Melanie Simms
email: melanie.simms@glasgow.ac.uk

Dennis Eversberg

Camille Dupuy

Lena Hipp

ORGANIZING YOUNG WORKERS UNDER PRECARIOUS CONDITIONS: WHAT HINDERS OR FACILITATES UNION SUCCESS

Keywords: Unions, organizing strategies, precarious work, young workers, qualitative case studies, Europe, US

Under what conditions do young precarious workers join unions? Based on case studies from France, Germany, the UK and US, we identify targeted campaigns, coalition building, membership activism, and training activities as innovative organizing approaches. In addition to traditional issues such as wages and training quality, these approaches also featured issues specific to precarious workers, including skills training, demands for minimum working hours, and specific support in insecure employment situations. Organizing success is influenced by bargaining structures, occupational identity, labor market conditions, and support by union leaders and members. Innovative organizing tends to happen when unions combine new approaches with existing structures.

Introduction

Around the world, many unions face challenges in engaging, recruiting, and representing young people. In Europe and North America, the general decline of union membership and density has drawn particular attention to overcoming these challenges as a possible route to renewal. These challenges have always existed because young people often have periods of instability within the labor market before they settle on a particular job and sector [START_REF] Corral | Working conditions of young entrants to the labour market[END_REF]. In the context of post-1945 economic prosperity, this labor market instability was not a major concern for unions, largely because many, or even most, workers ended up in organizations and occupations that did have union representation.

This has changed dramatically in many countries. Not only do young workers tend to spend far longer moving between precarious jobs in the formative stages of their working lives [START_REF] Allmendinger | Atypical employment in Europe 1996-2011[END_REF][START_REF] Bradley | Fractured transitions: Young adults' pathways into contemporary labour markets[END_REF], but they are also less likely to come into contact with unions in the course of these unstable trajectories [START_REF] Bryson | Why Have Workers Stopped Joining Unions? The Rise in Never-Membership in Britain[END_REF]. In consequence, unions' membership base is aging and increasingly concentrated in those sectors where stable jobs still dominate, which further challenges their vitality and survival [START_REF] Pignoni | La syndicalisation en France dans la fonction publique et les entreprises[END_REF].

How can unions respond to this situation? Is there something particular about young people's experiences of work and the labor market that means unions have to develop new approaches to engaging these workers? If so, what evidence is there of innovative union campaigns that successfully target young, precarious workers? And what factors facilitate or hinder the success of innovative organizing activities in different national settings? This article addresses these questions by presenting evidence from seven innovative organizing projects targeting young workers in four countries: the USA, UK, France, and Germany.

Although precarity varies between different national and sectoral contexts, we argue that young workers today generally experience deeper and more extensive precariousness than preceding generations. As a result, unions are facing an ongoing need to innovate in order to engage and represent them effectively. This article therefore explores in more detail the claim made in the introductory article of this special issue (Tapia and Turner, this volume) that "precarity breeds innovation." We illustrate how precarious working conditions can facilitate union innovation and what barriers unions and workers face in trying to organize in precarious contexts.

Before advancing the article's core argument, we need to define two key terms: young workers and union innovation. The precise definition of who counts as "young" is far from consistent, both among the unions investigated here and in the literature on union organizing. Far from solely being a biological or statistical category, youth is also a social experience, which in recent years has increasingly been characterized by precarity. The link between young people and precarity is further explored in later sections, but it is important here because it is central to the argument that youth is a relative rather than an objective category. Young workers are thus defined in relation to other generations rather than by a clear age boundary. In line with the focus of our research project, the case studies documented here all draw the line for young at 35 years of age.

Innovation is even more difficult to define. In line with definitions in the management literature (see [START_REF] Baregheh | Towards a multidisciplinary definition of innovation[END_REF] for an extensive discussion), this article takes innovative practices to be activities that markedly depart from customary practices and approaches within that specific union and sector. Such innovative practices spread across and between unions nationally and internationally [START_REF] Dörre | Organising-A strategic option for trade union renewal[END_REF].

In this article, we therefore focus on the adoption of different types of practices in settings in which they have not previously been used. This focus allows us both to identify different types of practices and to highlight limitations of previous union activities seeking to organize young workers in precarious jobs. Innovation from below (i.e., driven by workers or members) can be taken as an indication that existing structures and practices are not delivering for prospective members. Innovation driven from above (i.e. by a union driving new practices) indicates that unions have identified a potential weakness or limitation of their established structures and practices. Both forms of innovation therefore allow insight into the (perceived) weaknesses and limitations of existing activities.

Changes in union membership and support for unions among young workers

In recent decades, union membership has declined in most of the industrialized world. In OECD countries, the proportion of workers organized in a union is on average less than half what it was in the early 1960s. In the UK, where union density peaked at nearly 50 percent in the early 1980s, this proportion has decreased to around 25% today. Likewise, in France, union density decreased from around 20% to less than 10%, in the US from around 25% to around 10%, and in Germany from around 35% to less than 20% percent today (OECD & Visser, 2017). . While a decline in unionization can be observed in all industrialized countries, including the ones in this study, it has been particularly notable among young workers. Across Europe, with the exception of the UK in recent years, the greatest decline in unionization has been among young workers [START_REF] Serrano Pascual | Young People: The Labour Market and Trade Unions[END_REF]. In 2015, around 13% of workers aged between 18 and 35 years in Germany were union members (ISSP, 2015, own calculations). In the UK, this rate was at around 18%, in France around 8%, and in the US around 7%. Unionization rates of young workers were therefore well below the average unionization rate in all of these countries (see Figure 1).

FIGURE ONE ABOUT HERE

As [START_REF] Bryson | Why Have Workers Stopped Joining Unions? The Rise in Never-Membership in Britain[END_REF] have shown, lower unionization rates among young workers are not necessarily an expression of a lower desire for unionization. On the contrary, an examination of recent data provided by the International Social Survey Program from the year 2015 shows that, with the exception of Germany, slightly more young workers agree or even strongly agree with the statement that "unions are good for workers" than the workforce as a whole (see [START_REF] Givan | Public Perceptions of Union Efficacy:A Twenty-Four Country Study[END_REF][START_REF] Hipp | What Do Unions Do? A Cross-National Reexamination of the Relationship between Unionization and Job Satisfaction[END_REF] for international comparisons of attitudes towards unions more generally). Young workers in the UK and the US, hold particularly positive opinions about unions. In France, support does not significantly differ between young workers and the general working population. Only in Germany is there lower support for unions among young workers than among the wider working population (see Figure 2).

FIGURE TWO ABOUT HERE

Precarity: Deteriorating Labor Market Opportunities for Young Workers

Central to why union membership and union density tend to be lower among young workers are structural labor market changes. In particular, one reason for the low union membership among young workers is the prevalence of non-standard employment among labor market entrants and the emergence of new jobs in low-skill service occupations [START_REF] Oliveira | Youth and precarious employment in Europe[END_REF][START_REF] Vandaele | Youth representatives' opinions on recruiting and representing young workers: A twofold unsatisfied demand[END_REF]. In the countries under consideration here, the number of open-ended full-time positions declined in all labor market groups, while precarious employment in various forms has continued to expand [START_REF] Allmendinger | Atypical employment in Europe 1996-2011[END_REF]. Forms and consequences of non-standard employment vary across national, sectoral and occupational settings [START_REF] Armano | Mapping Precariousness, Labour Insecurity and Uncertain Livelihoods: Subjectivities and Resistance[END_REF][START_REF] Hipp | Institutions and the prevalence of nonstandard employment[END_REF].

In this study, we focus on the employment relationship, although we acknowledge that young people often experience multiple forms of precarity that do not exclusively result from their paid work. We understand employment precarity as resulting primarily from the destabilizing of open-ended, full-time employment contracts. Despite variations between national contexts, fixed-term employment tends to be the most important form of nonstandard work among young people in most European countries (Hipp et al., 2015). In the US, young people also tend to be employed on fixed-term contracts twice as often than working population in general (CPS 2005 supplment, see Flood, King, Ruggles, & Warren, 2017 for further information) What is clear is that the structural shifts in employment contracts and labor markets disproportionately affect young workers because they are new entrants, and this was taking place even before the labor market restructuring that occurred in the wake of the 2008 financial crisis. As a result, even those young workers who are successful at entering the labor market face far greater risks of short-term and flexible employment than recent generations.

Despite the growth in precarious employment and the related precarity it inevitably brings to wider social life, the issue of economic security and job stability is of paramount importance to young workers. Not only is a smooth transition from school to work related to future occupational success (Brzinsky-Fay, 2017), the prospect of having a secure job is also subjectively very important. In a recent survey on work orientations, more than 90 percent of young workers in France, Germany, the UK, and the US said that having a job was (very) important to them personally. However, when the same people were asked whether their jobs actually were secure, only around a third of young workers under the age of 33 in France and the UK and around 40 percent in Germany and the US were convinced their jobs were secure (International Social Survey Programme, 2015, own calculations).

