Pavel A Dub

Oleg A Filippov

Natalia V Belkova

Jean-Claude Daran

Lina M Epstein

Rinaldo Poli

Elena S Shubina

Hydrogen bonding to carbonyl hydride complex Cp*Mo(PMe 3 ) 2 (CO)H and its role in proton transfer

The interaction of the carbonyl hydride complex Cp*Mo(PMe3)2(CO)H with Brønsted (fluorinated alcohols, (CF3)nCH3-nOH (n = 1-3), and CF3COOH) and Lewis (Hg(C6F5)2, BF3•OEt2) acids was studied by variable temperature IR and NMR ( 1 H, 31 P, 13 C) spectroscopies in combination with DFT/B3LYP calculations . Among the two functionalities potentially capable for the interactioncarbonyl and hydride ligandsthe first one was found to be the preferential binding site for weak acids, yielding CO•••HOR or CO•••Hg complexes as well as the CO•••(HOR)2 adducts. For stronger proton donors ((CF3)3COH, CF3COOH) hydrogen-bonding to the hydride ligand can be revealed as an intermediate of the proton transfer reaction. Whereas proton transfer to the CO ligand is not feasible, protonation of the hydride ligand yields an ( 2 -H2) complex. Above 230 K dihydrogen evolution is observed leading to decomposition. Among the decomposition products compound [Cp*Mo(PMe3)3(CO)] + [(CF3)3CO •2HOC(CF3)3] -resulting from a phosphine transfer reaction was characterized by X-ray diffraction. Reaction with BF3•OEt2 was found to produce [Cp*Mo(PMe3)2(CO)BF4] via initial attack of the hydride ligand.

Introduction

Being among the earliest discovered organometall i c complexes, metal carbonyls continue to play one of the central roles in organometallic chemistry. [START_REF] Elschenbroich | Organometallics: A Concise Introduction[END_REF][START_REF] Horwit | [END_REF] Addition of the hydride function to a transition metal carbonyl complex results in the formation and coexistence of three potentially basic centers inside the carbonyl hydride complex: the metal atom, the hydride ligand and the CO-oxygen atom. With the exception of bridging carbonyl ligands, [START_REF] Kristjánsdóttir | Transition Metal Hydrides[END_REF] protons from strong acids usually add to the metal atom or to the hydride ligand. 4 On the other hand, much bulkier Lewis acids generally add to carbonyl oxygens. [START_REF] Horwit | [END_REF] Weak proton donors (mostly alcohols) interact with either the hydride ligand or the CO-oxygen. 4,5 Very recently we have reported the results of the protonation of the carbonyl hydride complex Cp*Mo(PMe3)2(CO)H (1) by the strong acid HBF4. At low temperatures proton transfer in THF yields the dihydrogen complex [Cp*Mo(PMe3)2(CO)( 2 -H2)]BF4 as a sole observable protonation product, however when the same reaction is carried out in CH2Cl2 the only product of the reaction is the cationic dihydride complex

[Cp*Mo(PMe3)2(CO)(H)2]BF4. 6 The stability of the [Cp*Mo(PMe3)2(CO)H2]BF4 ion pair and the classical/nonclassical equilibrium appeared to be fine tuned by the series of cation-anion-s olvent non-covalent interactions. In order to get further insight into the proton transfer mechanism we performed a more detailed study of the interaction between 1 and series of Brønsted acids (fluorinated alcohols (CF3)nCH3-nOH (n = 1-3) and trifluoroacetic acid) , as well as a few selected Lewis acids.

Results

Compound 1 was first described by Lapinte et al 7 and obtained in a pure form by Parkin et al upon reaction of Cp*Mo(PMe3)3H with CO. 8 It possesses catalytic activity in the decomposition of HCO2H to CO2 and H2 and was found to react with RCOOH (R = H, Me, Et) at room temperature, leading to 55 dihydrogen evolution and Cp*Mo(PMe3)2(CO)(η 1 -O2CR). 8 This result suggests the presence of the ( 2 -H2) complex along the reaction coordinate. The initial attack by fluorinated alcohols was therefore expected to take place at the hydride site yielding a dihydrogen bonded complex as a reaction 60 intermediate. 9,10 Dihydrogen-bonding is known 11 to induce a high field shift of the 1 H NMR hydride resonance and a decrease of its longitudinal relaxation time (T1min), whereas in the IR spectra it causes a low frequency shift of the M-H band and a high frequency shift of the characteristic bands of all 65 other ligands, such as e.g. CO. 11 To our surprise the addition of 2 equiv CF3CH2OH (TFE) to the solution of 1 in CD2Cl2 at 200K resulted in a low field shift of the hydride resonance (Δδ = 0.19 ppm; Δδ ≡ δ(1 + ROH) -δ( 1)), see Fig. 1. The measured value of T1 (930 ms at 500 MHz, 200 K) was a bit higher than 70 that of the starting material (860 ms under the same conditions ). Further TFE addition (4.5 equiv total) led to a greater shift of the resonance (Δδ = 0.29 ppm) and T1 = 900 ms. The direction of the signal shift together with nonlinear T1 dependence suggest that the hydride ligand is not interacting with TFE.

75

The Cp* and PMe3 proton resonances of 1 showed practically no change following the addition of excess TFE in this solvent (Table 1). However, the 31 P{ 1 H} signal is sensitive to the presence of the alcohol, shifting by  = -0.4 (with 2 equiv TFE) to -0.6 ppm (4.5 equiv TFE). а An invariant 2 JHP of 77 Hz was measured under all conditions; b An invariant 2 JНP of 8 Hz was measured under all conditions ; с An invariant 2 JCP = 28 Hz was measured under all conditions; d at 298K. Addition of TFE (3.7 equiv) to the solution of 1 in the les s polar toluene-d8 at 200K causes again a low field shift (Δδ = 0.28 ppm) of the hydride resonance and a high-field shift (Δδ = -0.9 ppm) of the 31 P{ 1 H} signal. In this solvent the Cp* and PMe3 proton resonances are quite sensitive to the acid addition, each shifting to higher field by Δδ = -0.08 ppm (Table 1).

The dependence of all the chemical shifts on the TFE amount suggests the presence of equilibrium in the system. Indeed, when the temperature of the toluene solution of 1 with 3.7 equiv TFE was raised to 290K, all the resonances  values decreased, in agreement with an equilibrium shift toward starting material upon warming (ΔδH = 0.10 ppm, ΔδCp*H = ΔδPMe3 = -0.02 ppm, ΔδP = -0.4 ppm), see Table 1. The proces s is completely reversible, the signal shifts restoring their values upon cooling back to 200 K.

Finally, the low temperature (200K) 13 C{ 1 H} NMR study in toluene-d8 revealed that the characteristic triplet of the CO resonance of 1 ( 2 JC-P = 29 Hz) at δ 253.1 (Fig. 2) shifts downfield by 6.1 ppm in the presence of 2 equiv of (CF3)3CO H (PFTB). Other signals have ΔδC(C5Me5) = 1.1 ppm, ΔδC(PMe3) = -0.7 ppm, ΔδC(C5Me5) = -0.5 ppm). Like in the case of ΔδH and ΔδP, the ΔδC depends on the acid amount, the temperature and the solvent. In more polar CD2Cl2, use of TFE at 200K gives ΔδCO = 2.3 ppm (3.3 equiv) and 2.5 ppm (4.5 equiv). Attachment of an acceptor to the oxygen end of the CO ligand is known to shift the 13 CO resonance downfield. [START_REF] Horwit | [END_REF] Together with the IR spectroscopic results (see below), this allows to sugges t the formation of a CO•••HOR F hydrogen bond. 45 band intensity and the appearance of two new bands, CO(1a) at 1736 and CO(1b) at 1720 cm -1 (Fig. 3). When the temperature was raised to 250K the 1720 cm -1 band completely disappeared and only traces of the 1736 cm -1 band remained, while the intensity of the band of the starting material 50 increased. Cooling back to 200 K reproduced fully the initially observed picture, demonstrating the reversibility of the process . More acidic proton donors, (CF3)2CHOH (HFIP) and PFTB, forming stronger hydrogen bonds, give more intense bands (with lower intensity of the CO(1)) at even lower frequencies 55 (Fig. 3, Table 2). These changes indicate a progressive shift of the hydrogen bonding equilibrium to the right with an increas e of the alcohol proton donating ability. The low frequency CO shift in the presence of proton donors is evidence for hydrogen bonding with the carbonyl ligand. 12,5 Appearance of a second CO band, shifted by -11 to -18 cm -1 in respect to CO(1a), indicates formation of yet another H-bonded complex, which we believe contains CO•••HO bond with the alcohol dimer (Scheme 1). Note that binding of the second alcohol molecule at the hydride ligand would shift the CO band to higher frequency relative to CO(1a). 5 The nature of the hydrogen bonded species 1a and 1b was further assessed with the help of DFT calculations. Geometry optimization and frequency calculations were performed for hydride 1 and for its hydrogen bonded complexes with CH3OH, CF3OH and CF3COOH at DFT/B3LYP level. 20 Coordination of proton donors at the carbonyl and hydride ligand sites was considered as well as possible complexes with two proton donor molecules. The structures obtained are exemplified for the case of CH3OH in Fig. 4, interactio n energies and spectroscopic parameters are gathered in Table 3, 25 and selected geometrical characteristics are given in the ESI † (Table S2). Formation energies (Table 3) obtained for complexes with one HOR molecule show preference for classical CO•••HOR bonding over non-classical MoH•••HOR interaction for methanol, which becomes less significant for 30 stronger proton donors. BSSE correction reduces formation energies by 14-35% but the EBSSE pattern is similar to that of E's (Table 3). Table 3. Formation energies (relative to the separated reactants, in kJ mol -1 ) and frequencies (in cm -1 ) calculated for H-complexes between 1 and proton donors in gas phase.

