L Doyen
email: luc.doyen@u-bordeaux.fr

C Armstrong

S Baumgärtner

C Béné

Abdoul Fabian Blanchard

Ahad Cisse

R Cooper

L Dutra

A Eide

Debora M De Freitas

P.-Y Hardy

From no whinge scenarios to viability tree

Keywords: Minimal whinge, safe operating space, scenarios, ecological economics, modeling, sustainability, viability kernel

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

The need for no-whinge spaces Achieving the 'Millennium Development Goals' (United Nations, 2012) and operationalizing a sustainable development reconciling human well-being, food security, economic performance, the production of ecosystem services with biodiversity conservation and limited pollution is among the greatest challenges of the century, especially in the face of global changes including climate change and the world demographic transition [START_REF] Godfray | Food Security: The Challenge of Feeding 9 Billion People[END_REF]. The creations of the IPCC and IPBES (International Panel for Biodiversity and Ecosystem Services) at the interface between decision support and scientific knowledge are in direct line with these concerns. In that respect, the production and assessment of model-based scenarios and policies such as those examined in [START_REF] Leadley | Biodiversity Scenarios: Projections of 21st century change in biodiversity and associated ecosystem services[END_REF] constitute major scientific challenges.

Regarding biodiversity management, the heterogeneity of stakeholders contributes to the complexity in the evaluation and design of public decision making policies, management and scenarios [START_REF] Doyen | Ecological-economic modeling for the sustainable management of biodiversity[END_REF]. Stakeholders including fishers, farmers, hunters, conservation and regulation agencies, consumers, tourists or NGO's can indeed differ in their interpretations of the world, preferences, strategies, levels of information and inputs in the dynamics of systems. In this context, what may be more readily available to regulating agencies and decision makers is information about what is acceptable / non-acceptable to the various stakeholders. In other words, a realistic goal for management may often be to avoid undesirable states that lead to non-compliance or rejection from stakeholders. Such a goal function has been called the minimum sustainable whinge (MSW) in [START_REF] Pope | Fisheries resource management theory and practice[END_REF]. Interestingly the word 'sustainable' is used in MSW because piecemeal management measures to silence short term complaints (for example, about subsidy) may lead to worse long term complaints. This MSW can be illustrated for fisheries with objectives of current yield, profit, employment and biodiversity conservation where:

• Profits are seen as the difference between revenues derived from fishing and cost of efforts (days at sea) and are required to remain non negative or above current values. Stakeholders interested in such a goal are mainly producers (fisherfolks).

• Employment concerns local people involved in processing and trading sectors, as well as the local politicians and decision makers.

• Catches are required to stay above the volumes required for food security. Stakeholders concerned here are typically consumers.

• Biodiversity metrics are required to be not too altered. Related stakeholders include environmental NGOs and conservation agencies.

To meet the challenge of sustainable development, Rockstrom et al. (2009) also proposed a framework based on boundaries that define the safe operating space (SOS) for humanity, associated with the planet's biophysical subsystems or processes. The framework relies on the idea that most environmental systems react in a nonlinear, often abrupt, way, and are particularly sensitive around tipping levels of certain key variables. If these thresholds are crossed, then these systems could shift into a deleterious state or collapse with disastrous consequences for humans. By contrast, as long as the thresholds are not crossed, humanity has the freedom to pursue long-term social and economic development1 . Similarly the concept of Safe Minimum Standards (SMS) as in [START_REF] Ciriacy-Wantrup | Resource Conservation[END_REF] relates to tipping thresholds and risky areas. Such a geometrical approach has also strong connections with the so-called Goldilocks zone used in astro-physics as in [START_REF] Powell | A Parade of New Planets[END_REF] where the emphasis is put on feasibility through acceptability windows in a dynamic context instead of optimality or stationarity. Tolerable Windows Approach (TWA) as introduced in Petschel- [START_REF] Petschel-Held | The Tolerable Windows Approach: Theoretical and Methodological Foundations[END_REF] also relies on safe boundaries and feasibility regions for the assessment of climate policies.

We here first argue that the MSW, SOS, SMS, TWA or Goldilocks, all constitute instances of a relevant approach to ecosystem and biodiversity management for sustainability, if they also account for the interactions between ecological and economic systems. In particular, Rockstrom et al. (2009) define a SOS for humanity. Yet, the way how humanity moves in this operating space is to a large extent determined by policymakers and economic leaders in industrialized countries. To transform the concepts of MSW, SOS, SMS, TWA or Goldilocks into a relevant and comprehensive approach for the sustainable management of ecosystem and biodiversity, boundaries related to human needs have to be included as well. For instance, food security and poverty lines combined with demographic, economic growth and technological progress must be considered when designing policies for sustainable development and the management of socio-ecosystems. This is the idea underlying the 'Oxfam doughnut' [START_REF] Raworth | A Safe and Just Space for Humanity: Can We Live Within the Doughnut? Oxfam Discussion Paper[END_REF], with an explicit focus on the social justice requirements underpinning sustainability used at more local or regional scale [START_REF] Dearing | Safe and just operating spaces for regional social-ecological systems[END_REF]. Moreover, interactions and feedbacks between ecological and economic systems, which often are non-linear, need to be taken into account.

We thus propose to rely on ecological economic approaches, modeling, dynamics and constraints to operationalize sustainable development and MSW. In particular, we argue that the use of viability kernels [START_REF] Aubin | A survey of viability theory[END_REF][START_REF] Béné | A viability analysis for a bio-economic model[END_REF][START_REF] Baumgartner | Ecological-economic viability as a criterion of strong sustainability under uncertainty[END_REF][START_REF] Schuhbauer | Economic viability and small-scale fisheries -A review[END_REF] makes possible the delineation of 'no whinge spaces' for sustainable management, policies and scenarios of ecological-economic systems.

Ecological-economic constraints and interdependencies

In order to sustainably manage ecosystems, an ecological-economic viewpoint bringing together ecological considerations and human well-being is needed. The underlying public policies must be part of a general prospect of sustainable development, reconciling environmental, economic and social requirements with a perspective of equity within and between generations. Bio-economics or ecological economics can contribute to such an objective [START_REF] Clark | Mathematical Bioeconomics[END_REF][START_REF] Costanza | The value of the world's ecosystem services and natural capital[END_REF][START_REF] Doyen | Ecological-economic modeling for the sustainable management of biodiversity[END_REF]. The aim of these disciplines is basically to investigate the control of the environmental-economic systems at play by analyzing and designing decision strategies, policies and scenarios for biodiversity together with the commodities, provisioning services and other ecosystem services it provides.

Bio-economic boundaries and whinge thresholds are tightly coupled and concentrating effort on any one of them in isolation is not sufficient because its dynamics is likely to affect the others. If one constraint is violated, then other boundaries are also under serious risk. For instance, significant land-use changes in developing countries influence water resource and local consumption relying on land-use such as farming. The climate-change boundary of SOS depends on staying on the safe side of the freshwater, land, aerosol, nitrogen-phosphorus, ocean and stratospheric thresholds. Not complying with the nitrogenphosphorus constraint can affect negatively some marine ecosystems, potentially reducing their capacity to absorb CO2 and thus altering the climate boundary but also their ability to provide food. In other words, there is a need to draw on a more integrated, systemic and holistic approach to cope with these ecological-economic constraints and dynamics and to anticipate whinges related to the violation of bioeconomic boundaries due to interrelated dynamic processes and feedbacks in the joint ecological-economic system. Uncertainties underlying ecological-economic system dynamics exacerbates and complexifies such a challenge.

