Adrien Lagarde

Abdoul Ahad Cisse

Sophie Gourguet

Olivier Le Pape

Olivier Thébaud

Nathalie Caill-Milly

Gilles Morandeau

Claire Macher

Luc Doyen

Gretha

Ahad-Cissé Abdoul

Gourguet Sophie

Le Pape Olivier

Thébaud Olivier

Caill-Milly Nathalie

MACHER Morandeau Gilles

How MMEY mitigates bio-economic impacts of climate change on mixed

Keywords: bio-économique, scénarios, changement climatique, pêcheries, soutenabilité, golfe de Gascogne bio-economics, scenarios, global warming, fisheries, sustainability, Bay of Biscay

Introduction

Marine biodiversity and ecosystem are under extreme pressure worldwide because of the intensification of fishing methods and an overall increase of seafood demand. Thus, according to FAO (2014), around 80% of worldwide commercial fish species are overexploited or fully exploited. Climate change complicates and exacerbates the issues by inducing new, or intensifying existing, risks, uncertainties and vulnerabilities.

In that context, the European Union explicitly accounts for the objectives of mitigating and adapting to the effects of climate change in the area of maritime spatial planning and integrated coastal zone management 1 .

The Common Fisheries Policy (CFP -Reg. UE 1380/2013 11/12/2013) reaffirms the obligations associated with the international commitments of a sustainably management of fisheries and strengthens the existing arrangements. It also puts forward a more regional approach for optimizing the various devices in order to reach, in particular, the maximum sustainable yield by 2020. In the meantime, the Marine Startegy Framework Directive 2 (MSFM) aims at protecting and conserving the marine biodiversity. Positive economic and social benefits as well as food security are also targeted.

In the Bay of Biscay, a warming of ∼ = 0.2 °C / decade between the surface and 200 m depth has been observed for the period 1965[START_REF] Pikitch | Ecosystem-based fishery management[END_REF][START_REF] Decastro | Present warming within the context of cooling-warming cycles observed since 1854 in the Bay of Biscay[END_REF]. Such climate changes already impact some fish species. For flat fish species including Sole with a wide distribution around the Bay of Biscay, recent studies have shown correlations between the abundances of these species and the increase in temperature [START_REF] Hermant | Impact of warming on abundance and occurrence of flatfish populations in the Bay of Biscay (France)[END_REF]. For boreal species, abundance decreases in the Bay while for southern species it increases. Recruitment appears to be the main impacted process [START_REF] Koutsikopoulos | Temporal trends and spatial structures of the sea surface temperature in the bay of biscay[END_REF]. As the definition of boundaries and access rights is a particular issue for fisheries resources, climate change poses a new challenge and new conceptions, in particular institutional ones [START_REF] Badjeck | Impacts of climate variability and change on fishery-based livelihoods[END_REF]. Variations in the spatial distribution of species indeed question the revision of fishing rights and the geographical redeployment of fleets [START_REF] Rajudeen | Expansion of lophius piscatorius distribution in iceland: exploring the ecological and economic viability for establishing sustainable monkfish fisheries in northwestern iceland[END_REF].

As a consequence, designing management tools and public policies that ensure the long-term bioeconomic sustainability of marine fisheries has become a major challenge (FAO, 2014). In response, there has been a growing need of integrated assessment tools to support management advices [START_REF] Thébaud | Building ecological-economic models and scenarios of marine resource systems: Workshop report[END_REF] such as ecosystem-based fishery management (EBFM), [START_REF] Pikitch | Ecosystem-based fishery management[END_REF][START_REF] Link | Keeping humans in the ecosystem[END_REF][START_REF] Doyen | Ecoviability for ecosystembased fisheries management[END_REF]) that aims at integrating the ecological and economic complexities of fisheries, instead of focusing on isolated target species.

However, the way to operationalize the EBFM approach remains challenging [START_REF] Sanchirico | An empirical approach to ecosystem-based fishery management[END_REF][START_REF] Doyen | Ecoviability for ecosystembased fisheries management[END_REF], especially from the bioeconomic viewpoint. New models are needed, notably to integrate the multiple complexities at play [START_REF] Plaganyi | Models for an ecosystem approach to fisheries[END_REF]. These models are expected to account for the multispecies and multi-fleets nature of fisheries, for the multiple ecosystem services they provide as well as for climate impacts. They should also help evaluating the bioeconomic effectiveness and sustainability of current regulatory instruments such as fishing quotas or financial incentives, and designing relevant ecosystem-based management tools [START_REF] Patrick | Myths that continue to impede progress in ecosystem-based fisheries management[END_REF].

Many fish stocks are currently managed to reach their maximum sustainable yield (MSY), through limitations on fishing quotas or efforts [START_REF] Mace | A new role for MSY in single-species and ecosystem approaches to fisheries stock assessment and management[END_REF]. At MSY, catches are maximized at levels where the stock can regenerate. This strategy has been set as the main reference point of many world fisheries, and has been introduced in the US' Magnuson-Stevens Act (NOAA, 2007) and consequently is one of the key objective of the new CFP. However the sustainability of this monospecific strategy in multispecies contexts is disputed [START_REF] Larkin | An epitaph for the concept of maximum sustained yield[END_REF]. In particular, applying MSY policies from single-species assessments in multispecies communities with trophic interactions has been shown to induce biodiversity losses [START_REF] Walters | Mixed gill-netters 12-18 m (f = 11) 6 145 1466 Mixed gill-netters 18-24 m (f = 12) 9 210.3 1500 Mixed gill-netters >24 m (f = 13)[END_REF]. Instead of MSY, many resource economists advocate the use of maximum economic yield (MEY) targets, at which profits are maximized [START_REF] Dichmont | On implementing maximum economic yield in commercial fisheries[END_REF]. Harvesting at MEY is notably known to favor higher biomasses than harvesting at MSY [START_REF] Clark | Mathematical bioeconomics: the mathematics of conservation[END_REF]. In a singlespecies context, harvesting at MEY is thus a more profitable and viable strategy than maximizing sustainable yield. In that perspective, the MEY management strategy has been chosen as a reference point for Australian fisheries, although its implementation remains difficult [START_REF] Dichmont | On implementing maximum economic yield in commercial fisheries[END_REF]. However, maximizing profits from a single stock can also induce overexploitation and extinction, provided its price is higher than the cost of depleting the stock [START_REF] Clark | The economics of overexploitation[END_REF]. Moreover, in a dynamical context, extinction can follow from maximization of present value, whenever discount rates are sufficiently high [START_REF] Clark | The economics of overexploitation[END_REF].

To account for the multispecies nature of fisheries, multispecies reference points and targets are now proposed [START_REF] Moffitt | Moving towards ecosystem-based fisheries management: Options for parameterizing multi-species biological reference points[END_REF]. However, the potential bioeconomic consequences of such multispecies harvesting policies remain largely unknown. There have been attempts at designing multispecies MSY (MMSY) policies, at which total catches are maximized [START_REF] Mueter | Using multi-species surplus production models to estimate ecosystem-level maximum sustainable yields[END_REF]. But in mixed fisheries where technical interactions occur, that is when one fishing fleet harvests different species, maximizing total yields has been suggested to endanger some species [START_REF] Ricker | Maximum sustained yields from fluctuating environments and mixed stocks[END_REF][START_REF] Legovic | Impact of maximum sustainable yield on independent populations[END_REF][START_REF] Guillen | Estimating MSY and MEY in multi-species and multi-fleet fisheries, consequences and limits: an application to the Bay of Biscay mixed fishery[END_REF]. Potential consequences of multispecies MEY (MMEY), at which total profits are maximized, have also been investigated [START_REF] Aglia | Analysis of open-access commercial exploitation and maximum economic yield in biologically and technologically interdependent fisheries[END_REF]. As in the single-species case, MMEY is found to be more profitable than MMSY [START_REF] Guillen | Estimating MSY and MEY in multi-species and multi-fleet fisheries, consequences and limits: an application to the Bay of Biscay mixed fishery[END_REF], however, it has been suggested that a combined MEY is susceptible to induce the overexploitation of stocks with low value [START_REF] Chaudhuri | A bioeconomic model of harvesting a multispecies fishery[END_REF][START_REF] Guillen | Estimating MSY and MEY in multi-species and multi-fleet fisheries, consequences and limits: an application to the Bay of Biscay mixed fishery[END_REF]. In other words, if a multispecies fishery is seen as a portfolio of natural assets, maximizing total profits could neglect the conservation of inferior assets, thus inducing biodiversity losses.

The aim of this paper is to evaluate and compare the bioeconomic merits of multispecies MSY and MEY policies respectively as well as to question their relevance for operationalizing ecosystem-based management for mixed fisheries facing climate change in the Bay of Biscay.

More specifically, this paper examines the impact of climate warming on the bioeconomic performances of Bay of Biscay mixed fisheries and gives insights into the most sustainable management strategy for coping with global warming. To achieve this, a multi-class, multi-fleets and dynamic model for Sole (solea solea) and European hake (merluccius merluccius) is developed and calibrated using ICES and IPCC data. Our study deals with the mixed fisheries of the Bay of Biscay operating in divisions VIIIa and VIIIb according to the International Council for the Exploration of the Sea (ICES) sections (Figure 1). Regarding fish species, we focus on common sole and hake.

