THE EMERGENCE OF A NEW POLITICAL ECONOMY OF GAZE

It draws its materials from a segment of my Thesis work (Bordreuil 1987): La Production de la Centralité Urbaine, covering the pre-Haussmannian moment of Paris's centrality. Beneath the rendering of this singular and quite striking urban moment there lies a more theoretical topic: in order to grasp what was then taking place, rather than choosing in between some macro or micro approaches, there is no better way than to crisscross their respective scopes.

1. A fleeing urban center as a harbinger of the advent of a more liquid society.

Our paper draws on already collected archival materials to address the historicity of the "city center" as a prototypical urban form. The bulk of my material -popular literature and iconography, statistics, geographical data, as well as analysis written by "experts" mandated by civic bodies -focuses on the pre-Haussmanian moment (1830-1850) in Paris 's city life, precisely that critical period of the "displacement of the center" from the city's heart to its outer limits (on the Grand Boulevards built, as we know, on the very rubble of the city walls). The issue of the city center's migration to its very northwestern edge hovered above the local civic mindedness of the time (see Pierre Lavedan's work (1969) on this "question"1 ). This historical moment proved worrisome for its contemporaries precisely because they attributed this spatial drift to an underlying logic of growing importance according to which the future would belong to more fluid societies. As they redefined the center as "a center of movements", the somewhat threatening prospect of a future center "landing" in places where movements would be given freer reins -that is, beyond the old city -made sense to them.

Further -to the extent that "le mouvement va au mouvement" --this prospect was compounded by the notion that not only a new dynamic was taking hold in Paris Northwestern fringes, but that it belonged to the category of the "self generating" ones (the "snow ball effect" trope being used at length by the commentators of the time) 2 . In other words, this recursive loop (in that case a positive feed back loop) was granted an ominous strength that will subdue urban life as it used to be. This rather somber scenario -it must be said -was mostly promoted by citizens (landlords, merchants) living on the opposite side of the city (that is in the southeastern quarter of the city)3 . And it is also true that this "question of the displacement of the center" lost some of its criticality in the following years. Yet, if this alarm was tamed down, it was precisely because the new spatial features their "prophets" had come to elicit as keys to modernity (the easiness of movements) were soon enough implemented, but in all directions. In other words, rather than an exiting centrality, what took place was an enlargement of the urban centrality -a Haussmannian enlargement, to name it. Therefore, there is no reason to overlook the rhetoric they used in their campaign; to the contrary and on two accounts. The first one is that the rationale their analyses had constructed was endorsed in the guidelines of the Haussmann enterprise. The second, being that these analyses -their emphasis on a "mobility turn" of the city life -appear well grounded in what historical approaches tell us about that moment in urban life. Indeed, those were times of growing mobility (re traffic flows), of a re assemblage of the city material infrastructure so as to give free way to these growing flows and, finally, of a close policing of "urban manners", city dwellers being a quintessential component of this new liquid order.

Traffic wise, we should not underestimate the swelling of it during the first half of the century, a period during which Paris's population doubled (from half a million of inhabitants to one million). If we focus on passenger transportation, a turning point seems to be the launching in 1828 of "Omnibus" services: 30 thousand users at the beginning of it; but a doubling up of this number in just one year (60 thousand). To these numbers one should add the customers of "voitures de place" (our modern "cabs") whose number doubled in between 1819 and 1853 (whereas omnibus would carry 14 passengers, "voitures de place" could cater for 12). All in alland not withstanding the weight of the "voitures bourgeoises, the heir of "aristocratic equipages" -this translated into a huge pressure on the uses of city streets, a contested turf in between pedestrians and their rolling others4 .

So much so that it comes as no surprise that the rulers of the city tried to put some order in the ensuing "traffic jams", mostly in allotting separated paths to the two halves of humanity 5 : sidewalks, to name them, was the desired solution. The increase of the overall length of this new amenity is quite striking: 16 kms in 1833, 240 in 1848, 1 100 in 1870. Yet, this was not enough, since the use of sidewalks should be itself rendered more liquid. Here we turn towards the last layer of implemented urban policies. At stake, there were not only material fixtures that should be realigned so as to ease the flow (signboards flattened on the walls rather than sticking out of them in order to catch the attention of coming passers by; but, too, all sorts of protuberances, be they benches or all kinds of corbelled constructions). Further, human beings and their active propensities had also to be smoothed out of the flow. A constant thread of "police orders" running during this first half of the century, would prevent any kind of laboring activities on the sidewalks. To which must be added that, eventually, these "police orders", widened their target, beyond the busy endeavors, to any kind of sedentary use of the street (that is sidelining the "Casaniers" in favor of the "Mobiles" 6 ) of the street. In 1862, one ordinance was edited which would go further than the aim of regulating moves as they come. It stated that the movement itself was the root principle of law: consequently, the right to station, to stay in place, became subordinated to the superior public good of moving.

In one word, the increasing fluidity of urban life was both a driving force of urban changes and an imperative worthy of abiding by.

Effects of growing urban fluidity on the daily management of gaze.

The driving hypothesis of this presentation is that the gathering of all the elements that enter in the framing of whatever "sense of place" (sketches, panoramas, tourist guides, ads, and "talk of the town" columns of a flourishing new press) was made possible by a critical shift in the delicate exercise of the reciprocal gaze in public space. How can we draw this link tying fluidity to visibility (or, so to speak, gaze-ability)?

Addressing that question by its geographical dimension we propose to consider that this (gaze) shift found some grounds in the deterritorialization of pre-existing urban social spaces; that is, through a progressive but steady "disowning" of urban places and their reassembly from a juxtaposition of turfs into the seamless texture of a public space. Indeed, integral to the ideal of "public space" is the notion of freedom granted to whoever comes and goes in it; in short "universal accessibility" constitutes its ruling principle. Thus one can get a sense of this reshuffling of urban social spaces in the ways it came to be welcomed by commentators of the time. For instance, they would marvel at the new afforded possibility to go through and through in Paris 7 . Another critical symptom of this reshuffling was the terminological battle that raged about the proper naming of its urban ways: "Streets" or "Roads"? That is, should the circulating component be emphasized in naming those ways as "road"? Or not? Then, is a street something else than a circulation way? To be sure, it was, in the 18 th century, and it continued to be so in neighborhoods not yet affected by the rising liquidity of urban life. Arlette Farge 8 convincingly describes the street as a locus where not only people would live outdoors (working, chatting), but would make a living out of their street lives (sometimes barely surviving). The typical profile of the "petit commissionaire" helps to grasp some features of this local life (and its relationships with "non local" life, that is the rest of the urban world). Very often he is young and his business is to offer assistance to those foreign to the neighborhood. Commentators describe him as being both generous (he offers services, often of the piloting kind) and mischievous (he can trump his clients, force their hand or, worse, react back if he judges that he was not "tipped" enough 9 ). In any case there is some sense in framing him as a "gate keeper".

Then, one understands well that the "watch out" side of his activities was vital for his prosperity; and this will afford us a transition to the second plane of the analysis -shifting, so to speak, from the goings to the gazing. Indeed, commentators describe that type of "petit industriel en plein 6 To refer here to the wording of the time.

7 « Aller par monts et par vaux » ; see Walter Benjamin quote in the "Arcades Project" (M3, 2). To which must be added that this new possibility was taken as a mark of the superiority of Paris in comparison with others cities. 8 Farge, Arlette : Vivre dans la rue à Paris au 18 ème siècle. Paris, Gallimard, 1992. 9 See : l'Illustration, February 10th, 1843. In this article the « petit commissionnaire » is dubbed « le petit boueur ».

vent" as masterful in the art of "voir venir" (of seeing who is coming); as "subtle observers10 ", skilled at guessing from where and for what needs a coming body was entering his turf. Conversely the entering person was well aware that he or she was placed under close scrutiny and would be better off behaving accordingly. In that kind of circumstances the exchanges of gaze could trigger some significant consequences. To which must be added that, to the extent that street life was very often both unpredictable and unstable11 , that it could at any moment nurture some local turbulence or turmoil and these consequences could always turn bad. In short, the prospect of being embroiled in situations that could go awry was part of the urban experience.

Having placed the growing fluidity of urban life upon that (schematized) backdrop, we then can more easily delineate how it impacted the texture of social spaces. On the one hand, it tended to erase, or smoothen, the gate component of the urban structure -the disappearance of urban thresholds was both pointed to and celebrated by the contemporaries12 -thereby weakening the grip of the gate keeper's gaze mentioned above. On the other, it prompted in every local site a more intricate overlapping of different urban dwellers trajectories. Specifically, whether these citizens would come from nearby or would be transiting from far away, they would fray the same arteries, thus becoming hard to discern from each other. Therefore, local communities being routinely crisscrossed by non-locals, the profile of the outsider lost some of its salience, giving place to the anonymous "passer by". In other words, the meaning of "strangeness" shifted from being foreign to the place, to that of simply being an unknown among others equally unknown to each other. Hence, the topos of the (more or less) "Lonely Crowd" (David Riesman) could enter the place.

To go back to the theme of "gaze ability", our proposed analysis would be that, given the new mobile circumstances, the complex economy of rights and duties that tied the onlookers to each other (both as subject and object of gaze) and ruled their reciprocal involvements as such, was displayed in a renewed fashion. One instance of that shift would be that the moral economy commanding some discretion re the doings of others was loosing some of its grip. For a part, given the fleeting side of urban encounters, the possibility of correctly inferring the whole of an ongoing activity from a segment of behavior could no longer be presumed. For another, we have to keep in mind that the monitoring of one's doing is critical but only (or all the more) when it is enacted by "significant others": meaning ones being known to the person under consideration and, as such, appearing as possible "book keepers" and/or custodian of a reputation13 . To the extent that local life would be progressively drowned under trans local flows, the corporation of those "local book keepers" would shrink and/or be sidelined.

To sum it up, the reshuffling of social spaces considered as fields of gaze-ability, would bear upon the following features:

A sidelining of the monitoring of "activities in course"; that is the visual tracking of walkers in order to know more about what they were "up to". And also that this visual tracking would lose some of its undesired and perilous consequences, thereby granting onlookers the possibility of a more innocuous exercise of their scoping abilities.

Obviously, this is not to say that they could walk "eyes wide shut", so to speak. Rather, it meant that the irreducible task of monitoring other's acts tended to fold back to the strict control of their walking ways. Or to put it otherwise, the visual monitoring would stem from within a "walker's perspective" and embrace the other as such -that is another walker coming by, thus leaving aside any kind of attention as to what his or her walk would conceal.