Viability kernels

Viability modeling is now recognized by a growing number of scientists and stakeholders as a relevant framework for sustainability and the management of ecological-economic risks and vulnerabilities [START_REF] Jennings | Indicators to support an ecosystem approach to fisheries[END_REF][START_REF] Cury | Viability theory for an ecosystem approach to fisheries[END_REF][START_REF] Thébaud | Building ecological-economic models and scenarios of marine resource systems: workshop report[END_REF][START_REF] Krawczyk | Computation of viability kernels: A case study of by-catch fisheries[END_REF][START_REF] Schuhbauer | Economic viability and small-scale fisheries -A review[END_REF][START_REF] Doyen | Ecoviability for Ecosystem Based Fisheries Management[END_REF][START_REF] Oubraham | A survey of applications of viability theory to the sustainable exploitation of renewable resources[END_REF]. In the context of dynamic systems, the aim of the viability approach is to explore states and controls that ensure the 'good health' and safety of the system [START_REF] Béné | A viability analysis for a bio-economic model[END_REF] over time. By identifying the viability conditions that allow various constraints to be satisfied throughout time, considering both present and future states of a dynamic system, the viability approach conveys information on sustainability and especially strong sustainability [START_REF] Baumgartner | Ecological-economic viability as a criterion of strong sustainability under uncertainty[END_REF]. It accounts for dynamic complexities, uncertainties, risks and multiple sustainability objectives. In that sense, links with the MSW, TWA and SOS are very strong. The approach has already been successfully applied to socio-ecosystem management in several contexts including land-use issues [START_REF] Mouysset | Co-viability of farmland biodiversity and agriculture[END_REF], biodiversity valuation [START_REF] Béné | Contribution values of biodiversity to ecosystem economic and ecological performances: a viability perspective[END_REF] or fisheries management [START_REF] Maynou | Coviability analysis of Western Mediterranean fisheries under MSY scenarios for 2020[END_REF][START_REF] Gourguet | The Cost of Co-viability in the Australian Northern Prawn Fishery[END_REF][START_REF] Schuhbauer | Economic viability and small-scale fisheries -A review[END_REF][START_REF] Doyen | Ecoviability for Ecosystem Based Fisheries Management[END_REF]. In relation to food security, [START_REF] Hardy | Food security-environment conservation nexus: a case study of Solomon Islands' small-scale fisheries[END_REF] or [START_REF] Cissé | Ecoviability for small-scale fisheries in the context of food security constraints[END_REF] provide useful bio-economic insights in the context of developing countries under strong demographic pressure.

The viability kernel provides a solid and rigorous mathematical tool to address viability issues [START_REF] Aubin | A survey of viability theory[END_REF]De Lara & Doyen, 2008). In short, viability kernels are 'spaces' (states of the socio-ecosystem) from which starts at least one policy trajectory satisfying the whole set of constraints throughout time. The viability kernel is both a conceptual tool to represent sustainability and a computing tool for the applications relying on dynamic programming methods. The viability kernel gives major insights into MSW as illustrated by Figure 1. In that very stylized and simplified bio-economic example inspired by hunting or fishing, a renewable resource is harvested to provide food security. Assume that the initial whinge constraints are lower bounds such as biodiversity minimal level x lim and guaranteed consumption c lim . Viability kernel as displayed in blue in the right-hand side of Figure 1 is characterized by an upper boundary related to harvesting quotas (the upper curve of the viability kernel) and a more demanding resource ceiling x pa larger2 than the initial conservation threshold x lim . This is explained in more mathematical terms in the the Box A.

=⇒

Figure 1: Bio-economic whinge and viability kernel. Left-hand side: In red, the whinge space: In the present case, (for sake of simplicity) we only represent two constraints: the ecological and food security ones (the economic constraint is omitted). These constraints are indicated on the diagram by the two dotted lines and the associated two thresholds: x lim and c lim . Below these two thresholds the constraints are violated, the system is in crisis and whinges occur. In yellow, viability occurs in a static way. Right-hand side: In blue the viability kernel represents the set of initial conditions of the system which ensures that the controlled dynamics (illustrated by the system trajectories) will satisfy the viability constraints at any time. Above the viability kernel boundary, initial conditions are viable at t = 0 but the dynamics of the system is such that future crisis and whinges can not be avoided. Only within the viability kernel is the system without whinge and will remain so at any time in the future.

Box A: A stylized bio-economic model

A controlled dynamic system under constraints. We consider a simple controlled dynamic system under constraints including the following ingredients:

• A non linear population dynamics with harvest in discrete time: a stock (biomass, abundance of a species) x(t) evolves according to renewable mechanisms f (typically Verlhust growth) and catches c(t)

x(t + 1) = f ( x(t) ) -c(t), t = 0, 1, . . . ( 1
)
The control of the system is the catch c(t) while x(t) stands for the state of the system.

• A first no whinge constraint relates to ecological conservation with a safe stock threshold:

x(t) ≥ xlim, t = 0, 1, . . . ( 2
)
• A second no whinge constraint relates to food security with a basic need threshold:

c(t) ≥ clim, t = 0, 1, . . . ( 3
)
Viability kernel or no whinge kernel. A more demanding space is provided by the following set of states from starts at least one harvest trajectory c(.) avoiding whinge situations over time in the sense of constraints ( 2), (3): Consequently, viable catches lie within the corridor

Viab = { x0 ∃ c(t)
[clim, cpa(x)]
where the precautionary quota cpa(x) is defined by cpa(x) = f (x) -f (xpa) + clim. Flexibility and adaptiveness emerge from this set of viable controls as several no-whinge options can be selected among the interval [clim, cpa(x)].

Minimal whinge kernel: from a mine pit to a tree of viability

More generally, considering any social-ecological system impacted by several anthropogenic pressures and lower tolerable boundaries on both the states and controls of these socio-ecosystems, we conjecture that the viability kernels look like in Figure 2. Such an outcome expands the stylized bio-economic model associated with Figure 1. The mathematical model underpinning such a figure is portrayed in Box B. This geometric shape of kernels is suggested by different applied modeling works as in [START_REF] Hardy | Food security-environment conservation nexus: a case study of Solomon Islands' small-scale fisheries[END_REF][START_REF] Mouysset | Co-viability of farmland biodiversity and agriculture[END_REF][START_REF] Cissé | Ecoviability for small-scale fisheries in the context of food security constraints[END_REF][START_REF] Gourguet | The Cost of Co-viability in the Australian Northern Prawn Fishery[END_REF]. We termed such domains no whinge kernels for several reasons. First, they are based on feasibility boundaries in line with MSW principle. Second, we called them kernels because they relate to the computation of viability kernels. On the left hand side of the figure 2, the yellow zone depicts combinations of states, controls and indicators remaining above several whinge thresholds including food security, guaranteed profitability or conservation of biodiversity and other ecosystem services (typically cultural or regulating). This zone can also be termed the no whinge space. The consistency between these whinge constraints and the dynamics at play restricts the no whinge domain to the blue area where upper bounds emerge as in Figure 1. This zone related to the viability kernel can also be termed the no whinge kernel because it exemplifies the need for anticipating and avoiding whinge crisis. The shape of a 'tree of life' which emerges is well aligned with the objectives of sustainability scenarios and policies for biodiversity and ecosystem services. This is why we name them viability trees.

Interestingly, the general MSW shape identified for these spaces can be decomposed in three distinct areas. A first zone in red shaped here as a circle3 exemplifies the need to remain above boundaries to avoid ecological-economic whinges. Such a 'pit mine' indeed captures crisis situations where at least one bio-economic requirement, including basic protein needs or minimal wetlands or forests, or necessary biodiversity is violated. But as said previously, it is not enough to stay above the unsustainable (red) pit to remain safe over time since upper boundaries are needed as upper limits for consumption of renewable resources illustrated in Figure 1. In that case, the TWA metaphor strongly applies. By contrast, the other branch without upper limits and related to biodiversity illustrates the fact that some states do not need to be bounded from both side in the context of sustainable development. Examples of such a result are biodiversity levels from the ecological side which can potentially rise without altering the system sustainability as a whole. To go further, the unbounded potential of biodiversity state captures the idea that the renewable nature underlying biodiversity is a pivotal element of sustainability of the whole socio-ecosystem by providing renewable resources, services, welfare and well-being over time. Such a configuration stresses a significant difference with the planetary space approach, the 'Oxfam doughnut' or TWA where upper limits and ceilings are major elements. In contrast our no whinge kernels and viability trees make possible more open spaces and more room for manoeuvre.

Whinge

The selection of no whinge thresholds

A key challenge of the minimal whinge approach involves turning broad conceptual objectives into quantifiable and measurable management constraints or targets, against which the performance of management strategies and scenarios can be assessed [START_REF] Thébaud | Viability trade-offs in the evaluation of strategies to manage recreational fishing in a marine park[END_REF]. This specification of operational sustainability objectives requires three elements: performance indicators that specify the quantities of interest, targets for the performance indicators, and measures of tolerance or acceptance that the indicator must achieve, usually specified as probabilities. However, given the uncertainty typical of most ecosystem management problems, and the diversity of stakeholders usually encountered in resource management systems, identifying performance indicators, targets and tolerance levels often proves a challenge in itself. Stakeholders may be unsure about which indicators, thresholds and tolerance levels should be retained, or they may disagree with respect to what these should be.

We propose here to use levels of status quo or 'Business as Usual' as indicators to specify the initial whinge boundaries and determine the 'mine pit' to avoid. Such an approach has already been applied in several viability works including [START_REF] Mouysset | Co-viability of farmland biodiversity and agriculture[END_REF]; [START_REF] Gourguet | The Cost of Co-viability in the Australian Northern Prawn Fishery[END_REF]; [START_REF] Hardy | Food security-environment conservation nexus: a case study of Solomon Islands' small-scale fisheries[END_REF]; [START_REF] Cissé | Ecoviability for small-scale fisheries in the context of food security constraints[END_REF]. Such a framework to design MSW thresholds has strong connections with no-regret strategy as introduced in [START_REF] Bell | Regret in decision making under uncertainty[END_REF]. The various metrics used in the viability works previously mentioned combine indicators of biodiversity, catches and profits. Regarding profitability, the constraint has sometimes been relaxed to simply include positive rents. With regards to biological constraints, thresholds such as the ICES limits for fisheries can also be used. However it is clear that the governance of the socio-ecosystems at play requires flexibility and bargaining space between stakeholders to specify the whinge limits. We here argue that the flexibility should rely on tolerance values such the level of risks stakeholders are ready to cope with [START_REF] Thébaud | Viability trade-offs in the evaluation of strategies to manage recreational fishing in a marine park[END_REF].

Box B: A general model

A multi-state multi-pressure dynamic model under uncertainty. The social ecological system is described by a set of n stocks impacted by m distinct agents. The n stocks whose states at time t are denoted by xi(t) are governed by the following controlled and uncertain dynamic equations:

xi(t + 1) = fi ( x(t), c(t), ω(t) ) , (4)
for initial time t = t0 to temporal horizon t = T . The global state x(t) representing the ecosystem state is the vector of stocks x(t) = (x1(t), . . . , xn(t)). The vector c(t) = (c1(t), . . . , cm(t)) is the control of the system. The variables ω(t) = (ω1(t), . . . , ωp(t)) represent the uncertainties (stochasticities) affecting the dynamics of the system. The functions fi may account for various complexities including inter-specific competition or habitat impact.

The no whinge constraints. The viability approach focuses on the consistency of controlled dynamics of the system with respect to constraints capturing a no whinge space. These no whinge constraints can involve ecological thresholds as in population viability analysis (PVA). Economics constraints can also be integrated thus allowing for multi-criteria and bio-economic analyses as follows: (5)

       Biod ( x(
The total harvest h(t) stems from catch of stocks i by agents j and depend on actions through production function

hij (t) = hj ( xi(t), cj (t), ω(t) ) . ( 6
)
The food security constraint in (5) refers to some basic need standard denoted by h lim (t) which may be time-dependent typically because of demographic growth. The economic profit πj (x(t), c(t), ω(t)) of each agent j is computed as the difference between the revenues derived from provisioning services hj (t) and operating costs associated with the decision cj (t); Note that these values are assumed to be random because of market price and cost (e.g. fuel) uncertainties. Ecological indicators Biod(x(t), ω(t)) correspond to biodiversity or biological metrics. They can also be uncertain because of stock measurement errors. In that context the threshold Biod lim can stand for an ecological tipping point.

Similarly ecosystem services indicators Serv(x(t), ω(t)) encompass regulating and cultural services induced by the ecosystem and the states of the system.

In contexts where uncertainties have a probabilistic nature, no whinge situations and bio-economic viability can be defined as the fulfillment of constraints with a high enough probability (De Lara & Doyen, 2008); namely

P ( Constraints (5) are fulfilled for t = t0, .., T ) > β (7)
where β corresponds to a confidence rate (99%, …) while probability P is computed with respect to uncertainty ω.

The no whinge kernel. A key mathematical tool is a again provided by the viability kernel. The viability kernel corresponds to a safe space within the initial set of constraints where the system needs to remain to be viable and to remain so in the future. It exemplifies the need for anticipating whinge crisis. In our stochastic context, this reads

Viab β = { x0
there exist a strategy of feedback controls c(t, x)) such that dynamics (4) and constraints (7) hold true

} . ( 8
)

Conclusion

This paper stresses the interest of both the no whinge approach and viability modeling for the sustainable management of ecosystems and socio-ecosystems. First, we argue that the no whinge approach (or minimal sustainable whinge) based on thresholds of ecological and socio-economic acceptability is in line with numerous approaches for operationalizing sustainability including SOS, SMS and TWA. Second we point out that the use of viability kernels makes possible the delineation of no whinge policies and scenarios accounting for the ecological-economic interactions and dynamics. The trees of viability derived from the viability kernel in our bio-economic context capture the idea that a bounded use of the different ecosystem services mixed with the unbounded renewal potential of biodiversity are pivotal ingredients for the sustainability of socio-ecosystems and the identification of no whinge policies avoiding the rejection of these policies from the different stakeholders involved. Of interest are also the room for manoeuvre, the adaptiveness and the open shape underlying the viability tree as opposed to the planetary space approach, the 'Oxfam doughnut' or TWA which are characterized by more closed and limited forms.

Our approach rests on three domains of scientific advances in the fields of global change, sustainability and conservation. The first questions how human action, decision making and policies affect ecosystems and natural resources. The second aims at understanding ecosystem functioning, dynamics and processes driving ecosystem services for human well-being and development. The third field of enquiry is research into resilience and viable regulations of socio-ecosystems, emphasizing standards, constraints, guidelines, flexibility, governance and room for manoeuvre. These three domains constitute significant features of the ecological economics and conservation research agenda.