Bay of Biscay Case Study

Sole

Common sole (solea solea) is a benthic species whose distribution extends from the West African coasts to the Baltic. In the Bay of Biscay, common sole is in the center of its range area (average latitude 44.5 °N); the work carried out by [START_REF] Hermant | Impact of warming on abundance and occurrence of flatfish populations in the Bay of Biscay (France)[END_REF] did not reveal a clear trend in the evolution of its distribution.

The sole stock appears as a fragile stock and is subject since 2002 to a management pattern aiming at restoring the spawning biomass at its level of precaution (Bpa). This goal was finally reached in 2009 (Figure 2). However, due to surprisingly low recruitment in 2010, the stock is again in decline. Consequently, since 2016, a 10% reduction in total allowable catches (TAC) compared to 2015 and 2014 has been imposed (ICES, 2017) by the European commission. Thus, French fleets are forced to harvest a maximum of 3420 tons since this year (European [START_REF] Union | Council regulation (EU) 2016/72 of 22th[END_REF]. Although the spawning biomass of sole came back up for three years, it still remains dangerously below the sustainable reference point (Bpa = 13000 tons) since 2013 (ICES, 2017).

Hake

Distributed in the North-East Atlantic, European hake (merluccius merluccius) is present along the coasts of Norway to Mauritania. Temperature is a factor that affects the early stages of hake life. Experiments in a controlled environment for the development of hake eggs at different temperatures showed significant mortalities outside the optimal range 10-13° [START_REF] Guevara-Fletcher | Effect of temperature on the development and mortality of european hake (merluccius merluccius L.) eggs from southern stock under laboratory conditions[END_REF]. Studies in the Mediterranean using habitat models show that nurseries require stable background temperatures (11.8-15 °C), low background velocities (<3.4 cms-1) and productive plankton fronts [START_REF] Druon | Modelling of european hake nurseries in the mediterranean sea: An ecological niche approach[END_REF]. Growth or survival of hake juveniles is increased with the availability Figure 3: Historical evolution of the spawning biomass for the European hake. The dashed line refers to the precautionary threshold (Bpa) estimated at 46200 tons (ICES, 2016) of adequate feeding. Changes in ocean conditions may affect prey availability and thus affect migration behavior and hake growth [START_REF] Benson | Changes in pacific hake (Merluccius Productus) migration patterns and juvenile growth related to the 1989 regime shift[END_REF]. Goikoetxea and Irigoien's work (2013) in the Northeast Atlantic on hake highlighted the role of the North Atlantic Oscillation (NAO) in the success of recruiting hake for several years (Figure 3).

More specific informations about species ecology can be found from the bibliographical synthesis made by Caill-Milly et al. (in press).

An economical interest for Hake and Sole fisheries

Hake and common sole are among the first four species in terms of economic values on the Atlantic coast. In 2015, on the Atlantic coast, Hake represents 20% of the overall production in values while Sole reaches 7% for only 2% of the volume (IFREMER, 2017). For the national landings, hake represents 7% of the total value while the sole landings' rise to almost 11% [START_REF] Gourguet | Managing mixed fisheries for bio-economic viability[END_REF]. The sole is less abundant than hake which is the dominant species for fisheries in the European Union [START_REF] Eumofa | Etude de cas : La transmission des prix dans la fili'/ere du merlu frais en espagne[END_REF]. Indeed, although hake has suffered from severe overexploitation with a fall in its recruitment in the 1990s (Figure 3), the stock has observed a recovery of its spawning stock (ICES, 2016) in the following years strongly induced by better recruitments and by the European mono-specific management plan set up at that time (MSY). The price per kilo of the sole is much more important than hake. In 2015, it is worth almost 12 € per kilo with a 60 million € market against 3€ per kg for hake within a market which represents almost 45 million €. These values can be explained by the marked preference of consumers for sole. Consequently, the high abundance of hake and the high price of sole implies that both species are among the most economically and ecologically interesting.

The main French fleets, presenting here and targeting those two species, may be divided into three groups of vessels based on their main gear: various fish trawlers, sole gill-netters and various fish gill-netters. These three fleets can then be separated into 13 sub-fleets ranked by size with a population of more than 400 vessels across the Bay of Biscay [START_REF] Gourguet | Managing mixed fisheries for bio-economic viability[END_REF].

Data Sources

Recruitment and spawning biomass estimation of the sole and hake have been extracted from population models by the ICES on an annual basis for sole and quarterly for hake3 from 1991 to 2013. Sole data are derived from a population dynamics model named XSA (Extended Survivors Analysis -Shepherd, 1999) while hake data have been estimated through the SS3 (Stock Synthesis 3) model based on commercial catches and on abundance data (ICES). Economic data and transversal data of effort and production by fleet and gear have been collected from the Fisheries Information System of IFRE-MER and the French Directorate for Fisheries and Aquaculture (DPMA) through the European Data Collection Framework (DCF). Sea Surface Temperatures (SST) arise from a project led by the European Union called OpEc4 which aimed at rebuilding the history of all marine ecosystems, biological and historical data such as water temperature, oxygen, salinity. . . The geographical coordinates used in this study are: latitude (43.75, 47.39) and longitude (-6.90, -2.77). They do not refer to the entire bay of biscay but only to our two ICES studying areas. For the SST projections over 2100, we rely on the more recent IPCC (Intergovernmental Panel on Climate Change) report which provides, according to four emission scenarios (RCP)5 , many environmental forecasted data. In this paper, we choose to focus on the worst and best climate scenario, respectively, RCP 8.5 and RCP 2.6.

The bio-economic model

We rely on a multi-species, multi-class, multi-fleets and dynamic model in discrete time inspired by [START_REF] Quinn | Quantitative fish dynamics[END_REF], [START_REF] Doyen | A stochastic viability approach to ecosystem-based fisheries management[END_REF] and [START_REF] Gourguet | Managing mixed fisheries for bio-economic viability[END_REF]. Environmental, biological and economic components of the model are described in figure 4. These relations highlight how the different interactions occur: SST impact recruitment through the specific responses of the SSB with respect to the environmental

Multi-species age-class dynamic model

For each species, population dynamics described on a yearly basis in discrete time by age group is first characterized by natural and fishing mortality mechanisms as follows :

N s,a (t + 1) = N s,a-1 (t)exp(-M s,a-1 -F s,a-1 ) N s,As (t + 1) = N s,As-1 (t)exp(-M s,As-1 -F s,As-1 ) + N s,As (t)exp(-M s,As -F s,As ) (1)
where N s,a (t) stands for the abundance of the exploited species, s = 1, 2 (Sole, Hake respectively) at age a = 2, . . . A s at time t. The age class starts at two because the first one stands for recruitment. Thus, abundances of species N s,a (t) evolves according to both natural M s,a and total fishing F s,a (t) mortality of the species s at age a. Furthermore, the total fishing mortality F s,a (t) is derived from the sum of the fishing mortality of the 13 fleets f at year t 0 = 2014 described such as:

F s,a = 13 f =1 u f (t)F s,a,f (t 0 ) (2)
where u f (t) stands for the fishing effort multiplier of the fleet f at time t. The initial fishing mortality, F s,a,f (t 0 ), is related to catchability, effort and number of boats as follows :

F s,a,f (t 0 ) = q s,a,f e f (t 0 )K f (t 0 ) (3)
with e f (t 0 ) is the mean value of fishing effort by vessels of sub-fleet f expressed in number of days at sea, K f (t 0 ) is the number of vessels by sub-fleet f both for the baseline year 2014 and q s,a,f the catchability of fleet f on species s at age a.

Stock-recruitment dynamics

The spawning biomass SSB s (t) is described by:

SSB s (t) = As a=1 γ s,a υ s,a N s,a (t) (4)
where γ s,a stands for the proportion of mature individuals of species s at age a and υ s,a represents the weights (in tons) of individuals of species s at age a and a = 1, . . . , A s .

We assume that the recruitment dynamics depends on both SSB and sea surface temperature (θ) in a stochastic way as follows:

N s,1 (t + 1) = f (SSB s (t -∆ s ), θ(t -∆ s ), ε s (t -∆ s )) (5)
Here N s,1 (t) represents the recruits as explained previously and θ(t) stands for the sea surface temperature at time t while ε s (t) captures the environmental stochasticity affecting the recruitment. In the meantime, ∆ s is a lag with respect to the time necessary for the egg to become a catchable recruit (about two years for the sole s = 1 and about one year for hake s = 2). This allows us to reach a better representation of the stock recruitment relationship. We will discuss and explain these lags in the results section. The integration of environmental factors in recruitment is in line with recruitment [START_REF] Cushing | Climate and Fisheries[END_REF][START_REF] Glantz | Climate Variability, Climate Change, and Fisheries[END_REF][START_REF] Laevastu | Marine climate, weather and fisheries[END_REF]. Recruitment may be affected by sea temperature through many behavioral and physiological processes during spawning and larval phase such as metabolic cost of spawners, natural mortality of eggs and larvae, food availability etc.. [START_REF] Hermant | Impact of warming on abundance and occurrence of flatfish populations in the Bay of Biscay (France)[END_REF].