As we know well, this sort of close range mutual checking relies also and entirely on a quick exchange of gazes; and we owe Erving Goffman a nice analysis of its ritual delicacies. As a reminder, "Civil Inattention", to name this basic ritual display, is a two steps process. It starts with an exchange of gaze between passers by, so as to ensure each of them that they are mutually aware of their common presence (a "notification of presence", so to speak). Then, it closes by a sort of "dimming of the light" (using here a Goffman's metaphor), that is a synchronized withdrawal of gazes, so as to signify to the other that he or she is no longer the subject of any kind of lasting attention14 . Although we will not apply in the following this micro analytical "lens", it is worth mentioning it, insofar as this helps us to better locate our focus of attention re the scoping abilities of urban dwellers. Indeed what was at stake in the reshuffling of the urban visual field relates mostly to the "long range" component of their visual attentions. That is, the ability to observe the observing behaviors of others (third parties, as it were) and in so doing, not risking (or risking less) to be trapped in some sort of embroilment.

In this regard, the flourishing of urban depictions centering their scope on what uses fellows urbanites would make of their eyes is quite striking; to the point that some typologies were built up according to differences re these uses. A case in point is the opposition in between the "badaud", the "physionomist" and the "flaneur" types. In short, a "badaud" could be defined as an avid observer, so much engulfed in its activity that he would forget that he could himself be a subject of observation. To the contrary, the "physionomist" and the "flaneur" belonged to the category of the "unobserved observers". These two socio types, in turn, gave ways for some nuances, the physionomist's gaze being more of the staring kind, whereas the flaneur's one could indulge in some sorts of musing digressions15 .

The license to watch: Balzac and the new field-stations for the study of social order

But what were the cognitive effects of the gaze thus transformed? We turn here to two essays in which Balzac, as a third observer, privileges sidewalks as "field-stations" for understanding the very stuff of the social order. In so doing, he endorses the position of the "unobserved observer" proper to the flaneur.

I rely on two non-fiction texts by Balzac (1966), "Le traité de la vie élégante" and la "Théorie de la Démarche". I have chosen them because they make ample room for visual observations; particularly some insights caught from some cafés terraces, on the Boulevards. Placing Balzac in its own time and space gives some credence to his treatment as a forerunner of the Social Sciences to come. Time wise (and I am following here Robert Nisbet's Sociological Tradition) Balzac's post-revolutionary era progressively yielded a sense of society as a "sui generis" entity (to use Durkheim's words); a meaning that was accessible to ordinary actors. Spatially speaking, the local concentration of so many socio-types, coming from so many different back grounds, the profusion of new commodities ("all that is elsewhere is here", to quote a W Benjamin quote in his "Arcades Project"), nourished the sentiment that the whole of society was "out there", graspable by an encompassing gaze. We know that Balzac took Cuvier's tables of comparative anatomy as a model, striving for its synoptic ideal as applicable to social matters. His aim was to reveal the laws that rule the social order (as Cuvier had done for the natural order). And his method was similar: to spot some tiny material details that would allow for a classification in all certainty of the bearers of that mark in that (sub) species rather than in another one. Thus, in so doing, his "enquiries" in social life deliberately remained at the level of appearances; Balzac's take was that by simply sojourning in the realm of what is left to see (and to be observed) the underlying layer of the overall picture of the different social conditions was entirely accessible and readable.

The two texts develop the rationale for such a "superficial" gaze. In one word, the surface, the clothes, the gestures tell the truth of their bearers. Here an underlying ontology is at play: one that seizes material forms as crystallizations of embedded values. To cite Balzac, those values are akin to a "vapor" that cannot help to perspire at a behavioral level. To the extent that for Balzac the social order is a hierarchical one (and this appears to him as a good thing!), one can read those two texts as attempts to ground the superiority of the elites by demonstrating the superiority of their "body language". To be able to spell out the code of their "elegance" and demonstrate its intrinsic superiority suffices to warrant the inherent superiority of the members of these elites.

The reader can nevertheless note that, in order to spell out this code of excellence Balzac the observer devotes much attention to those who, on the sidewalks of the city, try to "pass", fabricating or counterfeiting their true social identity. But his stance consists at poking fun at them -"outsmarting" them by simply detecting the ensuing lack of unity of their performances ("unity" for him is a keyword to define what "elegance" is about). Thus, two conceptual schemata appear at work in Balzac's dialectic of appearances and essence: the first is akin to the modern one of "embodiment" (social values, habitudes making their ways through bodies). The second relates to the trope of fetishism. That is to the gullibility of masses encaged as they were in their "home made" fabrications; or fooled as they were by their own duperies. But obviously for Balzac the question was not, as it would be the case in a Marxist vein, to help those masses to get rid of their illusions. The bad thing for him (and one not to be corrected) was not that they were entertaining fictions but more likely that they opted for the wrong ones: "contra natural" kind of fictions.

To conclude in an interrogative mode: one finds in Balzac some conceptual tools that became central in our critical "tool box"; yet he devised them in a markedly different spirit. The depictions of city life at that time were a fertile ground for their elaboration; but to clarify those concepts is a task contemporary sociologists are still busy at.

Jean Samuel Bordreuil, March 2015

Bibliography

Honoré de Balzac, « Traité de la vie élégante », La comédie humaine 7, Paris, Editions du Seuil, 1966 (pp.564-580). Honoré de Balzac, « Théorie de la démarche », La comédie humaine 7, Paris, Editions du Seuil, 1966 (pp. 580-596)

See LAVEDAN, P., La question du déplacement de Paris et du transfert des Halles au Conseil municipal sous la monarchie de juillet, Paris, Imprimerie municipale, 1969. For a more in depth approach of that critical moment the main sources are the numerous « Mémoires » written by experts and/or by members of civic bodies, in between 1835 and 1845 (JS Lanquetin, A Rabusson, Perreymond, Victor Considerant being the most noticeable among them).

See: Michel Serres book: Feux et signauxde brume -Zola, Paris, Grasset, 1975. A book where the recursive loop is a central trope, as it was for the experts of the « question of the displacement of the center of Paris ».

Lanquetin, one of the spearheads of this alarming campaign was himself a wine merchant from Bercy.

On this topic Alfred Martin's works are very useful. See: MARTIN, A., Etude historique et statistique sur les moyens de transport dans Paris, 1894, BHVP, 30881.

Quoting Balzac : « Les pauvres vont à pied, les riches en équipages ».

« Subtils observateurs, ils pouvaient voir si vous veniez de Londres ou bien d'Anvers ». in L. GOZLAN, Le triomphe des omnibus, poème héroico-comique, 1828, BHVP.

To quote Arlette Farge, op cit

See Walter Benjamin (« Arcades Project ») on this theme of the erasement of thresholds in the urban life.

What is a Biography? At one point in Stigmas, Erving Goffman answers « along the lines of » nothing else than a white page « darkened » or written over by others.

InGOFFMAN E. (1963), p. 83 -88.

See Walter Benjamin notes about this difference in the flaneur section of the Arcades Project.


João Barbosa

Mário Florido

Vítor Santos Costa

Typed Unification: when failure may not be wrong

In a previous paper, we have investigated unification extended with regular types restricted to an Herbrand interpretation of term functors. Here we extend this investigation allowing arbitrary interpretations for functors. This new approach leads to a new typed unification algorithm which returns most general unifiers, failure or a wrong value denoting a type error during the unification process.

Introduction

Unification has been widely applied to different scenarios in computer science. Examples include database processing systems, natural language processing, computer vision, expert systems, computer algebra, programming language implementation, automated theorem proving, type inference and logic programming.

This paper is concerned with unification in typed first order theories, where types are described by regular grammars [START_REF] Zobel | Derivation of polymorphic types for Prolog programs[END_REF][START_REF] Mishra | Towards a theory of types in Prolog[END_REF][START_REF] Yardeni | A type system for logic programs[END_REF][START_REF] Frühwirth | Logic programs as types for logic programs[END_REF][START_REF] John | Abstract Domains Based on Regular Types[END_REF]. It is inspired by the fact that in logic programming many practical applications of variables, and the domains of functions, are naturally restricted to specific semantic domains which are subsets of the whole semantic universe.

With this motivation in mind we present a term unification algorithm which will return one of three different results: a most general unifier, failure or wrong. This last value wrong is inspired by a similar notion used by Robin Milner to denote run-time type errors [START_REF] Milner | A theory of type polymorphism in programming[END_REF] and, in our framework, it corresponds to the unification of terms that can never belong to the same semantic domains.

This paper generalizes a typed unification algorithm previously defined by the authors in [START_REF] Barbosa | Typed SLD-Resolution: Dynamic typing for logic programming[END_REF] that was used in the dynamic typing of logic programs. In [START_REF] Barbosa | Typed SLD-Resolution: Dynamic typing for logic programming[END_REF] functions symbols f of arity n, had co-domains which were always sets of terms of the form f (t 1 , . . . , t n ) where the arguments t i belong to the corresponding domain of f . This basically induced a partition of the Herbrand domain into sets of trees. We extend this notion by enabling the use of arbitrary semantic domains and co-domains for functors. A function now, may map integers to integers, integers to lists, floats to lists of integers, and, thus, semantic domains are no longer confined to be sets of trees.

Example 1: Let cons be the list constructor with type cons :: α × list(α) → list(α), where list(α) = nil + cons(α, list(α)) (+ denotes disjoint type union). Suppose we have terms t 1 = cons(1, X) and t 2 = cons(Y, 1). These terms unify using first-order (untyped) unification, but they do not have a correct type, since the second argument of cons must be a list. This illtyping is captured by our new typed unification algorithm that, in this case, outputs wrong.

The most obvious related work is many-sorted unification [START_REF] Walther | Many-sorted unification[END_REF], though many-sorted unification, in general, assumes an infinite hierarchy of sorts and we do not assume a hierarchy of types. The most related unification problem is many-sorted unification with a forest-structured sort hierarchy [START_REF] Walther | Many-sorted unification[END_REF], but even compared with this strong restricted unification problem, our work gives easier and nicer results, mostly due to the use of an expressive universe partition based on regular types but with no underlying hierarchy on the domains.