In particular, urgent research regarding ecological economics, no whinge and viability approaches relates to both applied and theoretical issues. From the applied viewpoint, the implementation on field with stakeholders using participatory approaches for fisheries, farming, forestry or water management is a major challenge. More specifically, the test of the 'viability tree hypothesis' put forward in the paper and its title should constitute a generic and transversal perspective across these different case studies. From the theoretical viewpoint, the development of the ecosystem approach accounting for complex ecosystem dynamics including spatial and/or network components is a main challenge. Operationalizing sustainability and resilience by developing methods eliciting endogenous viability standards in a context of uncertainty including shocks, extreme events, stochasticities or ambiguities is also a key challenge. Which governance for the sustainability and resilience of ecosystems and socio-ecosystems is also a crucial task given the high heterogeneity of stakeholders involved in the environmental systems and the lack of coordination underlying the so-called tragedy of the commons ? In that respect, linking game theory, multi-agent methods with viability framework is a pivotal methodological task.

  and x(t) starting from x0 satisfying dynamics (1) and constraints (2), (3) ∀t ∈ N } Two contrasted cases. Assume that the population dynamics f satisfies the following conditions: fx > 0; f (0) = 0; f (K) = K; fxx < 0. We can distinguish between two contrasted configurations for the viability kernel as follows:• No viability: the viability kernel is empty: Viab = ∅ if the basic need clim is too demanding in the sense that clim > cMSY where cMSY is the Maximum Sustainable Yield defined by cMSY = max x≥0 f (x) -x • Partial viability: the viability kernel is nonempty when the basic need clim is not too demanding in the sense that 0 ≤ clim ≤ MSY. In that case the viability kernel reads Viab = [xpa, +∞[ where the precautionary stock threshold xpa is defined by xpa = min ( x, x ≥ clim, x ≥ xlim f (x) -clim = x ) No whinge catches. As proved in De Lara & Doyen (2008), determining viable controls consists in maintaining the state x within the viability kernel Viab. For the previous model, this corresponds to the constraint f (x) -c ≥ xpa.

Figure 2 :

 2 Figure2: From the no whinge space (in yellow) to the no whinge kernel (in blue) taking the form of a tree of life or tree of viability.Most indicators or states (typically consumption on top) need to evolve within a safe corridor or window while biodiversity state just needs to have a viability floor.

At this stage, it may be worth distinguishing between global and local/regional systems or scales. The existence of thresholds or tipping points, so that there is a desirable domain of the system and an undesirable one, is most obvious and well evidenced for a number of small-scale, local systems. At the global scale, it is a hypothesis that Planet Earth exhibits this kind of threshold-behavior, but evidence is much weaker. Insofar as we rather address local systems here, e.g. fisheries, we are on the scientifically safe side when using such arguments.

The notation pa has links with the precautionary limits (Bpa) of the ICES approach for fisheries.

We implicitly assume that the level of floor thresholds are normalized.

Acknowledgments

This work has been carried out with the financial support of the Belmont Forum through the SEAV-IEW network (ANR-14-JPF1-0003). The role of the research projects ACROSS (ANR-14-CE03-0001), NAVIRE (Cluster of Excellence COTE, ANR-10-LABX-45) and OYAMAR (FEDER) as well as the support from IHP (Institut Henri Poincaré) in Paris during the trimester "mathematics of bio-economics" in the framework of Mathematics of Planet Earth 2013 initiative were decisive.


Laszlo Laszlo S Toth
email: .toth@univ-lorraine.fr

Viet Q Vu

Satyaveer Singh Dhinwal

Yajun Zhao

Roxane Massion

Cai Chen

Casey F Davis

Terry C Lowe

The Mechanics of High Pressure Compressive Shearing with Application to ARMCO ® Steel

Keywords: High Pressure Compressive Shearing, Shear testing, Severe Plastic Deformation, Plasticity, ARMCO ® Steel, Ultra Fine Grain Structure

In High Pressure Shearing (HPS) a flat sample is subjected to large shear strain by displacing one of its flat surfaces with respect to the other under high compressive force. The process is driven by non-sliding friction provided by the large compressive stress. The HPS process was first proposed by Fujioka and Horita [Materials transactions, 50 (2009) 930-933], and was applied to aluminum strips; upscaling was also developed. In the present work we introduce a modified version of HPS where the compression strain is relevant, so the new process is called High Pressure Compressive Shearing (HPCS). In HPCS, the experiments are conducted similarly to HPS but with the addition of a confining-pressure and more compression strain to increase the shear strain. This paper presents the mechanical analysis of stress and strain states during HPCS and demonstrates in situ measurements of stress-strain relations in ARMCO ® steel subject to HPCS at room temperature up to an equivalent strain of 33.3. The strain hardening characteristics, the texture and the microstructure were analyzed. It has been shown that compression during HPCS results in a non-hydrostatic stress state and that ultimate steady state of grain fragmentation can be readily reached by HPCS.

Introduction

The application of very large strains to metals can enhance their mechanical properties, especially increasing their yield stress. A large number of Severe Plastic Deformation (SPD) processes have been developed for this purpose over the past four decades [START_REF] Estrin | Extreme grain refinement by severe plastic deformation: A wealth of challenging science[END_REF]. The main characteristics of SPD processes are the achievement of extremely high plastic strain with the assistance of high hydrostatic pressure and little or no change in the sample dimensions. After SPD, the deformed metal can display ultrafine-grained microstructures, in the submicron or even nanostructured range (for alloyed metals). Nevertheless, the amount of strain needed to reach the smallest possible grain size is quite large, and can be estimated to be between equivalent strains of 10 and 50.

The most widely studied SPD processes are ECAP (Equal Channel Angular Pressing [START_REF] Segal | Materials processing by simple shear[END_REF]), HPT (High Pressure Torsion [START_REF] Zhilyaev | Using high-pressure torsion for metal processing: Fundamentals and applications[END_REF]) and ARB (Accumulated Roll Bonding [START_REF] Saito | Novel ultra-high straining process for bulk materials-development of the accumulative roll-bonding (ARB) process[END_REF]). ECAP is suitable for large size specimens but many passes are needed to reach the steady state grain size regime. HPT can readily reach the steady state in a single step, however, the samples are very small thin disks.

Unfortunately, when the thickness of the disk is increased, strain localization can readily happen in its middle section [START_REF] Panda | Analysis of heterogeneities in strain and microstructure in aluminum alloy and magnesium processed by high-pressure torsion[END_REF]. Finally, ARB is a test which can be readily industrialized, nevertheless, the process is quite elaborate and requires at least 7 ARB cycles to approach the steady state. A variant of the HPT testing was proposed in 2009, called HPTT (High Pressure Tube Twisting [START_REF] Tóth | Severe plastic deformation of metals by high-pressure tube twisting[END_REF]), which is an SPD technique suitable for deforming short tubes. HPTT permits deforming relatively large samples, larger than in HPT. Its only disadvantage is that the length of the tube is limited; the tube length should not normally exceed the diametric dimension of the tube.

Interestingly, both HPT and HPTT produce samples in which there is a significant gradient in strain: in HPT the strain is proportional to the distance from the central rotational axis, while in HPTT the strain is about four times larger at the internal surface of the tube compared to the external diameter. The HPTT strain gradient was successfully modeled in [START_REF] Pougis | Stress and strain gradients in high-pressure tube twisting[END_REF].

Still another variant of HPT was proposed by Fujioka and Horita [START_REF] Fujioka | Development of high-pressure sliding process for microstructural refinement of rectangular metallic sheets[END_REF] which consists in simply shearing a strip parallel to its flat surface. They named the process High Pressure Sliding (HPS).

Numerous studies have been carried out to explore the capacity of this process [START_REF] Lee | Strengthening of AA7075 alloy by processing with high-pressure sliding (HPS) and subsequent aging[END_REF][START_REF] Masuda | Extra strengthening and superplasticity of ultrafine-grained A2024 alloy produced by high-pressure sliding[END_REF][START_REF] Takizawa | Incremental feeding high-pressure sliding for grain refinement of large-scale sheets: application to Inconel 718[END_REF][START_REF] Takizawa | Superplasticity of Inconel 718 after processing by high-pressure sliding (HPS)[END_REF][START_REF] Tazoea | Application of high-pressure sliding for grain refinement of Al and Mg alloys[END_REF][START_REF] Horita | Severe plastic deformation under high pressure for production of superplastic materials[END_REF][START_REF] Masuda | High-pressure sliding using rod samples for grain refinement and superplasticity in Al and Mg alloys[END_REF] and HPS was scaled up to deform samples with dimensions of 100 mm long and 10 mm wide [START_REF] Takizawa | Scaling up of high-pressure sliding (HPS) for grain refinement and superplasticity[END_REF].

