths can be used.

APPENDIX A EXPRESSION GOVERNING THE RESONANT ANGULAR FREQUENCIES OF THE BAND-STOP RESONATORS

As presented in Section II, the resonant angular frequencies of the stopband parts of an N-band bandpass resonator (MBPR) are the positive solutions of [START_REF] Mokhtaari | Couplingmatrix design of dual and triple passband filters[END_REF]. IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

In the case of an N-band bandpass filters, the coefficients n p of the U (N ) (ω) function [see [START_REF] Xu | Compact microstrip dual-/tri-/quad-band bandpass filter using open stubs loaded shorted stepped-impedance resonator[END_REF]] are given by [START_REF] Chen | Dual-band and triple-band substrate integrated waveguide filters with chebyshev and quasi-elliptic responses[END_REF] in the case of an odd value of p ( p = 2q -1, q ∈ [1; N]).

For q = N and so p = 2N -1, this equation leads to

n 2N -1 = - ω 0 b 0 (41)
knowing that P 0 W (1→N -1) = 1 as defined in Section II-A. So, [START_REF] Chen | Dual-band and triple-band substrate integrated waveguide filters with chebyshev and quasi-elliptic responses[END_REF] becomes

n p = n 2q-1 = (-1) N -q n 2N -1 P N -q W (1→N -1) . ( 42
)
Applying (42) with q = N -1 (p = 2N -3) leads to

n 2N -3 = -n 2N -1 P 1 W (1→N -1) . ( 43
)
Considering any angular frequency of the stopband parts ω k , k ∈ [1; N -1], this expression can be written as follows:

n 2N -3 = -n 2N -1 P 1 W (1→N -1) {k} + ω 2 k (44) which implies that n 2N -1 P 1 W (1→N -1) {k} = -n 2N -3 + n 2N -1 ω 2 k . ( 45
)
In the same way, for q = N -2 (p = 2N -5), (42) becomes

n 2N -5 = n 2N -1 P 2 W (1→N -1) = n 2N -1 P 2 W (1→N -1) {k} + ω 2 k P 1 W (1→N -1) {k} = n 2N -1 P 2 W (1→N -1) {k} -n 2N -3 ω 2 k + n 2N -1 ω 4 k (46) which leads to n 2N -1 P 2 W (1→N -1) {k} = n 2N -5 + n 2N -3 ω 2 k + n 2N -1 ω 4 k . ( 47
) Continuing thus for the other values of q from N -1 to 1, one can show that n 2N -1 P N -q W (1→N -1) {k} = N -q r=0 n 2(q+r)-1 ω 2r k .

(48)

Let's focus now on the case q = 1 (p = 1). First, using (42), we have

n 1 = (-1) N -1 n 2N -1 P N -1 W (1→N -1) = (-1) N -1 ω 2 k n 2N -1 P N -2 W (1→N -1) {k} . ( 49
)
Using (48), this expression becomes

n 1 = (-1) N -1 ω 2 k 1 (-1) N -2 N -2
b 1 = ω 1 (ω 2 1 -ω 2 2 )(ω 2 1 -ω 2 3 ) Y 3 ω 4 1 + Y 2 ω 2 1 + Y 1 (57) b 2 = ω 2 (ω

Fig. 1 .

 1 Fig. 1. From a low-pass prototype to an N -band bandpass filter.

  is the transformation function, N the number of bands of the MBPFs, and ω the de-normalized angular frequency (associated with the MBPFs), ω 0 and b 0 are the resonant angular frequency and the susceptance slope parameter of the wide bandpass resonator, respectively, ω k and b k (k ∈ [1; N -1]) are the ones of the stopband resonators. The goal of the synthesis process is to link the 2N output parameters of the transformation function (the resonant angular frequencies ω k and the slope parameters b k , k ∈ [1; N -1]) to the input ones, i.e., the low and high cut-off angular frequencies of the N bands (ω L i and ω H i , i ∈ [1; N]) as defined in Fig. 1.

Fig. 2 .