Language

The alphabet for the language of types includes an infinite set of type variables TVAR, a finite set of base types TBASE, an infinite set of type function symbols TFUNC, an infinite set of type symbols TSYM, parenthesis, and the comma. Then, a type term is either a type variable α ∈ TVAR, a base type b ∈ TBASE, a type function symbol f ∈ TFUNC applied to an n-tuple of type terms f (τ 1 , . . . , τ n ) or a type symbol σ ∈ TSYM applied to an n-tuple of type terms σ(τ 1 , . . . , τ n ). If a type function symbol has arity 0, we call it a type constant. If the arity of a type function symbol is larger or equal to one, we call it a complex type term. In a complex type term we call the type function symbol that starts it the principal type functor. We use the same notation for complex type terms and complex terms, but from context it should be obvious to which one we are referring in each case.

Type symbols are defined by type definitions of the form:

σ(α 1 , . . . , α n ) = τ 1 + • • • + τ n .
The type variables on the left-hand side of the type definition are called parameters and correspond to all the variables that occur on the right-hand side. The sum τ 1 + • • • + τ n is a union type, describing values that may have one of the types τ 1 , . . . , τ n , called the summands. The '+' is an idempotent, commutative, and associative operation. Type definitions are called deterministic if no two type terms on the right-hand side have the same principal functor, and no type terms on the right-hand side start with a type symbol. Deterministic type definitions correspond to tuple-distributive types [START_REF] Zobel | Derivation of polymorphic types for Prolog programs[END_REF] and from now on we assume type definitions are deterministic.

We assume the universe of semantic values is separated into several disjoint domains

U = D 1 ∪ • • • ∪ D n .
Each domain is associated with a type. Note that since the domains are disjoint, if we have a certain type symbol σ defined by the type definition σ(α) = nil + cons(α, σ(α)), we are assuming that no other type exists that uses the same type function symbols nor the same constants. We also assume that there is a one-to-one correspondence between function symbols and type function symbols, i.e., for each function symbol f there is a correspondent type function symbol f . Let t ∈ τ mean that the semantic value denoted by t belongs to the domain associated with type τ . Let t be a ground term. If t is a constant c, then the type for c :: τ , where τ is a base type b or a type symbol σ, if the type constant associated with c is a summand in the definition of σ. We say the principal type functor of c is τ .

If t is a complex term f (t 1 , . . . , t n ), then we need to highlight the possibility of the term having a type error. In general, the type for f is f ::

τ 1 × • • • × τ n → τ ,
where n is the arity of f , and τ is a base type, a complex type term, or a type symbol. If all t i are such that t i ∈ τ i , then there is no type error. But if at least one t i is such that t i / ∈ τ i then we say that there is a type error. If τ is a type symbol σ then in the type definition of σ there is a summand of the form f (τ 1 , . . . , τ n ). We also assume that the type for a variable is the top, a type representing the sum of all types, but we want to distinguish between different variables, and, more importantly, relate different variables, so we assume that each variable X i is associated with a type variable α i , that can be instantiated by any type, or remain as a variable, which can be interpreted as "any type".

Typed Unification Algorithm

We have three types of constraints. Equality between terms t 1 = t 2 , equality between types τ 1 . = τ 2 , and membership of terms in types t ∈ τ . Suppose we want to unify two terms t 1 and t 2 . Let the type for the principal functor of t 1 and t 2 be τ 1 and τ 2 , respectively. We build an initial tuple of sets of constraints of the form ({t

1 = t 2 }, {t 1 ∈ τ 1 , t 2 ∈ τ 2 , α . = τ 1 , α . = τ 2 })
, where α is a fresh type variable. Then the following rewriting system rewrites the tuple (C, T ), where C are equality constraints that are generated in the algorithm, and T is a set of type constraints, both equality between types and membership of terms in types. The rules for the rewriting system are meant to be applied in order, i.e., if rule n and n + k can both be applied, we apply n. They are as follows:

1. (C, {f (t 1 , . . . , t n ) ∈ τ } ∪ Rest) → (C, {t 1 ∈ τ ′ 1 , . . . , t n ∈ τ ′ n , τ ′ 1 . = τ 1 , . . . , τ ′ n . = τ n } ∪ Rest)
, where the type for f is f ::

τ 1 × • • • × τ n → τ ,
and τ ′ i are the types for the principal functors of t i , respectively

(C, {X

i ∈ α i } ∪ Rest) → (C, Rest) 3. (C, {c ∈ τ } ∪ Rest) → (C, Rest) 4. (C, {f (τ 1 , . . . , τ n ) . = f (τ ′ 1 , . . . , τ ′ n )} ∪ Rest) → (C, {τ 1 . = τ ′ 1 , . . . , τ n . = τ ′ n } ∪ Rest) 5. (C, {τ . = τ } ∪ Rest) → (C, Rest) 6. (C, {f (τ 1 , . . . , τ n ) . = g(τ ′ 1 , . . . , τ ′ m )} ∪ Rest) → wrong, if f ̸ = g or n ̸ = m 7. (C, {τ . = α} ∪ Rest) → (C, {α . = τ } ∪ Rest),
) = g(s 1 , . . . , s m )} ∪ Rest, T ) → f alse, if f ̸ = g or n ̸ = m 13. ({t = X} ∪ Rest, T ) → ({X = t} ∪ Rest, T ), t is not a variable 14. ({X = t} ∪ Rest, T ) → ({X = t} ∪ Rest[X → t], T ), if X does not occur in t 15. ({X = t} ∪ Rest, T ) → f alse, if X occurs in t.
We show a few examples of the application of this algorithm to pairs of terms.

Example 2: Suppose f has type f :: int × int → int, and 1 and 2 have type int. Then let

t 1 = f (1, f (X, 1)) and t 2 = f (Y, f (2, Y ))
. The generated tuple of constraints is:

({f (1, f (X, 1)) = f (Y, f (2, Y ))}, {f (1, f (X, 1)) ∈ int, f (Y, f (2, Y )) ∈ int, int . = α, int .
= α}. Applying the rewriting rules, we get to:

({f (1, f (X, 1)) = f (Y, f (2, Y ))}, {f (1, f (X, 1)) ∈ int, f (Y, f (2, Y )) ∈ int, int . = α} → * ({f (1, f (X, 1)) = f (Y, f (2, Y ))}, {X ∈ α 1 , Y ∈ α 2 , α 2 . = int, α 1 . = int, int . = int, int . = α} → * ({f (1, f (X, 1)) = f (Y, f (2, Y ))}, {α 2 . = int, α 1 . = int, α . = int} → * ({Y = 1, X = 2}, {α 2 . = int, α 1 . = int, α . = int}
The algorithm stops without failure and the most general unifier is [Y → 1, X → 1].

Now an example where we get the result f alse.

Example 3: Suppose f has type f :: int × int → int, and 1 and 2 have type int. Then let

t 1 = f (1, f (1, X)) and t 2 = f (Y, f (2, Y ))
. The generated tuple of constraints is:

({f (1, f (1, X)) = f (Y, f (2, Y ))}, {f (1, f (1, X)) ∈ int, f (Y, f (2, Y )) ∈ int, int . = α, int .
= α}. Applying the rewriting rules, we get to:

({f (1, f (1, X)) = f (Y, f (2, Y ))}, {f (1, f (1, X)) ∈ int, f (Y, f (2, Y )) ∈ int, int . = α} → * ({f (1, f (1, X)) = f (Y, f (2, Y ))}, {X ∈ α 1 , Y ∈ α 2 , α 2 . = int, α 1 . = int, int . = int, int . = α} → * ({f (1, f (1, X)) = f (Y, f (2, Y ))}, {α 2 . = int, α 1 . = int, α . = int} → * ({1 = 2}, {α 2 . = int, α 1 . = int, α . = int}
The algorithm halts and outputs the result f alse.

And, finally, an example where we get the result wrong.

Example 4: Suppose f has type f :: int × int → int, g has type g :: int → f loat, and 1 and 2 have type int. Then let

t 1 = f (1, f (1, X)) and t 2 = f (Y, g (2)
). The generated tuple of constraints is:

({f (1, f (1, X)) = f (Y, g(2))}, {f (1, f (1, X)) ∈ int, f (Y, g(2)) ∈ int, int . = α, int .
= α}. Applying the rewriting rules, we get to:

({f (1, f (1, X)) = f (Y, g(2))}, {f (1, f (1, X)) ∈ int, f (Y, g(2)) ∈ int, int . = α} → * ({f (1, f (1, X)) = f (Y, g(2))}, {X ∈ α 1 , Y ∈ α 2 , α 2 . = int, α 1 . = int, int . = int, f loat . = int, int . = α} → * ({f (1, f (1, X)) = f (Y, g(2))}, {α 2 . = int, α 1 . = int, f loat . = int, α .
= int} The algorithm halts and outputs the result wrong.

Termination

Here we show that, for any input, the unification algorithm always terminates.

Theorem 1 -Termination: For any t 1 and t 2 , let (C, T ) be the set of generated constraints from the terms. The rewrite system always terminates, returning a most general unifier, false, or wrong.

Proof. We divide the proof into three parts. Firstly, we prove that the number of membership constraints reduces to zero, from rules 1 through 3. Afterwards, we prove the termination of the rewriting algorithm for the type equality constraints, from rules 4 through 9. Then we prove the termination of the rewriting algorithm for equality constraints, from rules 10 through 15. Since the algorithm uses the rules in order and no membership constraints are generated after we start applying steps 4 through 9, such as no type equality constraints are generated after we start applying rules 10 through 15, we prove the termination of the algorithm.

The proof for the first part follows a usual termination proof approach, where we show that a carefully chosen metric decreases at every step.

The metric is the number of symbols on the left-hand side of constraints. All rules 1 through 3 decrease this metric.

The second and third parts are each the Martelli-Montanari algorithm for its corresponding kind of constraints, type equality and equality, respectively. Therefore they also terminate, since the Martelli-Montanari algorithm terminates.

Another important property is that the algorithm is sound. We conjecture this property, because we have not completed the prove yet (it is ongoing work).

Theorem 2 -Soundness:

1) Suppose our algorithm returns a unifier µ. Then µ is a most general unifier and any substitution θ = µ • λ that grounds terms t 1 and t 2 is such that θ(t 1 ) = θ(t 2 ) and θ(t i ) is well-typed.

2) Suppose our algorithm outputs wrong. Then, there is no substitution θ that grounds terms t 1 and t 2 such that θ(t 1 ) and θ(t 2 ) have the same type.

