Sophie Van Meyel

Séverine Devers

Joël Meunier
email: joel.meunier@univ-tours.fr

Earwig mothers consume the feces of their juveniles during family life

Keywords: Cooperation, Coprophagy, Dermaptera, Family life, Parental care

Many animals consume the feces of their conspecifics. This allo-coprophagy can have benefits, such as access to nutrients and symbionts, but also risks for consumers, mainly due to direct contact with pathogens that develop on feces. In the European earwig Forficula auricularia, mothers and juveniles live in nests lined with their feces. This surprising habit allows juveniles to consume the feces of their siblings during family life and provides them with nutritional benefits when mothers provide low care. However, it was unclear whether earwig mothers also practice allo-coprophagy, and whether this behaviour is motivated by their nutritional needs.

Here, we set up four types of experimental families in which we manipulated the nutritional needs of mothers and/or juveniles and measured the effects on the production of feces by the juveniles, and the consumption of these feces by the mothers. Our results first show that fed juveniles produced more feces pellet in presence of fed compared to food-deprived mothers.

We also found that, overall, about 50% of the mothers consumed juveniles feces. This consumption was both more likely and larger when the feces were produced by fed compared to food-deprived juveniles, while the proportion of feces pellets eaten was larger in fooddeprived compared to fed mothers. Overall, our results reveal that allo-coprophagy involves every family member and suggest that it can have both nutritional and non-nutritional benefits for earwig mothers. Allo-coprophagy could thus favour the maintenance of mothers in the nest and, more generally, promote the early evolution of family life.

INTRODUCTION

The consumption of feces produced by conspecifics is a taxonomically widespread phenomenon in animals [START_REF] Weiss | Defecation behavior and ecology of insects[END_REF]. When present, its expression is generally associated with three main types of benefits. First, it may provide an additional source of food (including enzymes, metabolites and undigested particles) to consumers [START_REF] Nilsson | Coprophagy in larval Culiseta bergrothi (Diptera: Culicidae)[END_REF][START_REF] Nalepa | Detritivory, coprophagy, and the evolution of digestive mutualisms in Dictyoptera[END_REF]. This can be decisive to improve resistance against starvation when other food sources are absent, as reported in the German cockroach Blattella germanica [START_REF] Kopanic | An adaptive benefit of facultative coprophagy in the German cockroach Blattella germanica[END_REF]. Second, the consumption of feces may improve the efficiency of food utilization. This is because the microorganisms and microbes that rapidly colonize feces often initiate the digestion of organic compounds (e.g. cellulose), detoxify allelochemicals and soften the substrate [START_REF] Nalepa | Detritivory, coprophagy, and the evolution of digestive mutualisms in Dictyoptera[END_REF][START_REF] Weiss | Defecation behavior and ecology of insects[END_REF], which overall make coprophagy a source of enzymes and metabolites that would be otherwise difficult to access. This phenomenon is called the "external rumen" and has been reported, for instance, in Passalidae beetles [START_REF] Halffter | Subsocial behavior in Scarabaeinae beetles[END_REF][START_REF] Dillard | Effects of breeding ecology on parental care and family living in passalid beetles[END_REF]. Finally, coprophagy may ensure the transfer of gut mutualistic microorganisms among conspecifics. A growing body of literature indeed demonstrates that the acquisition of these gut microbes can be essential for recipients [START_REF] Moran | Evolutionary and ecological consequences of gut microbial communities[END_REF], as they are involved in numerous physiological, reproductive, and behavioural functions of the host.

For instance, the gut microbiota can affect the digestion of specific food sources [START_REF] Cleveland | Symbiosis between termites and their intestinal protozoa[END_REF][START_REF] Mirabito | A double-edged sword? The cost of proctodeal trophallaxis in termites[END_REF], hormone signalling, metabolism and ageing [START_REF] Broderick | Gut-associated microbes of Drosophila melanogaster[END_REF], immunocompetence [START_REF] Wei | Insect pathogenic fungus interacts with the gut microbiota to accelerate mosquito mortality[END_REF][START_REF] Itoh | Detoxifying symbiosis: Microbe-mediated detoxification of phytotoxins and pesticides in insects[END_REF][START_REF] Wang | Changes in microbiome confer multigenerational host resistance after sub-toxic pesticide exposure[END_REF], general activity [START_REF] Hosokawa | Symbiont acquisition alters behaviour of stinkbug nymphs[END_REF], and behavioural tasks [START_REF] Jones | The gut microbiome is associated with behavioural task in honey bees[END_REF]. This transfer of gut mutualistic micro-organisms could be particularly important in insects, as their inherent moulting events typically lead to a repeated defaunation of their gut, which thus requires simple mechanisms (such as coprophagy and mouth-to-anus contacts) allowing regular reinfections by the microbiota -a process that could promote the general evolution of complex forms of social life such has eusociality [START_REF] Nalepa | Origin of termite eusociality: trophallaxis integrates the social, nutritional, and microbial environments[END_REF][START_REF] Nalepa | Origin of Mutualism Between Termites and Flagellated Gut Protists: Transition From Horizontal to Vertical Transmission[END_REF].

Whereas allo-coprophagy should be facilitated when producers and consumers share a nesting area and thus have direct access to each other's feces, this phenomenon seems to be rare in group-living species [START_REF] Weiss | Defecation behavior and ecology of insects[END_REF]. The main reason is that a wide range of pathogenic bacteria and fungi are known to use feces as a substrate for their development [START_REF] Bailey | The epidemiology and control of nosema disease by the honey-bee[END_REF][START_REF] Bucher | Disease of the larvae of tent caterpillars caused by a sporeforming bacterium[END_REF] and the associated risk of contact and infection for the host dramatically grow when host density is high and/or living space is confined [START_REF] Weiss | Defecation behavior and ecology of insects[END_REF][START_REF] Jackson | Does sanitation facilitate sociality?[END_REF][START_REF] Schmid-Hempel | Parasites and Their Social Hosts[END_REF]. As a result, many group-living species do not show coprophagy and/or have evolved sanitation behaviors to limit contacts with feces from conspecifics [START_REF] Meunier | Social immunity and the evolution of group living in insects[END_REF]. Classical examples of these behaviors are expelling feces out of the nest [START_REF] Thomson | Some adaptations for the disposal of faeces. The hygiene of the nest in Australian birds[END_REF][START_REF] Weiss | Good housekeeping: why do shelter-dwelling caterpillars fling their frass?[END_REF][START_REF] Biedermann | Larval helpers and age polyethism in ambrosia beetles[END_REF] and restricting defecation to a single location [START_REF] Dethier | The world of the tent-makers: a natural history of the eastern tent caterpillar[END_REF][START_REF] Farji-Brener | Social life and sanitary risks: evolutionary and current ecological conditions determine waste management in leaf-cutting ants[END_REF].

In the European earwig Forficula auricularia, individuals actively maintain their feces in the nest. In this subsocial insect, mothers remain with their clutch of eggs during winter and then stay with the newly emerged juveniles (called nymphs) during the first two weeks following egg hatching [START_REF] Lamb | Parental behaviour in the Dermaptera with special reference to Forficula auricularia[END_REF][START_REF] Kölliker | Benefits and costs of earwig (Forficula auricularia) family life[END_REF][START_REF] Van Meyel | Love them all: Mothers provide care to foreign eggs in the European earwig Forficula auricularia[END_REF][START_REF] Tourneur | Variations in seasonal (not mean) temperatures drive rapid adaptations to novel environments at a continent scale[END_REF]. During this post-hatching family life, mothers and nymphs actively maintain their feces in the nest and line its walls and ground with this material [START_REF] Körner | Growing up with feces: benefits of allocoprophagy in families of the European earwig[END_REF].

Mothers provide multiple forms of post-hatching care to their nymphs, including fierce protection against predators, grooming behaviours and food provisioning through regurgitation [START_REF] Staerkle | Maternal food regurgitation to nymphs in earwigs (Forficula auricularia)[END_REF][START_REF] Mas | Differential effects of offspring condition-dependent signals on maternal care regulation in the European earwig[END_REF][START_REF] Ratz | The population determines whether and how life-history traits vary between reproductive events in an insect with maternal care[END_REF] and they typically favour nymphs in good compared to poor conditions [START_REF] Mas | A chemical signal of offspring quality affects maternal care in a social insect[END_REF][START_REF] Mas | Differential effects of offspring condition-dependent signals on maternal care regulation in the European earwig[END_REF].

Nevertheless, earwig nymphs exhibit early foraging capabilities and are thus relatively independent of maternal care [START_REF] Kölliker | Benefits and costs of earwig (Forficula auricularia) family life[END_REF][START_REF] Thesing | Short-term benefits, but transgenerational costs of maternal loss in an insect with facultative maternal care[END_REF][START_REF] Vogelweith | Age, pathogen exposure, but not maternal care shape offspring immunity in an insect with facultative family life[END_REF]. Some studies even showed that nymphs survive better in the absence compared to the presence of their mothers when the environment contains limited food resources (Meunier & Kölliker, 2012;[START_REF] Kramer | When earwig mothers do not care to share: Parent-offspring competition and the evolution of family life[END_REF], raising fundamental questions about why earwig mothers stay with their juveniles.

Recent works propose that the maintenance of feces in the nest could mediate a form of social immunity and/or allow the expression of allo-coprophagy, which could overall promote family life for earwig mothers and juveniles. A previous study showed that the feces of both mothers and nymphs exhibit antimicrobial properties, their maintenance in the nest could thus be a form of social immunity preventing the development of mould and other potential pathogens in the offspring vicinity [START_REF] Diehl | Feces production as a form of social immunity in an insect with facultative maternal care[END_REF]. This phenomenon has also been reported in the subterranean termite Coptotermes formosanus [START_REF] Chouvenc | Extended disease resistance emerging from the faecal nest of a subterranean termite[END_REF] and the burying beetle Nicrophorus vespilloides [START_REF] Rozen | Antimicrobial strategies in burying beetles breeding on carrion[END_REF]. On the other hand, some studies have shown that the feces produced by earwig nymphs are consumed by their siblings during family life [START_REF] Falk | Sibling cooperation in earwig families provides insights into the early evolution of social life[END_REF]. This consumption increases when mothers reduce their investment in maternal care [START_REF] Kramer | Negative association between parental care and sibling cooperation in earwigs: a new perspective on the early evolution of family life[END_REF][START_REF] Kramer | When earwig mothers do not care to share: Parent-offspring competition and the evolution of family life[END_REF] and the feces produced by earwig nymphs exhibit nutritive value that promotes sibling survival in absence of any food source or in presence of maternal feces only [START_REF] Körner | Growing up with feces: benefits of allocoprophagy in families of the European earwig[END_REF]. Hence, the maintenance of feces in the nest could (also) serve as food stock for nymphs (that is self-protected from the competition with microbes via its antimicrobial properties), which may help to mitigate the risks associated with food deprivation and/or a lack of maternal food provisioning. Whether this food stock can also be used by mothers and thus whether it can encourage mothers to stay with their (foraging) offspring remained untested.

In this study, we investigated whether and why mothers consume feces produced by their juveniles in the European earwig F. auricularia L. Using a 2x2 full factorial experiment, we tested whether 1) earwig mothers consume feces pellets produced by their nymphs, and whether 2) mothers adapt their level of coprophagy to their own nutritional needs and/or the nutritional condition of the feces-producing nymphs (which may, for instance, shape the nutritional quality of the feces ; [START_REF] Körner | Growing up with feces: benefits of allocoprophagy in families of the European earwig[END_REF]. We also investigated whether offspring could actively assist their mothers via allo-coprophagy by testing whether 3) nymphs adapt their production of feces (e.g. via a modified foraging activity and/or food intake) to the nutritional needs of their mother. We set up four types of experimental families in which we provided food either to the mothers, the nymphs, both mothers and nymphs, or none of them.

We then counted the number of feces pellets produced by nymphs and the occurrence and level of maternal consumption of these nymphal feces pellets. If mothers exhibit allocoprophagy, we predicted that mothers would consume at least some feces pellets produced by their nymphs. If the level of allo-coprophagy is driven by the nutritional needs of the mother, we predicted feces consumption would be higher in food-deprived compared to non-food deprived mothers. Finally, if nymphs actively promote allo-coprophagy when tended by food-deprived mothers, we predicted that nymphs would produce more feces when they were tended by a food-deprived compared to a fed mother.

MATERIALS AND METHODS

Experimental setup

The experiment involved 95 earwig families produced by 95 F. auricularia females (from the clade B of this species; [START_REF] Wirth | Molecular and reproductive characterization of sibling species in the european earwig (Forficula auricularia)[END_REF][START_REF] González-Miguéns | Speciation patterns in the Forficula auricularia species complex: cryptic and not so cryptic taxa across the western Palaearctic region[END_REF]) field-sampled in June 2018 in Pont-de-Ruan (France) and then maintained under standard laboratory conditions until egg-laying (Nov-Dec 2019) and then egg hatching (Jan-Feb 2019; details of the standard rearing conditions are provided in Meunier et al., 2012). One day after egg hatching, each family was randomly assigned to one of four treatments consisting of the provisioning of food to either i) the mother (N=24), ii) the nymphs (N=24), iii) both the mother and the nymphs (N=23) or iv) none of them (N=24). To this end, the number of nymphs per family was first standardized to 25 (initial mean ± SE number of nymphs per clutch = 34.52 ± 0.57; nymphs were never mixed between clutches) and the resulting experimental family transferred in Petri dishes (10 cm diameter) lined with moistened sand and maintained under 20°C and 12h:12h light: dark cycle.

The manipulation of food provisioning was done by temporarily separating mothers and nymphs in two small Petri dishes (5 cm diameter) lined with moistened sand, then providing them with an ad libitum amount of food or no food (depending on the treatment) for two hours, and finally returning mothers and their nymphs to their original large Petri dishes. Food provisioning was manipulated two, four, six and eight days after egg hatching. Earwig family life typically lasts 14 days and nymphs can survive up to 15 days without food access [START_REF] Körner | Growing up with feces: benefits of allocoprophagy in families of the European earwig[END_REF]. Mothers were fed with a standard food preparation (mostly containing pollen, carrots, and cat food; see detailed food composition in Meunier et al., 2012) on each of these days. By contrast, nymphs were fed with the standard food preparation on days two and four, whereas they were fed with green-coloured pollen pellets on day six and eight [START_REF] Kramer | Negative association between parental care and sibling cooperation in earwigs: a new perspective on the early evolution of family life[END_REF]. The consumption of green pollen allowed the subsequent production of green feces pellets by juveniles, which facilitated both their counting by the experimenter and the discrimination between nymph and (when present) mother produced feces [START_REF] Falk | Sibling cooperation in earwig families provides insights into the early evolution of social life[END_REF].

Immediately after our last manipulation of food provisioning on day eight, 15 nymphs per family were isolated in a new container to allow feces production. Fourteen hours later, we removed the nymphs, counted the number of feces pellets in the container (without touching them as they are very fragile and cannot be removed and/or moved) and then transferred each mother to the container with the feces of their own nymphs. These 14 hours of isolation are unlikely to mask the (potential) effect of mothers' nutritional condition on nymphs feces production, because (1) the effect of maternal condition on nymph behaviours is known to operate during at least 1 day without direct contacts with mothers [START_REF] Wong | Cues of maternal condition influence offspring selfishness[END_REF] and because (2) feces production is the outcome of a long physiological process, which thus here, is more likely to reflect the impact of the 8 previous days of direct contacts with a mother rather than the 14h of isolation.

Two hours later, we counted the number of remaining feces pellets in the container.

The very few feces pellets produced by mothers during these two hours (when present) were typically non-green and larger than the ones produced by nymphs and were thus excluded from this counting. To minimize observer bias, the feces counting was done blindly regarding the treatment applied to the mothers and their nymphs. For feces production (day 8) and feces consumption (day 9), family members were transferred to new Petri dishes (10 cm diameter) lined with a piece of white, moistened filter paper (Macherey-Nagel GmbH & Co. KG, Düren, Germany) to facilitate the counting of green feces pellets.

We defined the number of feces pellets eaten by a mother as the number of green feces pellets present in the container before minus after maternal introduction. Somewhat surprisingly, this number was negative in 4 of the 95 replicates (4.2%). A likely explanation for these values is that even if mothers typically produce feces pellets that are non-green and larger compared to the feces of their nymphs, a few mothers might have produced some 'small' indistinguishable feces pellets during the experiment and this production happened to be larger than the number of nymph's feces pellets they have possibly consumed in these four replicates. Among these four females, two belonged to the treatment where both females and nymphs had no food access (our calculation led to -1 and -1 eaten feces pellets, respectively), one to the treatment where both females and nymphs had food access (-2 eaten feces pellets) and one to the treatment where only the nymphs had food access (-1 eaten feces pellets). For consistency, we removed these four values in all our statistical analyses. Nevertheless, either changing these values to zero, removing or maintaining them in our analyses do not qualitatively change the results (see details in Table S1). Note that only a few replicates contained the typical 'large' non-green maternal feces pellets, which suggests that maternal feces production was overall very limited during the 2h of our experiment. Thus, the potential limit of the method discussed above is unlikely to affect our general conclusions.

Statistical analysis

We analyzed the effects of nymphs' and mothers' food status on the production of feces pellets using a linear general model (LM), in which the number of feces pellets produced by the fifteen nymphs was entered as a response variable, while nymphs' food status, mothers' food status and the interaction between these two factors were entered as explanatory factors. Because many mothers did not consume any fecal material produced by their nymphs (see results and Figure S1), this data was analysed using two successive approaches. We first analyzed the effects of nymphs' and mothers' food status on the consumption of at least one nymphs' feces pellet by a mother using a generalized linear model (GLM) with binomial error distribution. In this model, whether a mother was or was not coprophagous (1 or 0, respectively) was entered as a response variable, while nymphs' food status, mothers' food status and the interaction between these two factors were entered as explanatory factors. Second, we analysed the effects of nymphs' and mothers' food status on the level of feces consumption by coprophagous mothers using an additional GLM (with binomial error distribution corrected for overdispersion) and an additional LM, in which the proportion and the absolute number of feces pellets eaten by a coprophagous mother, respectively, were used as a response variable, while nymphs' food status, mothers' food status and the interaction between these two factors were entered as explanatory factors. The proportion of feces pellets eaten by the coprophagous mothers was entered using the cbind function in R, in which we used the number of feces pellets eaten and non-eaten. To fulfil model assumptions and thus ensure both homoscedasticity and a Gaussian distribution of LM residuals, we log-transformed both the number of feces pellets produced by nymphs and the absolute number of feces pellets eaten by mothers. When required, we conducted pairwise comparisons between treatments using estimated marginal means of the LMs and GLM, and Tukey-adjusted P-values. All these models were conducted with R v4.0.2 (R Core Team, 2017) loaded with the packages car and emmeans.

RESULTS

Overall, the groups of nymphs produced from 2 to 194 feces pellets (mean ± SE = 48.32 ± 4.97; Figure S1A), which corresponds to an average of 3.2 feces pellets per nymph per 14h. The consumption of at least one nymphs' feces pellet by the mother was present in 46 of the 91 (50.5%) trials. This maternal coprophagy was overall more frequent when the feces were produced by fed compared to food-deprived nymphs (Figure 1A; LR χ 2 1 = 5.99, P = 0.014), whereas it was independent of mothers' food access and of an interaction between mothers' and nymphs' food access (Table 1).

In the 46 coprophagous mothers described above, the absolute number of nymphs' feces pellets eaten by each female varied from 1 to 125 (mean ± SE = 12.98 ± 3.28; Figure S1B).

These numbers correspond to 1 % to 75 % (mean ± SE = 20.05 ± 2.80 %) of the total number of nymphs' feces pellets available during the tests, indicating that coprophagous mothers never consumed all the feces pellets produced by their nymphs. The proportion of feces pellets eaten by these coprophagous mothers was higher when mothers were starved compared to fed (LR χ 2 1 = 20.65, P < 0.0001), whereas it was independent of nymphs' access to a food source and of an interaction between nymphs' and mothers' food access (Table 1). By contrast, the absolute number of feces pellets eaten by coprophagous mothers was higher when the feces were produced by fed compared to food-deprived nymphs (Figure 1; F 1,42 = 12.28, P = 0.001), whereas it was independent of mothers' food access and of an interaction between nymphs' and mothers' food access (Table 1).

The number of feces pellets produced by the nymphs was shaped by an interaction between nymphs' and mothers' food access (Figure 2; F 1,91 =7.57, P = 0.007; see table 1 for full model description). Contrary to our predictions, however, this interaction reveals that the fed nymphs produced more feces when they were tended by fed compared to food-deprived mothers (pairwise comparison: t 91 = -3.37, P = 0.006) and that this effect was absent in the food-deprived nymphs (pairwise comparison: t 91 = 0.50, P = 0.959).

DISCUSSION

Whereas earwig juveniles actively consume the feces produced by their sibling during family life [START_REF] Falk | Sibling cooperation in earwig families provides insights into the early evolution of social life[END_REF], whether earwig mothers may also exhibit allo-coprophagy remained unknown. Here we first show that about 50% of the mothers indeed consumed feces produced by their juveniles. The likelihood that a mother became coprophagous was higher in the presence of feces produced by fed compared to food-deprived nymphs. Similarly, we found that the level of feces consumption by coprophagous mothers was higher when the feces were produced by fed compared to food-deprived nymphs (for the absolute number of feces eaten, only), and higher when mothers had no access compared to previous access to a food source (for both proportion and absolute number of feces eaten). Finally, our data show that the fed nymphs did not increase their production of feces when their mothers were food-deprived (i.e.

no sign of offspring assistance), and instead that the fed nymphs produced more feces when they were tended by fed compared to food-deprived mothers.

Our results first confirm that earwig mothers can consume the feces of their juveniles. In the context of family life, allo-coprophagy typically serves to clean the nesting area and/or to provide nutritional benefits to consumers [START_REF] Nilsson | Coprophagy in larval Culiseta bergrothi (Diptera: Culicidae)[END_REF][START_REF] Kopanic | An adaptive benefit of facultative coprophagy in the German cockroach Blattella germanica[END_REF][START_REF] Weiss | Defecation behavior and ecology of insects[END_REF][START_REF] Mirabito | A double-edged sword? The cost of proctodeal trophallaxis in termites[END_REF]. In earwigs, feces consumption is unlikely to be a simple nest cleaning behaviour, as nests' walls and grounds are actively covered with mothers and nymphs feces and these feces have antimicrobial properties [START_REF] Diehl | Feces production as a form of social immunity in an insect with facultative maternal care[END_REF][START_REF] Körner | Growing up with feces: benefits of allocoprophagy in families of the European earwig[END_REF]. Our findings also suggest that the nutritional benefits of allo-coprophagy only partly determine its expression by earwig mothers. This is because we found that the proportion of feces consumed was overall higher in coprophagous mothers that were previously starved compared to wellfed, but also that the nutritional needs of mothers did not shape their likelihood to become coprophagous nor the absolute quantity of feces they have eaten. Allo-coprophagy might thus have other (additional) functions in earwig mothers, among which access to specific nutrients, mutualists symbionts and/or immune components are likely candidates [START_REF] Diehl | Feces production as a form of social immunity in an insect with facultative maternal care[END_REF][START_REF] Arcila | Friend or foe? The apparent benefits of gregarine (Apicomplexa: Sporozoa) infection in the European earwig[END_REF][START_REF] Van Meyel | Alteration of gut microbiota with a broad-spectrum antibiotic does not impair maternal care in the European earwig[END_REF].

Our data also reveal that fed nymphs (and only fed nymphs) produced more feces when their mothers were already well-fed. Under standard pathogen-free environments, increased production of feces typically results from a higher food intake. Our results thus suggest that fed nymphs received an additional amount of food from their fed mothers, probably during the periods of family interactions. Interestingly, this pattern was absent with starved nymphs: these nymphs did not increase feces production in presence of fed mothers. This implies that fed mothers did not provide food to starved nymphs and thus that mothers control this food transfer. This might be the case if mothers provide food to their offspring mostly via regurgitation [START_REF] Staerkle | Maternal food regurgitation to nymphs in earwigs (Forficula auricularia)[END_REF] and not via allo-coprophagy on maternal feces.

Overall, our findings also emphasize that fed mothers provided food to nymphs that were already well-fed and thus likely to be of good condition, and not to nymphs that were starved and in poor condition. This is in line with previous works showing that earwig mothers increase their investment into care (i.e. foraging activity, allocation of food resources and grooming behaviours) when they are exposed to chemical cues extracted from well-fed compared to poorly-fed juveniles [START_REF] Mas | A chemical signal of offspring quality affects maternal care in a social insect[END_REF][START_REF] Mas | Differential effects of offspring condition-dependent signals on maternal care regulation in the European earwig[END_REF]. These results support a general prediction stating that parents preferentially feed offspring of higher reproductive value to maximize their fitness return on parental investment [START_REF] Haig | Brood reduction and optimal parental investment when offspring differ in quality[END_REF]. More generally, they highlight that earwig mothers do not provide unconditional care during family life (Meunier & Kölliker, 2012;[START_REF] Kramer | When earwig mothers do not care to share: Parent-offspring competition and the evolution of family life[END_REF] and may neglect juveniles that are the most in need of care (see also [START_REF] Mas | A chemical signal of offspring quality affects maternal care in a social insect[END_REF]).

To conclude, our experiments revealed that earwig mothers can consume the feces of their juveniles during family life, that this consumption is only partly driven by the nutritional needs of the mother and that juveniles do not adapt their feces production to the nutritional needs of their mothers. These findings call for future studies investigating the effects of nymphs starvation on the quality of their fecal pellets and exploring the non-nutritive drivers of maternal consumption of juveniles feces, among which the transfer of gut microbiota could be a prime candidate [START_REF] Van Meyel | Alteration of gut microbiota with a broad-spectrum antibiotic does not impair maternal care in the European earwig[END_REF]. They also call for studies disentangling whether this behaviour is a mere product of selection for immediate food access with no further implication in the early evolution of family life, or whether it may have such an implication by selecting for the maintenance of mothers in the vicinity of their foraging juveniles and thus, by ultimately promoting the emergence and consolidation of facultative family living from a solitary state.

Figure 1 -

 1 Figure 1 -Effect of juveniles' or mothers' access to food during family life on the (a) likelihood

Figure 2 -

 2 Figure 2 -Effects of juveniles' or mothers' access to a food source during family life on the

Table 1 -

 1 Statistical overview of the effects of juveniles' or mothers' access to a food source during family life on (a) the number of feces pellets produced by nymphs, (b) the likelihood of maternal coprophagy, as well as (c) the proportion and (d) number of feces pellets eaten by mothers. Significant p-values are in bold. LR = Likelihood Ratio.

		(a) No. feces	(b) Likelihood	(c) Proportion of	(d) No. nymphs'
		pellets produced	of maternal	nymphs' feces	feces eaten by
		by nymphs	coprophagy	eaten by mothers	mothers
		F 1,91	P	LR χ 2	1	P	LR χ 2	1	P	F 1,42	P
	Juveniles	16.25 0.0001	5.99 0.014	1.38	0.240	12.28 0.001
	Mothers	4.07	0.047	0.39 0.532	20.65 <0.001	1.57 0.217
	Interaction	7.57	0.007	1.50 0.220	3.18	0.075	3.87 0.056

ACKNOWLEDGEMENTS

We thank Maximilian Körner, Marlène Goubault, Florent Figon, Louis Pailler, Fanny Ruhland, Lucas Sire, Glauco Machado and anonymous reviewers for their comments on previous versions of this manuscript. We also thank Laurent Ardouin for granting us access to his orchards for earwig field sampling.


Tom Druet

Bastien Chapuis

Manfred Jules

Guillaume Laffont

Emmanuel Moulin

NDT AND QUALITY ASSURANCE OF WELDED PRODUCTS V-1670-15 PASSIVE SHM SYSTEM FOR CORROSION DETECTION BY GUIDED WAVE TOMOGRAPHY

Guided elastic waves emitted by a sensor and propagating to another one are often used as the physical way of detecting the defect in Structural Health Monitoring (SHM) systems of plate like structures. However, the implementation of SHM systems is restricted in many situations by the necessity to store or to harvest the electric energy necessary to emit the waves. A promising way to overcome this constraint is to use techniques based on the cross-correlations of the ambient noise in place in the structure. The idea is to take advantage of the elastic noise naturally present in the structure (due to engine vibrations or aero-acoustic turbulences on the fuselage of an aircraft for example) in order to avoid the emission of the elastic waves by the SHM system. The complexity of the embedded SHM system is therefore reduced. We present here studies of noise cross-correlation techniques that have been conducted with the aim of doing passive guided wave tomography of extended defects (such as corrosion) using an array of piezoelectric (PZT) transducers. Noise is generated by spraying compressed air on the surface of a thin aluminum plate. Passive measurements are compared to active signals to demonstrate the effectiveness of the cross-correlation technique. Experimental results which come from tomographic time-of-flight imaging algorithms will also be described. Finally, an extension of this technique using Fiber Bragg Gratings (FBG) optic sensors will be presented.

IIW Keywords:

NDT, non-destructive testing, structural health monitoring, SHM, tomography

ABSTRACT

Guided elastic waves emitted by a sensor and propagating to another one are often used as the physical way of detecting the defect in Structural Health Monitoring (SHM) systems of plate like structures. However, the implementation of SHM systems is restricted in many situations by the necessity to store or to harvest the electric energy necessary to emit the waves. A promising way to overcome this constraint is to use techniques based on the cross-correlations of the ambient noise in place in the structure. The idea is to take advantage of the elastic noise naturally present in the structure (due to engine vibrations or aero-acoustic turbulences on the fuselage of an aircraft for example) in order to avoid the emission of the elastic waves by the SHM system. The complexity of the embedded SHM system is therefore reduced. We present here studies of noise cross-correlation techniques that have been conducted with the aim of doing passive guided wave tomography of extended defects (such as corrosion) using an array of piezoelectric (PZT) transducers. Noise is generated by spraying compressed air on the surface of a thin aluminum plate. Passive measurements are compared to active signals to demonstrate the effectiveness of the cross-correlation technique. Experimental results which come from tomographic time-of-flight imaging algorithms will also be described. Finally, an extension of this technique using Fiber Bragg Gratings (FBG) optic sensors will be presented.

IIW-Thesaurus keywords: NDT, non-destructive testing, structural health monitoring, SHM, tomography 1 Introduction

The Structural Health Monitoring (SHM) consists of embedding sensors in a structure such as an aircraft or a naval ship in order to detect defects (for example cracks or corrosion in metallic materials or delamination in composite materials) before a serious fault occurs in the structure. Guided elastic waves emitted by a sensor and propagating to another one are often used as the physical way to detect defects. In aeronautics, the classical approach generally aims at minimizing the number of sensors to limit the embedded mass as well as the sensors intrusiveness within the structure. Comparisons between current signals and baseline signals are often performed in order to reveal the presence of defects [START_REF] Croxford | Strategies for guided-wave structural health monitoring[END_REF]. However, this method may not be robust under certain conditions such as changes in temperature, stress, sensors aging, etc.

A possible strategy to avoid the use of baseline signals can consist in increasing the number of sensors and performing guided wave tomography. Indeed, more relevant physical information is obtained from the structure, making the diagnosis more robust. Moreover tomography algorithms produce images that are much easier to interpret than temporal signals. Implementation of SHM systems with a large number of sensors could be intrusive with the use of piezoelectric transducers. Optical fiber sensors using Fiber Bragg gratings (FBGs) for dynamic strain measurements should allow multiplexing capabilities with low intrusiveness in the structure. However, FBG are generally used only as sensors but not as a source of elastic waves. A promising way to tackle these constraints is to use techniques based on cross-correlation of the ambient acoustic noise present in the structure. It has been shown that, under certain conditions, transient responses between two sensors can be passively estimated from the cross-correlation of ambient noise [START_REF] Weaver | Ultrasonics without a Source: Thermal Fluctuation Correlations at MHz Frequencies[END_REF][START_REF] Sabra | Using cross correlations of turbulent flow-induced ambient vibrations to estimate the structural impulse response. Application to structural health monitoring[END_REF][START_REF] Moulin | Applicability of acoustic noise correlation to Structural Health Monitoring in non-diffuse field conditions[END_REF]. The idea is to take advantage of the elastic noise naturally present in the structure (due to engine vibrations or aero-acoustic turbulence on the fuselage of an aircraft for example) in order to avoid the need for emission of the elastic waves by the SHM system.

This paper shows an active tomography image which has been performed with a time-of-flight tomography algorithm using experimental data produced by piezoelectric transducers. Furthermore, a comparison between active signals and passive signals which comes from cross-correlation of ambient

[Type text] [Type text]
noise produced by spraying compressed air on an aluminum plate shows that it is possible to detect time-of-flight passively. In Section 2 the tomography algorithm used for this work is presented. Then, in Section 3 the experimental setup used to acquire the data necessary to obtain the tomography images is described. Finally, in Sections 4 and 5 the experiments are presented showing that it is possible to use cross-correlation of ambient noise present in the plate-like structure to get the data required for the tomography algorithm. Several configurations of sensors were studied using piezoelectric transducers, Fiber Bragg Gratings and a combination thereof.

Time-of-Flight Tomography Algorithm

Images shown in this paper are obtained with a time-of-flight tomography algorithm which uses the Simultaneous Iterative Reconstructive Technique (SIRT) [START_REF] Kak | Principles of Computerized Tomographic Imaging[END_REF]. Straight ray assumption is taken within this framework. Straight ray tomography does not take into account refraction and diffraction. By ignoring diffraction, only defects bigger than the first Fresnel zone [START_REF] Williamson | A guide to the limits of resolution imposed by scattering in ray tomography[END_REF] and varying slowly are correctly reconstructed. By ignoring refraction the algorithm is limited to low contrast flaws. Better algorithms that take into account refraction and diffraction exist [START_REF] Huthwaite | High-resolution guided wave tomography[END_REF] and will be studied in future work. A simulation respecting these assumptions (i.e. no refraction and diffraction) was conducted (see Figure 1). The configuration on Figure 1a shows 30 sensors (which are depicted by yellow points) and three different defects. The image values matrix has a size of 23 x 23 pixels. Input data is a set of time-of-flight values deduced from dispersion curves of the A0 mode at 30 kHz propagating in a 2 mm thick plate. The absolute group velocity is 1447 m/s when waves propagate in the healthy part of the plate and 1300 m/s, 1200 m/s and 1056 m/s when they propagate in defect zones (i.e. zones of reduced thickness which can illustrate corrosion phenomenon). In practice, image smoothing is interesting since corrosion and pixels will unlikely be superimposed on each other. That is why Figure 1 and the experimental image on Figure 3b are smoothed by performing interpolation.

Experimental Setup

The time-of-flight algorithm described in the previous section helped, among others things, to design an experimental setup (see Figure 2). The setup is composed of a computer running LabView which controls an oscilloscope and a multiplexer; a generator which is synchronized with the oscilloscope by a trigger; an amplifier; a filter and finally the piezoelectric transducers attached to the 2 mm thick aluminum plate. Phenyl salicylate (SALOL) is used to bond localized thin calibrated aluminum layers on the plate. For experimental purposes, these layers will serve as an easily removable defect with a somewhat similar effect than corrosion on wave velocity due to thickness change.

Passive experimental results

The input data necessary to obtain the tomography images is a set of time-of-flight values, as presented in previous sections. This Section aims at assessing the retrieval of this kind of data from passive (1)

Piezoelectric sensors (PZT)

A comparison between active and passive signals, obtained thanks to cross-correlation of ambient elastic noise, is presented in Figure 4. This is a reason to think that passive tomography is promising. We can notice in the passive signals some parasite wavelets caused by the spatial distribution of the ambient noise [START_REF] Chehami | Experimental study of passive defect detection and localization in thin plates from noise correlation[END_REF][START_REF] Colombi | Green's function retrieval through crosscorrelations in a two-dimensional complex reverberating medium[END_REF]. Indeed, the more the ambient structural noise is spatially uniformly distributed the better the convergence towards Green's functions will be.

Figure 5 shows that it is possible to obtain time-of-flight on passive signals for higher frequencies, of at least 250 kHz (for the ambient noise created by spraying the air on the aluminum plate). Here, the S0 as well as A0 modes can be detected in Figure 5 but it is not possible to correctly identify the time-of-flight for A0. Indeed, S0 reflections arrive at the same time as the first A0 wave packet.

Fiber Bragg Gratings (FBG)

It has already been demonstrated that FBGs are able to detect guided waves emitted by piezoelectric transducers [START_REF] Betz | Acousto-ultrasonic sensing using fiber Bragg gratings[END_REF][START_REF] Botsev | Structural Health Monitoring using an embedded PZT-FBG ultrasonic sensor array[END_REF]. Here we perform passive experimental measurements using only fiber Bragg gratings sensors. With such passive measurements, PZT transducers for elastic wave emission would no longer be necessary, reducing the intrusiveness of the sensors in the structure. Elastic noise is generated in the structure using the same setup as in previous sections: by spraying compressed air on the aluminum plate surface. Figure 6 shows the passive reconstruction of the A0 mode at 40 kHz between FBGs bonded on the surface of a 2 mm thick aluminum plate and spaced 400 mm apart from each other. Comparison between active and cross-correlation signals.

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This demonstrates the feasibility of passive reconstruction using FBGs only. Indeed, the theoretical timeof-flight values perfectly match the maximum of the first wave packet (A0 mode). Figure 7 shows the passive reconstruction (in red) between a FBG and a PZT for several distances between both sensors. The PZT is used here only as sensor of the elastic noise, not to emit the guided waves. It has been chosen for convenience in this experiment because it is easier to move than the FBG. It confirms that the first wave packet is well reconstructed and that it is possible to correctly identify its time-of-flight with passive measurements using only FBG. In Figure 7, in the same way as in Figure 4, we can see that the first wave packets are adequately superimposed on passive and active signals.

Conclusion

This paper has shown that it is possible to image defects experimentally by using piezoelectric transducers and a time-of-flight tomography algorithm without the need of baseline signals. Active experimental tomography using guided wave data was performed in this paper. It was also shown that it is possible to detect time-of-flight by using cross-correlation of ambient noise with piezoelectric transducers as well as with fiber Bragg gratings. Passive experimental tomography is the next step of our study by merging the results obtained in these two domains.

  As a contributor to the IIW website you implicitly accept the terms of RD023: The author declares that the uploaded document(s) is(are) not covered by a copyright other than the author. If not, the document must not be uploaded on the website without the authorization of the copyright holder (RD024 or RD025). [Type text] [Type text] Passive SHM system for corrosion detection by guided wave tomography Tom Druet (a), Bastien Chapuis (a), Manfred Jules (a), Guillaume Laffont (a), and Emmanuel Moulin (b) (a) CEA LIST (France) (b) Université de Valenciennes (France)

Figure 1 :

 1 Figure 1: (a) Configuration; (b) Result after 1 iteration.

[

  Type text] [Type text]

Figure 2 :Figure 3 Figure 3 :

 233 Figure 2: (a) Experimental setup; (b) 30 piezoelectric transducers (Ø 18 mm) stuck on the 2 mm thick aluminum plate.

[

  Type text] [Type text] signals. Equation 1 shows the cross-correlation formulation with 𝑢 ⃗ the displacement field at the point 𝑥 and 𝑣 the displacement field at the point 𝑥 ′ ⃗⃗⃗ . Every passive signal obtained in this Section results from cross-correlation of ambient noise recorded during the amount of time necessary (10 seconds in our experiments) for it to converge towards Green's functions. 𝐶 𝑢 ⃗ ⃗ ,𝑣 ⃗ (𝑡, 𝑥 , 𝑥 ′ ⃗⃗⃗ ) = lim 𝑇→∞ 1 𝑇 ∫ 𝑢 ⃗ (𝜏, 𝑥 )𝑣 (𝑡 + 𝜏, 𝑥 ′ ⃗⃗⃗ ) 𝑇 𝑑𝜏 𝑇 0

Figure 4 :

 4 Figure 4: Comparison between active and cross-correlation signals. Eight cycles tone-burst at 20 kHz were generated in order to obtain the active signals shown in Figure 4. The elastic noise used for the passive signals was produced by spraying compressed air on the aluminum plate surface. The absolute values of those signals are plotted on Figure 4 for several distances between sensors. This way, the propagation of the first wave packet is shown for each active signal. Passive signals show a satisfactory reconstruction of Green's functions at least for the first wave packet. Indeed, the result is symmetrical (because of acoustic reciprocity between the two observation points) and first wave packets are adequately superimposed on passive and active signals. It is therefore possible to get time-of-flight from passive signals. This is a reason to think that passive tomography is promising. We can notice in the passive signals some parasite wavelets caused by the spatial distribution of the ambient noise[START_REF] Chehami | Experimental study of passive defect detection and localization in thin plates from noise correlation[END_REF][START_REF] Colombi | Green's function retrieval through crosscorrelations in a two-dimensional complex reverberating medium[END_REF]. Indeed, the more the ambient structural noise is spatially uniformly distributed the better the convergence towards Green's functions will be.

[

  Type text] [Type text]

Figure 5 ;

 5 Figure 5; Reconstruction of passive signals for high frequenciestheoretical time-of-flights for S0 and A0 at 250 kHz.

Figure 6 :

 6 Figure 6: Reconstruction of passive signals with 2 fiber Bragg Gratings bonded on the surface of a 2 mm thick aluminum plate.

Figure 7 :

 7 Figure 7: Reconstruction of passive signals between a piezoelectric transducer and a fiber Bragg grating.Comparison between active and cross-correlation signals.


A Lesaine

D Bonamy

G Gauthier

C L Rountree

V Lazarus

Highly porous layers of silica nano-spheres sintered by drying: Scaling up of the elastic properties from the beads to the macroscopic mechanical properties

Keywords: numbers: 82.70 Dd, 62.20.de, 81.16.Dn Keywords: Multi-scale homogenization approaches, mechanical properties, highly interconnected porosity, linear elasticity, particles sintering, drying of colloidal dispersions

Layers obtained by drying a colloidal dispersion of silica spheres are found to be a good benchmark to test the elastic behaviour of porous media, in the challenging case of high porosities and nanosized microstructures. Classically used for these systems, Kendall's approach explicitely considers the effect of surface adhesive forces onto the contact area between the particles. This approach provides the Young's modulus using a single adjustable parameter (the adhesion energy) but provides no further information on the tensorial nature and possible anisotropy of elasticity. On the other hand, homogenization approaches (e.g. rule of mixtures, Eshelby, Mori-Tanaka and self-consistent schemes), based on continuum mechanics and asymptotic analysis, provide the stiffness tensor from the knowledge of the porosity and the elastic constants of the beads. Herein, the self-consistent scheme accurately predicts both bulk and shear moduli, with no adjustable parameter, provided the porosity is less than 35%, for layers composed of particles as small as 15 nm in diameter. Conversely, Kendall's approach is found to predict the Young's modulus over the full porosity range. Moreover, the adhesion energy in Kendall's model has to be adjusted to a value of the order of the fracture energy of the particle material. This suggests that sintering during drying leads to the formation of covalent siloxane bonds between the particles.

I. INTRODUCTION

Porous materials formed by cohesive beads are commonly found in nature (sandstones, sedimentary rocks, opals, soils...) and in industry (ceramics, pharmaceutical pills, filter cakes, photonic materials, paintings...). Sintering between beads, whether resulting from evaporation, heat or compression, confers an overall cohesion and solid behavior to the material. For engineering purposes, it is of utmost importance to relate the mechanical properties at the macroscale to microscale behavior, whatever the constitutive relations of the components (elastic, plastic, viscoplastic...). This constitutes a broad field [START_REF] Willis | Homogenization Techniques for Composite Media[END_REF][START_REF] Castaneda | Heterogeneous materials[END_REF][START_REF] Dormieux | Microporomechanics[END_REF]; this paper only addresses the linear elastic part.

When looking for the equivalent elasticity of a packing of cohesive grains, a first possibility is to exploit the analogy between scalar elasticity and scalar electricity [START_REF] Jernot | [END_REF] and sketch the material as a network of resistances. Effective medium theory [5][6][7] then permits, from the particle coordination number and the density probability function of contact resistance, to compute the effective resistance and subsequently the elastic modulus of the packed system. However, these two parameters are difficult to assess, and this approach fails to take into account the inherent tensorial nature of elasticity. Even in the simple case of an isotropic solid, two parameters (e.g. bulk and shear moduli) are necessary to fully describe the elastic behavior of the material. Therefore, it is more appropriate to use homogenization methods rigorously derived within the framework of continuum mechanics and multi-scale asymptotic analysis [START_REF] Willis | Homogenization Techniques for Composite Media[END_REF][START_REF] Castaneda | Heterogeneous materials[END_REF][START_REF] Dormieux | Microporomechanics[END_REF]. Finally, when the particle size becomes submicrometric, adhesive surface forces are expected to become relevant. Kendall's approach explicitely takes these forces into account. Herein, it proves to be a relevant framework to cast the problem into. To the best of our knowledge, only a few papers report quantitative comparisons between these theoretical approaches and experimental ones (see e.g. [8][9][10] for past attempts).

The study herein proposes nanoporous materials obtained by drying a monodisperse aqueous colloidal suspension consisting of nanometer-sized silica spheres (Ludox HS-40) as a benchmark medium to test models against. During drying, water evaporation brings the particles into contact and transforms the initially liquid dispersion into a solid layer constituted of self-organized sintered particles (Sec. II). Controlling the drying rate provides a simple way to modulate the porosity of the dried material [11]. Both bulk and shear elastic constants are measured by ultrasound methods. Since the goal here is to bridge the continuum mechanics and soft matter communities [START_REF] Goehring | Desiccation cracks and their patterns: Formation and modelling in Science and Nature[END_REF][START_REF] Birk-Braun | [END_REF][14], we review the basic ingredients involved in classical homogenization schemes: the rule of mixtures and the Eshelby, Mori-Tanaka and selfconsistent schemes (Sec. III), and in Kendall's approach (Sec. IV). Sec. V compares the theoretical predictions with the experimental data and Sec. VI discusses the results.

The self-consistent approach accurately predicts both the shear and bulk moduli with no ajustable parameters, as long as the porosity is sufficiently small (less than 35%). Conversely, Kendall's approach predicts the variations of the Young's modulus with porosity over the full range, provided that the adhesion energy is properly adjusted. The fitted value is found to be surprisingly high when compared to the values usually considered in this kind of problem [START_REF] Birk-Braun | [END_REF]15]: It falls very close to the fracture energy, i.e. the energy required to break the covalent siloxane bonds, showing that drying colloidal suspensions enables the formation of strong covalent siloxane bonds between particles. This apparent discrepancy with the literature is discussed. The study herein is a follow-up of a previous study [11]. It uses Ludox HS-40, an aqueous dispersion (initial mass concentration φ m ∼ 40%) of silica spheres, commonly used as a model colloidal suspension [START_REF] Birk-Braun | [END_REF]14,[16][17][18]. According to SAXS measurements of the form factor, silica beads have an average diameter a = 16.2 nm with a relative polydispersity σ = 14%. In all the study the same batch (number STBF8427V) of Ludox was used. The porous material is obtained from 25 g of suspension poured into a Petri dish (3.5 cm radius), which corresponds to an initial liquid height of 5 mm. The system is then left to dry at room temperature (T = 25 ± 2 • C) in an enclosure with constant relative humidity (RH) until the evaporation ceases. This process typically lasts between 2 and 14 days depending on the RH value. As water evaporates, the beads come in contact, sinter and form a porous solid material of final thickness h ∼ 2 mm. Some desiccation cracks [START_REF] Goehring | Desiccation cracks and their patterns: Formation and modelling in Science and Nature[END_REF] appear during this drying process (Fig. 1A). These fractures are due to the shrinkage of the layer induced by evaporation and impeded by the substrate (i.e. bottom of the Petri dish) [19]. These cracks divide the layer into smaller morsels [20,21]. The size of the morsels increases with higher RH, that is slower drying.

II. EXPERIMENTAL METHODS

A. Sample preparation

B. Properties at the bead scale

Due to the nanometric size of the beads, their density (ρ s ) and bulk (k s ) and shear (µ s ) elastic moduli (or equivalently Young's modulus (E s ) and Poisson's ratio (ν s )) cannot be directly measured. Thus, the bead properties are assumed to be equivalent to the bulk properties of pure silica. Specifically, the spec values of pure silica (Corning 7980 standard grade) are used for these parameters [22]: ρ s = 2.20 g/cm 3 , k s = 35.4 GPa, µ s = 31.4 GPa, E s = 72.7 GPa, ν s = 0.16.

