Jean-Michel Servet

Hadrien Saiag

HOUSEHOLD OVER-INDEBTEDNESS IN NORTHERN AND SOUTHERN COUNTRIES: A MACRO-PERSPECTIVE

fruits of economic growth' have been distributed in an increasingly unequal manner. While this period has been the 'golden age' for financial markets, 3 it could also be called the 'calamitous years'. Unemployment rates, in particular, were much higher from 1980-2000 than 1950-60. The purpose of this chapter is to show that the rise of household debt is not unique to 'developed' countries. In the North, indebtedness is a well-documented fact; it has been quantified and its contribution to macroeconomic relationships has been studied. However, little data is available from the South. This could lead one to think that rising household debt is not an issue. Yet, the contributions in this volume indicate the opposite. In fact, the South is also affected by over-indebtedness. There is a growing mismatch between monetary income and cash needs, the source of debt.

Our argument draws on two kinds of sources. Concerning the North, we rely primarily on the work of the French school of regulation. In the South, rising household debt has not, to our knowledge, been the object of thorough macroeconomic analysis. Nor is it measured by statistics, due to the extent of informal financial practices. In this chapter, we aim to shed light on the gap created by the slow growth of most households' cash incomes and their growing financial needs. Therefore, we rely on official reports (mainly from the International Labour Organization, concerning remuneration and employment trends) and statistics (from World Bank, concerning growth of cash needs).

. The second part examines the rise of indebtedness in the South. To grasp its magnitude, we revisit the process of financialization. It is based on a growing monetization of social relationships. As in the North, the slowdown of growth in workers' earnings has been accompanied by a sharp rise in social inequality. Incomes are 3 Term used by the Observatoire de la Finance in Geneva.

Introduction

Neo-liberal policies have shaped modes of production, trade and financing for the last thirty years and the crisis of 2007-8 revealed the untenable nature of household indebtedness in the United States. Apologists of neo-liberalism are convinced that such policies enabled two decades of prosperity, following the 1973-4 crisis. The approach has become hegemonic, explicitly opposing interventionist Neo-Keynesian policies under which public and private debt drive demand for investment and consumer goods, through deficit spending and money creation. This makes it possible to anticipate demand and make it 'effective', because it is solvent. By contrast, neo-liberals believe deficits should be banned; for them, a balanced budget is a precondition of equitable distribution. To the extent that deficits are considered the culprits of rising prices, they favour debtors at the expense of creditors. Although neo-liberal morality (if we can call it that) opposes this type of imbalance, growing levels of household indebtedness reveal that such opposition is naive. China and India notwithstanding, 1 overall growth rates were on average two times higher during the so-called 'Keynesian' period (the post-WWII boom from the 1950s to early 1970s) than they were during the neo-liberal years that followed 2 and precarious working conditions have greatly increased. Since the 1980s, the 1 Massive government intervention in these countries has fueled growth; their governments do not comply with neo-liberal dogma.

2 Pollin (2003: 131) shows that the overall growth rate of low-income and middle-income countries (excluding China) were 5.5 per cent between 1961-80, and only 2.6 per cent during the neo-liberal era . Growth rates per capita fell from 3.2 per cent to 0.7 per cent during the same period.

3 evolving in a way that is incompatible with cash needs, which are growing as home consumption decreases due to urbanization; the widespread desire to imitate the consumption patterns of others due to the potential equality between individuals (see [START_REF] Dumont | Homo aequalis -Tome 1 : Genèse et épanouissement de l'idéologie économique[END_REF]; and the rise of informality and irregular income flows. Household over-indebtedness stems from this contradiction. Finally, the third part discusses a key element of over-indebtedness: inflation. If social relations would allow it, modest inflation would be a relatively peaceful way to solve the problem of growing private and public debt in a way that favours lenders.

I. Household debt in the North: the heart of an ephemeral accumulation regime

To understand the rise of household indebtedness in the North, we must first reconstruct debt's role in macroeconomic dynamics. The latter have been studied in depth by the Regulation School. Between 1950-70, according to this school of mainly French economists, Northern countries experienced different forms of a single accumulation regime4 called 'Fordism'. It was characterized by a conflictual, albeit stable compromise between capital and labor concerning the distribution of productivity gains (the Fordist wage labour nexus) resulting from the 'rationalization' of production. As economies were rarely open to international trade at the time, wage gains engendered an increase in effective demand.

Moreover, government intervention helped support economic activity (through budget deficits and advances to the Treasury) and a relatively equitable distribution of wealth. Added to this was constant inflation, whose negative real interest rates were advantageous to borrowers.

The crisis that hit in the 1970s fundamentally threatened the institutions underpinning these macroeconomic relationships: the rise of real wages collided with an opening of economies [START_REF] Boyer | Théorie de la régulation[END_REF]. The institutional forms 5 that provided the scaffolding of the accumulation regime were overturned. Thus, since the 1980s, governments have scaled back direct intervention in the economy: they now create institutions designed to 'deregulate' modes of production, trade and financing. They have provided the impetus to significantly develop financial markets (particularly those dealing with public debt) and dismantle the Fordist wage labour nexus. In the North, the result has been the development of new forms of work on the margins of wage labour 6 (short-term, part-time, student jobs, sub-contracting, etc.). These new forms of employment have driven down the salaries of unskilled workers and increased insecurity (due to poor benefits packages). Consequently, social inequality has skyrocketed (Feller and Stone, 2009; Picketty and Saez, 2003 7 ).

Despite the escalation of social inequality caused by these changes, this new institutional arrangement has sometimes given way to macroeconomic coherence, as in the case of the United States and, to a lesser extent, the United Kingdom. This appears paradoxical, since the dismantling of the Fordist wage system and the increasingly inequitable distribution of wealth has depressed effective demand. As a result, households are heavily indebted, through various forms of consumer credit (particularly credit cards). But the novelty of this finance-led accumulation regime8 does not lie herein. Rather, it is the way it disconnects consumer spending and income generation. Indeed, the advent of loan securitization (especially mortgages) has fostered a new form of household debt: the main limitation is no longer anticipated household income, but the valuation of the securities that serve as collateral. In other words, the only limit to household indebtedness is the valuation of real estate and financial securities. As the values and volume of these securities increase, so does debt. As a result, massive household indebtedness now underpins the American economy, household debt at one point reaching 96% of GDP in 20099 . In 1954, total domestic debt in the United States was only US$ 500 billion. In the first half of 2009 alone it exceeded US$ 50,000 billion10 .

The United States' unique position in the global financial system has made this situation possible. At a global level, this headlong rush was facilitated by trade surpluses from countries like China and oil exporting countries with small populations. Their investments appeared safe, because they were made in one of the world's most developed countries, and because the dollar became the financial world's reserve currency following World War II. Thus, a considerable amount of money flowed into the United States. This accumulation regime is based on a reversal of roles whereby the productive sector must now answer to the financial sector. According to Robert [START_REF] Boyer | Is a Finance-led growth regime a viable alternative to Fordism? A preliminary analysis[END_REF], the break with Fordism is obvious. In Fordism, the wage labour relationship prevailed, insofar as it allowed the synchronization of mass production and consumption by institutionalizing a distribution of productivity gains from the 'productive' sphere. In the finance-led accumulation regime, there is a reversal of the hierarchical relationship between institutional forms-finance and the wage relation-that is advantageous to the financial sector, which now controls the economy. Indeed, stock prices simultaneously determine investments decisions, household debt and consumption choices, fiscal policy (the tax burden has been shifted to employees to enable the development of capital markets), and finally, monetary policy (aimed at financial stability). Finance no longer relies on production and trade; it has reversed the relationship at a scale never before seen. While financiers claim to serve society and create prosperity, finance has become increasingly predatory [START_REF] Servet | Genève dans l'empire de la liquidité[END_REF], destroying wealth through the constant processes of valuation/devaluation that ensure its development and result in precarious living conditions for the immense majority of the population.

This accumulation regime has really only flourished in the United States and the United Kingdom. The supposed virtuous circle of debt, financial markets and household consumption has not been observed elsewhere. In other words, the modes of regulation in Northern countries are very diverse. Michel Aglietta (1998 11 ) has argued that the equity-based ('patrimonial') growth regime would become a permanent fixture in continental Europe. The argument is consistent with the increasingly widespread use of household debt to offset loss of income. But we cannot conclude from this alone that worldwide economic growth is characterized by a finance-led accumulation regime. Indeed, according to the typology created by one regulationist theorist [START_REF] Boyer | Théorie de la régulation[END_REF], an accumulation regime must be capable of reproducing institutional forms consistently over a relatively long period. Currently, debt allows households to significantly boost their consumption by increasing the value of their assets in order to compensate the loss of wage income. This implies that household wealth is increasingly based on asset-backed securities. Robert [START_REF] Boyer | Is a Finance-led growth regime a viable alternative to Fordism? A preliminary analysis[END_REF] has showed that this only holds true in the United States and possibly the United Kingdom 12 . Elsewhere, the model has not taken hold. Thus, although the financial sector now has considerable influence over economic developments in continental Europe, 13 we have not yet seen an accumulation regime that is capable of generating steady growth since Fordism.