3) Suppose our algorithm outputs f alse. Then, there is no substitution θ such that θ(t 1 ) = θ(t 2 ), but there is a substitution θ′, such that θ′(t 1 ) and θ′(t 2 ) have the same type.

* This work was financially supported by: Base Funding -UIDB/00027/2020 of the Artificial Intelligence and Computer Science Laboratory -LIACC -funded by national funds through the FCT/MCTES (PIDDAC).


Being an Airbnb host, becoming an entrepreneur? Empirical evidence from Lyon

Guillaume Barral Université Jean Moulin Lyon 3 "…there is an evolution of the type of users on all platforms. Even the platforms which were built up to link peers to peers are repositioning on a larger intermediation role and welcome professionals too (to expand their service range)"1 . This is the case of Airbnb, founded in 2008 to be a peer-to-peer accommodation platform. At first, there were no enterprises nor companies. It allowed people to rent out an underutilized space of their accommodation or an underutilized accommodation for a while. But since, hosts' representations have changed, and their practices have got more and more professional. Airbnb has reshaped socio-spatial relationships towards property, to such an extent that hosts has become more and more calculating. This system spurs people on marketing themselves, their accommodation(s) and their services to be more competitive… in short, to become micro-entrepreneurs [START_REF] Stabrowski | People as businesses': Airbnb and urban microentrepreneurialism in New York City[END_REF].

These micro-entrepreneurs, also called "professionals" because they earn their living off their accommodations [START_REF] Ke | Sharing Means Renting ?: An Entire-marketplace Analysis of Airbnb[END_REF][START_REF] Moreno | Pro vs Joes: Agent Pricing Behavior in the Sharing Economy[END_REF][START_REF] Stulberg | Airbnb Probably Isn't Driving Rents Up Much, At Least Not Yet[END_REF], are made up of multi-listers. They represent 6% of hosts in New York in 2014 and control "one in five unique units in New York City booked on Airbnb as private short-term rentals" (Schneiderman, 2014, 10) or 10% in Paris where they control 30% of the short-term rentals' market 2 . Their possessions are located in the main touristic areas, and therefore the most profitable ones: for example, in Paris, these rentals have exploded in the IVth and the VIth districts 3 . With the growing number of such listings in city centers, the Airbnb syndrome contributes to the dwindling number of flats for rent on local real estate markets and, therefore, to the rent increase [START_REF] Samaan R | Airbnb, Rising Rent, and the Housing Crisis in Los Angeles, Laane, a new economy for all[END_REF]. To fight back, municipalities implemented laws and asked for studies to understand and control the development of the short-term rentals market.

In this context, I wanted to conduct research about hosts' professionalisation that could be of use to the Métropole de Lyon, which is trying to understand and "limit the professionalisation of this activity and to allow everybody to rent out their accommodation for a while, in the original spirit of collaboration" 4 . This academic study attempts to fill this gap by discussing multi-listers' practices in Lyon, a town where the multi-listers represent 8% of the hosts and control 19% of the short-term rentals' market in July 2017, according to the data extracted by Tom Slee on Airbnb 5 . Are they acting like entrepreneurs, professionals, managers, intermediaries…? According to what kind of indicators can they be called so: the number of short-term rentals they control, the income they earn, the time during which they rent out their accommodations, their business strategy…? In which way do they contribute to the airbnbfication of the city center of Lyon? Are they even the only ones to do it? Since Airbnb spurs each user on being competitive and calculating on this entire-marketplace [START_REF] Ke | Sharing Means Renting ?: An Entire-marketplace Analysis of Airbnb[END_REF], I consider that some of the hosts who rent out just one accommodation must follow similar strategies to those followed by multi-listers -at a different level at least. And yet, this phenomenon has received little attention. This is the reason why it is necessary to run a qualitative study on hosts' profiles in order to show that multi-listers are not the only ones "reflecting the growing professionalisation of this sector of the sharing economy" (Stabrowski, 2017, 333). This project aims to qualify and criticize the statement defended by Airbnb that most hosts "use their Airbnb income to help pay their mortgage or rent"6 and to distinguish "ordinary individuals that want to do collaborative economy from those, more professional, that are doing of it a well-established business"7 . To this end, I discuss the "logiques d'action", as defined by Philippe Bernoux [START_REF] Amblard | Les nouvelles approches sociologiques des organisations[END_REF], which lead people to rent out an accommodation on Airbnb in Lyon. At last, I question the policy the Métropole de Lyon is due to implement as of February 1rst 2018 in order to frame the Airbnb phenomenon and to help the Métropole face the transformations initiated by this platform on the real estate market.

My objective is not to obtain a representative sample of the Lyon hosts, but rather to understand their actual professionalization. This is the reason why it seemed appropriate to run in-person interviews. This pilot study analyses the motivations and characteristics of Airbnb hosts. In doing so, I investigate four key questions:

-Which indicators can we use to identify professional hosts? -Who are the professional hosts in Lyon and how did they choose to do this work? -In which way do they use their skills and their (social, cultural…) capitals to found a sustainable profit? -To what extent do they differ from those of want to play a part in sharing economy, in the spirit and practices?

The interviews already conducted with owners and concierge companies as well as the exploratory exploitation of a database allow me to formulate the hypothesis that hosts in Lyon register on Airbnb out of opportunism, but that the platform drags them and trains them to intensify their practice and to earn more and more benefits from their accommodation(s).

Translated from the French by Guillaume Barral : « …l'on assiste sur toutes les plateformes à une évolution du type d'utilisateurs. Même les plateformes qui s'étaient construites sur une mise en relation de particuliers à particuliers se repositionnent sur un rôle d'intermédiation plus large et accueillent également des professionnels (pour élargir leur gamme de services) »(Terrasse,

2016, 15). 2 Source: https://www.lesechos.fr/19/02/2016/LesEchosWeekEnd/00019-012-ECWE_airbnb--quand-un-canapecache-un-professionnel.htm (last checked: January 2018) 3 3 Source: https://www.mediapart.fr/journal/france/310715/comment-airbnb-squatte-la-france?onglet=full (last checked : December 2017)4 Translated from the French by Guillaume Barral: « limiter la professionnalisation de l'activité tout en permettant à chacun de louer de manière occasionnelle son logement, dans l'esprit collaboratif originel de ces plateformes ». Source: http://www.met.grandlyon.com/airbnb-co-de-nouvelles-regles-pour-un-developpementmaitrise-a-lyon/ (last checked: March 2018) 5 Source: http://tomslee.net/airbnb-data-collection-get-the-data (last checked : September 2017)

Source: http://blog.airbnb.com/economic-impact-airbnb/ (last checked: October 2017)

Translated from the French by Guillaume Barral: « les simples particuliers voulant faire de l'économie collaborative, de ceux, plus professionnels, qui en font un business bien rodé »(Deligia, 2018, 58)


Dirk Steglich
email: dirk.steglich@hereon.de

Jacques Besson

Inken Reinke

Heike Helmholz

Monika Luczak

Vasil M Garamus

Björn Wiese

Daniel Höche

Christian J Cyron

Regine Willumeit-Römer

Strength and ductility loss of Magnesium-Gadolinium due to corrosion in physiological environment: Experiments and modeling

Keywords: Biodegradable magnesium, Finite elements, Damage modeling, Corrosion localisation, Mass loss, Mg-10Gd

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.

Introduction

Magnesium (Mg) and its alloys are becoming increasingly popular as alternatives to permanent implant materials due to their biodegradability, biocompatibility and ability to promote bone growth. Not only is the proven biocompatibility beneficial for tissue regeneration and consequently medical applications but also the documented osteoinductive and anti-inflammatory effects [START_REF] Lin | Biodegradable mg-based alloys: biological implications and restorative opportunities[END_REF]. Mg alloys have better mechanical properties and a better degradation behaviour than 5 biodegradable polymers. Since the typically used alloying element Aluminium can be potentially neurotoxic, researchers been published to date, including physical, phenomenological, analytical, and machine learning-based models. Dahms et al. [START_REF] Dahms | A simple model for long-time degradation of magnesium under physiological conditions[END_REF] presented a simplified model based on the mass conservation law, but did not include chemical and electrochemical processes due to the analytical nature of the approach. Other studies used continuum damage mechanics (CDM) models to describe the effect of Mg degradation on a phenomenological level. Gastaldi et al. [START_REF] Gastaldi | Continuum damage model for bioresorbable magnesium alloy devices -application to coronary stents[END_REF] and Grogan et al. [START_REF] Grogan | A corrosion model for bioabsorbable metallic stents[END_REF] used CDM models to simulate the diffusion-controlled degradation of Mg-based absorbable stents. The CDM approach allows for the study of the micro-scale geometric discontinuities induced by degradation on the mechanical integrity of the overall specimen without explicitly modeling the moving boundary. For addressing the latter, it is required to include the corrosive medium in the analysis, see e.g. [START_REF] Grogan | A physical corrosion model for bioabsorbable metal stents[END_REF]. In any case, these approaches do not consider the electrochemical processes that occur during degradation, which may vary locally. As a result, CDM model needs to be calibrated for each combination of corrosive environment and material composition. A method for experimentally generating the required data was described by Boland et al. [START_REF] Boland | Mechanical and corrosion testing of magnesium we43 specimens for pitting corrosion model calibration[END_REF] for the magnesium alloy WE43. The authors successfully predicted the strength reduction of tensile samples as a function of the mass loss caused by in vitro degradation. In the same spirit, the present work aims at extending the predictive capabilities of numerical tools towards the ductility quantified by the failure strain as a function of the corrosion time. Corrosion is regarded as initiation for strain localisation and damage initiation. In the following, a mass diffusion type analysis is utilised to model the degradation of a Mg-10Gd alloy. Similar to the procedure described in [START_REF] Hermann | Combining peridynamic and finite element simulations to capture the corrosion of degradable bone implants and to predict their residual strength[END_REF], a concentration field representing Faraday's dissolution-related phase change of magnesium is simulated by a mass loss analysis and subsequently mapped to a variable affecting the local strength in the domain. Based on such a configuration, a finite element simulation of compression tests and tensile tests is performed to analyse the global mechanical strength of the sample that remains at the given point in time due to the corrosion up to that point. In contrast to the mentioned work, here, the local strength loss and the damage variable are considered in general as independent. While the first one is directly taken from the mass loss analysis, the latter evolves during loading in the mechanical simulation. This allows for the realistic incorporation of surface defects resulting from corrosion into a simulation model, facilitating the analysis of their effects on the mechanical behaviour of standard samples.

