R González-Martínez

J Sertucha

CIRIMAT Jacques Lacaze
email: jacques.lacaze@ensiacet.fr

Laine -3 68%

Si Laine

Si 35%

Avery -Simo

Shiao

wt.% Si Kobayashi

alloy Kobayashi

The mechanism of intermediate temperature embrittlement of cast irons by magnesium

Keywords: Spheroidal graphite cast iron, intermediate temperature brittleness, magnesium, oxygen

Metallic alloys such as steels and nickel-based alloys are potentially vulnerable to high-temperature brittleness in the range of 500°C to 900°C that results in a loss of ductility. Ferritic spheroidal graphite cast irons show similar loss of ductility at temperatures between 350°C and 500°C, suggesting the term intermediate temperature brittleness. Previous works have shown that sulfur and magnesium are detrimental while phosphorus added at a reasonable level is beneficial in preventing loss of ductility in ferritic spheroidal graphite castings.

In order to clarify the role of magnesium, a ferritic spheroidal graphite cast iron with high silicon content and low sulfur and phosphorus contents was subjected to tensile tests between room temperature and 650°C. It was found to be susceptible to embrittlement at intermediate temperatures around 400°C. MgO precipitates were observed on the fracture surface, which increased in size with test temperature, confirming the role of the environment that was reported by previous researchers. This finding is cross-checked with the results of the literature and allows us to propose a scheme for intermediate temperature embrittlement of ferritic spheroidal graphite iron.

Introduction

Nickel-based alloys and steels exhibit a ductility trough between 500°C and 1000°C, as studied by Lei Zheng et al [START_REF] Lei Zheng | Mechanism of intermediate temperature embrittlement of Ni and Ni-based superalloys[END_REF] for nickel-based alloys and Lankford [START_REF] Lankford | Some considerations of strength and ductility in the continuouscasting process[END_REF] and Mintz and Crowther [START_REF] Mintz | The influence of composition on the hot ductility of steels and to the problem of transverse cracking[END_REF][START_REF] Mintz | Hot ductility of steels and its relationship to the problem of transverse cracking in continuous casting[END_REF] for steels in connection with continuous casting. Steels also exhibit low temperature brittleness with a transition temperature from ductile to brittle fracture (TTDB) that varies from about -100°C to about +100°C depending on the alloy composition and process parameters. In particular, it is known that the segregation of sulfur and phosphorus at the grain boundaries results in a shift of the TTDB to higher temperatures, making many steels susceptible to temper embrittlement. This segregation decreases grain boundary strength that favors intergranular fracture. In the high-temperature regime, the ductility trough is related to intergranular fracture, but is generally not due to grain boundary segregation, but depends on precipitation.

Ferritic spheroidal graphite cast irons (SGI) are similar to steels in that they exhibit low temperature brittleness with the TTDB located between -50°C and room temperature (RT) or slightly above, and a ductility trough in the temperature range of 300°C to 500°C. The latter is associated with the occurrence of intergranular fracture (IG) and has been referred to as 673 K embrittlement by Yanasigawa and Lui [START_REF] Yanagisawa | Influence of the structure on the 673 K embrittlement of ferritic spheroidal graphite cast iron[END_REF] and high temperature embrittlement by Wright and Farrell [START_REF] Wright | Elevated temperature brittleness of ferritic ductile iron[END_REF]. In the present work, we will adopt the name intermediate temperature embrittlement (ITB) which has also been proposed and is more appropriate for the temperature range involved. Wright and Farrell [START_REF] Wright | Elevated temperature brittleness of ferritic ductile iron[END_REF] demonstrated that ITB in SGI increases with increasing sulfur and magnesium content and suggested a mechanism based on sulfur diffusion at grain boundaries. In this temperature range, phosphorus also tends to segregate at grain boundaries in SGI, as confirmed by Chen et al [START_REF] Chen | The effect of phosphorus segregation on the intermediate-temperature embrittlement of ferritic, spheroidal graphite cast iron[END_REF][START_REF] Chen | The role of magnesium-containing spheroidizer and counteraction of misch metal in the intermediate temperature intergranular embrittlement of ferritic nodule iron[END_REF] using Auger electron spectroscopy. However, these authors came to the conclusion of the role of MgO particles on the fracture, in agreement with other works [START_REF] Yanagisawa | Influence of the structure on the 673 K embrittlement of ferritic spheroidal graphite cast iron[END_REF][START_REF] Lui | High-temperature properties of ferritic spheroidal graphite iron[END_REF][START_REF] Kobayashi | 673K embrittlement of ferritic spheroidal graphite cast iron by magnesium[END_REF][START_REF] Shiao | Eutectic cell wall morphology and tensile embrittlement in ferritic spheroidal graphite cast iron[END_REF]. In addition, phosphorus has been shown to be beneficial in decreasing the incidence of ITB in alloys with magnesium/phosphorus ratios less than 1.5 [START_REF] Kobayashi | 673K embrittlement of ferritic spheroidal graphite cast iron by magnesium[END_REF][START_REF] Takanezawa | Fracture mechanism and suppression method of 673 K embrittlement in spheroidal graphite cast iron[END_REF]. Similarly, ferrite grain size refinement is effective in reducing the risk of brittle fracture and loss of ductility [START_REF] Yanagisawa | Influence of the structure on the 673 K embrittlement of ferritic spheroidal graphite cast iron[END_REF][START_REF] Chao | The effect of microstructure of ferritic spheroidal graphite cast irons on intergranular fracture at intermediate temperatures[END_REF].

Experimentally, ITB is best evidenced by the abrupt decrease in the temperature range 300-500°C in the elongation at rupture, A, or the decrease in the reduction area at rupture, RA. It has been shown quantitatively by Iwabuchi and Kobayashi [START_REF] Iwabuchi | Prevention of elevated temperature brittleness in ferritic ductile iron by phosphorus[END_REF][START_REF] Iwabuchi | Suppression of elevated temperature brittleness in spheroidal graphite cast iron by increasing phosphorus content[END_REF] that these two quantities are nearly equal and thus can be considered equivalent in practice, which will be used here to report data from various sources.

Interestingly, although most often studied by tensile testing, this embrittlement also plays an important role in cyclic loading, as revealed by low-cycle fatigue and rotating bending fatigue data [START_REF] Wright | Elevated temperature brittleness of ferritic ductile iron[END_REF][START_REF] Kobayashi | 673K embrittlement of ferritic spheroidal graphite cast iron by magnesium[END_REF][START_REF] Garcia Trelles | Experimental characterization of the short crack growth behavior of a ductile cast iron (DCI GJS-500) affected by intergranular embrittlement at temperatures nearby 400°C[END_REF]. Not all ferritic SGIs are sensitive to ITB as shown in Figure 1 using data from Wright and Farrell [START_REF] Wright | Elevated temperature brittleness of ferritic ductile iron[END_REF] for SGIs with 2.2-2.6 wt% Si (more details are given in the Appendix A). At 400-425°C, Figure 1 shows that RA falls below 10% for alloys B, E, and G but not for alloy H (solid triangles). This alloy H had been spheroidized with low magnesium addition, leading to a residual magnesium content of 0.023 wt%, less than half the amount of the other three alloys.

Figure 1. Evolution with temperature of the reduction of area at rupture for the alloys B, E, G and H studied by Wright and Farrell [START_REF] Wright | Elevated temperature brittleness of ferritic ductile iron[END_REF]. Whenever there were duplicate values, this is the mean value that is plotted.

Wright and Farrell confirmed that embrittlement is associated with IG fracture and showed that it initiates on micro-porosities. The authors also noted that low ductility at 425°C, i.e., a low RA value, is associable with high magnesium and sulfur contents, and drew a parallel with ferritic steels by considering sulfur to be the embrittler. On the other hand, Iwabuchi and Kobayashi [START_REF] Iwabuchi | Suppression of elevated temperature brittleness in spheroidal graphite cast iron by increasing phosphorus content[END_REF] concluded that IG fracture occurs due to segregation at the grain boundaries and associated it with a %Mg/%P ratio greater than 1.5 as proposed by Takanezawa et al. [START_REF] Takanezawa | Fracture mechanism and suppression method of 673 K embrittlement in spheroidal graphite cast iron[END_REF], where %Mg and %P are the magnesium and phosphorus contents of the alloy expressed in wt%.

Most of the studies on ITB in spheroidal graphite cast irons were concerned with alloys that had been heat-treated to ensure a fully ferritic matrix. The quality of the spheroidizing treatment has been checked by light optical metallography on one tensile specimen tested at RT. Figure 2 shows a fairly even distribution of well-rounded graphite spheroids. Finally, one piece of this tensile specimen was used for chemical analysis whose results are listed in Table 1.

Results.

The tensile tests curves of duplicates were quite reproducible and one for each of the test temperatures has been plotted in Figure 3. The values of UTS and A for all tests, including the duplicates, are listed in Table 2. In some cases, the rupture occurred outside the grips of the extensometer in which case A was not reported. The alloy showed the maximum UTS values at room temperature (RT) while the A values at 8.8% and 13.1% are in the high range of previously reported data on SGI with about 5 wt.% Si [START_REF] González-Martínez | Effects of high silicon contents on graphite morphology and room temperature mechanical properties of as-cast ferritic ductile cast irons[END_REF]. When the testing temperature increased to 100°C, the UTS values decreased a little and the A values increased slightly over the RT results at 14.3% and 15.9% that possibly indicates that the chemical ordering of ferrite already started to decrease. At 350 °C and above, the UTS values continuously decreased while A was observed to fall below 5% from 350°C to 500°C and then to increase to quite high values at 650°C. The alloy thus appeared sensitive to ITB with a ductility trough between 350°C and 500°C with a possible minimum at 475°C. In Figure 4, A values from the present work are plotted as a function of the testing temperature with solid symbols that illustrate the ductility trough between 350°C and 500°C. For comparison with literature, data for SGI with silicon content higher than 3.5 wt.% were selected [START_REF] Kobayashi | 673K embrittlement of ferritic spheroidal graphite cast iron by magnesium[END_REF][START_REF] Shiao | Eutectic cell wall morphology and tensile embrittlement in ferritic spheroidal graphite cast iron[END_REF][START_REF] Laine | The mechanical properties of ductile iron at intermediate temperatures: The effect of silicon content and pearlite fraction[END_REF], including a molybdenum alloyed SiMo SGI [START_REF] Avery | Effect of temperature cycle on thermomechanical fatigue life of a high silicon molybdenum ductile cast iron[END_REF].

As seen with Figure 1 for low silicon SGI, it is noted that some high silicon alloys are immune to ITB. These alloys are those with low magnesium content and, generally, a low %Mg/%P ratio in agreement with results by Takanezawa et al. [START_REF] Takanezawa | Fracture mechanism and suppression method of 673 K embrittlement in spheroidal graphite cast iron[END_REF] and Cheng et al. [START_REF] Cheng | Effect of residual magnesium contento n termal fatigue cracking behavior of high-silicon spheroidal graphite cast iron[END_REF]. In the same line, it is noticeable that Chen et al. [START_REF] Chen | The role of magnesium-containing spheroidizer and counteraction of misch metal in the intermediate temperature intergranular embrittlement of ferritic nodule iron[END_REF] suppressed ITB in SGI containing 3.9 wt.% Si by spheroidizing with Ce and no Mg, and that even a 0.011 wt.% residual Mg was sufficient to generate ITB. [START_REF] Shiao | Eutectic cell wall morphology and tensile embrittlement in ferritic spheroidal graphite cast iron[END_REF] and Laine et al. [START_REF] Laine | The mechanical properties of ductile iron at intermediate temperatures: The effect of silicon content and pearlite fraction[END_REF], as well as on a SiMo SGI [START_REF] Avery | Effect of temperature cycle on thermomechanical fatigue life of a high silicon molybdenum ductile cast iron[END_REF].

An interesting feature that Wright and Farrell [START_REF] Wright | Elevated temperature brittleness of ferritic ductile iron[END_REF] reported is that the ductility loss is much less apparent when considering the evolution of UTS as function of temperature because it continuously decreases. This means that UTS values of samples showing or not IG fracture should be comparable. In order to compare literature results for cast irons with differing silicon contents, Figure 5 shows the evolution of UTS after normalizing each series of data with their maximum value at room temperature. Results for low and high silicon contents are reported with open and solid symbols, respectively, with literature data from Wright and Farrell [START_REF] Wright | Elevated temperature brittleness of ferritic ductile iron[END_REF],

Iwabuchi et al. [START_REF] Iwabuchi | Suppression of elevated temperature brittleness in spheroidal graphite cast iron by increasing phosphorus content[END_REF] and Laine et al. [START_REF] Laine | The mechanical properties of ductile iron at intermediate temperatures: The effect of silicon content and pearlite fraction[END_REF]. Data for the SiMo SGI studied by Avery et al. [START_REF] Avery | Effect of temperature cycle on thermomechanical fatigue life of a high silicon molybdenum ductile cast iron[END_REF] are plotted with a large plus sign as in Figure 4.

It is seen in Figure 5 that all UTS values show the same temperature evolution within a reasonable scatter band, with some of the values of the present study appearing however quite low. Though not further investigated, this could be related to the fact that there was no heat-treatment applied to the present material. It is noticeable that UTS processes in two steps at increasing temperature, with a slow and then steeper decrease below and beyond a critical temperature of about 300°C. Such a behavior had already been noticed for a SiMo SGI [START_REF] Hervas | Damage analysis of ferritic SiMo ductile cast iron submitted to tension and compression loadings in temperature[END_REF] and is here extended to the whole silicon range used in SGI. Farrell [START_REF] Wright | Elevated temperature brittleness of ferritic ductile iron[END_REF], Iwabuchi et al. [START_REF] Iwabuchi | Suppression of elevated temperature brittleness in spheroidal graphite cast iron by increasing phosphorus content[END_REF], Laine et al. [START_REF] Laine | The mechanical properties of ductile iron at intermediate temperatures: The effect of silicon content and pearlite fraction[END_REF] and Avery et al. [START_REF] Avery | Effect of temperature cycle on thermomechanical fatigue life of a high silicon molybdenum ductile cast iron[END_REF]. Normalization was carried out using the RT value for each of the series of results.

SEM characterization of the rupture surfaces was systematically carried out and Figure 6 presents a representative micrograph for six test temperatures. At room temperature, Figure 6a shows that the alloy exhibited mainly cleavage fracture with some areas of intergranular fracture such as the one indicated with the arrow. At 100 ºC, the fracture surface was very similar to that at room temperature, only a slight increase of the ductile fracture area was observed. On the fracture surface of the material tested at 350 °C (Figure 6b), both an increase of the area of intergranular fracture and a change from cleavage to ductile fracture are observed. With further increase in testing temperature up to 450°C (Figure 6c), 475°C (Figure 6d) and 500°C (Figure 6e), the intergranular fracture area increased further and occupied most of the surface at 475°C. At 500ºC, the fracture started switching from intergranular to ductile fracture with the apparition of areas covered with small cupules (see later, Figure 8). Finally, the samples tested at 650°C exhibited a fully ductile fracture (Figure 6f) with much larger cupules containing one or more graphite spheroids. Detailed observation of the ruptured surfaces evidenced the presence of particles on the IG facets as exemplified with the few micrographs in Figure 7. After testing at RT, the particles that are observed are very small; see Figure 7a and the enlarged view in Figure 7b. Already after testing at 100°C, Figure 7c and 7d, many more and larger particles could be noticed. The surface of the sample tested at 450°C was found to be covered by numerous hemispherical aggregates, typically 0.5 µm in size, as seen in Figure 7e and 7f. Note that care was taken to have the micrographs in Figure 7a, 7c and 7e at exactly the same enlargement so as to evidence the dramatic increase in the size of the precipitates as the test temperature increased. EDS spectra (see examples in Appendix B) showed these particles to contain high levels of Mg and O, and thus certainly to be MgO precipitates such as those reported by several authors, e.g. Chen et al. [START_REF] Chen | The effect of phosphorus segregation on the intermediate-temperature embrittlement of ferritic, spheroidal graphite cast iron[END_REF]. However, the present results show that these precipitates are very small and few in number in the as-cast state. This is only when the material is heated that numerous MgO particles precipitate and grow to a significant size in relation with the test temperature. This implies diffusion of oxygen from the ambient air into the material, certainly mostly through grain boundaries as the process is anyway quite rapid. It will be seen in the discussion that this finding is much in line with other experimental observations reported in the literature.

Figure 8a and the enlargement in Figure 8b show details of the ductile area in the sample tested at 500°C. As noted above, the cupules are very small and are seen here to be associated with particles closely resembling MgO precipitates, which is confirmed by the EDS spectrum in Figure B3 in Appendix B. This strongly suggests that the rupture there started from a grain boundary covered with MgO particles, but evolved as apparently a ductile fracture because of the increased softness of the matrix at that temperature. A trial at slightly higher temperature, which was not carried out, would certainly show that an extended part of the rupture surface would be similar with small cupules containing MgO particles.

Discussion

This discussion has two aspects. First, it describes an attempt to rationalize the possible effects of S, Mg, and P by confronting numerous data from the literature.

Second, experimental evidences are detailed that lead to propose a mechanism for ITB in SGI.

Review of S, Mg and P effects on ITB

Instead of using the quantity (%Mg+4.5•%S) previously proposed to account for the detrimental effect of both sulfur and magnesium [START_REF] Li | Brittleness at medium temperature of spheroidal graphite, mixed graphite, and compacted graphite high-silicon molybdenum cast irons[END_REF], the product %Mg•%S was considered. More precisely, it was realized that the curve separating alloys showing ITB from those that do not in the work by Wright and Farrell [START_REF] Wright | Elevated temperature brittleness of ferritic ductile iron[END_REF] can be represented as %Mg•%S=2.3•10 -4 (wt.%) 2 for sulfur content between 0.001and 0.01 wt.%. On the other hand, the ratio %Mg/%P appears well suited for rationalizing the beneficial effect of phosphorus over magnesium [START_REF] Kobayashi | 673K embrittlement of ferritic spheroidal graphite cast iron by magnesium[END_REF][START_REF] Takanezawa | Fracture mechanism and suppression method of 673 K embrittlement in spheroidal graphite cast iron[END_REF]. Figure 9 -Wright and Farrell [START_REF] Wright | Elevated temperature brittleness of ferritic ductile iron[END_REF], selecting only those alloys that had graphite with high nodularity (see Appendix A for details).

-Iwabuchi and Kobayashi [START_REF] Iwabuchi | Suppression of elevated temperature brittleness in spheroidal graphite cast iron by increasing phosphorus content[END_REF].

-other works, meaning results from Chen et al. [START_REF] Chen | The role of magnesium-containing spheroidizer and counteraction of misch metal in the intermediate temperature intergranular embrittlement of ferritic nodule iron[END_REF], Yanagisawa and Lui [START_REF] Yanagisawa | Influence of the structure on the 673 K embrittlement of ferritic spheroidal graphite cast iron[END_REF],

Kobayashi et al. [START_REF] Kobayashi | 673K embrittlement of ferritic spheroidal graphite cast iron by magnesium[END_REF], Takanezawa et al. [START_REF] Takanezawa | A new heat resistant cast iron for thin wall exhaust manifold[END_REF] and Li and Sloss [START_REF] Li | Brittleness at medium temperature of spheroidal graphite, mixed graphite, and compacted graphite high-silicon molybdenum cast irons[END_REF].

To classify the literature data, IG has been considered when elongation at rupture or the reduction of area falls below 10%, while a mostly ductile fracture was assumed if above. It is clear that the transition may be at up to 12-13%, see for example the results by Chao et al. [START_REF] Chao | The effect of microstructure of ferritic spheroidal graphite cast irons on intergranular fracture at intermediate temperatures[END_REF]. On the whole, Figure 9 confirms that both low %Mg/%P and %Mg•%S values are favorable for making SGI immune to ITB. This has been emphasized by drawing the rectangle at the bottom left of the graph in which mostly alloys that show no IG at 400-425°C are found. The right boundary of the rectangle was set at the critical value of 1.5 that was mentioned above, and the upper boundary at 2.3•10 -4 (wt.%) 2 in agreement with Wright and Farrell data. There are however some observations to be made about this graph:

-a number of results from Wright and Farrell are outside to the right of the rectangle, which may be due to the selected value of 10% that might be underestimated.

-In the other way but not appearing in figure 9, Cheng et al. [START_REF] Cheng | Effect of residual magnesium contento n termal fatigue cracking behavior of high-silicon spheroidal graphite cast iron[END_REF] and Yanagisawa et al. [START_REF] Yanagisawa | 673 K embrittlement of ferritic spheroidal graphite cast iron with low residual magnesium content[END_REF] reported results suggesting a lower critical value of %Mg/%P at 1 to 1.1.

-The horizontal arrow points to alloy i of Yanagisawa and Lui [START_REF] Yanagisawa | Influence of the structure on the 673 K embrittlement of ferritic spheroidal graphite cast iron[END_REF] that showed IG when having large ferrite grains and no IG with small ferrite grains. This alloy is located far from the origin of the graph but it was noticed it contained quite a large amount of manganese at 0.39 wt.%. Manganese may have tightened up much of the sulfur as MnS, which means that this point should probably be better located closer to the horizontal axis. This suggests that the sulfur content should be corrected to account for MnS precipitation in a more refined analysis. Kobayashi [START_REF] Iwabuchi | Suppression of elevated temperature brittleness in spheroidal graphite cast iron by increasing phosphorus content[END_REF], and other works (see text).

Mechanism for ITB

In addition to an extended series of tensile tests under air, Wright and Farrell [START_REF] Wright | Elevated temperature brittleness of ferritic ductile iron[END_REF] carried out a few tests under a 10 -3 Pa vacuum at 425°C and observed no IG fracture for alloys that were sensitive to ITB under air, namely their alloys E, B and G as described in Appendix A. The authors recognized they could not explain this result with the embrittling mechanism assuming sulfur diffusion that they considered based on its known effect in steels at low temperature. Further, the alloys studied by Wright and Farrell contained also phosphorus that is known to compete for grain boundary segregation with sulfur [START_REF] Briant | Competitive grain boundary segregation in Fe-P-S and Fe-P-Sb alloys[END_REF]. Hence, if the sulfur content is increased, it may be expected it will overcome the repelling effect of phosphorus, and vice versa. As a matter of fact, what we can conclude is that not only sulfur but also magnesium and phosphorus do not embrittle SGI on their own in between 350°C and 500°C.

A further remark concerns magnesium that is expected to be one of the most potential elements to decrease the strength of grain boundaries according to the theoretical approach of Seah [START_REF] Seah | Adsorption-induced interface decohesion[END_REF] that has already been referenced by Hung-Mao Lin et al. [START_REF] Lin | Effect of maximum temperature on the cyclic-heating-induced embrittlement of high-silicon ferritic spheroidal-graphite cast iron[END_REF] in the context of SGI. However, magnesium is not known to segregate to grain boundaries, i.e., elemental magnesium at grain boundaries has not been reported at level that would suggest grain boundary segregation. In agreement with this latter statement, all observations of magnesium rich areas at the grain boundaries or fracture surface of SGI show it associated with other elements, in particular oxygen and phosphorus [START_REF] Chen | The effect of phosphorus segregation on the intermediate-temperature embrittlement of ferritic, spheroidal graphite cast iron[END_REF][START_REF] Yanagisawa | 673 K embrittlement of ferritic spheroidal graphite cast iron with low residual magnesium content[END_REF][START_REF] Lekakh | Improving hightemperature performance of high Si-alloyed ductile iron by altering additions[END_REF].

Based on the above remarks and the results of this work, it is quite straightforward to propose a mechanism for ITB of as-cast SGI. During tensile or other mechanical testing under air, oxygen penetrates through the grain boundaries leading to precipitation of MgO. As the temperature of the test increases, the amount of oxygen that enters the material and diffuses into it increases, and the size and number of MgO particles increase accordingly. When the test temperature has reached 300-350°C, the size of these particles gets higher than the critical size for initiating grain boundary defects and the ductility starts decreasing with the appearance of IG fracture. As the temperature increases, IG failure becomes more and more frequent up to 500°C; beyond that, the decrease in ferrite grain strength does not allow the critical stress for IG failure to be reached so that ductility is recovered.

The above mechanism is sustained by two further observations reported in the literature. First, ITB is sensitive to the strain rate, the ductility minimum being displaced to higher temperature when the strain rate is increased [START_REF] Kobayashi | 673K embrittlement of ferritic spheroidal graphite cast iron by magnesium[END_REF][START_REF] Wright | Elevated temperature brittleness of ferritic ductile iron[END_REF][START_REF] Iwabuchi | Suppression of elevated temperature brittleness in spheroidal graphite cast iron by increasing phosphorus content[END_REF]. As pointed out by Wright and Farrell [START_REF] Wright | Elevated temperature brittleness of ferritic ductile iron[END_REF], this is a strong indication that the phenomenon is, at least partly, controlled by diffusion. While Wright and Farrell considered sulfur as the diffusing species, it appears more probable that this is oxygen as discussed above. Also, Yanagisawa and Lui [START_REF] Yanagisawa | Influence of the structure on the 673 K embrittlement of ferritic spheroidal graphite cast iron[END_REF] studied the effect of the ferrite grain size by varying the casting dimensions. The elongation at rupture increased with the grain size from RT to 200°C, but then decreased and at 400°C the two casts with largest ferrite grain size showed IG whilst those with the smallest ferrite grain size did not (their figure 5). This could possibly be attributed to a blocking effect of the grain boundaries, but could as well be indicative of grain boundary diffusion that is retarded by smaller grain size because of the lengthening of the diffusion path.

It has been noted that most of the literature results discussed in this work refer to materials that have been heat treated to ensure a fully ferritic matrix. It is quite possible that oxygen entered these materials during the heat treatment, leading to the Mg-rich precipitates that were observed. In turn, heavy section castings in which large stresses develop during cooling could be susceptible to ITB as the shake-out operates in the 300-500°C temperature range. Such an effect could be more pronounced in high silicon SGI because the chemical ordering of ferrite begins in the same temperature range.

Conclusion

The present study showed the presence of MgO precipitates on the intergranular facets of the fracture surface of tensile specimens tested between room temperature and 650°C, obtained from an as-cast high-silicon spheroidal graphite cast iron. The size of these precipitates increases significantly with test temperature, indicating that IG fracture-related embrittlement is controlled by oxygen ingress from ambient air. This analysis is supported by a number of other observations reported in the literature, such as the fact that ferritic SGI are immune to embrittlement when tested in vacuum. Further studies in this area should be devoted to the beneficial effect of phosphorus on the intermediate temperature embrittlement of ferritic SGI. Whenever there were duplicate values, this is the mean that is plotted.

Appendix B

Attempt has been carried out to characterize the composition of the particles using EDS. Owing to their small size, at most 200 nm when the electron beam of the SEM is of the order of 1 µm in diameter, the EDS spectra were expected to show signal coming from both the pointed particle and the matrix around it. The spectra shown in

RE=0. 37and

 37and Fe balance, wt%) was added before pouring the melt. After cooling of the castings, the keel-blocks were removed from the molds and cleaned, and two tensile specimens were machined out from the bottom leg (total length of 115 mm, gage of 50 mm in length and 10 mm in diameter).The tensile parameters, namely the ultimate tensile strength (UTS) and the elongation at rupture (A) were measured using an MTS 810 tensile testing equipment at a constant displacement rate of 0.90 mm/min, thus corresponding to an initial strain rate of 3•10 -4 s -1 . Duplicate tensile tests were performed at RT, 100°C, 350C, 450°C, 475C, 500C and 650°C. For temperature testing, use has been made of a MTS 653 furnace attached to the tensile equipment, and the samples were hold for 10 min at the selected temperature before testing. Three thermocouples located along the gage section were used to monitor the sample temperature and limit the temperature fluctuations within 3°C. Once the tensile test had been completed, one of the fracture surfaces was analyzed by scanning electron microscopy (SEM) using a Quanta 200 Carl Zeiss Ultraplus operated at 20 kV. Local chemical microanalyses were carried out with an energy dispersive spectrometer (EDS) X-max 20 Oxford that the SEM is equipped with.

Figure 2 .

 2 Figure 2. Light optical micrograph of the microstructure of the cast alloys.

Figure 3 .

 3 Figure 3. Tensile strain-stress curves at the various testing temperatures.

Figure 4 .

 4 Figure 4. Variation of A with temperature for the present study and works on high silicon SGI (with silicon content higher than 3.5 wt.%) by Kobayashi et al. [10], Shiao et al.[START_REF] Shiao | Eutectic cell wall morphology and tensile embrittlement in ferritic spheroidal graphite cast iron[END_REF] and Laine et al.[START_REF] Laine | The mechanical properties of ductile iron at intermediate temperatures: The effect of silicon content and pearlite fraction[END_REF], as well as on a SiMo SGI[START_REF] Avery | Effect of temperature cycle on thermomechanical fatigue life of a high silicon molybdenum ductile cast iron[END_REF].

Figure 5 .

 5 Figure 5. Variation with temperature of the normalized value of UTS for low (open symbols) and high (solid symbols) silicon SGI. Literature data are from Wright and

Figure 6 .

 6 Figure 6. SEM micrograph of the fracture surface at room temperature (a), 350C (b), 450C (c), 475°C (d), 500C (e) and 650°C (f). The arrows indicate intergranular facet.

Figure 7 .

 7 Figure 7. SEM micrographs of IG facets after fracture at room temperature (a), 100°C (c) and 450C (e). Note that the micrographs a, c and e are at the same enlargement, and that b, d and f are details of a, c and e, respectively. The arrow in a and c points to the precipitate seen in the enlarged view, b and d respectively.

Figure 8 .

 8 Figure 8. SEM micrographs of the fracture surface of a sample tested at 500C. b shows detail of a.

  is a plot of %Mg•%S values versus %Mg/%P values for a number of literature results with solid and open symbols representing respectively alloys showing or not IG fracture at 400-425°C. These results are from:

Figure 9 .

 9 Figure 9. Plot of %Mg•%S versus %Mg/%P of SGI alloys showing (open symbols) or no (solid symbols) IG fracture. Data from Wright and Farrell [6], Iwabuchi and

Figure A1 .

 A1 Figure A1. Evolution with temperature of the UTS (a) and of the reduction of area at rupture RA (b) for the alloys B, E and G studied by Wright and Farrell [6]. Open symbols are for test under ambient air and solid symbols for trials under vacuum.

Figuresb

  Figures B1 (sample ruptured at 100°C), FigureB2(sample ruptured at 450°C) and FigureB3(sample ruptured at 500°C) do all show a peak corresponding to O and a small one associated to Mg in addition to the peaks of C, Fe and Si. It has been verified that the Mg peak does not appear when the EDS spectra is accumulated on the matrix while the O peak is generally smaller in those locations. It can thus be claimed with some confidence that the particles were MgO precipitates.

Figure B2 .

 B2 Figure B2. SEM micrograph and EDS spectrum with the beam focused on the particle in the center of the field. Sample ruptured at 450°C.

Figure B3 .

 B3 Figure B3. SEM micrograph and EDS spectrum with the beam focused on the particle in the center of the field. Sample ruptured at 500°C.

Table 1 .

 1 Chemical composition (wt.%) of the cast alloy.

	C Si Mn P	S	Cu	Cr	Ni	Ti	Mg	Ce	La	Sb

3.00 4.99 0.20 0.012 0.007 0.036 0.040 0.061 0.016 0.038 0.0066 0.0033 0.0059

Table 2

 2

	Test Temperature (°C)	UTS (MPa)	A (%)
	RT	700	13.1
	RT	691	8.8
	100	652	15.9
	100	658	14.3
	350	425	---
	350	501	3.4
	450	490	---
	450	490	4.2
	475	230	---
	475	361	---
	500	328	4.5
	500	328	---
	650	77	23.1
	650	73	20.5

. Results from duplicate tensile tests (UTS and A) conducted at room temperature (RT), 100°C, 350°C, 450°C, 475°C, 500°C and 650°C.

Funding

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

Data availability statement

Raw data (strain/stress curves) are available on request to the authors

Appendix A. Details about the data from Wright and Farrell.

Amongst the many alloys studied by Wright and Farrell [WR85], those with fully spherical graphite as indicated in their Table 2 were selected for figure 9. These are alloys with a silicon content varying from 2.12 wt.% to 3.30 wt.%. All alloys had been submitted to tensile testing at room temperature and at 425°C, at a strain rate of 8.3•10 -5 s -1 up to yielding, followed by a strain rate of 8.3•10 -4 s -1 to fracture. The


domain: sdv sde

text:
Machon Nathalie
email: nathalie.machon@mnhn.fr

Gallant-Canguilhem Matthieu

Ropars Lise

Jeanne Vallet

Temporal analysis of plant community changes documenting climate and urban practice impacts in cities. The case of Paris (France) over a 136-year period. Running title: Comparison Parisian plant communities 1884-2021

Keywords: Functional traits, streets, docks, squares, Joseph Vallot, Ellenberg indicators, Jaccard's distances

 Background and Aims: Besides its intrinsic value, urban biodiversity participates to city dwellers' better health and quality of life. One method to define which factors are driving biodiversity is to determine the changes in communities between two time-periods and confront them with the environmental and/or practice changes that have occurred at the same time.

 Methods: We compared the changes that occurred in the Paris (France) plant communities at a 136-years interval, thanks to two inventories performed in exactly the same streets, wharves and squares, according to exactly the same protocol. We examined which species disappeared, endured and appeared from the 19 th century onwards, determined the traits of these different groups of species and deduced which factors could have driven the shifts in the composition of the Parisian weed communities.

 Key Results: The total number of species (about 190 species) remained relatively similar over a 136-year timespan. Nevertheless, a turnover of about 50% of the species happened.

The differences between the 1884 and 2021 floristic data mostly followed climatic and human activity changes i.e. the current flora seems to be better adapted to a warmer climate and shifts in soil qualities. The differences in weeding practices also modeled the new plant communities.

 Conclusions: This study is documenting changes in plant communities over such a long period. Generally, comparable lists of urban data are scarce because former naturalists rarely studied city areas and seldom detailed reproducible protocols. In our case, the protocol was strictly reproducible and accurate enough to enable a comparison. The results inform on the impact of global and local changes on biodiversity. It also provides important clues about how to better integrate biodiversity in urban projects for the ecosystem to function better and to improve future city dwellers' well-being.

Introduction

Urbanization is one of the principal reason for land use changes worldwide to bring about biodiversity alterations [START_REF] Maxwell | Biodiversity: The Ravages of Guns, Nets and Bulldozers[END_REF]. However, biodiversity is present in cities.

Besides its intrinsic value, it also participates to city dwellers' better health and quality of life [START_REF] Aerts | Biodiversity and Human Health: Mechanisms and Evidence of the Positive Health Effects of Diversity in Nature and Green Spaces[END_REF]. Territorial collectivities have to preserve it in order to participate to the general conservation of biodiversity and to give citizens the best possible life-conditions. For these reasons, it is necessary to determine what the main drivers of the composition of natural urban communities are, and particularly the composition of plant communities, which are key elements for biodiversity. Their state shapes the quality of all networks and food chains interactions. In cities, they are composed of both cultivated and spontaneous species, and among them, native and exotic ones. Cultivated species are species planted by humans for food or ornamental purposes. Spontaneous ones are those that grow naturally, originated locally or that landed by themselves from outside the city, or even spilled over from gardens.

As conceptualized by [START_REF] Aronson | Hierarchical Filters Determine Community Assembly of Urban Species Pools[END_REF], spontaneous urban vegetation is the result of processes by which species filter which bare human disturbances and city environment. Most plant species in cities dwell in diverse habitats [START_REF] Lundholm | Habitat Origins and Microhabitat Preferences of Urban Plant Species[END_REF] including parks, public gardens [START_REF] Shwartz | Local and Management Variables Outweigh Landscape Effects in Enhancing the Diversity of Different Taxa in a Big Metropolis[END_REF], flowerbeds, lawns, river banks, railways and green spaces at business sites [START_REF] Serret | Potential Contributions of Green Spaces at Business Sites to the Ecological Network in an Urban Agglomeration: The Case of the Ile-de-France Region, France[END_REF]. Among these habitats, public spaces such as streets, squares and riverbanks offer a multitude of micro-green spaces that can harbor a number of plant species, especially when management and trampling are limited [START_REF] Pellegrini | Streets as New Places to Bring Together Both Humans and Plants: Examples from Paris and Montpellier (France)[END_REF]; [START_REF] Schmidt | Effects of Urban Structure on Plant Species Richness in a Large European City[END_REF]. In streets of cities like Paris -except for alignment trees, which are planted mainly to decorate and provide shade in summer plants are rather wild and grow spontaneously in every crack in pavements, walls and roads.

Up till now, few studies have described the species that grow in such public spaces and have determined the drivers of their communities' compositions [START_REF] Omar | Drivers of the Distribution of Spontaneous Plant Communities and Species within Urban Tree Bases[END_REF] [START_REF] Lundholm | Habitat Origins and Microhabitat Preferences of Urban Plant Species[END_REF].

One method to define which factors are driving biodiversity is to determine the changes in communities between two time periods and confront them with the environmental and/or practice changes having occurred at the same time, and examine how adjustments may have taken place [START_REF] Grimm | Global Change and the Ecology of Cities[END_REF]. Thus, comparing urban plant communities, observed at more than a one-century interval, make up extremely informative case studies of the effects of local and global changes on biodiversity (Salinitro et al. 2019a).

Trait-based approaches can even more accurately provide general rules for community ecology [START_REF] Cadotte | Beyond Species: Functional Diversity and the Maintenance of Ecological Processes and Services: Functional Diversity in Ecology and Conservation[END_REF], as well as a more mechanistic understanding of community assembly and disassemblyparticularly in the context of global climate change [START_REF] Mcgill | Rebuilding Community Ecology from Functional Traits[END_REF], [START_REF] Williams | Coral Reef Ecology in the Anthropocene[END_REF]. Thus, in our study, reproductive and ecological traits have been investigated to explain interspecific variations at temporal scale, in response to environmental and anthropogenic changes as well.

Among ecological traits, Ellenberg's indicator values (EIV) [START_REF] Ellenberg | Biological Monitoring: Signals from the Environment[END_REF] describe in which environmental conditions plant species grow in Central Europe [START_REF] Diekmann | Species Indicator Values as an Important Tool in Applied Plant Ecology a Review[END_REF].

They can be used as a numerical system to classify species' habitat niches [START_REF] Silvertown | Absence of Phylogenetic Signal in the Niche Structure of Meadow Plant Communities[END_REF] i.e., what their peak of occurrence are in terms of light availability (L), temperature (T), atmospheric (Ma) and soil moisture (Ms), soil reaction (R), soil fertility (N) and salinity (S).

Furthermore, dependence of plant species on pollinators is also a trait that can characterize their ecological needs. For each of these traits, their means per community can be calculated.

Comparison in time or space can characterize changes among communities. This paper presents the changes that occurred in the Paris plant communities at a 136-year interval, thanks to the existence of an inventory published in 1884 in a book entitled "Essay on the flora of the Paris pavement limited to outer boulevards, or Catalog of plants that grow spontaneously in streets and on wharves" [START_REF] Vallot | Essai Sur La Flore Du Pavé de Paris Limité Aux Boulevards Extérieurs[END_REF], by a well-known French botanist Joseph Vallot. In 2021, we followed in his footsteps, using the same inventory protocol at exactly the same places. We used the trait-based approach to understand how environmental changes may have affected biodiversity from that time onwards. (Salinitro et al. 2019b) have listed different studies published on the same subject in other European cities. None of them held such precise data, performed exactly the same protocol in exactly the same places and used species' characteristics to inform the recorded changes and thus characterize global changes' impacts on communities. Our study is more precise than previous ones in that it was based on data related to the abundance of species by their presence in a large sample of streets, wharves and squares in Paris, and not just in general, as was the case, for example, in Italy in Bologna (Salinitro et al. 2019b), Brussels in Belgium [START_REF] Godefroid | Temporal Analysis of the Brussels Flora as Indicator for Changing Environmental Quality[END_REF], Plzen in the Czeck Republic [START_REF] Chocholoušková | Changes in Composition and Structure of Urban Flora over 120 Years: A Case Study of the City of Plzeň. Flora -Morphology, Distribution[END_REF] or Zurich in Switzerland [START_REF] Landolt | Some Results of a Floristic Inventory within the City of Zürich[END_REF].

[Figure 1] Comparing both available datasets, the present paper reports the changes that happened over 136 years in the plant communities found in 52 public spaces, all found in Paris, France: streets, squares and Seine river's wharves. The study only focused on the spontaneous flora and excluded cultivated/ornamental vegetation. The habitats sampled all belonged to the micro-structures that support vegetation, i.e., cracks in asphalt, the base of alignment trees, the feet of walls, the spaces between paving stones… We examined which species were growing in the Parisian public spaces in the late 19 th century, which ones disappeared, endured, and emerged from that time on. We determined the traits of these different groups of species and deduced which factors could have driven the shifts in Parisian weed communities' compositions.

MATERIALS AND METHODS

Study site

The study is located in the capital city of France (the World Geodetic System 1984 reference is 48° 50′ 26.91″ N, 2° 23′ 17.46″ E), within the limits Paris used to have just before the annexation of the bordering cities, which took place in 1860 (map fig 1, list Annex 1). The area covered an area of approximately 3.5 km², in the city's densest districts.

During the 19 th century, on emperor Napoleon III's command, the Paris center was deeply modified by Baron Haussmann [START_REF] Haussmann | Mémoires Du Baron Haussmann I Avant l'Hotel de Ville[END_REF]. He opened wide avenues, created many grounds and green spaces, built pavements for pedestrians, planted trees along the main streets and erected the famous Paris Haussmann buildings. He resurfaced the streets, wharves and squares by re-paving and asphalting the main roads. By 1884, the work was completed.

In 1884, the human density was approximately 80,000 habitants/km². It was lower in 2021, when there were around 20,500 inhabitants/km² (according to INSEE i.e. the French National Institute for Statistics and Economic Studies, 2017, http://www.insee.fr).

In 2021, the heritage of Baron Haussmann is still significant. Urbanization has not changed much in 136 years. Districts have more or less the same shape today as they used to at the end of the 19 th century. All the streets that existed in late 19 th century still exist, except one (rue Bunant).

Floristic data 1884's inventories

In 1884, Joseph Vallot published the list of the plants growing in 52 public spaces (PS): 35 streets, 15 riverbanks and 2 squares in Paris (France), comprised in the study site. He collected the data in all the habitats in streets, squares and river wharves: 1° in the cracks between cobblestones in streets and quays, on banks; 2° stones on quays; 3° plants on small walls; 4° the rare bare soils; 5° the soil around tree bases along boulevards and avenues.

Among the plants inventoried, we removed all trees and shrubs because we considered they had probably been planted. We also removed the only mention of a non-angiosperm plant (i.e., Equisetum arvense). We have updated each species' names according to the French list TAXREF 15.0 (https://inpn.mnhn.fr/telechargement/referentielEspece/taxref/15.0/menu).

Only one species namely Carex muricata L., observed by Vallot, remains questionable as it is a historical name covering several taxa (Carex otrubae, divulsa, leersii, pairae, spicata…).

Joseph Vallot did not mention when he carried out his inventories. As he did not record any spring or autumn species, we assumed that he collected his data over one growing season between May and August, over a single year, around 1884.

2021's inventories

We researched the names and locations of each public space (PS) (street, wharf and square) visited by Vallot. Some of them had changed name since then, but only one street had to be discarded (Rue Bunant), as it no longer existed in 2021. We thus carried out our inventories in all streets but one of them within the same limits as Vallot's.

We listed all the herbaceous angiosperm species between May and August 2021, except ornamental species. When necessary, we used the app Pl@ntNet® to ensure the validity of our identifications.

A matrix of the presence/absence of each species, in each of the sampled locations, at each period: in Annex 1, it is presented with species in the rows; and in the columns, the streets/wharves/squares as found in 1884 or 2021.

Regarding the native status, we used the database from the National Botanical Conservatory of Parisian Basin CBNBP (http://cbnbp.mnhn.fr/cbnbp/), which indicates whether each species is considered as native or exotic in France.

Ellenberg indicator values and species dependency on pollinators

For each species, seven Ellenberg indicator values were collected using the R package TR8 [START_REF] Bocci | TR8: An R Package for Easily Retrieving Plant Species Traits. Edited by Richard Fitzjohn[END_REF]: temperature, light, nutrient, soil reaction and humidity, atmospheric humidity, and salinity indicator values from the French database Baseflor (Julve Philippe 1998), and also their dependence on pollinators for their reproduction (SPD). Ellenberg's temperature indicator values are species-specific indices describing preferences along climatic gradients [START_REF] Ellenberg | Biological Monitoring: Signals from the Environment[END_REF]. They characterize the optimum temperature class for a species' growth and survival, among 9 classes ranging from "alpine to nival, altimediterranean floor" (T-number = 1) to "thermomediterranean to subarid floor" (T-number = 9). The soil reaction gradient ranges from 'high acidity, never moderately acidic or alkaline' (R-number= 1) to 'alkaline and calcareous conditions, only calcareous soils' (Rnumber = 9). The soil moisture covers species growing on dry soils (Ms-number = 1) to species growing on wet, hypoxic soils (Ms-number = 9). EIVs for nitrogen/nutrients span species growing on poor soils (N-number 1) to species growing on nutrient-rich soils (N-number = 9). Light species growing in deep shadow have an EIV of (L-number = 1), while species growing in full light have an EIV of (L-number = 9). For all these indicators, we calculated the community-mean preferences per street in 1884 and in 2021 (hereafter CMP) i.e., the mean of EIV of all species inventoried in each street [START_REF] Garnier | Plant functional markers capture ecosystem properties during secondary succession[END_REF].

We also classified species on the basis of their dependence level to pollinators for their reproduction. Following the methods of [START_REF] Duchenne | European Plants Lagging behind Climate Change Pay a Climatic Debt in the North, but Are Favoured in the South. Edited by Regan Early[END_REF], plants were scored as more or less pollinator dependent. Pollen vectors (i.e., insects, wind or selffertilization) were retrieved from the Baseflor (Julve Philippe 1998) Ecoflora (http://ecoflora.org.uk/ Fitter and Peat 1994) and BiolFlor (www.biolflor.de Klotz, Kühn, and Durka 2002;Kühn, Durka, and Klotz 2004) trait-bases. To integrate the sometimescontradictory data indicated by the different databases, the authors calculated a single variable as the percentage of times "insects" proved to be a pollen vector for a given species, across all databases. Their "species pollinator dependence" (SPD) was 100% when they were reportedly completely pollinator dependent, i.e., allogamous and self-incompatible. It was 0% when they were reported as only reproduced by selfing and/or pollinated by the wind. For each street, wharf and square, we also calculated the mean dependence to pollinators and compared them between the two periods.

All the data were treated using R packages rstatix (https://rpkgs.datanovia.com/rstatix/) and (http://haozhu233.github.io/kableExtra/). The tests performed on mean Ellenberg indicator values and mean dependence to pollinators by public space were Wilcoxon's paired series median comparison tests, since both normality and homoscedasticity hypotheses were not validated for the 1884 data set.

Floristic distances among streets and among wharves between the two periods

To properly characterize inventories in relation to each other, we calculated pairwise Jaccard's distances between the 1884 and 2021 street floras, using the R vegan package (https://github.com/vegandevs/vegan). With those distances, we drew the two dendrograms using the R dendextend package (https://cran.r-project.org/package=dendextend) with Ward's method, which minimizes the intraclass inertia and maximizes the interclass inertia in order to obtain the most homogeneous closures and subgroups possible. In the 1884 pool of species, 10.70 % were exotic (i.e., arrived in France after 1492). In 2021, this rate was 23.53%.

RESULTS

General

The mean number of species growing in these streets and the wharves increased a lot, from 5.23 sp and 30.52 sp respectively in 1884 to 32.05; and 40,47 sp in 2021. In both squares, species richness remained relatively identical, with a mean of 31 and 31.5 species.

Mean trait analyses

[Table 1] [Figure 3]
Analyses of comparisons of mean Ellenberg indicator values of plants in the Parisian public spaces detected significant differences between street, wharf and square communities between the two periods (Fig 3). In particular, they revealed a significant increase in temperature preference between the one in the 19 st century to the other in 21 st (the mean EIV passing rising from 5.21 to 5.69, p-value < 0.001). In the same way, analyses revealed that 21 st century plants preferred drier (EIV passing from 4.75 to 4.73, p-value=0.013) and more alkaline soils (mean EIV passing from 5.82 to 6.24, p < 0.001). Dependence on pollinators also seemed to have increased between these periods (the mean dependence increasing from 36.39% to 45.17%). Preferences for light, salinity, atmospheric humidity and nitrate did not seem to have changed significantly between these two periods (tab 1 and figure 3). Except for one street inventoried in 2021, we found a clear differentiation of floristic composition between these two periods in streets and Seine wharves (Figures 4 and5). Each street and wharf were floristically more similar to the other streets and wharves of its period than to itself at a 136-year interval. The distribution of species is completely changed between the two periods. The streets that were floristically similar are not the same currently. The average floristic distance is greater in streets in 1884 than in those in 2021, probably due to the low number of species per street found in the 19 th century.

Discussion

The inventory performed by Vallot in 1884 followed an intense urbanization process that occurred over the 19 th century and completely changed the shape of the city. Our inventories occurred in a very similar territory, from an urbanistic point of view. Thus, differences between the 1884 and 2021 floristic data mostly followed climatic and human activity changes. Contrary to other studies on other cities conducted worldwide, the present results pointed out that the total number of species (about 190 species) remained relatively similar over a 136-year timespan. Nevertheless, there was a turnover of about 50% of species, only half of them identical, and the other half exclusive to one of the two period inventories.

However, even if some of those exclusive species already existed or still exist in Paris, they were only found in one of the two inventories. To explain such a turnover, it should be noted that beyond their mere presence, species abundance, and thus detectability, may have changed with time. Usually, variations of species abundances are due to stochastic processes or are linked to environmental changes or practices. In any case, a study of the databases containing all the data from inventories carried out in Paris over time from the 17 th century onwards, suggested this: 18 species became extinct in Paris; 39 (species) emerged during the 20 th century and 7 in the 21 st century. In particular, the number of alien species doubled (representing from 10.70% to 23.53% of total species). This 50% turnover is lower than turnovers found in Bologna and in Plzenˇ in Czech Republic (Salinitro et al. 2019b) and [START_REF] Chocholoušková | Changes in Composition and Structure of Urban Flora over 120 Years: A Case Study of the City of Plzeň. Flora -Morphology, Distribution[END_REF] where similarity indices were only 35% and 43% in 120 years and in 60 years exceeded the ones in Brussels (a 79.48% similarity) [START_REF] Godefroid | Temporal Analysis of the Brussels Flora as Indicator for Changing Environmental Quality[END_REF].

In contrast to these previous studies in other cities, we directly compared the plant community indices in public spaces between the two time periods. This provides more accurate information than simply comparing species' average preferences in the overall lists. Actually, we incorporated variations in species abundance at the city level and highlighted more subtle processes than just the appearance or disappearance of species.

Floristic distances among streets and wharves during both periods revealed high communities' differentiation over more than one century (fig 3 and4). The flora in each space resembles that in other spaces of its time, much more than does its own flora at 136-year intervals.

The most obvious sign of the current flora's adaptation to a warmer climate is the increase in mean EIV values concerning plant communities' temperatures (Fig 2). Some species, because they were well-suited to cold climates, disappeared from Paris (e.g., Carum carvi or Turritis glabra) or displayed reduced abundance. For example, Centaurea jacea and Fallopia convolvulus decreased their occurrences by 2.5 between 1884 and 2021 (3,9 ؉ to 1.6 ؉ ).

Conversely, species well-suited to hot climates arrived recently from warmer regions in the world (e.g., Senecio inaequidens, from South Africa) or their abundance increased (e.g., Hordeum murinum multiplied its occurrences by 2.5 between these two periods).

Changes in city planning also impacted street plant communities. For example, communities' better tolerance to drier soils is observed, though annual rainfall amounts increased very slightly over one century, from 605 to 615 mm/year on average, between 1884 and 2021, according to MétéoFrance (i.e. the French organism for climate survey https://meteofrance.com/) data. However, maximum rainfall also increased (from 29.33 to 40.63 mm in 136 years, according to MétéoFrance), as did soil impermeability, with paved areas being transformed into asphalted areas. This transformation mainly followed the May 1968 demonstrations, during which students used paving stones as projectiles to attack law enforcement officers. As a result, more rainwater ran off into sewer systems and rivers, probably leaving little water available for soil and plant hydration. Wharves have also been channelized, leading to the disappearance of more or less aquatic species, like Rorippa x anceps and amphibia, or Rumex hydrolapathum. Recently, Portulaca oleracea and Senecio inaequidens, plants particularly adapted to dried soils, have invaded the French capital's pavements.