Inspired by the HPS technique, this paper proposes a variant of the HPS process, called HPCS (High Pressure Compressive Shearing), illustrated in Fig. 1. In the present realization, we employed an existing three-axes ECAP machine to impart linear shear. Two samples (with the same dimensions) are positioned in the horizontal channel on the two opposite surfaces of a plunger. In the vertical channel a punch (Punch 3) applies a large compressive force on the ensemble. The two horizontal punches displace the plunger horizontally; Punch 1 is the driving punch applying force FD while Punch 2 exerts a confining force FBP. The two samples are sheared by the friction forces Ft. To enhance the friction and avoid slipping (or 'sliding'), both the sample surfaces and the other surfaces in contact with the samples are sand-blasted before testing. The purpose of applying a confining force is to increase the hydrostatic stress on the contact surfaces between the samples and the die, to avoid sliding. The back-pressure also helps stabilize the process (e.g. avoiding fracture of the plunger). Fig. 1a shows the sample shapes that one would ideally expect if only simple shear takes place, with no compression. So the shapes in the schematic in Figure 1a are oblique-angled parallelograms after shearing. However, during deformation the large compression force progressively mashes the samples, so they become thinner and elongate along the shear direction. So finally the samples take the form as shown in Fig. 1b. Apart from some differences at the left ends of the samples, the shapes of the two samples are the same after testing. There appear some small variations in their thickness, which is due to some small misalignments of the plunger.

An important element of this HPCS setup is the possibility to determine the stress and strain in situ by measuring the displacements and the forces directly during the processing. In this work, the mechanics of HPCS are analyzed and the equations necessary to convert the forces into stress and the displacements into strain are presented. It is generally difficult or even impossible to measure the material strength during SPD processing because the sample is fully constrained.

HPTT is a process where it is readily possible [START_REF] Pougis | Stress and strain gradients in high-pressure tube twisting[END_REF]. In HPT, Chinh et al. [START_REF] Chinh | A new constitutive relationship for the homogeneous deformation of metals over a wide range of strain[END_REF] and Wetscher et al. [START_REF] Wetscher | Strain hardening during high pressure torsion deformation[END_REF] presented empirical approaches to derive the flow stress variations during HPT. Another difficulty is the proper calculation of the magnitude of the hydrostatic stress produced by the normal pressure. It was shown for HPTT that the axial pressure produces a fully hydrostatic stress state [START_REF] Pougis | Stress and strain gradients in high-pressure tube twisting[END_REF]. For HPT, it can only be valid if the test is done under full constraints conditions.

In the present work, an analytic formula is obtained for the hydrostatic stress state for HPCS.

The mechanics of HPCS

In order to set up the stress and strain equations, first we need to examine the experimental shape changes during HPCS. Fig. 2 shows schematically how the sample changes its form during shearing. We start with a rectangular shape with initial dimensions of 0 t , 0 l and 0 w , for the thickness, length and width, respectively. The dimension 0 w does not change during shearing because the sample is fully constrained in the width direction between the walls of the channel.

However, the thickness of the sample decreases during shearing due to a combined effect of the shear and the large normal pressure. Indeed, the deformed sample has a nearly rectangular shape, as seen in the example in Fig. 1b. In the following first we analyze the strain, and then the stress state due to HPCS. The reference system is defined in Fig. 2.

Strain state:

During deformation, the sample dimensions 0 t and 0 l change into t and l due to the applied displacement s. At any time, the length of the sample is:

o l l s = + . (1)
The thickness of the sample is also known from the volume constancy condition; we obtain:

0 0 l t t l = . ( 2
)
The strain state is plane strain with the following strain rate tensor:

2 0 2 0 0 0 0 ε γ ε γ ε -     =       , ( 3
)
where ε& is the thickness strain rate (it is negative), and γ& is the shear strain rate. The accumulated normal strain is:

0 0 0 0 0 ln ln t t l dt t d t t l s ε ε ε = = = = + ∫ ∫ . ( 4
)
The increment of shear strain is defined as:

ds d t γ = . ( 5
)
Using Eqs. (1-2), we get:

0 0 0 ( ). . l s ds d t l γ + = . ( 6
)
The accumulated shear strain is:

2 2 0 0 0 0 0 1 ( ) 2. . d l s l t l γ γ γ   = = + -   ∫ . ( 7
)
For the strain rates, we get them from Eqs. ( 4) and [START_REF] Pougis | Stress and strain gradients in high-pressure tube twisting[END_REF]:

0 s l s ε = -+ & & , 0 0 0 1 .( ). . l s s t l γ = + & & . (8a,b)
The equivalent strain rate is defined according to von Mises:

. 3

eq ij ij ε ε ε = & & & . ( 9
)
Using Eqs. (8a,b) as well as ( 4) and ( 6) in Eq. ( 9) we obtain:

2 0 2 2 2 0 0 0 ( ) 4 3.( ) 3. . eq l s s l s t l ε + = + + & & . ( 10
)
After integration, we get the following formula for the equivalent strain:

( )

1 1 ln ln 3 2 3 eq a b a c b c a b a c a ε  - - = -+ -   + +   , (11)

Stress state:

For the present experimental setup (Fig. 1), the stress tensor is defined as:

11 22 33 0 0 0 0 σ τ σ τ σ σ     =       . ( 12
)
The 2 2 σ component is obtained directly from the experimental data. It is defined by: ( )

22 0 / . /[ .( )] y y F w l F w l s σ = = + . ( 13
)
The shear stress is calculated from the difference in horizontal forces applied by the pressing and confining punches: giving the net tangential force x F acting on the individual samples:

0 0 0 0 .( ) 2. .( ) x D BP F F F w l s w l s τ - = = + + . ( 14
)
There are still two other components of the stress tensor that need to be determined: 11 σ and 33 σ .

They can be obtained with the help of an assumed yield function and the associated flow rule.

For the yield function ( ( )

f σ ), the
f σ σ σ σ σ σ σ τ σ   = - + - + - + =   . ( 15
)
The application of the flow rule provides the following relations:

11 11 22 33 11 11 ( ) ( ) f ε λ λ σ σ λ σ σ ε σ ∂ = = - - - = - ∂ (16) 22 11 22 22 33 22 ( ) ( ) f ε λ λ σ σ λ σ σ ε σ ∂ = = - - + - = ∂ (17) 33 22 33 33 11 33 ( ) ( ) 0 f ε λ λ σ σ λ σ σ σ ∂ = = - - + - = ∂ .6 2 f γ ε ε λ λ τ τ ∂ = = = = ∂ (19)
From Eq. ( 18), we obtain:

11 22 33 2 σ σ σ + = . ( 20
)
Eqs. ( 16) and ( 17) provide:

11 22 2 3 ε σ σ λ = - & & . ( 21
)
Eq. ( 19) defines the value of λ & :

12 γ λ τ = & & . ( 22
)
In this way, we obtain the following formulas for the two initially unknown normal stress components: 11 22 8 .

τ ε σ σ γ

= - & & , 33 22 4 . τ ε σ σ γ = - & & . (23a,b)
Substitution of the stress components into the flow function (Eq. 15) provides the von-Mises equivalent stress:

2 . 3 48 eq ε σ τ γ   = +     & & . ( 24
)
Using this formula and the equivalent strain from Eq. ( 11), a stress-strain curve can be constructed.

The hydrostatic stress:

Knowing all normal components of the stress tensor, it is possible to obtain the hydrostatic stress:

11 22 33 22 4 3 h σ σ σ ε σ σ τ γ + + = = - & & . ( 24
)
This formula shows that the applied compressive stress is not entirely converted into a hydrostatic stress state. The difference is in the second member on the right side of Eq. ( 24), in which both ε& and 22 σ are negative. Therefore, the applied normal stress contributes to the plastic deformation of the samples. The application of HPCS on ARMCO ® steel that we present in the following sections will show to what extent the normal pressure contributed to the plasticity of the samples.