Proton donor were optimized leading to stable energy minima. The formation energy for the latter complex (Table 3) is equal to the sum of two interactions (CO•••HOR and MoH•••HOR) for methanol, whereas mutual weakening of the two hydrogen bonds (anticooperativity 15 ) was found in complex E for TFA (by 5 kJ mol -1 ). A cooperativity effect is observed for the formation of As can be seen from the Table 3, hydrogen bonding to the CO ligand leads to a low frequency shift of the СО band, whereas Н•••Н bond formation leads to a high frequency СО shift. In agreement with our experimental data, CO increas es with the proton donor strength. According to the calculations , the experimentally observed CO(1b) band could certainly be assigned to species C. The CO(1a) band, on the other hand, could be generated not only by complex A, but also by complex E. Dihydrogen bonded complexes of B or D type should appear at frequencies higher than CO (1).

H-complex notation H-bond type E ΔEZPVE ΔEBSSE H (OH) a CO (CO) b OH c CH3OH A CO•••HOR -25 -19 -22 -13 1813 (-50) -142 B H•••HOR -20 -15 -15 -20 1865 (+2) -261 2CH3OH C CO•••(HOR)
Cp*MoH(CO)(PMe3)2 + H + → [Cp*MoH(COH)(PMe3)2] + (1) Cp*MoH(CO)(PMe3)2 + H + → [Cp*Mo( 2 -H2)(CO)(PMe3)2] + 2)
Protonation of the carbonyl ligand (Equation 1) was found less exothermic by E = 85 kJ mol -1 than protonation of the hydride ligand (Equation 2). Proton affinities (PA) -H298K = 936 kJ mol -1 and 1027 kJ mol -1 were calculated for carbonyl and hydride ligands, respectively. The latter is slightly lower than PA values calculated for other transition metal hydride complexes (1074-1132 kJ mol -1 ). 4 When CF3COOH (PA = 1338 kJ mol -1 ) was used as proton source the [Cp*MoH(COH)-(PMe3)2] + OCOCF3 -ion pair product could not be optimized, all attempts leading back to the hydrogen bonded complex. Thes e data are in support of the notion that the CO protonation pathway is not feasible. The formation enthalpy H for [Cp*Mo( 2 -H2)(CO)(PMe3)2] + OCOCF3 -ion pair is -328 kJ mol -1 relative to the isolated ions in the gas phase, which is higher than the PA difference for these ions (PA = 311 kJ mol - 1 ).

The energy profiles for proton transfer from CF3COOH to the hydride ligand of 1 were explored starting from the dihydrogen bonded complexes B and D, leading respectively to B' and D' as final products (Fig. 5). The structures of thes e products are depicted in Fig. 6. Cooperative enhancement of the dihydrogen bond strength in complex D relative to B lowers the barrier substantially to only about 8 kJ mol [START_REF] Elschenbroich | Organometallics: A Concise Introduction[END_REF] . Taking solvent effects (under the CPCM approach) into account makes the proton transfer to the hydride ligand even more favorable, providing additional stabilizatio n of the hydrogen bonded ion pair product (Fig. S1). This stabilization is more evident for higher polarity solvents such as THF or CH2Cl2. The occurrence of proton transfer to the hydride site was confirmed by experimental studies. Addition of 2 equiv of CF3COOH to 1 in hexane at 200 K generates new CO bands at1804 and 1816 cm -1 , assigned respectively to the dihydrogen 70 bonded complex and [Cp*Mo(CO)(PMe3)2(η 2 -H2)] + . Corresponding bands of the hydrogen bonded acid (CO = 1788 cm -1 ) and of the CF3COO anion ( as OCO = 1660 cm -1 ) are als o observed. Use of excess acid makes the IR spectra uninformative. The 1/CF3COOD interaction was also studied 75 by [START_REF] Elschenbroich | Organometallics: A Concise Introduction[END_REF] H NMR in THF-d8. The acid activity is reduced in THF, thus a stronger excess of the acid is necessary to promote proton transfer. A wide 1 H resonance at  -4.5, assigned to the (η 2 -HD ) complex, appeared only in the presence of 10 equiv of acid at 200 K (Fig. S2), accompanied by Cp*Mo(PMe3)2(CO)D (1-d) 80 formation by H/D exchange. Corresponding 31 P triplet appears at  32.35 and shifts to  32.28 with the increase of the acid amount (Fig. S2) due to the CO•••HO bond formation ( 31 P resonance of 1 shifts from  32.0 to 31.9).

Experimental evidence for the formation of the dihydrogen 85 bonded complex could only be obtained by IR spectroscopy on concentrated toluene solutions of 1 (of the order of 10 -2 M instead of 10 -3 M) in the presence of PFTB, as evidenced by a high frequency shoulder on the CO band (CO = 18 cm -1 at 250 K, Fig. S3). This complex contains most probably two 90 alcohol molecules. The NMR monitoring in toluene-d8 at 200 K (Fig. S4) shows, in addition to a downfield shifted hydride resonance, the presence of a broad peak at δ -4.10 with T1 = 20 ms, characteris tic for a dihydrogen complex. The dihydrogen complex is unstable and looses H2 upon warming, as already observed in the case of BF4 -anion. 6 NMR monitoring of the room temperature reaction between 1 and 5 equiv of PFTB in benzene shows slow dihydrogen evolution with 46% and 68% conversion after 4 h 20 min and 49 h 40 min, respectively. At the same time the broad resonance of the alcohol proton shifts to lower fields indicating further involvment of the alcohol into hydrogen bonding. The 31 P NMR spectrum of this solution features peaks of different products , one of which was crystallized and analyzed by single crystal Xray diffraction (Fig. 7). 16 It is also related to the reaction between FeH2(HPPh2)4 and Ph3C + BF4 -, yielding [FeH(HPPh2)5] + through the initial formation of the 16-electron [FeH(HPPh2)4] + intermediate. 17 We have shown recently that the primary product of H2 loss from [Cp*Mo( 2 -H2)(CO)(PMe3)2] + [BF4] --Cp*Mo(PMe3)2(CO)FBF3 -is not stable at ambient temperatures and slowly decomposes. 6 One of the products was 6 In this case BF3 presumably reacts with the hydride ligand giving initially the BHF3 -anion, which however exchanges readily with the Et2O•BF3 excess yielding 55 BF4 -. A similar observation was reported for Cp2Nb(CO)H 18 and for the reaction of Cp2TiBH4 with Et2O•BF3 to give Cp2Ti( 2 -BF4). 19 Thus, the reactivity pattern for Lewis acids is the same as for Brønsted ones: weak acids coordinate to the CO ligand whereas strong ones interact with the hydride and 60 ultimately lead to hydride abstraction.

Discussion

The hydrogen bonding and proton transfer equilibria described above for the Cp*MoH(CO)(PMe3)2/ROH system can be summarized as shown in the Scheme 2, illustrating the variety The CO•••HOR interaction for group VI-VII hydride-free transition metal carbonyl complexes such as Cp*M(CO)2(N 2 ) (M = Mn, Re), 12 (C5Et5)M(CO)3 (M = Mn, Re), ( 6 -C6H3Me3)M(CO)3 (M = Cr, Mo), 20 and the related NO•••HOR interaction has been documented for CpMo(CO)2(NO). 20 Both interactions have been proposed to be the first step in the proton transfer to the metal atom. 20 The data described in this contribution, together with those obtained for other transition metal hydride complexes, CpReH(CO)(NO), 21 ReH2(CO)-(NO)(PR3)2, 22 and CpRuH(CO)(PCy3), 5,23 show that CO•••HOR or NO•••HOR bonding is a "dead-end pathway", not leading to the proton transfer. In contrast, the H•••HOR bond formation is the actual first reaction step. Assisted by the binding of the second proton donor molecule, which increases the hydrogen bond strength and lowers the proton transfer barrier, it leads to the ( 2 -H2) complex formation. The results acquired up to date for molybdenum hydride complexes Cp*MoH(CO)(PMe3)2, Cp*MoH(PMe3)3 24 and Cp*MoH3(dppe) 15 allow explaining the different stability of ( 2 -H2) complexes and reactivity pattern depending on the ligands (Fig. 9). Fig. 9. Potential energy curves (E, in kJ mol -1 ) for the non-classical to classical transformations calculated at the B3LYP level in the gas phase for [Cp*MoH2(PMe3)3] + , 24 [Cp*MoH4(dppe)] + [((CF3)2CHO)2H] -, 14 and [Cp*MoH2(CO)(PMe3)2] + BF4 -. 6 In agreement with the established mechanism of proton transfer to transition metal hydrides 9 the protonation proceeds via dihydrogen bond formation in all three cases. For the mos t electron reach Cp*MoH(PMe3)3 the minimum of ( 2 -H2) is 35 very shallow with negligible (of 0.8 kJ mol -1 ) barrier for isomerization to classical dihydride. 24 The latter is stabilised by 54.6 kJ mol -1 and thus can not convert back to the non-class ical isomer in order to evolve H2. The classical -non-class ical interconversion is very easy in the case of protonated 40 Cp*MoH3(dppe). 15 Corresponding minima are very close in energy (-18.5 kJ mol -1 for the isolated cations and -6.3 kJ mol -1 for the [Cp*MoH4(dppe)] + [((CF3)2CHO)2H] -ion pair) and separated by a very low barrier (Fig. 9). Therefore, the nonclassical [Cp*Mo( 2 -H2)H2(dppe)] + intermediate could not be 45 observed experimentally, but the classical tetrahydrid e complex is not thermally stable and loses dihydrogen quite easily. The [Cp*Mo(H)4(dppe)] + stability could be controlled by temperature and by the strength of the cation-anio n interaction within the ion pair. 25 The [Cp*Mo( 2 -H2)(CO )-50 (PMe3)2] + /[Cp*Mo(H)2(CO)(PM e3)2] + pair of isomers is separated by a much higher barrier, whereas the relative depth of the two minima is strongly affected by the environment. 6 As the result, both could be observed experimentally but the interconversion is possible and both isomers evolve H2 when 55 the barrier for the latter reaction could be overcome (above 230 K).