Despite the different trajectories of 'developed' countries, debt expansion has been the common vector, draining resources to the benefit of the financial sector. Hence, tremendous financial growth (such as the sextupling of stock market capitalization between 1990-2007 and the boom in securitized financial products) frequently goes hand in hand with outflows from the 'productive' sphere (just look at the financial sector's profits and extraordinary salaries 14 ). Here too, the United States is stands apart: 'The enormous financial growth that has accompanied market deregulation and globalization suggests that the process could not go on forever. In 1980, the profits generated by the American financial sector represented 10 per cent of total private sector profits; in 2007, it was 40 per cent. The same year, the financial sector employed only 5 per cent of the salaried workforce of the private sector, and contributed only 15 per cent of value added. The financial sector has thus sucked out the value created by the economy; the value has disappeared in exorbitant salaries for traders, extravagant commissions for capital restructuring operations, securitization and asset management. Market capitalization of the financial sector collapsed in 2008 because it had inflated without restraint, increasing six fold; the financial sector's share grew from 6 per cent in 1980 to 19 per cent in 2007.' (Aglietta and Rigot, 2009: 19) In other words, the United States and perhaps the United Kingdom notwithstanding, growing household debt has not been able to stave off the decline in effective demand. There are several reasons for this. The dismantling of the Fordist wage regime may be justifiable from a 12 For more on the unequal distribution of financial assets in France, see [START_REF] Duigou | Faut-il patrimonialiser les droits sociaux?[END_REF]. 13 See Coriat (2008) concerning France. 14 There is a point of consensus among economists who study this issue. For a detailed calculation, see publications of the Political Economy Research Institute (PERI) of the University of Massachusetts Amherst (http://www.peri.umass.edu/). For a sociological analysis of the rise in household debt in France, and its draining effect, see the review Sociétés contemporaines, no. 46 (2009), particularly articles by Hélène Ducourant and Ana Perrin Heredia. For Germany, see [START_REF] Haas | Overindebtedness in Germany[END_REF] microeconomic perspective, but it reveals pressure to drive down global demand on a macroeconomic scale, all things being equal. The surge in income inequalities has stripped wealth of its reproductive capacity, because households' marginal propensity to consume is decreasing in relation to the level of income: wealthy people's financial investments do not create effective demand because they do not stimulate production. In addition, increased international competition for domestically produced goods has reduced fiscal pressure on the private sector in the name of competitiveness. Customs duties collected by the governments have decreased considerably. And since public expenditures have not been cut, these tax revenues have been recovered through taxes on salaries, and very occasionally and to a much lesser extent, on real estate, assets, and luxury goods. These direct and indirect taxes have pushed down incomes of the majority at the same social benefits have come under threat; health care and education costs are eating up increasingly large portions of family budgets, leading to an overall pressure to reduce other expenses. Thus, the current crisis is very much endogenous to modes of production, trade and financing based on growing household debt. It reveals the contradictions inherent to the current phase of capitalism. The finance-driven accumulation regime can no longer reproduce itself. It has effectively shot itself in the foot: over-indebted households can no longer honor their debts.

The financial sector has sucked its victims dry. The corollary of diminished income is increased debt, which augments outflows from the productive sector to the financial sector, thereby increasing inequality, which depresses demand and income, which in turn tends to increase indebtedness. In the medium term, this translates into healthy financial institutions on the one hand, depressed incomes for the majority with rising unemployment on the other (a 'stagboom') (Servet, 2010a). But ultimately, these outflows end up 'suffocating' those they 'feed'. The crisis of 2008 erupted because the amounts drained by the financial sector surpassed the surplus actually generated. The race for debt has produced unsustainable disequilibrium. It has undermined the balanced reproduction of modes of production, trade and financing. Extend the analogy of David Ricardo's position on the limits of the capitalist system of production in terms of increased rents, and Karl Marx's regarding falling rates of profit, and it would appear that the limit of the neo-liberal accumulation regime is the extent to which the financial sector drains value from all other productive activities (capitalist or otherwise), public resources and salaried income.

II. Southern countries and financialization: exacerbating the gap between needs and income

According to statistics from international organizations, household debt does not appear to be a problem in the South. True, the portion of private sector macroeconomic debt from financial institutions compared to GDP has grown significantly between 1980 and2007 (see Table 1.1), doubling in Africa and the Middle East, and increasing by almost 150 per cent throughout the Asia and Pacific. Only Latin America and the Caribbean seem to be an exception, with a mere 29 per cent increase during the period. This increase is partly due to consumer credit, which has grown considerably in the South in recent years 15 . Nevertheless, the private sector debt-to-GDP ratio remains much lower in the South than in the North: in Latin America and the Caribbean, this ratio for accounts 26 per cent of that of developed countries, 52 per cent for Asia and the Pacific, 34 per cent for the Middle East and 39 per cent for Africa. Adults in developed countries are four times more likely to have a loan from a bank than in developing countries. Banks in developing countries mainly lend to the rich, who supposedly offer better guarantees: in the North, the average loan to a household represents 53 per cent of national income. In developing countries, the rate is 128 per cent16 . These figures might suggest that debt problems are relatively foreign to southern countries... Note: Domestic credit to private sector refers to financial resources provided to the private sector, such as through loans, purchases of nonequity securities, and trade credits and other accounts receivable, that establish a claim for repayment. For some countries these claims include credit to public enterprises.

However, such data tend to underestimate household debt. It is difficult to quantify precisely: statistic bureaus only take into account indebtedness to financial institutions when in fact, a large part of household debt in the South comes from 'informal' practices17 , with large national and regional differences regarding the weight and the forms. As this book demonstrates, informal financial practices are as diverse as the contexts in which they evolve.

Besides rotating savings and credit associations where debt and credit balance out over time, there exist many other forms of associations and community organizations [START_REF] Servet | Community relations, individual, social and economic constraints in the saving and loans associations[END_REF].

They cover funeral expenses, local celebrations, community works, and infrastructure development. There are also a multitude of private intermediaries. These include formal or informal specialized financial providers that may or may not lend against collateral: shopkeepers who offer cash advances; traders who extend credit based on standing crops; employers or intermediaries who offer salary advances; merchants selling goods on credit that they immediately buy back at lower prices; in-kind advances by suppliers to small shopkeepers, artisans and itinerant traders; loans from a former employer for apprentices who want to start their own business, etc. Savings collectors still exist: shopkeepers and acquaintances who allow the unbanked to save in a safe place rather than hoard at home. These 'moneykeepers' often lend, as well. The repayment schedule varies according to the collector's own collection schedule-weekly, monthly, harvest time. In some cases, the creditor may only demand interest payment and not the principle, much as a landlord who collects rent does not try to sell the property to the tenant.

To understand the rise of debt in the South, let us clarify what we mean by financialization.

Financialization cannot be reduced to the expansion of financial markets (as is often the case).

We distinguish three levels [START_REF] Servet | Banquiers aux pieds nus : la microfinance[END_REF](Servet, : 35-60, 2010a)). At the base is monetization of the economy. Monetization is due to the increased commodification of the everyday conditions of domestic reproduction. It is the result of a diminution of economic organization based on the household and public institutions. This has led to a decrease in home consumption and increased monetization of family budgets. For the first time, money is needed to meet the most basic needs: it is no longer reserved for special occasions [START_REF] Coppet | Une monnaie pour une communauté mélanésienne comparée à la nôtre pour l'individu des sociétés européennes[END_REF]. Next, comes financial intermediation, as it becomes increasingly necessary to use an intermediary to access loans, payments, transfers, insurance and savings 18 ; the intermediary, however, is not necessarily a financial institution. At the highest level are financial speculative markets. The summit (speculation) cannot exist without the base (monetization). But at the heart of the current financialization, the former dominates the latter through a process of exploitation. As in any exploitative relationship, the 'superior' feeds off the 'inferior'. Speculative markets are thought to be indispensable to the very existence of monetization, especially since high finance has become absolutely necessary for people to 'make ends meet' and acquire the production means for the majority. Thus, although it is impossible to accurately quantify the second level of financialization in the South (which includes household debt), it is possible to identify its source of growth: the growing gap between needs and cash incomes.

Income transfer programs implemented in so-called 'developing' countries are really part of the process of financialization. These programs aim mainly to cover the health care and education costs of low-income families. They are particularly common in Latin America. The best documented example is probably that of Mexico-the Progresa program, now known as

Oportunidades-but such programs can also be found in other latin-american countries, Africa and Asia [START_REF] Handa | The Experience of Conditional Cash Transfers in Latin America and the Caribbean[END_REF]. These programs encourage the monetization of expenditures, as opposed to transactions in kind. Such monetization may also encourage debt, by increasing financial flows at the local level to the benefit of microfinance institutions and local moneylenders. In Mexico, for example, these subsidies are very popular with small businesses, as they enable clients to reimburse purchases made on credit (fiado) (Morvant-Roux, 2006: 225).

Various factors underscore the growing gap between incomes and needs. First of all, as in the North, the slow progression of incomes in the South has been accompanied by inequitable income distribution, in a context marked by sluggish and more volatile economic growth (ILO, 2008: 3). On average, wages grew more slowly between [START_REF] Easterly | The middle class consensus and economic development[END_REF]and 2007than between 1995and 2000. Between 2001and 2007, they increased an average of only 0.3 per cent per year in Latin America and the Caribbean and 1.8 per cent in Asia. In addition, there appears to be a leveling-off effect during economic crises, particularly in Latin America: wages are hit hard during declining growth in this region, and they rarely recover their pre-crisis level (ILO, 2008: 3-12). Between 1990 and 2006, wages grew less quickly than productivity, resulting in 18 See for example the boom in mobile payments in the South. a shift in the distribution of value added to the benefit of capital (ILO and IILS, 2008: 7).

Inequalities in income distribution have thus increased: the gap between the incomes of the poorest 10 per cent of the population compared to the richest 10 per cent has widened from 1995-2000 and from 2001-2006 in most Southern countries where data are available (ILO, 2008: 24). Similarly, the Gini coefficient (which measures inequalities in income distribution) has increased in most countries of Asia and the Pacific, Latin America and the Caribbean, and Central and Eastern Europe. Africa seems to be an exception: the Gini coefficient has declined in most countries where statistics are available (ILO and IILS, 2008: 11-12).