Material

The Mg alloy with 10 wt% Gd (Mg-10Gd) was produced by direct chill casting followed by solid solution heat treatment and in-direct extrusion as described by [START_REF] Harmuth | Wide range mechanical customization of mg-gd alloys with low degradation rates by extrusion[END_REF]. In contrast to the previous work, a round bar of 6 mm was extruded. The parameters were chosen so that the properties are comparable to the material extruded at 400 °C with an exit speed of 3.3 m/min. The exit speed was identical to that of 10 mm rods and adjusted due to the different extrusion ratio. Therefore, this material was extruded at 0.8 mm/s with an extrusion ratio of 1:69 at 400 °C. Run-in and run-out of the extrudate were cut and not used in the following.

Experiments

To quantify the corrosion and its effect on the mechanical behaviour, two different sample geometries were considered: prismatic bars (4 mm×4 mm×8 mm), and prismatic sticks (2 mm×2 mm ×50 mm), similar to those used in [START_REF] Boland | Mechanical and corrosion testing of magnesium we43 specimens for pitting corrosion model calibration[END_REF]. The first one served two purposes: it was used during immersion tests to determine the mass loss as a function of time and to do the subsequent compression testing. The latter served as a tool to characterise the tensile properties of the corroded material. Some samples were further examined before mechanical testing by computed tomography (CT). The samples are referred in the following as immersion bars (IB) and tensile sticks (TS), respectively. All samples were machined from the center of the extruded rod with identical machining parameters to achieve reproducible and comparable surface quality. This state is referred to as "as-received" in the following. It is the typical status prior to mechanical testing, but not suited for corrosion experiments. The latter required a more specific surface preparation, see the following section.

A subset of IB samples were quasi-statically compressed up to a strain of 10% to investigate the effect of plastic deformation on the corrosion rate. These samples are referred to as "predeformed" in the following. Table 1 summarises the number of samples and the different stages of immersion considered.

Corrosion

The immersion tests were performed according to NACE/ASTM G31-12a [START_REF] Nace | Standard guide for laboratory immersion corrosion testing of metals[END_REF] standard as described by [START_REF] Nidadavolu | On the determination of magnesium degradation rates under physiological conditions[END_REF]. Prior to immersion, the materials were treated according to [START_REF] Gawlik | The effect of surface treatments on the degradation of biomedical mg alloys-a review paper[END_REF][START_REF] Gawlik | Acetic acid etching of mg-xgd alloys[END_REF] to obtain a homogeneous surface. At first, all types of material were cleaned with organic solvents for 40 min in an ultrasound bath in n-hexane. This procedure was followed by etching with acetic acid (150 s in 125 g glacial acid per 0.5 l ultrapure water type I). The material was neutralised and cleaned by short dipping in 1 mol/L NaOH, ultrapure water type I and acetone. Finally, the material was cleaned again for 40 min in n-hexane before drying and weighting. This state is referred to as "as prepared", see table 1. The weights of the immersion bricks were recorded at specific points in time to calculate the average degradation rates based on mass and a moderate localisation of corrosion and degradation with statistically distributed pitting events. In addition to that preparation, the ends of the tensile sticks were protected with 5 mm heat-shrink tubing before sterilisation and immersion.

The immersion test was performed under sterile and physiological conditions at 37 (Thermo Fisher Scientific Inc.). To optimise the immersion vessel regarding the specimens' geometry, the immersion bricks were placed in 12-well microtiter plates in 3 mL; the tensile sticks were placed in T25 cell culture flasks in 8 mL.

As the immersion test was pursued in a semi-static manner, the medium was changed twice a week for one to five weeks.

The pH value and osmolality were monitored during media exchange. Additionally, the released Mg ion concentration was determined in the immersion medium supernatant by atomic absorption spectroscopy as described in Globig et al.

[18] using a flame AAS (Agilent 240 AA, Agilent Technologies) at a wavelength of 285.2 nm. A calibration curve ranging from 0.05 to 1.00 mg/L Mg was applied for quantification.

After the immersion test, the degradation products of the immersion bricks were removed with chromic acid (immersion for 10 min 180 g/L Chromium(VI) oxide in ultra pure water) to calculate their mass loss. Before and after this procedure, the weights of the immersion bricks were measured with an electronic balance (SBA 32, accuracy 0.1 mg, SCALTEC Instruments GmbH). In addition to the mass loss method, a uniform-type corrosion rate was calculated by dividing the mass loss Δ [g] through the product of the surface area , the density of the sample and the time of immersion . In case of the immersion bricks, an initial surface area of = 160 mm 2 and a density of = 1.876 10 -3 g/mm 3 were taken.

J o u r n a l P r e -p r o o f

Journal Pre-proof (Volume Graphics GmbH, Heidelberg, Germany). The software Simpleware (Synopsys, Inc.) was used to further edit and segment the reconstructed CT scans. Before segmentation, all scans were aligned and cropped to reduce the background.

To segment the scans, multi-level Otsu segmentation was used that selects an optimal threshold with discriminant analysis applied on the histogram [START_REF] Otsu | Threshold selection method from gray-level histograms[END_REF]. Three masks were chosen for segmenting the tensile stick scans, see Fig. 1. These masks represented the corroded sample (Mg-10Gd) (yellow), the corrosion product (green) and the surrounding area (turquoise).

For the immersion bricks, only two masks were chosen because the corrosion product was removed from them. Subsequently, an FE discretisation was created in Simpleware with the mask of the corroded sample (Mg-10Gd). The FE model was exported to Abaqus via a standard input file (*.inp).

J o u r n a l P r e -p r o o f

Journal Pre-proof

Mechanical Testing

The tensile tests were conducted on a Zwick Roell Z0101 machine at a speed of 2 mm/min, with a maximum force of 2500 N and a preload of 0.3 N. While the force signal was acquired by the load cell, the displacement signal was recorded with the help of Electronic Speckle Pattern Interferometry, with a gauge length of approx. 6.5 mm. Additionally, the cross-head displacement was recorded. In the case of IB compression tests, a Schenck servo-hydraulic 1000 kN machine was used. Load was applied via mirror-polished parallel plates in axial direction, with a cross-head speed of 0.4 mm/min.

Two extensometers were used to measure the relative displacement of the plates. The average of their readings was used to calculate the displacement along the compression direction. Stresses and strains reported in the following are understood as normalised forces (i.e., force divided by the initial cross-section) and strains as length change divided by the initial length. Hence, they constitute global quantities, characterising samples as a whole.

To verify that the tests performed with our special sample geometries were mechanically meaningful and delivered results comparable to those of tests with standard sample geometries, a comparison on the basis of the as-received material was performed. Standard samples (smooth round bars and compression cylinders) were machined from the extruded material. Figure 2 shows the results of tensile and compression tests performed on standard samples vs. the special samples used here. Apparently, the mechanical behaviour of both was nearly identical over a large strain interval. Since strain localisation is favoured in the case of TS and IB samples, these samples only failed at lower strains compared to the standard samples.

Modeling

Four subsequent steps were conducted to identify the data required to numerically predict the time-dependent strength loss. The strength loss is caused by uniform and localised degradation:

• Determination of the elastic-plastic behaviour of the as-received material by FE-simulations of tensile-and compression tests (no corrosion),

• • Simulation of the mechanical behaviour of tensile and compression samples on the basis of a damage model to predict the strength loss effect induced by corrosion.

For the two latter points, statistical effects were investigated by considering Weibull-distributions of the surface flux leading to localisation of mass loss, one cause for pitting corrosion [START_REF] Jafarzadeh | Computational modeling of pitting corrosion[END_REF]. The description of pitting corrosion of aluminium via Weibull statistics was discussed by [START_REF] Gazenbiller | Statistical analysis of AA-1050 localized corrosion in anhydrous ethanol[END_REF]. The flow of data described above is depicted in fig. 3.

Mechanical Behaviour: Plasticity and Damage

Tensile and compression tests of the as-received material were used to calibrate a plasticity model. Due to the textureinduced asymmetry in tension and compression, the Cazacu-Plunkett-Barlat (CPB)-model [START_REF] Cazacu | Orthotropic yield criterion for hexagonal closed packed metals[END_REF] was selected for describing the irreversible time-independent deformation. Since the extrusion direction (ED) was the main loading direction and no information on other deformation modes apart from tension/compression along the ED was available, the original isotropic 140 version of the CPB-model was used. The CPB-model was combined with a damage model on the basis of a weak coupling approach to predict damage nucleation and failure numerically. To this end, the effective stress of the CPB-model was used to define a yield surface by a plastic potential

Φ , , = 0. (1)

J o u r n a l P r e -p r o o f

Journal Pre-proof Here is the Cauchy stress tensor, the internal damage variable, and the flow stress of the undamaged (matrix) material, which changes in general with the accumulated plastic strain. The weak coupling between plasticity and damage 145 was achieved by defining the yield surface as

Φ = -(1 -) = 0 (2)
and using the following definition of an "equivalent stress"

= | 1 | -1 + | 2 | -2 + | 3 | -3 1∕ , ( 3
)
where 1 , 2 and 3 are the eigenvalues of the deviator of , controls the shape of the yield surface, and the strength differential effect (-1 < < 1). The flow stress of the material was expressed by a function of the accumulated plastic strain as:

150 ( ) = 0 + 1 (1 -exp -1 ) + 2 (1 -exp -2 ) . ( 4
)
Damage evolution was assumed to be controlled solely by the plastic strain via a damage nucleation function ̇ [START_REF] Chu | Void nucleation effects in biaxially stretched sheets[END_REF],

hence:

̇ = ̇ = √ 2 exp - 1 2 - 2 ̇ . ( 5
)
The modulus of elasticity 0 for isotropic elasticity was determined by using a resonant frequency and damping analyser on solution annealed Mg-Gd alloys with different Gd contents and assumed to be to 44.25 GPa [START_REF] Xu | The effect of solid solute and precipitate phase on young's modulus of binary mg-re alloys[END_REF]. Poisson's ratio was set to 0.3. The model parameters , , 0 , 1 , 1 , 2 , 2 , , , were calibrated by fitting stress-strain data of the finite-element simulations to the experimental results of the as-received (i.e. uncorroded) samples.

Diffusion-type Model for Mass Loss Prediction

The corrosion process was simulated by means of a three-dimensional diffusion-based model. The diffusion process representing the faradaic dissolution of magnesium samples was simulated through the finite element method according to Fick's second law

= Δ , ( 6
)
where is the magnesium concentration, the time, the coefficient of isotropic diffusion and Δ the Laplace operator.