Different recruitment functions f have been considered here including the Ricker (1958), [START_REF] Beverton | On the dynamics of exploited fish populations[END_REF] and Cushing models as displayed in Table 4. Most of these stock-recruitment models are derived from a generalisation of the Ricker and Beverton-Holt model [START_REF] Hilborn | Quantitative fisheries stock assessment: choice, dynamics and uncertainty[END_REF]. With these different formulations, we performed regressions6 of recruits over SSB and SST time series7 in order to find the model that best fits the data.

Economic scores

Assuming that discards are neglectable, landings equals catches and are defined by the Baranov catch equation:

C s,a,f (t) = N s,a (t)u f (t)F s,a,f (t 0 ) 1 -exp(-M s,a - N f f =1 u f (t)F s,a,f (t 0 ) M s,a + N f f =1 u f (t)F s,a,f (t 0 ) (6)
Incomes derived from catches reads as follows:

Inc f (t) = s As a=1 p s,a (ω s (t))υ s,a,f C s,a,f (t) (7)
where υ s,a,f is the mean weight of landed individuals of species s at age a. Price p s,a (ω s (t)) corresponds to the market value (euros by kg) of species s at age a for year t under the stochastic scenario ω s (t).

Profits π(t) as the difference between incomes and costs are defined by:

π f (t) = (Inc f (t) + α f u f (t)K f (t 0 )e f (t 0 )) (1-τ f )-V f p(t)e f (t 0 ) + c var f e f (t 0 ) + c f ix f u f (t)K f (t 0 ) (8
) α f corresponds to the income per unit of effort of sub-fleet f of other species caught. The dynamic will not be detailed but we assume to have constant values per unit of effort (VPUE) for other species, thus Inc f (t) is only a part of the global income. K f (t 0 ) represents the number of vessels by sub-fleets f , e f (t 0 ) stands for the mean value of fishing effort (i.e. days at sea) by vessels of sub-fleet f , τ f is the landing cost by sub-fleet as a proportion of the gross income, V f represents the volume of fuel used by fishing effort unit and c var f and c f ix f corresponds respectively to the variable 8 and annual 9 (fixed) cost by a vessel of sub-fleet f . These parameters are based on economic data available for 2008 (IFREMER, SIH, DPMA10 , Table 9 and10). The price of fuel is considered constant over time, set at a price of 0.5 € per liter.

Fishing strategies

We here consider three fishing strategies in order to compare them in terms of bioeconomic outcomes: Status-Quo (SQ), Multi-species Maximum Sustainable Yield (MMSY), Multispecies Maximum Economic Yield (MMEY).

Status-Quo Strategy:

The first fishing strategy entitled Status-Quo (SQ) maintains fishing efforts constant throughout the period of interest such as:

u SQ f (t) = 1 ∀f = 1, .
. . , 13 and ∀t = t 0 , ..., T with T = 80

Multi-species Maximum Sustainable Yield (MMSY) Strategy: The second strategy aims at reaching a maximum sutainable yield over all species considered, that is to say, to maximize long-term landings of the differents fleets. The objective is to find the best constant effort multiplier vector noted . u M M SY f that maximizes total catches. The mean total catches over time is defined as the average of the total catches over the entire temporal horizon and where E corresponds to the expectations with respect to the parameters such as:

C M M SY ( . u f ) = E   1 T T t=1 2 s=1 As a=1 13 f =1 C s,a,f (t)   (9)
with T = 80. Once we have the total catches, we are looking for the combination of the best fishing effort multipliers noted .

u M M SY f that maximize the previous metrics.

C M M SY ( . u M M SY f ) = max u f C M M SY (u f ) (10)
As explained in the introduction, by adopting a multi-species point of view, the MMSY management model seems more relevant mainly because most fleets do not target (voluntarily or not) only one species. Thus, this management model could potentially offer a better management from an ecosystem and multi-species perspective than a single-species point of view [START_REF] Voss | Regional trade-offs from multi-species maximum sustainable yield (mmsy) management options[END_REF].

Multi-species Maximum Economic Yield Strategy:

The third strategy we consider consists in maximizing the Net Present Value (NPV):

N P V ( . u f ) = E   T t=t 0 1 (1 + r) t 13 f =1 π f (t)   (11)
with r = 4% the discount rate. E corresponds again to the expectations with respect to the stochastic parameter ω and prices p included in the profit formula (equation 8). Maximizing the NPV is equal to reaching the maximum economic yield for both species and so finding the best effort vector noted .

u M M EY f . N P V ( . u M M EY f ) = max u N P V ( . u f ) (12)
To compute the optimal solutions, we used the SCILAB software.

Climate Scenarios

In our study, we consider two climate scenarios (best scenario: RCP 2.6 and worst scenario : RCP 8.5) illustrated in the figure 5. We notice an upward trend for historical temperatures and a recent and sharp increase for the last few years. Indeed, from 2007, after a fall of more than 0.5°C, the Bay of Biscay is getting warmer with a rise of nearly 1.5°C in just 6 years. This constat is the result of an increase in warming of 0.06/0.07°C per year over the last 30 years [START_REF] Le Treut | Les impacts du changement climatique en Aquitaine: un état des lieux scientifique[END_REF]. We must precise that inter-annual variations induced by atmospheric flux and ocean currents [START_REF] Michel | Evolution of upper layer temperature in the Bay of Biscay during the last 40 years[END_REF] are the main causes of uncertainty and are very difficult to predict even with complex climate models. Yet, the accuracy of climate models is steadily increasing since the 1990s because of the advancement of research, more available data and also due to some major technological discoveries (IPCC, 2013). Therefore, even if these models cannot predict what the temperature will be to the tenth of a degree in 80 years, they are getting closer to reality by relying on verifiable physical principles and on emission scenarios more than likely due to our human activities (IPCC, 2013). These projected temperature data are integrated each year in the recruitment formula and impact it which itself affects the SSB, so it is the species stock and by extension the economy that undergoes consequences as a whole. Undoubtedly, the extent of the impact of water temperature differ depending on the climate scenario and species. The conclusions emerging from this difference can then highlight the danger of potential temperature increase. That is what we strive to show in the next section.

Results

This section presents the merits of integrating a temperature-dependent stock-recruitment model into our bioeconomic model in order to determine the fishing strategy, that best

Impact of warming on stock-recruitment model

In the tables 1 and 2, we present the main results of the regressions. The equations 13 and 14 notably highlight the importance of lags between recruitment and SSB and SST in this paper. They are lagged proportionnaly to the time necessary for the species to become an egg, a larvae, a juvenile then a catchable recruit : two years for the sole et one year for hake in average. We know that a recruitment model only driven by SSB is likely to appear less explanatory than a model which includes an environmental factor [START_REF] Cury | Resolving hjort's dilemma: How is recruitment related to spawning stock biomass in marine fish?[END_REF]table 4). This is highlighted in table 4. Moreover, all estimated coefficients (a, b, c) are statistically significant at the 5% level (Table 1 and2). For both species, the Ricker model turns out to be more relevant than the Beverton-Holt, Cushing or Cobb-Douglas model and all coefficients are statistically significant. These conclusions are consistent with the study carried out by [START_REF] Anneville | Stock-recrutement et espace. approche par la modélisation individus centrée[END_REF] which explains that the Ricker model is "the best pattern [...] because it ensures a much stronger regulation". [START_REF] Beverton | Stock, recruitment and moderating processes in flatfish[END_REF] also confirms that the Ricker model is the best pattern to explain the stock-recruitment relationship especially if the effect of temperature is integrated. We see the influence of temperature on recruitment in Table 1 and2. Explained by the coefficient c, the impact is negative for both species. However, some species are much more impacted by warming than these ones [START_REF] Hermant | Impact of warming on abundance and occurrence of flatfish populations in the Bay of Biscay (France)[END_REF].

Sole's SR model N 1,1 (t + 1) = aSSB 1 (t -1)e -bSSB 1 (t-1)-cθ(t-1) 2 + ε 1 (t -1) (13)

Table 1: Parameters and standard errors of the estimated Ricker model (equation 13) accounting for environmental factor for sole.

Hake's SR model

Hake's model is first built with quaterly data. Given that no recruits is observed by the ICES during the winter quarter so the first three quarters are equals to an entire year. Thus, to harmonize it, we sum all the three quarters to transform the quarterly model into a yearly model such as:

N 2,1 (t + 1) = N 2,1 (t 1 + 1) + N 2,1 (t 2 + 1) + N 2,1 (t 3 + 1) with t = t 1 + t 2 + t 3 and N 2,1 (t i + 1) = aSSB 2 (t i -2)e -bSSB 2 (t i -2)-cθ(t i -2) 2 + ε 2 (t i -2) ∀i = [1, 2, 3]
so the yearly basis model is described as: The solid lines within colored fields display the average of these 500 trajectories with respect to the climate scenario.

N 2,1 (t + 1) = 3 i=1 (aSSB 2 (t i -2)e -bSSB 2 (t i -2)-cθ(t i -2) 2 + ε 2 (t i -2)) (14)

Status-Quo : not ecologically and economically viable

Figure 6 describes the estimated 11 bio-economic performances of the SQ strategy under the two climate scenarios: best scenario/RCP 2.6 and worst scenario/RCP 8.5 over the period 2014-2088. On the top, are plotted the SSB of Sole and Hake while profits are 11 500 trajectories are represented via an uncertainty factor in the recruitment formula such as : Ns,1(t) = Ns,1(t) + εs(t) with εs∈ [-σs, σs] where σs represents the standard error of the regression of the species s displayed on the bottom.