These income inequalities are part of a broader context characterized by persistent exclusion, marginalization and discrimination of social groups. In this sense, inequality is not, fundamentally, a consequence of poverty due to inadequate resources. Conversely, poverty should be seen as the result of inequality, particularly structural inequalities and discrimination that cause exclusion and marginalization of ethnic, linguistic, caste and some immigrant groups [START_REF] Diop | Poverty vs. Inequality[END_REF]. The probability of being poor varies significantly and it is difficult to attribute it to a particular social class. It is probably no coincidence that nearly 44 per cent of the world's 'poor' is concentrated in South Asia: 80 per cent of humanity considered to be living 'in absolute poverty' is in India, Bangladesh, Pakistan, Sri Lanka, Bhutan, Nepal and the Maldives, although the Gini coefficient in these countries is not higher compared with other regions of the world. In analyzing the caste system in India as a social structure of accumulation, Barbara Harriss-White (2003: 176-199) highlights its key role in perpetuating inequalities, particularly by fragmenting the labour market. Countries like the Indian Union, Pakistan, Bangladesh and Nepal have much higher poverty rates than most other countries because of the persistence in these societies of untouchability-one of the most abject forms of inequality-and the high incidence of debt bondage (Breman, Guérin and Prakash, eds., 2009). More generally, William [START_REF] Easterly | The middle class consensus and economic development[END_REF] has shown a close correlation between high poverty rates and strong social fractures. According to Easterly, this is because the dominant groups do not seek to build the capacity of excluded groups, who may challenge and weaken their power. Thus, there is greater consensus around sharing resource in countries where social divisions are less.

Parallel to the slowdown in income growth in most households, there has been an exacerbation of cash needs. This is partly due to the decline in consumption of food and clothing from one's own production. There is little data available on this subject; we have identified only two recent household budget surveys 19 per cent ten years earlier (p.70). Most importantly, this survey shows that home consumption is much higher in rural than in urban areas 21 . Thus, the decline in home consumption is related to growing urbanization rates. As Table 1.2 shows, growth rates throughout the South increased considerably and steadily from 1960 to 2008. Consequently, monetary needs have increased substantially, in order to access to food 22 . Note Urbanization refers to the ratio of the number of people living in urban areas (as defined by national statistical offices) to the number of residents in each country.

The growth of urbanization rates has also increased cash needs in other ways. When slum dwellers are expelled from their homes, for example, they are often forced to move to farther away from the urban areas that provide them jobs and cash income, thereby decreasing their monetary resources. Urban sprawl in some cities in the South has also made it particularly difficult for small farmers to maintain their activities: streams used for irrigation are diverted, 19 The authors would like to acknowledge Nicolas Brice of CIRAD for informing us of these studies. 22 The current situation differs from the little-monetized African economies of the late 1960s, described by [START_REF] Hallu | Réalités africaines et enquêtes budget-consommation[END_REF] and [START_REF] Servet | Des ethnies aux paysanneries noires (insertion de l'agriculture africaine sub-saharienne dans le marché mondial et genèse des paysanneries noires[END_REF].

water increasingly polluted, and distances to walk made even longer due to new fences set up around housing estates. The capitalist regime, under the pretext of spreading prosperity, relies on the development of private property rights ( [START_REF] Soto | The other path: the invisible revolution in the Third World[END_REF], in other words, exclusive access to previously public goods. The poor were once thought of as a kind of 'industrial reserve army', driving salaries downward and, in periods of expansion, providing a labour pool to meet needs. This downward pressure has not disappeared in all countries, but contemporary forms of economic development tend to make poor redundant.

Globalization has also led to an explosion of needs; and, in absence of a sharp rise in disposable income or a new ascetic morality, the inevitable consequence is significant subjective impoverishment. In some ways, the wealth of some creates the poverty of others, not necessarily because the poor are pillaged, in a quasi-mercantilist sense, but because poverty is exacerbated by the desire to imitate the consumption patterns of others. In ancient hierarchical societies, where social status determined consumption capacity and ostentation, the desire to imitate was limited, as the social order was consensual. The destruction of these hierarchies led to a widespread desire to access the same goods and services (see Ortega y [START_REF] Ortega Y Gasset | The Revolt of the Masses [original: La rebelión de las masas[END_REF][START_REF] Ortega Y Gasset | The Revolt of the Masses [original: La rebelión de las masas[END_REF]). The potential equality between individuals created by the 'flattening' of the values of hierarchical societies has been accompanied by a rise of a society of envy (Dupuy 1992) in which needs are exacerbated. One cannot need something that one does not know exists, or considers to be part of a world that is not one's own. Most economists have long thought scarcity as the cause, the motive, the underlying assumption of rational economic behavior to meet universally felt needs [START_REF] Rist | The Delusions of Economnics, The Misguided Certainties of a Hazardous Science[END_REF]; all one needs to do is adjust one's scarce means to meet their desires. However, scarcity is not a cause. Rather, it is a consequence of economic behavior (Salhins, 1972). Yet, the needs that have emerged with the flattening of hierarchies are not mere 'whims': they are a way for people to integrate into society. Indeed, the observations of Maurice Halbwachs (1933: 106) Note : Description: Passenger cars refer to road motor vehicles, other than two-wheelers, intended for the carriage of passengers and designed to seat no more than nine people (including the driver).

Finally, the last thirty years have seen a rise in so-called 'informal' jobs. 23 In the late 1980s, studies on informality showed it was not in regression; it was more than a remnant destined to disappear with economic growth. Rather, it was a structural element at the heart of accumulation strategies in developing countries (Portes, Castells and Benton, eds. 1989;Bernard, ed. 1991 on the Maghreb; on India, see Srivastavan, 2012;Harriss-White and Sinha, eds., 2007). The phenomenon is difficult to quantify, given the many possible indicators and definitions of 'informal' jobs. Jacques [START_REF] Charmes | Concepts, measurement and trends[END_REF] nevertheless makes an attempt: he quantifies the growth of informal employment in all nonagricultural employment between 1997-2007 in 47 countries. He defines informal employment by the absence of a written contract and social protection. This amounts to including informal self-employment, domestic labour and paid jobs without social protection, whether the company is declared or not. In North Africa and Sub-Saharan Africa, informal employment as a portion of total employment outside agriculture has grown gradually. It even accelerated starting in the 1990s 24 . South Asia and Southeast Asia saw a similar trend for the years 1985-1989 25 (no data available before this period). Finally, in Latin America, the growth of informal employment has been constant since the 1990s 26 (again, no data is available prior to this period).

The rise of the 'informal employment supports our argument in two respects. First, let us underline the singular relationship of 'informality' to time 27 . Income from the so-called 'informal' economy is generally more volatile than income from 'formal' jobs: job changes are more frequent [START_REF] Arias | Informalidad, protección social y mercado de trabajo en la Argentina[END_REF]) and many of these jobs are outside the wage system. This last point is particularly important, because the growth of 'informal' jobs has gone hand in hand with a rise in independent workers in all regions of the South between [START_REF] Servet | Community relations, individual, social and economic constraints in the saving and loans associations[END_REF]-2006, except in Africa (ILO, 2008: 10). The temporal nature of productive activities outside of wage employment differ greatly from those specific to wage employment. For example, a significant portion of revenues for informal activities come from expenses incurred by salaried employees. These expenses are not spread evenly throughout the year: there are strong seasonal fluctuations. Therefore, summer months are often synonymous with school holidays and reduced activity, as are some religious festivals. Moreover, expenditures are not distributed evenly over the month: when incomes are limited, the bulk of spending takes place once or twice a month. Indeed, low-income wage earners incur a significant portion of their expenses a few days after being paid 28 . In other words, a significant portion of income earned 24 This growth is noticeable in countries where we have enough data over time. This includes Algeria, Morocco, Tunisia, Mauritania, Mali, Kenya, Guinea and Chad. Egypt has taken a slightly different path: the share of informal employment declined sharply between 1975-89 and 1985-89, before starting to grow. 25 Thailand is an exception. 26 Chile and Mexico notwithstanding. 27 Space does not permit us to expand on this argument. For more details, see Saiag (2011: 25-27). 28 In Argentina, relatively poor households often make their food purchases once every two weeks (when the head of family receives his pay). This practice helps limit spending so as not to go into debt in order to access food. in jobs that fall outside of wage employment depends on the expenditures of the owners of capitalist production means (wages), which control the creation of money based on their anticipation of future revenue 29 . The temporality of the 'informal economy' is not monthly: supplier relations, 'informal' finance, investment needs, etc. arise on a weekly basis at best. The poor manage their household budgets on a day-to-day basis, incurring substantial indebtedness primarily due to cash flow difficulties.

The rise of informal employment in Latin America has been accompanied by a decline in social protection systems since the 1990s 30 . Indeed, in most countries across the region, social coverage depends on contributions made during one's working life. And yet, one of the characteristics of 'informal' employment is the absence or very low level of income [START_REF] Saavedra-Chanduvi | Informaity, social protection and antipovrety policies[END_REF][START_REF] Rofman | Pension systems in lain America: concepts and measurements of coverage social protection discussion[END_REF][START_REF] Apella | Los trabajadores independientes y el sistema de seguridad social. El caso del Gran Buenos Aires[END_REF]. In most countries in the region, there was a sharp decline between 1990-2000 in the share of the workforce contributing to the pension system 31 . Who contributed to the system also becoming more unequal: during the same period, contributions to pension schemes from the first four quintiles of the population declined sharply as the inputs of the fifth quintile grew or remained relatively stable 32 . It is true that type of employment is less an element of hierarchization in social protection systems than in the past, thanks to assistance programs for the poor and nonjob related social protection programs set up since the 2000s in the region. However, these changes have occurred with the deregulation of the labour market and overall weakening of social protection systems since the 1980s, resulting in less social protection for the whole population [START_REF] Barrientos | Labour markets and the (hyphenated) welfare regime in Latin America[END_REF]. In Africa, the situation is different. There is a decline in family solidarity because of the increasing individualization of society. This decline is not being offset by the institutionalization of social protection.. These developments have led to new cash needs (and expenditures) for the affected populations: to protect themselves against precarious and unstable employment (and unemployment) and health care issues (or the death 29 According to the terms of creation and destruction of payment means as they apply to 'modern' credit currencies. We acknowledge [START_REF] Benetti | Marchands, salariat et capitalistes[END_REF] for proposing a distinction between workers and capitalists based on the conditions that govern access to means of payment. 30 Brazil is exceptional in this respect. 31 Argentina, Brazil, Uruguay, Chile, Costa Rica, Venezuela, Colombia, Paraguay and Bolivia. In Mexico, the rate remains stable. It has increased slightly in Nicaragua and Guatemala, and more significantly in El Salvador and Peru. 32 The only exceptions were Peru, Guatemala, El Salvador and Mexico. of a relative), it is necessary to mobilize savings or use debt. These 'unexpected' events can have a significant impact on income-generating activities and plunge the vulnerable into debt.