Multiple passes:

Eq. ( 11) gives the strain value that can be reached in one pass. The strain depends largely on the shear displacement. It is, however, not desirable to apply too large shear displacement because the sample length is increasing during the test, thus, its surface on which the compression is applied increases, which leads to a decrease of the compression stress under constant compression force. There might be also technical limitation in the displacement, which is the case for the present setup (Fig. 1). One way to increase the strain is by cutting the sample into half and shear one part again. Such a test is an interrupted test and can be performed with unlimited repetitions, so extremely large strains can be reached in a small number of steps. An application of Eq. ( 11) for such HPCS multipass deformation process is presented in Table 1 for the initial geometry of 0 t =2 mm, 0 l =10 mm and a shear displacement s =10 mm.

Experiments

HPCS experiments were conducted on ARMCO ® steel comprised of soft 99.8% iron supplied by Goodfellow Cambridge. The as received samples were in rolled and annealed condition, with an average grain size (surface weighted average) of 38 µm. Two identical samples with dimensions 0 1.9 t mm = , 0 9.6 mm l = and 0 20 mm w =

were positioned on the two sides of the HPCS apparatus plunger. They were sand blasted on their flat surfaces before testing. The corresponding surfaces were also sand blasted on the punch, on the die, and on the plunger. The applied normal loading was set to 35 tons. The resulting normal stress decreased progressively because of the increase of the sample surface area during shearing. The confining force of 8 tons was maintained constant during testing. All punches had a cross section of 20 x 20 mm². The loading forces were continuously recorded together with the displacements of the punches. As the punch displacements were not measured directly at the sample but at the other ends of the punches, the recorded displacements were corrected for the elastic deformations of the punches using Hooke's law with a Young's modulus of 200 GPa. The pushing (driving) punch was moved with a velocity of 0.5 mm/s. The experiments were conducted at room temperature.

Multiple-pass experiments were conducted up to three passes, according to the procedure presented in the preceding section; the experimental conditions are given in Table 2. Two firstpass experiments were carried out: test no. 1(a) was done to verify the homogeneity of the deformation process, and the multi-pass testing 1(b)-2(b)-3(b) was made to reach extreme large strains, well into the steady state. In order to verify that no sliding occurred during testing, two fine line-markings were placed perpendicular to the shear direction on the opposite flat surfaces of the samples: they were found to be shifted to each other during shearing exactly with the imposed shearing distance.

After the first and third passes, the samples were sectioned in their mid-width side along the deformation direction (longitudinal cross-section) for both optical microscopy and electron backscattering diffraction (EBSD) investigations. After sectioning, the samples were mechanically polished with 600, 1200, 1800 and 4000 SiC grit papers. For optical microscopy, the samples were further polished with 9 µm, 6 µm diamond suspension and final polishing was performed with colloidal silica suspension (Struers' OP-S solution). To reveal microstructure for optical microscopy, these samples were etched with Nital solution (3% HNO3 and 97% C2H5OH). For EBSD, the samples were electrolytically polished with TENUPOL's A2 solution consisting of 700 ml ethanol (C2H5OH), 100 ml butyl glycol (C6H14O2) and 70 ml perchloric acid (60%) (ClO4). The operating parameters for electrolytic polishing were: 60V, 3.5A for the first pass

Experimental results

The initial microstructural state of the as received steel is shown in a longitudinal cross section in Fig. 3. The grain morphology was nearly equiaxed and the texture presented the characteristics of a bcc rolling texture with moderate intensity. During each test the sample thickness decreased significantly, see Table 2, where the accumulated compression, shear and equivalent strains were calculated form Eqs. ( 4), [START_REF] Pougis | Stress and strain gradients in high-pressure tube twisting[END_REF], and [START_REF] Takizawa | Incremental feeding high-pressure sliding for grain refinement of large-scale sheets: application to Inconel 718[END_REF], respectively. The equivalent stress was obtained from Eq. ( 24) and the stress-strain response is plotted in Fig. 4. Fig. 1b shows the optical image of the two deformed samples in their longitudinal cross sections after the first pass.

EBSD studies were performed at the central parts of the specimens. For the first pass (test no.

1(a) in Table 2), inverse pole figure maps (IPFs) were measured at three positions across the thickness to reveal possible deformation and microstructure heterogeneities: near to the top and bottom of the sample, as well as in its middle part. Large maps were obtained with dimensions at least 110 µm on each side. Fig. 5a shows the IPF, only for the middle area, for the simple reason that the other two maps (top and bottom areas) were very much similar. Fig. 5b displays the IPF map after the third pass. Grains were detected by imposing a minimum of 5° disorientation across boundaries excluding grains with three pixels and below. For better visualization, the IPF maps were superimposed with the grain boundary networks. For the lower strain, the grain structure was slightly inclined with respect to the shear direction and the grains were elongated. The inclination angle of the stringers corresponds to the imposed shear value (Fig. 5a). At the higher strain, the stringers are no longer distinguishable (Fig. 5b).

The grain-ellipticity distributions in Fig. 5c cover the whole spectrum for both strain levels. At the lower strain, the maximum probability is at about 0.3 for the top-area and of 0.6 for the middle and bottom areas. At the higher strain it is at about 0.4. Near-equiaxed grains had relatively low occurrence probability for both strains.

The grain sizes were calculated as the equivalent circle diameters of the grains. Their

distributions show a single central peak, see Fig. 5d for the area-frequency distribution. The average grain sizes are displayed in Table 3. As can be seen, the grain sizes were submicronic and decreased only slightly from top to bottom across the sample thickness during the first pass.

After the third pass, the grain size is only about 200 nm; the difference between the number and surface weighted average sizes is relatively small (Table 3).

The disorientation distributions obtained between neighboring grains are presented in Fig. 5e-f. These distributions were computed between neighboring grains by assigning to a grain its average quaternion orientation in the ATEX Software. It is important to emphasize that they are not pixel-to-pixel disorientations. (The differences between the two kinds of distributions have been discussed in [START_REF] Toth | Ultrafine-grain metals by severe plastic deformation[END_REF].) As can be seen, there are little differences between the disorientation distributions of the three IPF maps after the first pass. They show double peaks, one at about 13° and another at about 47°. The high angle boundary (more than 15°) fractions were 75%, 77%, and 77%, in the top, middle and bottom regions, respectively. After the third pass (Fig. 5f) the disorientation distribution is approaching the random one but significant differences can be observed below 25° and in the range 35°-55°. The relative frequency of the high angle boundaries is more than 92% after the third pass.

The crystallographic textures show the characteristic bcc simple shear texture features (Fig. 6).

They are displayed here in form of {110} pole figures as well as in ODF form in two sections of Euler space: in the φ2=0° and 45° sections. While the pole figures display all orientations, the ODF sections contain the main ideal components; listed in Table 4 (for the 45° section). Their positions are also indicated in the pole figures. Note that these textures are statistically well established as they were plotted from large grain number populations obtained by EBSD: more than 61000 grains from the mid-thickness region of the sample for the first pass, and from about 5000 grains from the third pass.

Discussion

In the present study the mechanical analysis of the HPCS has been presented and an application was carried out on an ARMCO ® steel. In this section we first discuss the main characteristics of the mechanics in terms of stress, strain, and strain hardening. Then the microstructure will be briefly discussed focusing on the grain fragmentation phenomenon. Finally, the crystallographic texture will be analyzed and the future prospects of HPCS will be presented.