The SQ strategy is not ecologically or economically viable. Sole biomass trends negatively, even with a favorable climate change scenario, and collapses under the pessimistic climate scenario. Hake biomass remains above its Bpa under both scenarios, given the high initial levels of abundance observed in the fishery. While the best climate scenario allows the fishery to remain viable, the worst scenario leads to negative profits in the fishery.

More globally, the SQ strategy highlights the fact that if fishing effort is not adjusted, global warming will amplify the current fall of the sole SSB and will lead to an economic collapse also to an collapse of Sole (under the worst climate scenario). Therefore, management strategies are required that adjust fishing effort in order to moderate the impact of global warming on bio-economic outcomes. The MMSY strategy performs better ecologically and economically than the SQ strategy. As expected, the more extreme the climate scenario is, the more negative the impact is.

MMSY : not ecologically viable but economically viable

The decline of sole SSB below its Bpa appears still inevitable but is clearly mitigated by this strategy. Under the worst climate scenario, the weakness of the fishing effort multipliers first generates a recovery of this stock which collapses however on the long run because of the too high rise of temperatures. Under the best climate scenario, the sole stock decreases initially but then stabilizes after 15 years at around 10 000 tons. The Hake stock displays similar trends as in the SQ scenario, with lighter values regardless of climate scenarios.

Interestingly, although the purpose of this strategy is not to maintain the SSB above Bpas, it significantly improves biological outcomes. Indeed, maximizing landings can not be dissociated from sustainably high level of stock. Therefore, the MMSY strategy, which requires explicit landing goals, implicitely accounts for ecological objectives through the fishing effort structure, thus performing better than the SQ strategy. The MMEY strategy displays better bio-economic performances than the SQ and MMSY strategy.

MMEY : ecologically viable and economically viable

Regardless of the climate scenario, the sole and hake SSB display the same trend than in the MMSY strategy (figure 8) but at higher levels for Sole. Nevertheless, for hake, the SSB remains at lower values regardless of the climate scenario. Under the best climate scenario, profits are multiplied by more than two, reaching 60 million of euros per year. Under the worst climate scenario, profits are lower (Figure 8).

This could be explained by the weak price of hake which leads the MMEY strategy to focus much more on Sole's profits and conservation. On the contrary, the MMSY strategy which aims at maximizing catches has a wider interest to protect hake, taking advantage of its high abundance. Thus, the MMEY strategy emerges as the best way to mitigate climate change effects. We elaborate on the reasons for such outcomes related to the structure of fishing effort multipliers in the next section.

MMSY, MMEY efforts : an activity reduced for sole gill-netters

Table 3 displays the different MMSY and MMEY fishing effort multipliers as well as the economic outcomes i.e mean landings and the NPV of the entire temporal horizon. We first observe that, globally, the MMSY and MMEY strategies imply a important mitigation of the number of boats as almost all the optimal multipliers ( .

u M M SY f , . u M M EY f
) are smaller than 1. This is in line with the actual mitigation of the number of vessels on the Atlantic coast due to the CFP. (SOURCCCCCCE)

Such an outcome indicates that climate change significantly affects the performances of these strategies as well as the computation of optimal management. Indeed, in figure 3, we ascertain that NPV and landings are lower under the worst climate scenario/RCP 8.5 with respect to each strategy. Furthermore, the MMEY strategy displays a better NPV while the MMSY strategy observes higher landings. This is consistent with the purpose of each strategy. [START_REF] Gourguet | Managing mixed fisheries for bio-economic viability[END_REF].

Type of fleet (number of vessels -

K f (2008)) RCP 8.5 RCP 2.6 . u M M SY f . u M M EY f . u M M SY f . u M M
We also notice that, regardless of the climate scenarios, MMEY multipliers are globally higher than MMSY multipliers for mixed trawlers and mixed gill-netters. By contrast, MMEY multipliers for sole gill-netters are lower than MMSY multipliers for all climate scenarios 12 . The choice of strategy has thus a major impact on the sole stock. In particular, the MMSY fishing effort multipliers of sole gill-netters plays a pivotal role for the sole stock. have two species) means a more diversified ecosystem. By contrast, if the Simpson's index tends to one it means we have a lower level of diversity. The figure shows a wide heterogeneity of the ecosystem which may be explained by a domination of one species on another (hake on sole) or a simple extinction of one species (in this case, Sole).

Bioeconomic synthesis displays the MMEY as the best strategy

Here, the values of Simpson index are weak <1.2 indicating that diversity is at stake. This is due to low abundances of Sole, which ever collapses under the worst climate scenario for the SQ strategy.

Discussion

In this last section, we discuss the benefits of managing a mixed fishery with the MMEY strategy and adress the first question asked in the title by the figure 9.

MMEY as an ecological and economic win-win strategy

Figure 9 shows that the MMSY and the MMEY strategies improve the ecological state and economic performance of the fishery as compared to the SQ strategy. Furthermore, the MMEY strategy yields bio-economic gains as compared to MMSY. This ranking SQ<MMSY<MMEY (in the pareto sense) hold true for the two climate scenarios: is in this case a win-win strategy.

Nevertheless, there is no guarantee that the profit of each sub-fleet remains positive because we maximize the aggregated profits of all fleets. Indeed fishing effort multipliers will be higher for the more profitable sub-fleets while less profitable fleets will see their effort reduced in the MMEY strategy. [START_REF] Clark | The economics of overexploitation[END_REF] explains that maximizing NPV with no biological or social constraint may deplete the stock. Indeed, if the discount rate (r) exceed the intrinsic growth rate of the fishery, it is possible to have an 'optimal' extinction. Fortunately, this result does not apply in our case, mainly because of the recovery of hake in the past few years. Therefore, our economic strategy generates a positive effect on the two fish stocks.

However, the ecological gains of MMEY are strongly altered by high climate change effects. Under the highest climate change scenario, the simpson index gain is very limited . This is due to the fact that the sole stock is strongly alterered under this scenario as illustrated by figure 8 (top left).

Diversification of fleets produces greater benefits in the face of climate change

Comparing the multipliers of the MMSY and MMEY strategy, we observe that both strategies imply low fishing effort multipliers for sole gill-netters compared to other fleets, regardless of the climate scenario. This entails a reduction of fishing effort for the fleets which have a greater contribution to the overall fishing mortality of Sole (Figure 10). This observation can be explained in two ways: fishing effort multipliers major in hake14 or they diversify their targets. We can not really elicit any responses regarding this assumption mainly because we are only focusing on two species. Figure 12 seems showing that fleets are not majoring in one species but on the contrary, they diversify their targets so as to avoid the plummet of the sole's stock and to still be able to make profit with this benthic species. Furthermore, we notice that fishing effort multipliers of the mixed trawlers and mixed gill-netters for the MMSY strategy are as a whole lower than those in MMEY strategy. That could explain why profits are higher for this strategy compared to the MMSY strategy. On the contrary, fishing effort multipliers of the sole gill-netters are globally lower for MMEY than for MMSY, which explains why the sole SSB is higher under this strategy for both climate scenarios.

From the economic viewpoint, given that increased SST effect on SSB is more accentuated on sole, we can hypothesize that fleets which have a strong dependency on this species in their turnover would be the most sensitive to an increase of this negative climate impact. As we are almost sure that global warming should persist in the next years (IPCC, 2013) regardless of the climate scenario, fishing effort should orientate towards fleets whose contribution to the fishing mortality will be less important. In fact, as shown by [START_REF] Macher | Optimal selectivity and effort cost a simple bioeconomic model with an application to the bay of biscay nephrops fishery[END_REF], the optimal selectivity of a fleet depends negatively on the level of effort. Thus, lower fishing effort such as sole gill-netters fishing efforts favour a diversification of species caught by these fleets regardless of the climate scenario.

Therefore, many fleets relying on the sole will have to diversify their activities and change their targets especially if TAC of sole and therefore landings continue to decrease as they have been for almost 20 years (Figure 11). The price of sole has rised by 80% between 1994 and 2015 (Figure 13). This explains why sales in value remain high despite their limited share in volume15 which is steadily decreasing [START_REF] Aglia | Analysis of open-access commercial exploitation and maximum economic yield in biologically and technologically interdependent fisheries[END_REF]. Moreover, because of negative warming effects on targeted species, new commercial strategies and a reorganisation of the sector might be observed in the next years [START_REF] Lagiere | Evolution du systeme de gestion de la pecherie de sole du golfe de Gascogne et analyse preliminaire de la dynamique des flottilles[END_REF]. This sectoral change may be restricted by a number of factors : on the one hand, the French fleet is aging16 and the cost of renewing is high17 for new operators whose number has significantly decreased in the last years (Figure 14) and on the other hand it is much more difficult for large vessels operators especially for sole gill-netters to adapt their fishing gears [START_REF] Lagiere | Evolution du systeme de gestion de la pecherie de sole du golfe de Gascogne et analyse preliminaire de la dynamique des flottilles[END_REF]. Conversely, small vessels are already using 2 to 3 different gears per year. With the introduction of European regulations on discards, using case-by-case solutions for fleets to deal with discarding and gearing patterns appears to be one key of success [START_REF] Morandeau | Why do fishermen discard? Distribution and quantification of the causes of discards in the southern Bay of Biscay passive gear fisheries[END_REF].