These developments lead us to refine our notion of savings, even for China. Presently a land of savers, China's savings fund Western countries. This is true from a macroeconomic standpoint. But refine the analysis by social groups, and it becomes clear that it would be wrong to think in terms of averages and imagine that every Chinese family has the ability to save. Even with an annual growth rate of 8-9 per cent that allows formal businesses to absorb 12 million additional workers, an equivalent number of people appear to be supernumerary [START_REF] Monteil | Emploi informel et gestion des inégalités en Chine urbaine. Les politiques de promotion de « l'emploi communautaire » parmi les « groupes vulnérables » à Chengdu[END_REF]11). To survive, they engage in small informal activities. This category (known as ruoshi qunti in official Chinese terminology) appears unable to save. As elsewhere, they face the hard knocks of life through debt 33 . It is estimated that the country has 130 million urban informal workers [START_REF] Monteil | Emploi informel et gestion des inégalités en Chine urbaine. Les politiques de promotion de « l'emploi communautaire » parmi les « groupes vulnérables » à Chengdu[END_REF]12), plus another 150 million rural supernumeraries. In the survey conducted for her thesis (EHESS, Paris, 2010), Amandine Monteil reveals that 43 per cent of respondents in six districts of the cities of Chengdu and Leshan claim they are unable to save (p. 66, 83, this percentage is even higher among urban residents over 30). While official propaganda stresses autonomy and the virtue of not going into debt (p. 365), three-quarters of those interviewed said they had already borrowed: 49 per cent to cover medical expenses (this rate is higher among migrants: 52 per cent, see p. 39-40, 61), 39 per cent for children's education, 23 per cent to cover daily needs, 27 per cent for business purposes. And 7 per cent indicated having done so often. To borrow, they resort to family (54 per cent), friends (44 per cent), and neighbors (19 per cent), while only 11 per cent use financial institutions. Amandine Monteil also reports (p. 78) on the practice of gambling (mahjong, a high-stakes card game), which illustrates, as revealed in the case of Haiti and borlette [START_REF] Servet | Inadequate growth, exclusion and indebtedness[END_REF], another expedient way to make ends meet. These urban poor are the outright opposite of China's economic and financial miracle. Moreover, the weak social protection in China, which explains the population's high saving rate, reveals a universal consequence for the most vulnerable: debt (albeit at much lower rates than what is observed in many other parts of the world). There is probably a relationship between this indebtedness among a fraction of China's population and the rise of pawnbrokers, astutely analyzed by Thierry Pairault (2002[START_REF] Pairault | Maisons de prêt sur gage en Chine: une visite de Chongqing[END_REF], and the low growth of microfinance [START_REF] Pairault | Pratiques populaires et microfinancières chinoises[END_REF]. 33 Amandine Monteil clarifies that the sums are small and quickly reimbursed, but it is still debt.

As in North, the rise of household debt in the South implies that households are increasingly subject to outflows of income. Contrary to popular belief, these practices do not always involve so-called 'usury' interest rates. However, shops that keep customers loyal by selling on credit generally have higher prices than businesses that do not. Similarly, merchants who advance credit during the lean season or for the purchase of inputs in exchange for unharvested crops get produce at a better price than if they had to purchase it at harvest time.

Consequently, the disposable income of the most vulnerable households is whittled away by the seller's profit margin, which is in fact a hidden interest rate. When the cost of credit exceeds the surplus created by the activities financed, the borrower is ultimately impoverished, which explains some cases of over-indebtedness.

It is against this backdrop that an increasing number of microcredit institutions have entered crises: in 2008-2009Nicaragua, Morocco, Bosnia-Herzegovina, Pakistan, and, in end-2010, the Indian states of Karnataka and Andhra Pradesh [START_REF] Servet | La crise du microcrédit en Andhra Pradesh[END_REF]. It is tempting to attribute the rise in arrears not to the global crisis itself but to the poor management of microfinance institutions and uncoordinated competition34 . Of course, if we consider these crises as the results of the financial crisis created by developed countries (and primarily the United States), which has led to an almost universal crisis in production and trade, then the microfinance institutions are not to blame. This said, the direct or indirect impact on the decline in migrant remittances has affected some institutions (Servet 2010a) In Pakistan, 21 per cent nationally and 30 per cent of the regions were affected by a default movement [START_REF] Chen | Growth and Vulnerabilities in Microfinance[END_REF]. Although Peru is not yet in this situation, it is not far. The same is true for some parts of India.

The analysis put forth in this chapter suggests a different interpretation than that of poor governance of microfinance institutions. It explains why this default movement in Nicaragua, Andhra Pradesh and Pakistan has been organized by lobby groups-political in the first two cases and Islamist in the third. Of course clients took advantage of the microfinance institutions' inability to limit their loans and to assess their own institutional weaknesses. But if these customers sought out credit (not just with microfinance institutions but also with socalled 'usury' lenders), it was because they had to, in order to live and make payments on previous loans. The managers of microfinance institutions have often assumed that clients return for larger loans because they are satisfied with the services offered. They did not consider that repeated borrowing could be the sign of a growing addiction to credit, which would eventually lead to over-indebtedness.

III. Overcoming over-indebtedness with inflation

At the first level of financialization (Servet 2010a) where the crisis has erupted, there is an economy of debt [START_REF] Graeber | Debt : The First Five Thousand Years[END_REF]. It has different terms and conditions depending on the country, but everywhere, indebtedness has appeared to compensate for the relative impoverishment caused by slower growth in countries with neo-liberal economic policies and unequal income distribution. As Warren E. Buffett said, 'There's class warfare, all right, but it's my class, the rich class, that's making war, and we're winning'36 . It is common to understand unequal distribution of income and the exploitation associated with it from a perspective of wage relations or subordination in the workplace.

But reducing exploitation to the direct exploitation of labour is limited, because the dependencies and transfers from one sector to another and between social groups can also be achieved through forms of debt. These debts, like any subordinate relationships in the productive sector, allow capital to exploit labour, thanks to transfers that occur via usurers.

The draining of financial resources through various forms of debt can be interpreted as a particular form of the capital-labour relationship. The way this relates to exploitation is not through the face-to-face relationship of an employee paid by the hour or on a piece basis (and then selling products that appear to be handicrafts)37 . But generally speaking, paying interest to develop productive or trading activities is the equivalent of paying a levy on the income from this activity. There is no capital-labour relationship at an interpersonal level but on a global scale, we can identify transfers from one sector to another. By decreasing the income earned through interest and repayment on the principal, inflation can contribute to reducing exploitation.

When debt becomes so unbearable that it endangers the harmonious interactions between social groups and the reproduction of society itself, the economic system can create an escape valve through reforms or debt reduction thanks to price increases. Since the 1970s, there have been four types of policy to deal with public debt. In 1998, Russia declared default and its public debt was rescheduled through re-negotiation. A similar situation has been observed in many developing countries over the past two decades. Public spending in the United Kingdom in the early 1980s and in Sweden in the 1990s was reduced by deep economic reforms. In the 1970s Japan and the United States turned to innovation and growth by stimulating investment.

Finally, Western Europe and Argentina used monetary weapons-devaluation and inflationto reduce their debts, somewhat moderately in Europe in the 1970s and more radically in Argentina in 2002. History is full of examples, dating back to the Antiquity, that show how price increases have been a way of reducing debt in a way that allows society to 'settle its accounts' more or less peacefully, provided social and political relations allow for it. Many ancient cities even practiced debt forgiveness (under popular pressure), to the detriment of creditors, who were forced by the political establishment to accept the terms. A moderate price increase of 4-5 per cent per year would, over the course of one decade, depreciate borrowed capital by half. If the nominal interest rate is fixed, and is less than 2 per cent, the real interest rate appears negative. Naturally, reversing the trend in terms of the distribution of income from capital and income from labour implies a simultaneous adjustment to real wage and quasi-wage income and the income of small producers, so that they do not see their purchasing power drastically reduced. But, regardless, the actual amount of repayment of interest and principal will be reduced by inflation anyway.

Inflation can also alleviate the debt burden borne by households through the management of public debt. Over the last thirty years, public debt in many countries has increased inequalities in terms of income redistribution (both the cause and consequence of household overindebtedness). It is paradoxical, since unlike private debt, public debt can have an equalization effect: its cost can be borne by the affluent even while public expenditures benefit everyone, especially the poorest (Barba and Pivetti, 2009). But this implies a political will to raise taxes. This has not been the case for nearly thirty years. In fact, the term 'paradox' is too weak, as we are at complete odds with the situation described by Aldo Barba and Massimo Pivetti: public debt has benefited the richest, because it represents a portion of tax they did not have to pay, even though they benefited from the remuneration of public debt, while the poorest are the first to suffer from the reduction of public expenditure38 . In other words, for more than a quarter of a century, the part of the budget that the richest have not wanted to tax becomes a collective debt that benefits the most affluent and tends to decrease the well-being of the poorest, and even the majority of the population. Perhaps because of the political causes of this debt, we will one day speak of it as a 'debilitating debt', which would justify a collective movement to default on it.