 2 Fig. 2. Transformation of a low-pass prototype capacitance into a multiband bandpass resonator (MBPR).

Fig. 3 .

 3 Fig. 3. Schematic representation of the proposed MBPF.

Fig. 4 .

 4 Fig. 4. MBPRs with parallel LC resonators.

Fig. 6 .

 6 Fig. 6. Layout of the fabricated tri-band bandpass filter. tri-band bandpass filter and Fig. 7 a picture of the fabricated filter together with the quad-and quint-band ones. It consists in three-branched stars, allowing to keep the filter compact.Each star is made of a tri-band resonator together with its previous and next inverters. A tri-band resonator consists in one bandpass element, i.e., a short-circuit stub of length λ/4 at ω 0 , and two stopband ones consisting of a short-circuit stub of length λ/4 at ω 1 and ω 2 , respectively, and separated from the connection point by another quarter-wavelength line at the same respective frequency having the J-inverter role. Compared with the method described in[START_REF] Ma | A design method of multimode multiband bandpass filters[END_REF], our proposal allows a direct determination of the values of circuit elements from the transformation formulas. For implementation, as mentioned in Section III, we have chosen suitable values to avoid too narrow (less than 0.1 mm) or too wide (more than 4 mm) linewidth. All lengths and widths of the resonators and the inverters are summarized in TableII. The tri-band filter is 72.1 × 68.5 mm 2 without taking account for the two 50 access lines.Fig.8presents the EM-simulated and measured frequency responses of the fabricated tri-band bandpass filter. The correlation between both responses is very good over the entire frequency band. The measured RL are 12.3, 14.9, and 14.1 dB from the first to the last bandpass and the measured

Fig. 7 .

 7 Fig. 7. Photograph of the three fabricated filters: tri-band (top), quad-band (bottom left), and quint-band (bottom right).

Fig. 8 .

 8 Fig. 8. EM-simulated (dotted lines) and measured (solid lines) S-parameters of the triple-band bandpass filter.

  951 GHz, b 0 = 5.251, b 1 = 6.381, b 2 = 5.697, and b 3 = 8.066. An ideal LC structure implementation is then made as in the tri-band case and all the resonators are again fixed identical by taking all the C L P i equal. All the resonator component values and J-inverter ones are summarized in Table

Fig. 9 .

 9 Fig. 9. Ideal frequency response of the specified quad-band bandpass filter.

Fig. 10 .

 10 Fig. 10. Layout of the fabricated quad-band bandpass filter.

  137 GHz, b 0 = 5.421, b 1 = 6.075, b 2 = 5.735, b 3 = 8.323, and b 4 = 10.050. An ideal LC quint-band bandpass filter is also implemented to validate the method. The resonator component and J-inverter values are summarized in Table V and the frequency responses shown in Fig. 12. Once again, all the bands are in perfect agreement with the specifications. Fig. 13 presents the layout of the fabricated quint-band bandpass filter and a picture is also shown in Fig. 7. Three stars separated by J-inverters are obtained, each star including seven branches with an additional stopband one compared to the quad-band case. All lengths and widths of the resonators and the inverters are summarized in Table VI. The quint-band filter is 91.5 × 77.3 mm 2 without taking into account the two 50 access lines.

Fig. 12 .

 12 Fig. 12. Ideal frequency response of the specified quint-band bandpass filter.

Fig. 13 .

 13 Fig. 13. Layout of the fabricated quint-band bandpass filter.

Fig. 14 .

 14 Fig. 14. EM-simulated (dotted lines) and measured (solid lines) S-parameters of the quint-band bandpass filter.

Fig. 15 .

 15 Fig. 15. Ideal frequency response of the specified quint-band elliptic bandpass filter.

  obtain the equation governing the resonant angular frequencies of the band-stop resonatorsN -1 r=0 n 2r+1 ω 2r k = 0. (51)This equation has 2(N -1) solutions, one half being positive and the other half negative. Of course, in this case, only the positive solutions are retained.APPENDIX B RESONANT ANGULAR FREQUENCIES AND SLOPE PARAMETERS FOR A QUAD-BAND BANDPASS FILTER

TABLE I COMPONENT

 I VALUES OF THE IDEAL LC TRI-BAND THIRD-ORDER BANDPASS FILTER

	ω 2 =	-n 3 + n 2 3 -4n 5 n 1 2n 5

TABLE II MICROSTRIP

 II

TRI-BAND THIRD-ORDER BANDPASS FILTER: LENGTH AND WIDTH DIMENSIONS (IN MM)

TABLE IV MICROSTRIP

 IV

QUAD-BAND THIRD-ORDER BANDPASS FILTER: LENGTH AND WIDTH DIMENSIONS (IN MM)

TABLE VI MICROSTRIP

 VI QUINT-BAND THIRD-ORDER BANDPASS FILTER: LENGTH AND WIDTH DIMENSIONS (IN MM)

ACKNOWLEDGMENT

The authors would like to thank Christophe Guitton (University of Rennes 1, Rennes, France) and Gilles Picoult (INSA Rennes, Rennes) for the fabrication of the printed circuit board.

This work was supported in part by the European Union through the European Regional Development Fund (ERDF) and in part by the French Region of Brittany, Ministry of Higher Education and Research, Rennes Métropole and Conseil Départemental 35, through the Contrat Plan État Région (CPER) Project SOPHIE/STIC & Ondes.

Yi Wu (Student Member, IEEE) was born in Tai'an, Shandong, China. He is currently pursuing the Ph.D. degree at the National Institute of Applied Sciences (INSA Rennes), Rennes, France.

His research interests include the design of filters for microwave applications and the optical instruments for space applications.


Adel Tayeb

Di Cesare

Jean-Benoit Le

Identification of Hyper-viscoelastic models from one heterogeneous test for elastomers

Identification of Hyper-viscoelastic models from one heterogeneous test for elastomers

A. Tayeb 1a , N. Di Cesare 2 and J.B. Le Cam 1

1 Univ Rennes, CNRS, IPR (Institut de Physique de Rennes) -UMR 6251, Rennes, France, 2 Univ. Bretagne-Sud, UMR CNRS 6027, IRDL, F-56100, Lorient, France.

a adel.tayeb@univ-rennes1.fr

Abstract. In this work, the identification of Hyper-viscoelastic models of elastomers is performed from only one heterogeneous relaxation experiment performed with a cruciform specimen. The identification procedure is based on the so-called Finite Element Model Updating (FEMU) technique. Even though the experiment induces heterogeneous kinematic fields, only the measured and predicted global force at the specimen edges has been used in the residual to be minimized. The minimization problem has been solved by using of the Inverse-PageRank-Particle Swarm Optimization (I-PR-PSO) algorithm. It has been found that such identification procedure coupled with the heterogeneous relaxation experiment were sufficient to identify both hyperelastic and viscoelastic parameters. The capability of the proposed methodology has been assessed by comparing both kinematic fields and global forces obtained experimentally and by using of the identified parameters. It has been found that the identified parameters predicted well the experiments, even the ones not used in the identification procedure.

Introduction. Thanks to their high elasticity, high damping and high elongation at failure, elastomers are widely used in several engineering applications such as automotive, aerospace and civil engineering applications.

The mechanical behaviour of such material is modelled generally within the framework of hyperelasticity. The time-dependent character of the behaviour is modelled by associating a viscoelastic part to the instantaneous hyperelastic response; see [START_REF] Tayeb | On the nonlinear viscoelastic behavior of rubber-like materials: Constitutive description and identification[END_REF] and references therein. This viscoelastic part is represented in terms of a Prony series. Classically, the hyperelastic parameters are identified from homogeneous experiments namely equibiaxial tensile, pure shear and simple extension. In the recent years, a new method has been applied to perform the identification of hyperelastic models. It is based on performing one heterogeneous experiment, which induces several loading cases with a large strain distribution for each loading case. The identification of the hyperelastic parameters are identified either by FEMU [START_REF] Tayeb | Sensitivity based virtual fields for identifying hyperelastic constitutive parameters[END_REF] or Virtual Fields Method (VFM) [START_REF] Bastos | Inverse identification of hyperelastic parameters by metaheuristic optimization algorithm[END_REF].

On the other hand, the viscoelastic parameters are identified by using stress relaxation experiments at low strain levels following the assumption that nonlinear viscoelastic models should coincide with the linear viscoelastic model of Boltzmann in the small strain domain. In the present work both hyperelastic and viscoelastic parameters are identified from only one heterogeneous test of equibiaxial tensile of a cruciform elastomer specimen. The identification is performed with the FEMU method by using the I-PR-PSO optimization.

Methods. The experimental setup is reported in Fig. 1. It is composed by a biaxial tensile machine, a CCD optical camera, the cruciform elastomeric specimen and LED lighting system to ensure homogeneous distribution of cold lighting during the test. An equibiaxial displacement of 70 𝑚𝑚 is applied to the four branches of the cruciform specimen at the same loading rate of 500 𝑚𝑚 𝑚𝑖𝑛 ⁄ . This loading was followed by a 60 𝑠 relaxation in order to highlight the viscoelasticity effects. During the experiment both the global forces and kinematic fields were measured. The latter were measured using DIC with 7D © Software. The FEMU method used in this work is based on the minimization with respect to the constitutive parameters of the global force between the experiment and numerical simulation of the experiment. During the identification process, the constitutive parameters are modified by the PSO following the objective function formulated in terms of the difference of the global forces. The objective function used in this work reads as follow:

∑ ( 𝐹 𝑒𝑥𝑝,𝑖 -𝐹 𝑛𝑢𝑚,𝑖 𝐹 𝑒𝑥𝑝,𝑖 ) 2 𝑛 𝑖=1 , (1)
where the subscript (𝑒𝑥𝑝) and (𝑛𝑢𝑚) corresponds to experimental and numerical data, respectively. At the end of the optimization process the retained parameters are the ones minimizing the objective function of Eq. 1.

Figure 1: experimental set-up

Results. The results of the identification procedure are the hyeperelastic and viscoelastic constitutive parameters. The accuracy of the identified parameters to predict the experiments is assessed through the comparison of the global response force and kinematic fields. Fig. 2 shows these results. The relative error for the force is inferior to 2 % and under 1 %for the kinematic fields. These results have been obtained with simulated data and will be applied to real experiment data. It can be concluded that this identification procedure enable us to identify both hyperelastic and viscoelastic parameters with only one experiment and without full field measurement. Conclusion. The identification of the hyperelastic and viscoelastic parameters from only one heterogeneous relaxation experiment is performed by using only the general force response. The method has been applied to both generated and experimental data and showed good results in terms of the ability to predict both the general force response and kinematic fields.

Figure 2 :

 2 Figure 2: comparison of the obtained results with the reference data: (a) global force, (b) relative error of the global force, (c) relative error of the first in-plane principle stretch and (d) relative error of the second in-plane principle stretch


J B Le Cam
email: ajean-benoit.lecam@univ-rennes1.fr

Revisiting the deformation mechanisms in rubbers from surface calorimetry-based energy characterization

Complete energy balances carried out during cyclic deformation of rubbers (especially filled rubbers) show that viscosity is not systematically the preponderant contribution to the hysteresis loop: the mechanical energy brought to the material is not entirely dissipated into heat but can be predominantly used by the material to change its microstructure. A ratio in terms of energy has been defined to fully characterize the energetic behaviour of rubbers and opens new way of interpretation of the resistance of rubbers through their ability to store mechanical energy.

Introduction

Elastomers are widely used in many industries such as automotive, nuclear, or civil engineering for their high deformability, high damping and resistance to fatigue. Such properties are generally obtained by adding fillers in the rubber matrix, which induces or increases the hysteresis loop in the mechanical response. Classically, the mechanical energy involved in the hysteresis loop is assumed to be mainly dissipated into heat. Nevertheless, several observations question this assumption: (i) the mechanical response of some elastomers exhibits a hysteresis loop only when strain-induced crystallization (SIC) occurs. In this case, no self-heating accompanies the mechanical cycles [START_REF] Le Cam | Energy storage due to strain-induced crystallization in natural rubber: the physical origin of the mechanical hysteresis[END_REF] (ii) the mechanical hysteresis can be not time-dependent [START_REF] D'ambrosio | A phenomenological model for healing and hysteresis in rubber-like materials[END_REF][START_REF] Dorfmann | Modelling dilatational stress softening of rubber[END_REF][START_REF] Rey | Hyperelasticity with rate-independent microsphere hysteresis model for rubberlike materials[END_REF][START_REF] Vandenbroucke | A Hyperelasto-Visco-Hysteresis model for an elastomeric behaviour: Experimental and numerical investigations[END_REF] (iii) if all the energy contained in the hysteresis loop were due to viscosity, the self-heating would be much higher than that observed experimentally. One can therefore wonder about the nature and the time dependency of the phenomena involved in the formation of the hysteresis loop and about the real contribution of the intrinsic dissipation to it. Three recent studies investigate the energetic behaviour and the energy storage during deformation of rubbers [START_REF] Le Cam | Energy storage due to strain-induced crystallization in natural rubber: the physical origin of the mechanical hysteresis[END_REF][START_REF] Lachhab | Energy stored during deformation of crystallizing tpu foams[END_REF][START_REF] Loukil | Stored energy accompanying cyclic deformation of filled rubber[END_REF]. The present paper presents and discusses on the results reported in these studies.

Energy balance

The energy balance is carried out from the mechanical and the calorific responses obtained during mechanical cycles. The mechanical response provides the energy and its rate corresponding to the hysteresis loop. Integrating the heat source with respect to time over one cycle gives the mean intrinsic dissipation:

The difference between and the intrinsic dissipation gives the energy rate stored at each cycle:

To further discuss on the relative contribution of the energy stored in the hysteresis loop of rubbers, a ratio gse has been proposed. It is written in terms of energy as follows:

When tends to 0, no energy is stored during the deformation, when tends to 1, the whole hysteresis loop is due to energy stored and no intrinsic dissipation is detected.

Experimental setup

Three materials were considered: an acrylonitrile butadiene rubber (NBR) unfilled and carbon black filled, an unfilled natural rubber and a polyurethane (TPU). Further information on the materials and the specimens used is provided in [START_REF] Loukil | Stored energy accompanying cyclic deformation of filled rubber[END_REF], [START_REF] Le Cam | Energy storage due to strain-induced crystallization in natural rubber: the physical origin of the mechanical hysteresis[END_REF] and [START_REF] Lachhab | Energy stored during deformation of crystallizing tpu foams[END_REF], respectively. The mechanical tests consist in applying sets of several cycles at an increasing maximum stretch lmax. The number of cycles at a given maximum stretch is chosen in such a way that the last one is mechanically and thermally stabilized. Several loading rates were applied. Temperature measurements were carried out by using FLIR infrared cameras.

Results and discussion

The heat source s (in W.m -3 ) is calculated for each stabilized cycle at different maximum stretches. A diagram is proposed to summarize the energetic behaviour of the three materials considered. It will be fully presented and discussed during the talk.

Concluding remarks

Recent studies reviewed in this paper investigate the energetic behaviour and the energy storage during deformation in different types of elastomers. A ratio gse has been proposed to relativise the energy stored at each cycle. Results show that viscosity is not systematically the preponderant contribution to the hysteresis loop: the mechanical energy brought to the material is not entirely dissipated into heat and can be mainly used by the material to change its microstructure. This ratio is of a first importance for designing rubber parts as it characterizes the greater or lesser ability of the rubber to absorb mechanical energy without damaging, typically in the case of fatigue [START_REF] Ruellan | Fatigue of natural rubber under different temperatures[END_REF].