Perspectives

In the medium to long-term vision, it will probably be a large part of the French fleet which will have to adapt to biological changes induced by global warming. The large level of investment needed to rejuvenate the fleet and the decrease of quotas for some species is creating and is likely to create major problems in the future. Even if the increase in prices sustains profits and if the external or internal demand follows, governments or European institutions will have a crucial role to play. [START_REF] Poloczanska | Responses of marine organisms to climate change across oceans[END_REF] are already observing a movement of marine species sensitive to warming towards the poles. Fortunately, in our two ICES studying divisions, the sole stock is not supposed to be affected by this effect mainly because it is a subtropical species with its centre of distribution in the Bay of Biscay [START_REF] Desaunay | Changes in occurrence and abundance of northern/southern flatfishes over a 20-year period in a coastal nursery area (Bay of Vilaine) and on the eastern continental shelf of the Bay of Biscay[END_REF]. However, the processes implicitly determining in recruitment assessment are not fully understood yet such as the processes in survival success during the juvenile phase [START_REF] Le Pape | The food limitation hypothesis for juvenile marine fish[END_REF].

Nevertheless, we can now assume that global warming alters sustainable gains that could be earned by management strategies in an ecosystem approach. The results for our case study illustrate the potential of strategies aimed at MMEY to entail greater adaptation capacity in the face of climate change. This result is to place in the context of policy objectives which at the moment largely focus on mono-MSY or MMSY. So as to avoid harmful effects, global changes have to be mitigated by management structures which are already set up but which could be more efficient (almost half of mondial stocks are managed with the MSY method) as we have proved it in our study with the MMEY method.

With this study, we underline the importance to integrate both multi-species, multifleet nature of fisheries as well as upstream temperature and more specifically SST in recruitment models and in management models of fisheries. This in line with [START_REF] Hughes | New paradigms for supporting the resilience of marine ecosystems[END_REF] who claim : "restoring marine [...] ecosystems after they have degraded is much more difficult than maintaining them in good condition". Such an ecosystem policy has already been tested by the Pacific Fishery management Council in 1998 in the management of sardine stocks (Sardinops Sagax ). The council adopted a control of fishing variable and fluctuating depending on temperature increase, however the temperature index used in this model (SPSST18 ) was not truly reflecting the temperature in the studying area, namely the area of the greatest spawning activity [START_REF] Hill | Assessment of the pacific sardine resource in 2011 for us management in[END_REF].

From a policy perspective, such results would lead to suggest the need for management strategies to periodically adapt fishing effort regulations based on the economic, social context and especially on the local and global environnemental changes.

Figure 1 :

 1 Figure 1: Map of the Bay of Biscay and ICES divisions. The studying area is in blue.

Figure 2 :

 2 Figure 2: Historical evolution of the spawning biomass for the common sole. The dashed line refers to the precautionary threshold\Bpa. (ICES, 2017)

Figure 4 :

 4 Figure 4: Relations existing between environmental, biological and economic factors within the bio-economic model. Arrows stand for the interactions between variables while figures between brackets refer to the equations/models that link the various factors within the bio-economic model.

Figure 5 :

 5 Figure 5: Evolution of historical SST from 1991 to 2013 then SST trajectories according to the two climate scenarios (RCP 2.6 and 8.5) from 2014 to 2100 (IPCC, 2013)

Figure 6 :

 6 Figure 6: Strategy Status-Quo -Sole (top left) and Hake (top right) SSB trajectories under the three climate scenarios in thousands of tons. The black solid line represents the precautionnary threshold of the species' stock (Bpa). The third figure (bottom)represents the total profits over all thirteen fleets. The field of the 500 simulated trajectories under the best climate scenario is in green and in red under the worst. The solid lines within colored fields display the average of these 500 trajectories with respect to the climate scenario.

Figure 7 :

 7 Figure 7: Strategy MMSY -Sole (top left) and Hake (top right) SSB trajectories under the two climate scenarios. The black solid line represents the precautionnary threshold of the species' stock (Bpa). The third figure (bottom) represents the total profits of all thirteen fleets. The field of the 500 simulated trajectories under the best climate scenario is in green and in red under the worst. The solid lines within colored fields display the average of these 500 trajectories with respect to the climate scenario.

Figure 8 :

 8 Figure 8: Strategy MMEY -Sole (top left) and Hake (top right) SSB trajectories under the two climate scenarios. The black solid line represents the precautionnary threshold of the species' stock (Bpa). The third figure (bottom) represents the total profits of all thirteen fleets. The field of the 500 simulated trajectories under the best climate scenario is in green and in red under the worst. The solid lines within colored fields display the average of these 500 trajectories with respect to the climate scenario.

Figure 9

 9 Figure 9 synthesizes the bio-economic outcomes of the three management strategies through the average NPV of the entire temporal horizon on the Y-axis versus the Simpson's index of diversity 13 on the X-axis. A Simpson's index close to 2 (because we 12 Except for the two smallest sub-fleets of the sole-gill netters fleet under the worst climate scenario. That is explained by their weaker contribution of the sole mortality (figure 10) than the two biggest sub-fleets 13

Figure 9 :

 9 Figure 9: Simpson's Index of Diversity versus Average Net Present Value.

Figure 12 :

 12 Figure 12: Fishing effort multipliers (left axis) and contribution to the overall fishing mortality of Sole in percentage (right axis). Fleets (X-axis) are ranked by contribution to sole mortality with SGN=Sole gill-netters, MGN=Mixed gill-netters and MT=Mixed Trawlers. The first figure accounts for the best climate scenario (RCP 2.6) and the second (bottom) for the worst climate scenario.

Figure 14 :

 14 Figure 14: Numbers of sailors in the Atlantique facade[START_REF] Aglia | Analysis of open-access commercial exploitation and maximum economic yield in biologically and technologically interdependent fisheries[END_REF]

Table 2 :

 2 Parameters and standard errors of the estimated Ricker model (equation 14) accounting for environmental factor for hake.

Table 3 :

 3 Optimal fishing effort multipliers for MMEY and MMSY strategies with respect to the two climate scenarios. Numbers between brackets refer to the number of vessels in 2008

	EY
	f

Table 10 :

 10 Mean reference costs of the thirteen sub-fleets. Source: Ifremer, SIH, DPMA, 2008

In the report made by the ICES, the sea council hypothesizes that no recruits is observed in the fourth quarter, the sum of the three previous quarters represents here the annual and actual spawning stock (ICES)

Operational Ecology (End date : 31/12/2014) -http://marine-opec.eu/

Representative Concentration Pathways

Ordinary Least Squared for the log-linearised model of the sole with 22 observations and autoregressive process of order 1 for the log-linearised model of the hake to correct the autocorrelation of its errors with 66 observations

7 By using the Scilab software and one of its econometric modules named GROCERhttp://dubois.ensae.net/grocer

.html 8 The variable cost includes oil, supplies, ice , bait, gear, and

equipment costs[START_REF]Intergovernmental Panel on Climate Change[END_REF] The annual cost includes maintenance, repair, management and crew costs, fishing firms, licenses, insurances and producer organisation. Those costs date from 2008

DPMA stands for Direction des Peches Maritimes et de l'Aquaculture which corresponds to the Directorate for Sea Fisheries and Aquaculture at the French Ministry of Agriculture and Fisheries. SIH means Systeme d'Informations Halieutiques, the fisheries information system monitored by Ifremer, the French Research Institute for the Exploitation of the Sea (http://wwz.ifremer.fr/institut_eng)

The fleets targeting strongly the sole decrease their effort in order to increase the effort of fleets which mainly targets hake

Sales in Volume = landings -unsold

The average age of French vessels is over 25 years old in 2012. Still in 2012, only 20% of the fleet was under 15 years. In constrast, almost 57% of the ships had more than 25 years[START_REF] Aglia | Analysis of open-access commercial exploitation and maximum economic yield in biologically and technologically interdependent fisheries[END_REF]

Furthermore for big vessels

Scripps Pier Sea Surface Temperature

Acknowledgment

This work has been carried out with the financial support of COTE LabEx, through the NAVIRE project (SceNArios of bioeconomic VIability and REsilience for ecosystembased fisheries management in Aquitaine) in association with GRETHA (CNRS-Univ. of Bordeaux), IRSTEA, IFREMER and Campus DoMar (Spain). This study also relates to the network SEAVIEW (Scenario, fishEry, ecologicAl-economic modelling and VIability nEtWork).