It is difficult to compare debt reduction during growth years (even low ones) and the period post-2008, because economic stimulus policies appear less effective and credible while policies to reduce public spending play a deflationary role for all production and trade activities. Since declaring default ultimately leads back to debt, due to the high level of growth that potential creditors will require later, a solution that calls for the return of inflation appears the most likely scenario, even most desirable in the medium term, to overcome massive public and private debt. Admittedly, at a time (September 2010) when large firms are borrowing long term (40-50 years) at near-zero rates 39 , proposing to return to price increases may seem preposterous. But, given the experience of the last hundred years, is it not the same thing as making an investment at a very low interest rate, which would suppose a stability of the prices for several decades? Maybe this is how we should interpret the increase in fuel and food prices in 2007. Although price increases were influenced by the shift in the object of speculation following the collapse of the derivatives market based on American subprime mortgages, they acted as a kind of systemic reality check, by depreciating real estate and financial assets compared to real goods.

Similarly, in late 2010, there were once again new price hikes for some commodities, not to mention a rise in the price of gold, especially sought after by the Chinese. From this standpoint, the turnaround in food and fuel prices in 2008 can be seen as the effect of deflationary pressures related to the beginning of the crisis and its subsequent development as a 'stagboom'. But, the dominant monetary ideology among financial authorities and interest groups that adhere to it to advance neo-liberal policies have prevented and continue to prevent that the system purge itself through price increases 40 .

As economies are currently dealing with the consequences of monumental debt, it is surprising to hear so many experts propose to revive production and trade by... private debt from financial markets. Among the many statements along these lines include this quote from Microsoft's loan of 4.75 billion dollars is practically ephemeral (three years), but the rate is well below the increase in prices: 0.875 per cent per annum. 40 Robert [START_REF] Boyer | Feu le régime d'accumulation tiré par la finance[END_REF]33) posits a deflationary depression scenario (the excess of debt leads to a cumulative fall in prices) by looking at the hypothesis of Irving [START_REF] Fisher | The Debt-Deflation Theory of Great Depressions[END_REF]. Similarly Paul Krugman, in an article in The New York Times, says that the risk of deflation is much greater than that of inflation, despite strong growth in government deficits. He draws on the example of Japan between 1997-2003, when the Bank of Japan bought huge amounts of bank debt, while consumer prices continued to fall (Paul [START_REF] Krugman | The Big Inflation Scare[END_REF], 'The Big Inflation Scare' The New York Times, May 28, 2009).

The more that is done to clean the banks, to enable them to perform their functions best the more robust the recovery will be.' 41 However, if the dominant forms of private credit are at the root of the crisis, it is hard to imagine that they will be able to solve it. It is as if a therapist who, after forcing a patient to become an alcoholic to attain illusory happiness, proposes a treatment that involves drinking more wine.

  that allow us to quantify changes in home consumption: one conducted in Mexico in 2008 (Dirección de Análisis Económico y Social, 2008) and in Tanzania in 2001 (National Bureau of Statistics Tanzania, 2001). In the first, non-cash income in 2008 accounted for 19.9 per cent of all household income, whereas this proportion was 21.4 per cent eight years earlier. Similarly, in Tanzania, subsistence agriculture covered 28.6 per cent of total food expenditures in 2000-01. 20 , compared to 35.5

  and the decline in resources from tourism and multinationals with subsidiaries in the South has caused a drop in the volume of activities of microenterprises and workers' wages, thereby decreasing the volume of loans granted by microfinance institutions that serve these clients. But what is really at issue here is the rush to extend larger loans to more people, without ensuring microcredit's effective contribution to income-generating activities 35 . Of course, if only the managers of these microfinance institutions had been more vigilant, if only they had more effective information systems, they would have better anticipated portfolio risk, having identified clients with multiple loans from different organizations. In Nicaragua, 40 per cent of borrowers had loans from more than one institution in 2009. In Morocco, the portion was 40 per cent in 2007, 39 per cent in 2008 and 29 per cent in 2009. In Bosnia-Herzegovina, it was 40 per cent in 2009.

  Marek Belka, Executive Director of the IMF's European Department: 'I think what is most important for the pace of economic recovery in the euro zone is how decisive policies are toward revitalizing the banking sector […] So much depends on the health of the banks […]

Table 1 : Domestic credit to private sector, as % of GDP, 1980-2007 1

 1 World Bank national accounts data, OECD National Accounts data files (GDP), IMF and Word Bank (credit). Author's calculations.

	Regions	1980	1994	2007	variation 1980-2007
	Africa	31,96	54,71	65,55	+ 105,13
	Asia and Pacific	34,88	69,68	86,44	+ 147,79
	Latin America and Caribbean	34,25	46,34	44,09	+ 28,71
	Middle East	28,61	43,48	56,23	+ 96,56
	Developed economies and European Union	86,25	127,83	166,72	+ 93,30
	Source :

Table 2 : Urbanization rates, 1960-2008

 2

	Regions

Source: World Development indicators; authors' calculations.

  For example, between 2002-07, the number of households owning a car doubled in the Asia and Pacific, and shot up elsewhere (57 per cent, 56 per cent and 35 per cent respectively for the Middle East, Africa and Latin America -table 1.4). The social nature of these goods is ambivalent: their need is exacerbated by a desire to imitate (Guérin and Selim 2012), but at the same time they are essential to a growing number of income-generating activities, including in rural areas.

	in his study L'évolution
	des besoins de la classe ouvrière on the nature of social needs are still relevant: through
	consumption, the working class participates in a civilization that is beyond themselves.
	For this reason, we have recently seen a boom in consumer goods in the South. Subscriptions
	to telephone lines (mobile and landlines) skyrocketed between 1990 and 2007 (table 1.3),

growing by over 4000 per cent in Asia and the Pacific, nearly 3000 per cent in Africa, and about 700 per cent in the Middle East. Similar growth can be observed in Internet access and computer ownership. Consumption of other goods has also risen sharply, although the proportions are not as impressive.

Table 3 : Subscription of mobile and fix line (% 1990-2008)

 3

	Regions

Table 4 ; Number of households possessing at least one passenger car

 4

	Regions	2002	2007	Variation %
	Africa	9297703971	14523157030	(+) 56,20
	Asia and the Pacific	22864456152	45529094925	(+) 99,13
	Central and south-eastern Europe and CIS	38491499429	50059394164	(+) 30,05
	Latin America and the Caribbean	18547572289	24974251553	(+) 34,65
	Middle East	784975315,8	1233140122	(+) 57,09
	Source: International Road Federation and United Nation Populations Division. World Development Indicators.
	Authors' calculations.

In the terminology of regulation theory, accumulation regimes are the result of interactions between key institutions that underpin the economic process. They last several decades, the time it takes to generate a surplus without being interrupted by a crisis that calls into question its main characteristics.

The term 'institutional form' refers to the social relations key to the regulation of economies.Regulation theory identifies five main forms: the monetary regime, the wage-labor nexus, forms of competition, state intervention and the international regime(Boyer, 2004: 39-41).

Vatin (ed.), 2007. See alsoILO and IILS (2008; 119).ILO (2009) discusses the rise of temporary work in the North.

For updated data from[START_REF] Piketty | Income inequality in the united states, 1913-1998[END_REF]., see http://emlab.berkeley.edu/~saez/index.html

Robert Boyer's expression (2000).

http://www.peri.umass.edu/236/hash/c054892e7a23115bfbd0c22c9e90f57c/publication/432/

This was possible because China, Japan, Hong Kong and Taiwan accumulated 1.65 trillion dollars in U.S. Treasury bonds.

[START_REF] Clévenot | Les difficultés à nommer le nouveau régime de croissance[END_REF] discusses this article and internal controversies among regulation theorists over the nature of this new economic structure in the United States.

For data on Argentina, see de[START_REF] Nigris | La bancarización en Argentina[END_REF]. For Peru, see[START_REF] Aspec | Crédito y sobreendeudamiento de los consumidores en el Peru[END_REF].

For a detailed look at access to banking services for different countries see data from[START_REF] Kendall | Measuring access to financial services around the World[END_REF].

For example in Andhra Pradesh (India) the majority of debt of rural households comes from practices related to the informal sector. According to Johnson and Meka (2010) 93% of households in the state are in debt; 57% of these households have borrowed with interest from friends while 17% used an informal lender.

We would like to underscore the importance of these inverted commas: this is not our category. We have used it here to support our argument, but it is not within the scope of this article to conduct a critical analysis. For an analysis of the multiple relationships between formal and informal, and a study of informal organizational arrangements, seeLautier (2004).

See comparative analysis done by[START_REF] Chen | Growth and Vulnerabilities in Microfinance[END_REF], and studies specific to each of these countries referenced in the bibliography of this article.

See the critical analysis conducted by Jude (2006),[START_REF] Servet | Banquiers aux pieds nus : la microfinance[END_REF] 2010c) and[START_REF] Bateman | Why doesn't microfinance work ? The Destructive Rise of Local Neoliberalism[END_REF].

Warren E. Buffett, quoted by Richard H. Thaler, 'What the Rich Don't Need', The New York Times, Sept. 25, 2010.

Ancient societies have many examples like this, where exploitation does not occur via direct employment of labour. This applies, for example, to domestic work that confers the worker the status of craftsman; it is the case for all rents based on advancing seed or lending land.

In this case, income redistribution is more unequal after taxes than before. The example of the United States is striking(Aglietta and Berrebi, 2007, pp. 139-143).

On September 14, 2010, EDF borrowed one billion pounds over

years; GDF Suez borrowed 700 million pounds for 50 years, while their German rival RWE simultaneously launched a 'perpetual' bond issue for 1.75 billion euros. UK pension funds appear to be the buyers of these securities.


Emeric Plancher

Pouya Tajdary

Thierry Auger
email: thierry.auger@ensam.eu

Olivier Castelnau

Véronique Favier

Dominique Loisnard

Jean-Baptiste Marijon

Claire Maurice

Vincent Michel

Odile Robach

Julien Stodolna

Validity of crystal plasticity models near grain boundaries: a contribution of elastic strain measurements at the micron scale

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

Introduction

The buildup of intergranular microstresses (type-II internal stresses) and intragranular stresses or residual stresses (type-III internal stresses) upon plastic deformation is a central issue in mechanics of materials [1]. The development of local stresses is generally due to incompatibilities between the anisotropic (elastic and plastic) deformation on both sides of grain boundaries. Another contribution to internal stresses is associated with spatial localization patterns at the microstructural scale of plastic deformation. Since plasticity precedes many other metallurgical phenomena such as recrystallization, phase transformations or damage, its correct understanding (i.e. the ability to reliably predict local stress/strain fields in a plastically deformed material) has major implications. For example, the simulation of intergranular stresses is critical to describe environmental effects such as stress corrosion cracking or liquid metal embrittlement, because intergranular stresses drive the dynamics of fracture [2,3,4]. During recrystallization, the stored elastic energy associated with the dislocation network governs the microstructure evolution [5,6]. It is therefore desirable to quantify the degree of accuracy on local stress/strain fields one can reach through simulation, by confronting experimentally accessible stress and strain fields to modelling results in polycrystals.

The usual measurement of internal stresses is performed with X-ray or neutrons diffraction, by using beam cross section significantly larger than the grain size. Those techniques often lead to a reasonably accurate but averaged estimation of micro-strains [7] that compares well with plastic texture simulations [8,9,10]. Recently, the advent of local probes such as digital image correlation (DIC) [11], high angular resolution electron backscatter diffraction (HR-EBSD) [12,13] or synchrotron Laue microdiffraction [14,15] has opened the way to a new class of measurements that can directly map mechanical fields to a high degree of accuracy, down to a micrometer spatial resolution. Therefore, full field measurements can be carried out at the same scale than microstructural models featuring crystal plasticity constitutive laws, and paves the way to advanced validation of scale-transition modelling.

One of the key questions raised when using crystal plasticity models is whether grain boundaries can be treated as sole geometrical interphases, as thin surfaces with specific properties and special kinematics conditions [16], or as non-local microstructural constituents with special properties per se [17]. Grain boundaries are obstacles to dislocations slip but, depending upon their nature or upon the misorientation between adjacent grains, they may still transmit some amount of plastic deformation from one grain to the other [START_REF] Priester | Grain boundaries, From theory to engineering[END_REF]. Conventional crystal plasticity models, such as the one employed in this publication [START_REF] Teodosiu | Large Plastic Deformations[END_REF][START_REF] Méric | [END_REF], do not define any particular behavior for the grain boundary, an assumption that may not be realistic enough.

In this work, a concentrated austenitic solid solution, typical of 316L stainless steel, is employed as a model alloy. We report here an original comparison between two mechanical fields measured at similar and fine spatial resolution, i.e. elastic and total (elastic + plastic) strain fields versus the prediction of a local crystal plasticity simulation [START_REF] Teodosiu | Large Plastic Deformations[END_REF]. First, the crystal plasticity parameters were fitted for the alloy and the set of parameters was validated by considering experimental fields developing in a single crystal [21,22]. Then, an in situ bending experiment was performed at the European ESRF synchrotron to investigate how both strain fields localize near a general grain boundary in a large bi-crystal. Strain measurements were performed near the grain boundary at several loading steps with optical DIC (total strains) and Laue microdiffraction (elastic strains). The agreement between the prediction of the finite element simulation and the experimental data is discussed for both strain fields and the local activation of slip systems.

Experimentals 2.1 Materials

Macroscopic single crystals of high purity austenitic stainless steel were grown by Bridgman directional solidification in a horizontal furnace with an argon cover gas. For tensile and bending experiments, single crystal samples were cut by spark erosion from a macroscopic single crystal, with the crystal axes aligned with the sample axes within 3°. A bi-crystal bending sample was also cut out from a macroscopic oligocrystal obtained in a separate crystal growth attempt. The grain boundary was selected for its position, considering geometrical requirements for the bending sample, and thus its character was random. Therefore, the chemical composition of the bicrystal and the single crystals slightly differs (see table 1). The gauge section of the two tensile specimens was 8x3x0.5 mm 3 .

The shape of the single crystal bending specimen was 30x4.3x0.46mm 3 . The shape of the bicrystal bending specimen was 30x4.8x0.48mm 3 . All samples were carefully polished to a 1µm grade or below and the preparation quality checked using channeling contrast imaging in a scanning electron microscope. Details of the single crystal bending and tensile experiments have been reported in [22]. The elastic constants of the 316L oligocrystal were measured by Resonant Ultrasound Spectroscopy and reads: C11=202 GPa, C12=130 GPa and C44=128 GPa (uncertainty on the values is evaluated below +-1GPa

Laue microdiffraction and DIC under synchrotron radiation

In situ Laue microdiffraction experiments were carried out at the French CRG beamline BM32 of the European Synchrotron (ESRF, Grenoble, France), described in reference [23]. The incoming polychromatic X-ray beam (energy range 5-23 keV) was focused down to ~800x600 nm 2 by means of Kirkpatrick-Baez mirrors. Laue pattern were recorded using a MARCCD ® detector (pixel size: 79.14 microns, 2048x2048 pixels, 16 bits dynamic) positioned 60mm above the specimen surface. A strain-free Ge wafer was used to calibrate the setup geometry (detector orientation, etc.). A custom-designed 4-point bending rig was installed on the translation stage pre-tilted at 40°. The device was equipped with a load cell with a capacity of 20N and an extensometer sensitive to displacements below 1µm to measure the macroscopic force/displacement imposed on the sample.

An optical microscope available on beamline BM32 (Allied Vision Technology GiGE ® , Objective Mitutoyo 10x/0.28, f=200) was employed in situ to acquire optical images used for the measurement of the total strain field on the specimen cross-section surface by the DIC technique. A MoS2 powder (MOLYKOTE ® Microsize, with submicron grain size) was deposited on the sample surface to create the random speckle pattern. When a DIC image is to be acquired, the optical microscope has to be positioned such that its focal plane covers the sample cross-section surface. Therefore, measurements were carried out sequentially, the microscope being brought in and out between loading steps. Further details of the experimental setup can be found in [22].

Processing of the Laue microdiffraction data was performed with the software LaueTools (see e.g. [START_REF] Robach | Chapter 5 : Laue microdiffraction at ESRF[END_REF]) to extract the deviatoric part of the elastic deformation tensor [START_REF] Chung | [END_REF]. In this work, the full elastic strain tensor was recovered assuming a vanishing stress vector on the specimen surface, which is a sound approximation due to the small attenuation length of X-rays (65 microns in Fe at 22 keV) and the lack of in-depth microstructure gradients in the specimen. The plane stress hypothesis was used to determine the hydrostatic part of the stress tensor and then, using generalized Hook's law, the hydrostatic part of the elastic strain tensor was extracted. Laue patterns consisted mostly in a single diffraction figure, except on the grain boundary where two overlapping figures could be detected. In that case, the figure with the more intense diffraction peak was indexed and associated to the relevant grain. The area investigated on the sample was approximately 400µmx600µm, centered on the grain boundary with a step size of 25µm. Orientation angles use the Bunge convention in degrees throughout the paper (𝜑 $ , Φ , 𝜑 ' ). The maximum measured lattice rotation between the initial state and the loaded state reaches 2.5° point to point (in the scanned area).

Modelling

Formulation

The kinematics of the crystal plasticity model relies on the finite transformation framework (small elastic distortions but large lattice rotations) first proposed by Lee where the deformation gradient tensor is built as the product of an elastic part with a plastic part [26]. The elastic part embodies the anisotropic elastic stretch and the grain rotation while the plastic one accounts for crystallographic slipping along specific slip systems (up to 12 for a fcc crystal). The constitutive laws rely on dislocation densities resolved on each slip system as internal variables 𝜌 )* whose evolution model strain hardening in metals and alloys. At each time increment of the computation, the local critical shear stress 𝜏 ,

) is computed with the updated dislocation densities using the forest hardening interaction matrix resolved on every slip system (via the 𝑎 )* interaction matrix between slip systems (s) and (u) including diagonal self-interaction terms) [27]:

𝜏 , ) = 𝜏 / + 𝜇𝑏3∑ 𝑎 )* 𝜌 )* *5$,$' (1)
where µ is the isotropic shear modulus, b the norm of the Burgers vector and 𝜏 / the lattice friction stress. The onset of slip is triggered according to the Schmid law. The rate-dependent slip rate prescription proposed by Pierce [28] for single crystals is used. It is approximated using a power law:

𝛾 ) ̇= 𝛾 / ̇8|: ; | : < ; = > 𝑠𝑖𝑔𝑛(𝜏 ) ) 𝑖𝑓 |𝜏 ) | > 𝜏 , ) , 𝛾 ) ̇= 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (2)
where 𝛾 / ̇ is a reference shear rate and 𝑛 is the stress sensitivity parameter. These material parameters are to be fitted to reproduce the correct flow stress rate dependence. Following Teodosiu's law with the hypothesis of a constant mobile dislocation density [START_REF] Teodosiu | Large Plastic Deformations[END_REF], the rate of evolution of the stored dislocation density is governed by a dislocation production term and is balanced by a dislocation annihilation term taking into account the dynamic recovery during deformation (related to the annihilation distance of dislocation dipoles 𝑦 , ):

𝜌 ) ̇= |N ; | Ȯ P 3∑ Q R RS; T -2𝑦 , 𝜌 ) W (3)
where K and 𝑦 , are material's parameters that are usually fitted to reproduce stage I and stage II on a single crystal.