Joining a union and paying membership dues may not appear attractive to young workers whose experience of the labor market may well include periods of not having a job, being employed on a temporary basis, and/or feeling insecure about their job. At the same time, union membership tends to be dominated by core workers [START_REF] Pignoni | La syndicalisation en France dans la fonction publique et les entreprises[END_REF], that is, typically older, male, and high-skilled workers. In practice, these workers often enjoy greater protections in law or collective agreements against job loss and low wages at the expense of labor market outsiders, who are more likely to be young, female, and less skilled workers. In some contexts, this may make unions seem to be less natural allies for young, precarious workers and may even mean they are seen as contributing to labor market precarity by regulating and enforcing protections for core workers [START_REF] Barbieri | Italy: No country for young men (and women): The Italian way of coping with increasing demands for labour market flexibility and rising welfare problems[END_REF][START_REF] Rueda | Social democracy and active labour-market policies: Insiders, outsiders and the politics of employment promotion[END_REF]. Despite this, [START_REF] Benassi | Straight to the core -explaining union responses to the casualization of work: The IG Metall campaign for agency workers[END_REF] show that unions have often effectively responded to pressures for segmentation by pursuing inclusive strategies that target core and peripheral workers equally.

Researching union innovation in representing young workers

The qualitative data analyzed in this study were collected between 2014 and 2016 as part of a four-country study into young workers and unions funded by the Hans Böckler Foundation in Germany (for details see Tapia and Turner in this volume). For the analyses, the authors selected those initiatives from the total of 24 case studies that 1) focused on young workers in precarious situations and 2) could help illuminate the factors that constrain or facilitate union innovation when targeting these groups. Based on these two criteria, we found seven cases that were of particular interest because of specific dynamics and issues they raised relating to these themes. Key features and description of these campaigns are highlighted in Table 1. After defining the innovative practices found in each case, our analyses focused on the factors that contributed to or hindered their success. Each case was analyzed with these themes in mind, and we use them to structure the discussion below.

TABLE ONE HERE

Evidence of innovative campaigns

It is first necessary to illustrate the kinds of innovative practices that are evident in these campaigns. Following the definition above, the focus here is on practices that break with existing ways of organizing and representing young workers in a particular sector or union.

The campaigns identified have all been selected because they have led to significant organizing successes among young workers and constitute new additions to unions' tactical repertoire. This raises the question of whether there is something particular about (these groups of) young workers that requires or facilitates innovation. Here, we return to the centrally important experience of precarity faced by young workers in these cases. Table 2 summarizes the innovative aspects of the campaigns, along with the factors that have facilitated and impeded that innovation and the successes observed.

TABLE TWO HERE

Of course, precarious employment and high levels of insecurity in transitions towards adulthood are not new, but the current generation of young workers is experiencing precarity in very different ways [START_REF] Armano | Mapping Precariousness, Labour Insecurity and Uncertain Livelihoods: Subjectivities and Resistance[END_REF]. Specifically, precarity is at the heart of a collective experience for young workers, which destabilizes the relationship between individuals' current positions and actions and their future life trajectories. While workers of earlier generations also often faced precarity at labor market entry, most of them were able to actively construct a more stable career at later life stages. Today, the trajectories of young workers are less clearly defined. For unions, this represents a fundamental challenge.

The normalization of precarious employment for these workers brings two, potentially contradictory dynamics. On one hand, precarious workers may increasingly consider it too risky to organize, for example, because their incomes are contingent on getting favorable assessments from managers or because they live in constant fear of losing their jobs. On the other hand, in some instances precarious workers may also be more likely to organize.

Precarization has undermined the previous industrial consensus that paid employment should ensure economic independence and stability, which may incentivize young people to act. In this scenario, unions could provide structures within which collective interests coalesce to inhibit the spread of precarious employment. While it is clear that the former response empirically dominates the landscape, the cases examined in this article are examples of the latter. This link between the experience of precarity and the spread of innovative union practices is at the heart of this analysis. The unions in these cases have all attempted to engage with the forms of precarious employment facing young workers whom they are seeking to organize and represent. These efforts to organize and engage young people have presented the unions involved with specific challenges. The examples illustrate the tensions between the opportunities and constraints in overcoming two of the central drivers of innovation: identifying issues of importance to precarious workers and overcoming the limitations of historical structures of union activism that fail to meet the needs of these members.

In an effort to identify and give voice to the issues that are important to young precarious workers, one of the most important innovations is the attempt to develop narratives about the benefits of acting rather than acquiescing to the precariousness these workers are experiencing. This challenges unions to demonstrate their effectiveness. In a context where achieving access to a standard employment is unrealistic -and may even be seen by workers as undesirable -the unions in these cases have focused on more achievable issues such as raising wages, providing access to training, improving contracts, and reducing discrimination. This has, in turn, allowed the unions to question the normalization of practices of precarious employment. Importantly, however, there is little evidence that unions have tackled the broader conditions that create precarious jobs, such as shareholderdominated business models, low-price competition, or exploitative supply chain practices.

Nonetheless action by union leaders, by activists and workers is important in identifying relevant issues and developing a collective response from the union. In many of these initiatives, therefore, innovation can simply mean extending union activity and representation to target young, precarious workers.

Moreover, because the established organization models developed by unions over decades tend to rely on ongoing involvement of workers in union activities, there has had to be degree of innovation in approaches to representation in these initiatives. Established union structures often do not fit well with the lives of young, precarious workers. High labor turnover, which are in evidence in our cases from the fast food and reality TV sectors, imply that turnover among activists is equally high. Training and developing activists is a resourceintensive activity for any union and can increase the risk of these campaigns for unions. To some degree this is an inevitable consequence of targeting young workers, but it raises important questions about how and by whom the sustainability of some of these initiatives can be ensured. It is often argued that reliance on paid union officers to sustain organizing activity is undesirable and probably unfeasible given the scale of the challenge facing unions [START_REF] Heery | Partnership versus organising: alternative futures for British trade unionism[END_REF]. These cases show that sustaining innovation requires some stability of leadership and support that is not always feasible among a precarious target group.

The objective of the following sections is therefore to explore some of these innovative practices and to identify factors that facilitate and constrain innovation. Throughout the analyses, our intention is to illustrate the dynamic tension between structure and agency, between institutions and actors. What becomes clear is that both are important in explaining the successes and failures of these initiatives. Institutional frameworks, labor market conditions, and union action can all work to support or hinder these innovations. Furthermore, we wish to devote special attention to the specific actions undertaken by workers, activists, and union leaders to shape these initiatives in particular directions.

Factors that facilitate innovation in organizing young, precarious workers

The cases provide evidence of factors that can facilitate and impede the success of new approaches to organizing among young precarious workers. These can be grouped under three main headings: 1) the institutional context, 2) the labor market, and 3) the level of union support. In each area, we see how the positive and negative dynamics mentioned above play out to establish conditions and actions that can either promote or constrain activity when organizing these precarious workers. This emphasis on the interactions of these influences and, specifically, on the fact that they can both promote and constrain innovation and effectiveness, is important because it shows how these initiatives are always in a state of flux, with an end point that is often unclear and uncertain.

What may be surprising is that many of these favorable conditions are neither new nor specific to precarious employment. Our findings underscore the argument made by [START_REF] Simms | Mobilising contingent workers: An analysis of two successful cases[END_REF] that structures of collective bargaining and interest representation are essential for effectively organizing and representing the interests of precarious workers. In the cases here, there is an opportunity to explore the agency of workers, members, activists, and union leaders in more detail. Additionally, the cases highlight how particular labor market conditions can facilitate or constrain action, how the presence of occupational identities can serve as a resource, and how processes of "mimetic isomorphism" [START_REF] Dimaggio | The Iron Cage Revisited: Institutional Isomorphism and Collective Rationality in Organizational Fields[END_REF]) allow unions to draw on others' experiences and adapt successful practices developed in different national or sectoral contexts.

The institutional context: existing bargaining structures

Several of the cases offer examples of how existing bargaining structures and established rights for interest representation can be used as a starting point from which unions can seek new ways to bring in young workers. Here, innovation here has meant two things: the extension of bargaining arrangements and of representational capacity. In case of the first, unions have tried to extend existing bargaining arrangements from their core areas of activity into fields of more precarious work. In the case of BECTU, this involved efforts to extend the collective bargaining coverage established at one London branch of Picturehouse Cinemas to other branches of the chain [START_REF] Simms | An innovative campaign for low paid, young workers: the challenges of spreading activism within union structures[END_REF]. Similarly, the "Real Deal" campaign aimed to establish the structures of interest representation common to other fields of TV scriptwriting in the reality TV sector [START_REF] Alvarez | The Real Deal: Organizing Young Workers in Reality Television in New York City. Case study prepared for the research project Young Workers and the Labor Movement[END_REF]. Another case in point is the East German auto-supply firm Automax [START_REF] Thiel | Disillusioned but prepared for conflict: A new generation of East German workers rediscovers interest representation. Case study prepared for the research project Young Workers and the Labor Movement[END_REF]. The first step there was to exercise workers' right to elect a works council with the hope of eventually achieving a regular collective agreement. In all of these cases, the potential benefits of these institutional arrangements (works council representation, collective bargaining) were evident to workers because similar established structures existed for comparable workers, so the differences between represented and unrepresented groups were clearly visible in everyday experience.

A second innovative approach to extending structures of interest representation can be seen in projects that operate primarily within workplaces already covered by institutionalized interest representation, which then use these structures to specifically address the interests of less well-organized groups of young workers. The clearest example is Praktisch. Besser. Jetzt., in which youth representatives in hospitals and elder care homes were contacted and empowered as a first step to organizing apprentices [START_REF] Behrend | Young Care Workers in Germany and ver.di's[END_REF]. In the second step, youth representatives were supported in bringing the project to workplaces and vocational schools, where they held days of action, conducted surveys, and identified issues crucial for representation. Again, these issues -mostly relating to shortcomings in on-the-job training due to time pressure and personnel shortages -were not in themselves "new" but they were innovative in the sense that they reflected the specific concerns of these young workers and emerged from the forms of precariousness they experience at work.