Stress and strain state in HPCS

The main purpose of the present study is to establish the mechanics of the HPCS process. The strain state as well as the stress state have been analysed and each component of the stress and strain tensor defined by analytical expressions. For the calculation of the 11 σ and 33 σ normal stress components of the stress tensor, the material behavior was assumed to be von Mises isotropic. This was necessary for obtaining the hydrostatic component of the stress state. Indeed, only the deviatoric stress produces plastic flow of the material, which is the difference between the applied stress and the hydrostatic stress state. The mechanical isotropy hypothesis is acceptable for mild textures, which are actually weak in large strain SPD processes.

The evolution of the hydrostatic and the applied normal stress is displayed in Fig. 7a for the first pass as a function of the equivalent strain together with the strain hardening curve and the evolution of the shear stress component. As can be seen, at the starting point, the normal stress components are equal, so they produce a fully hydrostatic state. As soon as plastic deformation begins, the three normal components evolve differently. σ is due to the increase in the surface area of the sample as the compression force was kept constant throughout the HPCS process (35 tons). As can be seen in Eq. 24, the evolution of the linearly as a function of the equivalent strain. Inspection of Eqs. 4, 7 and 11 shows that these strain quantities are not actually linear functions of the total equivalent strain. The strain rates also vary as a function of the equivalent strain: the compression strain rate decreases while the shear strain rate increases. These functions are also not strictly linear, only they appear so in Fig.

7b

. The evolution of the hydrostatic stress is related to the ratio of the two strain rates, ε γ

& & ,
which is also plotted in Fig. 7b. This ratio shows significant evolution during HPCS: it decreases from about -0.2 at the beginning to -0.05 at the end of the test. Now, in order to estimate the relative contributions of the applied compressive force and the shear force to the plastic yielding of the material, we first calculate the deviator tensor starting from its definition:

ij ij h ij S σ σ δ = - . ( 25
)
With the help of the results obtained in Section 2, we obtain:

4 0 4 0 0 0 0 S ε τ τ γ ε τ τ γ   -       =           & & & & . ( 26
)
This deviatoric stress tensor has two normal and two shear components. The normal stress components originate from the applied compression stress while the shear components arise from the shear forces generated by friction. Their respective contribution to the total flow stress can be represented by their ratio. We define the following ratio for the evaluation of their relative contribution to the total equivalent stress:

11 12 4 S S ε ρ γ = = - & & . ( 27
)
Thus, it turns out that the strain rate ratio perfectly defines the ratio of the magnitudes of the two kinds of deviator stresses. The ρ quantity is also plotted in Fig. 7a as a function of the equivalent strain. It starts at about 0.8 and decreases down to 0.2 by the end of the test during the first pass.

Thus, the contribution from the compression force is not far from the shear force at the beginning. Even at the end of testing of the first pass, at the strain of 3.55, the contribution from the compression force remains significant (0.2). The change of stress state is visualized in the S11-S12 section of the von Mises yield surface in Fig. 7d where the initial and final stress states are indicated. Note that the effect of strain hardening on the size of the yield surface was omitted in Fig. 7d to better show the relative changes in the stress state. This figure shows also the evolution of the combined stress state for the second and third passes. As can be seen, the stress state approaches the simple shear stress state during the HPCS process. This stress state variation is clearly due to the decrease in the thickness of the sample so that the compression strain becomes nearly negligible with respect to the shear strain during a multiple-pass HPCS testing.

Strain hardening

One of the main advantages of the present HPCS process is the possibility to measure strain hardening during straining. The stress-strain responses displayed in Fig. 4 during the three passes show that the material hardens progressively in each pass. Finally, during the third pass, a steady state regime is attained at a flow stress of 1150 MPa. It is also evidenced by the hardening curves in Fig. 4 that the yield stress is lower at the beginning of subsequent passes. This effect can be attributed to a dislocation recovery process which is activated during the unloading process at the end of the previous pass. Namely, it has been shown experimentally during HPT testing, that there is a large decrease in the vacancy distribution during unloading [21,22] because vacancies are less mobile under high hydrostatic stress, so that they activate dislocation annihilation processes during unloading. Therefore, during subsequent reloading, there is less number of dislocations in the material which leads to a lowering of the flow stress. Hosakawa et al. [START_REF] Hosokawa | Microstructure and Magnetic Properties in Nanostructured Fe and Fe-Based Intermetallics Produced by High-Pressure Torsion[END_REF] estimate linear work hardening during HPT in pure iron up to a shear strain of 57 (von Mises equivalent strain of 32.9) and found a flow stress level of 1130 MPa during HPT which is nearly the same as in our experiments. (Note that Hosokawa et al. presented their results using the Hencky strain measure which is shown to be invalid for simple shear in [START_REF] Hosokawa | Microstructure and Magnetic Properties in Nanostructured Fe and Fe-Based Intermetallics Produced by High-Pressure Torsion[END_REF][START_REF] Hosokawa | Work hardening and microstructural development during high-pressure torsion in pure Iron[END_REF] . Nevertheless, the Hencky strain can be converted back into shear strain using the formula presented in the Appendix, so comparisons are possible.)

Once the steady state regime is reached in strain hardening, the average grain size remains constant. It has been shown by Pougis et al. [START_REF] Pougis | Extension of the Derby-relation to metals severely deformed to their steady state ultra-fine grain size[END_REF] that the steady state in SPD processes corresponds with an extension of the so-called Derby-plot [26,27] . This point lies exactly on the Derby-line (see the Derby figure extended for the UFG domain in Ref. [START_REF] Pougis | Extension of the Derby-relation to metals severely deformed to their steady state ultra-fine grain size[END_REF]). The occurrence of dynamic recrystallization is evidence of the steady state, not only from the agreement with the Derby-plot but also by examining the microstructure in the steady state, see the IPF map in Fig. 5b. The very small grain size and the large deviation from the expected very elongated grain shape provide clear evidence of grain subdivision and grain boundary migration. The existence of a small peak on the hardening curve in Fig. 4 at the beginning of the third pass is also an indication of the occurrence of dynamic recrystallization; typical in strain hardening curves when DRX becomes the major mechanism of the microstructure evolution.

It is well established that strain hardening can be analyzed by plotting the rate of hardening as a function of the stress, called also the Kocks-Mecking plot, as shown in Fig. 7c. (The derivatives of the hardening curves were obtained by first fitting the hardening curves by a polynomial function.) Fig. 7c shows clearly Stages III and IV during which the strain hardening rate decreases. During the third pass, it even reaches the zero value, i.e. the steady state. In each pass both stages III and IV appear, each time shifted parallel to a higher stress level. This is the consquence of the unloading and reloading sequences between passes. As discussed above, a dislocation annihilation process takes place when the large hydrostatic stress is removed from the sample. A consequence of this dislocation density reduction is that the microstructure is no longer in its equilibrium state for remaining in the preceding Stage IV state, so that to reach again Stage IV, first the dislocation density has to be recovered by further strain hardening, necessarily in Stage III. Stage IV is reached in each pass; the longest one in the third pass.

Finally, Stage IV ends and the steady state stage is reached. This last stage is called Stage VI in Ref. [START_REF] Beygelzimer | Some physical characteristics of strain hardening in severe plastic deformation[END_REF] because to arrive at a constant stress level there has to be a transition stage between the linear and no hardening stages; which is called Stage V in Ref. [START_REF] Beygelzimer | Some physical characteristics of strain hardening in severe plastic deformation[END_REF].

Grain refinement and microstructure

The main characteristics of the microstructures obtained by SPD processes is their extreme refinement. In this study, the average grain size of the steel was refined by the HPCS-induced equivalent strain of 33 from the as received number average size of 31.1 µm to 180 nm, a reduction by approximately 170 times. By comparison Pippan et al. [START_REF] Pippan | The Limits of Refinement by Severe Plastic Deformation[END_REF] report 150 nm grain size at the same equivalent strain in ARMCO ® steel. Hosokawa et al. [START_REF] Hosokawa | Microstructure and Magnetic Properties in Nanostructured Fe and Fe-Based Intermetallics Produced by High-Pressure Torsion[END_REF] report at about the same strain value a grain size of 300 nm after High Pressure Torsion to 0.85 mm thick electrolytic pure Fe discs with 12 ppm carbon.

Our study after the first pass shows that the degree of refinement was essentially uniform through the thickness, see Table 3. Furthermore, the angle of the elongated grain structure with respect to the shear direction was also identical in the top, middle, and bottom regions observed (only the middle region is shown in Fig. 5a as the others are very similar). Additional evidence of the uniformity of shear through the thickness of the strip can be found in disorientation distributions shown for top, middle, and bottom regions in Figure 5e.

Crystallographic texture

The crystallographic textures measured after one pass at three places in the cross section of the deformed sample appear to be characteristic bcc shear textures (only the mid-section is shown to save space in Fig. 6). It is also the case after the third pass. All ideal components appear, with different intensities with respect to each other. The texture intensity is relatively low which means that anisotropy is relatively small, supporting the use of the von Mises isotropic yield surface in the mechanical analysis. It is an interesting fact that the compression strain component during HPCS has no detectable effect on the texture evolution; the textures are really like ideal simple shear textures. This might be due to the relatively small value of the accumulated equivalent strain in compression (-0.6 during the first pass and -0.94 after the third) with respect to the equivalent strain corresponding to the simple shear component of the deformation (3.55 and 21.1, respectively). The deformation path also supports the tendency towards a fully simple shear deformation mode, which was evidenced by the movement of the loading point on the von Mises yield surface in Fig. 7d.

Perspectives

HPCS is shown to be an efficient way to deform metals to large strains. While the stress and strain rate state is multiaxial, it is readily possible to obtain equivalent stress-strain curves. As can be seen in Table 1, during multipass HPCS process the strain accummulates extremely rapidly, after only two passes, the strain is already 13.1. As a comparison, 16 passes would be needed to reach the same strain in the ARB process, or 11 in 90° ECAP.

It is also possible to change the strain path between passes by rotating the sample around either the shear direction or the plane normal. Thus, similar to ECAP, different routes can be established.

Conclusions

A new variant of the HPS process was established in the present paper, called HPCS. First the mechanics of the process were examined. All stress and strain components were expressed in analytical form (except the applied compression and applied shear forces, measured directly).

HPCS was applied to ARMCO ® steel in an apparatus in which the shear force, normal force, and shearing distance could be monitored to allow the stress-strain response to be measured in situ into the steady state work hardening regime. The crystallographic texture and the microstructure was examined by EBSD. The results presented substantiate the following main conclusions:

1. The HPCS deformation process was established experimentally in a setup different from the originally proposed HPS process [START_REF] Fujioka | Development of high-pressure sliding process for microstructural refinement of rectangular metallic sheets[END_REF] using a three axis extrusion machine. The new feature is a significant decrease in the thickness of the sample during large strain shearing. Both the stress and strain states are multiaxial in the proposed HPCS setup.

2. The applied compression stress does not result in a purely hydrostatic stress state in HPCS: an analytical formula was developed for the deviator stress tensor assuming von Mises isotropic flow. The deviatoric stress state progressively approaches the simple shear stress state as a function of strain.

3. Multi-pass experiments were carried out on ARMCO ® steel, well into the steady state of strain hardening at 1150 MPa which defines a point on the Derby-plot of dynamic recrystallization. The microstructure features also suggested that dynamic recrystallization took place in the steady state, at room temperature. 4. A characteristic bcc simple shear texture was obtained via HPCS of ARMCO ® steel.

5. The grain size was reduced by about a factor of 170 upon reaching steady state flow conditions. The grain size distributions, grain ellipticity, crystallographic texture, and crystallographic disorientation distributions were uniform throughout the thickness of the samples. (A2)

Using this formula, one can convert the Hencky strain into shear strain, so all published results that used the Hencky strain can be used in other works. Further, the shear value can be converted into the von-Mises equivalent strain:

3 vM ε γ = . ( A3
)
The difference between the two strain measures is very high for large strains. For example, in HPT testing, after one turn of a disk of 1 mm thickness and 20 mm in diameter, the simple shear formula gives at the outer radius a shear of 125.7, which is in Hencky strain

5.58

sample ( 1 .

 1 0 mm thickness) and 17V, 1.05A for the third pass sample (0.25 mm thickness) at a flow rate of 20 for 15 seconds at 15-20°C. The EBSD scans were performed by using JEOL JSM-6500F field-emission gun scanning electron microscope. The obtained raw data were processed with the ATEX software [19] to obtain grain sizes, disorientation distributions, crystallographic textures and Inverse Pole Figure (IPF) maps.

  constructed for dynamic recrystallization. In the Derby-plot the flow stress is normalized by the shear modules and plotted as a function of the average grain size normalized by the Burgers vector, in a double logarithmic presentation. For the steady state in the case of the present ARMCO ® steel we obtain the following Derby coordinates:

Fig. 1 .ShearingFig. 2 .Fig. 3 .

 123 Fig. 1. (a): Schematic figure for the experimental HPCS setup. (b): The two samples after the first pass deformation.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Stress-strain curves obtained by HPCS for ARMCO ® steel in three consecutive passes up to an equivalent strain of 33.34.

Fig. 6 .

 6 Fig. 6. The crystallographic textures in {110} pole figures, and the ODFs in two sections of the Euler orientation space (top row: φ2=0°, bottom row: φ2=45°). SPN and SD denote the orientations of the shear plane normal and the shear direction, respectively.

  The largest component is22

	σ , which is
	the applied compressive stress, 33 σ is smaller, and 11 σ is the smallest. The difference between
	the components is produced by the second terms that appear in Eqs. 23a,b. All normal stress
	components decrease first, then slightly increase from a strain of about 2.4. The decrease in 22

& & & & &

& & & & &

& & & & (18) 12 21

& & quantity in Eq. 23. To understand this effect, the evolution of the strain components and the ε γ & & ratio are plotted in Fig.7b. Both the compression and shear strains increase nearly

Acknowledgements

The authors acknowledge the help received from Prof. Yan Beygelzimmer (Donetsk Institute for Physics and Engineering, Ukraine) for the integration of Eq. ( 10) that led to Eq. (11)

This work was supported by the French State through the program "Investment in the future" operated by the National Research Agency (ANR) and referenced by ANR-11-LABX-0008-01 (LabEx DAMAS). Viet Q. Vu acknowledges the PhD scholarship awarded by the Vietnamese Government (Project 911).

Data availability statement:

The experimental data will be provided upon request.

Appendix Conversion of the Hencky strain into shear strain

During large strain simple shear, the grain shape changes from an initially spherical one into a very elongated ellipse. This shape change motivated the establishment of the so-called Hencky strain ( eq h ) which is the von Mises equivalent strain calculated from the two logarithmic principal strains of the two axes of the ellipse resulting from their length changes [START_REF] Onaka | Equivalent strain in simple shear deformation described by using the Hencky strain[END_REF]:

This strain measure, however, has been proven to be inappropriate to characterize the strain state during simple shear because it neglects the rotation of the ellipse [START_REF] Jonas | The Hencky equivalent strain and its inapplicability to the interpretation of torsion testing experiments[END_REF] , and does not satisfy the principles imposed by the group theory [START_REF] Beygelzimer | Some physical characteristics of strain hardening in severe plastic deformation[END_REF] . Nevertheless, because of the large number of papers that used the Hencky strain, it is useful to make the conversion between the Hencky strain and the correct equivalent strain.

From the above Hencky strain equation one can readily express the γ quantity: