Evžen Kočenda
email: evzen.kocenda@cerge-ei.cz

Mathilde Maurel
email: mathilde.maurel@univ-paris1.fr

Gunther Schnabl
email: schnabl@wifa.uni-leipzig.de

Short-and Long-Term Growth Effects of Exchange Rate Adjustment

Keywords: JEL Classification number(s): C54, E32, E42, F32, F33, N20 Exchange rate regime, crisis, shock adjustment, theory of optimum currency areas, Mundell, Schumpeter, Hayek, cointegration

 we show that the degree of business cycle synchronization with the (potential) anchor country matters for the impact of exchange rate flexibility on growth.

1

Introduction

The recent wave of financial, balance of payments and sovereign debt crises has revived the discussion about the appropriate adjustment strategy in the face of asymmetric shocks. In most crisis events such as the 1997/1998 Asian crisis, the 1998 Japanese financial crisis, the 1998 Russian flu, the 2001 collapse of the Argentine currency board and even the US subprime market crisis, the crisis countries embarked on monetary expansion and depreciation as crisis solution strategies. In contrast, originating in growing intra-European current account imbalances [START_REF] Arghyrou | Current Account Imbalances and Real Exchange Rates in the Euro Area[END_REF], during the most recent European sovereign debt crisis a set of European crisis countries opted for staying in the Economic and Monetary Union (EMU; the EMU crisis countries) or maintaining tight exchange rate pegs to the euro (the Baltic countries and Bulgaria). The consequence was a strong pressure to curtail government expenditure and to cut wages.

The different adjustment strategies in the face of crisis based on inflation or deflation are embedded into different theoretical frameworks. [START_REF] Keynes | The General Theory of Employment, Interest, and Money[END_REF] and [START_REF] Mundell | A Theory of Optimum Currency Areas[END_REF] favoured monetary expansion and depreciation to provide a quick fix for missing international competitiveness and high unemployment. In contrast, [START_REF] Schumpeter | Theorie der Wirtschaftlichen Entwicklung[END_REF] and [START_REF] Hayek | Monetary Nationalism and International Stability[END_REF] stressed the role of wage and price cuts to boost long-term growth via an increasing marginal efficiency of private investment. In the context of the choice of the exchange rate regime, based on [START_REF] Friedman | The Case of Flexible Exchange Rates[END_REF], [START_REF] Mundell | A Theory of Optimum Currency Areas[END_REF] modelled the benefits of exchange rate adjustment in the face of asymmetric shocks to stimulate (short-term) growth [START_REF] Tavlas | Optimum-Currency-Area Paradoxes[END_REF]. In contrast, [START_REF] Mckinnon | Optimum Currency Areas[END_REF] highlighted the role of fixed exchange rates for macroeconomic stabilization and therefore as a tool for preserving the (long-term) growth performance.

Empirical studies on the impact of the exchange rate regime on growth have come to mixed results. For instance, [START_REF] Levy-Yeyati | To Float or to Fix: Evidence on the Impact of Exchange Rate Regimes on Growth[END_REF] who examine the impact of the exchange rate regime on growth for a sample of 183 countries in the post-Bretton-Woods era based on a pooled regression framework find a negative impact of exchange rate stability on growth for emerging market economies. In contrast, [START_REF] Maurel | Keynesian and Austrian Perspectives on Crisis, Shock Adjustment, Exchange Rate Regime and (Long-Term) Growth[END_REF] static and dynamic panel estimations find a positive impact of exchange rate stability on growth for a set of 60 mostly emerging market economies.

We aim to augment this literature by isolating the long-term and short-term growth effects of exchange rate stability / flexibility based on a cointegration framework. This research allows us to reconcile [START_REF] Mundell | A Theory of Optimum Currency Areas[END_REF] and [START_REF] Mckinnon | Optimum Currency Areas[END_REF] view on the impact of exchange rate flexibility / stability on growth in the face of asymmetric shocks. Documents de Travail du Centre d'Economie de la Sorbonne -2012.69 (Version révisée)

Short-and Long-Term Growth Effects of (Non-) Exchange Rate Adjustment

From 2008 to 2012 the European debt crisis revealed the different adjustment strategies to asymmetric shocks and crisis. When many European periphery countries were hit by bursting bubbles, the reversal of capital inflows and (near to) unsustainable debt levels, the EMU membership barred the way towards depreciation as a quick fix for the adjustment of unit labour costs to regain international competitiveness.

The loss of independent monetary policy and the exchange rate as adjustment tools to asymmetric shocks made price and wage adjustments necessary, which amplified the crisis and triggered different policy responses. Whereas Ireland (like the Baltic countries and Bulgaria) embarked on doughty adjustment measures in the private and public sector, in Greece political resistance retarded reforms. The delayed reforms in Greece were reflected in substantial rescue packages and rising imbalances in the ECB's TARGET2 mechanism, which both provided a substitute for pre-crisis private capital inflows as financing mechanism for persistent current account deficits.

The issue of the appropriate adjustment mechanism for unsustainable current account deficits within the EMU is reminiscent of a discussion during world economic crisis in the 1930s. Whereas [START_REF] Keynes | The General Theory of Employment, Interest, and Money[END_REF] called for monetary expansion and depreciation to provide a short-term growth impulse, [START_REF] Hayek | Monetary Nationalism and International Stability[END_REF] -in the spirit of [START_REF] Schumpeter | Theorie der Wirtschaftlichen Entwicklung[END_REF] -stressed the need of price and wage adjustment. The controversy between [START_REF] Keynes | The General Theory of Employment, Interest, and Money[END_REF] and [START_REF] Hayek | Monetary Nationalism and International Stability[END_REF] reflected different attitudes concerning the role of the government for macroeconomic stabilization. Whereas [START_REF] Keynes | The General Theory of Employment, Interest, and Money[END_REF] stressed the need for a timely anti-cyclical public macroeconomic impulse, [START_REF] Hayek | Monetary Nationalism and International Stability[END_REF] believed in the self-healing forces of the market.

The real exchange rate adjustment, which is required to rebalance the current account position of crisis countries, is framed by the theory of optimum currency areas. Building upon [START_REF] Friedman | The Case of Flexible Exchange Rates[END_REF] advocacy of flexible exchange rates, [START_REF] Mundell | A Theory of Optimum Currency Areas[END_REF] assumed that countries with a high probability of asymmetric shocks better preserve the exchange rate as an adjustment mechanism to stabilize growth, if labour markets are rigid. This reflects the Keynesian assumption of (short-term) price and wage rigidity and the crucial role of the government for macroeconomic stabilization. In contrast, [START_REF] Mckinnon | Optimum Currency Areas[END_REF] argued that in small and open economies a fixed exchange rate serves as a macroeconomic stabilizer by absorbing nominal shocks to promote growth. To maintain a fixed exchange rate, sufficient price and wage flexibility is necessary, which is line with [START_REF] Hayek | Monetary Nationalism and International Stability[END_REF] (1911) notion that declining prices and wages are the prerequisite for a robust recovery after crisis. 1 The growth effects of the crisis adjustment strategies based on exchange rate or wage adjustment have a goods market and a capital market perspective. [START_REF] Keynes | The General Theory of Employment, Interest, and Money[END_REF] stressed the short-term dimension with a focus on goods markets. In his view depreciation in the case of crisis helps to "jumpstart" the economy as real depreciation restores instantaneously the international competiveness. Real wages decline without cumbersome wage negotiations as inflation increases. [START_REF] Hawtrey | The Gold Standard[END_REF], who dedicated his academic work to the deflationary consequences of the return to the gold standard, pioneered the financial market based arguments in favour a monetary expansion during crisis. He assumed that low-cost credit during crisis helps to prevent a credit crunch (which is triggered by increasing risk perception in the private banking sector) and a dismantling of investment projects. New investment is encouraged which speeding up the recovery. As a fundamental restructuring process in the economy is circumvented, dire wage cuts become dispensable, and this helps to maintain economic activity via the consumption channel.

In contrast, [START_REF] Hayek | Monetary Nationalism and International Stability[END_REF] and [START_REF] Schumpeter | Theorie der Wirtschaftlichen Entwicklung[END_REF] help to understand the negative longterm growth effects of crisis therapy via monetary expansion and depreciation as restructuring is postponed. In Schumpeter's (1911: 350) real overinvestment theory the recession is a process of uncertainty and disorder, which forces a reallocation of resources on the enterprise sector ("cleansing effect"). The reallocation is a pre-requisite for long-term growth as speculative investment is abandoned, inefficient enterprises leave the market, the efficiency of the remaining enterprises is strengthened and new enterprises, products and production processes emerge. Exchange rate depreciation during crisis is an impediment to long-term growth as the "unadapted and unlivable" persists (Schumpeter 1911: 367). 2 In this context the 1 Although the Keynesian notion of discretionary policy making is matched here with the need for exchange rate flexibility and Hayek's belief in price and wage adjustment as crisis solution strategy is linked to exchange rate stability, this does not exclude that both authors made different policy propositions concerning the choice of exchange rate regime. [START_REF] Keynes | Activities 1940-44 -Shaping the Postwar World: The Clearing Union[END_REF] Bancor was equivalent to a fixed exchange rate regime, whereas [START_REF] Hayek | Monetary Nationalism and International Stability[END_REF] denationalization of money implies a flexible exchange rate regime. In detail [START_REF] Hayek | Monetary Nationalism and International Stability[END_REF] embraced fixed exchange rates for a gold standard, but not for a fiat money-based international monetary system. As Kindleberger (1985: 1) puts it the "dichotomy is not between any particular views of those great economists. It is rather more general, between one school worried about inflation and deflation of prices and the quantity of money, and the other about output and employment. 2 Schumpeter's (1911) argument, which has been designed for the private enterprise sector, can be applied for the government sector as well. A strong recession will trigger only structural reforms if there are restrictions on fiscal and monetary expansion in place. exchange rate regime matters, as a fixed exchange rate (or membership in a monetary union) imposes the need for structural reforms.

Documents de

The monetary overinvestment theory of [START_REF] Hayek | Monetary Nationalism and International Stability[END_REF] allows approaching the long-term growth effects of monetary expansion and depreciation from a capital market perspective.