In sum, nothing in our cases undermines Simms and Dean's argument (2015) that, as with any other group, successful organizing among precarious young workers is strongly influenced by the availability of institutionalized rights and bargaining structures. Even the French case of ASSO, which may at first sight appear to represent a counterexample, confirms this on closer examination. Despite their skepticism concerning established unions and bureaucratic procedures, ASSO's founders quickly realized that what they needed was indeed a union rather than some kind of informal body, and they describe their present situation as a struggle to find ways to become accepted as a partner for a more conventional form of collective bargaining [START_REF] Dupuy | Origins and functioning of a young workers' trade union: ASSO and nonprofit organization employees. Case study prepared for the research project Young Workers and the Labor Movement[END_REF].

Labor markets: sector, region and occupation

Labor markets clearly matter in organizing these young, precarious workers. The dominance of some forms of labor practices in sectors such as retail present particular challenges to unions (discussed in the next section), while, by contrast, the dynamics in some labor markets facilitate innovation in organizing. In the German cases in particular, labor markets provide some young workers with conditions that make it more attractive -or at least less risky -for them to collectively stand up for their interests by providing realistic employment alternatives that reduce the individual risks associated with collective action.

In the case of East German automotive supply firm Automax, this is mainly due to the export-based strength of the German manufacturing sector. Even in a region like Saxony, which has suffered widespread deindustrialization since the 1990s and a substantial oversupply of labor in the corresponding sectoral labor market for many years, labor market changes have led to increased choices for young workers. Demographic changes and internal migration, combined with the manufacturing boom, meant that skilled workers no longer perceive their current jobs as the only option [START_REF] Thiel | Disillusioned but prepared for conflict: A new generation of East German workers rediscovers interest representation. Case study prepared for the research project Young Workers and the Labor Movement[END_REF]. That said, precarity is rife. Temporary agency work and subsequent fixed-term contracts are the basis for the typical, prolonged entry route into the core workforces. Despite this, young workers know that their skills are in short supply, and that if they lost their job they would not remain unemployed for long. Under these conditions, bottom-up organizing emerged in which workers themselves initially decided they wanted to organize and then actively approached the union for support in founding a works council. Importantly, this is not an isolated case, but represents a broader trend that is observed by trade unionists at the local level in many regions of East Germany [START_REF] Goes | Gewerkschaften im Aufwind[END_REF].

The Praktisch. Besser. Jetzt. case shows how the incredibly poor working conditions due to personnel shortages in the German care sector prompted apprentices to raise their problems despite widespread precarity and low wages. Here, an important dynamic is evident. Labor shortages not only give rise to relative labor market confidence that facilitates organizing; they also underpin the conditions that young workers are protesting against. In the case of the Praktisch. Besser. Jetzt. project this was the lack of mentoring and frequent moves between wards [START_REF] Behrend | Young Care Workers in Germany and ver.di's[END_REF]. This highlights the much wider challenge facing unions as they seek to address the problems raised by precarious work.

Often those problems can only be resolved by engaging in a much more fundamental questioning of the business model being used. Addressing that wider question is much more challenging for unions, but without it, continued precarity is almost inevitable.

These examples point to how tight labor markets in particular occupations, regions, or sectors can influence the conditions for organizing among young precarious workers. Labor market conditions can create issues for organizing, but they can simultaneously increase workers' confidences and their readiness to organize. Tight labor markets also bring into focus the weaknesses of business models that rely on precarious employment.

Support of the union: sectoral organizing and occupational identity

Some of the cases also demonstrate that unions have often found it much more effective to organize workers on a local, regional, or sectoral basis, rather than company-by-company. This is particularly evident in sectors dominated by highly fragmented workforces and individualized working practices, where there are very practical barriers to working collectively to fight for workers' interests. Creating spaces for workers to meet, get to know each other, and to prepare action on a local or regional level has been shown in these cases to be an effective strategy. These structures differ in their formality across the cases, but they generally serve two functions. First, they provide a platform for workers to support each other's struggles, e.g., when the UK bakers' union mobilized young workers to picket each other's workplaces [START_REF] Simms | An innovative campaign for low paid, young workers: the challenges of spreading activism within union structures[END_REF]. Second, they foster the emergence of a community of people doing the same or similar jobs, which can create empowerment by helping workers to develop a common occupational identity, such as in the RAP project [START_REF] Fullin | The Retail Action Project: Mobilizing Young Workers in American's Most Unequal City. Case study prepared for the research project Young Workers and the Labor Movement[END_REF].

The example of ASSO is important here [START_REF] Dupuy | Origins and functioning of a young workers' trade union: ASSO and nonprofit organization employees. Case study prepared for the research project Young Workers and the Labor Movement[END_REF]. It has provided a focus for employees from different professional sectors whose only point in common is that they work for nonprofit and third sector organizations (community associations, political parties, unions and foundations). It has since developed into a community of employees and is increasingly formalizing and coalescing into an organization that has features of a formal trade union. A similar dynamic can be seen in the UK bakers' union, which actively developed local sectoral organizing during the Fast Food Rights campaign (Holgate et al 2017). It has also been observed in the case of New York's Retail Action Project. Here, organizers explicitly intended to create a space that would be perceived as "a cool organization for young people" [START_REF] Fullin | The Retail Action Project: Mobilizing Young Workers in American's Most Unequal City. Case study prepared for the research project Young Workers and the Labor Movement[END_REF]. This included both establishing the worker center as a place where workers could simply hang out and setting up committees for art, social media, and communication, which have proved very attractive, especially to young workers.

Importantly, the attempts to create a broader occupational identity were not restricted to providing such spaces for social interaction. In many of the campaigns, there is clear evidence of unions attempting to build sectoral and/or occupational initiatives with the explicit objective of improving young precarious workers' terms and conditions of work. An example is the provision of training for job-specific skills by the affiliated Center for Frontline Retail (CFFR) in the context of the Retail Action Project [START_REF] Fullin | The Retail Action Project: Mobilizing Young Workers in American's Most Unequal City. Case study prepared for the research project Young Workers and the Labor Movement[END_REF].

Examples from the US Writers' Guild's Real Deal initiative include the organization of workshops and the provision of counseling and networking opportunities by the US Writers' Guild's Real Deal initiative [START_REF] Alvarez | The Real Deal: Organizing Young Workers in Reality Television in New York City. Case study prepared for the research project Young Workers and the Labor Movement[END_REF]. In the UK, BECTU's approach to organizing young precarious workers in broadcasting relies heavily on providing job-specific skills training [START_REF] Simms | An innovative campaign for low paid, young workers: the challenges of spreading activism within union structures[END_REF]. By contrast, in the German cases [START_REF] Behrend | Young Care Workers in Germany and ver.di's[END_REF][START_REF] Thiel | Disillusioned but prepared for conflict: A new generation of East German workers rediscovers interest representation. Case study prepared for the research project Young Workers and the Labor Movement[END_REF], unions can draw on the much stronger sense of occupational identity resulting from its vocational training system. This provides a relatively solid foundation in young workers' mindsets that unions can readily appeal to.

Sectoral organizing, and the inevitable effort invested in building an occupational and/or sectoral worker identity as the basis for organizing activity, is clearly an important factor in building effective union representation, especially where workplaces and work identities are highly fragmented and individualized. Unions in these cases have grasped these challenges, and the case studies demonstrate they have found effective responses to them. This effort to invest in unions' strategic capacities to strengthen occupational identity helps build a platform, which can be used to organize around the particular issues facing an occupational group. In a context where many (young) workers are precarious, this helps reinforce ideas that the union is relevant to that occupational group and is challenging some of the issues associated with precarious work.

Support of the union: learning from other unions

A final facilitating factor to consider is that new kinds of organizing practices can be facilitated by a kind of transversal institutional learning in which successful practices transfer between institutional contexts (see DiMaggio & Powell, 1983 on such "mimetic isomorphism). [START_REF] Fine | Worker centers : organizing communities at the edge of the dream[END_REF] For instance, the UK Bakers' union developed its "Fight for £10" campaign from the "Fight for 15"of its US partner SEIU [START_REF] Simms | An innovative campaign for low paid, young workers: the challenges of spreading activism within union structures[END_REF]. Another example is Automax [START_REF] Thiel | Disillusioned but prepared for conflict: A new generation of East German workers rediscovers interest representation. Case study prepared for the research project Young Workers and the Labor Movement[END_REF]. Here, the IG Metall union official used a variant of a well-established tactic used by the services union Ver.di, namely the concept of "condition-based organizing" (Dribbusch, 2016: 358;Pernicka, Glassner, & Dittmar, 2016: 88-9). The condition presented to workers in this case was to say: If you achieve a 40% unionization rate, the union will support you in founding a works council. If you achieve 70% unionization, the union will bargain for a collective agreement. This approach has been adopted by many IG Metall organizers at the local level (Schmalz & Thiel, 2017: 477) and is explicitly informed by Ver.di's experiences of organizing young precarious workers in the service sector.

