Somlanare Romuald Kinda
email: skinda@univ-ouaga2.

Climatic shocks and food security: The role of foreign aid

Keywords: food prices vulnerability, foreign aid, food security, climatic shocks JEL classification: F35, Q17, Q18, Q54

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Introduction

According to several reports [START_REF] Christensen | Climate change 2007: The physical science basis[END_REF], greenhouse gases emissions are related to planet warming, and changes in precipitation. This could negatively affect developing and developed countries. There is increasing evidence that greenhouse gases have already begun to warm the planet [START_REF] Christensen | Climate change 2007: The physical science basis[END_REF]. This in turn will likely cause changes in precipitation patterns as warming trends continue [START_REF] Christensen | Climate change 2007: The physical science basis[END_REF]). That could have significant negative impacts in both developed and developing countries. For instance [START_REF] Mitchell | Adapting to Climate Change: Challenges and opportunities for the development community[END_REF] conclude that southern Africa will probably be 10-20 per cent drier than the previous 5 decades. Moreover climate change will negatively affect agricultural production and food security.

Several authors have analysed relationship between climate change and food security. [START_REF] Christensen | Climate change 2007: The physical science basis[END_REF] conclude that food production is highly vulnerable to weather conditions. [START_REF] Haile | Weather patterns, food security and humanitarian response in sub-Saharan Africa[END_REF] and [START_REF] Dilley | Natural Disaster Hotspots: A Global Risk Analysis[END_REF], and Ringler et al. (2010) confirm that food crises and malnutrition in Africa are related to climate events. In addition, [START_REF] Lee | Impact of Climate Change on Agricultural Production in Asian Countries: Evidence from Panel Study[END_REF] show that climate variability (such as high temperature and precipitation) raises agricultural production in Asian countries while high drop temperatures are detrimental. [START_REF] Badolo | Climatic shocks and food security in developing countries[END_REF] show that the impact of climatic variability on food security in developing countries is worsed in presence of civil conflicts or countries that are vulnerable to food price shocks.

Despite these previous studies, little is known about the role that foreign aid flow can play in dampening the effects of climate change. There are several reasons behind the interest in the effect of foreign aid. One of the main motivations is that foreign aid flows are a major source of revenue for developing countries (particularly least developing countries (LDCs)). They can constitute an important mechanism mitigating the effect of climatic shocks on food security because they allow receiving countries to mobilize or stabilize resources for the financing of agriculture production or food import. In the economic literature, several papers have analysed the importance of aid flows as shock absorbers in developing countries. Most of them focus on the effects of foreign aid on macroeconomic stabilizations and coping with various types of shocks. [START_REF] Collier | Aid, shocks, and growth[END_REF] show that aid flow mitigates the negative effect of commodity prices. [START_REF] Chauvet | Aid, Volatility, and Growth Again: When Aid Volatility Matters and When It Does Not[END_REF] and [START_REF] Guillaumont | Aid and Performance: A Reassessment[END_REF] find that aid is efficient for countries that face economic vulnerability-external shocks (trade, food prices shocks) and natural disasters. However, a limitation of these studies is that they do not analyse the potential role of foreign aid in mitigating the adverse effects of climatic shocks on food security.

This paper explores the role of aid in the mitigation of the effects of climatic variability in the recipient economies. Using panel data for 71 developing countries over the period 1970-2008, results indicate that foreign aid significantly dampens the effect of climatic shocks on food security. More interestingly, it appears that the marginal effect of aid is high for countries that are more vulnerable to food price shocks.

The plan of the paper is as follows. Section 2 contains a discussion of the literature review on the relationship between climatic shocks, foreign aid, and food security. Section 3 discusses the econometric method, and Section 4 presents empirical results. The last section is devoted to concluding remarks and implications.

2

Literature review

Determinants of food security

2.1.1.Traditionnal factors

According to the economic literature, many factors may affect food security in developing countries. [START_REF] Badolo | Climatic shocks and food security in developing countries[END_REF] group the main factors of food security into three approaches: production-based approach, market-based approach, and institutional failures.

The production-based approach assumes that food security in developing countries can be explained by a decline of food availability. In the economic literature, food availability decline is the result of population growth and the scarcity of natural resources and food security [START_REF] Malthus | Essai sur le principe de population[END_REF], [START_REF] Ehrlich | The Population Explosion[END_REF], [START_REF] Ophuls | Ecology and the Politics of Scarcity Revisited: The Unraveling of the American Dream[END_REF]).

Contrary to production-based approach, the market-based approach considers that famine is not explained by food supply or availability but due to food access. As a precursor of this approach, Sen (1983a) puts in light the concept of entitlements. He suggests that people have an entitlement to food that depends on personal endowments and exchange conditions. The endowments can be defined as the combination of all resources owned by people (for instance land, equipment, animals, knowledge, skills, revenues of employment). Sen (1983a) shows a shift in exchange conditions and employment can reduce the ability of people to acquire food and be a source of insecurity in developing countries. In addition, several authors analyse the link between economic performance and food insecurity. [START_REF] Wiesmann | Global Hunger Index: A Basis for Cross-Country Comparisons[END_REF] and [START_REF] Smith | Explaining child malnutrition in developing countries: a cross-country analysis[END_REF] show that national incomes contribute to food security because they: (a) strengthen countries' health environments and services and women's education by rising government budgets, (b) raise the ability of countries to buy food in international markets. In addition, through poverty reduction, [START_REF] Smith | Explaining child malnutrition in developing countries: a cross-country analysis[END_REF] suggest that economic growth increases food access and security for households.

Final, institutional failures such the implementation of inappropriate policies, the failure of governments, and civil conflicts can affect food delivery (Keen (1994) and Sen 2000). According to Sen (2000), democracy can help to prevent famines. In democracy, leaders are incited to implement policies to fight against food insecurity. Indeed, by bringing out information, free press can favour the implementation of warning systems and policies for food security. [START_REF] Dreze | Hunger and Public Action[END_REF] and [START_REF] Smith | Explaining child malnutrition in developing countries: a cross-country analysis[END_REF] show that democratic institutions are an important determinant of food security. By encouraging human rights (including right to food) and community participation, Haddad and Oshaug (998) and Haddad and Oshaug 1998 conclude that democracy reduces child malnutrition.

The importance of climatic shocks

In addition to traditional factors, climatic variability can impact food security in developing countries through several channels [START_REF] Badolo | Climatic shocks and food security in developing countries[END_REF] that are agricultural production, households incomes, food prices, economic resources and civil conflicts.

First, climatic variability can impact food security through agricultural production. In short term, rainfall variability and extreme events (such droughts and floods) reduce farm yields and decrease agricultural income, and household, and national food availability. In the longer term, it favours underinvestment, agricultural stagnation and rural poverty in countries depending on rainfed agriculture [START_REF] Kydd | Agricultural development and pro-poor economic growth in sub-Saharan Africa: potential and policy[END_REF].

Second, rainfall variability can affect food security through household incomes. By decreasing agriculture production, it negatively impacts household income coming from the agriculture sector. According to some authors (for instance International Labour Organization , 2007), 36 per cent (66 per cent) of the world's (sub-Saharan Africa respectively) working population livelihoods depends on agriculture production. In addition to this sector, climatic variability can impact rural labour markets through incomes reduction and a decrease of demand for goods and services in affected areas. Therefore climatic shocks increase the vulnerability of households to food insecurity. For instance, in Africa, climatic variability increases food insecurity through adverse effects on farm revenues [START_REF] Nhemachena | Measuring the Economic Impact of Climate Change on African Agricultural Production Systems[END_REF]. Furthermore, climatic shocks can increase food insecurity through negative effect on economic growth. For instance, in developing countries [START_REF] Dell | Climate Change and Economic Growth: Evidence from the Last Half Century[END_REF] shows that climatic shocks have severe impact on economic growth through a decrease in total productivity, agricultural yields and investments. This reduce the ability of countries to buy food on international markets, to increase investments in infrastructures, services and technology that support food production and to finance public goods such health and education services.

Third, climatic variability affect food security through food prices. By declining food availability in markets, climatic shocks can increase food prices and reduce food accessibility. Using theoretical models, Ringler et al. ( 2010) conclude that climatic variability favours higher food prices, and childhood malnutrition in sub-Saharan Africa.

Fourth, climatic variability can be a factor of food insecurity by increasing the risk of civil conflicts. Indeed, over the long term, [START_REF] Buhaug | Implications of Climate Change for Armed Conflict[END_REF] assume that climate shocks will probably lead to substantial scarcity and variability of renewable resources. This will increase the risk of civil conflicts (Miguel et al. 2004) and favour food insecurity.

Foreign aid as a resilience factor to climatic shocks

This subsection is devoted to the analysis of ways to deal with climatic shocks by investigating the role of foreign aid as a risk coping mechanism.

The stabilizing effect of aid

Several arguments may justify the stabilizing effect of aid. First, we may consider that foreign aid has a stabilizing effect on economic growth and resources in countries affected by climatic vulnerability. A characteristic of many developing countries is that they have a disproportionate part of their GDP in climate sensitive sectors such as agriculture. Several authors such as [START_REF] Dell | Climate Change and Economic Growth: Evidence from the Last Half Century[END_REF] and [START_REF] Mendelsohn | The distributional impact of climate change on rich and poor countries[END_REF] have shown that climatic shocks have large and negative effects on economic growth in poor countries through reduction in agricultural production and exports [START_REF] Jones | Climate shocks and exports[END_REF]. Foreign aid can be countercyclical and, in presence of climatic shocks (and then revenue shocks), can reduce public spending volatility.

Second, the economic literature [START_REF] Chauvet | Aid, Volatility, and Growth Again: When Aid Volatility Matters and When It Does Not[END_REF][START_REF] Guillaumont | Aid and Performance: A Reassessment[END_REF] on aid and economic growth has shown that foreign aid is more efficient in vulnerable countries by allowing economic growth to become more stable. By stabilizing the economic resources (potentially affected by climatic shocks), foreign aid contributes to reducing the instability of public investment necessary to finance public services (health, education, infrastructures) that support food and agricultural production.

Third, foreign aid can mitigate the effect of climatic shocks on food availability in the recipient countries through the vulnerability of countries to food price shocks. Indeed, [START_REF] Combes | Are Foreign Aid and Remittances a Hedge against Food Price Shocks in Developing Countries?[END_REF] show that foreign aid has a dampening effect on the impact of food price shocks on both level and volatility of household consumption. Moreover climatic variability favours food insecurity through food price shocks vulnerability [START_REF] Badolo | Climatic shocks and food security in developing countries[END_REF]. Therefore, the dampening effect of aid may be high for countries that are more vulnerable to food price shocks. Moreover, aid may favour the implementation of economic reforms (Collier 1997;Morrissey 2004) reducing the high level and volatility of food prices in developing countries. Indeed better economic and trade reforms reduce the dispersion of food prices in non-integrated markets [START_REF] Aker | Information from Markets Near and Far: Mobile Phones and Agricultural Markets in Niger[END_REF][START_REF] Araujo | Devaluation and Cattle Market Integration in Burkina Faso[END_REF].

Aid and capacity building

Beyond the importance of more financial resources (foreign aid), developing countries need technical resources to strengthen institutions. Indeed, one problem that faces developing countries is the lack of capacity building to withstand climate change. According to UNDP (2009), capacity is 'the ability of individuals, institutions and societies to perform functions, solve problems, and set and achieve objectives in a sustainable manner'.

By providing education (skills), experience, and knowledge, external technical experts allow many developing countries to implement mitigation and adaptation policies against climate change.

According to [START_REF] Nelson | Investment in Humans, Technological Diffusion, and Economic Growth[END_REF], education increases people's ability to understand information, and adopt new behaviours. It allows people (agricultural households for example) to adopt new technologies such as drought and heat tolerance crop varieties. Foreign aid may facilitate the implementation of climate change adaptation policies in agricultural science and technology, water conservation, risk management, and capacity building (Huang 2013). Moreover, foreign aid might finance investments that increase the flexibility of the economy. By financing education, individuals can become more adaptable and less dependent on climate sensitive sectors (agriculture for example).

In other words, the capacity building efforts of foreign aid may potentially contribute to reduce the negative effect of climate change by improving the quality of human resources (through education), the transfer of new technologies and good practices from other countries (through international cooperation, networking, and information sharing), and by increasing income.

Empirical analysis

Empirical model

The aim of the paper is to analyse the role of aid in the mitigation of the effects of climatic shocks on food security. For this purpose, we follow the following steps:

First, we test the effect of climatic shocks on food security. The baseline equation is:

𝑌 𝑖,𝑡 = 𝛼 𝑖 + 𝛽 1 𝐶𝑆 𝑖,𝑡 + 𝜔𝑋 𝑖,𝑡 + 𝛾 𝑡 + 𝜀 𝑖,𝑡 (1)
With 𝐶𝑆 𝑖,𝑡 the variable of climatic variability (log) in a country i at the period t; is the error term, represents time fixed effect, and country fixed effects. 𝑋 𝑖,𝑡 is the matrix of control variables. They are population growth, income per capita (log), democratic institutions, arable land (log), cereal production land (log) and real effective exchange rate (log).

is the food

t i,  t  i  t i Y ,
supply. Because food security is a multidimensional concept, we use the proportion of undernourished people for robustness checks.

Second, we analyse the mitigating effect of Aid. To test this hypothesis, we adopt the following specification (equation 2): 𝑌 𝑖,𝑡 = 𝛼 𝑖 + 𝛽 1 𝐶𝑆 𝑖,𝑡 + 𝛽 2 𝐶𝑆 𝑖,𝑡 * 𝐴𝑖𝑑 𝑖,𝑡 + 𝛽 3 𝐴𝑖𝑑 𝑖,𝑡 + 𝜔𝑋 𝑖,𝑡 + 𝛾 𝑡 + 𝜀 𝑖,𝑡

With 𝐴𝑖𝑑 𝑖,𝑡 the amount of official development assistance (ODA) per capita. If this hypothesis is proven to be true, the coefficient associated to the interactive term (𝛽 2 ) will be positive. The hypotheses tested are : 𝛽 1 <0 𝛽 2 >0 and 𝛽 3 >=0

Estimation strategy

In order to estimate our models (equations ( 1), ( 2)), we may apply ordinary least square (OLS), fixed effects (FE) or random effect (RE). However OLS don't take into account unobserved heterogeneity of countries. Therefore it is adequate to apply fixed effects (FE) or random effect (RE). In order to choose the adequate specific effect, we will use the Hausman test.

Data sources and description of variables

Data cover the period from 1960 to 2008 for 71 developing countries and are compiled in fiveyear averages (1960-1964, 1965-1969…). The data on foreign aid, population growth, income per capita, and proportion of undernourished people are from World Bank Statistics (World Bank Group 2011). Those on democratic institutions, rainfall, and food supply are from respectively Polity IV ( 2011), Cerdi and Food and Agriculture Organization (2011).

Food security

According to the United Nations Development Programme (UNDP 1994), food security can be defined as "a situation that exists when all people at all times have physical, social and economic access to sufficient, safe and nutritious food that meets their dietary needs and food preferences for an active and healthy life". In the literature several indicators have be used because food security is a multidimensional and flexible concept. Similarly to [START_REF] Badolo | Climatic shocks and food security in developing countries[END_REF], we use two complementary indicators2 : proportion of undernourished people and food supply per capita. Food supply per capita is a simple average of food supplies of main cereals consumed in developing countries. There are maize, rice, sorghum, millet, and wheat, soybeans and sugar.

Rainfall variability

Climatic variability is measured by the standard deviation of the growth rate of rainfall, which is frequently used in economic literature. It is computed as the five-year rolling standard deviation of the growth rate of rainfall series.

Baseline results

Columns 1 and 2 of table 1 show the results of the effects of rainfall variability on food insecurity with fixed effects (FE) and random effects (RE) respectively. The Hausman test shows that the fixed effect model is more appropriate than the random effect model. Therefore we use FE. We find that rainfall variability reduces food supply in developing countries. This result is similar to [START_REF] Badolo | Climatic shocks and food security in developing countries[END_REF]. According to them, results are explained by several reasons. Climatic variability is a factor of uncertainty and raises fluctuation in agricultural production. climatic variability raises fluctuation in agricultural production and negatively impact household This effect may be higher for countries that depend on rainfall for agriculture production (rainfed agriculture). In addition, through agriculture production reduction, climatic variability can reduce agricultural incomes and economic growth [START_REF] Dell | Climate Change and Economic Growth: Evidence from the Last Half Century[END_REF]) in developing countries.

Therefore their ability to import food (from international markets) will be limited.

In next step, we include control variables to check the robustness of results (from column 3 to 5 of table 1). There are cereal production land, arable land and real effective exchange rate. The results conclude that rainfall variability has a negative impact on food supply. Moreover land use improves food supply and production. The real effective exchange rate has no impact on food supply per capita.

The role of aid

This section focuses on the importance of foreign aid. We assume that foreign aid can improve food security in developing countries. Moreover, we consider that the negative effect of climatic shocks on food security could be less high in the presence of foreign aid.

Table 2 presents the results of the non-linear impact of climatic shocks on food supply, depending upon the aid. Unsurprisingly, climatic shocks always seem to be associated with food insecurity in developing countries. The results (column 2 of Table 2) indicate that the coefficients associated with climatic shocks and to the interactive term (climatic shocks *Aid) are negative and positive respectively. This result reveals that the negative impact of climatic variability on food supply decreases with aid. In other words, climatic shocks are factor of food insecurity in developing countries although its effect is mitigated in the presence of aid. These results confirm the idea that foreign aid acts as climatic shock absorbers in the recipient countries. First, foreign aid can be countercyclical and, in presence of climatic shocks (and then revenue shocks), smooths public spending. Second, by stabilizing economic resources (potentially affected by climatic shocks), foreign aid contributes to reduce the instability of public investment necessary to finance public services (health, education, infrastructure) that support food and agricultural production.

Previous authors [START_REF] Guillaumont | Aid and Performance: A Reassessment[END_REF][START_REF] Chauvet | Aid, Volatility, and Growth Again: When Aid Volatility Matters and When It Does Not[END_REF] have concluded that foreign aid is more effective in vulnerable countries by allowing economic growth to become more stable. Moreover Badolo and Kinda (2012) conclude that the negative impact of climatic shocks are high for vulnerable countries that are less able to maintain food availability. We assume that the role of aid as climatic shock absorber will be high in the most vulnerable countries. In order to test this hypothesis empirically, equation ( 3) is estimated with the most vulnerable countries in the sample (above the 75th percentile of the distribution of the vulnerability index). Column 3 of table 2 shows that the negative effect of climatic shocks on food security increases when the sample is restricted to the countries above the 75th percentile of the vulnerability index. For this restricted sample, the aid has a powerful dampening effect on the negative impact of climatic shocks on food security in developing countries (column 4 of Table 2).

Conclusion

This paper investigates the role of aid in mitigating the adverse effects of climatic shocks on food security in developing countries. Using panel data for 71 developing countries over the period 1970-2008, our results show that foreign aid significantly dampens the effects of climatic shocks on food security. More interestingly, it appears that the marginal effect of aid is high for countries that are more vulnerable to food price shocks. In other words, when countries exhibit a high level of vulnerability to food price shocks, aid has a strong dampening effect on the impact of climatic shocks on food security.

Our results are important in terms of recommendations for economic policies. While the debate about the role of foreign aid (specially food aid) on food security, is still ongoing, this paper shows that foreign aid dampens the impact of climatic variability on food security. However foreign aid has no direct effect on food supply in developing countries. This result suggests an increase of foreign aid in developing countries. However the policy must be implemented to countries that vulnerable food prices shocks and affected by climatic variability.

Table 1 :

 1 Impact of rainfall variability on food supply per capita

	Dependent variable	Food Supply
		(1)	(2)	(3)	(4)	(5)
		FE	RE	FE	FE	FE
	Rainfall variability	-0.0179***	-0.0194***	-0.0167***	-0.0140**	-0.0196**
		(-2.802)	(-2.986)	(-2.899)	(-2.308)	(-1.976)
	Rainfall	0.148**	0.0209**	0.122**	0.118*	0.110
		(2.036)	(2.403)	(2.443)	(1.702)	(0.909)
	Income per capita	0.0557**	0.107***	0.0688***	0.0785***	0.0233
		(2.079)	(4.533)	(2.827)	(3.060)	(0.579)
	Population growth	-0.0300***	-0.0257**	-0.0165*	-0.0237**	-0.0547**
		(-2.816)	(-2.414)	(-1.694)	(-2.339)	(-2.241)
	Democratic institutions	-0.000663	0.000476	-0.00200	-0.000159	0.000497
		(-0.185)	(0.136)	(-0.617)	(-0.0467)	(0.0940)
	Cereal production land			0.259***
				(9.772)
	Arable land				0.269***
					(6.983)
	Exchange rate (REER)					-0.0242
						(-1.449)
	Intercept	4.238***	4.780***	0.796	3.754***	4.826***
		(7.994)	(13.81)	(1.335)	(7.391)	(5.606)
	Observations	517	517	517	517	517
	Countries	71	71	71	71	71
	R-squared	0.289		0.417	0.361	0.299
	Note: t-statistics are presented in parentheses under the estimated coefficients. ***, ** and * indicate significance of the estimated coefficient at 1, 5,
	and 10 per cent, respectively. The study period is 1960-2008.
	Source: Author's estimate.

Table 3 :

 3 Impact of climatic variability on food security: inertia of food supply and endogeneity of aid

	Dependent variable	Food Supply
		Fixed effects	GMM-system
			Two step
	Lagged food supply		0.0150
			(0.540)
	Rainfall variability	-0.0718***	-0.106*
		(-2.859)	(-1.743)
	Rainfall variability* Aid per capita	0.0430*	0.00212**
		(1.941)	(2.344)
	Aid per capita	0.000296	0.0334**
		(0.797)	(2.489)
	Rainfall	0.208	-0.106
		(1.567)	(-1.245)
	Income per capita	0.205***	0.0723
		(4.577)	(1.447)
	Population growth	-0.00438	-0.119**
		(-0.288)	(-2.191)
	Democratic institutions	-0.00175	0.00324
		(-0.316)	(0.645)
	Intercept	2.844***	1.513***
		(2.902)	(2.825)
	Observations	370	358
	R-squared	0.117
	Countries	67	67
	AR(1)		0.06
	AR(2)		0.95
	Hansen test		0.75
	Instruments		24

Note: t-statistics are presented in parentheses under the estimated coefficients. ***, **, and * indicate significance of the estimated coefficient at 1, 5, and 10 per cent, respectively. The study period is 1960-2008. Source: Author's estimate.

Table 4 :

 4 Rainfall variability, foreign aid, and proportion of undernourished people

	Dependent variable	Prevalence of undernourishment
		Full sample		Vul>50th per.
		(1)	(2)	(3)	(4)
	Rainfall variability	0.0721***	0.0704***	0.102***	0.107***
		(5.110)	(3.892)	(4.203)	(3.812)
	Rainfall variability* Aid per capita		-0.00681***		-0.0122***
			(-3.194)		(-4.012)
	Aid per capita		-0.0441***		-0.0564***
			(-3.405)		(-3.654)
	Rainfall	-0.000536	-0.000384	-0.000110	-0.000192
		(-1.050)	(-0.472)	(-1.148)	(-1.367)
	Income per capita	-0.0711***	-0.0917***	-0.0704***	-0.0872***
		(-4.873)	(-4.102)	(-3.744)	(-3.546)
	Population growth	0.632*	0.682	0.725*	0.916*
		(1.702)	(1.453)	(1.703)	(1.894)
	Democratic institutions	-0.171	-0.193	-0.206	-0.239
		(-1.103)	(-0.904)	(-1.006)	(-0.974)
	Intercept	-61.94***	-71.65***	-46.54**	-49.72**
		(-3.676)	(-3.041)	(-2.292)	(-2.044)
	Observations	314	216	232	173
	R-squared	0.134	0.194	0.199	0.346
	Countries	79	61	62	50

Notes: t-statistics in parentheses; *** p<0.01, ** p<0.05, * p<0.1 Source: Author's estimate.

bThe global hunger index is currently considered as the best indicator of food security. However it is not available over the long period of time.

Robustness checks

Inertia of food supply and dealing with endogeneity of aid

Is food supply characterized by inertia phenomena? In other words, it is important to know if the lagged level of food supply is a potential determinant of the current level of food supply. We include the lagged level of food supply in our baseline equation. The dynamic nature of our specified model requires the implementation of Generalized Method of Moments (GMM) estimation from [START_REF] Arellano | Some Tests of Specification for Panel Data: Monte Carlo Evidence and an Application to Employment Equations[END_REF], [START_REF] Arellano | Another look at the instrumental variable estimation of error-components models[END_REF], and [START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF].

Because aid aims to improve food security (to reduce poverty) in the receiving country, we may assume that countries with greater needs (food insecurity, macro vulnerability, etc.) will receive more aid. The econometric estimation of model facing reverse causality between aid and food security requires us to deal correctly with endogeneity. We use lagged variables (aid per capita) to control for endogeneity. The results in Table 3 show that the lagged level of food supply has no effect on its current level (columns 2). There is no inertia for food supply in developing countries. In addition, using lagged variables to control for endogeneity of aid does not alter the previous results.

Complementary indicator of food security

We consider the proportion of undernourished people as another indicator of food security. In column (1) of table 4, we find that rainfall variability has an effect on the proportion of undernourished people. Column (2) reveals that the impact of climatic variability on malnutrition decreases with aid. Column (3) shows that aid has a powerful dampening effect for countries that are highly vulnerable to food price shocks (above the 75th percentile of the vulnerability index).

Appendix


Precision mechanism is widely used for various industry applications. Quality inspection for precision mechanism is essential for manufacturers to assure the product leaving factory with expected quality. In this paper, we propose a novel automated fault detection method, named Tilear, based on a Deep Belief Network (DBN) auto-encoder. DBN is a probabilistic generative model, composed by stacked Restricted Boltzmann Machines. With its RBM-layer-wise training methods, DBN can perform fast inference and extract high level feature of the inputs. By unfolding the stacked RBMs symmetrically, a DBN auto-encoder is constructed to reconstruct the inputs as closely as possible. Based on the DBN auto-encoder, Tilear is structured in two parts: training and decision-making. During training, Tilear is trained with the signals only from good samples, which enables the trained DBN auto-encoder only know how to reconstruct signals of good samples. In the decision-making part, comparing the recorded signal from test sample and the Tilear reconstructed signal, allows to measure how well a recording from a test sample matches the DBN auto-encoder model learned from good samples. A reliable decision could be made. We perform experiments on two different precision mechanisms: precision electromotors and greasing control units. The feasibility of Tilear was demonstrated first. Additionally, performance of Tilear on the acquired electromotor dataset was compared with the state-of-the-art machine learning based fault detection technique, support vector machine (SVM). First result indicates that Tilear excels the SVM in terms of the Area Under the Curve (AUC) obtained from the Receiver Operating Characteristics (ROC) curve plot: 0.960 achieved by Tilear, while 0.941 by SVM.

Introduction

Precision mechanism is widely used for various industry applications, such as precision electromotor for industrial automation systems, greasing control units for microsystems, and so on. Quality inspection for precision mechanism is essential for manufacturers to assure the product leaving factory with expected quality.

Normally, quality inspection at the manufacturer side is performed by trained experts with different methods. Traditionally, it is accomplished by the experts listening to the sound emitted by the product under different conditions. This subjective assessment is expert individual dependent, and can be influenced by several factors. That brings in the variability in the quality inspection.

Certain algorithms and techniques have been developed to overcome the disadvantages introduced by traditional subjective assessment. These techniques roughly can fall into three categories: signal analysis based methods (SAMs), dynamic model based methods (DMMs), and knowledge based methods (KMs) [START_REF] Liu | Fault Detection and Diagnosis of Permanent-Magnet DC Motor Based on Parameter Estimation and Neural Network[END_REF]. With SAMs, experts directly analyze the characteristics of measured signal by performing certain time-frequency transforms, like Fast Fourier Transform (FFT). It is widely used by the industry, but it is always necessary to find a best signal feature before starting the threshold comparison. For some barely seen defect types, it takes a long time to select features. As for the DMMs, an accurate dynamic model for each specific mechanism model is required before performing the quality inspection. KMs have been widely studied recently with the development of machine learning algorithms. Most of these readily available techniques are on the basis of discriminative learning models. A certain amount of fault samples are required to perform the fault type classification [START_REF] Yang | Intelligent fault diagnosis of rolling element bearing based on SVMs and fractal dimension[END_REF][START_REF] Wu | Bearing Fault Diagnosis Based on Multiscale Permutation Entropy and Support Vector Machine[END_REF]. However, in practical applications, it is extremely difficult to get fault samples in abundance. What makes matters worse is that even a single type of defect typically has many different sensory manifestations.

Alternatively, we treat the fault detection problem as an anomaly detection problem, to overcome the scarcity of defective samples in the production line. The core of anomaly detection is to recognize the inputs that differ from those under normal conditions. Different anomaly detection techniques have been proposed [START_REF] Chandola | Anomaly detection[END_REF], such as classification based anomaly detection techniques or statistical anomaly detection techniques. J. McBain et al [START_REF] Mcbain | Feature extraction for novelty detection as applied to fault detection in machinery[END_REF] applied the boundary based method, namely Support Vector Data Descriptor (SVDP), with the high dimension signal features extracted by autoregressive model for motor fault detection. They tried to maximize the distance between the average distance of normal class and the average distance of the defective class. B. Zheng et al [START_REF] Zhang | Rolling element bearing feature extraction and anomaly detection based on vibration monitoring[END_REF] proposed to use different discriminative classification methods with extracted vibration signal features for bearing anomaly detection. These proposed methods are still discriminative based, which means the construction of the anomaly detector still needs the presence of defective samples, even if not a big number. Also, they usually train with carefully selected features, the choice of which may greatly influence the anomaly detector's performance. Rather than constructing the anomaly detector based on discriminative classifiers, Deep Belief Network (DBN) is selected since it is a generative model which has strong ability to perform fast inference and to learn features unsupervisedly [START_REF] Hinton | Training products of experts by minimizing contrastive divergence[END_REF][START_REF] Hinton | A fast learning algorithm for deep belief nets[END_REF][START_REF] Hinton | Reducing the dimensionality of data with neural networks[END_REF]. Firstly proposed by Hinton in 2006, DBN has attracted great attention from both academia and industry, and shown promising future in many tasks, such as real time speech translation and image recognition. To our knowledge, our project is the first time that deep belief networks is applied for machinery quality inspection.

The objective of this work is to develop a new automated fault detection system for precision mechanism inspection either using acoustic signals or vibration signals.

Treating the fault detection as an anomaly detection problem, this system is based on a Deep Belief Network (DBN) auto-encoder. It learns the sensory signals only from good samples, and makes decisions for test samples with the trained network.

2

Methods and Results

Theory basis: Deep Belief Networks

DBN is a probabilistic generative model, which employs a hierarchical structure constructed by stacking Restricted Boltzmann Machines (RBMs) [START_REF] Hinton | A fast learning algorithm for deep belief nets[END_REF][START_REF] Bengio | Learning Deep Architectures for AI[END_REF]. The RBM is a two layer neural network modeling the joint distribution of its inputs and outputs. To construct a DBN, a number of RBMs are stacked on top of each other. The hidden layers of lower level RBMs are the visible layers of the adjacent higher level RBMs. A greedy layer-wise training algorithm is applied to train the DBN, which is actually training the RBMs individually under the contrastive divergence rule [START_REF] Hinton | Training products of experts by minimizing contrastive divergence[END_REF]. Trained in this way, the DBN can perform a fast inference and extract high level representations, or features, of the input data. Thorough descriptions of DBNs' mathematical and technical details are available elsewhere [START_REF] Hinton | A fast learning algorithm for deep belief nets[END_REF][START_REF] Bengio | Learning Deep Architectures for AI[END_REF].

Proposed Method: Tilear

Taking the advantage of DBN auto-encoder's capability to reconstruct the input data as closely as possible, we propose an anomaly detection model, named Tilear which learns the input data tile by tile for the purpose of performing fast inference.

Tilear has two functions: "Teacher" for the training phase, "Tester" for the decision making phase. The biggest difference between Tilear and other teacher/tester systems is that during training phase (teacher), only input data from good samples will be learned by the auto-encoder of Tilear, while other systems usually need the presence of defective samples for teacher training. In the training phase of Tilear, small anomalies in the "good" data are tolerable variances. The scarcity of anomalies prevents the DBN from learning and reconstructing those. This property results in an additional reconstruction error for the data containing anomalies. Therefore, the higher the reconstruction error, the more anomalies the data sample contains. An anomaly detector thus can be made by comparing the reconstruction error with a threshold. In Tilear, the reconstruction error 𝑆 𝑖 , also named score, is the Root Mean Squared Error (RMSE) between the input data 𝐼 𝑖 and corresponding reconstructed data 𝑅 𝑖 , averaged over 𝑛 dimensions of the data, as expressed in Equation [START_REF] Liu | Fault Detection and Diagnosis of Permanent-Magnet DC Motor Based on Parameter Estimation and Neural Network[END_REF].

𝑆 𝑖 = √ ∑ (𝑅 𝑖𝑗 -𝐼 𝑖𝑗 ) 2 𝑛 𝑗=1 𝑛 (1)
The reconstruction error threshold 𝑆 𝑡ℎ demarcating the anomaly boundary is another model parameter. This is determined by searching the reconstruction error space of a validation dataset containing labeled good samples and defective samples with anomalies. With the selected 𝑆 𝑡ℎ , "Tester" can make a decision on the health status of test sample 𝑇 𝑖 by comparing its reconstruction score 𝑆 𝑖 to 𝑆 𝑡ℎ .

Experiments and Results

Experiments on two different precision mechanisms were accomplished: precision electromotors with a 2 stage planetary gearbox, and greasing control units.

For the precision electromotor, vibration signals were acquired from 36 samples including 21 good samples and 15 defective samples with missing gears on different stages. Cepstrograms of these signals were used as the input for Tilear. The distribution of the reconstruction errors is shown in Figure 2.

Fig. 2. Cumulative distribution of the reconstruction errors of the precision electromotor dataset.

The cumulative distribution of good samples is marked with green, while that of bad samples with red, which is flipped vertically to help examine the overlap between the two distributions.

The less overlapped they are, the better performance Tilear has. No overlap means the detector can always make the right decision. The threshold selected by Tilear for decision making is shown as a red dashed line along with its actual value.

From the above figure, it is observed that most of the defective electromotors with missing gears can be detected. Although there were few samples misclassified with the self-selected threshold, it is possible to filter out all defective samples with the price of some false negative samples.

In order to evaluate Tilear's performance, Area under Curve (AUC) obtained from the Receiver Operating Characteristics (ROC) curve plot is employed, due to the imbalanced class distribution in the electromotor dataset [START_REF] Chawla | Data mining for imbalanced datasets: An overview[END_REF]. SVM as a state-of-the-art machine learning algorithm was used as a comparison benchmark. LIBSVM [START_REF] Chang | Libsvm: A Library for Support Vector Machines[END_REF] As for the greasing control units, acoustic signals were acquired from 47 samples consisting of 24 greased ones, which were considered as good samples, and 23 non-greased ones, which were considered as defective samples. Spectrograms were used as the input of Tilear. AUC was again used as the evaluation metric. AUC for these control units was 0.995. Only 1 out of 23 non-greased samples was misclassified, while all greased ones were correctly classified.

Conclusion

Tilear, a new automated fault detection method for precision mechanism inspection, which firstly uses the Deep Belief Networks (DBN) based auto-encoder, was proposed. Tilear is trained to reconstruct the data only from good samples as closely as possible. By comparing the reconstruction errors, a decision can be made. The feasibility of fault detection using Tilear is verified with two different kinds of precision mechanisms. It is shown that Tilear has comparable performance with the state-of-art technique, Support Vector Machine, using the Area under the Curve as the performance evaluation metric. It is believed that DBN not only can be used for fault detection, but also has the potential in the fault classification area, on condition that enough defective samples are collected for training.

Fig. 1 .

 1 Fig. 1. Architecture of DBN based auto-encoder. Numbers in the blocks represent the number of nodes in each layer. Node number in the bottom layer represents the sampling points from the input data. Node number of the rest layers in the encoder represents the number of extracted highorder features for their respective input data. The number of nodes and layers are only examples. It is not required to have the same numbers in the experiments, or to be 2 𝑛 .By unfolding the stacked 𝑛 RBMs, an auto-encoder composed by (2𝑛 -1) RBMs is constructed. This (2𝑛 -1) directed auto-encoder can be fine-tuned with backpropagation[START_REF] Hinton | Reducing the dimensionality of data with neural networks[END_REF]. As shown in Figure1, the first n RBMs act as an encoder. High-level features of the input data are extracted by this encoder and stored at the hidden layer of the top RBM. The last n RBMs, including the top RBM of the encoder, form a decoder. This decoder reconstructs the input data with the extracted high-level features stored in the top RBM of the encoder. Generally speaking, a DBN based auto-encoder is to reconstruct training data as closely as possible.

Table 1 .

 1 was used to construct the SVM fault detector. Cepstrograms of the vibration signals were used as the input data for both Tilear and SVM. The comparison result is shown in table.1. It is observed that the AUC of Tilear is higher than that of SVM. This indicates that for Tilear had a better performance over SVM on this dataset, which has to be verified with further study. It is also worth pointing out that Tilear was faster for training compared to LIBSVM. The training time for Tilear was approximately 40 minutes, while LIBSVM always took at least several hours. In some sense, it is unfair to compare the training time here, since Tilear is developed to use Graphics Processing Unit (GPU) to achieve fast computation speed while LIBSVM not. The comparison of computational speed between Tilear and SVM on GPU platform is to be investigated in the future. Comparison of AUC between Tilear and SVM

		TILEAR SVM
	AUC	0.960	0.941


Pierre Colombé

Michaël Peigney

A Direct Method For Predicting The High-Cycle Fatigue Regime In SMAs: Application To Nitinol Stents

In fatigue design of metals, it is common practice to distinguish between high-cycle fatigue (occurring after 10000-100000 cycles) and low-cycle fatigue. For elastic-plastic materials, there is an established correlation between fatigue and energy dissipation. In particular, high-cycle fatigue occurs when the energy dissipation remains bounded in time. Although the physical mechanisms in SMAs differ from plasticity, the hysteresis observed in the stress-strain response shows that some energy dissipation occurs, and it can be reasonably assumed that situations where the energy dissipation remains bounded is the most favorable for fatigue design. We present a direct method for determining if the energy dissipation in a SMA structure is bounded or not. That method relies only on elastic calculations, thus bypassing incremental nonlinear analysis. Moreover, only a partial knowledge of the loading (namely the extreme values) is needed. Some results related to Nitinol stents are presented.

Introduction

This paper is concerned with the long-time behaviour of Shape Memory Alloys (SMA) bodies under prescribed loading histories. For elastic perfectly plastic bodies, a fundamental result is the Melan's theorem [1,[START_REF] Koiter | General problems for elastic solids[END_REF] which gives a sufficient condition for the energy dissipation to remain bounded with respect to time. That last situation is classically referred to as shakedown, and is associated with the intuitive idea that the body behaves elastically for time t sufficiently large, i.e. that the plastic strain tends to a limit as t → ∞. Melan's theorem has the distinctive property of being path-independent, i.e. independent on the initial state of the structure. Regarding fatigue design, shakedown corresponds to the most beneficial regime of high-cycle fatigue, as opposed to the regime of low-cycle fatigue which typically occurs if the plastic strain does not converge towards a stabilized value [3].

Much effort has been devoted to developing constitutive laws for describing the behaviour of SMAs. The phase transformation is typically described by an internal variable α which -depending on the complexity of the material model -may be scalar or vectorial. A fundamental observation is that the internal variable α must comply with some a priori inequalities that result from the mass conservation in the phase transformation process. As a consequence, the internal variable α is constrained to take values in a set K that is not a vectorial space (in most of SMA models, K is a bounded set). The presence of such constraints constitutes a crucial difference with plasticity a e-mail: colombep@eleves.enpc.fr b e-mail: michael.peigney@enpc.fr models, and calls for special attention when the structural evolution problem is considered [4][5][6][7].

It was recently proved that the Melan's theorem could be extended to SMAs [8,9]. When the shakedown limit provided by that theorem is exceeded, it was found that the large-time behaviour is dependent on the initial state: in the case of cyclic loadings, some initial conditions lead to shakedown whereas some others lead to alternating phase transformation. Such a feature is not found in standard plasticity. The shakedown theorem in [8,9] is pathindependent -in the spirit of the original Melan theorem [1,[START_REF] Koiter | General problems for elastic solids[END_REF] -and applies to a wide range of constitutive models of phase transformation in SMAs. As explained in this paper, the theorem in [8,9] can be used to obtain bounds on the loadings for which shakedown occurs, thus leading to a general method for the fatigue design of SMA structures.

Evolution of a SMA continuum

Constitutive laws

We consider constitutive SMA models which enter the framework of standard generalized materials [10]. The local state of the material is described by the strain and an internal variable α tracking the phase transformation. The variable α is constrained to take values in a given bounded set K. The rate-independent behavior of the material is determined by the constitutive relations (3)

σ = ∂w ∂ ( , α) , A = - ∂w ∂α ( , α), (1)
A = A d + A r , (2)
A r ∈ ∂I K (α). ( 4
)
where w( , α) is the free energy function and C is the elasticity domain of the material. In (3)-( 4), ∂ denotes the subdifferential operator [START_REF] Brézis | Opérateurs maximaux monotones et semigroupes de contractions dans les espaces de hilbert[END_REF] and I K (resp. I C ) is the indicator function of the set K (resp. C), i.e. the function that is equal to 0 in K (resp. C) and infinite outside K (resp. C).

In (2), A can be interpreted as the driving force for phase transformation. Eq. ( 3) is the normality flow rule. The term A r in (2-4) arises as a consequence of the constraint α ∈ K. We refer to [START_REF] Frémond | Non-smooth thermomechanics[END_REF] for a derivation of (1-4) from the principle of thermodynamics.

We assume that the sets C, K are convex and consider free energy functions w( , α) of the form

w( , α) = 1 2 ( -K•α) : L : ( -K • α) + f (α) ( 5
)
where L is symmetric positive, K is a given matrix, and f is a positive differentiable function (not necessarily linear nor convex). In such case, we have from ( 1)

σ = L : ( -K • α) , A = K t : σ -f (α)
where K t is the transpose of K and f is the derivative of f . The total strain is thus the sum of an elastic strain L -1 : σ (proportional to the stress) and an inelastic strain K • α (proportional to the internal variable α). The driving force A depends linearly on the stress σ, but may depend nonlinearly on the internal variable α.

Many existing models of shape-memory alloys fit in the format considered. For instance, a three-dimensional micromechanical model of single crystals used in the literature [4,6,7] is given by

w( , ξ) = 1 2 ( - n i=1 ξ i i ) : L : ( - n i=1 ξ i i ) + i=1 m i ξ i , C = [G - 1 , G + 1 ] × • • • × [G - n , G + n ], K = {ξ ∈ R n + : n i=1 ξ i ≤ 1}, ( 6
)
where n is the number of martensitic variants and i is the given transformation strain for each variant. The scalars 6) are all constitutive parameters of the model. The internal variable ξ = (ξ 1 , • • • , ξ n ) represents the set of volume fractions for each martensitic variant. Because of mass conservation in the phase transformation, the volume fraction of the austenite is equal to 1i ξ i , hence the constraint i ξ i ≤ 1 that is imposed on ξ. For ξ verifying 0 < ξ i (for all i) and i ξ i < 1, the term A r in ( 4) is equal to 0 and the relations (2-3) become

m i , G + i , G - i in (
ξi ≥ 0 if σ : i -m i = G + i , ξi ≤ 0 if σ : i -m i = G - i , ξi = 0 if σ : i -m i ∈ (G - i , G + i ).
Those relations define the evolution of phasetransformation in a way similar to crystal plasticity.

Phenomenological SMA models can also enter the format considered. An example is the model of [START_REF] Souza | [END_REF]14], defined by

w( , α tr ) = 1 2 ( -α tr ) : L : ( -α tr ) + a α tr + 1 2 h α tr 2 , C = {σ ∈ R 3×3 s : dev(σ) ≤ σ Y }, K = {α tr ∈ R 3×3 s : tr α tr = 0; α tr ≤ L }.
(7) where dev denotes the deviator. In the model (7), the internal variable is the macroscopic transformation strain α tr . The constitutive parameters h, L and σ Y are positive. The constitutive parameter a is positive in the superelastic regime, i.e. for sufficiently high temperatures. The norm . in ( 7) is the euclidean norm in the space of 3 × 3 symmetric matrices, i.e. τ = 3 i, j=1 τ 2 i j . In order to elucidate the meaning of the relations (2-4), consider the simplifying case a = h = 0 in (7). For α tr such that α tr < L , the term A r in ( 4) is equal to 0 so that (2-3) reduce to

αtr = λ dev(σ) with λ ≥ 0, λ(σ Y -dev(σ) ) = 0. (8)
The relation ( 8) corresponds the Von Mises flow rule for the internal variable α tr . In particular, when αtr 0 (i.e. phase transformation occurs), ( 8) can be equivalently rewritten as

dev(σ) = σ Y αtr αtr . (9)
For α tr such that α tr = L , the term A r in (4) takes the form A r = μα (with μ ≥ 0). Eqs (8)(9) have to be modified accordingly. In particular, (9) becomes dev(σ) = σ Y αtr αtr + μα.

Quasi-static evolutions

In the following we are interested in studying the evolutions of a continuum submitted to a prescribed loading history. The continuum occupies a domain Ω and is submitted to body forces f d . Displacements u d are imposed on a part Γ u of the boundary Γ, and tractions T d are prescribed on Γ T = Γ -Γ u . The given functions f d ,u d ,T d depend on position x and time t. The stress and state variables (σ, , α) in the continuum are also expected to depend on (x, t). In order to alleviate the expressions, this dependence will be omitted in the notations, unless in the case of possible ambiguities.

Quasi-static evolutions of the continuum are governed by the following system:

σ ∈ K σ , ∈ K , α ∈ K, α ∈ ∂I C ( A d ) , A r ∈ ∂I K (α), σ = L : ( -K•α), K t : σ -f (α) = A d + A r , (10)
where K σ and K are respectively the sets of admissible stress and strain fields, defined by

K σ = {σ|div σ + f d = 0 in Ω; σ • n = T d on Γ T }, K = { | = (∇u + (∇u) t )/2 in Ω; u = u d on Γ u }.
(11) Starting from a given initial state α(0), (10) determines the evolution of the stress and strain fields.

A general method for fatigue design 3.1 Shakedown theorem

We examine conditions under which the energy dissipation T 0 Ω A d • αdxdt remains bounded (with respect to time T ) for all solutions of the evolution problem (10). Such a situation is referred to as shakedown. As mentioned in the introduction, shakedown is related to the fact that the evolution becomes elastic in the large-time limit [8,15] and corresponds to high-cycle fatigue.

Let us introduce the so-called fictitious elastic response (σ E , E ) of the system, i.e. the response that would be obtained if the material was purely elastic. More precisely, (σ E , E ) is the solution of

σ E ∈ K σ , E ∈ K , σ E = L : E . ( 12
)
The central result is the following shakedown theorem [8,9]: If there exists m > 1, τ ≥ 0 and a time-independent field A * (x) such that

mK t : σ E (x, t) -A * (x) ∈ C ∀x ∈ Ω, ∀t > τ ( 13
)
then there is shakedown, irrespective of the initial condition. A proof of that theorem is presented in 3.2. We refer to [8,9] for more details.

There is a simple geometric interpretation of the condition (13): Consider a fixed location x and let Γ(t) be the curve described by K t : σ E (x, t). The condition [START_REF] Souza | [END_REF] means that, up to a time-independent translation, the curve Γ remains in the elasticity domain C (for time t large enough).

As an example, consider the material model (7). Using the presented theorem, it can easily be seen that shakedown occurs if ms E (x, t) -A * (x) ≤ σ Y where s E is the deviatoric part of σ E and A * (x) is an arbitrary deviatoric tensor. The obtained shakedown condition thus reduces to a restriction on the diameter of the curve s E (x, t), as for shakedown in linear kinematic hardening plasticity [15,16].

Observe that we did not assume the convexity of f . This is a welcome feature for the shakedown analysis of SMA bodies because the function f associated with some micromechanical SMA models is not necessarily convex [17][18][19][20]. We refer to [9] for a detailed discussion between the theorem (13) and the analog result in plasticity.

Finally, we note that the above theorem gives a sufficient condition for shakedown to occur, whatever the initial state is. When the loading is beyond the limit provided by the theorem, shakedown may still occur for some (but not all) initial conditions (see [8,9]) for some examples). In such case, the asymptotic behavior is strongly dependent on the initial state: For instance, in the case of periodic loading, some initial conditions leads to shakedown while others lead to alternate phase transformation (i.e. a periodic but non constant evolution of the phase transformation). Interesting, such dependence of the asymptotic regime on the initial state has also been observed in other nonlinear mechanical problems, such as contact with friction [21] and plasticity with temperature-dependent elastic moduli [22].

Proof of the theorem

Consider a solution ( , α, σ, A r , A d ) of the evolution problem (for some given initial condition) and let

D(t) = Ω A d • αdx
be the rate of dissipated energy. Assuming (m, τ, A r * ) satisfy (13), we show in the following that the total dissipated energy T 0 D(t)dt remains bounded as T → ∞. To that purpose, introduce the positive functional W(t) defined as

W(t) = Ω w( (t) -E (t), α(t)) dx.
We have

Ẇ(t) = Ω [(σ -σ E ) : (˙ -˙ E -K • α) + f (α) • α] dx. Since div(σ -σ E ) = 0 in Ω, (σ -σ E ) • n = 0 on Γ T and u -u E = 0 on Γ u , the principle of virtual power gives Ω (σ -σ E ) : (˙ -˙ E ) dx = 0. Hence Ẇ(t) = Ω [-K t : (σ -σ E ) + f (α)] • α dx
which using (10) can be rewritten as

Ẇ(t) = -D(t) + Ω [-A r + K t : σ E ] • α dx. Setting A d * = mK t : σ E (t) -A r * , we obtain Ẇ(t) = -D(t) + Ω [-A r + 1 m ( A d * + A r * )]. α dx. ( 14
)
The property [START_REF] Souza | [END_REF] shows that

A d * ∈ C for t > τ. Since α ∈ ∂I C (A d ) and C is convex, we have ( A d -A d * ) • α ≥ 0. ( 15
)
This last relation can be interpreted as the principle of maximum dissipation. Similarly, the relations A r ∈ ∂I K (α) and α ∈ K imply that

0 ≥ A r (t) • (α(t ) -α(t))
for any t . Taking the limit t -→ t with t < t, we obtain

A r • α ≥ 0 ( 16
)
where α is the left-time derivative. Combining ( 15)-( 16) with ( 14) yields

Ẇ(t) ≤ 1 -m m D(t) + 1 m Ω A r * • α dx.
Integrating with respect to time on [τ, T ] and noting that A r * is time-independent, we find (m -1)

T τ D(t) dt ≤ mW(τ) + Ω A r * • (α(T ) -α(τ)) dx ( 17
)
ESOMAT 2015

where the property W(T ) ≥ 0 has been used. Since K is bounded, there exists a positive constant K such that α ≤ K for any α ∈ K. Therefore

Ω A r * • (α(t) -α(τ)) dx ≤ 2K Ω A r * dx.
Combining that inequality with (17) gives

(m -1) T τ D(t) dt ≤ mW(τ) + 2K Ω A r * dx.
The right-hand side of that inequality is independent on T . Therefore T τ D(t) is bounded as T -→ +∞, which completes the proof.

Application to fatigue design

In practice, the shakedown condition ( 13) can be used to bound the loadings for which shakedown occurs. To illustrate this point, consider the simple situation of a proportional loading history: The functions f d ,u d ,T d that define the loading history take the form

f d (x, t) = λ(t) f d 0 (x), u d (x, t) = λ(t)u d 0 (x) , T d (x, t) = λ(t)T d 0 (x)
where λ(t) is a loading parameter, depending on time. The elastic response σ E in (12) can be written as

σ E (x, t) = λ(t)σ E 0 (x)
where σ E 0 (x) is the solution of the elasticity problem (12) for the loading ( f d 0 ,u d 0 ,T d 0 ). Therefore, the condition (13) becomes

mλ(t)K t : σ E 0 (x) -A * (x) ∈ C ∀x ∈ Ω, ∀t > τ. (18)
Assume that λ(t) varies between a minimum value λ min and a maximum value λ max . In such case, ( 18) is satisfied provided that

m(λ max -λ min )K t : σ E 0 (x) -A * (x) ∈ C (19)
for some A * (x) and some m > 1. It is necessary at this point to distinguish between the SMA models considered. For instance, for the micromechanical model ( 6) of single crystals, the condition ( 19) can be rewritten as

m(λ max -λ min ) i : σ E 0 (x) ≤ G + i -G - i . (20)
where we recall that i is the transformation strain for variant i. For a scalar m > 1 satisfying (20) at all point x to exist, it is necessary that

(λ max -λ min ){sup x∈Ω i : σ E 0 (x)} < G + i -G - i .
The conclusion is that shakedown occurs if the loading satisfies λ max -λ min < M HCF (21) where

M HCF = min i G + i -G - i sup x∈Ω i : σ E 0 (x) . ( 22
)
We can observe that the loading history only appears in (21) through the extreme values λ min and λ max : The detailed knowledge of λ(t) is not required. In particular, λ(t) does not need to be periodic. As an other example, consider the case where the elasticity domain C is a ball, such as for the phenomenological model (7). Then ( 19) can be satisfied by some A * (x) provided that

m(λ max -λ min ) K t : σ E 0 (x) ≤ diam C ∀x ∈ Ω (23)
where diam C is the diameter of C, i.e.

diam C = sup τ,τ ∈C τ -τ .

Shakedown occurs if (23) is satisfied by some m > 1 and for all x ∈ Ω, i.e. if

λ max -λ min < M HCF
where M HCF is now defined by

M HCF = diam C sup x∈Ω K t : σ E 0 (x) . ( 24
)
The conclusion is the following: If the amplitude λ max -λ min of the loading is smaller than M HCF , then the structure shakes down and therefore is expected to be in the high cycle fatigue regime. The scalar M HCF can thus be interpreted as a limit for high cycle fatigue. In practice, the calculation of M HCF only requires to solve a single problem of linear elasticity (so as to obtain σ E 0 ). The relation (21) can be used as a criterion for the fatigue design of SMA structures: If the amplitude of the loading is specified, the geometry of the structures and the choice of constitutive materials should be adjusted so as to satisfy (24). The latter indeed have a direct influence of the elastic response σ E 0 .

Application to Nitinol stents

In this section we discuss the application of the proposed method to Nitinol stents, that are used in the biomedical field for treating artery disease [23,24]. The tubular geometry of such devices (see Fig. 1) usually consists in the repetition of an elementary motif (a 'strut V' in the case of Fig. 1). In order to study the fatigue of stents, some extensive testing has been done in [25] on specifically designed diamond-shape specimen (a simplified model of such specimen is shown in Fig. 2). Those diamond shaped specimen consists in 2 'strut Vs' arranged in a symmetric fashion, so as to be easily fitted in a fatigue test machine. The specimen were submitted to displacement-controlled cyclic loading (along the horizontal direction in Fig. 2), with prescribed mean and strain amplitudes. The resulting S-N curves (strain amplitude vs number of cycles to failure) showed a low-to high-cycle fatigue transition occurring for a strain amplitude approximatively equal to 0.5%, without any clear influence of the mean strain: for a prescribed strain amplitude larger than 0.5%, failure typically occurred after a low number of cycles (10 3 -10 5 ), whereas for a prescribed strain amplitude smaller than 0.5%, the specimen survived after 10 5 cycles (see [25] for details).

It is interesting to see if the shakedown-based approach presented in Sect. 3 is consistent with then experimental results in [25]. To that purpose, linear Finite Element Analysis (FEA) was used to evaluate the fictitious elastic response σ E 0 . The geometry used in the computations is represented in Fig. 2(left). The maximum length along the horizontal direction (which is the direction of loading) is l =1.35 mm. The length in the vertical direction is equal to 3.104 mm and the thickness is equal to 0.2 mm. In Fig. 2(right) is shown the deformed configuration obtained by FEA for a reference value 0 of the imposed strain (arbitrary set equal to 0.055/l = 0.0407%). The corresponding stress distribution σ E 0 is shown in Fig. 3. The elasticity tensor L was chosen as isotropic with a Young modulus equal to 50 GPa and a Poisson ration equal to 0.35 [14]. Using the SMA model ( 7), the high cycle fatigue limit (24) becomes

M HCF = 2σ Y sup x∈Ω s E 0 (x)
.

Note that s E 0 (x) corresponds to the standard Von Mises stress. FEA shows that sup s E 0 (x) is approximatively equal to 1406 MPa. That value is reached in the knee area of the specimen (see Fig. 3 ). Using the constitutive value σ Y = 50 MPa [14], the shakedown limit is approximatively equal to 0.35% (strain amplitude), which -notably given the approximation on the geometry as well as on the constitutive parameters-is in satisfactory agreement with the experimental results of [25].

Concluding remarks

In this communication, we have presented a general method for the fatigue design of SMA bodies, based on shakedown theory. That method is direct in the sense that incremental nonlinear analysis -which is costly and somewhat tricky to perform for SMAs -is completely bypassed. Moreover, only a partial knowledge of the loading (namely the knowledge of the extreme values) is required. That method is very simple to use as it only relies on elastic calculations and can be implemented in the postprocessing stage. Further work is in progress to carry such an implementation is the general case of a parametrized loading history.

On a final note, we observe that all those simplifying features ultimately stem from a distinctive property of SMA models, namely the constraints that are placed on the internal variable. Compared to plasticity, such constraints makes solving incremental evolutions more complicated but they considerably simplify the shakedown analysis.

DOI: 10 α

 10 .1051/ C Owned by the authors, published by EDP Sciences, 2015 ∈ ∂I C (A d ),

Figure 1 .

 1 Figure 1. Geometry of a stent.

Figure 2 .

 2 Figure 2. Diamond-shape specimen used in unaxial traction tests: Reference configuration (left), Deformed configuration (right).

Figure 3 .

 3 Figure 3. Results of linear FEA (Von Mises distribution).