Appendix

Ns,1(t + 1) = aSSBs(t -∆s)e +bSSBs(t-∆s) + εs(t)

Ricker 2

(-3.37 | 0.66)

Ns,1(t+1) = aSSB1(t-∆s)e -bSSB 1 (t-∆s)-cθ(t-∆s) 2 +ε1(t-∆s)

B-H

B-H 2


Julien Bertolini

Bruno Castanié
email: bruno.castanie@isae.fr

Jean-Jacques Barrau

Jean-Philippe Navarro

Caroline Petiot

MULTI-LEVEL EXPERIMENTAL AND NUMERICAL ANALYSIS OF COMPOSITE STIFFENER DEBONDING PART 2 : ELEMENT AND PANEL LEVEL

Keywords: Skin/stringer debonding . Finite Element analysis. Postbuckling, fracture mechanics, VCCT

In the framework of test analysis pyramid, large specimens were studied. To represent the bending behaviour during postbuckling, specimens composed of a plate with a stiffener were supported on five points and loaded transversely by two points, thus being subjected to "seven point" bending. By modifying the positions of the two loading points, symmetrical and antisymmetrical buckles that led to interface failure between the flange and the skin could be simulated. First a global numerical model of the test was made in order to assess the efficiency of the approach developed in the first part of this study. Predictions were in accordance with experiments despite strong scatter. Then, a global/local test method was considered. In this method, the global model considered shell elements although the local model used volume elements. The onset of delamination was correctly predicted at local level. Finally, the method was applied to two large stiffened panels subjected to compression and shear.

-INTRODUCTION

In preparation for the introduction of composite structures in next generation-aircraft fuselages, a multi-level approach has been proposed for studying the failure of stiffened structures by debonding of the stiffener in the postbuckling phase. In the first part of this two-part article, the authors used 4-point bending tests on beam-type specimens with a flange [START_REF] Krueger | Testing and analysis of composite skin/ stringer debonding under multi-axial loading[END_REF], [START_REF] Meeksa | Stiffener debonding mechanisms in post-buckled CFRP aerospace panels[END_REF]. These tests completed the usual methods for interface characterization (DCB, ENF, MMB) and validated the various numerical approaches on a geometry closer to technological reality. This type of specimen was used to analyze the influence of local geometries and environmental effects on the debonding behaviour [START_REF] Bertolini | Contribution à l'analyse expérimentale et thèorique des ruptures de structures composites en postflambement par décollement de raidisseurs[END_REF]. At the global level, tests are usually done on panels having sizes of the order of a metre. The loads are compression ( [START_REF] Zimmermann | Buckling and postbuckling of stringer stiffened fibre composite curved panels -Tests and computations[END_REF]- [START_REF] Orifici | Degradation investigation in a postbuckling composite stiffened fuselage panel[END_REF]), shear exerted through a deformable square ( [START_REF] Farlay | In plane shear test of thin panels[END_REF]- [START_REF] Sheppard | Rib separation in postbuckling stiffened shear panels[END_REF]) and, more rarely, combined compressive/shearing loads with or without internal pressure ( [START_REF] Romeo | Buckling and postbuckling behavior of anisotropic plates under combined biaxial compression and shear loads[END_REF]- [START_REF] Castanié | Combined shear/compression structural testing of asymmetric sandwich structures[END_REF]). However, these tests are complex, costly and often perturbed by the boundary conditions. To correlate the tests, it is often necessary to consider the geometry and exact stiffness of the clamping design and also the initial defects of the specimens [START_REF] Bertolini | Contribution à l'analyse expérimentale et thèorique des ruptures de structures composites en postflambement par décollement de raidisseurs[END_REF]. Thus, in order to isolate the phenomena and ensure the reliability of the approach for validating a computing method [START_REF] Li | High Fidelity Failure Analysis for a Composite Fuselage Section[END_REF], it is desirable to perform tests on elements at an intermediate level. The authors used this type of element-level test to study the specificities of omega stringers [START_REF] Bertolini | An experimental and numerical study on omega stringer debonding[END_REF]. Three-or four-point bending tests were used but did not give an optimal simulation of the deformations obtained in postbuckling. So, at this level of analysis, a more specific test is required. Although rarely mentioned in the literature [START_REF] Meeksa | Stiffener debonding mechanisms in post-buckled CFRP aerospace panels[END_REF], the "7-point" bending test [START_REF] Van Rijn | A seven-point bending test to determine the strength of the skin-stringer interface in composite aircraft panels[END_REF] is proposed for validating the choice of computing method at this level .

From the modeling standpoint, the study on elementary specimens allowed the best suited methods to use in the aeronautical industry to be selected among those put forward in the literature. After a first selection in connection with the general context of the study, the interest of fracture mechanics ( [START_REF] Krueger | Testing and analysis of composite skin/ stringer debonding under multi-axial loading[END_REF], [START_REF] Krueger | Virtual crack closure technique: History, approach, and applications[END_REF], [START_REF] Krueger | Analysis of composite skin-stiffener debond specimens using a shell/3D modeling technique[END_REF]) and cohesive methods ( [START_REF] Camanho | Numerical Simulation of Mixed-Mode Progressive Delamination in Composite[END_REF], [START_REF] Turon | A damage model for the simulation of delamination in advanced composites under variable-mode loading[END_REF]) was compared. Both the methods tested use the Benzeggagh-Kenane mixed criterion [START_REF] Benzeggah | Measurement of mixed-mode delamination fracture toughness of unidirectional glass/epoxy composites with mixed-mode bending apparatus[END_REF]. It was shown that fracture mechanics allowed a saving of an order of magnitude in the computing time in order to determine the onset of debonding. This approach was also validated on elements with uniaxial tests for the specific geometry of omega stringers [START_REF] Bertolini | An experimental and numerical study on omega stringer debonding[END_REF]. However, the change to real structures means that local (debonding) and overall (postbuckling) effects have to be managed simultaneously or in parallel. Several approaches have been suggested for managing the problem on the two scales. In the Posicoss program [START_REF] Zimmermann | Buckling and postbuckling of stringer stiffened fibre composite curved panels -Tests and computations[END_REF], only the overall, non-linear response was analyzed and the sensitivity to boundary conditions determined. More recently, the Cocomat program took the local level into consideration through volume elements modelling the interface in the global model [START_REF] Degenhardt | Design and analysis of stiffened composite panels including posbuckling and collapse[END_REF] or by calculation of the Strain Energy Release Rates at the interfaces by post-processing [START_REF] Orifici | Degradation investigation in a postbuckling composite stiffened fuselage panel[END_REF]. Krueger and Minguet [START_REF] Krueger | Analysis of composite skin-stiffener debond specimens using a shell/3D modeling technique[END_REF] proposed a direct approach using a mixture of shell and volume elements. A geometrically non linear approach was developped by Cosentino and Weaver [START_REF] Cosentino | Approximate Nonlinear Analysis Method for Debonding of Skin/ Stringer Composite Assemblies[END_REF] using also the fracture mechanic to predict debonding. A global/ local approach has recently been developed by Orifici et al. [START_REF] Orifici | An analysis methodology for failure in postbuckling skin-stiffener interfaces[END_REF] which determines the onset of local debonding by means of a method of property degradation in elements. In the present study, the local method validated in the first part and in [START_REF] Bertolini | An experimental and numerical study on omega stringer debonding[END_REF] will be applied to a non-linear 3D local model of the 7-point bending test. The boundary conditions are provided by increments of the global 3D model. Once validated, this methodology will then be applied to real certification tests of two stiffened panels carried out at Airbus France under a current programme.

-SEVEN-POINT BENDING TEST

-Specimens and test description.

The geometry of the specimens is described in Figure 1. The specimens were flat panels to which a T-shaped stiffener was added in the longitudinal direction. The skin and the stringer were made of carbon/epoxy laminate and were co-cured without an adhesive film. The skin was composed of 16 crossed symmetrical plies. The same layup was used for the 14 plies composing the L-sections that formed the stringer. The skin/stringer interface was 90°/90° and the interface of the stringer ply immediately above was 90°/45°. A "nail head" was inserted at the base between the two back-to-back L sections so that this volume could be filled with resin. A special tool was used to hold the skin/stringer assembly in position during polymerization. A total of 10 specimens were made and tested. Strain gauges were used to monitor the test. Their positions were specific to the type of test and are given in Figure 1. The test rig was composed of two blocks (Figure 2). The principle of 7-point bending tests is given in Figure N° 2. Five points serve as supports for the skin part of the test piece. The supports used are adjustable and enable hyperstatism and flatness defects of the specimens to be managed. Moreover, each support includes a load sensor. The upper block is mobile and composed of two supports. A displacement sensor (LVDT) is installed on the mobile block so that its displacement can be known accurately. The supports in contact with the specimens are articulated to limit their punching effect during testing. A pre-sizing calculation was performed to position the supports in such a way as to avoid any fracture before debonding.

It is possible to simulate different fields of displacement by modifying the position of the movable points. The term chosen for this is symmetrical bending (Figure 3) for the type of buckles obtained in compression and antisymmetrical bending for the type obtained in shear (Figure 4). As the assembly is hyperstatic, the supports can be adjusted manually and incorpo-rate force sensors. It is thus theoretically possible to compensate for flatness defects in the specimens. Half of the 10 specimens were used for symmetrical bending tests and the other half for antisymmetrical bending tests.

-Test results and analysis.

The forces measured by the 7 sensors in the case of symmetrical tests are shown in Figure 5. Globally, the test seems fairly reproducible. The lower middle support (support 3) took up most of the force imposed by the upper supports. The scatter over 5 tests was limited to 14 %.

The behaviour and scatter were of the same type for the antisymmetrical tests, with a different distribution of forces on the supports. Debonding was detected at the first discontinuity of the force vs displacement curves. It occurred at the skin/stringer interface or at the interface of the ply immediately above (90°/45° interface) and propagated abruptly. The aspect of the debonded specimens is shown in Figure 6 and the morphology obtained by C-Scan is also given in the same figure.

The tests were reproducible but, although the scatter remained limited for the symmetrical tests (14%), it reached 32% for the antisymmetrical tests. Analysis showed that the lowest values and highest values were obtained for the same specimen batches, which appears to indicate that the source of scatter was not test related but should be found in the differences in the fabrication of the specimens.

-Reference model and test correlation.

A first reference model based on 3D shells was created using the Abaqus software (Figure 7). The supports were modelled as rigid bodies in contact with the skin. Each support was free in rotation and loading was obtained by the displacement of the upper supports. The lower supports were blocked in translation along the three directions. The symmetrical and antisymmetrical tests were simulated by simple repositioning of the supports and change of the contact zones. The elements between the stringer and the skin were coincident at the edge of the flange and refined in the zone where the debonding criterion applied. The VCCT technique was used and the Benzeggagh-Kenane criterion [START_REF] Benzeggah | Measurement of mixed-mode delamination fracture toughness of unidirectional glass/epoxy composites with mixed-mode bending apparatus[END_REF] was applied to one side only of the stringer so as to represent the lack of symmetry obtained during testing. The pre-crack was 0.5 mm long.

The response of the specimen in terms of machine force/displacement (Figure 8) shows that the simulated behaviour was globally in agreement with the experimental behaviour. The measurements showed that the central support took most of the force and the lateral supports became load-bearing beyond 1 mm. The simulation underestimated the force on the central support by about 15% at debonding. Analysis of the strains in the flange (S2, Figure 1) and under the skin (S3, Figure 1) indicated that the level of bending found experimentally was close to the simulated level (Figure N° 9). The same overall behaviour was obtained for the antisymmetrical case [START_REF] Bertolini | Contribution à l'analyse expérimentale et thèorique des ruptures de structures composites en postflambement par décollement de raidisseurs[END_REF]. The computed locations for the onset of debonding (Figure 10) agreed with those found experimentally (see Figure 6).

For the symmetrical case, the computed debonding load was within 7 % of the first fracture load observed in test E1 and within16 % of the mean fracture load. Considering the high scatter in the test results (about 14 %), this value is perfectly acceptable. For the antisymmetrical case, the computed debonding load was within less than 1% of the minimum debonding load obtained experimentally and within 19 % of the mean of the tests. Considering the high scatter in the test results (32%), this result is acceptable. The modal distribution at the failure load was of the same type for both tests. Mode I was preponderant and the ratio of GII/ (GI+GII) was 33% for the symmetrical test and 29% for the antisymmetrical test.

To sum up, the reference model enables the experimental behaviour of the specimens in 7-point bending tests to be correctly apprehended. The combined use of the Benzeggagh-Kenane criterion and the VCCT method again shows its efficiency with respect to a more complex problem. However, the main problem for this kind of modelling lies in the computing time, which is of the order of half a day because due to the mesh refinement needed for convergence. Using a global/local approach should lead to a distinct improvement in the time. It is the subject of the next section.

-Global/local approach.

The interest of this method lies in the fact that it dissociates the non-linear model required for the postbuckling computation from the model required for the debonding computation.

Local models that are refined but of small size can be used for detailed studies of specific zones and a model can be used as often as necessary at different places on the panel. The principle is based on the use of fields of displacement of a zone likely to undergo debonding, provided by the global model, by reinjecting these displacements at the boundaries of a local model better suited to the debonding analysis. As communication between the two scales only goes in one direction, any change in stiffness due to debonding in the local models will not be taken into account in the global model. In consequence, only the onset of debonding can be analyzed by this method. Nevertheless, this approach is consistent with the sizing principle for composite aeronautical structures, which does not tolerate any crack propagation. Moreover, tests at structure level show that failure is generally explosive, suggesting that the debonding phenomenon is unstable and that seizing the onset of debonding should suffice to compute the failure of the structure.

Two models were used in accordance with the process described above. The global model is presented in Figure 11. The skin, the flange and the web of the stiffener were modelled using reduced-integration linear plate elements (S4R). The link between the flange and the web of the stiffener was described by rigid links, as was the link between the web of the stiffener and the flange. So the radii between the web and the flange were not modelled. In accordance with the reference model used previously, loading was applied using seven supports, two of which (upper supports) were subjected to imposed displacements to produce the required loading. The global/local comparison for one increment of displacement is given in Table 1. The deflections obtained in the two cases were comparable and the difference was less than 3%, showing that the bends were practically identical. The local stress state could thus be correctly obtained directly and the energy release rates in mode 1 were identical to within 3%. The difference was greater in mode 2 and reached 15 % for the symmetrical case. The Benzeggagh-Kenane criterion was little affected, however, and the difference for both cases was less than 3.5%, which validated the global/local approach. The only notable difference between the two models came from the force exerted on the assembly to obtain the correct deflection (difference 12.5 % and 9%). This difference can be attributed to the rigidity of the links used between the skin and the stringer, which did not allow the displacements to be continuously represented and altered the global stiffness. To sum up, the global/local method has been validated by analysis of the 7-point bending tests and it is possible to move on to larger sized problems. Moreover, the objective of reducing computing time was achieved, i.e, a complete computation loop with the global/local approach takes about twenty minutes.

-PANEL ANALYSIS

-Introduction

The two tests presented in this part are not specific to this study but form part of the certification process for a current Airbus programme. Therefore, only the general methodology of the tests and numerical approach are given here. The aim is to validate the approach for a higher level and in an operational context. The problem of correlating with structural tests at these scales will not be presented here since it is long and complex and involves certain data that cannot be disclosed but the essential points can be found in [START_REF] Bertolini | Contribution à l'analyse expérimentale et thèorique des ruptures de structures composites en postflambement par décollement de raidisseurs[END_REF]. The present paper will limit itself to applying the global/local approach and the Benzeggah-Kenane criterion generalized to mode III for these tests on panels. It appears that the dominant debonding mode at these scales is mode III and the criterion is modified as follows, assuming by default that :

(eq 1)

Two uniaxial tests were analyzed: one under compression and one under shear.

-Tests description and failure analysis.

The two stiffened panels used in the programme were composed of a panel and three omega stringers made of carbon/epoxy laminate. No adhesive film was used for the fabrication and the stiffened panels were co-cured. The geometry of the panel under compression is given in Figure 13 (a). Two 50 mm resin blocks were molded at the ends of the panel so that the com- -------------------------------

G 2 C G 3 C = G C G 1 C G 2 C G 1 C - ( ) G 2 G 3 + G 1 G 2 G 3 + + -
-     η ⋅ + =
pressive load could be applied through the test equipment. The edges of the panel were stabilized by guides. The shear test was carried out using a deformable square [START_REF] Farlay | In plane shear test of thin panels[END_REF]. The panel under shear also had three stringers. The edges of the panel were reinforced by carbon/epoxy and glass/epoxy to avoid local failure of the panel near the loads introduction and to ensure the most uniform possible load transfer. Three lines of bolts were used. The first line was composed of titanium fasteners whereas the last two lines were composed of steel fasteners. This choice gave a better spread of the load among the fasteners.

For the compression test, the three stiffeners were cracked through but the breaks were not situated in the same plane (Figure 14). The skin was practically undamaged apart from local damage at one of the guides (possibility of secondary break). This implies that the overall buckling was not the source of the failure. Analysis of the stingers showed no visible debonding, which means that fracture was due to excessive stress in the stiffeners. Buckling was detected by a loss of linearity of the gauges at 24.3 kN and the breaking load was 106 kN.

Shearing failure was explosive and was caused by the debonding of the central stiffener (Figure 15). Buckling was detected by the loss of linearity of the gauges and by image correlation at 360 kN . The breaking load was 425 kN.

-Global/local finite element analysis of skin/stiffener debonding.

The global model for compression can be seen in Figure 16. The Abaqus software was used and plate elements (S4R) were chosen to model the structure as a whole. The resin blocks on either side of the panel were considered to impose the clamping conditions modelled by the rigid links between the (master) "steering" nodes and the (slave) nodes at the ends of the panel.

The anti-buckling guides were modelled by simple support conditions over the whole surface of contact. The internal radii of the stiffeners were not modelled but this hardly changed the overall stiffness of the panel. The interfaces between the skin and the flanges of the stringers were modelled by rigid links ensuring load transfer only.

The shear global model is shown in Figure 17. The composite panel was modelled in the same way but, to properly represent the postbuckling behaviour, it was necessary to represent the reinforcing plies, the frame itself (in volume elements) and the various attachment points.

Rotation of the different parts of the frame was allowed by controlling the local contacts. Force transfers between the frame and the stiffened panel were ensured by connectors (elements defined by 6 rigidities in translation and rotation). The shearing rigidities of the connectors representing the fasteners were given by Huth's model [START_REF] Huth | Influence of the fastener flexibility on the prediction of load transfer and fatigue life for multi-row joints[END_REF]. The model has 7 million degrees of freedom. The two models gave acceptable correlation with the tests and the details in terms of loads/displacements and local strains given by the gauges can be found in [START_REF] Bertolini | Contribution à l'analyse expérimentale et thèorique des ruptures de structures composites en postflambement par décollement de raidisseurs[END_REF].

The global/local approach presented and validated by the 7-point bending test was applied to both cases (Figure 18). The main modification was that the omega shape of the stringers made necessary the study of the debonding on both sides of the flange. To respect the St Venant hypothesis and thus avoid the boundary conditions at the edges of the local model perturbing the results, pieces of skin and stiffener web were modelled. The nodes in the thickness were steered in the same way as for the 7-point bending test, on the basis of the Kichhoff-Love hypothesis (Figure 12). The size of the local model was about a quarter of the length of the panel, which allowed the structure to be covered with a good compromise between computing time and accuracy [START_REF] Bertolini | Contribution à l'analyse expérimentale et thèorique des ruptures de structures composites en postflambement par décollement de raidisseurs[END_REF]. The local model was built up of 3D shells and the radius between the flange and the web of the stringer was modelled. The resin corners could vary from one stringer to another. It was therefore decided not to model them and to place the virtual crack at the flange/web radius (Figure 18). This hypothesis is very conservative, as shown in [START_REF] Bertolini | An experimental and numerical study on omega stringer debonding[END_REF].

In compression, the model predicted initial buckling of the panel at 20300 N while experiment recorded 24600 N, i.e. a difference of about 15 %. Although this discrepancy seems large, the theory of buckling shows that a variation of 56 hundredths of a millimetre in the thickness (ply thickness of 0.132 mm instead of 0.125 mm) would be enough to explain the difference considering the thickness of the panel. This hypothesis seems realistic as, in other tests carried out at Airbus where the actual thickness was measured, the true thickness of a ply was 0.13 mm.

Numerically, the behaviour of the panel in compression shows three distinct phases.

After the simple compression phase, a symetric buckling of the skin produces 11 waves. The amplitude of the buckles gradually increases up to about 108000 N where a change of mode occurs, introducing two additional waves. Similarly, the height of the buckles continues to increase until general buckling of the stiffened panel occurs at 153 800 N. The Tsai-Hill criterion is reached at the hat of the stiffener and at the failure locations identified in testing (Figure 19) for a load of 106 085N, i.e. within 1% of the measured failure load. The generalized Benzeggagh-Kenane criterion is reached at 144 900 N, which is well above the observed and computed failure load. The aspect of the debonded area in the local model is shown in Figure 19. If debonding had taken place, it would have occurred at the level of the free edges of the flange.

The calculations show that, once the debonding load is reached, the propagation of debonding is unstable. Debonding would thus lead to explosive failure. For the compression test, the numerical model is in agreement with the tests even if debonding of the stringer is not the source of the postbuckling failure of the stiffened panel.

For the shear case, the same numerical methodology was used. The calculated buckling load was 338 kN, as compared with the 360kN obtained experimentally. The error on the buckling load is thus about 10 %, which is acceptable considering the complexity of the test. After scanning the structure, the critical site was indeed found in the centre of the panel at the free edge of the flange of the central stringer, at the level of the largest buckle. Debonding was reached at the critical site for 469 KN, the error being 9.4 % with respect to the test result (425 KN). This error, which is reasonable considering the complexity of the test, could be due to •the non-linear behaviour of the panel is not perfectly represented in spite of the very fine modelling used,

•the value of the energy release in mode III is not identical to that of mode II although it is usual to make the approximation The most probable hypothesis is that both these aspects play a part in the forecasting error. Mode III has not been very well analyzed and the errors concerning the prediction of the field of displacements of the panel may lead to an overestimate of the debonding load.

The debonding modes at the critical sites are given in Figure 21. It is clear that mode 3 is dominant at failure for both loading cases. It is worth recalling that, despite the apparent similarity in terms of bending, mode 1 was dominant in the debonding zone for the 7-point bending test. While the study by Falzon and Steven [START_REF] Falzon | Buckling mode transition in hat-stiffened composite panels loaded in uniaxial compression[END_REF] shows that debonding tends to take place in the "belly" of the buckles where opening is maximum and mode III is negligible, the present study shows that this is not necessarily the case for panels stiffened with omega stringers, which themselves have thin flanges. Moreover, this presence of mode III brings out the need to develop standardized, reliable characterization methods in order to be able to study the often neglected effects of this mode in greater detail. This analysis also shows that debonding follows a quasi-exponential increase in the energy release rates at the stringer/skin interfaces. It is therefore possible to use strongly reduced values for the calculations without engendering serious conservativeness on the postbuckling margin [START_REF] Bertolini | Contribution à l'analyse expérimentale et thèorique des ruptures de structures composites en postflambement par décollement de raidisseurs[END_REF]. A preliminary calculation assuming that mode 1 is lowered by 30% and modes II and III by 50% shows that the debonding criterion is satisfied at 411 kN for the shear case, i.e. at a loading only 12.4 % lower than the calculation performed with the mean values of the critical energy release rates. For the compression case, the debonding load is found at 133 kN, a reduction of only 8.2 %. It is thus possible to use strongly reduced values when making the calculations without causing undue conservativeness

-GENERAL CONCLUSION

The multiscale analysis of stiffener debonding proposed in the first part has been applied at the "element" and "panel" levels. At element scale, the study concerned the debonding of stringers in 7-point bending. From a numerical point of view, a global/local approach was applied and enabled calculation of the loads corresponding to local debonding. At this stage, the mixed Benzeggagh-Kenane criterion was validated together with the computing method.

Moreover, in a certification programme, the methodology was applied to stiffened panels under compression and shear. The prediction of debonding at this scale is consistent with the tests. The dominant debonding mode for the technology studied is mode III. This observation should encourage more fundamental studies of this mode in composites as the available data are very patchy. To sum up, a multi-level experimental and simulation approach has been validated for static loads. If the mass of composite structures is to be optimized, it would be desirable to lower the threshold for postbuckling below the limit loads. However, it would then become necessary to use the same multi-level approach for fatigue. From a modelling standpoint, the non-linear analysis of strengthened panels in postbuckling proves extremely costly in terms of computing time and, if a full fuselage is to be treated, it will be beneficial to develop intensive computing methods.

FIGURES

Fixed support (5)

Loading points (2)

  The global computation was non-linear geometric. The local model comprised a skin part and a flange part. The skin and flange were modelled by 3D shells of coincident meshes. The displacements coming from the global model were imposed at the boundaries of the local model. The principle of reinjection is given in Figure 12. The boundary nodes of the global model were recreated in the local model at their geometric locations (master nodes). The nodes of the 3D shell elements of a section of the local model (slave nodes) were controlled geometrically by the 5 d.o.f. coming from the nodes of the plate elements following the Kirchhoff-Love hypothesis and via rigid elements. The choice of the size of the local model is studied in[START_REF] Bertolini | Contribution à l'analyse expérimentale et thèorique des ruptures de structures composites en postflambement par décollement de raidisseurs[END_REF] on a virtual test patch. The optimum size needed to verify the St Venant assumption is about half a wavelength. Therefore, the problem is more defined by the time needed to build the local model than by the minimum size. If the analysis area is too small, too many local models are needed. The "virtual crack" is 0.5 mm long. The VCCT technique is used to determine the energy release rates and the Benzeggagh-Kenane criterion is applied. The local calculation is non-linear requiring two or three steps.

  margin. This point should enable the number of tests required to characterize the interfaces to be made considerably smaller and allow conservative pre-sizing in the early stages of a programme.

Figure 1 :

 1 Figure 1: Specimen overall geometry and location of the strain gauges.

Figure 2 :

 2 Figure 2: Principle of 7-point bending test (symmetrical case).

Figure 3 :

 3 Figure 3: Symmetrical 7-point bending tests.

Figure 4 :

 4 Figure 4: Antisymmetrical 7-point bending tests.

Figure 5 :

 5 Figure 5: Force/displacement curves for symmetrical test cases.

Figure 6 :Figure 7 :

 67 Figure 6: Debonded patterns (a) : symmetric case, (b) : antisymmetric case.

Figure 9 :

 9 Figure 9: Test/simulation correlation (symmetrical case): curve of force/displacement at supports.

Figure 10 :D é f o r m a t io n s S 2 /S 3 / E f f o rt m a c h ine - 5

 10235 Figure 10: Debonding location and computed pattern.

Figure 11 :

 11 Figure 11: Global/local approach

Figure 12 :Figure 14 :

 1214 Figure 12: Principle of global/local boundary element displacement injection.

Figure 15 :

 15 Figure 15: Failure mode in shear.

Figure 19 :

 19 Figure 19: Finite element result of the compression case.

Figure 20 :

 20 Figure 20: Finite element result of the shear case.

Table 1 :

 1 TABLES Reference and global/local model comparison.

			Symmetric case	Antisymmetric case
		Reference model Global/local model Reference model Global/local model
	Loading point
	displacement	0,56	0,56	1	1
	(Normalized)
	Load (Normalized)	753	860	1000	1098
	Maximum deflection (Normalized)	0,8	0,78	1,64	1,61
	G 1 (J/m 2 )	223	216	272	264
	G 2 (J/m 2 )	109	92	123	114
	B-K Criterion	0,69	0,67	0,87	0,84

Master nodes

Inner radius pre-crack Skin/flange pre-crack