In order to scale up the properties of the beads at the macroscale, some knowledge of the particle arrangement at the microscale is necessary. Imaging the top of the morsels via an Atomic Force Microscope (AFM) provides the structure of the particle packing. Fig. 1B gives an example of the particle arrangement for RH = 50%. This image shows that the arrangement is on average homogeneous and isotropic along the surface, at larger scales. Henceforth, it is assumed that it remains true along the third direction. The agreement between the experiments and the models (Sec. V) will validate this hypothesis, a posteriori.

C. Sample properties at the macroscopic scale

The porosity, p, of the dried layer is a function of its density, ρ, as follows: p = 1-ρ/ρ s . Hence to determine p, we measure ρ using classical hydrostatic weighing techniques [11]. In the experiments herein, the prescribed constant relative humidity RH in the enclosure is varied between 10% and 95%, monitored within 3%, providing dried solid layers with increasing porosity, p, ranging between 0.34 and 0.45, respectively [11]. A larger range of porosities cannot be obtained without additional expensive heat treatments [23]. Moreover, heat treatments may change the physical properties of the system (e.g. by inducing grain growth). Figure 2 depicts the porosity as a function of RH, and Tab. I presents the corresponding tabulation of the values. Elastic properties (bulk, shear and Young's moduli and Poisson's ratio) are inferred via ultrasonic techniques [24]. Sound velocities for compression waves (c L ) and shear waves (c T ) were measured on the samples in ambient conditions, using a single transducter as described in the appendix. Bulk modulus (k) and shear modulus (µ) of the macroscopic sample can then be expressed as a function of ρ, c T and c L :

µ = ρc 2 T and k = ρc 2 L - 4 3 µ (1)
The material constants were obtained by averaging the values obtained with the transducers applied once at the top and once at the bottom of the sample. Also, as the resulting dried layers contain multiple fractured pieces (Fig. 1A), the measurements were conducted on two different morsels for each drying RH.

Figures 3A and3B present k and µ as a function of the RH value for two morsels, and Tab. II numerates the mean values. For humidities greater than ∼ 30%, k and µ are nearly the same in the two morsels; this highlights the homogeneity of layers dried under high humidity. At low humidities (i.e. RH < 30%, high evaporation rates), inhomogeneities arise, and thus, k and µ differ somewhat in the two morsels. The observation of a drying front crossing the layer during drying for RH = 15 and 20% suggests these low RH samples undergo directional horizontal drying. This leads to variations in the mechanical properties of these samples.

III. HOMOGENIZATION IN LINEAR ELASTICITY

Since the diameter, a, of the colloidal particles is several decades smaller than the thickness, h, of the layer (a/h ∼ 10 -8 /10 -3 ∼ 10 -5 ), one can define a Representative Elementary Volume (REV) of typical size such that a h (i.e. separation of length scales hypothesis). The scaling up of the elastic properties from the scale of the beads to the one of the layer is thus achievable by homogenization approaches [START_REF] Willis | Homogenization Techniques for Composite Media[END_REF][START_REF] Castaneda | Heterogeneous materials[END_REF][START_REF] Dormieux | Microporomechanics[END_REF] derived in the framework of continuum mechanics, in line with the seminal works of Eshelby [25], Hill [26], Budinsky [27] and Mori-Tanaka [28]. This section summarizes the underlying hypotheses and the main results of these approaches.

This paper approaches the problem as follows: the elastic behavior is first defined at the scale of the particles (Sec. III A). Then, the Representative Elementary Volume (Sec. III B) is introduced to provide a gateway between the micro-and macro-scales. The scaling up involves the resolution of the linear elasticity problem on the REV (Sec. III C). This problem is too complex to be solved analytically, and approximate results are instead invoked. Section III D summarizes the most commonly used rule of mixtures. As will be seen in Sec. V, this simple rule does not provide an accurate estimate of the elastic properties of the porous material.

The end of the section reviews Eshelby's (Sec. III E 1) and Mori-Tanaka's approximations (Sec. III E 2) as they are prerequisites for the self-consistent scheme (Sec. III F). A summary of the different models is given in Sec. III G.

A. Microscale

For any linear elastic material, the relation between the local strain and stress σ tensors can be written as:

σ(x) = C s : (x)
where C s is a proportionality constant expressing the elastic properties of the solid. Since and σ are secondorder tensors, this proportionality constant takes the form of a fourth-order tensor, called the stiffness tensor.

In the case of an isotropic material, C s reduces to:

C s = 3k s J + 2µ s K (2)
with J and K respectively the spherical and deviatoric parts of the fourth-order symmetric identity tensor I, given by:

J = 1 3 I ⊗ I and K = I -J (3)
with I the second-order identity tensor. This formulation presents the advantage of being compact, and it decouples the elastic properties into bulk k s and shear µ s contributions.

Notice combining the previous three equations equates to the more commonly used relation:

σ = k s (tr )I + 2µ s e
where e ≡ -1 3 (tr )I is the deviatoric part of . The above equations can be inverted, such that the local strain ( ) is a function of the stress (σ), and is written as follows:

= 1 + ν s E s σ - ν s E s (tr σ)I
where the following relations between k s , µ s , the Young's modulus (E s ), and the Poisson' ratio (ν s ) are invoked:

k s = E s 3(1 -2ν s )
and

µ s = E s 2(1 + ν s ) (4)
At the microscale, the system can be described as a composite material made of silica (stiffness C s ) and empty pores (stiffness C p ). In this instance, describing the elastic properties at the microscale requires a local fourth-order stiffness tensor: C loc (x) where C loc (x) = C s corresponds to the beads and C loc (x) = C p corresponds to the pores, such that:

σ(x) = C loc (x) : (x), ∀x ∈ REV (5)
Note, the stiffness of the empty pores is ideally zero. Yet in order to complete the calculations, it is necessary to assume a finite elasticity tensor for the pores. The final solution to the problem then corresponds to the limits k p /k s → 0 and µ p /µ s → 0, where k p and µ p denote the bulk and shear modulus of the pores [START_REF] Dormieux | Microporomechanics[END_REF]. Henceforth, the results are presented in this limit.

B. The Representative Elementary Volume as a gateway between the micro-and macro-scales

A classical way to proceed [START_REF] Castaneda | Heterogeneous materials[END_REF][START_REF] Dormieux | Microporomechanics[END_REF] is to isolate a Representative Elementary Volume (REV) and to prescribe some given displacement u at its boundary:

u = E.x, E x E x E x E x FIG. 4.
Representative Elementary Volume (REV) corresponding to a homogeneous matrix with spherical pores submitted to displacement Ex along its boundary, with E the strain tensor and x the position vector.

where x corresponds to the position vector (Fig. 4). The second-order strain tensor E corresponds to the macroscopic given strain tensor, which is assumed to be constant along the boundary. Using Gauss's divergence theorem, one can demonstrate that:

E = (x) (6)
where . ≡ 1 V REV . dV denotes the spatial average over the REV. The macroscopic stress Σ is then defined by its mean value:

Σ ≡ σ(x) (7)
in coherence with the Hill-Mandel's lemma [29,[START_REF] Mandel | Plasticité classique et viscoplasticité[END_REF]. At the macroscale, the porous material behaves as a linear elastic solid: thus, there is an effective stiffness tensor C such that

Σ = C : E (8)
If the microstructure arrangement is isotropic (i.e. there is no preferential orientation in the way the beads are packed), equivalent bulk (k) and shear (µ) moduli can be defined by:

C = 3kJ + 2µK (9)
The aim of homogenization is to derive the effective tensor C knowing C s and the microstructure arrangement. If isotropy is valid at all scales (the material components and their arrangement are isotropic), this is equivalent to look for the relations between (k, µ) and (k s , µ s ).

C. The strain concentration tensor, A

The local strain in the material (x) represents a solution of the problem of linear elasticity defined on the REV, hence it linearly depends on the macroscopic strain (E) prescribed at the boundary of the REV. In other words, there exists a fourth-order local tensor A, called the localisation or strain concentration tensor, such that:

(x) = A(x) : E, ∀x ∈ REV (10)
Plugging this last relation into (5), taking its mean value, implementing the definition of Σ (i.e. eqn ( 7)), and linking the result with the macroscopic constitutive relation (8) leads to:

C = C loc (x) : A(x) (11)
Thus to obtain C, it is necessary to determine A, which requires solving the elasticity problem for a REV composed of a beads assembly. This could, a priori, be done using computationally expensive numerical schemes. Alternatively, approximate solutions may be used. In what follows, three different estimations of C are provided: the upper Voigt bound and the Eshelby and Mori-Tanaka estimations as prerequisites for the self-consistent approximation.

D.

Voigt's upper bound [START_REF] Voigt | [END_REF]: the rule of mixtures A first possible approximation is to take A(x) = I that is (x) = E in all the REV, even in the pores. Then,

C = (1 -p)C s (12)
This approximation neglects the effect of the pores on the strain distribution. It corresponds to a material behavior in which the pores deform as if they were made the same homogeneous material as the solid phase, hence this estimation overestimates stiffness. One can rigorously demonstrate that it is an upper bound for the stiffness -the Voigt bound [START_REF] Voigt | [END_REF].

For an isotropic material, eqn ( 12) is equivalent to:

k/k s = 1 -p µ/µ s = 1 -p (13)

E. Pores embedded in a matrix

Another possible approach is to consider that the packing of beads is equivalent to an homogeneous solid matrix of elasticity C s , containing a repartition of spherical pores with the same porosity p (Fig. 4).

1. Weak porosity: Eshelby's approximation [25].

When the porosity is weak (p 1), the interactions between the pores can be neglected. Then A can be obtained from the analytical solution of the classical problem of Eshelby: an infinite elastic matrix of stiffness C s containing a single spherical inclusion of stiffness C p . It is then possible to show that: (14) where the expressions of α s and β s as a function of k s and µ s are given by:

k/k s = 1 -p 1-αs µ/µ s = 1 -p 1-βs
α s = 1 1+ 4µs 3ks and β s = 6(1+2 µs ks ) 5(3+4 µs ks ) (15)
It is noteworthy that α s and β s depend on k s and µ s only through µ s /k s = 3(1-2νs) 2(1+νs) (see eqn ( 4)), hence only on the dimensionless Poisson's ratio ν s .

2. High porosity: Mori-Tanaka's scheme [28].

For interacting pores, the Eshelby approach is not sufficient: a higher-order approximation is necessary. In this case, the Mori-Tanaka scheme is relevant. In this scheme, a single pore is isolated as an Eshelby problem, and a new boundary condition, u = E 0 .x, is adopted at infinity by choosing an auxiliary strain tensor E 0 which takes into account the interactions between the pores.

The detailed calculations are quite complex [START_REF] Dormieux | Microporomechanics[END_REF] and eventually lead to the following equations for k and µ: (16) with α s and β s given by eqn (15), so that the equations explicitly yield k ks and µ µs as a function of p, knowing k s /µ s , or equivalently, ν s .

k k s = 1 -p (1 -p) + p 1-αs µ µ s = 1 -p (1 -p) + p 1-βs
Notice that Eshelby's approximation ( 14) corresponds to the first order asymptotic expansion of the Mori-Tanaka scheme (16) with respect to p 1.

F. Self-consistent approximation [26] The Eshelby and Mori-Tanaka schemes assume a particular material geometry, in which individual, separate pores are surrounded by a solid matrix. But in the case of a sphere packing, pores are interconnected and percolate through the medium; thus, no surrounding matrix can be identified. Using a self-consistent scheme, which considers an Eshelby inclusion (solid or pore) in an equivalent homogenized medium of stiffness C, seems preferable. The approach used in solving this problem then invokes the one used for the Mori-Tanaka scheme, applied on the equivalent material instead of the solid phase. Again the derivations are quite complex [START_REF] Dormieux | Microporomechanics[END_REF] and only the final result is presented:

k k s = 1 -p 1 + ks k -1 α µ µ s = 1 -p 1 + µs µ -1 β (17)
with α = 1/(1 + 4µ 3k ) and β = 6(1 + 2 µ k )/5(3 + 4 µ k ) as in eqn (15). Equation 17corresponds to two nonlinear and coupled implicit equations on k and µ, which can be solved numerically, for instance using Matlab. Figure 5 compares the predictions of the above models for a solid phase made of silica (ν s = 0.16). As was expected, the rule of mixtures gives an upper bound, and is thus larger than the other predictions. All models predict a decrease in the effective stiffness with increasing porosity. Moreover, for a solid with negligible porosity (p ∼ 0), all models predict moduli equivalent to the bulk parameters (ie. k = k s and µ = µ s ).

One striking feature is that Eshelby's approximation and the self-consistent scheme yield nearly the same values. This is linked to the specific value of ν s used here. In the general case, the Eshelby's approximation and the self-consistent scheme give different predictions. Moreover, the Eshelby's approximation assumes non-interacting pores and thus should not be used on materials with porosities greater than a few percent. Thus the Eshelby's model will not be further considered in this paper.

The Mori-Tanaka scheme corresponds to a solid matrix containing non-connected pores. Such a system retains some stiffness even in the vicinity of p → 1, so that the Mori-Tanaka scheme predicts positive values of the moduli for 0 ≤ p < 1. On the contrary, the self-consistent scheme predicts that the stiffness vanishes for p ∼ 0.5. Mathematically, it is linked to the fact that the phases invert their roles at p = 0.5. Physically, this level of porosity is generally considered to correspond to the percolation threshold of pores [32].

IV. KENDALL'S TYPE MODELS

The above homogeneization schemes do not capture the effects of possible surface forces; these are expected to become significant for sub-micrometer particles, as this is the case here. It then becomes interesting to resort to Kendall's approach [START_REF] Kendall | Proceedings of the Royal Society of London[END_REF][START_REF] Kendall | Molecular Adhesion and Its Applications[END_REF]. This approach is based on the idea that the beads (of diameter a) are pressed together due to attractive surface forces, characterized by an interfacial energy, W . This energy induces a compressive force between two beads in contact [START_REF] Johnson | [END_REF] :

F adh = 3π 2 W a (18)
Using Hertz's theory on the contact of elastic spheres, Kendall relates the shrinkage displacement (δ adh ) between the centers of the beads to the adhesion force (F adh ):

δ adh = 3 32 1 -ν 2 s E s F adh a 1/2 2/3 (19)
Now, adding a perturbing force f F adh to the adhesion force gives F = F adh + f . To the first order, the perturbing force induces an additional displacement δ:

f = k n δ with k n = 9 16 πW E 2 s a 2 (1 -ν 2 s ) 2 1/3 (20)
The next step is to infer, from the stiffness of a single contact, the Young's modulus of the overall packing. Kendall first considers a simple cubic (sc) packing such that:

E sc = f /a 2 δ/a = k n a (21)
and the porosity of this packing is p = 1 -π/6 0.4764. Along the same lines, he then computes both the Young's modulus and the porosity for a variety of packing geometries (cubic-tetrahedral, tetragonal-sphenoidal and hexagonal) and E as a function of p fits well a (1 -p) 4 dependance on p:

E = A(1 -ν 2 s ) -2/3 (1 -p) 4 W E 2 s a 1/3 (22)
where A 16.1 is a fitting parameter [START_REF] Kendall | Proceedings of the Royal Society of London[END_REF]. Later, Thornton [36] provided some modifications of this approach: he used the theory from Johnson, Kendall and Roberts [START_REF] Johnson | [END_REF] (rather than Hertz theory) to estimate contact stiffness, and he considered a body centered orthorombic array (rather than simple cubic) as a reference packing. These two modifications reduce A by a factor of two. In a nutshell, A varies from 8 to 16.1, depending on the model and its assumptions.

V. EXPERIMENTAL RESULTS VERSUS THEORETICAL PREDICTIONS

A. Comparison with homogenization approaches

Figures 6A and 6B respectively report k/k s and µ/µ s as a function of p, where the datapoints corresponds to those of Fig. 2. These figures also present the predictions of the different models discussed in Sec. III. As anticipated in Sec. III, the rule of mixtures (eqn ( 13)) significantly overestimates the experimental data whereas the self-consistent model (eqn (17)) accurately predicts them. The Mori-Tanaka scheme (eqn (16)) is observed to fall in between.

The agreement between theory and experiment for both bulk and shear constants supports the hypothesis of isotropy made in the paper. If significant anisotropy existed in the packing, it would have resulted in the invalidation of eqn ( 9) and subsequent results. De facto, this gives some indication of the 3D packing structure which is difficult to access otherwise.

For the sake of completeness, Figures 7A and 7B present the Young's modulus (E) and Poisson's ratio (ν) as a function of p using the following equations [START_REF] Landau | Theory of Elasticity[END_REF]: Porosity is observed to affect only E, while ν remains constant within the errorbars. The latter can be interpreted in conjunction with recent works [START_REF] Rouxel | [END_REF]39] correlating ν with the short-to-medium range connectivity of the network at the microscale, which, hence, likely remains almost constant for the considered porosity range. [START_REF] Goehring | Desiccation cracks and their patterns: Formation and modelling in Science and Nature[END_REF][START_REF] Birk-Braun | [END_REF]15]).

E = 9kµ 3k + µ and ν = 3k -2µ 2(3k + µ) (23
Figure 8 compares the experimentally measured Young's modulus with the prediction of the selfconsistent scheme (eqns. ( 17) and ( 23)) and that of Kendall (eqn (22)). The latter better describes the dependency on p for p > 0.35, at the price of an additional fitting parameter: the adhesion energy W . Taking, as before, E s = 72.7 GPa and a = 16.2 nm, the fit of the experimental data with Kendall's equation gives W = 1.1 J/m 2 and W = 10.7 J/m 2 for A = 16.1 and A = 8 respectively. Now, two types of surface energies are potentially relevant in the problem, as the interfacial energy could be attributed either to the breaking of covalent bonds (here siloxane) or to a surface tension (here silica/air or silica/water) as in fluids [START_REF] Birk-Braun | [END_REF]15]. The macroscopic fracture energy for silica G c 8.2 J/m 2 [START_REF] Rountree | Method for measuring the tenacity of a material[END_REF] relates to the breaking of silica bonds, as does Griffith's energy [START_REF] Griffith | [END_REF] 2γ s ∼ 3.8 J/m

2 . This energy represents the work applied to break the bonds on a unit surface: this is inferred from the energy needed to break a single bond [START_REF] Luo | Comprehensive Handbook of Chemical Bond Energies[END_REF], 7.5 10 -19 J/bond (450 kJ/mol), assuming 5 bonds per nm 2 [START_REF] Iler | The Chemistry of Silica[END_REF] and no subsequent dissipation mechanisms such as plasticity or damage. The energies relating to the surface tension are considerably weaker: γ SV ∼ 0.3 -0.5 J/m 2 for silica/vacuum interfaces [START_REF] Sarlat | Vandembroucq[END_REF] and about six times less for silica/water interfaces [15].

The fitted value for W is several orders of magnitude larger than the ones associated with surface tension, but in line with the fracture energy of silica. This demonstrates that sintering during drying has probably led to the formation of covalent siloxane bonds between the particles.

VI. DISCUSSION

A. Competing models

The main outcomes from Sec. V are recalled and discussed:

• The self-consistent scheme provides the complete (tensorial) description of the elasticity, with no adjustable parameter. This for instance provides information on the isotropy of the material.