It is evident, therefore, that unions observe, mimic, and adapt successful tactics developed in other institutional and sectoral contexts. Although it is dangerous to assume that tactics and campaigns can simply be copied from other contexts, it is clear here that this is not what these unions are doing. They are observing and, crucially, adapting those tactics to their specific settings.

Factors that limit the extension of these innovations

Despite the positive outcomes of many of these initiatives, the cases also point to constraints limiting both the spread and the sustainability of outcomes in some circumstances. [START_REF] Simms | Accounting for greenfield union organizing outcomes[END_REF] argues that in order to build sustainable and effective representation of new groups of workers, unions must focus on two central objectives: building the representativeness of union structures and demonstrating the effectiveness of union influence in addressing issues relevant to the workers being targeted. This provides a helpful lens through which to identify some of those limiting factors in the cases under examination here.

Institutional context: the regulation of unions

Looking first at the representativeness of union structures, we see that in many national contexts the legal status of unions imposes strict constraints on their activities and structures. In all four countries under study, the cases provide evidence of limits imposed by the rules and laws about representativeness. When a clear definition of an independent union and of its legitimate activities is in place, the rights of the union and unionized workers are clearly established. But this also sets boundaries that formal union action cannot transgress.

The most notable example in this respect is probably the US Retail Action Project [START_REF] Fullin | The Retail Action Project: Mobilizing Young Workers in American's Most Unequal City. Case study prepared for the research project Young Workers and the Labor Movement[END_REF]. Here, the constraints imposed on the union made it effectively impossible for the union to formally engage in the kind of community-building work needed in the retail sector. As a result, RAP was formed as a formally independent structure -a worker centerthat was not subject to these strict limitations. In general, worker centers (and the alt-labor movement more broadly) can be interpreted as a reaction to the harsh constraints put on organized labor by US law [START_REF] Fine | Worker centers : organizing communities at the edge of the dream[END_REF]. In the UK, BECTU [START_REF] Simms | An innovative campaign for low paid, young workers: the challenges of spreading activism within union structures[END_REF] is also highly constrained by legal rules. Before taking strike action, members at the Picturehouse Cinema chain had to be balloted in a secret postal vote to secure a majority defined in law and had to give the employer notice of the strike action. Strike action is only legal over specific employment issues, and members' actions during the strike are closely regulated. In making their fight a union dispute, workers have therefore narrowed the range of issues they could legitimately address and limited the tactics they could use.

The French ASSO case [START_REF] Dupuy | Origins and functioning of a young workers' trade union: ASSO and nonprofit organization employees. Case study prepared for the research project Young Workers and the Labor Movement[END_REF] is also important. ASSO emerged precisely because activists were-and remain-skeptical about traditional unions and were looking for an alternative form of interest representation. They therefore initially chose to organize themselves outside established union structures, even though that prevented them being formally recognized as an industrial relations actor. The new organization thus succeeded in bringing together a group of workers around a common set of interests, but its effectiveness at representing those interests remained constrained because of the lack of access to formal structures of representation. Furthermore, ASSO's legitimacy could therefore be challenged by other, more conventional actors such as employers, employers' associations, and even unions. The tensions of this ambiguous status are clear to the activists, and are subject to on-going discussion and negotiation about the organization's future. At present, while having decided to take a step toward a more formal structure by joining the Solidaires union federation, the activists have also chosen to focus on other ways to get their voices heard through political and media actions. Focusing too much on this may risk a situation where their interests are sidelined further, in turn further undermining their scope to demonstrate effective representation.

Perhaps most tellingly, some of these initiatives have simply failed to generate sufficient support among the target groups to gain full institutional recognition. The German auto supplier case illustrates this particularly clearly [START_REF] Thiel | Disillusioned but prepared for conflict: A new generation of East German workers rediscovers interest representation. Case study prepared for the research project Young Workers and the Labor Movement[END_REF]. Here, activists within the firm did garner the backing expected by the union official as a precondition for successfully establishing a works council (40% membership), but they were unable to muster the membership quota he required of them before entering into the collective bargaining process (70% membership). As the union has a bargaining monopoly in the sector-based German industrial relations system, its decision to refrain from any bargaining attempts unless or until the 70% threshold is reached means that further progress can only be made through sustained in-firm organizing efforts (ibid.). It also means that workers have it in their own hands to change the situation.

The institutional contexts within which these initiatives develop are therefore extremely important in explaining some of the limitations of the effectiveness of these organizing efforts, but they do not entirely explain the outcomes. Two other factors are important: the churn of members and activists, and the challenges of integrating into the wider structures of the unions.

Labor markets: activist and member churn

In most cases, the precarious situation of young workers relates to high labor turnover in these workplaces, which in turn means that there is often churn in the membership and activist base. The previous section of this article has demonstrated that labor market conditions can facilitate organizing, but they can actually also constrain the outcomes. While churning is always a risk for any campaign, there is evidence that this problem has recently been exacerbated in precarious and youth-dominated labor markets in all four countries.

Young people have always moved between jobs more frequently than older workers [START_REF] Furlong | Young people in the labour market: Past, present, future[END_REF], but in comparison, the parents of today's young workers tended to experience a comparatively higher degree of stability in their choice of occupation, contractual position, and wider living situation than their children's generation today [START_REF] Rhein | Beschäftigungsdauer im Zeitvergleich: Bei Jüngeren ist die Stabilität der Beschäftigung gesunken[END_REF]. These aspects of stability have all been undermined by wider economic and social shifts, in particular the transition from an organized to a flexible model of capitalist accumulation [START_REF] Eversberg | Beyond individualisation: The German 'activation toolbox[END_REF]. For these projects and campaigns, the immediate challenge is to retain knowledge and expertise as the lives of these young activists change. In some cases, such as BECTU, this has been addressed both by ensuring a wide activist base, and by recruiting some activists to paid union positions [START_REF] Simms | An innovative campaign for low paid, young workers: the challenges of spreading activism within union structures[END_REF]. Given the importance of activist leaders in many of the projects and initiatives, there is always a risk that this expertise will be lost as life changes force young, precarious workers to move on. This inevitably impacts the likely effectiveness and sustainability of organizing outcomes.

Support of the union: the importance of effective representation

In part because of the challenges of a high labor turnover, the continued support and leadership from the union is particularly important in ensuring continuity of these campaigns targeting precarious young workers. The union structures are also essential to ensure effective representation of the interests of these workers both within the union and vis-a-vis management. Where the union has particular expectations about outcomes, tensions can emerge if these are not achieved. The German Automax case shows how even a strong and sustainable grassroots initiative that very deliberately used the formal structures of collective representation can fail to deliver its full potential without more active support from the wider union [START_REF] Thiel | Disillusioned but prepared for conflict: A new generation of East German workers rediscovers interest representation. Case study prepared for the research project Young Workers and the Labor Movement[END_REF]. Similarly, the US Retail Action Project case [START_REF] Fullin | The Retail Action Project: Mobilizing Young Workers in American's Most Unequal City. Case study prepared for the research project Young Workers and the Labor Movement[END_REF] illustrates that even where a campaign has been very successful in building occupational unionism and reaching out beyond established constituencies, constraints can emerge because of the limited funding and scope an established union can provide.

Almost inevitably these kinds of innovative approaches are seen as extensions to core union activities, and are funded and prioritized accordingly. This is not a criticism of the innovative and engaging work being done, but recognizes that it is resource-intensive and often risky even when there is "proof of concept". In effect, the wider uptake of innovative approaches can be constrained because individual union leaders need to be persuaded of the value of these activities.

In short, then, our cases illustrate some of the complexities that constrain innovation, the spread of innovation, and the sustainability of innovative tactics. These points highlight general challenges of union organizing. Our analyses clearly showed that there are limits to unions' efforts sustain organizing activity even when they have successfully met the challenge of reaching out to young precarious workers by implementing innovative strategies. This indicates the broader organizing issues that unions face today. Indeed, it is clear that sustainability of organizing outcomes is a major concern of all organizing activity [START_REF] Simms | Accounting for greenfield union organizing outcomes[END_REF]that requires continual reflection on the part of organizers, members and union leaders.

Conclusions

In this article we have explored innovative union campaigns in France, Germany, the UK and the US related to young precarious workers. The goal was to describe and analyze the ways in which unions innovate to attract these workers and the outcomes of these innovative Despite the variety of our seven cases in terms of country, sector, scale and dynamics, we can identify common factors that constrain and facilitate union organizing among young precarious workers. The institutional context, labor market, and union support all interact to explain both the successes and limitations observed.

A strong facilitating factor for innovative practices is the existence of established bargaining structures and rights for interest representation. Despite the lack of engagement of young precarious workers with unions, unions' presence, their hard-won power, and influence significantly improve the prospects for successful organizing strategies. At the same time, new practices do not normally originate from within the unions themselves, but are adapted from other collective movements. Furthermore, it comes as no surprise that organizing efforts usually prove more effective when unions can draw on a feeling of community based on a common occupational identity, and when labor market conditions are relatively favorable to young precarious workers' bargaining position. There are also constraints limiting the outcomes of innovative organizing. In some cases the formal rules for union representativeness can bar a group of young precarious workers from being formally recognized as bargaining partners. Further, the high costs and the intense efforts needed for successful organizing can create problems of legitimacy within the union, especially when there is a high level of churn among activists in high-turnover sectors.

Of course, it is also possible that counter-mobilization [START_REF] Kelly | Mobilization Theory: A New Paradigm for Industrial Relations[END_REF] may constrain the effectiveness of innovative approaches. Professional union busting has become a wellestablished business in countries such as the United States [START_REF] Hurd | Patterned responses to organizing: Case studies of the union-busting convention[END_REF] and it is quite probable that if some of these more innovative organizing approaches spread internationally, some companies may use established and new ways to hinder organizing.

Although there is little evidence of this in these cases (except for some attempts by the management of the east German auto supply firm to disrupt the establishment of a works council), it is certainly something to be mindful of in future research. And it is clear in these cases that there has been strong counter-mobilization by employers which has, at very least, made the campaigns and struggles of these workers harder and more protracted.

Our comparative research has highlighted that, far from remaining passive or contained by the interests of older workers, unions in all four countries are actively and innovatively working to develop new strategies of interest representation that can address the diverse situations of young precarious workers. Our results show that there is not necessarily a contradiction between the need to innovate and a concern for established union structures and institutionalized bargaining rights. In the end, successful organizing among young precarious workers, as among any other group, depends less on "new" or "different" organizing practices per se. Instead, it is important that unions manage to authentically relate to workers, are accepted as workers' legitimate representatives, and have the capacity to effectively advance workers' interests. If these conditions are met, the common assumption that unions are "outdated" and can no longer adequately serve the needs of young workers can certainly be proven as wrong today as it ever has been.

Figures and Tables

  practices. Unions across the global North have experienced an undeniable crisis of membership since the late 1970s. Their membership bases are shrinking and ageing. Most unions have few young members. This applies to both long-established and more recently founded unions, independent of their size and political orientation. The average age of members and union leaders is rising in all of the countries studied. Yet many unions have been hesitant to prioritize recruiting new, young members. The examples given above illustrate some cases where unions have accepted the challenge and actively addressed their shrinking and ageing membership profile. A common idea in most of the examples has been to expand the union's constituency by recruiting workers in labor market segments traditionally distant from unionism. In the course of this strategy change, young and precarious workers have become important target groups for unions formerly dominated by core workers.

Figure 1 :

 1 Figure 1: Union Membership in France, Germany, Great Britain, and the United States

Figure 2 :

 2 Figure 2: Proportion of workers saying that trade unions are good for workers

Table 1 : Overview of the case studies

 1

	Case	What is case about?	Sector	Tactics	Outcomes
	Praktisch. Besser. Jetzt.	Activating youth representatives and	Care sector -	Survey to identify relevant issues; campaigning	Boost to youth reps' confidence,
	Vereinte	organizing around the quality of	hospitals,	in vocational schools by youth representatives	establishment of active dialogue with
	Dienstleistungsgewerk-	vocational training -mostly problems due	nursing homes	and Ver.di officers; bargaining training for youth	apprentices. Detail of bargaining
	schaft (Ver.di), Germany	to personnel shortages (inadequate		reps	successes remains to be seen. Turnover
		mentoring, "ward hopping")			makes continuity among youth reps
					hard to achieve
	Automax	Bottom-up organizing process among	Automotive	Support for workers' initiative; assistance and	40% quota met within a few weeks;
	IG Metall (IGM), Germany	workers of an auto supply firm with	supply	training for young activists; quota of 40%	works council established; currently 50%
		support from IG Metall. Issues: low wages;		membership set for assistance in setting up a	membership. 70% quota for collective
		lack of recognition; co-determination in		works council, 70% for entering collective	bargaining not currently in sight
		the face of authoritarian and ineffective		bargaining negotiations
		management
	Living Wage campaign at	Campaign to achieve a Living Wage for	Entertainment	Strikes; use of social media, crowdfunding for	Campaign for Living Wage has turned
	Picturehouse Cinemas	cinema workers, strike action to establish	-Cinemas	strike action, support from public figures, wider	into a long-running dispute, outcome
	Broadcasting,	bargaining rights at non-unionized		community-building	open
	Entertainment,	cinemas
	Cinematograph and
	Technicians Union
	(BECTU), UK
	Fast Food Rights	Public campaigns and organizing among	Fast food and	Coalition building with international unions &	McDonald's has moved to offer workers
	campaign	fast food workers to fight precarity	coffee chains	other organizations, public action days at shops,	minimum hours; 300 new young
	Bakers, Food and Allied	through zero-hours contracts; active		affirmative action for young members in union	members; dramatic change to 'look and
	Workers Union (BFAWU),	strategy of 'rejuvenating' the union		bodies; bottom-up organizing at local sectoral	feel' of union gatherings
	UK			level, going for 'small, easy wins' as a first step
	Retail Action Project	Founding of a worker center to mobilize	Retail -	Systematic research on working conditions;	Key back-wage settlements reached;
	Retail, Wholesale and	and campaign against widespread sector-	primarily	coalition building; training, building skills and	some workplaces unionized; successful
	Department Store Union	specific problems such as wage-and-hour	fashion chains	occupational identity; community-building	in socializing young workers to the labor
	(RWDSU), US	violations, super-exploitation, erratic		among workers; employer-specific campaigns	movement; creation of a strong network
		scheduling, racial discrimination			among retail workers
	The Real Deal	WGAE's efforts to organize young workers	Entertainment	"Non-fiction writers' caucus" established as a	Six reality TV production companies
	Writers' Guild of America	in reality television in New York City.	-Reality TV	forum for community-building and job-skills	unionized; empowerment of young
	East (WGAE), US	Many young workers suffer from precarity	production	training provision; change of focus from	members in reality TV production,
		and devaluation of script		production companies to industry level; support	consolidation of an activist base
		writing/production work; outsourcing,		from political allies; legal pressure as a
				concomitant tactic


Jean-Baptiste Leblond

Pierre Mouro

Crack propagation from a pre-existing flaw at a notch root. I. Introduction and general form of the stress intensity factors at the initial crack tip

Keywords: Notch, pre-existing crack, mixed mode, stress intensity factors, homogeneity and continuity properties, universality properties

This paper and its companion are devoted to the study of crack kinking from some small pre-existing crack originating from a notch root (the notch root radius being zero). Both the notch boundaries and the initial crack are allowed to be curved; also, the geometry of the body and the loading are totally arbitrary. The ingredients required are knowledge of the stress intensity factors at the initial crack tip and use of a suitable mixed mode propagation criterion. This paper is devoted to the first point, and more specifically to establishing the general (that is, not yet fully explicit) form of the formulae giving these stress intensity factors. The method used is based on changes of scale (homogeneity properties of the equations of elasticity) on the one hand, and on continuity of the displacement and stresses at a given, fixed point with respect to the crack length on the other hand. The formulae derived for the stress intensity factors at the tip of the small crack are of universal value: they apply to any situation, whatever the geometry of the body, the notch and the crack and whatever the loading, the stress intensity factors depending always only upon the 'stress intensity factor of the notch' (the multiplicative coefficient of the singular stress field near the notch root in the absence of the crack), the length of the crack, the aperture angle of the notch and the angle between its bisecting line and the direction of the crack.

Introduction

Predicting crack growth from some notch root is a problem of obvious practical importance. If one adopts the classical theory of LEFM, however, one immediately encounters a problem : indeed crack propagation cannot occur directly from a notch root because the relevant energyrelease-rate is zero. One is therefore forced to postulate the presence of some pre-existing small flaw originating from the notch root, and study crack propagation from this initial crack.

We therefore consider the general problem depicted schematically in Fig. 1. An arbitrarily shaped isotropic elastic body under plane strain conditions is subjected to arbitrary prescribed displacements u p on the portion ∂ u of its boundary ∂ and to arbitrary prescribed tractions T p on the complementary portion ∂ T . This body contains a traction-free notch of apex O, with a zero root radius and an aperture angle 2(πψ) (so that ψ represents the angle between the bisecting line Ox of the notch, oriented towards the material, and the notch boundaries), with π/2 ≤ ψ ≤ π . The notch boundaries are allowed to be curved; we denote by C 1 the curvature of the upper boundary at the point O and by C 2 the curvature of the lower boundary at the same point. An edge traction-free crack (in full line in Fig. 1) of small length a originates from the notch apex O. It makes an initial angle ϕ (-ψ < ϕ < +ψ) with the bisecting line Ox of the notch. It is also allowed to be curved; we denote by C its curvature at the point O 1 . Since in general the crack is loaded in mixed mode I+II, subsequent propagation will occur along a deviated branch (in dotted line in Fig. 1). The purpose of this paper and its companion is to predict the beginning of the propagation path, and more specifically the kink angle χ, that is, the angle between the old and new tangents to the crack at its initial tip (see Fig. 1) 2 . This goal will be achieved through use of some appropriate mixed mode propagation criterion; we shall choose the well-known and widely accepted principle of local symmetry of [START_REF] Goldstein | Brittle fracture of solids with arbitrary cracks[END_REF] in Part II. However, prediction of the kink angle χ through this criterion of course requires knowledge of the stress intensity factors (SIFs) K p (a)(p = I, I I ) at the initial crack tip. Calculation of these SIFs, in the limit case a → 0, therefore appears as an indispensable prerequisite.