During the upswing low interest rates set by the central bank encourage investment with declining marginal efficiency. When rising inflation urges the central bank to lift interest rates again, investment projects with low marginal return have to be dismantled. The resulting cleansing effect is the prerequisite for a sustained recovery, as only dynamic investment persists. If, however, the central bank responds to the crisis by decisive interest rate cuts, investment projects with low marginal returns, i.e. a distorted production structure, are conserved. A structurally declining interest rate level deprives the interest rate of its allocation function (which separates high-return investment from low-return investment) thereby putting a drag on long-term growth [START_REF] Schnabl | Exchange Rate Volatility and Growth in Emerging Europe and East Asia[END_REF]. Furthermore, as stressed by [START_REF] Mundell | A Theory of Optimum Currency Areas[END_REF] the question of if the exchange rate regime has a positive or negative effect on the growth performance of countries -within an asymmetric world monetary system -hinges on the degree of business cycle synchronization with the anchor country. Because of underdeveloped goods and capital markets small and open economies have an inherent incentive to stabilize the exchange rate versus the currency of a large anchor country --usually the dollar or the euro [START_REF] Calvo | Fear of Floating[END_REF]. If business cycles are synchronized with the anchor country, the monetary policy of the anchor country will be in line with the macroeconomic needs of the small open economy. If, however, business cycles are idiosyncratic, there is a larger need to stabilize growth via exchange rate adjustment. The recent contributions of the exchange rate regime literature in the wake of the financial crisis are reviewed by Beckman et al., (2012).

Previous papers have tested for the overall average growth effects of exchange rate flexibility, partially contingent on business cycle synchronization. We augment this literature by separating between the long-term and short-term growth effects of exchange rate flexibility based on a sample of 60 small open emerging market economies with the help of an error correction framework.

Data, Flexibility Measures and Business Cycle Correlation

To trace the short-term and long-term impact of exchange rate flexibility on growth, we choose five country groups for which the choice of the appropriate exchange rate regime has been high on the political agenda. These country groups are the EU15, Emerging Europe, East Documents de Travail du Centre d'Economie de la Sorbonne -2012.69 (Version révisée) Asia, South America and the Commonwealth of Independent States (CIS). In the EU15 as well as in Central, Eastern and Southeastern Europe (Emerging Europe) the discussion about membership in the EMU and/or the optimum degree of exchange rate stability against the euro continues to be discussed controversially. 3 The discussion about the pro and cons of EMU membership was revived during the most recent crisis.

In East Asia and South America the optimum degree of exchange rate stability against the dollar continues to be discussed, in particular since the Asian crisis and drastic US interest rate cuts following the subprime crisis. Most recently, Japan, China and Brazil have been involved in a discussion on "currency wars" and competitive interest rate cuts. In the Commonwealth of Independent States, Russia's move towards a currency basket and the depreciation of the CIS currencies during the recent crisis have revived the question about the optimum exchange rate policy. In this context, the choice of the anchor currency and the degree of business cycle synchronization with the anchor country play an important role.

The five country groups include all countries of the respective region excluding microstates -which may bias the sample towards a very high positive effect of exchange rate stability on growth [START_REF] Frenkel | An Estimate of the Effect of Common Currencies on Trade and Income[END_REF] -and excluding countries with insufficient data. This brings us to a sample size of 60 countries. Table 1 provides an overview of all countries under research, grouped into regions. We also list the prevalent anchor currencies and thereby the reference countries for measuring business cycle correlation. For the countries in East Asia, South America and the CIS the dollar has been the prevailing target of exchange rate stabilization. Business cycle correlation is measured versus the US.

For the European countries before the introduction of the euro in 1999, the German mark has been the dominant anchor currency [START_REF] Gros | European Monetary Integration[END_REF]. Since then, the euro has become the natural anchor for the European non-EMU countries. Exchange rate flexibility is measured in terms of exchange rate fluctuations against the German mark before 1999 and against the euro after 1999. Once a country has entered the EMU the proxy for exchange rate flexibility is set to zero. Business cycle correlation in Europe is measured versus Germany, which is the largest European economy (and therefore a country with a high degree of business cycle correlation with the euro area). For Germany, France as the second largest European economy is used as a reference country to measure business cycle correlation.

The data source is the International Financial Statistics. Missing or inconsistent data were completed and cross-checked with national statistics, mainly by national central banks. Series used in estimations are at quarterly frequencies and are seasonally adjusted. Quarterly real GDP growth rates and inflation rates are calculated as year-over-year quarterly growth rates to filter out the seasonal pattern and lower the volatility of the transformed series (

ln ln

). The sample period starts in January 1994 for two reasons. First, to exclude the beginning of the 1990s, which for most of the Central, Eastern and Southeastern European and CIS countries implied high economic volatility and data uncertainty linked to the transition process. Second, to have a balanced sample the observations for all other countries start in 1994 as well. The time series end in 2010. We compute quarterly measures for trade integration as exports plus imports over GDP. Finally, we include the interest rate of the potential exchange rate anchor country as indicated in Table 1 as a proxy for external monetary conditions, which have been an important determinant of growth in emerging market economies [START_REF] Maurel | Keynesian and Austrian Perspectives on Crisis, Shock Adjustment, Exchange Rate Regime and (Long-Term) Growth[END_REF].

We use de facto exchange rate flexibility measures, because de jure flexibility measures are likely to be flawed by fear of floating. 4 Quarterly de facto exchange rate flexibility is measured by the standard deviation of monthly percent exchange rate changes of the respective quarter (σ) and the quarterly arithmetic average of monthly percent exchange rate changes (μ). Both measures are summarized by the z-score ( ) as in [START_REF] Schnabl | Exchange Rate Volatility and Growth in Emerging Europe and East Asia[END_REF] and [START_REF] Maurel | Keynesian and Austrian Perspectives on Crisis, Shock Adjustment, Exchange Rate Regime and (Long-Term) Growth[END_REF]. All three variables are calculated against the euro or the dollar, depending of the respective anchor currency as listed in Table 1.

Finally, as we aim to analyze the impact of exchange rate flexibility on growth contingent on business cycles synchronization we construct a dummy for business cycle synchronization (Dbcc) for every country based on the five country groups defined earlier. If business cycle correlation of specific countries with the reference country is lower than the country group average the dummy is set equal to one. The dummy is zero if the degree of business cycle correlation is above average.

Empirical Analysis

Given the different time dimensions of economic theories concerning the impact of exchange rate flexibility/stability on growth, the issue is an empirical one. The foregoing empirical analysis aims to disentangle the long-term and short-term effect of exchange rate flexibility 4 Calvo and Reinhart (2002) show that the official (de jure) classifications of the exchange rate regime by the IMF are not necessarily in line with the practiced (de facto) exchange rate strategies of countries. Therefore, following De [START_REF] Grauwe | Exchange Rate Stability, Inflation and Growth in (South) Eastern and Central Europe[END_REF]

where subscripts i and t represent country and time period indices, respectively, and α 0i and ε it are country-specific intercepts and error terms. The dependent variable w it represents the quarterly real growth rates from 1994 to 2010. Exchange rate flexibility (ERF it ) is to account for the long-run effect of exchange rate policy. We use three different measures of exchange rate flexibility / volatility as described above: standard deviation, mean of percent exchange rate changes against the anchor currency, and z-score. Further, we employ theoretically motivated explanatory variables that affect economic growth via direct or indirect channels.

Inflation (INF it ) serves as a control variable for macroeconomic instability. The interest rate of the anchor country (IR it ) accounts for the influence of the price of money with respect to growth. Finally, we include trade openness (TO it ) to account for the extent of the international economic integration, and a time trend (T t ) to reflect unobserved technological change.

We assume that the latter variables have an impact on the long-term economic performance while they do not matter in the short-term. There is a large number of other potential explanatory variables like investment, government spending, schooling, etc. which could increase the fit of the model. However, these variables would also generate endogeneity (for instance between investment and growth) and multicollinearity (for instance between government spending and inflation) as argued by De [START_REF] Grauwe | Exchange Rate Stability, Inflation and Growth in (South) Eastern and Central Europe[END_REF]. Therefore, we opt for a parsimonious specification, restricted to the variables mentioned above.

As a next step we modify our benchmark model into specification (2) in order to account for the effect of asymmetric countries:

(2)

In specification (2), we control for the fact that countries with asymmetric business cycles face bigger constraints to achieve the same growth when compared to countries without asymmetries. To do so we include additional variable that is constructed as ERF it multiplied by the dummy variable Dbcc capturing asymmetric business cycle (defined in section 3). Hence, the variable takes the value of the specific flexibility measure only for countries characterized by relative asymmetric business cycles, and does not enter estimation for countries that do not exhibit asymmetric business cycles. The inclusion of the interaction term ( ) aims to capture the short-term nature of asymmetric shocks.

Prior to estimation we test for the stationarity of the employed variables by several panel unit root tests (results not reported; available upon request). We find that the quarterly growth rate, inflation, and three measures of exchange rate flexibility are stationary, whereas trade openness, interest rate and by construction a trend, are integrated of order one. While the explained variable is stationary, the panel of explanatory variables combines stationary and nonstationary variables. This leaves open the possibility for the estimated specifications to be embedded in a dynamic error-correction model (ECM).

Similarly as in [START_REF] Elbadawi | Aid, Real Exchange Rate Misalignment, and Economic Growth in Sub-Saharan Africa[END_REF] we estimate an ECM version of the specifications ( 1) and ( 2) for our panel of 60 countries over the period 1994-2010. The details on the dynamic specifications ARDL are given in the Tables 234. Three econometric estimation frameworks are chosen for the ECM for panel data: pooled mean group, mean group, and dynamic fixedeffects estimators. The most restrictive is the dynamic fixed-effects estimator that assumes that all parameters are constant across countries, except for the intercept, which is allowed to vary across countries. The pooled mean group estimator is more general than the dynamic fixed-effects estimator as it imposes the restriction that all countries share the long-term coefficients. The mean group estimator is even more general as it assumes that economies differ in their short-term and long-term parameters.

The choice between the three estimators entails a tradeoff between consistency and efficiency. The dynamic fixed-effects estimator dominates the other two in terms of efficiency if the restrictions are valid. If they are not valid, the dynamic fixed effects will generate inconsistent estimates and is dominated by the pooled mean group and mean group estimates.

The pooled mean group estimator can be assumed to offer the best compromise between consistency and efficiency, because one would expect the long-term growth path to be driven by a similar process across countries while the short-term dynamics around the long-term equilibrium path differ because of idiosyncratic news and shocks to fundamentals. Following the above arguments we perform formal Hausman tests of the implied restrictions.

By estimating both specifications (1) and (2) our objective is threefold: to highlight the impact of exchange rate flexibility on growth, to disentangle the short-term versus long-term effect of exchange rate flexibility on growth, and to quantify the weight of countries with asymmetric business cycles (which we call asymmetric countries) in this impact. Our prior is that the impact of exchange rate flexibility should be positive in the short term, especially for asymmetric countries, but negative in the long term.

Estimation Results

The econometric estimation results are presented in Tables 2 to 4. The results are organized in a way that columns labeled as 1 contain an overall effect via coefficients from specification

(1), while columns labeled as 2 show the effects from specification (2) where we account for asymmetry in business cycle correlation. The null hypothesis of equality of coefficients between pooled mean group and mean group is in most cases rejected at 1 % level. The Hausman test favors the mean group model (Table 2) against the pooled mean group estimator (Table 3). 5 The dynamic fixed effects estimates are presented in Table 4. All estimates support our main assumptions regarding the long-versus short-run impact of exchange rate flexibility on growth and the more specific case of asymmetric countries.

The mean group results as presented in Table 2 suggest a negative effect of exchange rate flexibility on growth in the long term and a positive effect of exchange rate flexibility on growth in the short term. For all three flexibility measures (standard deviation, average yearly change, and z-score) exchange rate flexibility has are highly significant negative effect on growth in the long term, when controlling for interest rates in the anchor country, trade openness, and inflation. Both trade openness and the interest rate in the large reference country have the expected signs and are highly significant. Open economies grow faster and the gradual decline of the interest rate level in the large anchor countries (US, euro area, Germany) seems to have boosted growth in the emerging market economies. In the long term higher growth levels are linked to higher inflation levels. 6 The coefficient of the interaction term is significant for the z-score measure and insignificant for the other two flexibility measures. Negative coefficients for the interaction term indicate an additional negative effect of exchange flexibility for asymmetric countries in the long run.

In the short-run there is a positive effect of exchange rate flexibility (volatility) on growth for average change as flexibility measure, significant at 1% percent. In all other cases (five out of six), the overall coefficients of the short-term impact of exchange rate flexibility on growth are positive but not significant. Coefficients of the interaction term in the short run are positive but statistically insignificant. This indicates that positive impact of exchange flexibility is magnified for asymmetric countries in the short-run but statistical insignificance prevents us to draw a firm inference. In the short-run equation there is mostly no evidence for a significant positive impact of inflation on growth.

The results of our preferred estimates, obtained through the pooled mean group estimation, are presented in Table 3. They are mainly in line with the mean group estimation and therefore can be regarded as evidence for the robustness of the results. The long-run effect of exchange rate flexibility on growth is clearly negative, significant at the 1% level for all flexibility measures. For countries with asymmetric business cycles there is an additional, even larger negative effect of exchange rate flexibility on growth. In the pooled mean group estimations inflation has no significant positive long-run impact on growth. Openness and declining interest rates in the references country are clearly associated with higher growth.

In contrast, in the short run exchange rate flexibility mostly seems to be positively linked to higher growth, with average exchange rate changes being significant at 1% level and standard deviations at 5% level. 7 The interaction term identifies an additional and positive effect of exchange rate flexibility on growth for asymmetric countries in two out of three cases (standard deviation and z-score), suggesting that the effect of exchange rate flexibility on short-run growth is driven by the asymmetric countries. In contrast to the long-term coefficient a statistically significant positive link between inflation and growth is revealed.

The results of the dynamic fixed effects estimation are reported in Table 4. The negative long-term effect of exchange rate flexibility on growth remains highly significant for all three flexibility measures. There is no statistically significant long-term impact of inflation on growth. The highly significant impact of trade openness and interest rates changes in the anchor country on growth in our sample is confirmed. For countries with asymmetric business cycles an additional negative effect of exchange rate flexibility on growth is suggested. In the short-run a highly significant positive impact of exchange rate flexibility on growth is revealed. The estimates based on average change confirm that this positive impact is imputable to the presence of asymmetric countries. For the other two estimates, the positive effect of exchange rate flexibility on short-run growth applies to countries with or without comparatively synchronized business cycles The dynamic fixed effects estimation does not reveal any significant short-term link between inflation and growth.

All in all, the estimations provide evidence in favor of the hypothesis that in the short-run exchange rate flexibility helps to the smooth the business cycle, in particular for countries with idiosyncratic business cycles. In contrast, in the long-run the impact of exchange rate flexibility on growth is clearly negative. The positive impact of exchange rate flexibility on growth in the short-term supports the policy propositions of [START_REF] Keynes | The General Theory of Employment, Interest, and Money[END_REF], [START_REF] Hawtrey | The Gold Standard[END_REF],

and [START_REF] Mundell | A Theory of Optimum Currency Areas[END_REF], whereas negative long-run impact of exchange rate flexibility on growth is in line with [START_REF] Hayek | Monetary Nationalism and International Stability[END_REF], [START_REF] Schumpeter | Theorie der Wirtschaftlichen Entwicklung[END_REF][START_REF] Mckinnon | Optimum Currency Areas[END_REF]. Finally, positive impact of the short-run exchange flexibility is magnified for asymmetric countries and helps to smooth out asymmetric shocks in the short run.

Diagnostic checking

We complement our analysis by diagnostic checking; results are in Table 5. We perform panel-unit root tests designed by [START_REF] Choi | Unit root tests for panel data[END_REF] and a battery of Westerlund ( 2007) tests for cointegration. The results show that the residuals are stationary for both estimated specifications (with or without accounting for the business cycle asymmetry). Further, no matter what exchange rate flexibility measure is used (standard deviation, average, or z-score)

the results show that the variables are cointegrated; in all cases, two out of the four statistics proposed by [START_REF] Westerlund | Testing for Error Correction in Panel Data[END_REF] reject the null of no cointegration. Finally, it should be noted that [START_REF] Pesaran | Pooled Mean Group Estimation of Dynamic Heterogeneous Panels[END_REF] show that pooled mean group estimation does not require pretesting for unit roots and cointegration, and that pooled mean group estimation provides consistent and efficient estimates of parameters in a long-run relationship between stationary and integrated variables.

Conclusion

With the European sovereign debt crisis a controversial discussion concerning the appropriate monetary policy and exchange rate strategy to asymmetric shocks and crisis has reemerged.

We have aimed to derive from our econometrical exercise for a panel of 60 countries a policy recommendation for crisis countries. Our estimation results provide evidence that exchange rate adjustment stimulates growth in the short-term, but puts a drag on the long-term growth performance. As the overall effect is negative, the policy implication is to keep exchange rates stable to promote long-term growth via price and wage flexibility, in the spirit of [START_REF] Schumpeter | Theorie der Wirtschaftlichen Entwicklung[END_REF], [START_REF] Hayek | Monetary Nationalism and International Stability[END_REF][START_REF] Mckinnon | Optimum Currency Areas[END_REF]. In line with [START_REF] Schnabl | Exchange Rate Volatility and Growth in Emerging Europe and East Asia[END_REF] and based on our findings we recommend the crisis countries to proceed with structural reforms and real wage cuts. Painful restructuring and declining output today are likely to be rewarded with a robust economic recovery and rising income in the future. In contrast, monetary expansion and depreciation as a crisis solution strategy can be expected to provide short-term relief, but long-term pain. Mutual monetary expansion would lead into a wave of competitive depreciations and competitive interest rate cuts, and therefore global financial and economic instability. This would be an antidote of the coordinated structural reforms. Table 4 The Short-and Long-Run Effects on the aggregate output growth (dynamic fixed effects) Note: Columns denoted by 1 contain results from specification (1) and columns denoted by 2 contain results from specification (2) that includes exchange rate flexibility measure for asymmetric countries. Dynamic specifications are: ARDL (1,1,1,0,0,0) for specification (1), ARDL (1,1,1,1,0,0,0) for specification (2). Significance levels at 1, 5, and 10% are denoted by

  Travail du Centre d'Economie de la Sorbonne -2012.69 (Version révisée)

  ER flexibility denotes one of the three exchange rate flexibility measures used; e.g. average change, standard deviation, and z-score. Numbers in parentheses are standard errors. Brackets contain p-values for Hausman test.

  on growth, which may, possibly, reconcile both strands of the literature as presented in section 2 by attributing a time dimension to them. Our model below is not derived from a standard neo-classical analysis of growth. Rather, by relying upon[START_REF] Schnabl | Exchange Rate Volatility and Growth in Emerging Europe and East Asia[END_REF] or[START_REF] Maurel | Keynesian and Austrian Perspectives on Crisis, Shock Adjustment, Exchange Rate Regime and (Long-Term) Growth[END_REF] it principally correlates quarterly growth rates and exchange rate

	flexibility.
	4.1. Model Specification and Estimation Procedure
	We model changes in economic growth (w it ) as a function of exchange rate flexibility /
	volatility (ERF), inflation (INF), interest rate (IR), trade openness (TO) and trend (T) proxies
	for changes in technology. Equation (1) is our benchmark specification:
	,

we assume that exchange rate volatility is high in case of full exchange rate flexibility. This implies that low exchange rate volatility indicates exchange rate stabilization efforts by central banks. Documents de Travail du Centre d'Economie de la Sorbonne -2012.69 (Version révisée)

Table 1 Groups of countries Country group

 1 France, as the second largest country of the European Union, is used as a reference country for Germany.

		Anchor currency	Countries
		Reference country
	EU15	Euro/DM	Austria, Belgium, Denmark, Finland, France,
		Germany	Germany*, Greece, Ireland, Italy, Luxemburg,
			Netherlands, Portugal, Spain, Sweden, UK
	Emerging	Euro/DM	Albania, Bosnia-Herzegovina, Bulgaria, Croatia,
	Europe	Germany	Czech Republic, Estonia, Hungary, Macedonia,
			Latvia, Lithuania, Poland, Romania, Serbia,
			Slovak Republic, Slovenia, Turkey
	CIS	Dollar	Armenia, Azerbaijan, Belarus, Georgia,
		US	Kazakhstan, Kyrgyz Republic,
			Moldova, Russia, Ukraine
	East Asia	Dollar	China, Hong Kong, Indonesia, Japan,
		US	Malaysia, Philippines, Singapore,
			South Korea, Taiwan, Thailand
	Latin	Dollar	Argentina, Bolivia, Brazil, Chile,
	America	US	Columbia, Ecuador, Paraguay,
			Peru, Uruguay, Venezuela

* Documents de Travail du Centre d'Economie de la Sorbonne -2012.69 (Version révisée)

Table 2

 2 The Short-and Long-Run Effects on the aggregate output growth (mean group estimator) ER flexibility denotes one of the three exchange rate flexibility measures used; e.g. average change, standard deviation, and z-score. Numbers in parentheses are standard errors.

	Note: Columns denoted by 1 contain results from specification (1) and columns denoted by 2 contain results from specification (2)
	that includes exchange rate flexibility measure for asymmetric countries. Dynamic specifications are: ARDL (1,1,1,0,0,0) for
	specification (1), ARDL (1,1,1,1,0,0,0) for specification (2). Significance levels at 1, 5, and 10% are denoted by a , b , and	c , respectively.

Documents de Travail du Centre d'Economie de la Sorbonne -2012.69 (Version révisée)

Table 3

 3 The Short-and Long-Run Effects on the aggregate output growth (pooled mean group estimator) ER flexibility denotes one of the three exchange rate flexibility measures used; e.g. average change, standard deviation, and z-score. Numbers in parentheses are standard errors. Brackets contain p-values for Hausman test. Documents de Travail du Centre d'Economie de la Sorbonne -2012.69 (Version révisée)

	Note: Columns denoted by 1 contain results from specification (1) and columns denoted by 2 contain results from specification (2)
	that includes exchange rate flexibility measure for asymmetric countries. Dynamic specifications are: ARDL (1,1,1,0,0,0) for
	specification (1), ARDL (1,1,1,1,0,0,0) for specification (2). Significance levels at 1, 5, and 10% are denoted by a , b , and

c

, respectively.

  Documents de Travail du Centre d'Economie de la Sorbonne -2012.69 (Version révisée)

			Exchange Rate Flexibility Measures
	Variable	Average change 1 2	Standard deviation 1 2	1	z-score	2
	Long-run coefficients
	ER flexibility	-1.0176 a	-0.7831 a	-0.8444 a	-0.7437 a	-0.7056 a	-0.5959 a
	ER flexibility for	(0.1681)	(0.2064) -0.4942 c	(0.1983)	(0.2360) -0.2946	(0.1767)	(0.2274) -0.2807
	asymmetric countries		(0.2831)		(0.4120)			(0.2959)
	Inflation	0.0367	0.0391			0.0282	0.0303
	Interest rate	(0.0314) -0.6735 a	(0.0319) -0.6844 a	-0.5612 a	-0.5672 a	(0.0353) -0.5828 a	(0.0346) -0.5967 a
	Trade openness	(0.2013) 0.0617 a	(0.2007) 0.0614 a	(0.1890) 0.0518 b	(0.1888) 0.0519 a	(0.1941) 0.0502 a	(0.1910) 0.0509 a
	Trend	(0.0200) -0.0008 a	(0.0199) -0.0009 a	(0.0205) -0.0007 a	(0.0202) -0.0007 a	(0.0177) -0.0008 a	(0.0174) -0.0008 a
	Error correction	(0.0002) -0.2731 a	(0.0003) -0.2729 a	(0.0002) -0.2744 a	(0.0002) -0.2748 a	(0.002) -0.2740 a	(0.0002) -0.2748 a
	coefficients	(0.0184)	(0.0186)	(0.0181)	(0.0179)	(0.0183)	(0.0181)
	Short-run coefficients
	Δ ER flexibility	0.1372 a	0.0772 b	0.1122 a	0.0996 a	0.0805 a	0.0627 b
	Δ ER flexibility for	(0.0294)	(0.0314) 0.1143 b	(0.0423)	(0.0368) 0.0278	(0.0312)	(0.0321) 0.0427
	asymmetric countries		(0.0532)		(0.0719)			(0.0471)
	Δ Inflation	-0.0097	-0.0093	-0.0101	-0.0106	-0.0103	-0.0111
	Intercept	(0.0121) 0.0125 a	(0.0122) 0.0127 b	(0.0120) 0.0156 a	(0.0123) 0.0158 a	(0.0122) 0.0167 a	(0.0124) 0.0169 a
		(0.0052)	(0.0052)	(0.0052)	(0.0053)	(0.0049)	(0.0051)
	Hausman test (mean group	0.02	0.02	0.01	0.02	0.02		0.03
	against fixed effects)	[1.00]	[1.00]	[1.00]	[1.00]	[1.00]	[1.00]

[START_REF] Kočenda | Macroeconomic Sources of Foreign Exchange Risk in New EU Members[END_REF] show that new EU members should promote nominal and real convergence with the core EU members since both real and nominal factors impact the variability of their exchange rate risk. Documents de Travail du Centre d'Economie de la Sorbonne -2012.69 (Version révisée)

The mean group estimator of[START_REF] Pesaran | Estimating long-run relationships from dynamic heterogeneous panels[END_REF] is consistent but not a good estimator when either N or T is small. Hence, we consider results of the pooled mean group estimator on par with those of the mean group estimator.

Due to the lack of convergence during the estimation the inflation term is dropped from the equation in the specification with standard deviation as a proxy for exchange rate flexibility. Documents de Travail du Centre d'Economie de la Sorbonne -2012.69 (Version révisée)

[START_REF] Bubák | Volatility Transmission in Emerging European Foreign Exchange Markets[END_REF] show that exchange rate volatility increases in medium-term for some new EU countries with troubled financial sector development.[START_REF] Hanousek | Foreign News and Spillovers in Emerging European Stock Markets[END_REF] document capital market spillovers on the same set of countries. Documents de Travail du Centre d'Economie de la Sorbonne -2012.69 (Version révisée)

support by GAČR grant No. 403/11/0020. We also thank for support by Stiftung Geld und Währung. The usual disclaimer applies. Documents de Travail du Centre d'Economie de la Sorbonne -2012.69 (Version révisée)


Julien De Jaeger

Denis Solas

Thierry Baudin

Olivier Fandeur

Jean-Hubert Schmitt

Colette Rey

Eric S Huron

Roger C Reed

Mark C Hardy

Michael J Mills

J-H Schmitt

INCONEL 718 SINGLE AND MULTIPASS MODELLING OF HOT FORGING

Keywords: INCONEL 718, Forging, Dynamic Recrystallization, Crystalline Plasticity, Modelling

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

Introduction

INCONEL 718 is a nickel based superalloy hardened by nanometric precipitates such as γ'' (Ni3Nb) and γ' (Ni3 [Ti,Al]). This alloy also contains a δ (Ni3Nb) phase with an average size which is hundred times larger than the γ'' phase. This alloy is widely used for the aircraft engine applications.

The INCONEL 718 forging domain (Figure 1), is ranging between the melting point (T = 1150°C) determined by Lewandowski et al. [START_REF] Lewandowski | High temperature deformation behaviour of solid and semi-solid alloy 718[END_REF] and the γ'' phase solvus temperature (T = 940°C). Below the latter temperature, the nanometric precipitates harden the material, leading to excessive compression stresses during hot forging.

Within the optimal hot forging domain, the hardening phases are more or less dissolved according to their solvus temperatures (see Figure 1). The forging domain is composed of two sub-domains located on each side of the δ phase solvus temperature (T = 1025°C): the sub-δ-solvus domain below 1025°C, and the super-δ-solvus domain above 1025°C.

In the sub-δ-solvus domain the grain size is controlled by boundary pinning of precipitates. Forging in the sub-δsolvus domain leads to a final fine grain size, while coarse grains are observed in the super-δ-solvus domain. The forging domain is chosen with respect to the targeted final properties. In this study, the super-δ-solvus domain is only considered. During the hot forging, as soon as the δ phase is dissolved, the microstructure is only composed by the austenic matrix of nickel (FCC).

Through hot forging, dynamic recrystallization phenomena are overlapped by a classical hardening deformation process. For INCONEL 718, the dynamic recrystallization mechanism was characterized as a discontinuous process (see for instance, Wang et al. [START_REF] Wang | Microstructure evolution during dynamic recrystallization of hot deformed superalloy 718[END_REF]). Some authors [START_REF] Zhang | Mathematical modeling of the hot deformation behavior of superalloy IN718[END_REF][START_REF] Yuan | Effect of the δ phase on the hot deformation behaviour of Inconel 718[END_REF] T (°C) 850°C 940°C 1025°C γ' solvus γ'' solvus δ solvus subsolvus forging supersolvus forging 1150°C liquidus 664 report a necklace dynamic recrystallization process, depending on the deformation rate and the temperature compatible with a discontinuous behaviour of the dynamic recrystallization. Recrystallization is characterized by two processes: nucleation and growth. The nucleation process creates new grains with low dislocation densities. The grain growth process corresponds to grain boundaries motion, depending on the dislocation density and on misorientation between adjacent grains. These phenomena act to minimize the total stored energy in the system.

Usually, an industrial hot forging is a multipass process. In this case, additional parameters must be taken into account: the number of passes, the interpass time and the temperature evolution during the deformation steps and the interpass time. The number of parameters is reduced when isothermal hot forging is performed. The literature gives many results for INCONEL 718 behaviour during a single pass hot deformation in the super-δ-solvus domain [START_REF] Lewandowski | High temperature deformation behaviour of solid and semi-solid alloy 718[END_REF][START_REF] Wang | Microstructure evolution during dynamic recrystallization of hot deformed superalloy 718[END_REF][START_REF] Zhang | Mathematical modeling of the hot deformation behavior of superalloy IN718[END_REF][START_REF] Yuan | Effect of the δ phase on the hot deformation behaviour of Inconel 718[END_REF][START_REF] Guimaraes | Recrystallization and aging effects associated with the high temperature deformation of Waspaloy and Inconel 718[END_REF][START_REF] Medeiros | Microstructural modeling of metadynamic recrystallization in hot working of IN 718 superalloy[END_REF][START_REF] Thomas | High temperature deformation of INCONEL 718[END_REF]. In the case of the super-δ-solvus domain, two papers only refer to multipass hot deformation [START_REF] Mataya | Simulating microstructural evolution during the hot working of alloy 718[END_REF][START_REF] Park | Hot forging of a nickel-base superalloy[END_REF]: the first study [START_REF] Mataya | Simulating microstructural evolution during the hot working of alloy 718[END_REF] was realized with an initial grain size of 250 µm (1200°C / 7 hours) much larger than the present grain size. The second study [START_REF] Park | Hot forging of a nickel-base superalloy[END_REF] was performed in anisothermal conditions. For comparison, Ryan and McQueen [START_REF] Ryan | Hot strength and microstructural evolution of 316 stainless steel during simulated multistage deformation by torsion[END_REF] have studied an austenitic 316 stainless steel in multipass conditions.

The modelling proposed in this study takes into account the competition between hot hardening and dynamic recrystallization during both single pass and multipass processes. The evolution of the mechanical behaviour as well as the evolution of the microstructure of INCONEL 718, are computed. The numerical results are compared to the experimental observations. This study performed at the grain scale level, sequentially couples a deformation model (CristalECP) to a recrystallization model (CAReX [START_REF] Thébault | Polycrystalline modelling of Udimet 720 forging[END_REF][START_REF] Solas | Dynamic recrystallization modeling during hot forging of a nickel based superalloy[END_REF].

Experimental procedure

The laboratory samples were cut out from a 150 mmdiameter forged billet. Cylindrical specimens (12 mm in diameter and 18 mm long) were machined from the midradius of the billet.

Right before forging, the sample microstructure at high temperature (Figure 2 The thermo-mechanical paths had to comply with two main rules. Firstly, the sum of each macroscopic deformation step in the MPD and in the MPQ process (E2 + E3) had to be equal to the total deformation in the SP process (E1). Secondly, for the MPD and MPQ processes, the strain value E2 had to be higher than the strain value E3 (E2 > E3). For this study, the deformation rates were E1 = 0.70, E2 = 0.40 and E3 = 0.30 respectively.

For optical and Scanning Electron Microscopy (SEM) observations, each sample was then cut along their radius and mechanically polished. To reveal grain boundaries, a mechanical-chemical polishing was performed on the samples with MASTERMET ® (non-crystallized colloidal silica). The microstructures (mean grain size, twin fraction, sub-boundaries fraction…) were characterized at the end of each process, using an EBSD method coupled to a SEM FEG. A careful attention was paid to the location of the observed areas: all EBSD maps were performed on the sample core on a surface of 1 x 1mm 2 , with an analysis step size of 1 µm.

Experimental results

Mechanical behaviour

The mechanical behaviour during hot compression is plotted in Figure 4 for the three deformation paths. The compression curves (Figure 4) present an initial stress peak for the first pass (SP, MPQ and MPD) as well as for the second pass (MPQ and MPD). According to a previous study [START_REF] Guimaraes | Recrystallization and aging effects associated with the high temperature deformation of Waspaloy and Inconel 718[END_REF], these peak stresses result from the competition between the dislocation mobility and solute atoms. It should be noted that the three first pass curves are similar. For a strain value above 0.40, the SP, MPQ and MPD mechanical processes behave differently. Compared with the MPQ and MPD processes, the stress level corresponding to the SP process is smaller. Such SP curve corresponds to a maximal time of dynamic recrystallization and by extrapolation to a minimal dislocation density. To emphasize these behaviour differences, the hardening rate is plotted as a function of the true strain (Figure 5).

In the first pass, a transition between two linear ratings is observed. These ratings are represented by dashed lines in Figure 5. The first linear rating is identified as a work hardening and the second as a meta-stable equilibrium in the competition between the work hardening and the dynamic recrystallization. A previous analysis proved that this transition starts from E ≈ 0.06 for a lower deformation rate on the same material. It has been shown that this transition corresponds to the nucleation of recrystallized grains [START_REF] Jaeger | INCONEL 718 recrystallization in the delta supersolvus domain[END_REF]. This result will be used in the numerical part of this study. The same transition is observed during the second pass on the MPQ sample. On the contrary, the MPD sample does not exhibit this transition. From these observations, it can be concluded that the microstructure is reinitiated during the interpass of the MPQ process. Its mechanical behaviour is controlled by the work hardening, the nucleation of recrystallized grains and the grain growth.

The lack of transition in the MPD sample can be explained as follows: this interpass of ten seconds is too short to fully recrystallize the microstructure and nuclei are still present at the beginning of the second pass. As a consequence, the MPQ stress level is almost the same in both passes.

Microstructural evolution

The EBSD method characterization was performed on four different quenched samples at different stage of each process (Figure 6). 1. EBSD data corresponding to the grain size, the subboundary ratio and the twin ratio of the different quenched microstructures are given in Table 1. Quenching froze the microstructure and stopped grain boundary migration. It should be noted that the grain size of the as received state was 5 times lower than the dwell microstructure grain size. Moreover, the sub-boundary ratio was divided by a factor five, whereas the twin ratio was multiplied by a factor two. These analyses pointed out a growth phenomenon related to static recrystallization during the fifteen minutes dwell time before the first pass.

After a 0.70 total deformation, the final grain size and twin ratio decreased, compared to the initial microstructure. The twin ratio evolution means that twins cannot grow during dynamic recrystallization. This assumption agrees with the literature results. According to Gleiter's atomistic model [START_REF] Gleiter | The formation of annealing twins[END_REF], the formation of annealing twins can be explained by a nucleation process on the {111} planes on the growing grains. Nucleation twin process induces new crystallographic orientations within the microstructure. Two grains separated by a twin underwent different plastic and lattice rotations with increasing applied strain and the twin misorientation conditions progressively disappeared above a 0.70 strain. Such phenomenon explains the decrease of the twin ratio after the different processes. Moreover, observations showed that the final sub-boundary ratio was always higher after deformation. Mechanical behaviour of the final specimens being strongly dependent on the microstructure, a model is now proposed to understand the origin of these microstructures.

Numerical models

Two models are coupled and successively used: first, the polycrystalline model, implemented in a Finite Element code, computes the lattice rotation field, the local strain and stress fields and the dislocation density at each Gauss point of the grain meshing, for a fixed time increment. The new crystallographic orientation and dislocations densities within the grains are implemented within the cells (sites) of a 3D cellular automation. Next, the recrystallization model computes the probability for a site to be recrystallized (nucleation and growth) and determines the new crystallographic orientation and the new dislocation densities for recrystallized sites. These new data are implemented in the polycrystalline model for a new cycle. A step N is defined by a calculation with the polycrystalline modelling followed by a calculation with the recrystallization model.

Crystalline plasticity model: CristalECP

The polycrystalline model rests on the continuous theory of dislocations and on the framework of the finite transformations. The model is implemented in the Finite Element codes ABAQUS® and CAST3M®. Each grain of the polycrystal is meshed and each point of the meshing follows the hereafter crystalline plasticity laws.

Each grain of the polycrystal is considered as a single crystal. At any Gauss point of a given grain, the local plastic behaviour is described in terms of dislocation densities on the slip systems. For the chosen forging temperature, INCONEL 718 is a single phase material. The austenitic nickel matrix has a FCC structure and twelve conventional slip systems {111}<110>. The Schmid's criterion which rules the activation of the different slip systems s is given by:

s c s    (1)
where τ s is the resolved stress on the slip system s and τ c s the critical shear stress on the same slip system.

The constitutive visco-plastic law is given by:

m s c s s 1 0        (2)
where γ s is the slip rate on the slip system s, m is the strain rate parameter.

The critical resolved shear stress τ c s on the system s is given by: 667

   t st 0 a b t s c     (3)
where τ 0 depends on the lattice friction and on interactions with solute atoms, ρ t t is the total dislocation density on the twelve slip systems, µ, b are respectively the shear modulus, the Burgers vector magnitude and a st is a component of the interaction matrix between the slip systems s and t.

The dislocation density evolution corresponds to the Kocks and Mecking hardening law. It is composed of two terms: a production term based on Orowan's relationship (i.e. proportional to the mean free path of mobile dislocations L s ) and an annihilation term gc.ρ which takes into account the dynamic recovery during deformation.

K L with L b s t s s s s s            t c D 1 g -     1 1   (4)
Only self hardening a ss and latent hardening a st are considered here. The physical meaning of all the parameters set is detailed in [START_REF] Thébault | Etudes expérimentale et numérique des mécanismes de déformation et de recristallisation de l'UDIMET 720 lors du forgeage[END_REF]. D is the initial grain size assuming that a twin boundary is a classical boundary and L s is the mean free path and K is a material parameter.

At the forging temperature, material parameters b and a st were determined from the literature data for UDIMET 720 (Thébault et al. [START_REF] Thébault | Polycrystalline modelling of Udimet 720 forging[END_REF]). Parameters m and D were determined by inverse method from experimental tests. The τ 0, gc, ρ0, K and γ 0 parameters were identified by the Gauss-Newton algorithm [START_REF] Nocedal | Numerical Optimization[END_REF]. The material parameters used in this study are given by Table 2.

At each Gauss point of the meshing, for a given applied strain, the polycrystalline model gives the local stress, strain, lattice rotation fields and the dislocation density. The misorientation between two points of a same grain corresponds to deformation sub-grain boundaries.

Recrystallization model: CAReX

The above FE polycrystalline model does not contain any recrystallization phenomenon. To describe nucleation and grain growth, a second model is implemented in a 3D cellular automaton code and coupled to the deformation model. The recrystallization model uses the dislocation density computed by the deformation model. After each step of deformation, the FE polycrystalline model produces local gradient of dislocation density and intragranular crystallographic reorientations. It gives at each point of the FE meshing the total local dislocation density ρ. These data are implemented at each site of the cellular automaton.

Nucleation occurs at high stored energy sites. The density of stored energy is given by:

  t t stored b E   2 2 1 (5)
For dynamic recrystallization, the classical criterion, based on stored energy, gives a too large number of nuclei. In this paper a new criterion resting on nucleation probability Pn is proposed:

P P n n t t n                  exp 1 (6)
where ρn is a dislocation density parameter determined from the deformation model and n is a material parameter. ρn and n are identified (see Table 2). As soon as the criterion given in equation ( 6) is achieved, i.e the nucleation probability Pn is higher than a random number P, the site (or element) will obtain a minimal dislocation density ρ0 = 1.2 10 4 mm -2 and the site will keep its orientation. To grow, a nucleus or an already recrystallized site must be present. This condition is satisfied when the dislocation density is lower than a threshold value ρg (its identified value is given in Table 2). This value is larger than ρ0: a site which was recrystallized at N-1 cycle and submitted to low deformation at cycle N, can grow during the recrystallization at cycle N.

The velocity of the grain boundary between s and s' sites depends on the mobility M ss' of the interface. The growth rate is given by:

E M V SS    ' (7)
The boundary mobility between the site s and its neighbour s' is determined by:

                           3 ' ' 10 exp 1 SS m SS M M  (8)
The driving force ∆E corresponds to the variation in the system energy, between the initial and final configuration. The driving force consists of two terms:

stored gb E E E      (9)
where ∆E stored is the variation in stored energy obtained from the density dislocations and ∆E gb is the variation in boundary energy associated with the change of orientation associated to the migration of the grain boundaries:

668
where d ss′ is the distance between a site and its neighbors and γ ss′ is obtained from Read and Shockley equation: When the driving force is positive, the site retains its initial configuration (orientation and stored energy). When the driving force is negative, the probability of reorientation can be computed. This probability is the distance d traveled by the grain boundary within a cell d0 over an increment of time tinc:

                       * ' * ' * ' * ' ' ln 1           
P d t E M d Vt d d P inc SS inc SS        0 ' 0 0 ' ( 12
)
For each site, this probability is calculated with the six possible reorientations. The reorientation with the highest probability is chosen. The microstructure is updated once all probabilities have been calculated, and only those sites with probabilities over a random number are effectively reoriented. The Equation 12 parameters are determined by two methods which correspond to the deformation model results (ρn and ρg), and to the literature (Mm and γm) [START_REF] Thébault | Polycrystalline modelling of Udimet 720 forging[END_REF], as shown in Table 2.

Aggregate construction and boundary conditions

Two three-dimensional aggregate types are used in this study.

The first one is a cubic aggregate with random orientations (Euler's angles) determined from the equation ( 13):

    1 2 11 02 02 random , random , a cos random ,                  ( 13
)
This aggregate is composed of 64,000 elements and 1,000 grains. It gives the best trade off between the computation time and the computing storage. A simulated compression test up to E ≈ 0.70 can be performed in less than ten hours with a parallelization on eight processors. This aggregate is used in parameter identification. The second 3D aggregate is a Representative Elementary Volume (REV) of the material microstructure. It was obtained using the Rey et al.

procedure [START_REF] Rey | Grain boundaries and crystalline plasticity[END_REF]. The microstructure was obtained from EBSD mapping. Figure 7 shows a screenshot of these aggregates taken from the FE code CAST3M®. The objective of these aggregates is to construct a model of the forging process. It is possible to consider the forging as a simple compression test. A previous study led by Schwartz et al. [20] allows determining the best boundary conditions to apply in order to realize compression test (Figure 8).

Figure 8: Boundary conditions of the aggregates

The REV is composed of thirty layers of grains, with a 10 µm step between two layers. The total dimensions of this aggregate are 600x600x300 µm 3 divided into 75x75x30 elements with a total of 168,750 elements. The aggregate used contains 3,189 grains.

Modelling results

This coupled approach corresponds to a macroscopic dynamic recrystallization where deformation portion is ensured by the polycrystalline model and the recrystallization portion by the recrystallization model described above. For such a coupling, a coherent determination of a characteristic time for both models is

0 3 . 3 . 3 . 3 . 0 2 . 2 . 2 . 2 . 1 . 0 1 . ) ( ) (             H G F E D C B A BFCG AEDH u u u u u u u u L u u A D C B G b (nm) a st 𝜐 E (GPa) m D (µm) 𝜏 0 (MPa) gc (nm) ρ0 (mm -2 ) K γ (s -1 ) Mm (m 4. J -1 .s -1 ) γ m (J.m -2 ) ρ g (mm -2 ) ρ n (mm -2
) n γmatrix 0.26 8e -3 0.3 105 0.18 44 5 30 1.0e 7 21 3.3e -4 2e -10 0.05 1.3e 8 2.5e 8 7.5

1 2 3 O E F H 1 669
required. In the present case, for each simulated cycle, the deformation and recrystallization times (tinc) are similar.

The simulation results highlight three features: the comparison between experimental and numerical mechanical behaviour for SP process, the analysis of the dislocation density evolution during the modelling of SP process and the application for the multipass processes considering interpass effects. Only results obtained from the actual aggregate are presented, either on mechanical behaviour or on dislocation density evolutions.

Mechanical behaviour

Seventeen deformation-recrystallization cycles are performed up to a 0.70 final deformation; the first step corresponds to a 0.075 deformation increment, the others are similar with a 0.025 increment. It should be noted that the identification of the material parameters of the polycrystalline model were determined during the first deformation step, since it was previously proved that no dynamic recrystallization occurred for a deformation less than 0.06 [START_REF] Jaeger | INCONEL 718 recrystallization in the delta supersolvus domain[END_REF]. Experimental and numerical stress-strain curves are given in Figure 9. Ignoring the initial peak stress, the estimated stress is in good agreement with experimental results. The stress level, obtained during the seventeen numerical steps, oscillates around the experimental stress curve up to a 0.70 deformation.

Dislocation density evolution in the SP process

The distribution of the dislocation densities in the actual aggregate meshing is given in Figure 10. The four analysed steps correspond to the end of the first deformation step (ε1 ≈ 0.075), to the ends of the fifth step (ε5 ≈ 0.19), of the twelfth step (ε12 ≈ 0.43) and of the seventeenth step (ε17 ≈ 0.70).

Figure 10: Dislocation density distribution evolution obtained from the actual aggregate modelling the SP process

The probability curves show a wavy background and some major peaks located at the end of each step corresponding to a large amount of sites presenting large dislocation densities. The criterion used for recrystallization initiation rests on a critical dislocation density ρn = 2.5*10 8 mm -2 . For ε1, the major peak corresponds to a 2*10 8 mm -2 dislocation density and according to only equation ( 6), 100% of sites are potential nuclei.

For ε5 ≈ 0.19, the highest peak corresponds to the maximum stress of the stress-strain curve. This peak means that 30-40% of the sites present an average density dislocation ρ5 ~ 3*10 8 mm -2 with a start of recrystallization on some sites. Each small peak corresponds to sites having experienced a former recrystallization during the steps 1 to 4.

The step ε12 ≈ 0.40 corresponds to the end of the MPD and MPQ first pass (see Figure 4). The curve ε12, presents a constant background ending by one peak corresponding to a large amount of sites (30%) with an average value ρ12 ~ 3.5*10 8 mm -2 . It should be noted that ρ12 > ρ5. For SP process, the step ε17 is similar to the distribution curve ε12. After recrystallization, the local strain of the sites presenting large ρ values decreased. These results confirm the tendency for deformed sites to disappear in favour of recrystallized sites.

Multipass process

For the three processes, simulations of the evolution of the microstructures (Figure 11) and dislocation density distributions (Figure 12) are given for 5 steps: ε1 = 0.075, ε5 = 0.19, ε11 = 0.40, ε12 = 0.43 and ε17 = 0.70. Each step corresponds to one deformation followed by one recrystallization and is depending on results obtained from the former steps. The orientation maps are represented by a simplified microstructure within the initial configuration. In such a representation, a grain is sketched by a cube (or for 2D maps, represented by a square). Each cube contains 64 sites whereas each square contains 16 computation sites (or Gauss points). A nucleus is represented by a small square with similar or different crystallographic orientation according to the nucleation criterion. All maps correspond to the central layers of the aggregate. after the seventeenth deformation with the MPQ process and (i) after the seventeenth deformation with the MPD process These figures highlight the differences between the microstructures as well as the differences between the dislocation densities obtained after the SP, MPQ and MPD processes. As the nuclei orientation has been assumed to be the same as the parent grain, no distinct nucleus can be seen on the orientation maps.

The three processes having a common deformation path up to ε11 = 0.40, the microstructure evolutions are similar for ε1 to ε11 (Figure 11(a) to Figure 11(c)). For ε12 to ε17, large differences are obtained between SP process (0.25s interpass time), MPD process (10s interpass time) and MPQ process (15mn interpass time), as it can be seen in Figure 11(d), to Figure 11(i). For the first deformation step ε1, a gradient of orientation (represented by a gradient of colour) appears in some grains. See, for example, the grain at the bottom of Figure 11 671 deformation in most of the grains. New orientation of the nuclei (small squares within the grains) can be observed.

For the 11 th deformation step (ε11), the gradient of orientations inside grains are higher than previously (Figure 11(c)). A site with a new orientation can be seen at the bottom of the figure (pointed by black arrow); no nucleus being visible, the new site orientation results of the growth of a grain lying in the upper or lower layer. This result emphasises the 3D interactions between the grains of the aggregate.

The deformed grains develop a <110> fiber texture parallel to the compression axis. Nevertheless, as nucleation of annealing twins is not yet introduced in the model, the randomization of grain orientation associated to the evolution of recrystallized grains is not observed [START_REF] Humphreys | Recrystallization and related annealing phenomena[END_REF].

Let us come back to the SP process. Figures 11(c), 11(d) corresponding to ε11 and ε12 respectively show little evolution of the orientation maps. At the 17 th step (ε17) a duplex microstructure is pointed out. This microstructure is composed of grains with a large gradient of orientation (deformed grains) and grains without gradient (recrystallized grains). This result agrees with the experimental microstructure observed in Figure 6(b).

Figures 11(e) and 11(h) reveal the evolution of the microstructure due to the MPQ process for ε12 and ε17. Compared to Figure 11(c), a dwell time of fifteen minutes induces post-dynamic recrystallization and grain growth. Small differences are observed between ε12 and ε17. These numerical results point out that this MPQ process lead to a re-initialization of the microstructure before the second pass. For ε12 and ε17, the computed microstructures being close.

These numerical results point out that this MPQ process led to a re-initialization of the microstructure before the second pass. For ε12 and ε17, the computed microstructures being close. The dwell period has a significant effect on the distribution of dislocation densities of MPD and MPQ processes (Figure 12).

For MPQ process, simulations reveal a large decrease of dislocation density, when compared to SP process. At ε12 step, the sites corresponding the background can grow as soon as the growth criteria ρ g = 1.3*10 8 mm -2 is achieved.

For ε17, most of the sites can grow. For the MPQ process at ε12, the dislocation density distribution is similar to the one obtained after the ε5 step with the SP process. This confirms that the deformation level is low during the second pass of the MPQ process and the sample is partly recrystallized. As a conclusion, the simulation of the MPQ process agrees with experimental observations.

For the MPD process, the evolution of the crystallographic orientation is given in Figure 11(f) and in Figure 11(i). The microstructure obtained for ε12 step, is intermediate between the SP and MPQ microstructures. During the 10s interpass time, an important boundary migration occurs due to the meta-dynamic recrystallization phenomenon. According to Figure 12, almost 20% of the sites have a dislocation density above the growth criteria ρ g : the structure is not fully recrystallized.

As a consequence, after the twelfth deformation, the dislocation density distribution with the MPD process is intermediate between the SP and the MPQ processed.

During the short interpass time, the recrystallized sites can grow, but some deformed areas remain in the microstructure. These features explain the different work hardening rates observed with the three thermo-mechanical cycles. A new numerical simulation, compared with experimental results, is used to predict the mechanical behaviour and microstructure evolution of INCONEL 718 during multi-pass hot forging. This modelling is achieved by a coupling between a crystalline plasticity model and a recrystallization model. The numerical coupling enables the linking of the 3D interactions between grains of the aggregate. The numerical results are in good agreement with the experience. The approach enables to determining of the part of the different mechanisms (plasticity and recrystallization) at the origin of the different microstructure evolutions as well as the mechanical behaviour during complex multipass processes.

Figure 1 :

 1 Figure 1: Forging domain of INCONEL 718

Figure 2 :Figure 3 :

 23 Figure 2: Optical micrographs of the microstructures (the compression axis is vertical), (a) as received material and (b) initial microstructure obtained after 15mn dwell time Isothermal compression tests were performed on a hydraulic MTS press with a maximum loading of 50 T. This press was coupled to a lamp heating furnace allowing a maximum temperature of 1200°C. The temperature control was realized with a thermocouple welded on the sample surface. Starting from the microstructure of the Figure 2(b), an isothermal compression was performed at a 10 -1 s -1constant deformation rate. Three different deformation paths were investigated: a single pass process (SP), a multipass process with an intermediate dwell time of ten seconds (MPD), and a multipass process containing an intermediate quench, a reheating and a fifteen minute holding period before the second compression (MPQ). These thermo-mechanical paths are sketched in Figure3at T > T δsolvus .

Figure 4 :

 4 Figure 4: INCONEL 718, compression curve for a 10 -1 s -1

Figure 5 :

 5 Figure 5: Evolution of hardening rate depending of the true strain in INCONEL 718 (10 -1 s -1 )

Figure 6 :

 6 Figure 6: Material orientations maps from: (a) initial state, (b) SP process, (c) MPQ process and (d) MPD process The first microstructure (Figure 6(a)) was obtained before forging and corresponded to the initial microstructure (Figure 2(b)). The other three microstructures (Figures 6(b), 6(c), 6(d)) corresponded to the end of each process, i.e, at a total E = 0.70. The different microstructure features are given in Table1.

Figure 7 :

 7 Figure 7: Microstructures of the aggregates, (a) the random aggregate and (b) the actual aggregate

Figure 9 :

 9 Figure 9: Comparison between experimental and numerical mechanical behaviour obtained from the actual aggregate modelling the SP process

Figure 11 :

 11 Figure 11: Orientation maps from the modelling, (a) after the first deformation, (b) after the fifth deformation, (c) after the eleventh deformation, (d) after the twelfth deformation with the SP process, (e) after the twelfth deformation with the MPQ process, (f) after the twelfth deformation with the MPD process, (g) after the seventeenth deformation with the SP process, (h) after the seventeenth deformation with the MPQ process and (i) after the seventeenth deformation with the MPD process

  (a) (indicated by the black arrow). For the 5 th deformation step (ε5) corresponding to Figure 11(b), growth of few grains are observed for the first time. Moreover, the gradient of orientation increases with applied Superalloys 2012 : 12th international Symposium on superalloys Edited by Eric S. Huron, Roger C. Reed, Mark C. Hardy, Michael J. Mills, Rick E. Montero, Pedro D. Portella, Jack Telesman TMS (The Mineral, Metals & Materials society) 2012

Figure 12 : 3 /

 123 Figure 12: Comparison of the evolution of the dislocation density distribution at different stages of the SP, MPQ and MPD processes. Simulations are performed via the actual aggregate Conclusions An experimental study on INCONEL 718 allows stressstrain curves to be determined during three different hot forging processes. The microstructure evolution has been studied during these processes and linked with mechanical behaviour up to a 0.70 deformation: 1/ The single pass process exhibits the lowest final stress level and an heterogeneous grain size distribution. 2/ The multi-pass quench process shows the highest final stress level and an heterogeneous grain size distribution owing to the microstructure re-initialization during the interpass. 3/ The multi-pass dwell process leads to an intermediate final stress level and an homogeneous grain size distribution due to short interpass time keeping nuclei from the first pass.

Table 1 :

 1 Quantitative results obtained from EBSD data

		Grain Size	Subboundary	Twin
		(µm)	ratio (%)	ratio (%)
	As received	15	25	30
	Initial (Dwell)	95	5	55
	SP process	25	25	15
	MPQ process	55	35	15
	MPD process	20	10	25

Table 2 :

 2 Material parameters identified in this study taken from CristalECP and CAReX

	Superalloys 2012 : 12th international Symposium on superalloys

Edited by Eric S. Huron, Roger C. Reed, Mark C. Hardy, Michael J. Mills, Rick E. Montero, Pedro D. Portella, Jack Telesman TMS (The Mineral, Metals & Materials society) 2012

Acknowledgments

This work was supported by Forges de Bologne which has provided the material. The authors thank Nicolas Roubier from MSSMat/ECP for his many contributions during this effort.


J B Minne

L Desgranges
email: blionel.desgranges@cea.fr

V Optasanu
email: cvirgil.optasanu@u-bourgogne.fr

N Largenton
email: nathalie.largenton@edf.fr

L Raceanu
email: elaura.raceanu@u-bourgogne.fr

T Montesin
email: ftony.montesin@u-bourgogne.fr

Specific aspects of internal corrosion of nuclear clad made of Zircaloy

Keywords: Stress-diffusion coupling, Zircaloy oxidation, clad corrosion

In PWR, the Zircaloy based clad is the first safety barrier of the fuel rod, it must prevent the dispersion of the radioactive elements, which are formed by fission inside the UO 2 pellets filling the clad. We focus here on internal corrosion that occurs when the clad is in tight contact with the UO 2 pellet. In this situation, with temperature of 400 °C on the internal surface of the clad, a layer of oxidised Zircaloy is formed with a thickness ranging from 5 to 15 µm. In this paper, we will underline the specific behaviour of this internal corrosion layer compared to wet corrosion of Zircaloy. Simulations will underline the differences of stress field and their influences on corresponding dissolved oxygen profiles. The reasons for these differences will be discussed as function of the mechanical state at inner surface of the clad which is highly compressed. Differences between mechanical conditions generated by an inner or outer corrosion of the clad are studied and their influences on the diffusion phenomena are highlighted.

Introduction

Understanding the mechanisms of corrosion of zirconium alloys in the nuclear industry is an important issue because it is one of the parameters that affect the service life of fuel rods. It is generally accepted that, as long as the growing oxide layer remains protective, the diffusion of oxygen vacancies is the rate-limiting mechanism for zirconium oxidation; the oxide growth occurs at the metal-oxide interface and leads to a parabolic kinetic law. Several types of corrosion have been studied in order to represent the different oxidizing environments of the clad and lead to other mechanisms. Uniform waterside corrosion in reactor operating conditions (from 280 to 330 °C) is one of the most widely reviewed [1,2].

Internal corrosion at inner clad surface is another specific zirconium alloy oxidation in operating conditions which has been less studied. However, the evolution of this layer is of importance since the formation of an oxide layer leads to improve transfer of both heat and mechanical contact forces across the pellet/clad gap. As the burn-up increases, the gap between the UO 2 pellet and the clad decreases progressively to the closure. A tight pellet/clad contact then occurs, which generally corresponds to the initiation of the internal corrosion layer growth [3,4]. Moreover, this contact is related to a quick increase of the hoop stress in the clad, and also to a pellet/clad contact pressure.

Since the importance of stress generation in the oxide and the metal was often highlighted for a better understanding of the mechanisms involved in waterside corrosion [5], this paper underlines the influences of mechanical conditions generated by the inner and outer clad corrosion on the diffusion phenomena. We present first specific aspects of the growth kinetics and of the stress state of internal corrosion layers compared to waterside corrosion. Then, the use of a model which couples stress and oxygen diffusion enables us to appreciate the feedback of mechanical effects on the dissolved oxygen profiles.

Waterside corrosion vs. internal corrosion of Zircaloy clad

In water, for PWR conditions, clad oxidation occurs in two stages, referred to as pre-and posttransition regimes. In pre-transition regime, a black oxide film develops. It is characterized by a dense and protective layer of mixed tetragonal and monoclinic zirconia, the latter is the stable phase in this temperature range. Kinetics is rather closer to a cubic or quartic law than a parabolic one. The thickness of the layer controls the oxidation rate and is characterized by high compression stresses (1-2 GPa) near the oxide/metal interface.

After 2-3 µm, the layer degrades, cracks and becomes porous and a tetragonal-to-monoclinic transformation of zirconia occurs. The kinetic law switches to a pseudo-linear post-transition regime, which is constituted by the cyclic succession of several cubic regimes and accelerates the corrosion rate. The impact of the stress level in the zirconia appears here to be of great importance. The transformation from metal to oxide produces a high volume change, as the so-called Pilling-Bedworth ratio is 1.56 and leads to high residual stress in both the metal and the oxide layer. These stresses are responsible for these cracks in the waterside oxide layer and are often correlated to phase transformation in the zirconia [START_REF] Gibert | [END_REF].

. In the case of corrosion of the inner clad surface, conditions of oxide formation and growth kinetics are totally different, in comparison with waterside corrosion (see Fig. 1). Thus, internal zirconia comes as a fully tetragonal and dense layer, though this phase is stable at temperatures much higher than those of internal clad surface (340-400 °C) and at higher pressure. However, tetragonal zirconia could be induced at these temperatures by high stresses (> 4 GPa), or is stabilized by the addition of others elements (Y, Mg), or is produced by fission-induced phase transformation [START_REF] Gibert | [END_REF]. In this last case, the pressure of the high concentrations of interstitials (fission products) produced by elastic collisions would cause a denser material, thus a more symmetrical structure of zirconia.

It is generally reported that internal oxidation is observed for closed radial gap between the pellet and the clad, in operating conditions [3,4]. As the oxide layer grows first discontinuously in small patches, we can assume that the establishing of a local tight pellet/clad contact is needed to enable the oxygen diffusion from pellet to clad, which coincides with the mechanical conditioning of the rod. The growth rate is composed of 3 steps. The first 5 µm oxide thicknesses grow very quickly and correspond to the penetration length of fission products from pellet to clad. Moreover, it is well established that diffusion under radiation is drastically enhanced due to the ballistic migration of atoms [START_REF] Bérerd | [END_REF].

The layer grows then slowly up to 10-12 µm. Diffusion could be governed by a diffusion ratelimiting mechanism, like the "pre-transition" regime of waterside corrosion. An estimation of the parabolic diffusion rate constant leads to a value of the order of magnitude of 3.10 -19 m²/s, which is coherent with common values for this range of temperature [9]. At high burn-up, this thickness does not increase significantly, it does not exceed 15 µm and non-connected porosity is only observed near the oxide/metal interface after 3 cycles of irradiation. No residual gap is observed, which is related to a chemical interaction and to a good mechanical hang-up between the oxide layer and the pellet.

The development of a wrinkled surface can be observed at the interface between the pellet and the corrosion layer, linked with the high burn-up structure at the edge of the pellet. These wrinkles can be strongly marked with grooves and folds; the layer thickness can periodically double, which favour the good cohesion between the clad and the pellet.

Since the stresses were proved to have a non negligible feedback effect on the diffusion rate of oxygen in the metal [10], it would be interesting to know how the chemical diffusion process is affected by the stress field generated by the bonding and by the pressure of the pellet. For this purpose, we introduce the model used for our FEM simulations and the adopted methodology.

Model description of zirconium oxidation

The background theory which gives the influence of mechanical stress on the chemical diffusion was widely exposed in [10][11][12], and is based on previous works [13]. We recall here the main point used in our simulations. In the volume, the matter flux relative to the diffusing species (oxygen in our case), is given by a modified Fick's law [14] in which the diffusion coefficient D depends on the mechanical stress:

1 2 3 4 5 6 - = RT M D D ij ij σ η 0 0 1 (1)
where M 0 is the molar mass of the zirconium, η ij are the chemical expansion tensor coefficients, c is the oxygen concentration which is dissolved in zirconium, σ is the mechanical stress, T is the temperature and R is the universal gas constant. η ij are defined as follows:

c ox ch ij ij ∂ ∂ = . ε ρ η (2)
where ρ is the density of the zirconium and ox ch ij . ε are the chemical strain tensor coefficients, which are assumed to be isotropic in the metal [17]. For zirconia, chemical expansion coefficients are evaluated according to the chemical strain coefficients determined elsewhere [15]. Table 1 shows the parameters used for the model. The solids behaviours are elastic and isotropic. 3.55 × 10 -5 95 0.31 Table 1: Parameters used for the model.

We use here a strong coupling between concentration and mechanical stress. The stress locally modifies the diffusion coefficient and the oxygen concentration induces modifications in chemical strains, which produces additional mechanical stress. We assume that only oxygen is able to diffuse and produces one single phase change, that the chemical reaction of this phase change is always at the thermodynamic equilibrium, that the system is isothermal and that there is a perfect adhesion between the oxide and the metal, i.e., that there is a mechanical continuity at the oxide/metal interface.

Numerical simulation of oxygen dissolution into the metal

The numerical simulations were performed with the finite element CAST3M code applied to a 2 dimension axi-symmetric mesh which contains 4 domains (see Fig. 2): the edge of the pellet (500 µm length), the internal zirconia layer (5 and 15 µm thickness), the clad (570 µm length) and the external zirconia layer (2 µm thickness, which is the effective mean thickness that is involved in the pre-transition regime of external corrosion, and 6 µm thickness). The solubility limit of oxygen in the clad is set to 464

/ = ox met c
kg/m 3 [16]. The UO 2 pellet is under internal pressure p int and the clad is under external pressure p ext with p int = p ext = 155 bars.

We simulated the diffusion of oxygen in the clad for several thicknesses of internal and external zirconia layers during 3×10 7 s, which corresponds to a 10 µm penetration length of oxygen in the metal. No mobility of the interfaces was hence simulated. We focus here on the oxygen diffusion in the α-Zr(O) phase and on the bulk pressure in the clad, defined as (σ r +σ θ +σ z )/3. Comparisons between stress-diffusion coupling and standard Fick's diffusion are made on inner and outer sides of the clad for several zirconia thicknesses.

Results

Results are given in figures 3 and 4. The effect of the stress-diffusion coupling model is remarkable compared to the Fick's diffusion: the oxygen concentration can locally double (see Fig. 3). No significant differences between the inner and outer sides of the clad are noticed for both oxygen concentration profiles and bulk pressure.

As the adopted mechanical model is elastic, calculated compressive stresses in the α-Zr(O) phase are very high due to the isotropic chemical expansion coefficient. However, they are in the same order of magnitude of measured hoop stresses for Zircaloy-4 plate oxidized at 400°C in steam [START_REF] Godlewski | Proceedings of the 10 th International symposium on Zirconium in the Nuclear Industry[END_REF]. The bulk pressure in the clad, outside the α-Zr(O) phase, increases with the inner and outer oxide thickness, since the Young modulus of zirconia is greater than the clad's one, which tends to stiffen the clad.

Discussion

Our results show that, the more internal and external thickness increases, the more the bulk pressure increases in the clad and the less the oxygen diffuses in the metal. According to figure 4, the bulk pressure in the clad, outside the α-Zr(O) phase, increases with the inner and outer oxide thickness, since the Young modulus of zirconia is greater than the clad's one, which tends to stiffen the clad. As the oxide layer increases, on inner and outer clad surface, compressive stresses decreases, which is related to a decrease of the oxygen diffusion coefficient in the α-Zr(O) phase and explains the difference of concentration profiles. Actually, according to equation 1, increasing the tensile stress or decreasing the compressive stress allows the diffusion coefficient in the metal to decrease. Then, the feedback of the stress state on the corrosion kinetics lowers the rate diffusion of oxygen in the metal.

In service conditions, waterside corrosion thickness can reach a few dozen micrometers at high burn-up. However, only the last 2-3 µm oxide layer thickness in front of the zirconia/metal interface is involved in the rate-limiting mechanism for the clad oxidation. The rest of the layer is destabilized and degraded. This leads to the loss of protective properties of the zirconia layer and to stress relaxation in the clad. We can then conclude that the external penetration profile of oxygen does not change a lot during the lifetime of the rod even if oxidation kinetics is accelerated.

Unlike the outer zirconia, internal corrosion layer is formed under irradiation and fission products collisions. The stabilization of the tetragonal phase is then induced by radiation damage, and then no phase transformation can occur. For that case, the internal penetration profile of oxygen could evolve, as the oxide layer remains dense and protective.

VII Conclusion

The present work is a first approach of the modelling of the internal corrosion of nuclear clad. The phenomenology of the growths of internal and waterside corrosion, in service conditions, and the specific stress state of the pellet/clad system have been analysed in order to simulate the impact of mechanical stress on oxygen diffusion in the clad with the help of a stress-diffusion model. The main point of our results is that dissolution of oxygen in the clad can be interpreted in terms of external zirconia destabilization and internal zirconia stabilization. The future works will simulate the growth of zirconia layers in presence of mechanical stress.

Figure 1 :

 1 Figure 1: Comparison of internal [2] and waterside corrosion thickness [3] in PWR for UO 2 /MOX fuel and Zry-4/low Sn Zry-4 clad as function of the fuel rod burn-up.

Figure 2 :

 2 Figure 2: Geometry of the numerical model

Figure 3 :

 3 Figure 3: Oxygen concentration at the inner side of the clad. a) Fick's diffusion. b), c) and d) Stress-diffusion coupling for different values of thickness of internal and external zirconia.

Figure 4 :

 4 Figure 4: Bulk pressure (right) at the inner side of the clad. a) Fick's diffusion. b), c) and d) Stress-diffusion coupling.