• The self-consistent scheme predicts an almost linear dependence of elastic constants with porosity. This is in agreement with the experiments for p < 0.35. At higher porosities, a small discrepancy with the experimental data appears, which suggests a missing physical ingredient.

• Kendall's model succeeds in reproducing the behavior of the Young's modulus over the whole porosity range studied herein. Thus, the discrepancy between the self-consistent scheme and the experimental data is likely due to surface effects, which are accounted for in Kendall's model.

• The increased accuracy of Kendall's model comes at the cost of an additional adjustable parameter, the adhesion energy. Moreover, the value of the prefactor involved in the model (A in eqn ( 22)) depends on the precise assumptions made in the model [36] and has a tremendous effect (A 3 dependence) on the determination of the adhesion energy.

• Kendall's model only predicts the Young's modulus and cannot provide any information on the isotropy of the material.

It should be stressed that looking for elasticity around ponctual contacts would result at the macroscopic scale in a non-linear stress-strain relationship (hertzian theory). As a consequence eqn (10) (and thus eqn (8)) and eqn (22) remain valid provided that only small perturbations around an unloaded reference configuration with a finite contact area are considered. Finally, as the two approaches are not mutually exclusive, it may be possible to refine the self-consistent scheme to take into account the adhesive surface forces [32,45] as in Kendall's approach.

B. The physics of colloidal drying

This study involves particles with 15 nm diameter. It is expected that surface effects decrease with increasing particle size, hence the self-consistent scheme is expected to be more accurate (over a larger porosity range) as the particle size increases (and vice versa). Additional experiments with particles of different diameters should verify this hypothesis.

The occurrence of cracks during drying reveals the cohesive nature of the packing. These cracks are due to the combination of three effects: the adhesion between the particles and the substrate, the overall retraction induced by water evaporation, and the transmission of tensile stresses from the substrate throughout the whole layer via the adhesion between the particles. Interpretating the present data in light of Kendall's model provides an estimate of the adhesion energy.

• The value inferred here for the adhesion energy (W 1 -10 J/m 2 ) is on the order of the fracture energy commonly reported for silica [START_REF] Rouxel | [END_REF]. Hence covalent siloxane bonds were probably formed between the particles during the drying.

• This value is orders of magnitude larger than the one used in [START_REF] Birk-Braun | [END_REF]15] (W = γ 0.01 J/m 2 ) to describe the same system (dried Ludox HS-40). However their study concerns the onset of cracking, while our measurements concern dried samples. This suggests an significant evolution of W with time during the drying process.

• The value of the adhesion energy can be recast into a fracture toughness using Irwin's relation [46]:

K c = √ EW 0.15 -0.5 MPa.m 1/2
. This value measured at the end of drying is in line with the one estimated in [47] at the onset of cracking.

A challenge is to measure in situ and independently, from the onset of cracking to the end of drying, the evolution of both the elastic properties and the fracture energy. This would provide important information on how and when the covalent bonds are formed.

VII. CONCLUSION

To summarize, this paper uses a highly porous material formed by the drying of a colloidal suspension as a benchmark for homogeneization schemes of mechanical behavior. Using ultrasound measurements, it investigates the elastic properties of a dried layer of silica nanospheres. By modulating the drying rate, the influence of the porosity has been studied. It has been demonstrated that the self-consistent scheme accurately predicts both elasticity constants, with no ajustable parameters, as long as the porosity is small enough (less than 35% for 15 nm beads). For higher porosities, surface effects become visible; Kendall approach succeeds in taking them into account for a single elastic constant (Young's modulus), at the price of an additional ajustable parameter (adhesion energy). This adhesion energy is found to be on the order of the fracture energy in silica, implying that at the end of the drying the beads are probably linked by covalent bonds.

A surface energy equal to the fracture energy of the pure silica leads to a fracture toughness in line with that reported in [47] for this kind of system. A step forward in these works would consist of measuring the elastic properties during drying. This would provide information on the formation of covalent bonds. Further steps are (i) to test self-consistent and Kendall's predictions by varying the particle size as this is expected to change the contribution of the surface effects and (ii) to use Kendall's approach to interpret, in physical terms, recent mathematical refinements of self-consistent schemes [32,45].

VIII. ACKNOWLEDGMENTS

This work has benefited from the financial support of the Institut Universitaire de France, Triangle de la Physique (RTRA), Ile-de-France (C'Nano and ISC-PIF), LabEx LaSIPS (ANR-10-LABX-0040-LaSIPS) and PALM (ANR-10-LABX-0039-PALM) managed by the French National Research Agency under the "Investissements d'avenir" program (ANR-11-IDEX-0003-02). We thank D. Kondo and L. Dormieux for encouragements and helpful discussions about the homogenization schemes. We thank M. Impéror for the SAXS characterisation of the initial suspension. This appendix details methods for acquiring the elastic moduli and Poisson's ratio of the dried colloidal layers via ultrasonic techniques. As the morsels are thin (∼ 2 mm), this requires special signal processing techniques in calculating longitudinal (c L ) and transverse (c T ) wave speeds.

A single transducer is coupled to a face of the sample, using honey as the viscous couplant for pulse transmission. The transducer coupled with a pulse generator (Panametrics 5800) either provides a compression (Olympus Panametrics-NDT M116) or shear (Olympus Panametrics-NDT V222) pulse to the sample, the frequency of the pulse is 20 MHz in both cases. The ultrasonic system provides a controlled short pulse so as to control the wave introducted into the sample: ∼ 0.4 µs for both compressional and shear pulses.

The emitted pulse travels across the specimen, bouncing back and forth between the two opposite faces. Its successive passings at the specimen/transducer interface are detected at 500 MHz using a Tektronic TDS3054B oscilloscope. Figure 9 display the tension captured by the oscilloscope as a function of time. The thickness of the specimen is measured using a digital caliper with accuracy 0.01 mm. The material probed here is a sample dried at RH = 50% and the specimen thickness is 2.07±0.03 mm. For the sake of clarity, the signals are shifted by 3 V and -3 V respectively. The black box delimits the signal represented on Fig. 10.

t (µs)

In our experiment, the samples are thin (∼ 2 mm): the typical time between two successive echoes is ∼ 1 µs (resp. ∼ 1.6 µs) for compressional (resp. shear) pulses. In general, associating a single discrete time to the pulse arrival on the transducer proved difficult. This difficulty was overcome by the following processing scheme:

• For each pulse, the arrival times of three successive extrema of the signal are determined.

• For each series of matching extrema, the time delay between two matching extrema (ie extrema of the same color on Fig. 10) gives an estimate of the propagation time of the pulse through the sample.

• These estimates are evaluated between each pulse for each series of matching extrema, and subsequently averaged (Fig. 11).

Bulk modulus (k) and shear modulus (µ) can then be deduced from c L , c T and the density (ρ) using eqn (1). Tension (V) The black dotted lines represents the average of all estimated speeds for a given type of waves.

In order to validate this measurement procedure, we applied it to a block (5x5x25 mm 3 ) of pure silica glass (Corning 7980 standard grade). Measurements of the sound velocities were repeated on the two sets of opposite faces. As for the colloidal layers, density was determined by hydrostatic weighting. We measured ρ s = 2.20 ± 0.02 g/cm

FIG. 1 .

 1 FIG. 1. (A) Crack patterns of a dried colloidal layer. (B)Topographical AFM image of the top surface of the layer (scan size 500 × 500 nm 2 ). The sample pictured here was dried at RH = 50%.

FIG. 3 .

 3 FIG.3. Bulk modulus (k) and shear modulus (µ) as a function of relative humidity (RH). Error bars correspond to standard deviation. For each drying rate, the measurements were repeated on two morsels.

FIG. 5 .

 5 FIG.5. Predictions of the different schemes for νs = 0.16 (pure silica): rule of mixtures (red dotted line), Eshelby's approximation (brown dash-dot line), Mori-Tanaka scheme (green dashed line) and self-consistent scheme (blue solid line).

8 FIG. 9 .

 89 FIG. 9. Time evolution of the signal received at the transducer for compression waves (blue) and shear waves (bottom). The material probed here is a sample dried at RH = 50% and the specimen thickness is 2.07±0.03 mm. For the sake of clarity, the signals are shifted by 3 V and -3 V respectively. The black box delimits the signal represented on Fig.10.

4 FIG. 10 .FIG. 11 .

 41011 FIG.10. Two successive reflected pulses for the compression wave represented on Fig.9. Arrival times of matching extrema are indicated by vertical lines of identical color.

  FIG. 8. Normalized Young's modulus as a function of porosity. The numeric values used in Kendall's expression (eqn (22)) were Es = 72.7 GPa, a = 16.2 nm (measured values) and W = 5 J/m 2 (fitted value for A = 10.3, as used in

	B. Comparison with Kendall's type approach
		0.5
			Experimental data
			Self-consistent scheme
		0.4	Kendall's model
	s	0.3
	E/E	0.2
		0.1
		0
		0.3	0.35	0.4	0.45
			p

TABLE I .

 I Measured values for the porosity in each sample, using hydrostatic weighting in water and in ethanol. Due to water retention in the pores, actual values may be up to 1.5% larger[11].

	RH pw p eth
	11 0.343 0.334
	23 0.372 0.347
	36 0.363 0.383
	50 0.365 0.346
	65 0.36 0.371
	80 0.382 0.375
	90 0.391 0.396
	95 0.424 0.429

TABLE III .

 III 3 , in accordance with the value provided by the supplier. The corresponding values of k s and µ s are presented in Tab. III together with the values provided by the supplier. They are in good agreement with each other. This validates our characterisation methods. Elastic constants measured by ultrasounds on fused silica standard grade, Corning code 7980 (± provide the errorbars for one standard deviation).

		ks (GPa) µs (GPa)
	Spec.	35.4	31.4
	Meas. 1 35.9±0.3 31.0±0.1
	Meas. 2 35.8±0.8 31.4±0.2