Experimental

All manipulations were performed under an argon atmosphere.

60

All solvents were dried over appropriate drying agent (Na/benzophenone for toluene and THF; CaH2 for dichloromethane and hexane) and freshly distilled under an argon atmosphere prior to use. Benzene-d6 (Euriso-Top) was kept over molecular sieves and deoxygenated with an argon 65 flow before use. CD2Cl2, THF-d8, and toluene-d8 (Euriso-Top) were degassed by three freeze-pump-thaw cycles. Compound CH3OH and of free reactants were performed using pseudopotential-bas ed correlation consistent polarized valence triple-ζ basis set cc-pVTZ-PP 35 for Mo and 6-31G(d) basis set for P atoms. The differently optimized structures of 1 with the two above described basis sets do not differ by more than 0.020 70 Å for Mo-C bond distances and 0.008 Å for Mo-P, Mo-H and C-O bond distances. The formation energy of adduct A (see Table 3 for the legend) increases by only 1 kJ mol -1 while that of adduct B remains the same as do their geometries. The frequency shifts () resulting from hydrogen bond formation 75 differ by no more than 2%, except for OH of complex A which increases by 5%. The structures of the reactants, intermediates, transition states, and products were fully optimized without any symmetry restriction. Transition states were identified by 80 having one imaginary frequency in the Hessian matrix. No scaling factor was applied to the calculated frequencies since the optimization was run in the gas phase and the IR spectra were measured in solution.

Non-specific solvent effects were introduced through CPCM 85 continuum representation of the solvent by single-point calculations 36 on gas phase optimized geometries for toluene ( = 2.38), dichloromethane (ε = 8.93) and THF (ε = 7.58). The G solv values account for the solvation free energies, with inclusion of the solute free energy contributions G solv = E so lv + G gas -E gas , where E solv is the electronic energy plus the solvent entropy. 37 The optimisations of 1, CF3COOH and their dihydrogen bonded adduct B were also performed in THF, leading to only minor geometry changes for 1 and CF3COOH (Δr < 0.01 Å), while the H•••H bond length in B increased by 0.026 Å. Despite this lengthening, the interaction energy decreased by only 0.3 (ΔE solv ) and 0.1 kJ mol -1 (ΔG solv ). The same pattern was revealed for stronger dihydrogen bonded 10 complexes of anionic main group tetrahydrides optimisation s in solution. 38 Therefore the consideration of solvent effects was kept within the single point CPCM approach for the rest of the study.

Conclusions

We presented results of combined spectroscopic and theoretica l investigations of hydrogen bonding to Cp*MoH(CO)(PMe 3 )2 and its role in the proton transfer process. The formation of several hydrogen bonded complexes involving both carbonyl and hydride ligands was observed experimentally; their structures and spectroscopic features were also studied by DFT calculations. The CO

Captions for figures

Fig. 1 .

 1 Fig. 1. Hydride region of the 1 H NMR (500.3 MHz) spectrum of 1 alone(a), and in the presence of 2 equiv (b) or 4.5 equiv (c) of TFE. T = 200K, CD2Cl2, c(1) = 5.8•10 -2 M.

35

 35

Fig. 2 . 1 .

 21 Fig. 2. 13 C{ 1 H} NMR (125.8MHz, CO region) spectrum of 1 (a) and 1 in the presence of ca. 2 equiv PFTB (b). toluene-d8, 200K. c(1) = 1•10 -1 M, 1330 scans.The IR spectra in the TFE νOH region in CH2Cl2 showed the 40

Fig. 3 .

 3 Fig. 3. IR spectra (νCO region) of Cp*Mo(PMe3)2(CO)H (1) (0.005 M) (a) and in the presence of ca. 3 equiv. TFE (b), 3 equiv HFIP (c), 3 equiv PFTB (d). T = 200K, hexane, l = 1.2 mm.

  Scheme 1

Fig. 4 .

 4 Fig. 4. Optimized geometries of hydrogen bonded complexes between 1 and one and two CH3OH molecules.

-

  512 & -890a enthalpy values based on calculated OH: H (in kJ mol -1 ) = -75/( +720)13,14 ; b CO = CO(complex) -CO(1), CO(1) = 1863 cm -1 ; c OH = OH(complex) -OH(HOR); d TFA has the CO mode in this range, therefore the CO vibrations of 1 and TFA are mixed A In the case of interaction with two proton donor molecules (calculated for CH3OH and CfigurF3COOH), two adducts with the HOR dimer, CO•••(HOR)2 (C) and MoH•••(HOR)2 (D), as well as of complexes with simultaneous hydrogen bonding to carbonyl and hydride ligands, ROH•••(OC)MoH•••HOR (E),

  C and D [Ecoop = E(1•2HOR) -E(1•HOR) -E((HOR)2)], -7 and -13 kJ mol -1 for C and -8 and -9 kJ mol -1 for D with CH3OH and CF3COOH, respectively, showing the decrease of the relative impact of the cooperative effect into total formation energy (Ecoop/E(1•2HOR)) for stronger proton donor.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Proton transfer energy profiles calculated for the reaction of 1 with 60

Fig. 7 .

 7 Fig. 7. Molecular structure of [Cp*Mo(PMe3)3(CO)] + [(CF3)3CO•2HOC(CF3)3] -. For the sake of clarity, the cation and the anion are shown separately. Molecular ellipsoids are shown at the 30% probability level. The [Cp*Mo(PMe3)3(CO)] + cation results from a phosphine transfer reaction. This process is closely related to the CO 20

Fig. 8 .

 8 Fig. 8. IR spectra (νCO region) of Cp*Mo(PMe3)2(CO)H (1) (0.004 M) (a), 1 in the presence of ca. 5 equiv of Hg(C6F5)2 (b). Hexane, 200 K, l = 0.12 cm Use of Et2O•BF3 in toluene-d8 or THF-d8 gives an immediate 50

  Scheme 2.

Fig. 8 .

 8 Fig. 8. IR spectra (νCO region) of Cp*Mo(PMe3)2(CO)H (1) (0.005 M) (a) and in the presence of ca. 3 equiv. TFE (b), 3 equiv HFIP (c), 3 equiv PFTB (d). T = 200K, hexane, l = 1.2 mm.

Fig. 9 .Fig. 10 .

 910 Fig. 9. Optimized geometries of hydrogen bonded complexes between 1 and one and two CH3OH molecules.50

Fig. 6 .Fig. 7 .

 67 Fig. 6. Optimized structures of hydrogen bonded ion pairs as the products of proton transfer from CF3COOH to 1. Fig. 7. Molecular structure of [Cp*Mo(PMe3)3(CO)] + [(CF3)3CO•2HO-55

Fig. 8 .Fig. 9 .

 89 Fig. 8. IR spectra (νCO region) of Cp*Mo(PMe3)2(CO)H (1) (0.004 M) (a), 1 in the presence of 5 equiv of Hg(C6F5)2 (b). Hexane, 200 K, l = 0.12 cm Fig. 9. Potential energy curves (E, in kJ mol -1 ) for the non-classical 60

Table 1 .

 1 Selected NMR ( 1 H, 31 P{ 1 H}, 13 C{ 1 H}) parameters for complex 1 in different solvents in the presence of proton donors at 200 K.

	solv	ROH	[ROH]/[1]	H -а	H(Cp*)	H(PMe3) b	P	СО с
	СD2Cl2		0	-6.98 (t)	1.89 (s)	1.31 (d)	31.4 (s)	255.0 (t)
		TFE	2.0	-6.79 (t)	1.89 (s)	1.31 (d)	31.0 (s)
			4.5	-6.69 (t)	1.89 (s)	1.31 (d)	30.8 (s)	257.5 (t)
	tol-d8		0	-6.85 (t)	1.96 (s)	1.34 (d)	31.9 (s)	253.1 (t)
	tol-d8 d	TFE	3.7 0	-6.57 (t) -6.88 (t)	2.02 (s) 1.93 (s)	1.42 (d) 1.33 (d)	30.7 (s) 30.3 (s)	253.1 (t)
		TFE	3.7	-6.78 (t)	1.95 (s)	1.35 (d)	29.9 (s)
		PFTB	2.0	-6.38 (t)	1.82 (s)	1.21 (d)	30.1 (s)	259.2 (t)

Table 2 .

 2 Parameters of the CO bands for hydrogen bonded complexes of

	hexane	0	1791
	TFE	10	1791	1738	1720
	HFIP	5	1791	1732	1718
	PFTB	5	1791	1727	1716
	toluene	0	1771
	PFTB	5	1771	1707	1695

60 1 with alcohols at 200 K. Solv ROH [ROH]/[1] CO(1) CO(1a) CO(1b)

Table 4 .

 4 Crystal

		ROH	Mo Me 3 P PMe 3 H C	O	H	O ROH	Mo Me 3 P PMe 3 H C	O	H	O	O R H	Mo Me 3 P PMe 3 H C	O	H	O	O R H
							R							R	R
	Mo Me 3 P PMe 3 H C	O
		ROH
			O H R	Mo Me 3 P PMe 3 H C	O	ROH	R	R O H H O	Mo Me 3 P PMe 3 H C	O	R	R O H O	H H Me 3 P PMe 3 Mo C	O

  •••HOR type of bonding was found to dominate over the MoH•••HOR one, which is a rather rare occurrence. However, only the MoH•••HOR adduct is the intermediate of the proton transfer reaction yielding the nongrant 08-03-92506)) and support of the RFBR (08-03-00464) is gratefully acknowledged. Additional national support was obtained in France from the CNRS, the IUF, and in Russia from the Division of Chemistry and Material Sciences of RAS and the Russian Federation President grant (МК-380.2008.3). RP thanks the CICT for granting free computational time.

classical cation [Cp*Mo( 2 -H2)(CO)(PMe3)2] + .

Acknowledgment

Support of this work through France-Russia bilateral programs (PICS grant 2004-07, continuing within the framework of a GDRE (groupe de recherche européen) 2008-11 (RFBR-CNRS

Cp*Mo(PMe3)2(CO)H was synthesized according to the literature. 8 Et2OHBF4 (56%) was purchased from Sigma Aldrich and used as received. Et2OBF3 (48%) was purchas ed from Acros Organics and distilled before use.

Room temperature NMR investigations were carried out on Bruker DPX300 and Bruker AV300LiQ spectrometers operating at 300.1 MHz ( 1 H), 121.49 MHz ( 31 P { 1 H}) and 75.47 MHz ( 13 C{ 1 H}). Low-temperature 1 H, [START_REF] Hay | [END_REF] P{ 1 H} and 13 C{ 1 H} data were collected with Bruker AV500 spectrometer , operating at 500.3, 202.5 and 125.8 MHz, respectively. The temperature was calibrated using a methanol chemical shift thermometer; the accuracy and stability was ±1 K. All samples were allowed to equilibrate at every temperature for at least 3 min. The spectra were calibrated with the residual solvent resonance relative to TMS ( 1 H, 13 C), and with external 85% H3PO4 ( 31 P). The IR spectra were recorded on Specord M-82 and FT Infralum-801 spectrometers in CaF2 cells using home modified cryostat Carl Zeiss Jena for variable temperatur e measurements. The accuracy of the temperature adjustment was ±1 K.

X-ray Structure Determination

Crystals of [Cp*Mo(PMe3)3(CO)] + [(CF3)3CO•2HOC(CF3)3] - complex (νCO = 1836 cm -1 (neat)) were grown by diffusion of pentane vapor to the mixture of 1 with 5 equiv. of PFTB in benzene. A single crystal was mounted under inert perfluoropolyether at the tip of glass fibre and cooled in the cryostream of the Oxford-Diffraction XCALIBUR CCD diffractometer. Data were collected using the monochromati c MoKα radiation (λ= 0.71073).

The structure was solved by direct methods (SIR97 26 ) and refined by least-squares procedures on F 2 using SHELXL-97. 27 All H atoms attached to carbon were introduced in calculatio n in idealised positions and treated as riding models. The C and F atoms of one of the (CF3)3COH anions are disordered over two positions. This disorder was modelized and treated using the tools (PART, SASI, EADP) available in SHELXL-97. 27 The drawing of the molecules was realised with the help of ORTEP32. 28 Crystal data and refinement parameters are shown in Table 4. Crystallographic data (excluding structure factors) have been deposited with the Cambridge Crystallographic Data Centre as supplementary publication no. CCDC 742724.

Copies of the data can be obtained free of charge on applicatio n to the Director, CCDC, 12 Union Road, Cambridge CB2 1EZ, UK (fax: (+44) 1223-336-033; e-mail: deposit@ccdc.cam.ac.uk).

Computational details

All calculations were performed at the DFT level, by means of the hybrid B3LYP functional, 29 as implemented in Gaussian03. 30 The LanL2DZ basis set for the Mo and P atoms was that associated with the pseudopotential [START_REF] Hay | [END_REF]32 with a standard double-ζ LANL2DZ contraction, 30 supplemented in the case of P with a set of d-polarization functions. 33 The 6-31G(d) basis set was used for C atoms of Cp* ring. The 6-31++G(d,p) basis set was used for the CO group, the hydride ligand, the O and the acidic H atoms of proton donors. The 6-31G basis set was used for all other atoms. The basis set superposition error (BSSE) corrections to the gas phase complexation energies were introduced according to the counterpoise method of Boys and Bernardi. 34


Marie-Agnès Pallas

Adrien Le Bellec

Determination of a noise emission model for French medium-heavy vehicles

Keywords: Medium-heavy vehicle, noise emission, CNOSSOS-EU method I-INCE Classification of Subject Number: 13

The European Directives 2002/49/EC and 2015/996 provide a common evaluation method (CNOSSOS-EU) for Member States to produce noise maps assessing environmental noise exposure. A French transposition of the Directives specifies the corrective coefficients applicable for the French road network. Previous observations highlighted significant discrepancies between measurements carried out in France on medium-heavy vehicles (category 2) and CNOSSOS-EU. However, this category of vehicles was not specifically considered in the French method NMPB2008 and their actual noise emission is undocumented. Thus, noise emission measurements have been carried out on different types of vehicles in this category (van trucks, dump trucks, bus), exploring a wide range of controlled operating conditions to separate the contributions of propulsion and rolling noise. In a second step, two traffic scenarios of category 2 vehicles, with significantly different proportions of public transport vehicles, are used to determine average emission equations in accordance with CNOSSOS-EU modeling. The noise issue between these scenarios is linked to the propulsion noise contribution, which is of main interest at urban speeds but becomes insignificant at high speeds when rolling noise predominates. The two scenario models are confronted with CNOSSOS-EU variants, including the corrected model now recommended in the French regulation.

INTRODUCTION

European Directive 2002/49/EC of 25 June 2002 on the evaluation and management of the environmental noise [1] requires Member States to produce noise maps assessing exposure to environmental noise, particularly in large urban areas and around major roads, to be updated every 5 years. It was completed in May 2015 by Directive 2015/996

[2] specifying a common assessment method for all Member States, CNOSSOS-EU. Some corrections have been made in the corrigendum published at the beginning of 2018 [3]. The implementation of this new method is effective for noise mapping from the beginning of 2019. Concerning road noise, an adaptation of the model to the French road network and vehicle fleet was published in June 2018 [START_REF]relatif à l'établissement des cartes de bruit et des plans de prévention du bruit dans l'environnement[END_REF]. In accordance with the national NMPB2008 model, which relies on an extensive database of light and heavy vehicle pass-by measurements on French road surfaces, correction coefficients have been determined per vehicle category and road surface category [START_REF] Dutilleux | Matching directive 2015/996/EC (CNOSSOS-EU) and the French emission model for road pavements[END_REF].

The study presented in this paper specifically concerns the noise emission model of medium-heavy vehicles (vehicle category 2 of CNOSSOS-EU), i.e. with a gross weight exceeding 3.5T and equipped with two axles and twin wheels on the rear axle. This vehicle category is actually undifferentiated among heavy vehicles in the French method and no information is available on its specific acoustical behaviour on French roads. In a first step, statistical pass-by measurements on several roads were confronted to the prediction models, pointing out the misadaptation of the French NMPB2008 [START_REF] Besnard | Road noise prediction -1 Calculating sound emissions from road traffic[END_REF] and European CNOSSOS-EU noise emission models for representing this vehicle category [START_REF] Pallas | Experimental confrontation of medium-heavy vehicle noise emission to the CNOSSOS-EU prediction method[END_REF]. Thereafter, these have been completed by controlled pass-by measurements involving a sample of medium-heavy vehicles driving at constant speed on the same road surface type and scanning a wide range of real operating conditions, in order to separate the propulsion noise and the rolling noise components. Then, average emission equations have been determined by vehicle subcategories, gathering vehicles with similar acoustic behaviour and referred to by the generic names dump truck, van truck and bus. Traffic noise emission from vehicles of category 2 depends on the balance of these subcategories within the whole category traffic, and may vary according to the road location and area activity. By considering two scenario options of traffic composition of category 2, average noise emission models have been determined. This paper describes the approach and compares the scenarios to the European prediction model versions.

NOISE EMISSION MODELS

The method CNOSSOS-EU published in 2015 [2, 3] is currently the reference European version, used as the basis for an adaptation to French traffic and road network specificities through corrective coefficients. An underlying mismatch of the road emission model has recently been raised in 2018 and an updated version of the reference CNOSSOS-EU version is being considered. These various versions are summerised in this section.

Reference CNOSSOS-EU

The CNOSSOS-EU road noise emission model considers four vehicle categories, according to their mass and axle number [2]:

-category 1: light vehicles, ≤ 3.5 tons -category 2: medium-heavy vehicles, > 3.5 tons with two axles and twin tyres on rear axle -category 3: heavy vehicles, > 3.5 tons with three or more axles -category 4: powered two-wheelers Category 2, which is at the forefront of this paper, includes a wide variety of vehicles with various uses, bodies and equipment -i.e. delivery-, dump-and garbage trucks, tankers, buses and coaches -as long as they have only two axles. Any vehicle is modelled by a single omnidirectional point source, located 0.05 m above ground. Its acoustic power specified in the octave bands [63 Hz -8 kHz] under reference conditions (constant speed, flat and dry road, air temperature of 20 • C and a virtual road surface corresponding to an average of DAC 0/11 and SMA 0/11). If conditions deviate from these, correction coefficients shall be used.

In each octave band i, the total radiated power of the point source L W T,i is composed of a propulsion noise component L W P,i and a rolling noise component L W R,i -all depending on vehicle speed v -specified by four coefficients:

L W T,i (v) = L W R,i (v) ⊕ L W P,i (v) (1)
where the operator ⊕ stands for the energetic sum of both components and:

L W R,i (v) = A R,i + B R,i log v v ref (2)
L W P,i (v) = A P,i + B P,i v -v ref v ref (3)
The coefficients A R,i ,B R,i , A P,i and B P,i are given in Table F-1 of the Directive [2], for a reference speed v ref = 70 km/h. Corrective terms ∆L W R,road,i and ∆L W P,road,i are available for rolling noise and propulsion noise respectively, so as to take into account the effect of the road surface on vehicle noise emission:

∆L W R,road,i (v) = α i + β log v v 0 ( 4
)
∆L W P,road,i (v) = min{α i ; 0} (5)
the latter being designed to account for the effect of absorbing surfaces on propulsion noise. Values of the coefficients α i and β are given in Table F-4 of the Directive for each vehicle category on various Dutch road surfaces. For each road pavement, it should be noticed that the corrective coefficients for medium-heavy vehicles are always identical to those of the heavy vehicles, reflecting a similar noise level impact of a road surface change for both categories. The corrective coefficients α i and β are the means of action to adapt the model to the French road surfaces.

French adaptation CNOSSOS-FR

The French prediction method NMPB2008 was implemented as a transitional method for the previous rounds of strategic noise map production, pending the availability of the common European method. The propagation part of CNOSSOS-EU is quite similar to the French approach, but the respective road noise emission models differ with regard to the vehicle classification and to the sound power levels, among other things. Indeed, NMPB2008 identifies only two types of vehicles: light vehicles and heavy vehicles [START_REF] Besnard | Road noise prediction -1 Calculating sound emissions from road traffic[END_REF], the former matching with CNOSSOS category 1 and the latter with category 3. In addition, it clusters and ranks road surfaces into three groups, R1, R2 and R3, from the quietest to the noisiest, each one split in drainage or non-drainage surfaces. NMPB2008 was built on a wide set of national measurements and is considered as representative of the vehicle fleet driving on the French road network.

An adaptation of CNOSSOS-EU to the French context has been proposed by calculating correction coefficients α i and β that best match CNOSSOS to the NMPB2008 model, respectively for vehicle categories 1 and 3 and each road surface cluster [START_REF] Dutilleux | Matching directive 2015/996/EC (CNOSSOS-EU) and the French emission model for road pavements[END_REF]. For the undocumented French category 2, the adaptation has taken the same approach as Table F-4 of CNOSSOS-EU by replicating the correction coefficients available for category 3 [START_REF]relatif à l'établissement des cartes de bruit et des plans de prévention du bruit dans l'environnement[END_REF]. This French adaptation is named CNOSSOS-FR in this paper.

It should be noted that, in road and vehicle conditions similar to the reference conditions of CNOSSOS-EU, the use of uncorrected CNOSSOS-EU greatly undervalues French noise emission levels. This results in the production of quite significant correction terms, even for low-noise surfaces.

Modified version of CNOSSOS-EU

Some errors have recently been identified in the derivation of the sound power coefficients given in Table F-1 of CNOSSOS-EU, partly resulting from a mismatch between the respective propagation models of Harmonoise/IMAGINE and of CNOSSOS-EU [START_REF] Peeters | Correcting the CNOSSOS-EU road noise emission values[END_REF]. Among other issues, the need for an update of Table F-1 has been pointed out and discussed in 2018 within a European working group for being included in a revision of the Annex [START_REF] Kok | Obstacles to the implementation of CNOSSOS in the Netherlands[END_REF]. This affects the coefficient A for both the propulsion and rolling noise components, without changing the B coefficients and therefore the component dependence on speed [START_REF] Peeters | Correcting the CNOSSOS-EU road noise emission values[END_REF]. For vehicle categories 2 and 3, upgraded coefficients provide higher overall sound power levels by 2 to more than 3 dB(A) depending on speed. This modified CNOSSOS version is named CNOSSOS-2018 in the present paper.

Comparison of the models for vehicle category 2

By likening the reference surface of CNOSSOS-EU to the non-drainage group R2 of the French classification, the previous models can be compared for vehicle category 2, namely:

-CNOSSOS-EU in reference conditions -CNOSSOS-FR for a non drainage R2 surface -CNOSSOS-2018 in reference conditions For an easier comparison with the subsequent measurement data, the quantities represented are A-weighted maximum pass-by levels (L Amax ) on a receiver located at a distance of 7.5 m from the road axis and a height of 1.2 m, the source and the propagation being in accordance with the CNOSSOS-EU approach.

For a dense road surface, CNOSSOS-EU and CNOSSOS-FR noise emission models have identical propulsion noise components and differ by their rolling noise contribution. CNOSSOS-2018 is different from the preceding ones both in the propulsion and the rolling noise components. CNOSSOS-EU gives the lowest rolling noise contribution, while CNOSSOS-FR provides the highest one and a steeper increase. The total noise shows variable differences between models over the speed range, according to the predominance of the propulsion noise and the rolling noise contribution in the octave bands (Figure 1). From previous phase of the study, noise measurements performed on traffic of category 2 showed that CNOSSOS-EU undervalues actual noise levels, in global levels and in most octave bands. However, the model CNOSSOS-FR seemed to produce level predictions compatible with the measurements [START_REF] Pallas | Experimental confrontation of medium-heavy vehicle noise emission to the CNOSSOS-EU prediction method[END_REF]. With CNOSSOS-2018 as well, these models will be compared to measures taken from medium-heavy vehicles in controlled conditions and to specific models built on traffic composition scenarios.

DETERMINATION OF THE NOISE EMISSION EQUATION OF A CONTROLLED VEHICLE

All the controlled medium-heavy vehicles have been measured under very close conditions and similarly analysed. An instance is illustrated in this section.

Measurement

In this example, the vehicle is a van-type rigid with a gross weight of 12 tons, in good maintenance state and with a relatively low mileage considering common truck use. It was equipped with tyres of dimension 245/70 R 19.5 and a 6-speed gearbox. The gear ratios and the axle ratio are known. It was loaded with a mass placed on its back axle, corresponding to a mass to power ratio of 42.5 kg/kW.

The pass-by measurement procedure is based on standard NF S 31119-2 [START_REF]Acoustics -In situ characterisation of the acoustic performance of road surfaces -Pass-by measurements -Part 2 : Controlled vehicles[END_REF], similar to ISO 11819-1 [START_REF]Acoustics -Measurement of the influence of road surfaces on traffic noise -Part 1: Statistical Pass-By method[END_REF] but for controlled vehicles. Three microphones spaced by 10 m were located on each track side, at a distance of 7.5 m from the track axis and a height of 1.2 m. By each pair of facing microphones, an infrared cell provided information on the vehicle speed. The track surface was a dense asphalt concrete (DAC 0/10). The air temperature was variable from to 17 to 22 • C according to the mix of clouds and sun. No temperature correction has been made on the acoustic data.

Constant speed pass-bys were carried out in 5 or 10 km/h steps from 15 to 90 km/h. As far as possible two different engaged gears were tested for each speed setpoint, the adapted gear (most natural gear for the speed) and the inferior gear (one gear lower, thus involving a larger rpm). Besides direct but imprecise rev counter reading, knowledge of the powertrain mechanical characteristics makes it possible to calculate the engine rpm from the speed, in front of each microphone.

Determination of the vehicle noise emission equation

In accordance with CNOSSOS prediction model the analysis is carried out in octave bands from 63 Hz to 8000 Hz. By considering the valid constant speed measurements from all microphones, the vehicle noise emission equation in each octave band i is determined by optimising parameters of a noise level model L Amax,i,mod , using a least squares criterium between the set of measured levels L Amax,i,meas and the model L Amax,i,mod : min L Amax,i,meas -L Amax,i,mod 2

In this model, given by Equation ( 7), the total noise is the sum of a propulsion noise component depending on engine speed (Equation 8) and of a rolling noise component depending on vehicle speed (Equation 9).

L Amax,i,mod (N, v) = L A,prop,i (N ) ⊕ L A,roll,i (v) (7)
L A,prop,i (N ) = L 0,A,prop,i + α A,prop,i log N N ref (8)
L A,roll,i (v) = L 0,A,roll,i + α A,roll,i log v v ref (9)
where N ref and v ref are respectively the reference engine speed and the driving speed, and the operator ⊕ stands for energetic summation. The experimental investigation of more than one engaged gear at most speeds provides a wider engine rpm range for an improved optimisation. Except for the lowest octave band 63 Hz and also 125 Hz at a lower level, which include narrowband components switching from one octave to another with increasing speed, this model fits quite well to the octave band data and global data.

Thereafter, considering the common adapted gear driving conditions, a one-to-one relationship links vehicle speed to engine rpm N adapt (v). Thus, the noise emission equation of the vehicle at adapted gear depends on the sole variable v, using the previously optimised coefficients L 0,A,prop,i , α A,prop,i , L 0,A,roll,i and α A,roll,i (Equation 10).

L Amax,adapt,i (v) = L 0,A,prop,i + α A,prop,i log N adapt (v) N ref ⊕ L 0,A,roll,i + α A,roll,i log v v ref (10)
In a similar way an emission equation in inferior gear can be expressed for the selection of one gear lower. The global emission model, recomposed from the eight octave bands, is presented in Figure 2, in which pass-by measurement data for the various gear ratios are also plotted. The discontinuities observed on the propulsion noise and, consequently, on the overall noise are related to the gearbox shifts which imply large engine rpm changes.

The same operation has been performed with each vehicle tested. The propulsion and the rolling noise components of each vehicle in adapted gear will be used to determine average contributions from a set of vehicles. rolling noise propulsion noise @ adapted gear total noise @ adapted gear propulsion noise @inferior gear total noise @ inferior gear gear 1 gear 2 gear 3 gear 4 gear 5 gear 6

Figure 2: Noise emission of one vehicle of category 2, A-weighted global noise levels at 7.5 m -Model noise components in adapted gear and inferior gear driving conditions (lines) -Noise levels measured in the various gear ratios ( * )

NOISE EMISSION BY MEDIUM-HEAVY VEHICLE SUBCATEGORY

Several types of medium-heavy vehicles have been tested. They have been grouped in subcategories, designated with the generic names van truck, dump truck, bus. This classification was motivated by noticing some homogeneity within a subcategory but significant differences between them. First, a mean emission law is determined in each subcategory by energetically averaging propulsion noise components on the one hand and rolling noise components on the other hand per octave band, on the common speed test range. The average in global levels is obtained by summing the octave band averages. Then, in section 5 these will be combined according to traffic mix scenarios in order to derive an average noise emission of a vehicle of category 2. All vehicles had an internal combustion engine and were tested at one of two sites with a DAC 0/10 road surface.

The total number of vehicles tested is still limited, since seven vehicles could be assessed in depth. However, this is of the same order as the heavy vehicle sample used in its time for supporting the propulsion noise component of NMPB2008 [START_REF] Besnard | Road noise prediction -1 Calculating sound emissions from road traffic[END_REF].

Van trucks Intended for the delivery of packaged goods, box trucks form a significant part of medium-heavy vehicles. Detailed noise emission data is available from four trucks of two makes and different models, either with a rigid or a curtain sided load space. They have various characteristics in terms of gross weight, age and mileage. They are all equipped with a 6-speed gearbox, but with distinct ratios and therefore gear shifting at different speeds. The tyres are diversified in wear, dimensions and makes, including winter and retreaded ones. The vehicles were loaded such that the mass to power ratio was close to the recommendation of standard ISO 362 for noise certification, which requires 50 kg/kW. Information in the 8000 Hz octave was unavailable for one of them, without much significant impact on the recomposed global noise level.

Dump trucks Two dump trucks, used for the transport of bulk materials for public works, have been tested. They had different tyre makes and dimensions, engine powers, ages and both were equipped with a 8-speed gearbox. They were running empty. These vehicles are relatively noisy, mainly due to propulsion noise.

Bus Only one vehicle was available in this category, representative of a urban bus in new condition. It was equipped with a 6-speed gearbox and its maximum speed was limited to 70 km/h. It turned out to be relatively low-noise, behaviour which should be controlled with other buses or coaches.

Subcategory comparison

The average noise emissions in each subcategory are compared in Figure 3, in relation to the three prediction models described in section 2. Concerning propulsion noise, averaging tends to smooth out the vehicle-specific discontinuities spread over the speed range. Propulsion noise differences provide an obvious subcategory ranking dump trucks > van trucks > bus and may exceed 8 dB(A) between the average dump truck and the bus. Van trucks approximately lie at mid-range within this extent. Since the rolling noise components of the subcategories are relatively homogeneous, the same ranking is found on total noise at least up to 60 km/h beyond which rolling noise contributes to blur differences. Anyway, the bus remains the quietest over its whole operating speed range.

None of the three prediction models may adequately represent the total noise from the whole category, due to the wide discrepancies observed. Although one or the other prediction model may coincide with average van truck measurements in some octave bands, they overestimate van truck propulsion noise in global levels, and even more for CNOSSOS-2018. This has a corresponding impact on total noise levels. Dump truck propulsion noise outperforms predictions in most octave bands, at a lesser level with CNOSSOS-2018. In contrast the bus propulsion noise is lower than the prediction models in all octave bands, with the same behaviour transferred on total noise.

In octave bands, the relevance of either prediction model may vary with frequency. However in global levels, whereas CNOSSOS-2018 happens to be the most appropriate for dump truck subcategory, CNOSSOS-EU better reflects van truck behaviour below 50 km/h. Considering that rolling noise significantly contributes at high speeds, CNOSSOS-FR seems a relevant candidate prediction model for rolling noise.

Thus, the noise emission range -mainly controlled by propulsion noise -is quite wide between hypothetical extreme scenarios of a medium-heavy traffic, composed either only of dump trucks (for example near heavy industrial areas or quarries) or only of buses (in residential areas). An actual traffic composition lies somewhere in-between and the distribution between subcategories is essential for determining the representative average vehicle noise emission.

MEDIUM-HEAVY TRAFFIC NOISE EMISSION

The determination of the noise emission model representative of French vehicles of category 2 results from assumptions on the traffic composition, weighting the noise emitted by each subcategory. However, this composition may significantly vary with infrastructure, for instance a street in the city centre, an urban road-way or an interregional highway. Defining a noise emission model of an average vehicle relies on a compromise to represent the various realities.

Available traffic countings do not generally separate heavy vehicles from category 2 and 3 as defined in CNOSSOS, let alone subcategories. In order to perceive this traffic variability and its noise impact on the description of an average vehicle, subcategory counting has be punctually performed at four locations. These short-term daytime counts, carried out on working days, cannot definitely be representative of the Annual Average Daily Traffic on these road segments, neither of the traffic diversity on other roads and should be considered for indicative purposes only.

Approach

A traffic composition of category 2 is defined by: -a proportion n v of van trucks (including refrigerator trucks), -a proportion n d of dump trucks (including tank-, flatbed-, container-, refuse trucks), -a proportion n b of buses (urban buses and coaches)

with n v + n d + n b = 1.
For a traffic composition (n v , n d , n b ), in each octave band i the propulsion noise of an average vehicle L prop,i (v) is the weighted energetic mean of each subcategory propulsion noise, in the common speed interval. The next step consists in calculating the optimal coefficients ÂP,i and BP,i of an emission model given by the CNOSSOS equations, fitted to the noise level L prop,i (v) (Equation 11). The same approach is carried out for rolling noise and gives ÂR,i and BR,i from L roll,i (v).

ÂP,i , BP,i = min

A P,i ,B P,i

L prop,i (v) -A P,i -B P,i v -v ref v ref 2 (11) ÂR,i , BR,i = min A R,i ,B R,i L roul,i (v) -A R,i -B R,i log v v ref 2 (12)

Traffic scenarios and associated noise models

Road function turns out to be unimportant for the overall noise emission. From the four traffic counting sites, two groups stand out particularly, each including two sites with dissimilar functions. The key distinguishing factor between the two groups is the proportion of public transport vehicles, with a direct impact on the propulsion noise component but a lower one on rolling noise. Consequently, two typical scenarios are retained (Table 1) for propulsion noise while rolling noise component is common to both. The noise emission model associated with each scenario, extrapolated up to 110 km/h, is drawn in Figure 4. Since differences between scenarios affect propulsion noise only, total noise acoustically differs at low speed but not at high speed. With a high proportion of public transport vehicles, propulsion noise dominates up to 46 km/h, then rolling noise prevails. If the public transport ratio is low, rolling noise dominates from 52 km/h. This seems reasonable at the light of road noise knowledge. On the contrary, CNOSSOS-EU global rolling noise reaches propulsion noise contribution at about 100 km/h only and rolling noise is of the same order of magnitude as propulsion noise at the best, even at the frequencies where its contribution is known to be strong.

The propulsion noise difference between scenarios is almost constant over the whole speed range, equal to 1.6 dB(A) in global levels. This gives a varying difference on total noise, from 1.6 dB(A) at low speed and decreasing to become insignificant at high speed. It remains over 1 dB(A) up to 40 km/h (Figure 4, right).

Comparison to CNOSSOS

Considering the road surface of the test sites, both scenarios are compared to CNOSSOS-EU and CNOSSOS-2018 in reference conditions, and to CNOSSOS-FR with the road surface group R2 non-drainage (Figure 5). The relevance of the CNOSSOS versions with the average models associated to the two scenarios depends on trafic distribution and speed.

CNOSSOS-EU is well adapted to the scenario with a high public transport ratio up to 40 km/h but undervalues the other scenario by 1 to 1.5 dB(A). In both cases it undervalues noise emission at high speeds, up to 4 dB(A) at 110 km/h and road corrective terms are required.

CNOSSOS-2018 overestimates overall noise emission, up to 1.6 dB(A) at low and medium speeds with a low public transport ratio and up to 2.8 dB(A) if this ratio is high. Thus, its application would be tricky in this case since corrective terms provide no mean of reducing propulsion noise level contribution.

CNOSSOS-FR is now recommended for new noise mapping rounds. Its ability to correctly figure the noise situation is essential. On the present experimental test basis, for the road category R2 non-drainage, it happens to be an acceptable compromise for both scenarios in urban conditions. In a typical urban scenario with a high public transport ratio, overestimation is lower than 1 dB(A) at 20-30 km/h and smaller than 2 dB(A) over the whole speed range. If the public transport ratio is low the noise prediction error remains within ±1 dB(A) up to 50 km/h and does not exceed 1.3 dB(A) at medium and higher speeds.

CONCLUSIONS

The present study carried out detailed noise measurements on a series of controlled medium-heavy vehicles to document the French unreferenced noise emission of this vehicle category. By considering three subcategories, an average vehicle from two traffic composition differing by their ratio of public transport vehicles has been considered, with a noise issue occurring at urban speeds only. It turns out that the potentially updated CNOSSOS-EU version as discussed in 2018 overestimates noise emission up to 2.8 dB(A) if the bus proportion within category 2 traffic is high. For the road surfaces DAC 0/10 tested, the CNOSSOS version adapted and now recommended for French roads gives noise emission predictions within ±1 dB(A) up to 50 km/h if the bus ratio is low, up to 30 km/h if this ratio is high. At most speeds, it overvalues global noise levels without exceeding 1.8 dB(A).

