Juliette Tariel

Gary Longo

Angela Quiros

Nicole L Crane

Kimberly Tenggardjaja

Alexis Jackson

Bruce E Lyon

Giacomo Bernardi
email: bernardi@ucsc.edu

Gary Londo

Alloparental care in the sea: Brood parasitism and adoption within and between two species of coral reef Altrichthys damselfish?

Keywords: Altrichthys, behavior/social evolution, brood parasitism, damselfish ORCID

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

Introduction

The independent evolution of parental care in diverse taxa attests to the fitness benefits of providing care to offspring [START_REF] Clutton-Brock | The evolution of parental care[END_REF][START_REF] Gross | The evolution of parental care[END_REF][START_REF] Royle | The evolution of parental care[END_REF]. However, parental care also comes with costs, including lost opportunities for additional reproduction plus a reduced future survival or reproduction due to the price of investing in current offspring [START_REF] Trivers | Parental investment and sexual selection[END_REF][START_REF] Williams | The costs of reproduction, and a refinement of Lack's principle[END_REF]. In several taxonomic groups the evolution of parental care has been followed by the evolution of various forms of reproductive parasitism-reproductive strategies that allow individuals to gain the benefits of parental care without paying the costs. Depending on the taxonomic group, the 3/26 stage of reproduction at which the reproductive parasitism occurs, and how the mixed broods arise, these forms of reproductive parasitism have been called different things: brood parasitism, egg dumping, brood amalgamation, adoption, kidnapping of offspring or, most broadly, alloparental care [START_REF] Andersson | Brood parasitism within species[END_REF][START_REF] Eadie | Pre-hatch and post-hatch brood amalgamation in North American Anatidae: A review of hypotheses[END_REF][START_REF] Wisenden | Alloparental care in fishes[END_REF].

Alloparental care occurs when individuals provide parental care for offspring other than their own biological offspring. One key distinction between brood parasitism and other forms of alloparental care is whether the putative reproductive parasite provides subsequent parental care to its offspring. For example, nonparasitic forms of alloparental care include cooperative or social breeding where groups of parents reproduce socially or where nonbreeding helpers assist. In contrast, brood parasites donate eggs or offspring to the broods of others but provide no subsequent parental care-all parental care is provided by the host individuals. Two considerations make this otherwise clear distinction between parasitism and cooperative forms of alloparental care somewhat murky and, as a result, there is no universal agreement as to what falls under the label of brood parasitism. First, mixed parentage broods sometimes result from adoption of offspring after hatching and, in some cases, the adoption could be driven by the parasitic offspring rather than the adult parasite. Some consider these as examples of brood parasitism [START_REF] Eadie | Pre-hatch and post-hatch brood amalgamation in North American Anatidae: A review of hypotheses[END_REF] while others might not. Second, while the term parasitism implies a cost to hosts, providing care to other offspring is not always costly to hosts and, in some cases, may even provide a benefit [START_REF] Andersson | Brood parasitism within species[END_REF][START_REF] Canestrari | From parasitism to mutualism: Unexpected interactions between a cuckoo and Its host[END_REF][START_REF] Lyon | Conspecific brood parasitism in birds: A life-history perspective[END_REF]. The term brood parasitism is generally used to describe the reproductive strategy where adults deposit their eggs or offspring into the nests or broods of hosts, irrespective of costs to those hosts (e.g. [START_REF] Andersson | Brood parasitism within species[END_REF][START_REF] Lyon | An obligate brood parasite trapped in the intraspecific arms race of its hosts[END_REF], 2008). We use this terminology here and also note that it is important to understand the mechanisms by which these forms of alloparental care occur.

Brood parasitism and related forms of alloparental care have been documented in diverse groups, including birds [START_REF] Davies | Reproductive parasitism of broodcare helpers in a cooperatively breeding fish[END_REF][START_REF] Lyon | Conspecific brood parasitism in birds: A life-history perspective[END_REF][START_REF] Rothstein | A model system for coevolution: Avian brood parasitism[END_REF], fish [START_REF] Sato | A brood parasitic catfish of mouthbrooding cichlid fishes in Lake Tanganyka[END_REF], amphibians [START_REF] Brown | Phytotelm size in relation to parental care and mating strategies in two species of Peruvian poison frogs[END_REF] and arthropods, mostly insects [START_REF] Boulton | Brood parasitism among spiders: Interactions between Salticids and Diguetia mojavea[END_REF][START_REF] Holldobler | The ants[END_REF]. The specific parental benefits parasites gain by having their offspring raised by others varies depending on taxonomic 4/26 group, but invariably involves the provisioning of some costly form of care to offspring, such as nourishment, warmth or protection from predators [START_REF] Trivers | Parental investment and sexual selection[END_REF][START_REF] Williams | The costs of reproduction, and a refinement of Lack's principle[END_REF]. In most cases brood parasitism is a reproductive strategy pursued by the adult parasites, but in some related behaviours such as adoption, the reproductive parasitism might result from the behaviours of the adopted offspring. However, there are also examples where the adult parasites facilitate the adoption of their offspring: some female ducks take their ducklings into the territories of other females that then adopt the ducklings [START_REF] Eadie | Cooperation, conflict, and crèching behavior in goldeneye ducks[END_REF].

In some cases, brood parasitism involves parasitism of conspecifics (conspecific brood parasitism, CBP), which by necessity is a facultative strategy because there must be conspecific hosts to parasitize. Many brood parasites lay eggs in the nests of other species (interspecific brood parasitism, IBP), and in many cases the parasites have evolved to become obligate brood parasites where the entire species is dependent on hosts of other species for successful reproduction. Brood parasitism has been most widely documented and studied in birds and insects, so in these two taxa we have a better understanding in terms of why it occurs and how specifically brood parasites gain from parasitism. The evolutionary relationships between the two main forms of parasitism-conspecific and interspecific brood parasitism-have also been studied. This evolutionary relationship is fairly well understood in insects (interspecific parasites often evolve from close relatives; [START_REF] Buschinger | Evolution of social parasitism in ants[END_REF][START_REF] Buschinger | Social parasitism among ants: A review (Hymenoptera: Formicidae)[END_REF] but in birds the picture is far less clear and debated [START_REF] Kruger | Evolution of avian brood parasitism and phylogenetic history of brood parasites[END_REF][START_REF] Lyon | Mode of development and interspecific avian brood parasitism[END_REF][START_REF] Yom-Tov | Geographic variation in body size : The effects of ambient temperature and precipitation[END_REF]. Brood parasitism has also been observed in fishes, where IBP, easier to identify, has been studied more frequently [START_REF] Avise | Genetic mating systems and reproductive natural histories of fishes: Lessons for ecology and evolution[END_REF][START_REF] Davies | Reproductive parasitism of broodcare helpers in a cooperatively breeding fish[END_REF][START_REF] Taborsky | The evolution of bourgeois, parasitic, and cooperative reproductive behaviors in fishes[END_REF]. The best-known example is found in Lake Tanganyika, where the cuckoo catfish, Synodontis multipunctatus, is an obligate brood parasite of different species of mouth-brooding Trophein cichlids [START_REF] Blažek | Success of cuckoo catfish brood parasitism reflects coevolutionary history and individual experience of their cichlid hosts[END_REF][START_REF] Sato | A brood parasitic catfish of mouthbrooding cichlid fishes in Lake Tanganyka[END_REF].

IBP has been studied in both freshwater and marine substrate spawners, including many species of Cyprinid minnows that spawn in the nests of other species [START_REF] Baba | Brood parasitism and egg robbing among three freshwater fish[END_REF][START_REF] Johnston | The benefit to some minnows of spawning in the nests of other species[END_REF] as well as one marine species, the spinynose sculpin, Aemichthys, which spawns in buffalo sculpin Enophrys nests [START_REF] Kent | Interspecific nesting in marine fishes: Spawning of the spinynose sculpin, Asemichthys taylori, on the eggs of the buffalo sculpin, Enophrys bison[END_REF]. In contrast, CBP in fishes has rarely been documented. In cichlids, egg stealing and 5/26 brood adoption was shown to enhance the protection of a clutch from predators via dilution of eggs, rather than an attempt at parasitizing parental care (Mrowka, 1987a[START_REF] Mrowka | Egg stealing in a mouthbrooding cichlid fish[END_REF][START_REF] Wisenden | Intraspecific brood adoption in convict cichlids: A mutual benefit[END_REF]. Brood parasitism has not yet been reported in coral reef fishes, perhaps not surprisingly given their reproductive biology. Indeed, the majority of marine fishes exhibit a bipartite life history, with a sedentary adult stage and a dispersive pelagic larval stage [START_REF] Leis | The pelagic stage of reef fishes[END_REF].

Most families (57) are broadcast spawners, where gametes are released in the pelagic environment, thus lacking parental care altogether. In contrast, for those families ( 14) that are substrate spawners with demersal eggs, parental care is relatively common [START_REF] Barlow | Patterns of parental investment, dispersal and size among coralreef fishes[END_REF][START_REF] Leis | The pelagic stage of reef fishes[END_REF]. Such parental care, however, is usually restricted to the very early life stages that typically last only a few days (egg to hatching stage), for example tending, fanning and guarding eggs against predators in clownfishes (genera Amphiprion and Premnas) [START_REF] Fautin | Anemonefishes and their Host Sea Anemones[END_REF]. Indeed, after hatching, offspring become pelagic and remain in the water column for days to weeks. Given this pattern of reproduction, coral reef fishes have few opportunities to engage in brood parasitism.

In damselfishes (Pomacentridae), a family of substrate spawners that includes ~380 species that are mostly restricted to coral reefs, four closely related species lack a pelagic larval stage (apelagic species): the widespread spiny damselfish, Acanthochromis polyacanthus, and three species in the genus Altrichthys, a genus restricted to the Calamian Archipelago, Philippines [START_REF] Allen | Altrichthys, a new genus of damselfish (Pomacentridae) from Philippine seas with description of a new species[END_REF][START_REF] Bernardi | Monophyletic origin of brood care in damselfishes[END_REF][START_REF] Bernardi | The ecology of Altrichthys azurelineatus and A. curatus, two damselfishes that lack a pelagic larval phase[END_REF]Bernardi, Longo, & Quiros, 2017). These species are unique because, unlike most coral reef fishes, parents guard their brood for several weeks after hatching (biparental care system), until the offspring are large enough to escape predation [START_REF] Allen | Altrichthys, a new genus of damselfish (Pomacentridae) from Philippine seas with description of a new species[END_REF][START_REF] Bernardi | Monophyletic origin of brood care in damselfishes[END_REF][START_REF] Bernardi | The ecology of Altrichthys azurelineatus and A. curatus, two damselfishes that lack a pelagic larval phase[END_REF]Bernardi, Longo, et al., 2017;[START_REF] Kavanagh | Larval brooding in the marine damselfish Acanthochromis polyacanthus (Pomacentridae) is correlated with highly divergent morphology, ontogeny and life-history traits[END_REF][START_REF] Robertson | Field observations on the reproductive behaviour of a pomacentrid fish, Acanthochromis polyacanthus[END_REF]. In the Great Barrier Reef, the cost of parental care in Acanthochromis polyacanthus has been shown to be high [START_REF] Jordan | Rising costs of care make spiny chromis discerning parents[END_REF]. Indeed, parents aggressively defend their offspring and also partially feed them with the mucus produced by their skin (glancing behaviour) [START_REF] Jordan | Rising costs of care make spiny chromis discerning parents[END_REF][START_REF] Kavanagh | Notes on the frequency and function of glancing behavior in juvenile Acanthochromis (Pomacentridae)[END_REF][START_REF] Noakes | Parent-touching behavior by young fishes: Incidence, function and causation[END_REF][START_REF] Robertson | Field observations on the reproductive behaviour of a pomacentrid fish, Acanthochromis polyacanthus[END_REF]. Parents might also transfer important microbial symbionts to their offspring at that stage, as was shown in Discus cichlids (genus Symphysodon) [START_REF] Sylvain | Vertically and horizontally transmitted microbial symbionts shape the gut microbiota ontogenesis of a skin-mucus feeding discus fish progeny[END_REF]. Occasionally, broods suddenly increase in size 6/26 (number of offspring), and these jumps were interpreted as potential conspecific brood parasitism [START_REF] Jordan | Rising costs of care make spiny chromis discerning parents[END_REF]Thresher, 1985a[START_REF] Thresher | Distribution, abundance, and reproductive success in the coral reef fish Acanthochromis polyacanthus[END_REF]). Yet no genetic analysis was performed to ascertain that brood parasitism was actually present.

In this study, we focused on the two most abundant species of Altrichthys in the Calamian Archipelago, Philippines, A. azurelineatus and A. curatus [START_REF] Allen | Altrichthys, a new genus of damselfish (Pomacentridae) from Philippine seas with description of a new species[END_REF][START_REF] Bernardi | The ecology of Altrichthys azurelineatus and A. curatus, two damselfishes that lack a pelagic larval phase[END_REF]Bernardi, Longo, et al., 2017). Nests are laid deep inside the base of corals, where they cannot be observed directly. Both parents guard the nest and the resulting offspring after hatching. The nest is only guarded by the two parents, and we did not observe any additional satellite adults. Here, we report the occurrence of mixed parentage broods that we propose probably represent both IBP and CBP in Altrichthys. In a previous study in the Calamian Archipelago, where we observed 391 nests of A. azurelineatus and A. curatus [START_REF] Bernardi | The ecology of Altrichthys azurelineatus and A. curatus, two damselfishes that lack a pelagic larval phase[END_REF]Bernardi, Longo, et al., 2017), we noticed that some offspring looked different from most individuals, raising the question of whether brood parasitism was occurring. We now use a genotyping approach to investigate whether brood parasitism actually occurs in either of these two species.

There are two important considerations for studies that use genetic evidence alone to ascertain brood parasitism: (a) errors that come from false assignment or incorrectly sampling broods and (b) even with correct assignment and family sampling, correctly interpreting the biological mechanism that generated the observed genetic patterns. For the first issue, both false family assignment and inadvertent sampling of multiple broods would lead to genetic information that falsely indicates broods that are composed of offspring from more than one set of parents, a pattern consistent with brood parasitism.

The second issue concerns how to correctly interpret the biological mechanism that produces the genetic pattern of broods of mixed parentage. Above we discussed the two basic forms of alloparental care associated with mixed parentage broods: here we are concerned with the subset where a mixed parentage brood is tended by only one set of parents. [START_REF] Wisenden | Alloparental care in fishes[END_REF] provides an excellent overview of the mechanisms that can lead to these types of mixed parentage clutches and broods in fishes, while [START_REF] Eadie | Pre-hatch and post-hatch brood amalgamation in North American Anatidae: A review of hypotheses[END_REF] and [START_REF] Lyon | Conspecific brood parasitism in birds: A life-history perspective[END_REF] get themselves adopted, either by themselves or with assistance from their parents; such adoptions could be either parasitism or the result of accidental brood mixing.

Nonaccidental adoption has been suggested to be a post-hatch form of brood parasitism since many of the same benefits of parasitism through egg-laying could apply [START_REF] Eadie | Pre-hatch and post-hatch brood amalgamation in North American Anatidae: A review of hypotheses[END_REF], but this terminology has not been adopted universally. (c) Nest competition or takeover: two females (or mated pairs) compete for the same nest site, both lay eggs in the nest but eventually only one female (or pair) tends the nest [START_REF] Semel | Intraspecific parasitism and nest-site competition in wood ducks[END_REF][START_REF] Wisenden | Alloparental care in fishes[END_REF] or in other cases one pair takes over an occupied nest and usurps the original owner [START_REF] Robertson | Egg adoption can explain joint egg-laying in common eiders[END_REF]. (d) Nesting errors: females lay eggs in another's nests but the behaviour is not adaptive. This last scenario was long evoked for birds, but it is currently widely accepted that laying eggs in the nest of others is deliberate and adaptive in most cases [START_REF] Lyon | Conspecific brood parasitism in birds: A life-history perspective[END_REF]. (e) Egg stealing and offspring kidnapping: adults steal eggs or offspring [START_REF] Mckaye | Communal care and kidnapping of young by parental cichlids[END_REF][START_REF] Rohwer | Parent cannibalism of offspring and egg raiding as a courtship strategy[END_REF][START_REF] Wisenden | Alloparental care in fishes[END_REF], behaviours that can have several benefits for the thief and that probably do not benefit the pilfered offspring. This discussion should make it clear that different biological mechanisms can lead to identical genetic signatures of mixed parentage broods and hence that genetic information alone cannot resolve the underlying behavioural mechanisms.

Detailed behavioural observations are needed to fully distinguish among these five mechanisms, but such data are often difficult to obtain. Where such behavioural data are not possible, caution is required when interpreting genetic patterns.

With these theoretical expectations and potential caveats for genetic approaches in mind, we sampled broods of Altrichthys to determine if there was evidence for broods of mixed parentage, including mixed parentage within and across species. We then consider the possible mechanisms that could account for observed mixed parentage broods and conclude that both IBP and CBP occur in these two reef fishes.

8/26 2 Material & Methods

Sampling

We used clove oil solution and hand nets to sample broods while scuba diving, trying to collect all offspring in a brood (yet it is possible that occasionally a few individuals in a brood were not sampled). Care was taken to avoid sampling individuals from neighbouring broods, thus avoiding the possibility of sampling adjacent broods by mistake. A previous study showed that, on average, the distance between any Altrichthys nests was 3.9 m, and the distance between nests of the same species was on average 5.1 m [START_REF] Bernardi | The ecology of Altrichthys azurelineatus and A. curatus, two damselfishes that lack a pelagic larval phase[END_REF]Bernardi, Longo, et al., 2017), with a minimum distance between nests of 0.8 m. Sampled nests for this study were always at least 3 m from each other and no additional nests were within 3 m of the sampled nests. Predation levels are very high on coral reefs [START_REF] Cortesi | Phenotypic plasticity confers multiple fitness benefits 23/26 to a mimic[END_REF][START_REF] Feeney | Predation drives recurrent convergence of an interspecies mutualism[END_REF], and the Calamian Archipelago is no exception. Anecdotal evidence showed us that when guarding parents were removed, offspring in a brood would be consumed by predators (mainly serranids, Cephalopholis microprion, and wrasses, Halichoeres leucurus, Oxycheilinus celebicus) in a matter of seconds [START_REF] Bernardi | The ecology of Altrichthys azurelineatus and A. curatus, two damselfishes that lack a pelagic larval phase[END_REF]Bernardi, Longo, et al., 2017). It is difficult to exclude the possibility that some offspring may hide deep in the coral to avoid predation, although this was not observed. For logistical reasons (avoiding damaging corals where adults hide), presumed parents (pair of adults associated with the brood) were not sampled. Because nests are deep within the coral substrate, we also did not sample eggs to avoid damaging the habitat. We collected 21 Altrichthys azurelineatus broods comprising 414 individuals and 19 A. curatus broods comprising 305 individuals from 10 localities in the Calamian Archipelago, Philippines (Figure 1, Table 1; Table S1).

A previous study showed that some nests (7% of the observed nests) are composed of fry from different size classes [START_REF] Bernardi | The ecology of Altrichthys azurelineatus and A. curatus, two damselfishes that lack a pelagic larval phase[END_REF]. For this study, we measured the size of sampled individuals to the nearest millimetre and ranked them in three size classes:

1 (<7 mm), 2 (7-13 mm) and 3 (>13 mm). Within each cohort, there was very little variability in size between individuals (coefficient of variation was 0.546). S1). Samples were collected on the islands of Uson (USO), Sangat (SAN) and Culion (CUL), in the Calamian Archipelago, Palawan, Philippines.

9/26

Genotyping

Since we collected very small offspring, we first used a genetic barcode to identify each individual at the species level.

Microsatellites

To identify instances of conspecific brood parasitism, we genotyped each Altrichthys individual (previously identified by the barcoding approach described above) using microsatellites. Microsatellite primers originally described for the closely related Acanthochromis polyacanthus were used to genotype Altrichthys individuals (Miller-Sims, Delaney, Atema, [START_REF] Miller-Sims | DNA microsatellites in Acanthochromis polyacanthus[END_REF]. Specifically, we used loci AC42, AC37, AC45, POM3 and POM15 following protocols described in the literature [START_REF] Miller-Sims | DNA microsatellites in Acanthochromis polyacanthus[END_REF]. PCRs were carried out with an ROX500 size standard. Peaks were called and binned with the microsatellite plugin of GENEIOUS version 8.1 [START_REF] Kearse | geneious basic: An integrated and extendable desktop software platform for the organization and analysis of sequence data[END_REF].

Parentage analyses

To assign relationships among individuals of a given brood, we used the software COLONY [START_REF] Wang | Sibship reconstruction from genetic data with typing errors[END_REF], a likelihood-based program that provides the most likely family configuration. Most default (zero option) parameters were used. We also used the following options: Mating system I-female polygamy, male polygamy (results remained unchanged when changing the setting to monogamy); 12/26 parameters are commonly used in the literature because they generally provide robust results [START_REF] Reisser | Multiple paternity and competition in sympatric congeneric reef fishes, Embiotoca jacksoni and E. lateralis[END_REF]. colony identified families of full sibs (no halfsibships were detected). For this study, we assumed that the largest group of siblings in each nest was the host brood, and all other offspring not included in that group of siblings were considered parasitic. Amblyglyphidodon batunai and two Pomacentrus smithi (Table 1).

Results

Species identifications/barcoding

Interspecific mixed parentage broods

Interspecific parentage was found in eight of 40 broods (20%), but only corresponded to 26 of 719 individuals (3.5%) (Table S1). There was no statistical difference in the frequency of these mixed broods between A. azurelineatus and A. curatus broods (six of 21 broods were mixed in A. azurelineatus, two of 19 broods were mixed in A. curatus, χ 2 test, p = .15). We did not find a correlation between the size of the brood and the presence of interspecific offspring (df = 1, F-ratio = 1.35, p = .29). All of the foreign offspring were damselfishes and the majority of them were Amblyglyphidodon batunai (19 of 26). Adult Amblyglyphidodon batunai are common in the area where broods were sampled, but their density was not estimated. Two foreign offspring were identified as Pomacentrus smithi, and adults of that species are also very common. Finally, one A. curatus was present in an A. azurelineatus brood and four A. azurelineatus were found in an A.

curatus brood (Table 1).

Conspecific mixed parentage broods

Conspecific mixed broods were found in 30 out of 40 broods (75%), corresponding to 163 out of 719 individuals (22.7%) (Table 1, Figure 2). There was no statistical difference in the frequency of mixed broods between A. azurelineatus and A. curatus broods (15 of 21 broods were mixed in A. azurelineatus, 15 of 19 broods were mixed in A. curatus, χ 2 test, p = .58). Unlike the situation with interspecific mixed broods, brood size was correlated 13/26 14/26 with the presence of foreign conspecific offspring (df = 1, F-ratio = 38.69, p < .001). A large number of singletons (a single foreign individual from given alloparents) were found (41 individuals). On the other hand, mixed broods containing groups of siblings from the same alloparents, from two to 27 individuals, accounted for the remaining 124 foreign offspring. We did not find any half-sibs in our data set, meaning that there was no evidence of conspecific spawn sneaking behaviour. All raw data that support the findings of this study are provided in the Supporting Information.

Size ranges

We measured the sizes of individuals in broods and placed them in three size classes (Table 1). For A. azurelineatus, out of 20 broods, four were in class 1, six in class 2 and 10 in class 3. For A. curatus, out of 19 broods, three, eight, and eight were in size classes 1, 2 and 3, respectively. Broods containing different size classes of offspring were found in two out of five interspecific mixed broods and two out of two IBP broods for A. azurelineatus and A. curatus, respectively. In contrast, none of the other broods, including those affected by conspecific brood parasitism, contained more than one size class (Table 1). In addition, we did not find evidence of a relationship between size of the brood and incidence of mixed parentage broods.

Discussion

Neither adoption nor brood parasitism has been previously described in coral reef fishes.

Due to the peculiarities of coral reef fish life history (lack of parental care), very few species are susceptible to brood parasitism in the first place, yet, as we suggest, brood parasitism is indeed present in apelagic species. Here we show evidence of patterns that are consistent with both interspecific adoption and conspecific brood adoption and parasitism in Altrichthys.

Evidence for interspecific brood adoption

The frequency of interspecific mixed brood was low; it involved only 3.5% of individuals and in all cases the foreign offspring were damselfishes. However, we must distinguish two types of potential interspecific alloparental care that are very different. The most common type of mixed broods involved non-Altrichthys damselfishes, namely Amblyglyphidodon batunai and Pomacentrus smithi. These two species guard demersal 15/26 eggs (a trait shared by all damselfishes), which turn into pelagic larvae after hatching.

Larvae remain in the water column for 12-20 days in Amblyglyphidodon and 14-23 days in Pomacentrus [START_REF] Wellington | Planktonic larval duration of one hundred species of Pacific and Atlantic damselfishes (Pomacentridae)[END_REF]. Therefore, at recruitment time, larvae join Altrichthys broods, probably to gain protection provided by Altrichthys parents. We propose that these mixed broods are cases of adoption after hatching. We did not specifically quantify the aggressive behaviour of the parents, but when we could distinguish Altrichthys offspring from other species, we could see directed aggression towards the foreign offspring. While Altrichthys parents tend to be aggressive towards any potential threat to the brood, non-Altrichthys larvae generally benefit from this protection, even if sometimes being themselves targeted. Other species of damselfishes are very common on the studied reefs, in particular the very abundant Chrysiptera springeri and Pomacentrus stigma (the latter very closely resembles Altrichthys), although neither was found to have their offspring adopted by Altrichthys.

The relatively small number of these interspecific foreign offspring in host broods might suggest this form of alloparental care is so rare as to be unimportant to the adopted offspring, but our observations suggest otherwise. We did not follow the fate of specific broods over time, but we witnessed a massive recruitment event of Pomacentrus smithi that resulted in very large numbers of new settlers on the reef. In the course of few days, most settlers disappeared, presumably due to natural predation, which is consistent with the massive mortality incurred by coral reef fishes with a pelagic larval stage [START_REF] Almany | The predation gauntlet: Early post-settlement mortality in reef fishes[END_REF][START_REF] Goatley | Body size and mortality rates in coral reef fishes: A three-phase relationship[END_REF]. The remaining few visible ones were invariably seen in association with Altrichthys broods (two of them being identified in this genetic study), supporting the idea that an association with Altrichthys broods may increase survivorship for parasitic damselfishes. Moreover, although the total number of adoptees was small in terms of the fraction of total offspring in host broods, the fact that the only Pomacentrus recruits were seen in association with host broods indicates that adoption might be important for Pomacentrus smithi in the region of overlap with

Altrichthys. Interspecific brood associations can range from parasitic, commensal to mutualistic relationships [START_REF] Canestrari | From parasitism to mutualism: Unexpected interactions between a cuckoo and Its host[END_REF][START_REF] Johnston | The benefit to some minnows of spawning in the nests of other species[END_REF][START_REF] Mckaye | Cichlid-catfish mutualistic defense of young in Lake Malawi, Africa[END_REF]. The fact that the Altrichthys hosts in our study were often aggressive to the adopted offspring suggests that these adoptees might be costly to the hosts, and hence a form of reproductive 16/26 parasitism by the foreign offspring, but experimentally assessing the costs to hosts would be required to understand the nature of the relationship.

Our observations suggest that predation may be the key factor driving interspecific adoption in Amblyglyphidodon batunai and Pomacentrus smithi. Predation is known to be a potent factor favouring interspecific interactions in fish generally [START_REF] Hay | Mutualism and aquatic community structure: The enemy of my enemy Is my friend[END_REF] and evidence suggests that the benefits of protective associations may be strongest for smaller, younger age classes because these are particularly vulnerable to predation [START_REF] Feeney | Predation drives recurrent convergence of an interspecies mutualism[END_REF]. Predation is similarly thought to be the main factor in most of the previously studied cases of interspecific mixed clutches and broods in fish, most of which occur in freshwater systems [START_REF] Baba | Brood parasitism and egg robbing among three freshwater fish[END_REF][START_REF] Goff | Brood care of longnose gar (Lepisosteus osseus) by smallmouth bass (Micropterus dolomieui)[END_REF][START_REF] Johnston | The benefit to some minnows of spawning in the nests of other species[END_REF][START_REF] Mckaye | Cichlid-catfish mutualistic defense of young in Lake Malawi, Africa[END_REF].

However, predation was also thought to be a key factor in the one previous example of mixed interspecific clutches in a marine fish [START_REF] Kent | Interspecific nesting in marine fishes: Spawning of the spinynose sculpin, Asemichthys taylori, on the eggs of the buffalo sculpin, Enophrys bison[END_REF].

Protection from predation has also been suggested as an important factor in some forms of avian brood parasitism, mostly notably in the adoption that occurs in waterfowl species with precocial young that do not require food from the parents and for which protection from predators seems particularly important [START_REF] Canestrari | From parasitism to mutualism: Unexpected interactions between a cuckoo and Its host[END_REF][START_REF] Eadie | Pre-hatch and post-hatch brood amalgamation in North American Anatidae: A review of hypotheses[END_REF][START_REF] Eadie | Cooperation, conflict, and crèching behavior in goldeneye ducks[END_REF]. In these species, broods with mixed young can occur either by females laying eggs in the nests of other females or by adoption of offspring after hatching, which is sometimes facilitated by mothers that desert their offspring near females who will then adopt the offspring [START_REF] Eadie | Pre-hatch and post-hatch brood amalgamation in North American Anatidae: A review of hypotheses[END_REF]. There has been debate over whether this adoption is parasitic because host females could benefit through dilution of predation risk to their own offspring [START_REF] Andersson | Brood parasitism within species[END_REF][START_REF] Eadie | Pre-hatch and post-hatch brood amalgamation in North American Anatidae: A review of hypotheses[END_REF] In an experimental study of adoption in waterfowl, host females were very aggressive to ducklings that differed in size from their own offspring, sometimes even infanticidal, suggesting that adoption might not benefit the host [START_REF] Eadie | Cooperation, conflict, and crèching behavior in goldeneye ducks[END_REF].

The other type of interspecific mixed brood was even less common and involved individuals swimming together and being chased by some parents protecting their own brood. This is easy to identify in the field when the intruders have distinctly different sizes than the brood. It is less clear for us (and potentially for the parents) to identify intruders when their size is similar to the parents' own brood.

An alternative situation occurs when a large group of individuals is involved in the parasitic event, for example A. azurelineatus brood 1, and A. curatus brood 3 (Figure 2).

In those cases, broods are mainly composed of two genetic families of offspring that have similar sizes (brood 1:37, 27 individuals; brood 3:12, 11 individuals). Given the very high predation pressure on juvenile fishes, while very unlikely, it is not possible to entirely exclude that a group of offspring moved to another brood after hatching. However, a more plausible explanation is that a second pair of adults spawned in the nest and would have thus directly parasitized the parents. This is because the mortality of offspring swimming away from shelters and parental protection is likely to be extremely high; with such mortality, the possibility that large numbers of siblings successfully swim to a new nest site is remote. Importantly, regardless of the mechanism, mixed parentage broods are frequent and probably arise from both adoption of offspring and brood parasitism by parents. The large frequency of conspecific mixed broods is also consistent with the 18/26 observation that there is a significant relationship between mixed broods and brood size, since adoption and parasitism directly contribute to brood size. In avian studies, this same relationship between clutch size and parasitism has been observed and has been used as a method for detecting brood parasitism [START_REF] Eadie | Probing parentage in parasitic birds : An evaluation of methods to detect conspecific brood parasitism using goldeneyes Bucephala islandica and Bl. clangula as a test case[END_REF].

As with many genetic studies of parasitism, we have been able to detect the occurrence of brood parasitism (mixed clutches) and post-hatching adoption, but we currently lack the ecological and demographic information required to understand why this behaviour occurs and how the parasites benefit [START_REF] Andersson | Brood parasitism within species[END_REF][START_REF] Lyon | Conspecific brood parasitism in birds: A life-history perspective[END_REF][START_REF] Wisenden | Alloparental care in fishes[END_REF]. Parental care improves the survival of the offspring but it can also result in costs to parents [START_REF] Trivers | Parental investment and sexual selection[END_REF]. In Acanthochromis, there is a direct cost of producing mucus to feed and potentially transfer microbiomes to the offspring, as well as a chance of the adult being preyed upon while aggressively defending the brood [START_REF] Jordan | Rising costs of care make spiny chromis discerning parents[END_REF]. In an experimental setting at the Great Barrier Reef [START_REF] Jordan | Rising costs of care make spiny chromis discerning parents[END_REF], broods were manipulated to estimate parental cost and aggression towards potential intruders. After removing Acanthochromis offspring from a brood, mixed brood (host and parasites) as well as original broods were returned to the parents. The parents then showed more aggression towards parasites than their own brood, chasing the parasites far enough from the nest to be vulnerable to predation. In the Calamian Archipelago, we observed almost instant predation on offspring by several fish species (e.g. Cephalopholis microprion, Halichoeres leucurus and Oxycheilinus celebicus) when parents were removed [START_REF] Bernardi | The ecology of Altrichthys azurelineatus and A. curatus, two damselfishes that lack a pelagic larval phase[END_REF]Bernardi, Longo, et al., 2017). While we did not observe mucous feeding in offspring, we did observe parents chasing individuals from the brood away from the nests; these "offspring" could potentially represent parasites. Apparently, this chasing behaviour is not fully efficient, since a relatively large proportion of the broods were found with adopted offspring.

When CBP is frequent, as for Altrichthys, the amount of parasitism should not be so great as to jeopardize the integrity of the system. Another possibility is that the entire population size of either species of Altrichthys is relatively small (the entire range of the species is restricted to the Calamian Archipelago; [START_REF] Bernardi | The ecology of Altrichthys azurelineatus and A. curatus, two damselfishes that lack a pelagic larval phase[END_REF]. This may mean that individuals are related, thus allowing for some altruism to be present due to 19/26 genetic relatedness. Kin-selected aspects of brood parasitism have been suggested for some avian conspecific parasites as well [START_REF] Andersson | Brood parasitism within species[END_REF][START_REF] Lyon | Conspecific brood parasitism in birds: A life-history perspective[END_REF]. As mentioned above, in general, free larvae cannot swim long distances over reefs without incurring high levels of mortality. The distance between nests that we observed, ~5 m, may be structured by the dynamics of parasitism.

Conclusions

Our results are consistent with interspecific adoption and both CBP and adoption in two species of Altrichthys. A previous study on Acanthochromis polyacanthus suggests that CBP is probably present in that species as well. Indeed, on the Great Barrier Reef, a study aimed at estimating dispersal of Acanthochromis polyacanthus between reefs used an ecological approach by genotyping broods [START_REF] Miller-Sims | Dispersal in the spiny damselfish, Acanthochromis polyacanthus, a coral reef fish species without a larval pelagic stage[END_REF]. That study found that broods (groups of siblings) remain together from birth to being fully independent and beyond. In that study, 30 broods were genotyped, and one incongruent brood was discarded from the analysis. That discarded brood was the largest one, with 43 individuals, and included over 50% of individuals originating from different parents [START_REF] Miller-Sims | Dispersal in the spiny damselfish, Acanthochromis polyacanthus, a coral reef fish species without a larval pelagic stage[END_REF]. We think that this brood presents strong evidence for mixed broods in Acanthochromis. If this were the case, broods of mixed parentage would be present in all three tested species of apelagic fishes, Acanthochromis polyacanthus, Altrichthys azurelineatus and Altrichthys curatus.

Our study adds to the list of taxonomic groups that show some form of brood parasitism or adoption and, as such, increases our ability to understand which aspects of brood parasitism and adoption are general, and which are specific to particular sets of taxa. In fishes, interspecific mixed parentage broods appear to be much more common that conspecific mixed broods. It is unclear whether this pattern reflects the actual distribution of parasitism types because interspecific cases are considerably easier to detect because they are often apparent based on morphological differences between hosts and parasites whereas genetic methods or very careful demographic studies are required to detect conspecific cases. This same pattern and problem was observed for brood parasitism in birds-three decades ago the occurrence of IBP was accurately known (100 species) and was thought to be the dominant form of parasitism because few examples of CPB were known [START_REF] Andersson | Brood parasitism within species[END_REF][START_REF] Lyon | Conspecific brood parasitism in birds: A life-history perspective[END_REF]. The advent of modern genetic parentage 20/26 methods over the intervening years has revealed that CBP is actually more common than previously imagined, having now been documented in some 230 species [START_REF] Lyon | Conspecific brood parasitism in birds: A life-history perspective[END_REF]. Studies are now needed to determine whether this same trajectory will also apply to fish, or whether the CBP we have discovered is unusual.

Brood parasitism and adoption allows individuals to gain the benefits of parental care without paying the cost of care, but the nature of the relationship between host and donor should depend of the type of parental care. Some forms of care, like food for offspring, are "depreciable" [START_REF] Altmann | Two models for the evolution of polygyny[END_REF] in that the same unit of care cannot be spread among several offspring. Given this, family size is likely to be limited by the parental care and, accordingly, raising the offspring of others can impose high fitness costs to the hosts by reducing the number of host offspring produced. Other forms of parental care, such as protecting offspring from predators, are much more likely to be able to be shared among offspring; parasitism of this form of parental care may be less costly to hosts and, in some cases, could even benefit hosts where there is safety in offspring numbers [START_REF] Andersson | Brood parasitism within species[END_REF][START_REF] Johnston | The benefit to some minnows of spawning in the nests of other species[END_REF]. In birds, the type of brood parasitism is closely linked to the form of parental care-all but one of the 100 species of obligate interspecific brood parasites have altricial offspring that require large amounts of food [START_REF] Davies | Reproductive parasitism of broodcare helpers in a cooperatively breeding fish[END_REF][START_REF] Lyon | Mode of development and interspecific avian brood parasitism[END_REF]. In these species, costs to hosts are often high and these costs have fuelled coevolutionary arms races between defensive traits in the hosts and traits in the parasites to thwart the host defences [START_REF] Davies | Reproductive parasitism of broodcare helpers in a cooperatively breeding fish[END_REF][START_REF] Feeney | Advances in the study of coevolution between avian brood parasites and their hosts[END_REF][START_REF] Rothstein | A model system for coevolution: Avian brood parasitism[END_REF]. Although some forms of avian intraspecific brood parasitism also parasitize depreciable forms of care like food, intraspecific brood parasitism is particularly widespread in species with precocial offspring that feed themselves and for which protection from predation is a key aspect of parental care [START_REF] Andersson | Brood parasitism within species[END_REF][START_REF] Davies | Reproductive parasitism of broodcare helpers in a cooperatively breeding fish[END_REF][START_REF] Lyon | Conspecific brood parasitism in birds: A life-history perspective[END_REF]. The forms of interspecific brood associations widely reported in fishes [START_REF] Johnston | The benefit to some minnows of spawning in the nests of other species[END_REF][START_REF] Mckaye | Communal care and kidnapping of young by parental cichlids[END_REF][START_REF] Wisenden | Alloparental care in fishes[END_REF], and that we show here for both interspecific and intraspecific mixed broods in coral reef fish, probably share more in common with the brood parasitism seen in precocial birds.

  outline these mechanisms for birds. Five mechanisms can produce broods of offspring of genetic mixed parentage tended by only a subset of parents with offspring in the brood and, depending on the definition of brood parasitism used, 7/26 only the first or two of these are examples of brood parasitism. (a) Brood parasitism by adult brood parasites: a female (or a pair) lays eggs in the nest of another female (or pair in biparental species). (b) Post-hatching adoption: the offspring of one female (or pair)

Figure 1 .

 1 Figure 1. Sampling locations of Altrichthys broods (see also TableS1). Samples were collected on the islands of Uson (USO), Sangat (SAN) and Culion (CUL), in the Calamian Archipelago, Palawan, Philippines.

  Four primers were used together to amplify a 655-bp fragment of the mitochondrial COI gene: FishF1, 5′-TCAACCAACCACAAAGACATTGGCA C-3′; FishF2, 5′-TCGACTAATCATAAAGATATCGGCAC-3′; FishR1, 5′-TAG ACTTCTGGGT GGCCAAAGAATCA-3′; and FishR2, 5′-ACTTCAGG GTGACCGAAGAATCAGAA-3′[START_REF] Ward | DNA barcoding Australia's fish species[END_REF]. Each polymerase chain reaction (PCR) of 25 μl 10/26 contained 10-100 ng DNA template (0.5-2 μl), 100 nm each of the four primers, 200 mm each dNTP, 2.5 mm MgCl2, 50 mm KCl, 10 mm Tris-HCl pH 8.3 at 25°C and 1 unit of Taq polymerase. The reaction was amplified using the following programme: one cycle at 95°C for 2 min; 35 cycles at 94°C for 30 s, 54°C for 30 s and 72°C for 1 min; one cycle at 72°C for 10 min. The PCR products were visualized on 1% agarose gels. Each unexpected individual (from a different species than the majority of individuals within a brood) and a random selection of expected fish were sequenced with a combination of FishF1 and FishF2 primers. To positively identify the individual offspring, we also collected adults from several species of Amblyglyphidodon and Pomacentrus to match identifiable adults with sequences obtained from the offspring. Barcode sequences were also compared with BOLD (Barcode of Life) sequences to further confirm identification[START_REF] Ratnasingham | BOLD: The barcode of life data system (http://www.barco dingl ife.org)[END_REF].

Figure 2 .

 2 Figure 2. Composition of Altrichthys broods. Each bar represents a single brood (21 broods for A. azurelineatus, 19 broods for A. curatus). Numbers of individuals per brood are indicated on the y-axis. Colours represent families within each brood. For example, brood 5 in A. curatus corresponds to a single family (no parasites) with 21 individuals (a single colour), while brood 5 in A. azurelineatus includes two families (two colours) with a total of 30 individuals.

Table 1 .

 1 Altrichthys brood parasite counts. Columns correspond to: clutch identification number, sampling locality (see Figure1for abbreviation), number of individuals in the clutch, number of identified families in the clutch, number of parasitic individuals, number of extraspecific individuals,

	Mating system II-with inbreeding,

Full-likelihood; Likelihood precision-medium; Run specifications-Update allele frequency, no; Sibship scaling, yes, Number of runs, 2; Sibship prior-weak prior. These 11/26 and their identity. Extraspecific individuals were: AAZ, Altrichthys azurelineatus; ABA, Amblygliphidodon batunai; ACU, Altrichthys curatus; PSM, Pomacentrus smithi. Sizes of offspring, when available, are shown in three classes, given as 1 (<7 mm), 2 (7-13 mm) and 3 (>13 mm). Numbers of offspring belonging to each size class 1, 2 and 3 are given in that order.

  Altrichthys individuals. We only found one A. curatus individual in an A. azurelineatus brood and four A. azurelineatus in an A. curatus brood. The presence of the single A. Altrichthys species would be expected if conspecific parental care is superior to heterospecific care-if so, offspring should strongly prefer parasitizing their own species as hosts. The issue then is not lack of foreign offspring in Altrichthys generally, but the relative distribution of interspecific vs. conspecific adoptions. Adoption and parasitism is common but it mostly occurs as CBP rather than IBP.4.2 Conspecific brood adoption and parasitism (CBP)Conspecific adoption and brood parasitism was found to be rampant in Altrichthys, comprising over 20% of individuals. However, as with interspecific mixed broods, two patterns emerged that probably represent different forms of parasitism or adoption. In 41 broods, a single foreign individual was involved in the parasitism. In other broods, the parasitism involved groups of individuals, invariably siblings. A simple interpretation of this pattern is that, similarly to the interspecific examples described above, broods that involve single (or few) foreign individuals arise after hatching by the offspring swimming from one nest to another. In fact, we have occasionally observed groups of a few

	rarity of adoption between
	curatus parasitic individual may correspond to an individual swimming from one brood
	to another interspecific brood. The paucity of these observations, however, may indicate
	a strong cue and aggressivity towards congeneric offspring. Alternatively, the observed
	17/26

Acknowledgements

We thank Amalia Bernardi and Alessio Bernardi for help in the field. We thank the following: Bureau of Fisheries and Aquatic Resources (Commodity Clearance 2016-21/26 20091), Palawan Council for Sustainable Development (PCSD, GP 2016-03, Wildlife Transport Permit No. 2016-05-000062-DMO-Calamianes, Fisheries inspection and quarantine clearance Permit no. 009391) and Leonora Labrador and Eva Palanca of the Coron PCSD office. Ateneo de Manila University's Hendrik Freitag, Merab A. Chan, Jonathan Patrick H. Yan are thanked for the Memorandum of Agreement between UCSC and the Department of Biology at Ateneo de Manila. Busuanga Mayor (Samuel de Jesus) and Councilor (Ann D. Mercado), Coron Mayor (Clara Espiritu-Reyes) and Samuel Madamba, Culion Mayor (Emiliano Marasigan) and Councilor Leonilo Sarmiento are thanked for Prior Informed Consent (PIC), given in 2015. Hikari South Sea Pearl, Luna Sombrano, Agustin R. Badon II, Eric T. Dalisay and Pinky Perpetua helped in obtaining the PICs. We thank Dive Link Resort, Balinsasayaw Island, El Rio y Mar Resort and Dirk Fahrenbach at the Dugong Dive Center for letting us visit their reefs, and Gerald Allen and Kathryn Kavanagh for discussing our data and invaluable information on Altrichthys. We thank editor C. Riginos, W. Feeney and two anonymous reviewers for feedback that greatly improved the paper. This work was funded by a National Geographic Grant 9399-13, and a UCSC Committee on Research (COR) grant to G.B.

Data accessibility statement

All data that support the findings of this study are available as an electronic Supporting Information on the Wiley's website.

Supporting information

Additional supporting information may be found online in the Supporting Information section at the end of the article on the Wiley's website.


F Latourte

A Samida

A Chrysochoos

S Pagano

B Wattrisse
email: wattrisse@lmgc.univ-montp2.fr

An inverse method applied to the determination of deformation energy distributions in the presence of prehardening stresses

Keywords: digital image correlation, deformation energy distributions, pre-hardening stresses

This paper presents an experimental procedure to estimate the deformation energy distribution within plane samples submitted to mechanical loading. This procedure combined a digital image correlation (DIC) technique giving in-plane displacement fields with an identification method that separately provided fields of material properties and stress distributions developed during the loading. The method was first applied to simulated data to characterize the capabilities of the image processing. Finite element computations were first performed on a complex structure using a standard linear kinematical hardening model to generate multistage loadings leading to heterogeneous displacements and distributions of deformation energy. Loads and displacements were then used as inputs to check the robustness of the image processing by comparing the identified deformation energy fields with the computed ones. The procedure was then applied to experimental data. Tests were conducted under conditions similar to the numerical tests. The identification of a linear kinematical hardening model gave deformation energy patterns showing a good agreement with the simulated results, even in the presence of residual stresses induced by a prehardening.

INTRODUCTION

The conversion of mechanical energy into heat has been investigated for a wide range of materials by many researchers, including those in references [1] to [START_REF] Rittel | On the conversion of plastic work to heat during high strain rate deformation of glassy polymers[END_REF]. Using different experimental techniques (calorimeter [START_REF] Shenogin | Thermodynamics of the pre-yield deformation of glassy polymers: measurements with new deformation calorimeter[END_REF], thermocouples [START_REF] Zehnder | Hybrid method for determining the fraction of plastic work converted to heat[END_REF], infrared (IR) sensors [START_REF] Guduru | Dynamic full field measurements of crack tip temperatures[END_REF][START_REF] Chrysochoos | An infrared image processing to analyse the calorific effects accompanying strain localisation[END_REF]), these studies gave similar results, showing that a variable amount of mechanical energy is converted into heat during inelastic transformation. Such methods generally provide a macroscopic estimate of the Taylor-Quinney co-efficient that links mechanical and dissipated energies. Here, 'macroscopic' refers to the scale of the sample gauge part. Nevertheless, in the case of heterogeneous loading (e.g. necking, shear bands), global energy measurements are representative both of the material behaviour and combined structural effects. Under such conditions, the overall Taylor-Quinney coefficient can no longer be representative of the sole material behaviour. The material behavioural analysis then needs a local construction of the energy balance that requires an estimate of the deformation energy locally given to the material.

In this paper, a specific procedure is presented that is designed to estimate locally the deformation energy in the case of quasi-static, heterogeneous tests during elastoplastic transformations. Various causes can explain the response heterogeneity of a sample, like structural effects associated with the overall loading or with the boundary conditions, or material effects resulting, for example, from a nonhomogeneous initial state of the material. The latter situation is often encountered in particular mechanical situations, such as fatigue, for which the initial state plays a key role in the subsequent material evolution. The aim of this paper is to investigate such heterogeneous responses in order to verify the abilities of the developed protocol in order to distinguish structural and material effects better in terms of deformation energy.

Deformation energy constitutes the energy input provided during a mechanical test, and its assessment is necessary to perform a local energy balance.

The set-up combined an identification method providing stress and mechanical parameter fields with an imaging technique, namely digital image correlation (DIC), now commonly used to measure displacement and strain fields on the surface of thin specimens [START_REF] Chu | Application of digital-image-correlation techniques to experimental mechanics[END_REF].

In what follows, first the procedure allowing the estimation of the deformation energy fields associated with a heterogeneous material response is presented. Then, the method is tested on numerical examples to assess the performances and robustness of the method. Finally the identification results obtained on a dual-phase steel sample (commercial grade DP600) are discussed.

DEFORMATION ENERGY ESTIMATE

The local estimate of deformation energy requires stress and strain fields to be computed. These were obtained through a mixed approach combining experimental measurements and a numerical identification method. The strain distributions were derived from displacement fields obtained by a digital image correlation technique, while stresses were computed through the constitutive equations of an elastoplastic model previously identified on the basis of the displacement fields. In this work a J2 plasticity criterion associated with a usual linear kinematical hardening was chosen to describe the hardening effects in tension.

DIC

The in-plane components of the displacement vector were obtained by a DIC algorithm applied to the images recorded by a high-resolution camera (ATMEL Camelia 8M). The camera was located perpendicularly to the tested specimen fixed between the grips of a hydraulic testing machine (MTS). The average spot size of the speckle image recorded by the camera was estimated as the mean radius of the autocorrelation peak at its half height. In this case it was approximately 1 pixel.

The components of the displacement vector u(u x , u y ) were determined at a given point M(x, y) whose coordinates were given with respect to the frame of reference, by maximizing the normalized correlation coefficient defined for continuous signals by where I 1 stands for the reference initial image and I 2 for the deformed image, W 5 (W x (x, y), W y (x, y)) expresses for the local transformation between the two images, and CZ refers to the correlation zone. The local transformation can be associated, for example, with rigid-body translations [START_REF] Wattrisse | Analysis of strain localization during tensile tests by digital image correlation[END_REF], affine transformations [START_REF] Doumalin | Microextensometry by image correlation applied to micromechanical studies using the scanning electron microscopy[END_REF], or quadratic transformations [START_REF] Lu | Deformation measurements by digital image correlation: implementation of a second-order displacement gradient[END_REF]. Here the following polynomial transformation is identified

W x x, y ð Þ~X d i i~0 X d j j~0 a ij x i y j W y x, y ð Þ~X d i i~0 X d j j~0 b ij x i y j ð2Þ
where d i and d j are the degrees of the polynomial transformation along the x and y directions respectively. The correlation computation consisted of determining the set of transformation coefficients (a ij , b ij ) that maximize the correlation coefficient C(W). The dedicated optimization was performed iteratively: at the first iteration, the transformation was supposed to be null, whereas for the ith iteration, the in-plane displacement components u k x ,u k y of a set of points M k located in the vicinity of the point M were computed as

u k x ,u k y ~arg max a k 00 ,b k 00 ð Þ e C C a k 00 ,b k 00 À Á h i ð3Þ
C W ð Þ~Ð CZ I 1 x, y ð ÞI 2 xzW x x, y ð Þ, yzW y x, y ð Þ Â Ã dxdy ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ð CZ I 1 x, y ð Þ 2 dxdy q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ð CZ I 2 xzW x x, y ð Þ, yzW y x, y ð Þ Â Ã 2 dxdy q ð1Þ
Here a so-called classical normalized correlation coefficient was maximized, which was associated with a rigid-body translation motion between the initial image I 1 and the final image modified by the transformation computed at the previous iteration

I 2 xzW i{1 x x, y ð Þ, yzW i{1 y x, y ð Þ h i .
The maximization of e C C was performed using the discrete values of e C C(dx m , dy n ) computed from the multiple pixel virtual displacements (dx m , dy n ). These discrete data were interpolated in the vicinity of their maximum by quadratic functions giving a subpixel displacement resolution [START_REF] Oulamara | Subpixel speckle displacement measurement using a digital processing technique[END_REF].

The computed displacement components u k x ,u k y were approximated in the least-squares sense by polynomial expressions given in equation ( 2) to give the value of the local transformation coefficients at the current ith iteration

W i x x, y ð Þ, W i y x, y ð Þ h i
. This computation was repeated until convergence, and the convergence criterion was stated as the difference between the displacements computed at point M for two successive iterations.

For the applications presented in this paper, a biquadratic local transformation was chosen (i.e. d i 5 d j 5 2), and the convergence criterion was equal to 0.01 pixel. This relatively high value was chosen for all the computations (associated with both simulated and real images) in order to limit possible numerical oscillations in the iterative computation of the 18 parameters of the local transformation (a ij , b ij ). Furthermore, the quantity

I 2 xzW i{1 x x, y ð Þ, yzW i{1 y x, y ð Þ h i
was estimated using a bilinear interpolation of the grey level intensity in image I 2 .

The strain field was deduced by applying spatial differential operators on the displacement data. To improve the signal-to-noise ratio, the displacement field was filtered before being passed to the identification procedure. The chosen method for this operation was based on local least-squares fitting of the displacement data. The displacement field was locally approximated in the neighbourhood of each material point M (i.e. a material surface element) by a given function whose gradients can also be injected in a strain computation. Naturally, both the shape of the approximation function and the size of the approximation zone (AZ) may affect the efficiency of the filtering. A choose was made here to use biquadratic approximation functions in order to ensure the consistency with the correlation formulation. The errors on strains were estimated on computer-generated speckle images whose imposed deformation pattern was consistent with the one expected for the most strained state using the methodology described in reference [START_REF] Wattrisse | Analysis of strain localization during tensile tests by digital image correlation[END_REF]. These images were generated in order to present characteristics similar to the ones measured on real images (in terms of histograms, correlation radii, etc.). Furthermore, the acquisition noise was simulated on these images by adding a random distribution of grey levels whose amplitude was 5 per cent of the digitalization scale. The order of magnitude of the peak-to-peak error on the strain computed with the approximation functions was 10 23 , as shown in Fig. 1. The residual repartition plotted in this figure illustrates the efficiency of the filtering operation and particularly the absence of introduced bias by the filtering.

As will be presented in subsection 2.2.2, the strain field used in the identification procedure was not computed in this way, but directly deduced from the filtered nodal displacements and from the differentiation of the shape functions associated with the finite element description of the displacement field.

À Á ~ÐCZ I 1 x, y ð ÞI 2 xzW i{1 x x, y ð Þza k 00 , yzW i{1 y x, y ð Þzb k 00 h i dxdy ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ð CZ I 1 x, y ð Þ 2 dxdy q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Ð CZ I 2 xzW i{1 x x, y ð Þza k 00 , yzW i{1 y x, y ð Þzb k 00 h i 2 dxdy
r An important error source associated with DIC computations can also come from optical deformations of the recorded images due to out-of-plane movements or geometrical distortions. To limit the parasitic motions, the specimens were placed between grips that were rigidly mounted on the testing machine. The alignment was checked using the specific procedure described in reference [START_REF] Wattrisse | Analysis of strain localization during tensile tests by digital image correlation[END_REF]. The optical distortions were estimated using a parametric distortion model whose coefficients were identified on measured displacement fields corresponding to rigid-body motions. After being identified, the distortions were suppressed from all acquired images by applying the inverse transformation.

Identification procedure

The material behaviour was locally identified from in-plane displacement distributions obtained by DIC. The identification of material properties from field measurements is a particularly active research area that has led to the development of various approaches, with some of the most widespread being the finite element model updating method [START_REF] Meuwissen | Determination of the elasto-plastic properties of aluminium using a mixed numerical-experimental method[END_REF], the virtual field method [START_REF] Gre ´diac | Applying the virtual fields method to the identification of elasto-plastic constitutive parameters[END_REF], and the equilibrium gap method [START_REF] Roux | A stress scale in full-field identification procedures: a diffuse stress gauge[END_REF].

The framework chosen to state and solve the identification problem associated with an elastoplastic behaviour with positive linear kinematical hardening under the small strain hypothesis is now introduced.

Variational method

DIC provided in-plane displacement fields that gave, after derivation, fields of small strain e. In the most general case, these surface measurements do not allow identification of material properties on a whole volume. For this reason, work focused on thin flat samples for which the plane stress assumption was realistic. For each deformation increment, local measurements of three in-plane strain components allowed three material parameters at most to be determined locally [START_REF] Geymonat | Identification of mechanical properties by displacement field measurement: a variational approach[END_REF][START_REF] Chrysochoos | Local energy analyses of HCF fatigue using DIC and IRT[END_REF].

Here the identification of a linear kinematical hardening model under the small strain hypothesis was considered [START_REF] Latourte | Elastoplastic behavior identification for heterogeneous loadings and materials[END_REF]. An additive decomposition of the strain tensor into elastic and plastic parts was assumed. The chosen plane stress J2 plasticity model [START_REF] Simo | Computational inelasticity[END_REF] depended on two kinds of mechanical parameters: first, the two independent coefficients of the isotropic elastic behaviour (Young's modulus E and Poisson's ratio n) and then the two plastic parameters (yield stress s 0 and hardening modulus k). To identify this set of four parameters, a two-step method allowed first the two elastic coefficients in a pure elastic loading increment to be identified. The couple (s 0 , k) was then identified on the next elastoplastic loading increments.

In the following, the elastic identification step is first presented, followed by the elastoplastic one.

(a) Identification of the elastic constants. A variational method associated with an elastic energy functional defined in reference [START_REF] Geymonat | Identification of elastic parameters by displacement field measurement[END_REF] was used both to identify the field of elastic parameters and to determine the stress s locally. The energy functional F on an appropriate domain of interest V (typically the sample gauge part) reads

F t,B ð Þ~1 2 ð V e u ð Þ{B : tÞ : B {1 : e u ð Þ{B : t ð À Á dV ð4Þ
where the stress field t is statically admissible and the compliance tensor B is supposed to be symmetrical, non-negative, and definite. The functional F(t, B) is separately convex, positive, and null at point (s, A e ) if and only if (s, A e ) satisfies the constitutive equation. Therefore, the identification was performed by numerically minimizing the functional F(t, B). A minimization conducted by a relaxation method gave the solution stress field and the field of elastic parameters respectively.

During the tests the overall forces applied at the specimen boundary were recorded and passed to the identification procedure. An augmented formulation of equation ( 4) involving Lagrange multipliers was used to solve the static admissibility: the stress solution was compatible with the overall forces applied on some boundaries of V and also the stress-free boundaries.

For each loading step the variational approach provided a couple (s, A e ), where s was a stress tensor and A e a compliance tensor. This couple became a candidate solution to the elastic identification problem if it satisfied the comparison procedure on the local compliance tensor (i.e. the relative difference of two tensors A e determined between two consecutive loading steps is small enough). Otherwise the problem was assumed to be plastic, and an elastoplastic algorithm allowed the hardening modulus k and the yield stress s 0 to be identified separately.

(b) Identification of elastoplastic parameters. In such a case, a variational method based on two energy functionals was associated with two different descrip-tions of the mechanical problem [START_REF] Latourte | Elastoplastic behavior identification for heterogeneous loadings and materials[END_REF]: one based on the total displacement field, which is referred to here as 'standard formulation' and another one referred to as 'incremental formulation', based on an incremental displacement field. These formulations involved the secant and tangent plastic tensors (A s and A t ) associated with the chosen model both depending on the unknown plastic parameters and the already determined elastic ones. The energy functionals considered hereafter were the previously defined functional F associated with the 'standard formulation' and the functional G related to the 'incremental formulation'

G Dt,B ð Þ~1 2 ð V e Du ð Þ{B : Dt ½ : B {1 : e Du ð Þ{B : Dt ½ dV ð5Þ
where Dt represents the incremental stress field and e(Du) is the strain related to the incremental measured displacement. Functionals F and G correspond to constitutive error gaps written in a purely mechanical framework. They do not take the irreversibility induced by plasticity into account. Nevertheless, the plastic dissipation, included in the energy balance, can be estimated via the model and the identified parameters. Using F, G, and two successive plastic steps it became possible to find the incremental stress field and the plastic coefficients k and s 0 . More precisely, a fixedpoint algorithm was used to solve this problem: the functional (respectively F or G) was successively minimized on (Ds, k) at fixed s 0 and then minimized on (Ds, k), with k being fixed at its last identified value (obtained at the previous minimization step). To enforce the positive definitiveness of tensor B, the hardening modulus k was supposed to be positive. This hypothesis naturally excludes the investigation of softening behaviours. However, it was considered that this limitation did not affect the generality of the proposed procedure, the specimen softening being able to be interpreted as a structural effect associated with the development of strain localization, as shown in previous work [START_REF] Wattrisse | Kinematic manifestations of localisation phenomena in steels by digital image correlation[END_REF].

Numerical implementation and estimation of the deformation energy

From a purely numerical point of view, appropriate descriptions of the stress field, the material properties, and the displacement field in each finite element were required. To carry out the direct calculation, a formulation based on a naturally equilibrated Airy function was chosen. On each mesh element, the chosen Airy potential was a third-degree polynomial.

The material properties were considered to be constant within each finite element of the mesh. One possibility was to combine element subassemblies with constant material properties. To conclude with the finite element description, it was necessary to define the displacement interpolation. In this approach, the Q2 9 element, constituted of nine displacement nodes, was chosen, as described in references [START_REF] Batoz | Mode ´lisation des structures par e ´le ´ments finis[END_REF] and [START_REF] Latourte | Elastoplastic behavior identification for heterogeneous loadings and materials[END_REF]. The associated strain was quadratic, allowing the quadratic stress B:e(u) to remain close to the stress given by the Airy function.

By the previous identification procedure, knowledge at each loading step of both the incremental and total stress fields was assured. Furthermore, the quadratic strain associated with the measured inplane displacement could be computed in each finite element. The averaged deformation energy w def (V i , t) on a given stress element V i at time t is defined by

w def V i , t ð Þ~1 V i ð t t~0 ð x,y ð Þ[V i s x, y, t ð Þ : _ e e x, y, t ð Þdx dy dt ð6Þ
From a numerical point of view, the following discrete form of equation ( 6) was used

w def V i , t n ð Þ~1 2 1 V i X n{1 i~1 ð x,y ð Þ[V i s x, y, t iz1 ð Þ zs x, y, t i ð Þ ½ : e x, y, t iz1 ð Þ {e x, y, t i ð Þ ½ dx dy ð7Þ
where the spatial integration is exact, using the shape functions of the finite elements, and n stands for the number of load steps used for the identification.

APPLICATION EXAMPLE

The distribution of deformation energy was computed with the strain measurement fields and the computed stress fields derived from the identified model. For elastoplastic materials, the stress developed in a given situation directly depends upon the initial selfequilibrated residual stress pattern preceding the loading. Unfortunately, in most situations, this residual stress state cannot be easily determined.

For material behaviours like linear kinematical hardening, the presence of an initial residual stress state is directly translated as an offset of the current yield stress. The latter being one of the material parameters, it was then possible to account for the unknown initial state distribution using the identification procedure described in the previous section; the existence of a residual stress pattern was consequently translated as a distribution of yield stresses.

The aim of the proposed method was to determine the distribution of the deformation energy within a heterogeneously strained zone. Work first focused on an academic situation where the strain localization was related to a non-zero residual stress state and generated in such a way that the associated stresses were sufficiently important to be experimentally revealed. The main idea was to validate the approach on simple tests where heterogeneities could be easily tracked.

Here, the residual stress state resulted from a mechanical loading on a complex shape sample machined in an initially homogeneous material. The structural effects induced a heterogeneous hardening state of the material that generated a controlled and identifiable residual stress state after unloading. This final state represented the heterogeneous reference state for the subsequent loading step on which the deformation energy had to be estimated.

Numerical experiment

In a first step, the whole procedure was assessed on numerically obtained data in order to estimate its performance on 'perfect', noise-free data. By 'per-fect', it was assumed that the finite element description of the material behaviour was in complete agreement with the hypotheses conceded in the identification procedure: small perturbation hypothesis, plane stress assumption, and constitutive equations (isotropic linear elasticity and linear kinematic hardening).

General description of the simulations

The loading of the sample was simulated using a finite element code. The specimen geometry resembled the typical 'dog-bone' tensile specimen but with a trapezoidal area of additional width in the specimen centre, allowing the response of the specimen to be non-homogeneous. The loading was applied in four consecutive loading steps (see Fig. 2). The first two steps corresponded respectively to the loading and the total overall unloading of the sample, leading to the development of a heterogeneous inelastic stress state within the structure. The third step was related to the redistribution of the residual stresses associated with the 'machining' of the two triangular-shaped domains. This operation was intended to obtain the geometry of a standard tensile specimen, with a heterogeneous residual stress pattern. Numerically, this operation was done by changing the mechanical properties of the element set corresponding to these two zones and Fig. 2 Mesh and pre-deformed state of the sample computing the resulting self-equilibrated stress state (see Fig. 2, material 2); the rigidity of the material was divided by a factor 10 4 . Consequently, the mechanical contribution of these zones was negligible in terms of overall deformation energy. This operation allowed account to be taken of the nonuniform initial distribution of the hardening state variables, without any numerical difficulty. Finally, after the last simulated loading step, strain localization effects due to the heterogeneous pre-hardening state of the material were observed as expected.

Residual stress state

It has already been mentioned that the first three steps were only designed to obtain a heterogeneous residual stress state. At the end of the third loading step, the numerical simulation gave access to plastic strain and stress components. Figure 3 shows the distribution of both equivalent plastic strain and von Mises stress at this particular state. The strong similarity between the Figs 3(a) and (b) is explained by the fact that the initial state of the material (at the beginning of the first loading step) was homogeneous (i.e. p 5 0 and s 0 5 300 MPa).

The stress, strain, and deformation energy fields were computed with respect to this reference state. In order to illustrate the results associated with this pre-hardening state, in Fig. 4 the distribution of the axial stress and that of the deformation energy density (expressed in MPa, which is equivalent to 10 23 J/mm 3 ) given by the direct FE computation at the end of the fourth step were plotted. As expected, the incremental stress and deformation energy density developed in the two triangular-shaped zones were negligible, which is consistent with their negligible contribution to the overall stress pattern. The distributions observed in Fig. 4 are not completely symmetrical due to the boundary conditions simulating the grips: uniform pressure on one side and null displacement on the other.

Identification results

The overall forces coming from the numerical simulations were directly transferred to the identification procedure described in section 2. The inplane displacement field was interpolated, using the finite element (FE) shape functions, on a regular 27627 grid. Using this procedure, the mechanical properties (E, n, k, s 0 ) and the stress field were identified on 13613 elements.

Not surprisingly, the elastic property distributions were quite homogeneous and the mean values of the Young's modulus and of the Poisson ratio were equal to the imposed values. The local fluctuations of these identified data were associated with a standard deviation of 0. shows a high correlation level between the reference and identified distributions. The identified values were quantitatively consistent with the reference ones: the maximum error of about 3 per cent was located in the zones where stress gradients were the most important (in the four corners of the domain and in the middle of the free boundary). This repartition may be due to the stress elements (see section 2) that could not account perfectly for the spatial evolution of the stress. The slight uncertainties on the stress estimates affected the computing of the yield stress, and consequently the tangent modulus.

The identified quantities directly involved in the estimation of the deformation energy are plotted in Fig. 6. As already mentioned, they depended on the identified elastic parameters and thus integrated the discrepancy on these parameters. In particular, the axial component of the identified stress tensor is shown in Fig. 6(a). It also reveals strong stress gradients in the four corners of the domain and in the neighbourhood of the two free boundaries. The identified stress pattern was consistent with the reference one. Nevertheless, the important stress gradients were slightly underestimated with a maximum difference amplitude of around 10 per cent (see Fig. 6(c)). Here it is reiterated that this quantity was determined without any a priori knowledge of the prehardening state of the material.

The deformation energy density patterns are plotted in Fig. 6(b). The displacement field used for the identification was different from the one given by the finite element computation because it had been interpolated on a much coarser grid. Consequently, the shape functions used for the deformation computation within the identification procedure were also unsuited to perfectly represent the strain gradients. These results show, however, a good consistency with the reference values. It can be noticed that the profile of the discrepancy map of the axial stress fits with that of the deformation energy (see Figs 6(c) and(d)).

The specific influence of the superimposed noise on the displacement measurements on the identification results have already been presented in reference [START_REF] Latourte | Elastoplastic behavior identification for heterogeneous loadings and materials[END_REF]. It was observed that the noise did not affect the systematic patterns of the identification errors, but induced a supplementary random error distribution whose variance increased quasilinearly with its amplitude.

Real experiment

A similar experiment was carried out on a specimen machined in a 2.5 mm thick laminated sheet of DP600 steel. The initial geometry of the sample and the loading sequences were identical to the ones used in numerical simulations. The first loading step Fig. 5 Identified plastic parameters was a velocity-controlled tensile loading up to a macroscopic strain of around 1 per cent. Then, the sample was completely unloaded before removing its two triangular-shaped sides by careful machining. The sample was then set again between the grips of the testing machine. It should be remembered here that these operations were not mandatory to estimate the deformation energy patterns. Their only purpose was to obtain a relatively well-known heterogeneous initial state. Images of the deforming specimen were taken by the digital camera during all the different mechanical loading stages. In order to minimize the error sources due to mounting and unmounting the specimen, rigid joints were used between the grips and the testing machine.

By comparing an image of the specimen in its reference state and after machining, the residual strain field could be estimated. Figure 7 shows the measured and simulated equivalent plastic strain at the end of the third load step. Although the actual behaviour of the material was surely different from the one used in the simulation, it can be observed that the measured and simulated residual plastic strain patterns matched rather well. As this plastic strain field was directly related to the residual stress state, it was thus assumed that the residual stress pattern developed within the sample was similar to the one exhibited in the numerical simulations.

Experimental results

The macroscopic stress-strain response of the sample during the fourth load step is given in Fig. 8. It shows the three load levels used for the identification (one in Fig. 6 Identified mechanical fields Fig. 7 Measured and simulated equivalent plastic strain at the end of the third load step the elastic domain to identify the elastic parameters E and n, and two in the plastic domain to identify the two plastic parameters k and s 0 ). Naturally, the material response did not perfectly match the theoretical one associated with a linear kinematic hardening model. Nevertheless, it was possible to estimate the 'macroscopic' homogeneous parameters corresponding to this material behaviour, using this global response. Naturally, the plastic parameters identified in this manner did not account for the heterogeneity of the initial state. They are given here uniquely to illustrate the order of magnitude of the material parameters: E macro 5 194 GPa, n macro 5 0.3, k macro 5 2.4 GPa, s 0 macro 5 480 MPa.

Only the displacement fields measured at the three load levels represented in Fig. 8 were used for the material identification. They were computed on a regular virtual grid of 626682 measurement points, using correlation subsets of 40640 pixels and a biquadratic shape function for the subset transformation (as described in section 2).

The deformation field represented in Fig. 9 was computed using a sweeping local polynomial approximation method involving biquadratic approximation functions of the displacement field (which was consistent with the supposed subset local transformation). This procedure also allowed recovery of the fitted displacement data. The difference between the measured and the filtered axial displacement in the central part of the sample is plotted in Fig. 10. It shows that the data fitting suppressed a random Gaussian error component associated with a zero pixel mean and a 0.02 pixel standard deviation (see Fig. 10(b)), without introducing bias on lowfrequency gradients.

Deformation energy estimate

The filtered displacement data were then interpolated on a coarser 13613 grid before being used in the identification procedure in order to regularize the identification problem (see Fig. 11). They illustrate the symmetry of the measurements, which was consistent with the expected symmetry of the response of the specimen. From these displacement data the following elastic parameters were identified: E ident 5 192 GPa and n ident 5 0.26. They were close to the values deduced from the global stress-strain curve. These results also confirm the hypothesis of a non-damageable elasticity.

The plastic parameter distribution provided by the identification procedure is shown in Fig. 12. Their mean values were consistent with the ones derived from the analysis of the macroscopic stress-strain response of the sample: k ident mean ~1950 MPa and s 0mean ident ~480 MPa. As observed in the numerical results, the distribution of the identified yield stress s 0 was quite symmetric with respect to both vertical and horizontal axes. Furthermore, it matched rather well the distribution associated with the numerical results. It can clearly be observed that the four corners of the domain corresponded to higher values of yield stress and that the horizontal free boundaries, located in the direct neighbourhood of the removed part of the sample, had the smallest level of yield stress. In addition, the vertical boundaries showed high identified yield stress values, while the yield stress identified in the middle of the domain was slightly larger than its mean value.

Conversely, distribution of the identified hardening modulus k was slightly heterogeneous. The standard deviation on this parameter was around 120 MPa, which corresponds to approximately 6 per cent of k ident mean . This last result is also in good agreement with the expectation of a homogeneous distribution of this parameter.

The associated axial stress and strain distributions are plotted in Fig. 13. The patterns observed were The deformation energy density field was computed using the method described in the previous section. The estimate of the deformation energy density fields is given in Fig. 14 and is also in good agreement with the results derived from numerical simulations (see Fig. 6(b)). In particular, the smallest levels of deformation energy were observed in the

CONCLUDING COMMENTS

In conclusion, a procedure was designed allowing the measurement of deformation energy fields in specimens submitted to tensile mechanical loading. This procedure combines displacement measurements obtained by DIC with the stress fields given by a mechanical property identification method. Many technical difficulties were overcome and the first results presented in this paper are encouraging. Mechanical property identification procedures from full field measurements are most generally developed to characterize homogeneous materials. In some situations (e.g. necking and fatigue), the response of a sample to simple loadings is not homogeneous, which can be interpreted as a heterogeneous material behaviour, associated with a heterogeneous initial state. For that reason, an identification procedure was devised that was capable of characterizing heterogeneous material behaviours. Inverse problems such as identification are known to be very sensitive to perturbations, being most likely noise in experimental data. Therefore, the method was first tested on numerical data obtained by a direct finite element computation. The mechanical hypotheses associated with the simulation were identical to the ones conceded in the identification procedure, small perturbations, plane stress, isotropic elasticity, and linear kinematic hardening, thus minimizing the errors due to the inconsistency between the chosen computational framework and the identification formulation. The error sources were then principally related to (a) inconsistent stress and strain gradient estimates due to the use of a coarser finite element mesh in the identification procedure than in the direct computation and (b) the imperfect hypothesis of piecewise constant material properties.

However, the material heterogeneity imposed in the numerical simulation was correctly retrieved by the developed procedure, and the corresponding deformation energy was also correctly estimated either in its pattern or its amplitude. These results emphasized the potential of the procedure.

The method was also applied to a set of experimental data associated with a heterogeneous material comparable with the one studied in the numerical test. The obtained results were also very encouraging: the trends in the identified material parameters (hardening modulus and yield stress) and in the deformation energy distributions matched the ones noticed in the reference numerical test. The observed discrepancies could be related to the difference between the 'real' material behaviour and its most academic modelling used here in the identification. The next step of this work will thus enlarge the panel of more realistic material models considered by the identification procedure. An important effort will also lead to improvement of the temporal integration of the deformation energy.

Fig. 1

 1 Fig. 1 Residual on the strain field obtained by fitting the displacement field histogram of e filtered xx {e imposed xx for a simulated image, x being the loading axis

5

 5 MPa and 10 23 respectively. The identified plastic parameter distributions are shown in Fig. 5. It can be observed that the hardening modulus k was slightly overestimated (around 3 per cent) and that its spatial fluctuation corresponded to a standard deviation of 230 MPa, which represented 5 per cent of the imposed value (see Figs 5(a) and (b)).

Fig. 3

 3 Fig. 3 Mechanical fields at the end of the third load step

Fig. 8 Fig. 9

 89 Fig. 8 Macroscopic stress-strain curve associated with the fourth load step

Fig. 10

 10 Fig. 10 Residual on the displacement field after the fitting operation

REFERENCES1

  Taylor, G. I. and Quinney, H. The latent energy remaining in a metal after cold working. Proc. R. Soc. Lond., 1934, A1 143, 307-326.

Fig. 13

 13 Fig. 13 Identified axial stress component and axial strain used in the identification procedure