In the same way, the massive use of construction materials has caused the alkalization of (cement, concrete, etc...) soils. Plant communities seem to be more and more adapted to basic soils. Among the plants specializing in acid soils, Scleranthus annuus disappeared from Paris more than a century ago, and Rumex acetosella is now very rare in streets.

In contrast to what authors had found in the other cities [START_REF] Godefroid | Temporal Analysis of the Brussels Flora as Indicator for Changing Environmental Quality[END_REF], [START_REF] Chocholoušková | Changes in Composition and Structure of Urban Flora over 120 Years: A Case Study of the City of Plzeň. Flora -Morphology, Distribution[END_REF] and (Salinitro et al., 2019b) we did not find any changes in communities' tolerance for light. It should be noted that buildings did not change in size and streets kept the same width from the end of 19 th century onwards. Trees had also been planted before Vallot's inventories. Light has therefore probably not changed since then. However, as in other cities, no differences were found concerning communities' sensitivity to atmospheric humidity and salinity [START_REF] Godefroid | Temporal Analysis of the Brussels Flora as Indicator for Changing Environmental Quality[END_REF].

Community characterization in terms of pollinator dependence revealed a significant difference between the two periods (around 40% on average). According to [START_REF] Ropars | Wild Pollinator Activity Negatively Related to Honey Bee Colony Densities in Urban Context. Edited by Wolfgang Blenau[END_REF], cities harbor diverse plant species that flourish all year long because they apply low pesticide policies and are subject to heat island effect. Hence, they provide resources for pollinators throughout the year and create favorable conditions for the maintenance of diverse pollinator communities [START_REF] Harrison | Urban Drivers of Plant-pollinator Interactions. Edited by Karl Evans[END_REF]. We wish we could have compared these rates with other data for other cities or other environment types, but it seems they do not yet exist in the literature.

Here are the three specific human activities supposed to have modified public spaces' plant communities: means of transport; food trade; and weed control. In more than one century, horse-drawn vehicles were replaced by combustion engine automobiles. On the one hand, with the disappearance of horses from Parisian streets, their fodder and dung no longer provide abundant natural fertilizers. On the other hand, current car emissions tend to enrich the soil with nitrates. Thus, preference for soil richness does not seem to have changed over 136 years. Nevertheless, plants from horse forage seeds, such as Avena sativa and Phleum pratense, are no longer observed in Paris.

This species turnover detected in Paris over 136 years may also reflect the changes in food trade experienced as of the early 20 th century. It is estimated that 95% of the fruit and vegetables consumed in the capital were produced more or less locally. Indeed, the market garden production areas constituted a green belt, justified by the size of the city's population; hence of the Parisian market; and also by the use of the city's waste as fertilizer [START_REF] Moreau | Manuel Pratique de La Culture Maraichère de Paris. V. Bouchard-Huzard[END_REF]. The streets adjacent to these gardens probably received the seeds produced by crops there, thus extending their range to the most urbanized areas. These discharges from gardens may also have occurred during the transport of foodstuffs or seeds to points of sale such as markets or traders' carts. Thus, Vallot was, in his inventory, able to list lentils (Lens culinaris) or asparagus (Asparagus officinalis) in some of the streetsvegetables that we did not find during our inventorying.

Weeding and street cleaning are probably the practices with the greatest impact on biodiversity changes in Paris as of the 19 th century. Over paved streets, sweeping machines were used when possible, otherwise they were cleaned by hand, by teams of men and women equipped with efficient brushes or birch brooms, on an everyday basis [START_REF] Vallot | Essai Sur La Flore Du Pavé de Paris Limité Aux Boulevards Extérieurs[END_REF]) [START_REF] Haussmann | Mémoires Du Baron Haussmann I Avant l'Hotel de Ville[END_REF]. Since then, pavements have gradually been covered with bitumen or tar and the local authority has continued to weed streets and pavements with more or less drastic techniques, depending on financial and human resources, as well as inhabitants' and borough councils' sensitivities. Herbicides have been increasingly implemented. On 1 January 2017, the adoption of the Labbé law, which aims to better control the use of phytosanitary products on the French territory, greatly modified public spaces maintenance. Discontinuing the use of chemical products has led Paris to manage its spaces differently. According to Parisian public spaces' managers, weeding is more seasonal and therefore much less frequent (2 or 3 times a year on average) than in the 19 th century. It has become mainly mechanical, with the help of tools capable of tearing and brushing the pavement. Thus, our results showed that Parisian pavements and wharves hosted a richer flora in the 21 st century than they did in the 19 th , with more species depending on pollinators. We should specify, however, that inventories were carried out during the COVID-19 pandemic crisis, when street use and management may have had a specific effect on biodiversity's quality.

Nevertheless, since the abandonment of herbicides on cities' public spaces, the flora has already been found to be enriched.

Conclusion

This study is documenting changes in plant communities over such a long period. The existence of comparable lists of data are scarce because former naturalist rarely studied urban areas and seldom used or outlined reproducible protocols in detail. In our case, the protocol was strictly reproducible and sufficiently precise to allow comparison.

A welcome surprise: the present Parisian communities were richer than they seemed to be in the late 19 th century. It is probably due to changes in weeding intensity between these two periods. It is only recently that laws beneficial to biodiversity enacted against using pesticides in urban public territories have had an impact. As a result, the Parisian flora did not depauperate over time, which is heart-warming compared to what is observed in all ecosystems worldwide. The arrival of many exotic species offsets the loss of local ones (if they did not cause it) and enabled biodiversity to adapt to warming and other global changes experienced by our cities. Clearly, it enabled insect communities to endure and ensure pollination services.

We hope that other policy changes and a better integration of biodiversity in development projects will lead to even better biodiversity quality and better ecosystem functioning, thereby fostering future city dwellers' well-being.

Funding:

The research was funded by the National Museum of Natural History and the URBANATURE program.

List of figures

  282 species had been listed in Paris over both inventories (Fig 2). One third (95 sp.) were only found by Vallot in 1884 and not in the 2021 inventory. Among them, according to the French databases (MNHN & OFB (Ed) 2003), eighteen of them disappeared from Paris during the 20 th century. One third (94 sp.) were only found in 2021 and had not been seen by Vallot. Among them, 46 appeared in Paris during the 20 th (39 sp.) or 21 st centuries (7 sp.). And one third (93 sp.) were found in both inventories. The Dice coefficient (Dice 1945) i.e. their similarity was thus 0.496.

Figure 1 :Figure 2 :Figure 5 :

 125 Figure 1: Map of the 52 Paris's public spaces inventoried both in 1884 and 2021 (in red)

Table 1 :

 1 Wilcoxon's paired series median comparison tests: PD is the pollinator dependency; L light; T temperature; Ha atmospheric humidity; Hs soil humidity; pH soil pH; N nitrate and Sal salinity Ellenberg indicator values.

		1884	2021	u-stat	p-value
	PD (%)	36.39±20.6	45.13±4.56	434.0	0.02
	L	7.13±0.4	7.15±0.21	558.0	0.591
	T	5.21±0.27	5.69±0.16	34.0	<0.001
	Ha	4.68±0.55	4.65±0.18	756.0	0.545
	Hs	4.75±0.69	4.73±0.16	929.5	0.013
	pH	5.82±0.54	6.24±0.17	186.0	<0.001
	N	6.73±1.26	6.76±0.24	782.0	0.4
	Sal	0.23±0.27	0.16±0.08	698.5	0.743

Acknowledgement:

We thank Juliette Amara and Alice Amoureaux who initiated the study in 2020, during their internship at the National Museum of Natural History in Paris. We are also grateful to the specialist language editing service of Dominique Macabies for the language revisions.


Sonia Boscolo
email: s.a.boscolo@aston.ac.uk

John M Dudley

Christophe Finot
email: christophe.finot@u-bourgogne.fr

Modelling Nonlinear Propagation of Periodic Waveforms in Optical Fibre with a Neural Network

We deploy a neural network to predict the spectro-temporal evolution of a periodic waveform upon nonlinear fibre propagation and demonstrate efficient probing of the inputparameter space for on-demand comb generation or significant spectral/temporal focusing occurrence.

Introduction

Recent years have seen a growing interest in applying the techniques of machine learning to optical systems and, particularly, for the characterisation and control of ultrafast propagation dynamics [START_REF] Genty | Machine learning and applications in ultrafast photonics[END_REF]. We have successfully introduced the use of neural networks (NNs) as an efficient tool for substituting the nonlinear Schrödinger equation (NLSE) in the modelling of the shaping of ultrashort pulses that occurs upon nonlinear propagation in an optical fibre [START_REF] Boscolo | Artificial neural networks for nonlinear pulse shaping in optical fibers[END_REF][START_REF] Boscolo | Modelling self-similar parabolic pulses in optical fibres with a neural network[END_REF] or for predicting the generation of optical supercontinua [START_REF] Salmela | Predicting ultrafast nonlinear dynamics in fibre optics with a recurrent neural network[END_REF]. The fibre nonlinearity does not only affect the propagation of ultrashort pulses, but also a continuous wave modulated at some frequency will experience an energy exchange between the spectral lines making up its spectrum along with a change in the relative phase between the frequency components. New equally spaced frequency components will emerge giving rise to a frequency comb, while significant reshaping will take place in the time domain, generally leading to very high repetition rate pulse trains [START_REF] Pitois | Generation of 20-Ghz picosecond pulse trains in the normal and anomalous dispersion regimes of optical fibers[END_REF]. In this paper, we implement a NN to predict the spectro-temporal evolution of a periodic waveform in a fibre [START_REF] Boscolo | Predicting nonlinear temporal and spectral reshapings of periodic signals in optical fibre with a neural network[END_REF]. Both the normal and anomalous second-order dispersion regimes of the fibre are studied, and the speed of the NN is leveraged to probe the space of input parameters for the generation of on-demand target combs or the occurrence of significant temporal or spectral focusing.

Methods, Results and Discussion

The general problem studied in this work is the nonlinear propagation of two types of periodic waveforms, which have already been investigated in the context of linear shaping [START_REF] Finot | 40-GHz photonic waveform generator by linear shaping of four spectral sidebands[END_REF]: a continuous wave modulated at the frequency fm, yielding an optical spectrum made of a central component and two sidebands at ±fm, and a wave whose spectrum consists of four spectral lines without any continuous background. The data from numerical simulations of the NLSE based on the standard split-step Fourier algorithm is used to train a NN and validate its predictions. We employ a feedforward NN relying on the Bayesian regularisation back propagation algorithm and including three hidden layers. The NN learns the NLSE model from an ensemble of hundreds of thousands simulation data (real and imaginary parts of the spectral field) for the anomalous or normal dispersion regime of the fibre and corresponding to randomly chosen combinations of input parameters: amplitude ratio A of the central frequency component of the optical spectrum to the lateral sidebands, spectral phase offset 𝜑 of the sidebands relative to the central component, normalised propagation length 𝜉, and soliton-order number N. After training, the NN is tested on a distinct ensemble of tens of million data not used in the training step. The strength of the trained NN lies in its speed: in less than one minute, it can predict the output features of this large data set with high accuracy. Therefore, it can search the full 4D input parameter space for the optimum parameter sets meeting given targets without being trapped in local optima.

Figure 1 illustrates some examples of the NN's performance. In panels (a1) and (a2), the NN was asked to identify the input parameters that enable the formation of optical spectra made of nine spectral lines of equal intensity and of six equal intensity spectral lines but with the central component suppressed, respectively, when a three-frequency component initial condition is used at the input of an anomalously dispersive fibre. The predictions from the NN show good agreement with the results of the NLSE model. The scatter plot in panel (a3) highlights the existence of two distinct regions in the space of input parameters that support the formation of highly flat frequency combs. We also see that is possible to achieve a flat comb starting from both lower and higher intensity lateral sidebands as compared with the central component. As our NN accounts for both the spectral intensity and phase features of the generated comb, it is straightforward to reconstruct the temporal properties of the corresponding pulse train. In panel (b), the NN was asked to probe the input parameter space for the pulse train with the highest ratio of the pulse peak power to the average power. The temporal profile of the compressed waveform predicted by the NN is once again in very good agreement with that obtained from NLSE simulation, even when plotted on a logarithmic scale. We have also verified that the NN can efficiently reconstruct the longitudinal temporal evolution of the initial waveform. The results shown in panel (c) refer to the process of spectral focusing taking place in a normally dispersive fibre. Starting from three spectral lines of equal amplitude, the NN was able to identify a combination of spectral phase and propagation parameters leading to a remarkable inverse four-wave mixing. More than 80% of the total energy is concentrated in the central frequency component whereas the intensity level of the neighbouring components is more than 15dB lower. Further results including the generation of customised frequency combs and of optical undular bores [START_REF] Fatome | Observation of optical undular bores in multiple four-wave mixing[END_REF] in the normal dispersion regime will be presented at the conference.

Conclusion

We have demonstrated the ability of a trained NN to identify the input system parameters that are required to generate on-demand target frequency combs in a nonlinear optical fibre starting from periodic wave initial conditions with three or four frequency components. The accurate prediction of the longitudinal evolutions of the wave intensity profiles in the time and frequency domains by the NN for both the anomalous and normal dispersion regimes of the fibre has enabled replication of the processes of ultrashort pulse formation, spectral compression and undular bores that are involved in the NLSE.

Fig. 1 .

 1 Fig. 1. Various examples of use of the NN. (a) Generation of on-demand frequency combs in an anomalously dispersive fibre: (a1,a2) combs consisting of 9 spectral lines of equal amplitude and of 6 spectral lines of equal amplitude but with the central component cancelled, respectively; (a3) Regions in the input parameter space that enable the formation of high-flatness combs. (b) Temporal focusing in an anomalously dispersive fibre: generation of a pulse train with the highest pulse peak power relative to the average power (plotted on linear and logarithmic scales in panels 1 and 2, respectively). (c) Spectral focusing in a normally dispersive fibre: generated optical spectrum. The predictions from the NN (red circles) are compared with the results of NLSE numerical simulations (black diamonds or lines). Also shown are the initial conditions at the fibre input (blue crosses or lines).


Pierre Haessig

Economic consistency of salvage value definitions

Keywords: Tc_list=[10, 20, 40] plot_Cann_Tp0_i(Tc_list, i_max=0.20);

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

For simplicity, we start by analyzing a project with one single component. The investment cost of the component is . The component lifetime is . We also assume that the project horizon is shorter than the component lifetime ( ) so that there is no replacement cost. We consider zero operation and maintenance (O&M) costs in this analysis since they are just extras which are independant to this discussion. Therefore, from the component point of view, considering its own lifetime has the economic analysis horizon, the Net Present Cost of the component is just the investment:

Then, the annualized cost of the component over its lifetime is:

where is the Capital Recovery Factor (HOMER Software, 2023) which transforms a Net Present

Value of Costs into corresponding annuities:

Now we turn to the economic analysis of the project which includes this component. The Net Present Cost of the project over its lifetime is: where is the nominal salvage value of the component, which gets discounted since the salvage happens at the end of year . The corresponding annualized cost is:

(notice that annuities are computed over the project lifetime, i.e. )

The key subject of this report is: what should be value of the salvage value?

Properties of different salvage values definitions

Possible definitions of the salvage value

There are two possible definitions for the (nominal) salvage value:

a. the classical definition (citation...) which is proportional to the relative remaining lifetime of the component at the end of the project:

b. the economically consistent definition which we propose Remark: the two definitions are quite close, because so the economically consistent definition falls back to classical one for small discount rate , with a difference which is linear in (see illustration below and appendix).

Now we need to specify what we call "economically consistent"…

Economic consistency of salvage value

We say that a definition of the salvage value is economically consistent if it yields an annualized project cost which is equal to the annualized cost of the component. That is, there is no financial difference between the analysis of the component alone versus the same component embedded into the wider scope of the project.

This means that the economically consistent definition is the solution of the following equation:

Proof: see Appendix.

Illustration of the two definitions of the salvage value

S T p C ann,p = N P C p × CRF (i, T p ) T p S a = C c T c -T p T c S b = C c (1 + i) T c -(1 + i) T p (1 + i) Tc -1 S b ∼ S a + O(i) i i C ann,p  depends on S = C ann,c
and see appendix for the proof of the analysis of the difference between the two definitions for small discount rates.

Illustration of the economic inconsistency of the classical definition

Choosing the classical salvage value when computing the annualized project cost creates a positive bias (cost is overestimated compared to the annulized cost of the component alone).

The shape of this effect is more difficult to analyze compared to the shape of the salvage value. From the following plot we observe that:

the overestimation of annualized project cost is worst (strongest) for short projects ( ) it can reach +20% to +30% for the case below this overestimation decrease almost linearly with project horizon (and is of course zero for since salvage is zero)

the effect of the discount rate is complex: no overestimation when discount is zero (since in that case)

the overestimation grows quickly with discount rate, as long as it is "small enough" beyond a certain threshold of discount rate, the overestimation is near constant or even decreasing Further use of the economically consistent salvage value

In the above discussion, we have only studied the case a of project with an horizon shorter than the component lifetime . However, the economic consistency of the salvage value definition also applies to more complex cases. In particular, we analyze two useful cases:

1. The project horizon is longer than the component lifetime ( ) so that there is a replacement cost that needs to be accounted for 2. The project is (virtually) sold at mid-project term and bought back immediately after.

In both cases, we show (see appendix for mathematical prooves) that the annualized project cost in unchanged and equal to the annualized component cost

Project with replacements of the component with one replacement

With one replacement (occuring at the end of year ), the Net Present Cost of the project over its lifetime is:

To proove the economic consistency (annualized project cost = annualized component cost) we must of course assume that , the nomial replacement cost of the component, is equal to the initial investement cost: .

T p T c S b T p > T c T c T p N P C p = C c + R c - S 1 (1 + i) Tc 1 (1 + i) Tp R c R c = C c
For the salvage value , we use the above definitions except that the remaining life should be counted for the second component, which has only been used for time and which end of life should occur at

. This means that in the salvage value definitions become:

The classical salvage value definition becomes:

For the economically consistent definition, there is an ambiguity about which term of the numerator to update: should become , that is the component end of life happen twice later ? should become , that is update the usage duration of the component?

As prooved in the appendix, correct update of the definition is to update the usage duration ( becomes

):
With more than one replacement

If the component is replaced times, the salvage value should be updated by replacing by .

In the appendix, we have the proof for two replacements ( ) and we guess that it could be prooved for the general case .

Project is sold at mid-term and bought back immediately after

This case is useful when optimizing reinvestment within the project lifetime. Indeed in some context, it can be easier to model the complete sale of the project which is immediately bought after (possibly with different capacities) than to model the cost of the capacity change.

To proove the economic consistency (annualized project cost = annualized component cost) we must of course assume here that the project is bought back unchanged (no capacity expansion) even if the interest of this formulation is to allow capacity adjustment. For simplicity, we do not consider replacement (i.e.

).

We introduce the year of selling the project at mid project. There are two salvage values, one for the first sale at mid project ( ) and then the salvage at the real end of the project ( ) but considering only duration of that second sub-project for the aging of the component. With these notations, the Net Present Cost of a project, with mid-term sale and immediate buy back, over its lifetime , is:

The two economically consistent salvage values are the same as the case of a project without replacement, but with reduced project durations:

the mid project salvage uses as project duration the final salvage value uses as project duration

S T p -T c 2.T c S a = C c 2.T c -T p T c T c 2.T c T p T p -T c T p T p -T c S b = C c (1 + i) T c -(1 + i) T p -T c (1 + i) T c -1 n T p T p -n. T c n = 2 n ≥ 1 T p ≤ T c T mid ≤ T p S mid S f inal T p N P C p = C c - S mid + C c - S f inal 1 (1 + i) Tmid 1 (1 + i) Tmid 1 (1 + i) Tp T mid T p -T mid
Proof: see Appendix

Economic inconsistency of the classical definition with mid-term sale

To study the practical importance of using the above formulaes for salvage, we compute the annualized project cost when using the classical salvage values. Since there are two salvage events, this means using the following salvage values:

Like for the first graph of the economic inconsistency of the classical definition without mid-term sale, the effect is not so easy to analyze. We observe that:

the overestimation of annualized project cost is worst (strongest) when the mid-term sale happens at half-project life for "small enough" discount rates it can reach +13% the maximum is reached for mid-term sale happening before half-project in the case of "high" discount rates this overestimation is zero for the effect of the discount rate is complex, like for the case without mid-term sale:

no overestimation when discount is zero (since in that case)

the overestimation grows quickly with discount rate, as long as it is "small enough" (already +4%

for half-project sale when only)

beyond a certain threshold of discount rate, the overestimation is near constant or even decreasing Also, repeating the plot for different values of the component lifetime , it seems that the worst case overestimation is attained for a discount rate slightly below . This is similar (at least approximately) to the case without mid-term sale.