  Figure 1: Comparison of the overall noise emission models for vehicle category 2, Aweighted global noise levels at 7.5 m

Figure 3 :

 3 Figure 3: Average noise emission by subcategory and noise prediction models, Aweighted maximum global noise levels at 7.5 m -Propulsion noise (top left), rolling noise (top right), total noise (bottom left)

Figure 4 :

 4 Figure 4: Global noise emission of an average medium-heavy vehicle according to scenarios, A-weighted maximum global noise levels at 7.5 m -Left: noise model -Right: noise difference between scenarios

Figure 5 :

 5 Figure 5: Comparison of the overall noise emission models for an average vehicle of category 2, A-weighted global noise levels at 7.5 m

Table 1 :

 1 Proportion of vehicles in each subcategory

	Scenario	n v	n d	n b
	1	61 % 34 % 5 %
	2	27 % 20 % 53 %

marie-agnes.pallas@ifsttar.com

ACKNOWLEDGEMENTS

This project has been funded by the French Ministry for an Ecological and Inclusive Transition. It was also supported by the LabEx CeLyA of Université of Lyon, operated by the French National Research Agency (ANR-10-LABX-0060/ANR-11-IDEX-0007). The authors thank the entities who made some vehicles available.


A discrete

A discrete dislocation model of Stage I fatigue crack growth and an analysis of Mode I to Mode II transition at low �K1

V. Doquet

Laboratoire de Mecanique des So/ides, UMR 7649 du CNRS, Ecole Po/ytechnique, 91128 Palaiseau cedex, France Abstract.Simulations of dislocations nucleation and glide ahead of a crystallographic mode II crack are performed for push-pull and reversed torsion . An influence of the normal stress on the friction of crack flanks, as well as on the condition for dislocation emission is introduced.The crack growth rates are deduced from the dislocation fluxes at the crack tip. A comparison between the loading modes is made. Taking into account the presence of grain boundaries, the repeated decelerations and sometimes the arrest that characterise Stage I crack growth are described by the model. An analysis of the transition from mode I to mode II crack growth observed at low L'l.K1 in single crystals is proposed.

1.INTRODUCTION

In the first stage of their development, fatigue cracks are driven by the cyclic shear, but their growth rate is influenced by the normal stress: it is increased by an opening stress and reduced by a compressive one. The absence of an opening stress in torsion is often considered responsible for slower stage I growth and longer fatigue life compared to push-pull, for equivalent strain ranges [ 1 ]. The first aim of this work is to suggest mechanisms by which the normal stress could influence Stage I crack growth and illustrate them through micromechanical simulations. The second aim is to model the interactions of Stage I cracks with microstructural obstacles and the resulting irregularities in the crack growth kinetics. The third aim is to analyse the transition from stage II to stage I observed during push-pull tests on single crystals at low '1.K1.

2.A BRIEF DESCRIPTION OF THE MODEL FOR UNCONSTRAINED SLIP

Stage I fatigue cracks grow along localized slip-bands. They are submitted to either pure shear loading in torsion or combined opening and shear in push-pull. But plasticity -that is dislocation nucleation and glide -is related only to the shear components and is restricted to the slip band colinear to the crack, otherwise (that is : if noncoplanar slip is activated) there is a transition towards Stage II propagation.

In the simulations, only mixed mode 1+11 loading has been considered. The dislocations that arc emitted from the crack tip along the coplanar slip plane thus have a pure edge character. They are supposed to be perfect dislocations, but cross slip is not envisaged. In a first approach, decribed in details in [2], no obstacle to dislocation glide was considered. In a second phase, grain boundaries are introduced.

The cyclic loading path is followed by incremental time steps, '1.t, small enough for the velocity of each dislocation to be considered constant over .1.t. The nominal stress intensity factors for a Stage I crack of length 2a lying along the critical plane (i.e. the plane where the shear stress range is maximum) arc calculated at each time-step as:

where CTncp and 'tcp are the current values of the normal stress and shear stress on the critical plane . Since real Stage I cracks are not straight, allowance has to be made for asperity-induced friction that tends to reduce the crack driving force, and is either enhanced or reduced by a compressive (respectively tensile) normal stress . Experimental information concerning crack flanks frictional interactions for cracks loaded in mode II, in presence of a static normal stress, has been obtained through combined tension and torsion tests performed inside a scanning electron microscope on precracked tubular specimens of maraging steel (2]. An attempt is made here to reproduce qualitatively these experimental data through empirical equations. The effective mode II stress intensity factor is thus calculated, assuming that a uniform, normal-stress dependent friction stress, c, exists along the crack flanks:

with

K[[f = K!r°m ±cma (2) c = co.exp(-k+.Oncp) if Oncp �O c = co.exp(-k_.<Jncp) if O"ncp $ 0 (3)
co characterises the friction stress in the absence of any normal stress and is thus related to the tortuosity of the crack path . k+ and k_ are two constants, the latter connected with the friction coefficient of the material.

The simulations described below have been performed with k+ =0.057, L =0.014 (c is divided by one hundred for O"np = 80MPa but is multiplied by three only for O"np = -80MPa consistently with experimental data reported in [2]). The sign affecting the second term in (2) depends on the loading (-) or unloading ( +) situation. Figure 1 shows how K! 1 °m and Kf[ f vary in time during a reversed torsionanda�h-pull cycedesail:xrl respxtivdyby: 't = 'to.sinwt and cr = cro.sinrot with cro = 2-ro (equivalent stress ranges in the sense of Tresca) assuming that the crack lies along the maximum shear stress plane in each case. Note that .�.K!i°m is the same in both cases. In push-pull , KWmax is higher than in torsion, but the effective loading cycle is asymetrical, due to enhanced friction in the compressive part of the cycle. The critical mode II stress intensity factor for the emission of a coplanar edge dislocation from the crack tip, which depends on the mode mixity parameter \jl = arctan(Kn/K1) , is then calculated , using the expression given by Sun, Beltz and Rice [3]:

K ffcl = ,Y (:�) ["fus -0:.()1ls -"fu,)(� -\jl ) ] (4)
where a , 11ls and Yus are material parameters estimated by atomic models (density functional theory or embedded atom method) and tabulated in [3] for a few materials ('fusf'fus and o: are approximately 0.866 and 1.2, respectively, for Al and Ni). Since 'fus >'fus, equation ( 4) predicts a lower threshold stress intensity factor for dislocation nucleation when an opening stress is present. This is not the case in torsion, for which \jl = �• For push-pull, \jl = � during the tensile stage, but \jl = � in the compressive stage, since K I is then zero . For Al and Ni, equation ( 4) thus gives a nucleation stress intensity factor in the tensile stage of the push-pull cycle that is 92% of its value in torsion or in the compressive part of the cycle . The shear stress on each dislocation in the plastic zone is then evaluated as: in which the first three terms represent, respectively, the crack tip stress, the image stress and the stress of other dislocations. The last term is the far field term, negligible for dislocations located very close to the crack tip, but more important for leading dislocations, when the size of the plastic zone is a relatively large fraction of the crack size. In the following, however, the size of the plastic zone will never exceed 0.2a, so that equation ( 5) may be considered a reasonable approximation. The velocity of each dislocation is then calculated as: vi= v0.sign(bj).sign('tj).�-ril --rrY11 ( 6) where (x) is zero if x$0 and x otherwise. 't f is the resistance to dislocation glide when slip is not constrained by a grain boundary. For the simulations described below, 'tp varied between 20 and 90MPa. Typical values of vo and m for F.C.C. metals.(13msl and 0.88, respectively) were chosen.

The new position of each dislocation is then deduced, and annihilation criteria are checked: if a dislocation comes close enough to the crak tip or if the distance between two dislocations with opposite signs becomes less than 16nm, they are removed from the simulation.

The real mode II stress intensity factor, allowance made for crack tip shielding by the dislocation stress field [4], is then evaluated as:

The dislocation emission criterion:

K tip _ K ett � µ b i (?) II -II -� (1-V)'/21tXi K�� :::; -Kff1c1 or K�� � Kff1c1 (8)
is then checked to decide whether a negative or positive dislocation can be emitted.

This sequence is repeated until the second cycle is completed. Then, the crack growth rate per cycle is deduced from the dislocation flux as follows. The crack is considered to grow by one Burgers vector each time a pair of positive-negative dislocation has been emitted or when a dislocation returns to the crack tip. In the latter case, it is assumed that even though the crack tip geometry before the dislocation nucleation is, in principle, recovered when this dislocation comes back, the free surface increment created at nucleation, that has been exposed to environment and gas adsorption in the meantime, cannot be rewelded. Anyway, both events correspond to some cyclic plastic flow at the crack tip and should thus contribute to its growth.

3.RESULTS FOR UNCONSTRAINED SLIP

The dislocation dynamics during push-pull and reversed torsion loadings is analysed in details in [2]. Only global results will be given here. Simulations have been carried out for reversed torsion and push-pull for various loading ranges and various friction stresses co. The results are drawn on a bilogarithmic plot of da/dN versus &<:fr 0m on Fig. 2 (curves labelled B, E, F). For torsional loading, the effective loading is fully reversed, and no opening stress affects the threshold for dislocation emission, so that the calculated growth rates follow a very simple equation:

.fill_ = (1-v) ( � K eff2 -4 K n u c 1 2 ) (9) in which Kfiuc l is the emission threshold in pure shear. The threshold for propagation is thus 2Kr1 uc 1 in that case. Quite similarly to closure effects for mode I propagation, the friction stress along the crack flanks shifts the da/dN versus &<:fr0m curve compared to the "intrinsic" curve corresponding to zero friction.