This problem has never been envisaged previously with this degree of generality. However, partial results are available for various particular cases which all involve a notch with straight boundaries and a straight crack (C 1 = C 2 = C = 0), and angles ψ and ϕ taking special values. For instance, it is a classical result that for = π/2 and ϕ = 0 (surface crack originating perpendicularly from a regular, corner-free free surface), K I (a)

1.12σ ∞ yy √ πa and K I I (a) = 0, where σ ∞ yy denotes the remote tensile stress. The case where ψ = π but ϕ is arbitrary (semi-infinite crack with an arbitrarily kinked extension) has been extensively

1 Rigorously speaking, one should introduce parameters characterizing the geometry of the notch boundaries and the crack with a higher degree of accuracy than that envisaged here, that is, consider successive derivatives of the curvatures at the point O. It can easily be verified at the end, however, that this would not modify the results derived in any way.

2 χ is represented as negative in Fig. 1 because ϕ is taken as positive; we shall see in Part II that these angles are necessarily of opposite signs.

studied, for instance by [START_REF] Bilby | The crack with a kinked tip[END_REF], [START_REF] Wu | Elasticity problems of a slender Z-crack[END_REF], [START_REF] Amestoy | Déviation infinitésimale d'une fissure dans une direction arbitraire[END_REF], [START_REF] Amestoy | Crack paths in plane situations -II. Detailed form of the expansion of the stress intensity factors[END_REF]; accurate numerical results are available for that situation. The case where = π/2 and ϕ is arbitrary (oblique edge crack emanating from a regular free surface) has been considered by several authors [START_REF] Wilson | On Combined Mode Fracture Mechanics[END_REF][START_REF] Sha | Weight function calculations for mixed-mode fracture problems with the virtual crack extension technique[END_REF][START_REF] Aliabadi | Mixed-mode Bueckner weight functions using boundary element analysis[END_REF], whose (numerical) results have recently been compiled and compared by [START_REF] Fett | Stress Intensity Factors and Weight Functions[END_REF]. Finally, these authors have also considered the case where ϕ = 0 but ψ is arbitrary (crack lying along the bisecting line of an arbitrary notch), using previous works of [START_REF] Gross | Some Plane Problem Elastostatic Solutions for Plates Having a V-Notch[END_REF], [START_REF] Hasebe | A crack originating from a triangular notch on rim of a semi-infinite plate[END_REF] and [START_REF] Isida | Tension of a half-plane containing array cracks, branched cracks and cracks emanating from sharp notches[END_REF]; they have derived an approximate but accurate formula for K I (a)(K I I (a) being zero) in that case.

Treating the problem in its full generality (arbitrary geometry of the body, the notch and the small crack, arbitrary loading) obviously requires a radically new approach. That which will be used here, which is basically similar to that employed by [START_REF] Leblond | Crack paths in two-dimensional situations -I. General form of the expansion of the stress intensity factors[END_REF] and [START_REF] Amestoy | Crack paths in plane situations -II. Detailed form of the expansion of the stress intensity factors[END_REF] to study crack kinking in an arbitrarily shaped body subjected to an arbitrary loading, consists of two steps. The present Part I is devoted to Step 1. It consists of establishing the general form (that is, not yet fully explicit because it involves some unknown functions) of the asymptotic expression of the SIFs K p (a) for a small crack length a. The method is based on two ingredients : first, changes of scale and homogeneity properties of the equations of elasticity (invariance of these equations upon homothetical transformations), and second, continuity properties of the mechanical fields (displacement, stresses) at a given, fixed point of the body with respect to the crack length a for a = 0. The SIFs at the tip of the small crack are shown to depend on the 'stress intensity factor' κ of the notch (that is, the multiplicative factor of the singular stress field near the notch root, in the absence of the crack), the length a of the crack, the aperture angle 2(πψ) of the notch and the angle ϕ between its bisecting line Ox and the initial direction of the crack. The formulae derived are of universal value, that is, they apply to any situation, whatever the geometry of the body, the notch and the crack and whatever the loading, the SIFs at the tip of the small crack depending always solely upon the mechanical and geometrical parameters just mentioned.

The formulae obtained involve some (universal but) unknown functions of the angles ψ and ϕ. Although the approach used does not provide the values of these functions, it does yield a method for their determination through solution of a plane strain elasticity problem corresponding to some special case. Not surprisingly, this special case involves an infinite body containing a notch with arbitrary aperture angle 2(πψ) but straight (C 1 = C 2 = 0) boundaries and a straight (C = 0) crack of unit (a = 1) length making an arbitrary angle ϕ with the bisecting line Ox of the notch.

Step 2 will be envisaged in Part 2. It will consist of explicitly solving the elasticity problem just mentioned, using [START_REF] Muskhelishvili | Some Basic Problems of the Mathematical Theory of Elasticity[END_REF] complex potentials method. The determination of the kink angle χ of the future extension of the small crack, as a function of the angles ψ and ϕ, will be envisaged in conclusion.

The stress field near the notch root in the absence of a crack

The problem of finding the asymptotic form of the stress field near the apex of a notch with aperture angle 2(πψ) (π/2 ≤ ψ ≤ π) is a very classical one, which was solved for the first time by [START_REF] Williams | Stress singularities resulting from various boundary conditions in angular corners of plates in extension[END_REF]. He showed that for ψ < π, the stress field is of the form, in polar cordinates r, θ :

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ σ rr (r, θ) = -κ α 2 (2πr) α-1 (3 -α) sin((1 + α)ψ) (1 -α) sin((1 -α)ψ) cos((1 -α)θ) + cos((1 + α)θ) , σ θθ (r, θ) = κ α 2 (2πr) α-1 - (1 + α) sin((1 + α)ψ) (1 -α) sin((1 -α)ψ) cos((1 -α)θ) + cos((1 + α)θ) , σ rθ (r, θ) = κ α 2 (2πr) α-1 - sin((1 + α)ψ) sin((1 -α)ψ) sin((1 -α)θ) + sin((1 + α)θ) , ( 1
)
where α is the smallest positive solution (varying from 1 for ψ = π/2 to 1/2 for ψ = π ) of the equation

sin(2αψ) + α sin(2ψ) = 0 ( 2 )
and κ a constant (depending on the far geometry and loading) which we shall call the 'mode I stress intensity factor', or more briefly the 'stress intensity factor', 'of the notch'. The reason why κ is said to correspond to 'mode I' is that the associated stress field is symmetric with respect to the bisecting line θ = 0 of the notch (σ rr and σ θθ are even, and σ rθ odd, functions of θ). A 'mode II stress intensity factor of the notch' does also exist but it corresponds to some stress field which is less singular (that is, proportional to some greater power of r) than that associated to κ (except for ψ = π : limit case of a crack) and therefore negligible in the limit r → 0.

In fact, formulae (1) involve some degree of arbitrariness in that one can always multiply κ by some factor while dividing the rest of the expression of the stresses by the same factor. Choosing this factor (here as unity) means choosing a certain definition for the 'stress intensity factor of the notch'. The definition adopted here is the 'best' possible one, in the sense that it possesses the nice properties that κ reduces to the ordinary mode I SIF K I for ψ = π (case of a crack) and to the uniform tensile stress σ yy for ψ = π/2 (case of a regular, corner-free boundary). This is established in detail in the Appendix.

Continuity of the mechanical fields at a given point with respect to the crack length

We now wish to establish that the displacement and stresses at some given, fixed point of the body are right-hand continuous functions of the crack length a for a = 0 + .

Let us first consider the initial situation where the prescribed displacements u p are applied on ∂ u and the prescribed tractions T p on ∂ T , in the absence of any crack emanating from the notch root O. The displacement and stresses at the generic point M of the body are then denoted u(M) and σ (M). One can always suppose that a crack of length a in fact originates from the point O provided that one exerts on the lips of this crack some suitable closing tractions

T + (s) ≡ σ (s).n + (s), T -(s) ≡ σ (s).n -(s)
, where s denotes the curvilinear length along the crack from the point O and n + (s), n -(s) the unit normal vectors to the upper (+) and lower (-) lips at the point s, oriented towards the opposite lip. These tractions are O(s α-1 ) like σ (s).

Let us now consider the final situation where u p is still imposed on ∂ u and T p on ∂ T , but a traction-free crack of length a emanates from the point O; equivalently, the tractions T + (s), T -(s) applied on the crack lips in the initial situation are released. The displacement and stresses at the point M are then denoted u(M, a) and σ (M, a). If we take the difference between the final and initial situations, taking advantage of the linearity of the equations of elasticity, we obtain some Problem A where a zero displacement is imposed on ∂ u and a zero traction on ∂ T while tractions -T + (s), -T -(s) are exerted on the crack lips. The displacement and stresses at the point M are then u(M, a)u(M) and σ (M, a) -σ (M).