The approach is implemented in the Abaqus ® finite element code, using a UMAT subroutine [29].

Parameter identification

Crystal plasticity parameters for 316L stainless steels can be found in several works to date [4,30]. There are three categories of parameters. The first one comprises physical parameters that can be set based on the metallurgical state of the sample (burgers vector, initial dislocation density 𝜌 / and initial critical friction stress 𝜏 / ). (As said previously the 316L elastic constants are taken as: C11=202 GPa, C12=130 GPa, C44=128 GPa). The second set of parameters allows to describe the viscous or the rate dependent effects (𝛾 / ̇ and 𝑛). For 316L stainless steel, it is important to set carefully these parameters because austenitic steels are known to have a non-vanishing ratedependent flow stress that becomes negligible only below strain rate of 10 -2 s -1 [31]. For our purpose of testing at slow strain rate (i.e. lower than 10 -2 s -1 ), these parameters were constrained to correctly reproduce a quasi-static loading state with a minimized strain rate effect (𝛾 / ̇> 10 YZ 𝑠 Y$ and 𝑛 > 20). In order to perform this identification, tensile tests carried out on single crystals at a strain rate of 10 -4 s -1 along direction [100], as detailed in [22], were modelled by finite elements in Abaqus using C3D8 elements with a [100] crystal orientation. The parameters 𝜏 / , 𝛾 / ̇ and 𝑛 were then adjusted to fit the experimental curves (Figure 1). The smooth transition at the onset of plasticity given by the Pierce strain rate law is damped due to our choice of parameters. Experimental data is compared with the identified constitutive relation.

The third category of parameters (describing strain hardening) is composed first of the 𝑎 )* interaction matrix between the slip systems and the two material parameters K and 𝑦 , . The interaction matrix (a0 for self-interaction, a1 for collinear interactions, a2 for Lomer Cottrell locks, a3 for Hirth junctions, a4 for glissile dislocation interactions, a5=a0 for sessile dislocation interactions) is taken from the work of Devincre [32] which was later on already applied for the 316L(N) stainless steel [30]. The two remaining parameters were set to reproduce the slope of the single crystal tensile curves reported in Figure 1. It is known however that K and 𝑦 , may vary with the strain level and our choice may need to be refitted to better reproduce a larger amount of experimental data.

Overall, the resulting parameters are charted in table 2 along, when applicable, with their values as found in two other works in which a rate-dependent crystal plasticity approach was used [4,30]. The major differences between different parametrizations can be rationalized by being due either to a compositional difference with our alloys or to a different initial dislocation density (𝜌 / and 𝜏 / play in synergy to give a larger flow stress than our well annealed alloys). Reported, when the unit is not specified, the parameter is adimensional. * We note that there is an inversion between a2 and a3 in [30] compared to [32])).

This parametrization was validated against the measurement of the elastic strain and total strain fields, performed by Laue microdiffraction and DIC respectively along the specimen cross-section, in the bending experiment of a 316L single crystal (see [22]). For the simulation, the dimensions of the sample were 0.46×4.3×30 𝑚𝑚 3 and the [100] orientation was aligned with the loading axis. The four point bending test configuration was simulated by adding four rigid cylindrical shells (outer shells are fixed; inner shells move along the Ox axis). The contact interaction conditions between the four pins and the single crystal beam were frictionless sliding. The imposed boundary condition displacement rate used for the simulation was set to be in the quasi-static strain rate domain of the crystal plasticity model. The beam deflection was simulated up to the final measured one. One can see in figure 2 that the total strain and elastic strain profiles were reproduced rather accurately along the cross-section of the sample. The stress-strain curve of the single crystal oriented for multiple slips did not present stage I, but stage II. It reproduces, nicely the transition from elastic to plastic deformation together with the slope of elastic strain in the plastic regions associated with the crystal hardening. This gives us confidence that the crystal plasticity parameters are correctly set according to the fitting strategy that was used in this work.

Experimental and modelling setup for the bicrystal

A schematic view of the investigated bicrystal can be found in figure 3. The supposedly planar grain boundary was inclined with respect to the specimen axes as indicated. The misorientation angle between the two grains is 64° specifying a general grain boundary. The specimen was loaded in situ in the four-point loading device using the setup described in section 2.2. Sample deformation was carried out step by step and measurements were performed at 0 N (initial state), 3 N, 7 N, and then after complete unloading. After each load increment, the pin displacement was blocked, and we waited until the specimen relaxation was complete (~20 min). Then, a Laue map covering the area of the grain boundary was carried out. A DIC mapping measurement was also acquired at each Laue step while keeping the sample under load. The lower and the upper grains are respectively labelled Grain 1 and Grain 2. The modelling by crystal plasticity of the bicrystal bend test was performed similarly to the single crystal by simulating the entire bicrystal in Abaqus (dimensions 0.48x4.8x30 𝑚𝑚 3 ). The beam was split in two parts with a slanted surface representing the grain boundary (at an angle of 50° in the xy plane and 37° in the yz plane). The mesh for the finite element modelling was then generated using C3D8 elements with a fine grid close to the grain boundary and a coarser grid further away to save computational power. The fine/coarse separation was set manually to model with a coarse mesh the part of the beam that stays in the elastic regime so that the entire plastically deformed volume was covered with a fine mesh. The four point bending test configuration is simulated by adding four rigid cylindrical shells with identical boundary conditions and contact interactions. The displacement rate used for the simulation was small enough to be in the quasi-static loading state of the crystal plasticity model. The total strain was given as a result of the non-linear finite element modelling while the elastic strain was extracted at each step by elastically unloading the current stress using the generalized Hooke's law in the crystal reference frame. The elastic strain tensor was then rotated back to the sample reference frame to be readily compared with the measured one.

Results on the bicrystal:

After complete unloading, the pattern formed by the MoS2 deposit used for DIC at the surface of the bicrystal was cleaned in an ultrasonic bath. The sample surface was then observed using channeling contrast in a scanning electron microscope adapted for large field of view observations with low image distortions (TESCAN MIRA3 ® ). The backscattered electron image reported in figure 4 reveals the grain boundary and multiple slip traces observed in both grains. In grain 2, a slip trace analysis using the orientation data from Laue microdiffraction (𝜑 $ = -133°, Φ = 25°, 𝜑 ' = -163°) can provide the slip planes associated with the four families of traces observed. In grain 1 (𝜑 $ = -175°, Φ = 78°, 𝜑 ' = -107°) the slip traces associated with slip planes (-1-11) and (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11) are very close with a theoretical misorientation in the surface plane of only a few degrees.

To compare with the slip systems activated in the simulation, the area of interest was schematically sliced into 12 domains and the number of active systems tallied. The division was based on (i) approximate distance from the grain boundary and (ii) the nature of the applied stress (tensile, compressive or near neutral fiber). The activated slip systems and its corresponding dislocation density were extracted from the simulation as state variables. In the domains were a slip system is not activated (|𝜏 ) | < 𝜏 , ) or 𝛾̇< 10 Yd 𝑠 Y$ at all steps), the corresponding dislocation density variable remains constant. A visual one to one comparison between the slip traces in each domain from figure 4a and the activated slip systems given by a Schmid factor analysis was performed. It shows that in all the regions except 6 and 9 (which are closer to the grain boundary and are in the area which is subject to tensile stress) the simulation predicts correctly the type and number of activated slip systems. In the tensile part of the sample, according to the simulation, Grain 1 should be in dual slip mode (111)<01 ̅ 1> and (1 ̅ 11)<011> while grain 2 should be closer to single slip mode ( ̅ 1 ̅ 11)<101>. However in the experiment, due to internal stresses, not every predicted slip plane is activated. Grain 1 is rather deformed and grain 2 tries to accommodate. The experimental strain fields (total and elastic) have been extracted following the procedure described earlier respectively from the DIC and Laue diffraction data. They are plotted for the yy component (along the y axis of figure 3) of the total strain tensor (figure 5) and of the elastic strain tensor (figure 6). The total strain 𝜖 ff g distribution shows that the right part of the specimen is mainly in tension and the left part is mainly in compression as expected from bending. However, the grain boundary clearly disturbs the strain field. The strain magnitude is stronger in Grain 1 than in Grain 2. The simulation results for 𝜖 ff g are represented in the figure 5b. One can see that the main plastic activity domains are very well reproduced except at the top right corner. The normalized difference plot is shown in figure 5c. It reveals that the difference between the measurement and the simulation results over almost the entire analyzed surface, including next to the grain boundary area, remains below 20%. This level of agreement is considered as being rather good.

Figure 5: Total strain fields along y axis: experimental, modelling and difference plot at a loading of 7 N

The elastic strains 𝜖 ff h are shown in figure 6. One can see that the experimental elastic strains follow closely what is expected from a bending beam experiment. One sees clearly an area in tension at the right and an area in compression at the left consistent with the existence of a neutral fiber in a four-point bending experiment. The effect of the grain boundary can be seen weakly as some higher yy strain component appear as hot spots along the grain boundary. In figure 6b one can see the simulated corresponding elastic strain. Overall the shape looks similar but the intensity scale is significantly different and appears globally to be of higher magnitude than the experimentally measured one. The normalized difference plot is shown in figure 6c. The presence of a grain boundary can be easily inferred from this plot with two main sources of disagreement. The first one is the presence of an area at the right of Grain 1 (the area in tension) that seems to be frequently in disagreement between experimental data and simulation results. The other stunning characteristics is the clear distinction of the grain boundary as a major source of contribution of disagreement between experimental data and simulation results. This indicates that next to the grain boundary is a special area where crystal plasticity somehow overestimates the elastic strains induced by the plastic activity due to bending. The elastic strain from a large part of Grain 2 is well predicted, especially away from the grain boundary. This is expected as the area was only slightly plastically deformed (<0.5%). The same conclusion can be reached for Grain 1 close to the neutral fiber, but to a lesser extent.