Comparing the curves labelled A and B, it appears that even without friction, the crack growth rate is higher in push-pull than in reversed torsion, but the difference, appreciable for small loading ranges, vanishes as the loading range increases. This is probably because close to the threshold, the critical stage for crack propagation is dislocation nucleation, which is made easier by the opening stress, in push-pull, whereas for higher loads, nucleation is no more critical, but reverse dislocation glide, more natural under reversed torsion because of the symetry of the effective loading, is. According to this trend, the ratio between torsional and tensile fatigue lives should thus increase as the loading range decreases. This corresponds to experimental observations [l] and it is thus very encouraging. When crack flanks friction is introduced, the difference in Stage I kinetics between the two loading modes is amplified.

4.INTERACTIONS WITH MICROSTRUCTURAL OBSTACLES

4.1.Influence of an impenetrable obstacle

If there is an impenetrable obstacle ahead of the crack, the dislocations emitted pile up there, and as the crack propagates towards the obstacle, the shielding effect due to the pile up is stronger and stronger. The increment of K[{f necessary for the emission of a new dislocation is thus higher and higher. and there are fewer dislocations emitted, even though � Ki'i°m i na l increases, so that the crack decelerates. When the crack tip is finally so close to the obstacle (say, less than the core radius of a dislocation) that there is not enough space for a single dislocation loop to expand from the tip, the emission criterion of equation ( 8) is not applied any more. The crack is thus predicted to stop if the obstacle has not yet been overcome by dislocations when the crack tip is closer than a few nanometers .

Simulations of stage I growth for two constant stress ranges and thus increasing Mpf m were performed, in reversed torsion, for a crack approaching an impenetrable obstacle. The friction stress, 'tf was 30MPa Figure 3 shows the evolution of the calculated growth rate. For both cases illustrated, 'tcp, m ax > 't f, so that the crack-tip plastic zone hits the obstacle for each crack length considered. But for the smallest stress range, the number of dislocations emitted is small, so that down to 5µm distance from the obstacle, the rise of the shielding term in equation ( 7) is slower than the increase in K[ff , and the crack is able to accelerate, in spite of its constrained plastic zone. For the highest stress range , the rise of the shielding term compensates that of K[ff at the beginning, and then exceeds it, whence the constant and then decreasing growth rate.

4.2.Influence of grain boundaries

Grain boundaries (G.Bs.) are generally not impenetrable to dislocations. They can be sinks for dislocations as well as sources. Dislocations are assumed to pile-up on a G.B, until the shear stress on the leading dislocation reaches a value allowing slip transfer into the next grain, on a slightly misoriented plane. The tilt of the crack path beyond the G.B is not taken into account explicitely, but some of it is modelled through the roughness-induced friction stress. The critical shear stress for slip transfer, 'to.B .. should depend on the crystallographic misorientation of the adjacent grains, and on the angle of incidence of the slip plane on the G.B. plane.

Simulations of stage I growth at a constant stress range (�'t/2=36.5MPa) have been performed, in reversed torsion, for a crack of initial half length, a = I 05µm, in a polycrystal with a I 5µm mean grain size and a 30MPa friction stress. The calculated crack growth rates are plotted versus the crack length on Fig. 4, for two values of the critical stress for slip transfer at grain boundaries, 'to.B .. small enough to allow the crack to cross the grain boundaries. The size of the plastic zone, in microns, is indicated for each point . It can be seen that the condition for slip transfer beyond the first G.B.is met earlier and earlier in successive grains because of the increase in Af<f\0 m associated with an increase in the number of dislocations piled-up against the G.B .. But this does not produce a sudden acceleration, since slip transfer is progressive: the plastic zone may well extend over the G.B. but a substantial fraction of the emitted dislocations may still be withheld by the G.B. and have a strong shielding effect. For example, for 'to.B. =400MPa, the crack growth rate for a=l64µm is smaller than for a=I60µm, even though the plastic zone is larger, since in the former case, only 3 of the emitted dislocations have been transmitted to the second G.B and 8 of them remained stacked in a lµm wide zone ahead of the crack tip. It also appears that each time the crack tip enters a new grain, there is a sudden acceleration. The magnitude of this acceleration depends on 'to .B . . since the stronger 'to.B . is, the higher the fraction of emitted dislocations withheld by the first G.B., and thus the larger the decrease in dislocation shielding when the crack passes the G.B ..

Influence of the loading mode

It had been observed, on a Co45Ni alloy, that for equivalent stress ranges, it takes approximately twice as long for microcracks, once initiated, to cross the first G.B. encountered in reversed torsion than in push pull [1] but it was not clear whether this came merely from a slower transgranular growth rate, or if the interaction between microcracks and G.Bs. was stronger in torsion. The present simulations throw some light on that point. Figure 5 compares the stress concentrations due to the emitted dislocations, at lµm from the G.B. in the next grain, for a crack propagating toward a G.B., for reversed torsion and push-pull loadings of equivalent stress ranges. The stress concentration in the next grain is much higher in push-pull, because the opening stress increases KI{f and decreases Kf1 ucl, so that many more dislocations are emitted than in torsion. Slip transfer is thus easier, which obviously facilitates G.B. crossing, not to mention the possibility to activate non coplanar slip systems in the next grain, that would be able to shield the mode I singularity, contrary to coplanar slip. The slower development of microcracks in torsion thus probably results from stronger interactions with the microstructure as well as from the lower transgranular growth rate illustrated in Fig. 2.

5.TRANSITION FROM MODE I TO MODE II PROPAGATION

Several authors, among which Petit et al. [5] report transitions from mode I to mode II fatigue crack growth observed during push-pull tests on single crystals at decreasing AKr, when the mode I growth rate is about 10l OmJcycle. An equivalent transition from true mode I to "crystallographic mode I" is also observed in polycrystals close to the threshold. This transition is analysed below in terms of dislocation dynamics for the ideal case where mode I crack growth is due to dislocation emission on two symetrically inclined slip planes. For an FCC structure, this means that the crack plane is of (100) or (110) type, and that the angle between the crack plane and slip planes is 0 = 54°73 or 0 = 35°26 respectively. The threshold stress intensity factor for the emission of dislocations parallel to the crack front on inclined slip planes has been estimated by Sun et al. [3], neglecting tension-shear coupling ('Yll s='fus)

as:

16 µYu s • . o K r. emission=
that IS. 3.635Kn,emission for e = 54 73 and 2.76Kn,emission for (1-v)(l +cos0)sin20 0 = 35°26, where Kn.emission= � is the threshold for coplanar emission in mode II. Riemelmoser et al. [6] have shown that since mode I is not a reversed loading, Kr,max must exceed 1.3Kr,emission for some dislocations to come back to the crack tip, aided by the repulsive force from previously emitted dislocations. So, during push-pull tests at decrasing AKr, when Kr.max becomes less than l .3Kr,emission• the crack can no more propagate in mode I for lack of cyclic plasticity at its tip. But assume it bifurcates on either of the two active slip planes. The mode II stress intensity factor at the tip of the small branch, during the tensile stage of the cycle is, according to Amestoy et al. [7], k;r = F(0)Kr, where F(0)=0.358 for 0 = 54°73 and 0.272 for 0 = 35°26. Thus k;, =1.3F(0)K1,emission = l.284Kn,emission in both cases, which is largely sufficient for coplanar dislocation emission. In addition, contrary to mode I loading, the shear loading on the branch of length 2s is partially reversed: during the compressive stage of the cycle, the main crack is closed and Kr =0, but k;, can be roughly estimated as if the main crack did not exist as: k;, =crsin0cos0ms = 0.47crms, where cr is the applied compressive stress. This change in sign of k;, should allow some dislocations to come back to the crack tip and thus propagation along this slip plane.

6.CONCLUSIONS

Simulations of Stage I fatigue crack growth by dislocations emission/annihilation at their tip, taking into account their interactions with grain boundaries have been developped. The existence of a threshold for crystallographic mode II crack growth, even in single crystals, was predicted. The influence of friction stresses along the crack flanks was illustrated: it shifts the da/dN versus AKft0 m curve compared to the "intrinsic" curve without friction. The periodic decelerations, sometimes leading to crack arrest typical of microstructure-sensitive Stage I propagation were simulated. Stage I is predicted to be slower under reversed torsion than under push-pull for equivalent stress ranges in the sense of Tresca, because of slower transgranular growth rate and more difficult G.B. crossing. The ratio of torsional to tensile fatigue lives should depend on the roughness of Stage I cracks and the frictional properties of the material. An analysis of the transition from mode I to mode II propagation observed in single crystals as well as from true mode I to "crystallographic mode I" in polycrystals at low AK1 has been proposed.

dN 4 .

 4 µ .b:tf' II . 11

Fig. 1 Fig. 4

 14 Fig.1 Evolution of nominal and effective Kn for a crack lying along the maximum shear stress plane under reversed torsion and push-pull loadings of equivalent amplitude in the sense of Tresca.

. 6 .Fig. 2 Fig. 3 Fig. 5 .

 6235 Fig.2 Computed growth rates versus nominal ti.Kn for (A) push-pull, no friction (B) reversed torsion, no friction (C) push-pull, co=5MPa (D) push-pull, co=lOMPa (E) reversed torsion, co=5MPa (F) reversed torsion, co=lOMPa