Following the procedure outlined in [START_REF] Grogan | A corrosion model for bioabsorbable metallic stents[END_REF][START_REF] Marvi-Mashhadi | Simulation of corrosion and mechanical degradation of additively manufactured mg scaffolds in simulated body fluid[END_REF], nodal concentration fluxes on the surface of the samples were randomised with the use of the probability density function of the Weibull distribution and a scaling factor to simulate localised and pitting corrosion. The concentration fluxes act as Neumann boundary conditions of the diffusion problem. The negative sign represents the flow direction out of the solid:

= -( ) with (7)
( ) = -1 exp (-( ) ) , ( 8
)
with being the shape parameter of the distribution.

To achieve a prescribed mass loss at a given time in the simulation, the scaling factor was introduced in Eq. 7 and

calculated via ∑ = . ( 9
)
is the target overall flux preset and defines the integral/summation of all nodal fluxes. The concentration fluxes are connected to the change in concentration by Fick's law, Eq. ( 6). On this basis, the concentration at every integration point of each element can be calculated as a function of time. The global mass loss can be determined from the product of the concentration and volume in the integration points of the elements. The outcome is the amount of substance in this particular integration point. Summing up the amount of substance of each integration point results in the current amount of substance at time .

The corrosion process goes along with the formation of Mg(OH) 2 holding no mechanical relevance according to [START_REF] Marvi-Mashhadi | Simulation of corrosion and mechanical degradation of additively manufactured mg scaffolds in simulated body fluid[END_REF].

Therefore, concentrations less than MgGd of the initial molar density MgGd were not considered in the summation, that is, MgGd was a cut-off value. The initial molar density MgGd = 6.73610 -5 mol∕mm 3 was calculated from the molar mass of magnesium and the measured volume and mass of the IB sample, and was chosen as 0.4. The diffusivity in Eq. 6 is set to 10 -8 mm 2 ∕ [START_REF] Marvi-Mashhadi | Simulation of corrosion and mechanical degradation of additively manufactured mg scaffolds in simulated body fluid[END_REF]. At this point one needs to be aware not to over-interpret its physical effect. The rate of magnesium ion motion also depends on the (nodal) fluxes applied in the simulations. Hence the value of is somehow arbitrary. The required iteration of the fluxes to meet the measured mass loss might give the interpretation that the combination of both, diffusivity and fluxes, lead to the observed state of corrosion.

The total amount of substance of a body, ( ), is calculated as the sum of individual concentrations and volumes,

( ) = ∑ ( ) ( ) -MgGd , ( 10
)
with being the Heaviside function. The amount of substance at time = 0 is used as a reference and is denoted as start . The mass loss, Δ 0 ( ), is therefore calculated from:

Δ 0 ( ) = start -( ) start . ( 11
)
By iteratively simulating the mass diffusion process with different scale factors , a given mass loss can be achieved that depends on the Weibull distribution. This procedure was implemented using a Python code together with the Finite Element Suite ABAQUS/Standard [START_REF] Mangels | Simulating the influence of corrosion on the mechanical properties of a magnesium alloy for bone implants[END_REF].

Simulation of Strength Loss and Failure

This section describes a procedure to predict the strength loss of arbitrary three-dimensional solids due to mass loss.

It aims at replacing the scanning process described in chapter 3.2 by exploiting the diffusion model. As such, the spatial discretisation of the corroded sample becomes obsolete, and a mechanical simulation can be conducted on the basis of the initial geometry by using a damage model. The constitutive model described in section 4.1 will be applied here.

A key point is the translation of the progress of corrosion as captured by the diffusion simulations to the mechanical model, which is explained in the following. As introduced in section 4.2, the concentration (of the metallic phase) is described by a variable ∈ 0, MgGd . This concentration can then be related to the volume fraction of oxide which is assumed to be given by:

Ox = 1 - -MgGd (1-) MgGd for ≥ MgGd 1 for < MgGd . ( 12
)
This renders Ox = 0 for = MgGd , and Ox = 1 for ≤ MgGd . Assuming a simple rule of mixtures, the flow stress of the corroded material becomes

= ( , Ox ) = ( )(1 -Ox ) , ( 13
)
where the "pure" oxide material ( Ox = 1) is assumed to have no load-carrying capacity. In this equation, ( ) is still given by Eq. 4. A similar rule of mixtures was assumed for Young's modulus, that is, ( Ox ) = 0 (1 -Ox ).

Ox is assumed to take effect after a given period of corrosion but prior to the simulated mechanical test. That is, Ox pictures the mechanical effect of the corrosion product. By manipulating the result of the diffusion analysis by Eq. 12, the weakening of the material due to corrosion can be mapped to the relevant material points of the mechanical model. In the present case, concentrations given at nodal positions were interpolated to integration point values by a mapping tool available in the FE code Z-set [START_REF] Besson | Object-oriented programming applied to the finite element method part ii. application to material behaviors[END_REF] and subsequently read in as initial conditions.

Finite Element Models

For the simulation of the mass loss due to diffusion and for the mechanical analysis, identical spatial discretisations were used for each sample. The IB geometry was homogeneously meshed using 128,000 8-node diffusion elements (DC3D8 in ABAQUS) for the corrosion analysis and 8-node stress-strain elements with selective integration for the compression analysis with Z-set. For the mechanical analysis, displacements were prescribed on the quadratic faces of the IB parallel to the (x,y)-plane, and ′ , see Fig. 4. Reaction forces were recorded from these nodes. The strain was calculated from these displacements normalised by the initial length of 8 mm.

For the tension analyses, typical plane dog-bone samples were considered in the simulations. In the following, they are referred to as micro flat tensile samples (MFT). Note that their geometry was different from the one of the TS samples used in the experiments. The reason for this difference was that we sought to avoid numerically triggered localisation effects expected for exactly prismatic geometries. Shoulder regions ensure that load transfer is sufficiently uniform in the gauge section. Additional geometrical imperfections were not used. Diffusion and tensile analyses used discretisations with 71,660 8-node diffusion elements (DC3D8 in ABAQUS) and 8-node stress-strain elements with selective integration for the tension analysis. Since the shoulders of the MFT samples solely serve as load transmitters, their effect in the diffusion analysis is irrelevant. Therefore, nodal fluxes were only applied on the four faces of the prismatic section, see Fig. 4. Axial displacements were applied for the simulation of the tensile tests on the shoulder surfaces and ′ , see fig. 4. The reaction forces divided by the initial cross section constituted the stress signal. The strain was calculated from the displacement difference in the parallel section. The failure strain was defined as the strain at maximum global stress of each sample. Since the material does not show severe necking in tension, this definition leads to similar results than considering the elongation at fracture.

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Results

Assessment of experimental Results on Mg-10Gd Degradation

The morphologies and types of used specimens result in a different degradation behaviour. One representative immersion brick sample has been selected and is shown in Fig. 5. This sample is representative of most corroded immersion bricks as it has small pits all over its surface leading to a nearly homogeneous degradation after five weeks. For assessing the degradation behaviour of the tensile sticks qualitatively, the geometry as well as the ratio of surface to volume has to be considered. Opposite to the compact geometry of the immersion bricks, the tensile sticks are elongated with a larger surface area exposed to the immersion medium. Pitting corrosion and breaks of specimens were observed frequently, which prevented a comprehensive investigation.

The degradation of the immersion bricks was quantified via mass loss and release of Mg during degradation over five weeks. Fig. 6a shows the relative mass loss of immersion bricks increasing from a median of 0.8% after one week to a maximum of 3.5% over the corrosion time of five weeks. This was observed for both as prepared (ap, grey) and predeformed immersion bricks (def, red). Note that most recorded data points are close to each other in their respective group, which results in small standard deviations where these were not boosted by outliers, compare Fig. 6a. Very similar median values were observed for the as-prepared and predeformed immersion bricks.

In addition to the mass loss, the Mg release and the corrosion rate, which is based on the mass loss, are shown in of the degradation behaviour of the immersion bricks is confirmed by the quantitative analysis and can be described as relatively homogeneous with 3 outliers out of 50 specimens.

Parameter Calibration for the as-received Material

FE simulations were conducted for IB and MFT samples to describe the observed measured force-displacement behaviour of the as-received material. From the tensile test, at first the parameters of the hardening law, Eq. ( 4), were 250 calibrated. Second, the parameter describing the strength differential effect in Eq. ( 3) was determined on the basis of the compression test. Finally, the parameters of the damage model were set to meet the failure point observed in the tensile tests. Table 2 shows the parameters producing the simulation results depicted in Fig. 7. Since the heads of the MFT samples were included in the FE model, strain localisation occurs diagonally in the central region of the samples. This is caused by slightly elevated strain in the middle and subsequent damage at that position. For more details on the localisation 255 behaviour, readers are referred to [START_REF] Besson | Modeling of plane strain ductile rupture[END_REF], which analyses discretisation effects and model symmetries.

[MPa] 1 [MPa] 2 [MPa]

Tensile Behaviour of corroded TS-Samples

From the set of corroded TS samples, six individual samples were selected, which showed significant corrosion signs near or in the middle of the test section. The specimens were immersed for one week (two pieces), two weeks (one piece), three weeks (two pieces), and four weeks (one piece), see table 3. Prior to mechanical testing, they were scanned following the procedure described in section 3.2 to obtain the exact shape of the samples and of the corrosion product.

The magnesium material domain in the simulations was defined based on the image segmentation (yellow mask in Fig. 1).

Finite element simulations of the tensile tests were performed using the parameter of the as-received material, see table 2. For the discretisation, tetrahedral volume elements (C3D4 in ABAQUS) were used. The results of the simulations are depicted in Fig. 8a. The stress level of each individual sample could be met well. This indicates that the geometry of the corrosion pits was mapped sufficiently accurately to the finite element model for capturing the local reduction of cross section. It also confirms that the latter is primarily relevant for the strength loss caused by corrosion [START_REF] Van Gaalen | Automated ex-situ detection of pitting corrosion and its effect on the mechanical integrity of rare earth magnesium alloy -we43[END_REF]. The failure strain defined as the strain at maximum global stress, however, was consistently overestimated in the simulations, see dashed lines in Fig. 8a. This is, to a certain extent, expected since local porosities and micro-cracks caused by corrosion were not resolved in the CT scan. Hence, the damage nucleation strain had to be changed in order to capture these effects.

was therefore reduced compared to the value given in table 2 and set to 0.165, with the other parameters unchanged. This resulted in predictions represented by the solid lines noted "FEMmod" in Fig. 8a. A summary focusing on the measured From the simulations described in this section, it can be concluded that in samples that have undergone already some corrosion, the mechanical characteristics of the not yet corroded material are largely the same as those of the as-received material. Only the effects related to surface roughness and micro-cracks reduce the load bearing capacity. The (global) failure strain can be predicted with reasonable accuracy once micro-cracks on the surface are considered as damage initiators.