Figure 6: Elastic strain fields along y axis: experimental, modelling and difference plot at a loading of 7 N

Discussion

The in situ mapping of total and elastic strains by DIC and Laue microdiffraction allowed to characterize the evolution of these fields around a general grain boundary, with a model alloy of 316L, in a beam bending configuration. This constitutes a major step forward compared to similar studies aiming at the measurement of elastic strains, such as those carried out a few years ago which had a beamsize of 100 µm x 100 µm compared with 1 µm x 1 µm here [33] or what can be achieved with high energy X-ray diffraction [34]. The currently achieved spatial resolution with Laue microdiffraction at synchrotron facilities allows to tackle this issue with much higher accuracy.

The results of this work show that the heterogeneities in active slip planes are very well predicted with the formation of well identified plastic domains in the different parts of the area surrounding the grain boundary. While some localized discrepancies arise at one edge of the specimen (more specifically at the right-hand side in the tensile area), one can note that the difference between the predicted domains and the observed ones are not specifically related to the grain boundary. The total strain field is also very well predicted. It is mainly controlled by the macroscopic kinematics but the presence of grain boundary and the induced change in anisotropy clearly disturbs the field around it. This work brings another example to the well-known effect of grain boundaries on the formation of plastic heterogeneities or plastic glide domains [35,36]. It also illustrates the ability of standard crystal plasticity modelling to predict in a detailed manner these early stages of plastic deformation (the maximum total strain is here less than 2%). The measured elastic strain field has mainly the symmetry induced by the bending geometry but modified, to a lesser extent, by the presence of the grain boundary. Its prediction by conventional crystal plasticity is, on the other hand, much less accurate than for the total strain. Elastic strains and so internal stresses are sensibly overestimated in the neighborhood of the grain boundary and in the right-hand corner of the analyzed area (grain 1). Away from the grain boundary it is predicted correctly with an accuracy below 20% over more than 50% of the analyzed area.

Let us first discuss possible experimental artifacts. A possible source of discrepancy between the crystal plasticity model and experimental data could come from the relaxation of the specimen. Here, elastic and total strains were measured when relaxation was terminated and the applied force was stabilized. Typical stress drop between the end of the loading step and the relaxed state lies in the 5-20% range (see [22]). Relaxation is associated with limited glide of a small number of dislocations to adjust and minimize internal stresses (micro-plastic regime). It has therefore a limited influence on the total strain field but affects the elastic one in a significant manner. Relaxation is a standard concern for in situ stress measurements, observed on many specimens such as bulk polycrystals [37] or thin films [38] but also in 316L austenitic steel at room temperature [39]. In austenitic steels, it has been reported that the early stages of plastic deformation involve first the activation of lattice dislocations then followed by the activation of triple point sources (absent in our case) and the formation of dislocation pileups thereafter slip transmission across the grain boundaries [40]. Scanning electron microscope imaging at high resolution of the grain boundary area showed typical direct slip bands transfer across the grain boundary which would support this interpretation and would explain why relaxation occurs more specifically near the grain boundary. Recently, a polycrystal mean-field homogenization model coupled with a dislocation-based constitutive relation accounting for statistical distributions of internal stresses has also been proposed to reproduce the effect of relaxation (polycrystal average) on neutrons diffraction data [41]. This approach was however not concerned with the spatial resolution as the one used here with Laue experiment and full-field finite element modelling. A relaxation effect linked to grain boundaries was not envisioned and it relies on an additional parameter, the standard deviation of the distribution of resolved shear stress. On the experimental side, further investigation of this issue is required that should now be possible when using the new X-ray detector of BM32 operating at 10Hz. One could try to monitor the time evolution of elastic strains near a grain boundary for example. It might also be required to study at the micron scale from the interface the slip transfer such as carried out in several recent works [42,43].

Only the results concerning the yy component of the stress and strain fields are presented. Laue data were analysed using the standard method implemented in LaueTools software, in which the position of Laue spots on the X-ray detector screen were compared with the theoretical ones computed for an elastically strained Fe crystal lattice. As shown in [44,45,46], the digital image correlation technique can be used to estimate very precisely (accuracy of few hundredths of detector pixel) the spot displacement between two loading steps, leading to enhanced accuracy in the stress estimation. On ideal crystals (Si and Ge single crystals), stress accuracy of the order of 1MPa could be reach with this Laue-DIC technique. This constitutes another possible continuation of this work, allowing the investigation of the grain boundary effects on other components of the stress field. This will make it also possible to study lattice rotations with an accuracy better than 10 -4 rad.

The results presented here suggest that grain boundaries as microstructural objects need a special treatment in crystal plasticity modelling such as an accurate definition of slip transfer following [47], eventually complemented by the effect of incompatibility strains [48,42] and/or strain gradient effects [49]. Applying jump conditions for geometrically necessary dislocations may also be required [50]. Strain rate effects on the slip transfer could also be considered [51]. It could be required to treat the grain boundary as a thin surface with specific properties or special kinematics conditions allowing sliding and torsion [16]. Another class of interfaces treatment could be to integrate grain boundaries as a non-local microstructural constituent with special properties per se [17]. Going further would require to compare the results on our geometry with several of these approaches.

Conclusion

In trying to validate elastic strains (e.g. internal stresses) simulated by crystal plasticity with measured ones, it is shown that they are sensibly overestimated in the neighborhood of a general grain boundary, while there are better reproduced away from the grain boundary. A refinement of such type of experiment would allow to exclude artifacts or guide possible extensions of crystal plasticity models (inclusion of relaxation or slip transfer at grain boundaries). One could also explore the extension of modelling to various more sophisticated crystal plasticity models in order to be able to fully reproduce at least the bicrystal case. On the application side, this work demonstrated that the question of internal stresses due to incompatibility of the deformation at the grain boundary is not yet settled, hindering a thorough understanding of intergranular stress-controlled effects that remains a challenge in materials science. Experimental data is compared with the identified constitutive relation.

Figure 1 :

 1 Figure 1: Elasto-plastic response of a single crystal deformed under uniaxial tension along direction [100].Experimental data is compared with the identified constitutive relation.

Figure 2 :

 2 Figure 2: Elastic and total (i.e. elastic + plastic) strain evolution across the thickness of a [100] bent single crystal. The position of measurement points are along direction x on the surface normal to z (see inset). Experimental data from DIC and Laue microdiffraction (grey curves) are compared to the identified crystal plasticity model (blue and red curves).

Figure 3 :

 3 Figure 3: Schematic of the four-point bending experiment performed on a 316L bicrystal and details of the FEM mesh next to the grain boundary. Euler angles for grains G1 and G2 are (-175°, 78°, -107°) and (-133°, 25°, -163°) respectively.

Figure 4 :

 4 Figure 4: Backscatter electron image of the area of interest after complete unloading. (a) Slip traces patterns are detected in both grains revealing single or multiple slip activation. A slip trace analysis has been performed to identify the slip planes associated with the different families of slip traces. (b) Division of the area into 12 regions used to compare with the finite element model.

Figure 1 :

 1 Figure 1: Elasto-plastic response of a single crystal deformed under uniaxial tension along direction [100].Experimental data is compared with the identified constitutive relation.

Figure 2 :

 2 Figure 2: Elastic and total (i.e. elastic + plastic) strain evolution across the thickness of a [100] bent single crystal. The position of measurement points are along direction x on the surface normal to z (see inset).Experimental data from DIC and Laue microdiffraction (grey curves) are compared to the identified crystal plasticity model.

Figure 3 :

 3 Figure 3: Schematic of the four-point bending experiment performed on a 316L bicrystal and details of the FEM mesh next to the grain boundary. Euler angles for grains G1 and G2 are (-175°, 78°, -107°) and (-133°, 25°, -163°) respectively.

Figure 4 :

 4 Figure 4: Backscatter electron image of the area of interest after complete unloading. (a) Slip traces patterns are detected in both grains revealing single or multiple slip activation. A slip trace analysis has been performed to identify the slip planes associated with the different families of slip traces. (b) Division of the area into 12 regions used to compare with the finite element model.

Figure 5 :Figure 6 :

 56 Figure 5: Total strain fields along the y axis: experimental, modelling and difference plot at a loading of 7N

Table 1 :

 1 ). Chemical composition of the high purity alloys (single crystal and bi-crystal).

	Wt%	Fe	Cr	Ni	Mo	C	Mn, N, S, P, Si, Cu, O
	Monocrystal tensile	Bal.	17	14.6	2.3	<0.002	Not detected
	Monocrystal 4-pts	Bal.	17	14.6	2.3	<0.002	Not detected
	bending
	Bi-crystal 4-pts	Bal.	15.5	14	1.7	<0.002	Not detected
	bending

Table 2 :

 2 Crystal plasticity physical parameters used for the 316L stainless steel (N.A. Not Applicable, N.R. Not

	b (m)	𝜌 / (m -2 )	𝜏 /	𝛾 / ̇ (s -1 )	𝑛	K	𝑦 , (m)	𝑎 /	𝑎 $	𝑎 '	𝑎 [
			(MPa)

Acknowledgement C. Rey is warmly thanked by the authors for many fruitful discussions and for allowing us to use the CristalECP code. The "Laboratoire d'Imagerie Biomédical" from Sorbonne Université (Paris, France) is acknowledged for its help in measuring elastic constants by ultrasonic means. Beamtime allocation at the French Beamline BM32 CRG-IF at the European Synchrotron Facility (ESRF) is gratefully acknowledged (award No. HC/913 & HC/1449).