We further define some Problem B in the following way. Let us consider the same geometry as before, including a crack of length a originating from the point O. Let (e 1 , e 2 , e 3 ) denote an arbitrary orthonormal basis. We again impose a zero displacement on ∂ u and a zero traction on ∂ T and the crack lips, but now exert a unit point force parallel to e i on the point M. The resulting displacements on the upper and lower crack lips at the point s are denoted

v (i)+ (M, a; s), v (i)-(M, a; s).
Application of Betti's reciprocity theorem to Problems A and B yields

u i (M, a) -u i (M) = - a 0 T + (s).v (i)+ (M, a; s) + T -(s).v (i)-(M, a; s) ds . ( 3
)
Also, differentiating this equation with respect to the coordinates x j of the point M, we get

∂u i ∂x j (M, a) - ∂u i ∂x j (M) = - a 0 T + (s). ∂v (i)+ ∂x j (M, a; s) + T -(s). ∂v (i)- ∂x j (M, a; s) ds . (4)
The quantities (∂v (i)+ /∂x j )(M, a; s), (∂v (i)-/∂x j )(M, a; s) in this equation must not be interpreted as strains, since the coordinates with respect to which differentiation is performed are those of the point of application of the force, not those of the point of observation of the displacement. Rather, they should be interpreted as displacements on the upper and lower crack lips at the point s resulting from the application of a unit 'dipole' at the point M (that is, two opposite infinite forces parallel to e i applied on points separated by an infinitesimal vector parallel to e j , the product of the intensity of the forces and the distance between the points being equal to unity).

The displacements v (i)+ (M, a; s), v (i)-(M, a; s) are obviously bounded by some positive constant A. Similarly, (∂v (i)+ /∂x j )(M, a; s) and (∂v (i)-/∂x j )(M, a; s) being also displacements, as just noted, they are also bounded by some positive constant B. It then follows from Equations ( 3) and (4) that

|u i (M, a) -u i (M)| ≤ A a 0 T + (s) + T -(s) ds ;
(5)

∂u i ∂x j (M, a) - ∂u i ∂x j (M) ≤ B a 0 T + (s) + T -(s) ds . ( 6
)
Since the tractions T + (s), T -(s) are O(s α-1 ), it follows from inequalities ( 5) and ( 6) that

|u i (M, a)-u i (M)| and |(∂u i /∂x j )(M, a)-(∂u i /∂x j )(M)| are O(a α
). Since α is positive (see Section II above), this implies that the displacement and its gradient (and hence the stresses) at any given, fixed point M are continuous functions of a for a = 0 + , q.e.d.

Asymptotic form of the stress intensity factors at the tip of a small crack

We shall now examine the asymptotic form of the SIFs K p (a) at the tip of the crack originating from the notch root O for a → 0. We first consider the special case where the body considered is a circular disk of center O, radius R, endowed with a notch with apex at O and a crack originating from the same point, and subjected to some boundary traction field T (see Fig. 2). The geometric parameters of the problem are the radius R, the aperture angle 2(πψ) of the notch, the angle ϕ between its bisecting line Ox and the initial tangent to the crack, the curvatures C 1 , C 2 of the upper and lower boundaries of the notch, the curvature C of the crack and its length a. Thus the vector K(a) ≡ (K I (a), K I I (a)) of SIFs at the crack tip may be written symbolically as

K(a) ≡ L(R, ψ, ϕ, C 1 , C 2 , C, a).T , ( 7
)
where L is a (vectorial) linear functional, depending on the geometric parameters R, ψ, ϕ, C 1 , C 2 , C, a.

It is a classical property of the equations of elasticity that one can obtain a new solution from an old one by multiplying both the distances and displacements by some arbitrary positive factor λ while keeping the strains and stresses (and therefore the surface tractions) unchanged. In this operation, the parameters R, ψ, ϕ, C 1 , C 2 , C, a become λR, ψ, ϕ, C 1 /λ, C 2 /λ, C/λ, λa. Furthermore, since the SIFs are limits of certain stress components times the square root of some vanishingly small distance, they are multiplied by √ λ in the process. Thus the functional L satisfies the following 'positive homogeneity' property :

L(λR, ψ, ϕ, C 1 /λ, C 2 /λ, C/λ, λa).T = √ λ L(R, ψ, ϕ, C 1 , C 2 , C, a).T ⇒ L(λR, ψ, ϕ, C 1 /λ, C 2 /λ, C/λ, λa) = √ λ L(R, ψ, ϕ, C 1 , C 2 , C, a) (∀λ > 0) . ( 8
)
We now introduce the hypothesis that K(a) behaves like a certain power of the crack length a for a → 0; this implies that where β is an unknown exponent to be determined and L (β) (R, ψ, ϕ, C 1 , C 2 , C) some new linear functional1 . Inserting Equation ( 9) into Equation ( 8), one gets

L(R, ψ, ϕ, C 1 , C 2 , C, a) ≡ L (β) (R, ψ, ϕ, C 1 , C 2 , C)a β + o(a β ) , ( 9
)
L (β) (λR, ψ, ϕ, C 1 /λ, C 2 /λ, C/λ)(λa) β + o(a β ) = = √ λ L (β) (R, ψ, ϕ, C 1 , C 2 , C)a β + o(a β ) ;
identifying terms proportional to a β in this equation, one gets the following 'positive homogeneity' property for the functional L (β) :

L (β) (λR, ψ, ϕ, C 1 /λ, C 2 /λ, C/λ) = λ 1 2 -β L (β) (R, ψ, ϕ, C 1 , C 2 , C) (∀λ > 0) . ( 10
)
We now come back to the general case where the shape of the body is arbitrary. We consider, within , circular disks of center O and radius R (intended to be shrunk to zero at the end, but fixed for the moment), containing the entire crack (R > a): see Fig. 3. We denote by T (R, a) the traction field exerted on the boundary of the disk of radius R, when the crack length is a, as a result of the application of the prescribed loading (u p , T p ) on ∂ u and ∂ T . Clearly, the SIFs K p (a) are unchanged if one eliminates the exterior of the disk of radius R while preserving the traction field T (R, a) exerted on its boundary; thus Equation ( 7) holds for that loading :

K(a) ≡ L(R, ψ, ϕ, C 1 , C 2 , C,

a).T (R, a) .

(11)

Now we take the limit a → 0 in the previous equation. Then, R being fixed, T (R, a) tends towards the traction field T (R) exerted on the boundary of the disk prior to initiation of the crack, because of the property of continuity of the stresses at a fixed point with respect to a for a = 0 + established in the previous section. Thus, by Equation ( 9),

K(a) ≡ K (β) a β + o(a β ), ( 12
)
where

K (β) ≡ L (β) (R, ψ, ϕ, C 1 , C 2 ,

C).T (R) .

(13)

This equation shows that K (β) possesses the remarkable property of depending on the mechanical fields only through their value prior to initiation of the crack.

We shall finally show that they depend in fact only on the asymptotic value of the initial mechanical fields. In order to do so, we shall now let R → 0. Prior to doing that, however, we rewrite Equation ( 13), using Equation ( 10) with λ = 1/R, as

K (β) ≡ R 1 2 -β L (β) (1, ψ, ϕ, RC 1 , RC 2 , RC).T (R) . ( 14
)
Furthermore, we note that in view of Equations ( 1), the stresses prior to initiation of the crack are of the form

σ (r, θ) ≡ κr α-1 f(ψ; θ) + o(r α-1 )
where r and θ denote polar coordinates, κ the 'stress intensity factor of the notch' in the absence of the crack and α the 'exponent of the displacements' as in Section 2, and f(ψ; θ) some universal (depending only on ψ and θ) symmetric second-rank tensor. It follows that the traction field T (R) exerted on the boundary of the disk of radius R prior to crack initiation is of the form

T (R) ≡ κR α-1 {f(ψ; θ).e r (θ)} + o(R α-1 ) (15)
where e r (θ) denotes the unit radial vector and {T(θ)} the surface traction field defined by the traction T(θ). Inserting Equation (15) into Equation ( 14) and using linearity, we get

K (β) ≡ κR α-1 2 -β L (β) (1, ψ, ϕ, 0, 0, 0). {f(ψ; θ).e r (θ)} + o(R α-1 2 -β ) . ( 16
)
Now the left-hand side in this formula is in fact independent of R by definition (it is connected to the SIFs at the crack tip, which are governed only by the geometry of the body and the loading applied, not by the radius of the fictitious disk envisaged); it follows that the exponent α -1 2β in the right-hand side must necessarily be zero, which implies that

β = α -1 2 . ( 17
)
Since α > 1/2 except for ψ = π (case where the notch is in fact a crack), β > 0 except in that case, which means, by Equation ( 12), that the SIFs K p (a) tend to zero with a except for = π . Equation ( 16) may be rewritten, using Equation ( 17), as

K (β) = κL (β) (1, ψ, ϕ, 0, 0, 0). {f(ψ; θ).e r (θ)} + o(R 0 ) .
The left-hand side and the first term of the right-hand side here are independent of R. Hence the second term of the right-hand side must also be independent of R. Since it tends to zero with this parameter, it must be zero. Hence we finally get K (β) = κL (β) (1, ψ, ϕ, 0, 0, 0). {f(ψ; θ).e r (θ)} , (

that is,

K (β) ≡ L(ψ, ϕ)κ , (19)
where L(ψ, ϕ) is a universal vectorial function of the angles ψ and ϕ. The term universal means that Equation ( 19) holds for all envisageable geometries and loadings, K (β) being always given by the product of L(ψ, ϕ) and κ where the function L depends solely on the arguments ψ and ϕ (and is for instance completely independent of the curvatures C 1 , C 2 , C of the notch boundaries and the crack).

Practical calculation of the universal functions L p (ψ, ϕ)

The reasoning expounded in the preceding section has evidenced the existence of the universal vectorial function L(ψ, ϕ) but has not provided the values of its components L p (ψ, ϕ). We shall now see how this reasoning can be pursued to yield a practical method for effective determination of these components.

We start from Equations ( 18) and ( 19), which imply that

L(ψ, ϕ) = L (β) (1, ψ, ϕ, 0, 0, 0). {f(ψ; θ).e r (θ)} .
Now, by the very definition (9) of the functional L (β) , the preceding expression becomes

L(ψ, ϕ) = lim a→0 a -β L(1, ψ, ϕ, 0, 0, 0, a). {f(ψ; θ).e r (θ)} = = lim a→0 a 1 2 -β L 1 a
, ψ, ϕ, 0, 0, 0, 1 . {f(ψ; θ).e r (θ)}

(by Equation ( 8) with λ = a)

= lim R→+∞ R α-1 L(R, ψ, ϕ, 0, 0, 0, 1). {f(ψ; θ).e r (θ)}

(with R ≡ 1/a, and account being taken of Equation ( 17)). Using linearity, one therefore finally gets, for p = I, I I :

L p (ψ, ϕ) = lim R→+∞ L p (R, ψ, ϕ, C 1 = 0, C 2 = 0, C = 0, a = 1). R α-1 f(ψ; θ).e r (θ) . ( 20
)
What Equation ( 20) says is the following. Consider a circular disk of center O, radius R, endowed with a notch with apex at O, straight (C 1 = C 2 = 0) boundaries and aperture angle 2(πψ), and a straight (C = 0) crack originating from O of unit (a = 1) length, making an angle ϕ with the bisecting line Ox of the notch (Fig. 4). Enforce the 'fundamental' traction field R α-1 f(ψ; θ).e r (θ) on the boundary of that disk, the boundaries of the notch and the crack lips remaining traction-free. Then L p (ψ, ϕ) is identical to the p-th SIF at the tip of the crack, in the limit R → +∞. It is equivalent to saying that one must consider an infinite body with the following conditions on the notch boundaries and the crack lips and at infinity : (r, θ = ±ψ).e θ (θ = ±ψ) = 0, ∀r > 0 (conditions on the notch boundaries), σ (r, θ = ϕ ± ).e θ (θ = ϕ) = 0, ∀r, 0 < r < 1 (conditions on the crack lips), σ (r, θ) = (r, θ) + o(r α-1 ) for r → +∞, ∀θ, -ψ < θ < +ψ (condition at infinity), ( 21) where e θ (θ) denotes the unit orthoradial vector and { (r, θ)} ≡ r α-1 f(ψ; θ) the 'fundamental' stress field defined by Equations (1) except for the omission of the factor κ.

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ σ
Using a classical LEFM trick, one can remove the imposed stresses from infinity to the crack lips. The conditions on the notch boundaries and the crack lips and at infinity then become, instead of (21) : One will in fact need a more precise information on the behavior of the stresses at infinity than Equation ( 22) 4 . For r → +∞, the influence of the crack vanishes so that the stresses behave as if it were absent. Hence σ (r, θ) must behave as r α -1 where α is an exponent 'adapted' to the notched, crack-free geometry, and therefore satisfying one of the equations sin(2α ψ) + α sin(2ψ) = 0 , sin(2α ψ)α sin(2ψ) = 0 (see the Appendix); since σ (r, θ) is o(r α-1 ) for r → +∞, α must in fact be the greatest solution of one of these equations smaller than α. Since α is the smallest positive solution of one of these equations, α cannot be positive. Hence it is the greatest negative 1 solution. It follows that it must be equal to -α. Thus Equation (22) 4 may be refined in the following way :

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ σ (r, θ = ±ψ).
σ (r, θ) = O(r -α-1 ) for r → +∞, ∀θ, -ψ < θ < +ψ . ( 23
)
Part II will be devoted to the effective calculation of the functions L p (ψ, ϕ) through solution of the plane elasticity problem defined by Equations ( 22), ( 23).

Appendix: Definition of the stress intensity factor of a notch

The aim of this Appendix is to provide some justification for the definition of the 'stress intensity factor of a notch' κ implicitly implied in Equations (1) of the text.

It is first necessary to briefly recall some elements of Williams' (1952) classical treatment of the problem, based on the search of the form of the Airy stress function φ(r, θ). This function is looked for in the form φ(r, θ) ≡ r α+1 χ(θ) where α is the a priori unknown 'exponent of the displacements', that is, the exponent of r in their asymptotic expression for r → 0. Beltrami's equation φ(r, θ) = 0 yields the differential equation The condition for the first system to possess a non-trivial (non-zero) solution (A, C), that is, for its determinant to be zero, reads sin(2αψ) + α sin(2ψ) = 0 while the analogous condition for the second system is sin(2αψ)α sin(2ψ) = 0 .

χ iv (θ) + ((α - 1) 2 + (α + 1) 2 )χ (θ) + (α -1) 2 (α + 1) 2 χ(θ) = 0,
1 The solution α = 0 can be ruled out because it only represents some rigid-body (translatory) motion.

It is easily verified that for ψ < π, the smallest positive 1 value of α satisfying one of these equations, corresponding to the most singular stress field, is in fact a solution of the first equation. Provided that α takes this particular value, the coefficient A is arbitrary but C is tied to it through the relation To check that our definition of the 'stress intensity factor of the notch' κ is the best possible one, let us now consider the special situations where ψ = π (case of a crack) and ψ = π/2 (case of a corner-free boundary).

If ψ = π, α = 1/2 so that sin((1 + α)ψ) = -1, sin((1α)ψ) = 1, and the expressions of the stresses become 1 The 'displacement exponent' α must be positive because the 'stress exponent' α -1 must be greater than -1 in order for the total elastic energy to be finite.

Figure 1 .

 1 Figure 1. The general problem considered.

Figure 2 .

 2 Figure 2. The special case of a circular disk.

Figure 3 .

 3 Figure 3. Circular disk within an arbitrary body.

Figure 4 .

 4 Figure 4. The geometry to be considered to determine the functions L p (ψ, ϕ).

  e θ (θ = ±ψ) = 0, ∀r > 0 (conditions on the notch boundaries),σ (r, θ = ϕ + ).(-e θ (θ = ϕ)) = -(r, θ = ϕ).(-e θ (θ = ϕ)) = (r, θ = ϕ).e θ (θ = ϕ) ∀r, 0 < r < 1 (condition on the upper lip of the crack), σ (r, θ = ϕ -).e θ (θ = ϕ) = -(r, θ = ϕ).e θ (θ = ϕ)∀r, 0 < r < 1 (condition on the lower lip of the crack), σ (r, θ) = o(r α-1 ) for r → +∞, ∀θ, -ψ < θ < +ψ (condition at infinity) .

B

  the general solution of which is χ(θ) = A cos((α -1)θ) + B sin((α -1)θ) + C cos((α + 1)θ) + D sin((α + 1)θ) where A, B, C, D are arbitrary constants. The boundary conditions σ θθ (r, ±ψ) = φ ,rr (r, ±ψ) = 0, σ rθ (r, ±ψ) = -(φ ,θ /r) ,r (r, ±ψ) = 0 then yield the two systems of equations ⎧ ⎪ ⎨ ⎪ ⎩ A cos((α -1)ψ) + C cos((α + 1)ψ) = 0, (α -1)A sin((α -1)ψ) + (α + 1)C sin((α + 1)ψ) sin((α -1)ψ) + D sin((α + 1)ψ) = 0 , (α -1)B cos((α -1)ψ) + (α + 1)D cos((α + 1)ψ) = 0 .

  α) sin((1α)ψ) (1 + α) sin((1 + α)ψ) Awhile the coefficients B and D are necessarily zero. It follows that the Airy stress function is of the formφ(r, θ) = Ar α+1 cos((1α)θ) -(1α) sin((1α)ψ) (1 + α) sin((1 + α)ψ) cos((1 + α)θ) . Now define κ ≡ -2(2π) 1-α (1α) sin((1α)ψ) sin((1 + α)ψ) A;the expression of the stress function becomesφ(r, θ) = κ 2 (2π) α-1 r α+1 -sin((1 + α)ψ) (1α) sin((1α)ψ) cos((1α)θ) + 1 1 + α cos((1 + α)θ)and it follows that the stresses are given byσ rr (r, θ) = φ α-1 (3α) sin((1 + α)ψ) (1α) sin((1α)ψ) cos((1α)θ) + cos((1 + α)θ) ; σ θθ (r, θ) = φ ,rr (r, θ) = = κ α 2 (2πr) α-1 -(1 + α) sin((1 + α)ψ) (1α) sin((1α)ψ) cos((1α)θ) + cos((1 + α)θ) ; α)θ) + sin((1 + α)θ) ,that is, Equations (1) of the text.

The notation o(x γ ) is recalled to stand for a function of the form x γ f (x), with lim x→0 f (x) = 0.

they are identical to the classical formulae for a crack loaded in mode I, with κ ≡ K I .

If ψ = π/2, α = 1 so that sin((1α)ψ) = sin((1 + α)ψ) = 0, and we get the indeter-

in the expression of the stresses. To find the value of this indeterminate form, it suffices to note that since the value of α is such that the determinant of the system in A and C be zero,

for ψ → π/2. From there, one immediately derives the expressions of the stresses :

which correspond to some uniform stress field σ yy ≡ κ.

We have thus shown that with our definition of κ, this parameter possesses the nice properties of reducing to K I for ψ = π (case of a crack) and to σ yy for ψ = π/2 (case of a regular boundary), which justifies our choice.