S b,mid = C c (1 + i) Tc -(1 + i) T mid (1 + i) T c -1 S b,f inal = C c (1 + i) Tc -(1 + i) Tp-Tmid (1 + i) T c -1 S a,mid = C c T c -T mid T c S a,f inal = C c T c
i(i + 1) Tc (i + 1) T c -1
In [9]: simplify(Cann_p.subs({S:Sb}) -Cann_c)

Out[9]: 0 i S b S a S b ∼ S a + i + O(i 2 ) T p . (T c -T p ) 2.T c
The corrective term is zero, as expected for:

(when since the component is new)

(when since the component is at end of life) Series expension of around :

limit behavior of the difference with the classical definition :

Decrease of the economically consistent salvage value (definition b) for small project lifetime

Since the classical definition of salvage value is linear in project lifetime , its derivative is constant equal to .

Compared to this, the economically consistent salvage value (definition b) is nonlinear in . Its for small project values is less negative:

Economic inconsistency of the classical definition

T p = 0 S = 1 T p = T c S = 0 S b i = 0
In [10]: Sb_ser = series(Sb, i, n=2) simplify(Sb_ser)

Out[10]: T c (2 + O (i 2 )) + T p i (T c -T p ) -2T p 2T c
In [11]: SbTc_ser = series(Sb*Tc, i, n=2) simplify(SbTc_ser)

Out[11]:

-T p + + T c + O (i 2 ) T p i (T c -T p ) 2 S a
In [12]: Sa = (Tc-Tp)/Tc

In [13]: simplify((Sb_ser -Sa))

Out[13]: -T 2 p i + T c T p i + O (i 2 ) 2T c
In [14]: simplify((SbTc_ser -Sa*Tc))

Out[14]:

+ O (i 2 ) T p i (T c -T p ) 2 T p -1/T c T p
In [15]: Sb.diff(Tp).subs({Tp:0})

Out[15]:

log (i + 1)

(i + 1) T c -1
In [16]: simplify(series(Sb.diff(Tp).subs({Tp:0}), i, n=3))

Out[16]:

- + - - + O (i 3 ) 1 T c i 2 i 2 4 T c i 2
Injecting now the classical salvage value definition into the annual project cost:

Inconistency for short projects Focus on short projects (we need to take the limit , since the denominator is zero at ):

To analyze the effect of discount rate (which not monotonic as visualized with plot_Cann_Tp0_i ), we turn to a series decomposition: linear effect is positive: , which explaines the increase in overestimation for small discount rate quadratic effect is negative which explains the decrease when the discount rate gets "big enough"

And this series is maximal (zero of the derivative) at:

In [17]: Cann_p_rel = Cann_p.subs({S:Sa})/Cann_c Cann_p_rel = simplify(Cann_p_rel) Cann_p_rel

Out[17]:

(i + 1) -T c ((i + 1) T c -1) (T c (i + 1) T p -T c + T p ) T c ((i + 1) T p -1)
T p → 0 T p = 0

In [18]: Cann_p_rel_Tp0 = Cann_p_rel.limit(Tp, 0) Cann_p_rel_Tp0

Out [18]:

(i + 1) -T c (T c log (i + 1) + 1) ((i + 1) T c -1) T c log (i + 1) +T c /2 × i -T c /4 × i 2
In [19]: Cann_p_rel_Tp0_ser = simplify(Cann_p_rel_Tp0.series(i, 0, n=3)) Cann_p_rel_Tp0_ser

Out [19]:

1 + - - + O (i 3 ) T c i 2 T c i 2 4 T 2 c i 2 3
In [20]: Cann_p_rel_Tp0_ser.diff(i)

Out[20]:

- - + O (i 2 ) T c 2 T c i 2 2T 2 c i 3 In [21]: sympy.solve(Cann_p_rel_Tp0_ser.diff(i), i)[0] Out[21]: 3 (T c + O (i 2 )) T c (4T c + 3) i = 1 1 + 4/3.T c
In [22]: i_worst = 1/(1+4*Tc/3)

In [23]: simplify(Cann_p_rel_Tp0_ser.diff(i).subs({i:i_worst})) # = 0 + O(...)
However, the maximal value is not a constant:

To further analysis, we simplify the expression Cann_p_rel_Tp0 by recognizing a function evaluated at :

Check that the expression in is indeed equivalent to Cann_p_rel_Tp0 for :

Remark: Sympy doesn't see the equality between logs at the denominator, because it needs a positivity assumption:

Now looking at the derivative:

Out[23]: O ( ; T c → ∞) 1 T 2 c
In [24]: simplify(Cann_p_rel_Tp0_ser.subs({i:i_worst}))

Out [24]:

12 + 19T c + O ( ; T c → ∞) 1 T 2 c 4 ⋅ (4T c + 3) x ↦ C(x) x = (1 + i) T c
In [25]: x = symbols('x', positive=True) Cx = (1+log(x))*(x-1)/(x*log(x)) Cx

Out[25]: (x -1) (log (x) + 1)

x log (x)

x x = (1 + i) T c
In [26]: simplify(Cx.subs({x:(1+i)**Tc}) -Cann_p_rel_Tp0)

Out[26]: There is no analytical root that SymPy can find:

- - + 1 log ((i + 1) Tc ) (i + 1) -Tc log ((i + 1) Tc ) 1 T c log (i + 1) (i + 1) -Tc T
Only numerical root finding works:

converged: True flag: 'converged' function_calls: 9 iterations: 8 root: 6.00914294108186

And this is coherent with the numerical maximization of :

(1.2981372299707084, 1.298425606777053, 1.298198390999316) message: Solution found. success: True status: 0 fun: -1.298425607525638 x: 6.009143977032298 nit: 12 nfev: 12 1.793282305296451 Therefore, the worst case economic inconsistency happens for pairs which statisfies:

and this equation can be solved for the discount factor :

and since for most component lifetime values, we can linearize :

This observation is coherent the graph obtained with plot_Cann_Tp0_i : the worst case discount factor is about twice the inverse lifetime.

Proof of the economic consistency with replacements with one replacement

In [31]: # sympy.solve(num, x) # NotImplementedError

x * ≈ 6.0091

In [32]: opt.root_scalar(lambda x: 1 -x + log(x) + (log(x))**2, bracket = (5,10))

Out[32]:

C(x)
In [33]: Cx_fun = lambda x: (1+np.log(x))*(x-1)/(x*np.log(x)) Cx_fun(6.01-0.5), Cx_fun(6.01), Cx_fun(6.01+0.5) That is and assuming that replacement cost is equal to the initial investment.

Out

NPC becomes:

As mentioned in the section Project with one replacement cost, there is an ambiguity about which term of the numerator to update in the salvage value definition. So we create both possible expressions and inject them in the project annualized cost.

We show that replacing becomes (i.e. component end of life happen twice later) is incorrect. The correct update of the definition is to replace becomes (i.e. the usage duration of the component).

With two replacements

That is and again assuming that replacement cost is equal to the initial investment.

NPC becomes:

The definition of salvage value is updated by replacing by (i.e. the usage duration of the last component).

Proof of the economic consistency with a mid project sale and immediate buy back

We introduce the year of selling the project at mid project. There are two salvage values, one for the first sale at mid project ( ) and then the usual alvage at the end of the project.

NPC becomes:

i(i + 1)

Tc (i + 1) T c -1
In

i(i + 1) T p (-S(i + 1) -T p + 1 + (i + 1) -T c + (i + 1) -2T c ) (i + 1) T p -1 T p T p -2.T c
In for i in i_list:

  In [59]: i_list = [0.025, 0.050, 0.075, 0.10, 0.20] # list of discount rates plot_salvage_Tp(i_list, Tc=20); ]: #i_list = [0.025, 0.050, 0.075, 0.100, 0.200] # list of discount rates i_list = [0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.08, 0.10, 0.20] # list of many discount plot_Cann_Tp(i_list, Tc=20);

  ]: #i_list =[0.025, 0.050, 0.075, 0.100, 0.200] # list of discount rates i_list = [0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.08, 0.10, 0.20] # list of many discount plot_Cann_Tmid(i_list, Tc=20); In [81]: #i_list = [0.025, 0.050, 0.075, 0.100, 0.200] # list of discount rates i_list = [0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.08, 0.10, 0.20] # list of many discount plot_Cann_Tmid(i_list, Tc=10); #i_list = [0.025, 0.050, 0.075, 0.100, 0.200] # list of discount rates value discussed in the following is fact a relative salvage value (with true value ). Proof of the economic consistency of salvage value definition b Annualized cost of the component over its lifetime: We inject the proposed definition into the project annualized cost to show that it indeed gets equal to the component annualized cost: Behavior of the economically consistent salvage value (definition b) for small discount rate Observation: for small discount rate , tends linearly to : S = S relative . C c In [3]: i, S = symbols('i S') Tc, Tp = symbols('T_c T_p', positive=True) In [4]: Cann_c = CRF(i, Tc)

  c log (i + 1) In [27]: simplify(log(x**2) -2*log(x)) Out[27]: 0 In [28]: a = symbols('a') # without positive=True simplify(log(a**2) -2*log(a)) ]: num = Cx.diff(x)*x**2*log(x)**2 num = simplify(num) num Out[30]: -x + log (x

  [33]:In[34]: Cx_minus_fun = lambda x: -Cx_fun(x) res = opt.minimize_scalar(Cx_minus_fun, [2,6,10], bounds=[2,10]) Cx_max = -res.fun # 1.2984 Cx_argmax = res.x # 6.0091 print(res) In [35]: np.log(Cx_argmax)

  Sb_1rep_incorrect = ((1+i)**(2*Tc) -(1+i)**Tp)/((1+i)**Tc -1) In [41]: Sb_1rep = ((1+i)**Tc -(1+i)**(Tp-Tc))/((1+i)**Tc -1) * 1 Sb_1rep Out[41]: (i + 1

  [43]: simplify(Cann_p_1rep.subs({S:Sb_1rep}) -Cann_c) # == 0Out[43]: 0 T p ∈ [2.T c , 3.T c ]In[44]:NPCp_2rep = 1 + 1/(1+i)**Tc + 1/(1+i)**(2*Tc) -S/(1+i)In [45]: Cann_p_2rep = CRF(i, Tp) * NPCp_2rep Cann_p_2repOut[45]:

  fig, ax = plt.subplots() ax.axvline(0, color='gray') ax.axvline(Tc, color='gray') ax.plot(Tp, Sa, 'k--', label='Sa (classical)')

C c = 1

Appendix

Remark: the code cells above can only be run once the appendix is run.

In the following, we assume that the investment cost without loss of generality. Only, the salvage

In [75]: i_list = [0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.08, 0.10, 0.20] # list of many discount plot_Cann_Tmid(i_list, Tc=40);

In """Annualized project cost (relative to component's annualized cost), with respect to mid-term project sale (with immediate buy back), when using the classical salvage value definition, with assumption Tp=Tc (no replacement), for a given component lifetime Tc, for a list of discount rates.