The degradation layer does not need to be taken into account in this context.

Prediction of the Compression Behaviour (IB)

Simulations of IB compression tests were performed using the same discretisation as shown in Fig. 4a. Motivated by the appearance of the corroded surfaces (see Fig. 5), the parameter of the Weibull distribution, Eq. 8 was set to = 1.0, rendering a moderately smooth distribution of nodal fluxes in the mass diffusion analysis. The surface appearance is similar in this case to the experiment, compare Figs. 5b and9. The nodal fluxes were iterated to give the measured mass loss of 3.5% after five weeks, depicted in Fig. 6a. The calculated concentration fields were mapped to the oxide volume fraction Ox following the procedure outlined in section 4.3 for the points in time of one, two, three, four, and five weeks. For each of these stages, one simulation of the compression test was performed, rendering the results given in Fig. 10. While in the simulations a clear trend of stress reduction with immersion time prevails, this trend was less clear in the experiments.

In Fig. 10a, a single averaged stress-strain curve is presented for each corrosion stage to improve readability, whereas all

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Journal Pre-proof 8b for tensile specimens. Two power-law functions were fitted to visualise the decrease of maximum stress and failure strain (i.e. strain at maximum stress). The simulations meet the fitting functions very well. This indicates that the concept of sequential mass diffusion analysis and concentration-damage mapping is able to predict strength loss and reduction of failure strain in the time range considered.

It is worth mentioning that the change of the initial slope of the stress-strain curves of the corroded samples is caused by the roughness of the contacting surfaces, and is not related to a reduction of the net cross section of the samples. This effect could be captured in the simulations by adding two thin layers of elastic material to the surfaces exposed to compression load. Further, in these compression simulations, the damage nucleation strain, , was taken from the asreceived material, = 0.24, because in compression surface micro-cracks caused by corrosion do not play a role (crack closure effect). Micro-cracks are relevant for the ductility in tension, but not in compression.

Prediction of the Tensile Behaviour (MFT) as a Function of Corrosion Time

As already mentioned in section 5.1, the corrosion of TS samples was less homogeneous than in the case of IB samples.

Local attacks and pronounced pitting led to the breakage of many samples. After four weeks, only one sample could be taken for tensile testing, and after five weeks none of the initial five samples was left, see table 3. This is represented in Fig. 11, where stress and strain maxima experienced during tensile testing of the corroded samples are shown. To capture these results for the TS/MFT samples numerically, a high level of localisation for the mass diffusion analysis had to be chosen.

= 0.01 was selected here to seed fluxes on the sample surface. Further, the mass loss data experimentally acquired from the IB samples had to be adopted to the MFT samples. The relative volume losses were given as a function m a x . s t r e s s ( t e s t s ) f i t o f t e s t d a t a f a i l u r e s t r a i n ( t e s t s ) f i t o f t e s t d a t a s i m u l a t i o n s : m a x . s t r e s s ( a = 1 . 0 ) s i m u l a t i o n s : f a i l u r e s t r a i n ( a = 1 . 0 ) m a x . s t r e s s [ M P a ] t i m e [ w ] s t r a i n @ m a x . s t r e s s of the mean degradation depth

IB ∶ = 4 3 -8 2 + 5 (14)
MFT ∶ = 4 2 + 4 , ( 15
)
with being the characteristic length dimension of IB ( = 4 ) and MFT ( = 2 ). Provided that the mass density is the same for both samples, relative mass loss and relative volume loss are interchangeable. Solving ( 14) with = 0.035 for yields a degradation depth after five weeks of 0.028 for this sample. On the assumption that the 310 corrosion rates are identical for IB and MFT samples, inserting this into (15) renders an equivalent volume loss of 5.74%, which was iterated in the mass diffusion analyses once the MFT sample is considered. Note that here only the four outer surfaces were mapped with nodal fluxes, since the corrosive medium can only attack from these sides. The results of one representative mechanical analysis per point in time are illustrated in Fig. 11. The reduction in maximum stress was reasonably well captured by the simulations, with deficits, however, for longer corrosion times. The reduction of ductility 315 was captured well for all corrosion times, see Fig. 11b.

Discussion

There are different techniques and methods to determine the corrosion rate of metals. Considering the benefits of Mgbased biomaterials for medical applications, the description of the degradation behaviour under physiological conditions is particularly relevant [START_REF] Esmaily | Fundamentals and advances in magnesium alloy corrosion[END_REF]. However, the selection of physiological test conditions and media increases the complexity of the system. The immersion in a cell culture medium provides small negatively and positively charged ions, small organic compounds as well as large peptides and proteins. All these components relate to different effects on the Mg corrosion behaviour. A high -content accelerates the corrosion, 3-bind free 2+ ions and form less soluble complexes [START_REF] Agha | Magnesium degradation influenced by buffering salts in concentrations typical of in vitro and in vivo models[END_REF]. Most importantly, the interaction of material surface with proteins, which are introduced by the fetal serum, resulted in the complexation of 2+ and the formation of a more or less stable protection layer which may reduce the degradation rate [START_REF] Helmholz | In vitro investigation on degradable mg-based biomaterial under the impact of the serum glycoprotein fetuin[END_REF][START_REF] Hou | Different effects of single protein vs. protein mixtures on magnesium degradation under cell culture conditions[END_REF]. Due to such complex interactions, the sensitivity of the test system to variations of the degradation conditions is higher compared to single salt solutions and ambient conditions. Standards were defined for testing the immersion behaviour of metallic medical devices in electrolytes as ISO 10993-15 standard [START_REF]Iso 10993 biological evaluation of medical devices -part 15: Identification and quantification of degradation products from metals and alloys[END_REF], ASTM NACE/ASTM G31-12a standard, and ISO10933 part 12 for non flat, irregular samples. In the present study, the conditions were selected accordingly and slightly adapted. Immersion conditions were kept constant and equalled an electrolyte volume to test specimen surface area ratio of 0.125 ∕ 2 for the IB samples, and to 0.020 ∕ 2 for the TS samples. This represents a compromise between covering the whole sample body with a sufficient amount of immersion medium and meeting the recommendations of the mentioned guidelines. Nevertheless, differences in the corrosion behaviour between the IB and TS samples were observed.

The more pronounced local degradation of the TS compared to the IB might be attributed to the different volume to surface ratio. In semi-static degradation tests, less volume per surface (ratio between test solution volume and test specimen surface area) leads to a faster increase in pH, which promotes dense deposit layer formation and subsequently should lower the degradation rate [START_REF] Zhao | Influence of ph and chloride ion concentration on the corrosion of mg alloy ze41[END_REF]. As pit initiation statistics very much vary with the alloying element distribution, grain boundary effects as well as surface defects, it is still possible to observe high degradation rates despite electrolyte and deposit effects. Within this study it was observed that corrosion was suppressed in 3.5% NaCl solutions at pH values higher than 12.

Localised corrosion did not appear as already shown by Zeng et al. [START_REF] Zeng | Review of studies on corrosion of magnesium alloys[END_REF]. However, the cited work showed an influence of pH values on the distribution and morphology of corrosion pits. High pH values lead to more local corrosion with pit formation, limited by pH=12, above which corrosion was suppressed. This effect seems to be also existent in the present work, where the rapid increase in pH in the case of TS samples lead to severe local corrosion.

Due to the described differences in corrosion localisation, it was not expected that an identical Weibull distribution of fluxes, as considered in the case of IB samples, would produce the same results as the mechanical analyses of the TS/MFT samples. Apparently, such reuse of Weibull parameters is generally not possible even if the degrading systems are comparable. Due to the high sensitivity of the pit initiation mechanisms to small variations within the surface defect morphology as well as to the local electrolyte conditions, the application of IB/TS calibrated values to the MFT samples is, in general, not possible. If the degradation , in particular, becomes more heterogeneous, this aspect has to be kept in mind.

As a consequence and as described in section 5.5, = 0.01 was chosen to adequately simulate the localised corrosion scenario of the tensile samples. To quantify the effect of the Weibull statistics, a numerical study was performed. An MFT sample was seeded for mass diffusion (weight loss) analyses with four different Weibull distributions, namely with = 1.0, = 0.5, = 0.1, and = 0.01. For each case, 10 realisations were generated, leading to 40 simulations in total. The fluxes were iteratively determined such that they resulted in the same global mass loss in all cases. Only the parallel sections of the sample were considered to be affected by corrosion. One realization for each of the above four Weibull distributions is depicted in Fig. 12. Although the position of the nodal fluxes was randomly chosen in the parallel section, the simulations are comparable in the sense that the corrosion conditions were identical. The results of the subsequent mechanical analyses are depicted in Fig. 13. With increasing localisation, = 1.0 → 0.01, the maximum stress decreases by 52%. At the same time, the variation increases. Even more pronounced is the reduction of ductility, expressed by the average strain at maximum stress. Even for the "smooth" Weibull distribution of = 1.0, an average reduction of ductility compared to the one of the as-received material of 49% is calculated. = 0.01 renders a reduction of 91%. In essence, a high localisation level of corrosion leads to a moderate stress reduction, but to a drastic loss of ductility. In its principle, this is neither a contradiction nor surprising: Due to the low strain-hardening of the material, small changes in stress lead to large variations in strain. This is common for as-received material, but it also applies to corroded samples with non-uniform cross-sections.

The effect of the damage variable d in the analyses of failure is illustrated in Fig. 14. The mapped field of corrosion product Ox (see Fig. 14a) has two main effects on mechanics. First, it reduces the transfer of stresses via Eq. 13. Second, due to the in-homogeneity of the field, it triggers strain localisation in the remaining metallic material. The latter acts on the damage evolution, Eq. 5, and finally causes cracks to develop from the surface, see Fig. 14b. By this interaction, the effect of corrosion pitting can be accounted for in the mechanical model. Although the described model is of phenomenological nature, it appears to be versatile and hence useful for the assessment of medical implants undergoing corrosion, provided that the results of this in vitro study can be generalised towards in vivo problems. In the current mathematical formulation, it is capable of covering the whole range from perfectly uniform degradation to statistically distributed pitting corrosion and its superposition. Intergranular effects as well as effects related to hydrogen uptake are not explicitly modeled, but they are intrinsically included by the domain damage approach.

Commonly, FE analyses based on coupled damage models leading to material softening as utilized herein, show a mesh dependency. Therefore, in the computational study a dependency of results on the discretisation is expected. With increasing mesh refinement for a given element formulation, the localisation zone becomes smaller. Hence, the dissipated energy decreases with decreasing element size. While this effect has significant impact in the case of fracture toughness simulations, the influence in the case of tensile samples is less pronounced [START_REF] Besson | Modeling of crack growth in round bars and plane strain specimens[END_REF]. Further, for the diffusion analysis it is expected that a higher density of nodes affects the degree of localisation of corrosion. To quantify both effects for the current model, simulations of mass diffusion and tension are conducted for a mesh with 563,400 elements (instead of the 71,660 elements for the simulations described above) for one realisation of the Weibull distribution with = 0.5. The result of the mechanical analysis is included in Fig. 13a by green open symbols. The stress-strain behaviour is identical to the one obtained with the coarser mesh. The predicted ductility falls in the strain interval given by the other simulations.

It can therefore be concluded that the conditions of equal mass loss lead to comparable results in case the discretisation is changed.

The damage model, Eqs. 2 and 5, is kept as simple as possible here. The assumed damage nucleation function solely depends on the plastic strain. The damage growth rate does not depend on the stress triaxiality. Consequently, the model does not consider the growth and coalescence of voids and defects. The well-known triaxiality-dependence of failure in ductile metals [START_REF] Hancock | On the mechanisms of ductile failure in high-strength steels subjected to multi-axial stress-states[END_REF] cannot be described. This simplification, however, appears acceptable, since failure initiation of corroded samples happens at the surface where the triaxiality equals 1∕3. It is further in line with observations made for other magnesium wrought alloys, for which strain-based failure models revealed quantitatively good results [START_REF] Steglich | Prediction of deformation and failure anisotropy for thin magnesium sheets under mixed-mode loading[END_REF][START_REF] Jeong | Modelling-assisted description of anisotropic edge failure in magnesium sheet alloy under mixed-mode loading[END_REF]. The parameter calibration presented here is robust, since it is possible to predict failure both in tension and compression. Note that the latter is triggered by a plastic instability, which depends on the length-to-width ratio of the compressed sample (IB: 2:1).

The application of the model to other Mg alloys is possible. It has to be considered that the intensity of pit formation depends on the alloying elements (see e.g. [START_REF] Kannan | In vitro degradation and mechanical integrity of calcium-containing 535 magnesium alloys in modified-simulated body fluid[END_REF]), and its distribution. For example, calcium addition enhances the pitting corrosion resistance of magnesium alloys significantly, and the surface properties as well [START_REF] Boland | Mechanical and corrosion testing of magnesium we43 specimens for pitting corrosion model calibration[END_REF]. In parallel, a comparison of grinded and not grinded samples of the same alloy revealed a delay of corrosion by a factor of 50 times for grinded samples. This sensitivity of pit formation is very challenging, but not impossible to model on the macroscale. As each Mg processing and treatment has a certain impact on secondary phases (distribution and impurities), it can be assumed that J o u r n a l P r e -p r o o f

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Conclusions

In this paper, a computational framework was developed to study the effect of corrosion on the mechanical behaviour of Mg samples. Corrosion is modeled by a mass-diffusion type model accounting for, in general, parallel occurrence of uniform and localised corrosion. To model the statistical effects governing localised corrosion attacks, Weibull statistics is used. In our corrosion model, the overall loss of mass of a sample has to be prescribed (e.g., based on experimental data) to normalise the spatially distributed mass fluxes in the corrosion model properly. The mechanical behaviour of Mg is modeled by a state-of-the-art CPB plasticity model with a coupled damage model. This allows us to study how Mg degradation in immersed samples reduces the mechanical strength over time. We performed a large number of corrosion experiments and mechanical tests to validate our computational framework in a laboratory setting. It turned out that our framework could predict for both tension and compression tests the experimentally observed loss of mechanical strength due to corrosion when using extremely sharp Weibull distributions representing pitting corrosion. In addition to the strength loss due to corrosion, the reduced ductility could be numerically predicted.

Our study confirmed the importance to consider the surface/volume effect of the samples during corrosion testing and numerical analysis. It was shown that the conversion of the volume mass loss data via the surface related mean corrosion depth can be an appropriate way to do so. With consideration of calibrated Weibull statistics (e.g. = 1 approximates uniform-like corrosion, < 0.1 approximates strong pitting events) the macroscopic mechanical behaviour of the implant material can be adequately computed. Adaption of the approach to other Mg alloys and materials suffering pitting corrosion is possible but requires careful Weibull parameter calibration. A general limitation of our current computational framework is the need to prescribe the overall mass loss to calibrate the boundary conditions of the corrosion model. To overcome this limitation, in the future more sophisticated corrosion models are required that can predict the rate of mass loss if the geometry and material of the sample as well as the chemical properties of its environment are given. Such a more advanced corrosion model could, for example, rely on a phase-field approach or a peridynamics approach [START_REF] Hermann | Combining peridynamic and finite element simulations to capture the corrosion of degradable bone implants and to predict their residual strength[END_REF][START_REF] Shojaei | Dirichlet absorbing boundary conditions for classical and peridynamic diffusion-type models[END_REF].

Figure 1 :

 1 Figure 1: Applied segmentation masks: yellow -metallic part of the sample, green -corrosion product, turquoise -surrounding volume)

Figure 2 :

 2 Figure 2: Results of the tensile-and compression tests: standard samples (symbols) and TS/IB geometries (lines).

  135

Figure 3 :

 3 Figure 3: Sequence of activies for the modeling.

Figure 4 :

 4 Figure 4: Finite-element models of IB (left) and MFT (right). Nodal fluxes are prescribed on the outer surfaces appearing in green.

Fig. 6b .Figure 5 :

 6b5 Fig. 6b. Fig. 6b illustrates the fast degradation during the first week followed by an almost constant release up to the fifth week. The variance of values increased within the progress of the immersion test. Altogether, the qualitative evaluation

5 CFigure 6 :

 56 Figure 6: Experimental results of the IB immersion tests (1 to 5 weeks)

Figure 7 :

 7 Figure 7: Finite-element simulations of compression-and tension tests compared to the averaged TS and IB experiments

Figure 8 :

 8 Figure 8: Simulation results of individual TS samples in comparison to tensile tests

Figure 9 :

 9 Figure 9: Mass loss analysis of IB ( =1.0): Magnesium concentration after five weeks

  r e f e r e n c e 0 w (a) Stress-strain behaviour of IB-samples

Figure 10 :

 10 Figure 10: Results of IB compression tests in comparison to simulation results

  i m u m s t r a i n T e s t s S i m u l a t i o n s a = 0 . 0 1 , M L = 5 . 7 % (b) Failure strains

Figure 11 :

 11 Figure 11: Experimental (TS) and numerical (MFT) results for Δ ∕ 0 = 5.7% and = 0.01

J 1 0

 1 o u r n a l P r e -p r o o f e s s [ M P a ] s t r a i n s y m b o l s : m e s h r e f i n e d (a) Simulated macroscopic stress-strain curves r e f a = 1 . 0 a = 0 . 5 a = 0 . 1 a = 0 . 0 a i n @ m a x l o a d c o r r o s i o n l o c a l i s a t i o n l e v e l o a d [ M P a ] (b) Mean values and standard deviations of maximum stress and failure strain

Figure 13 :

 13 Figure 13: Numerical study to quantify the effect of Weibull statistics on the tensile behaviour of MFT samples: = 1.0, = 0.5, = 0.1, and = 0.01

Figure 14 :

 14 Figure 14: Mechanical analysis of MFT under tension (a=0.5)

Table 1 :

 1 Test matrix (w: week)

	Sample type	State	Comment	Corrosion time [w] Number of samples
	Immersion Bricks (IB) as-received as prepared predeformed compression prior immersion 1, 2, 3, 4, 5 carefully machined -etched for immersion 1, 2, 3, 4, 5 Tensile Sticks (TS) as-received carefully machined -as prepared etched for immersion 1, 2, 3, 4, 5	6 5 per corrosion time 5 per corrosion time ⇒ in total 56 IBs 5 5 per corrosion time

⇒ in total 30 TSs loss during immersion. These values can be used in the simulations assuming a negligible initiation phase contribution

Table 2 :

 2 Calibrated model parameters of the plasticity and damage model plasticity 0

Table 3 :

 3 The tensile sticks' test and simulation history (TS: tensile stick)

	ID	Corrosion time [w] Tested Simulated	ID	Corrosion time [w] Tested Simulated
	TS-03 TS-04 TS-25 TS-26 TS-46 TS-07 TS-08 TS-29 TS-30 TS-48 TS-11 TS-12 TS-21	1 1 1 1 1 2 2 2 2 2 3 3 3	✓ ✓ ✓ ✓ ✓ ✓ x ✓ ✓ x x x ✓	✓ x ✓ ✓ x x x x x x	TS-22 TS-33 TS-34 TS-50 TS-54 TS-15 TS-16 TS-38 TS-39 TS-52 TS-42 TS-43	3 3 3 3 3 4 4 4 4 4 5 5	✓ ✓ x ✓ ✓ ✓ x x x x x x	x x ✓ ✓ ✓ x x x x x x

Journal Pre-proofFigure 12: Surface views of Ox for different configurations rendering the same mass loss (initial configurations prior to tensile testing)

Acknowledgments

The authors like to thank Danny Mangels for his diligent scripting and model calibration, Jürgen Markmann and Kay Erdmann for their assistance during mechanical testing, Mohammad Mavi Mashadi for providing example input files out of his previous work, and Rainer Falkenberg for fruitful discussions. The advices of Annette Havelberg on proofreading are very much appreciated.

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Declaratio if ioterettt

The authors declare that they have no known competng fnancial interests or personal relatonships ☒ that could have appeared to infuence the work reported in this paper.

The authors declare the following fnancial interests/personal relatonships which may be considered ☐ as potental competng interests: